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Biographies

  1. Kuczma biography
    • For example in the student years he published papers such as: (with Stanislaw Golab and Z Opial) La courbure d'une courbe plane et l'existence d'une asymptote (1958), On convex solutions of the functional equation g[a(x)] - g(x) = j(x) (1959), On the functional equation j(x) + j [f (x)] = F(x) (1959), On linear differential geometric objects of the first class with one component (1959), Bemerkung zur vorhergehenden Arbeit von M Kucharzewski (1959), Note on convex functions (1959), and (with Jerzy Kordylewski) On some functional equations (1959).
    • On 1 December 1966, in addition to these position which he continued to hold, Kuczma became head of the Department of Functional Equations at Katowice.
    • From the founding of the new University, Kuczma became the head of the mathematics section and head of the department of functional equations.
    • In fact he published around 30 papers during the 1980s despite his severe disability and in [Selected topics in functional equations and iteration theory, Graz, 1991 (Karl-Franzens-Univ.
    • The first of these Functional equations in a single variable appeared in 1968 and was the first book to be written on this topic.
    • This is the first book ever published on functional equations in a single variable ..
    • The related questions of commuting functions, continuous iteration, and Schroder's and Abel's functional equations are also treated.
    • Kuczma's second book was An introduction to the theory of functional equations and inequalities.
    • Cauchy's equation and Jensen's inequality published in 1985.
    • Probably even the most devoted specialist would not have thought that about 300 pages can be written just about the Cauchy equation (and on some closely related equations and inequalities).
    • In the opinion and experience of this reviewer this is a very useful book and a primary reference not only for those working in functional equations, but mainly for those in other fields of mathematics and its applications who look for a result on the Cauchy equation and/or the Jensen inequality.
    • His final book Iterative functional equations was written jointly with Bogdan Choczewski and Roman Ger who had worked for his doctorate with Kuczma at the Silesian University of Katowice, graduating in 1971.
    • The book is a cohesive and exhaustive account of contemporary theory of iterative functional equations.
    • Fundamental notions such as existence and uniqueness of solutions of equations under consideration are treated throughout the book as well as a surprisingly wide scale of examples showing applications of the theory in dynamical systems, ergodic theory, functional analysis, functional equations in several variables, functional inequalities, geometry, iteration theory, ordinary differential equations, partial differential equations, probability theory and stochastic processes.
    • the paramount achievement of Professor Marek Kuczma was the creation and development of a systematic theory of iterative functional equations and founding a mathematical school centred around the seminar conducted by him since October 1964.
    • Kuczma was considered an outstanding mathematician highly esteemed by the international community of specialists; among the functional equationists he had commonly been treated as one of the informal leaders.

  2. Riccati biography
    • Rather the reverse, he was interested in all scholarly subjects as Sergio Bittanti points out in [S Bittanti, A J Laub, J C Willems (eds.), The Riccati equation (Springer Verlag, Berlin, 1991), 1-10.',6)">6]:- .
    • Sergio Bittanti writes [S Bittanti, A J Laub, J C Willems (eds.), The Riccati equation (Springer Verlag, Berlin, 1991), 1-10.',6)">6]:- .
    • While in the Val di Sole Riccati met with Nicolaus(II) Bernoulli and they had mathematical discussions regarding solving differential equations.
    • Riccati's life-long passion for studying methods of solving differential equations using separation of variables came through his reading of this book.
    • The subject of this scientific exchange is, first, a method of Riccati for separating the indeterminates in some differential equations, and then a question on the lunules quarrables which was disputed between Suzzi, one of Riccati's young disciples, and Daniel Bernoulli in 1724.
    • In fact Riccati produced detailed lecture notes (consisting of 154 pages) for teaching Suzzi and da Riva which were subsequently published as Delia separazione delle indeterminate nelle equazioni differenziali di prima e di secondo grado, e della riduzione delle equazioni differenziali del secondo grado e d'altri gradi ulteriori (On the separation of variables in differential equations of first and second order, and on the reduction of differential equations of second order and higher orders).
    • However, he is best known for his work on solving differential equations.
    • In the study of differential equations his methods of lowering the order of an equation and separating variables were important.
    • He considered many general classes of differential equations and found methods of solution which were widely adopted.
    • He is chiefly known for the Riccati differential equation of which he made elaborate study and gave solutions for certain special cases.
    • Although he probably began studying the equation in 1715, the first written record of the differential equation seems to be in a letter he wrote to Giovanni Rizzetti in 1720.
    • The equation was discussed by Riccati in the 1722-23 lecture notes we mentioned above [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • In expounding the known methods of integration of first-order differential equations, Riccati studied those equations that may be integrated with appropriate algebraic transformation before considering those that require a change of variable.
    • He then discussed certain devices suggested by Johann Bernoulli and expounded the method used by Gabriele Manfredi to integrate homogeneous equations.
    • Of these, one involves the reduction of the equation to a homogeneous one, while another more interesting method is that of "halved separation," as Riccati called it.
    • In the first, the entire equation is multiplied or divided by an appropriate function of the unknown so that it becomes integrable; second, after this integration has been carried out, the result is considered to be equal to a new unknown, and one of the original variables is thus eliminated; and finally, the first two procedures are applied to the result until a new and desired result is attained.
    • His work had a wide influence on leading mathematicians such as Daniel Bernoulli, who studied the equation in his Exercitationes quaedam mathematicae, and Leonard Euler who extended Riccati's ideas to integration of non-homogeneous linear differential equations of any order.
    • Bittanti describes the end of Riccati's life [S Bittanti, A J Laub, J C Willems (eds.), The Riccati equation (Springer Verlag, Berlin, 1991), 1-10.',6)">6]:- .

  3. Mikhlin biography
    • The book [Multidimensional singular integrals and integral equations, International Series of Monographs in Pure and Applied Mathematics 83 (Pergamon Press, Oxford-London-Edinburgh-New York-Paris-Frankfurt, 1965).',1)">1] is dedicated to her memory.
    • In 1966 Miklin and Maz'ya launched a Tuesday seminar on integral and partial differential equations at MatMekh.
    • The first one is based upon the so-called complex Green's function and the reduction of the related boundary value problem to integral equations.
    • As a result of his study of this problem, Mikhlin also gave a new invariant form of the basic equations of the theory.
    • Perhaps, his most important contributions are his works on the theory of singular integral operators and singular integral equations: he is one of the founders of the multi-dimensional theory, jointly with Francesco Tricomi and Georges Giraud.
    • Mikhlin was the first to develop a theory of singular integral equations as a theory of operator equations in function spaces L2.
    • He established Fredholm's theorems for singular integral equations and systems of such equations under the hypothesis of non-degeneracy of the symbol.
    • He also proved that the index of a single singular integral equation in the Euclidean space is zero.
    • In 1961 Mikhlin developed a theory of multidimensional singular integral equations on Lipschitz spaces which are widely used in the theory of one-dimensional singular integral equations.
    • Namely, he obtained basic properties of this kind of singular integral equations as a by-product of the Lp-space theory of these equations.
    • A complete collection of his results in this field up to 1965, as well as contributions by Francesco Tricomi, Georges Giraud, Alberto Calderon and Antoni Zygmund, is contained in the monograph [Multidimensional singular integrals and integral equations, International Series of Monographs in Pure and Applied Mathematics 83 (Pergamon Press, Oxford-London-Edinburgh-New York-Paris-Frankfurt, 1965).',1)">1].
    • Mikhlin's multiplier theorem is widely used in different branches of mathematical analysis, particularly in the theory of differential equations.
    • Four Mikhlin papers, published in the period 1940-1942, deal with applications of the method of potentials to the mixed problem for the wave equation.
    • In particular, he solved the mixed problem for the two-space dimensional wave equation in the half-plane by reduction to the planar Abel integral equation.
    • For a planar domain with a sufficiently smooth curvilinear boundary, he reduced the problem to an integro-differential equation, which he is able to solve when the given boundary is analytic.
    • In 1951 Mikhlin proved the convergence of the Schwarz alternating method for second order elliptic equations.
    • He also applied methods of functional analysis, at the same time as Mark Vishik but independently of him, to the investigation of boundary value problems for degenerate second order elliptic partial differential equations.
    • Mikhlin also studied the finite element approximation in weighted Sobolev spaces related to the numerical solution of degenerate elliptic equations.
    • The fourth branch of his research in numerical mathematics is a method for solving Fredholm integral equations which he called the "resolvent method".
    • This eliminates the need to construct and solve large systems of equations.

  4. Cercignani biography
    • The papers Elementary solutions of the linearized gas-dynamics Boltzmann equation and their application to the slip-flow problem and Solutions of linearized gas-dynamics Boltzmann equation and application to slip-flow problem appeared in print in 1962 while, in the following year, he published further papers, including two written jointly with his physics advisor Sergio Albertoni, namely Numerical Evaluation of the Slip Coefficient and Slip-coefficient expression is derived using an exact analytical solution of the slip flow problem.
    • As a mathematician, Cercignani obtained important results in the theory of Partial Differential Equations, Semigroup Theory, Monte-Carlo Methods, Spectral Theory, Riemann-Hilbert Problems, Fourier Analysis, and Functional Analysis.
    • Cercignani devoted a particular attention to the H-theorem, and to the problem of how macroscopic, irreversible evolution equations can follow from microscopic, reversible motion equations.
    • Cercignani has been one of the most active researchers of the Boltzmann equation and related topics.
    • In 1975 he published his most famous treatise 'The Boltzmann equation and its applications', which collects and unifies numerous results on the Boltzmann equation previously scattered in hundreds of references.
    • During the 1980's, he studied the evaporation-condensation interface between a gas and a liquid, stating an important conjecture for the long-time behaviour of the Boltzmann equation solutions.
    • Let us note the diverse fields to which he contributed: to the kinetic theory of rarefied gases, models of turbulence, the transport of neutrons and of semiconductors, the Boltzmann equation and its applications which have proved useful in nanotechnology.
    • Almost all of these show Carlo's passion for the Boltzmann equation, and more generally everything related to Boltzmann.
    • In 45 years of research, the Boltzmann equation led Carlo to work in theoretical mechanics, partial differential equations, numerical analysis, semigroup theory, spectral theory, Riemann-Hilbert problems, Fourier analysis, and many other areas.
    • A few years ago, a collaboration with Sasha Bobylev on self-similar solutions of the Boltzmann equation even led him to a new pretty formula for the inversion of the Laplace transform.
    • An early classic was Theory and Application of the Boltzmann Equation (1975).
    • The Boltzmann equation connects the discrete motion of the individual particles in a gas with the continuous motion of the gas as a whole.
    • As with many "basic" equations, its Achilles' heel is its complexity.
    • He updated this book thirteen years later, publishing it under the slightly revised title The Boltzmann Equation and its Applications (1988).
    • This book gives a complete exposition of the present status of the theory of the Boltzmann equation and its applications.
    • The Boltzmann equation, an integro-differential equation established by Boltzmann in 1872 to describe the state of a dilute gas, still forms the basis for the kinetic theory of gases.
    • But the main exposition is tied to the classical equation established by Boltzmann.
    • I believe that Cercignani has done us all an enormous service by providing books with a high level of clarity such as this one, and I recommend it for anyone interested in the Boltzmann equation.
    • The scientific work of Carlo Cercignani in fluid mechanics is dominated by a significant contribution to the kinetic theory of gases and the properties of the Boltzmann equation.

  5. Bezout biography
    • As we have indicated Bezout is famed for being a writer of textbooks but he is famed also for his work on algebra, in particular on equations.
    • His first paper on the theory of equations Sur plusieurs classes d'equations de tous les degres qui admettent une solution algebrique examined how a single equation in a single unknown could be attacked by writing it as two equations in two unknowns.
    • It is known that a determinate equation can always be viewed as the result of two equations in two unknowns, when one of the unknowns is eliminated.
    • Of course on the face of it this does not help solve the equation but Bezout made the simplifying assumption that one of the two equations was of a particularly simple form.
    • For example he considered the case when one of the two equations had only two terms, the term of degree n and a constant term.
    • Already this paper had introduced the topic to which Bezout would make his most important contributions, namely methods of elimination to produce from a set of simultaneous equations, a single resultant equation in one of the unknowns.
    • He also did important work on the use of determinants in solving equations.
    • This appears in a paper Sur le degre des equations resultantes de l'evanouissement des inconnues which he published in 1764.
    • As a result of the ideas in this paper for solving systems of simultaneous equations, Sylvester, in 1853, called the determinant of the matrix of coefficients of the equations the Bezoutiant.
    • These and further papers published by Bezout in the theory of equations were gathered together in Theorie generale des equations algebraiques which was published in 1779.
    • The degree of the final equation resulting from any number of complete equations in the same number of unknowns, and of any degrees, is equal to the product of the degrees of the equations.
    • By a complete equation Bezout meant one defined by a polynomial which contains terms of all possible products of the unknowns whose degree does not exceed that of the polynomial.
    • nor could he even label his equations with a suffix notation.
    • History Topics: Quadratic cubic and quartic equations .

  6. Ladyzhenskaya biography
    • It was here where she first started studying algebra, number theory and subsequently partial differential equations.
    • She became interested in the theory of partial differential equations due to the influence of Petrovsky as well as the book by Hilbert and Courant.
    • Being a talented student, the authorities often ignored absences at compulsory lectures while she attended research seminars including the algebra seminars of Kurosh and Delone and the seminar on differential equations headed by Stepanov, Petrovsky, Tikhonov, Vekua and their students and colleagues.
    • At the end of her fourth year she organized a youth seminar to study the theory of partial differential equations and persuaded Myshkis, a student of Petrovsky, to go with her to ask Petrovsky to chair the seminar.
    • Find the least restrictive conditions on the behaviour of parabolic equations under which the uniqueness theorem holds for the Cauchy problem.
    • For hyperbolic equations, construct convergent difference schemes for the Cauchy problem and for initial-boundary problems.
    • It was also here that she was strongly influenced to study the equations of mathematical physics.
    • In 1949 Olga defended her doctoral dissertation (comparable to an habilitation) which was on the development of finite differences methods for linear and quasilinear hyperbolic systems of partial differential equations, formally supervised by Sobolev though in practice it was Smirnov.
    • Her first book published in 1953 called Mixed Problems for a Hyperbolic Equation used the finite difference method to prove theoretical results, mainly the solvability of initial boundary-value problems for general second-order hyperbolic equations.
    • As in the previous decade, during the 1960s she continued obtaining results about existence and uniqueness of solutions of linear and quasilinear elliptic, parabolic, and hyperbolic partial differential equations.
    • She then studied the equations of elasticity, the Schrodinger equation, the linearized Navier-Stokes equations, and Maxwell's equations.
    • The Navier-Stokes equations were of great interest to her and continued to be so for the rest of her life.
    • Many papers written jointly by Olga and Nina Ural'tseva were devoted to the investigation of quasilinear elliptic and parabolic equations of the second order.
    • At the start of the last century Sergei Bernstein proposed an approach to the study of the classical solvability of boundary-value problems for equations based on a priori estimates for solutions as well as describing conditions that are necessary for such solvability.
    • From the mid-1950's Olga and her students made advances in the study of boundary-value problems for quasilinear elliptic and parabolic equations.
    • They developed a complete theory for the solvability of boundary-value problems for uniformly parabolic and uniformly elliptic quasilinear second-order equations and of the smoothness of generalized solutions.
    • One result gave the solution of Hilbert's 19th problem for one second-order equation.
    • When Olga first started to work on the Navier-Stokes equation, she was unaware of the work of Leray and Eberhard Hopf.

  7. Li Zhi biography
    • This was a notation for an equation and, although the work of Li Zhi is the earliest source of the method, it must have been invented before his time.
    • Here the numbers which in our notation correspond to the coefficients of the equation are placed above each other so that the coefficient of x is placed above the constant, the coefficient of x2 is placed above the coefficient of x etc.
    • Unlike most western algebraists, Li Zhi never explains how to solve equations, but only how to construct them.
    • But he does not limit his reflections to equations of degree two or three; for him, the fact that polynomial equations of arbitrarily high degree are involved is of little importance.
    • Moreover, he never explains what he understands by an equation, an unknown, a negative number, etc., but only describes the manipulations which should be carried out in specific problems, without worrying about arranging his text in terms of definitions, rules and theorems.
    • To solve the above equation Li Zhi would bring the leading coefficient to -1 and then give the solution; in this case 20.
    • The type of problem which worried mathematicians in Islamic countries, and in Europe, concerning the solution of cubic, quartic, and higher order equations did not seem to arise in China.
    • Li Zhi seems happy with equations of any degree and, although methods to solve equations do not appear explicitly, one has to assume that he used methods similar to those Ruffini and Horner discovered over 600 years later.
    • If we examine Li Zhi's solution closely we see a remarkable depth of understanding of equations.
    • The problem leads to a quartic equation with a factor x + 16.
    • Li Zhi goes through the detailed, and quite hard, argument which leads to the quartic equation .
    • The central theme is the construction and formulation of quadratic equations.
    • Some of these equations are solved by the "coefficient array method" described above, but some are formulated using the tiao duan or "method of sections".
    • This older geometric style method of solving equations was used by Chinese mathematicians before Li Zhi and so the New steps in computation gives historians a unique opportunity to see the new coefficient array method beside the older method of sections.
    • This is the quadratic equation we wrote in Li Zhi's coefficient array method above.

  8. Lions biography
    • Lions has made some of the most important contributions to the theory of nonlinear partial differential equations through the 1980s and 1990s.
    • Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be.
    • The sources of partial differential equations are so many - physical, probalistic, geometric etc.
    • - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods.
    • 70 (2) (1996), 125-135.',3)">3] is his work on "viscosity solutions" for nonlinear partial differential equations.
    • The method was first introduced by Lions in joint work with M G Crandall in 1983 in which they studied Hamilton-Jacobi equations.
    • Lions and others have since applied the method to a wide class of partial differential equations, the so-called "fully nonlinear second order degenerate elliptic partial differential equations." The problem that arises is decribed in [Notices Amer.
    • such nonlinear partial differential equation simply do not have smooth or even C1 solutions existing after short times.
    • Another equally innovative piece of work by Lions was his work on the Boltzmann equation and other kinetic equations.
    • The Boltzmann equation keeps track of interactions between colliding particles, not individually but in terms of a density.
    • There are many nonlinear PDEs that are Euler equations for variational problems.
    • The first step in solving such equations by the variational method is to show that the extremum is attained.
    • The first volume covered his work on Partial differential equations and Interpolation, the second volume contained Control and Homogenization, and the third volume Numerical analysis, Scientific computation and Applications.
    • The models used in that field consist of complex sets of partial differential equations, including the Navier-Stokes equations and the equations of thermodynamics.
    • In spite of what Lions himself liked to call the 'truly diabolical' complexity of the set of partial differential equations, boundary conditions, transmission conditions, nonlinearities, physical hypotheses, etc., that appeared in those models, Lions, in collaboration with Roger Temam and Shou Hong Wang, was able to study the questions of the existence and uniqueness of solutions, to establish the existence of attractors, and to do a numerical analysis of these models.
    • Finally, with Vivette Girault, he worked until January 2001 on perfecting a finite element method using two meshes, one 'rough' and one 'fine', for the numerical simulation of the Navier-Stokes equations.

  9. Kellogg Bruce biography
    • His thesis was Hyperbolic Equations with Multiple Characteristics and he published a paper with the same title in the Transactions of the American Mathematical Society in 1959.
    • While working at Westinghouse's Bettis Atomic Power Laboratory, Kellogg published papers such as: (with L C Noderer) Scaled iterations and linear equations (1960); Another alternating-direction-implicit method (1963); Difference equations on a mesh arising from a general triangulation (1964); An alternating direction method for operator equations (1964); and (with J Spanier) On optimal alternating direction parameters for singular matrices (1965).
    • Another major theme of his research was the behaviour of solutions to partial differential equations near corners and interfaces.
    • His 1976 regularity result (with John Osborn) for the Stokes equations in a convex polygon is still frequently referenced today.
    • An alternating direction iteration method is formulated, and convergence is proved, for the solution of certain systems of nonlinear equations.
    • D F Mayers writes in a review of On the spectrum of an operator associated with the neutron transport equation (1969):- .
    • The results also prove that the transport equation itself has a unique solution for the boundary conditions considered.
    • The author determines the behaviour of the solutions of second order elliptic differential equations in two independent variables at points where two interface curves cross, where an interface curve meets the boundary, or where an interface or boundary has a discontinuous tangent.
    • The mathematical foundations of the finite element method with applications to partial differential equations (1972) begins:- .
    • We consider interface problems for an elliptic partial differential equation in two independent variables.
    • Kellogg's own summary to Discontinuous solutions of the linearized, steady state, compressible, viscous, Navier-Stokes equations (1988) is as follows:- .
    • The compressible steady state viscous Navier-Stokes equations in two space dimensions are considered.
    • The equations are linearised around a given ambient flow field.
    • It is shown that the linearised equations are not, in general, elliptic.
    • Some boundary value problems for a second-order elliptic partial differential equation in a polygonal domain are considered.
    • The highest order terms in the equation are multiplied by a small parameter, leading to a singularly perturbed problem.

  10. Miranda biography
    • His thesis, on singular integral equations of the first and second kind with non-symmetric kernel, and the related question of the integral representation of a square integrable function, was inspired by the research of Tage Gills Torsten Carleman.
    • Examples of his work around this time are: Su un problema di Minkowski (1939) which considers the problem of determining a convex surface of given Gaussian curvature; Su alcuni sviluppi in serie procedenti per funzioni non necessariamente ortogonali (1939) which examines expansion theorems in terms of the characteristic solutions of an integral equation whose kernel, although symmetric, involves the characteristic parameter; Nuovi contributi alla teoria delle equazioni integrali lineari con nucleo dipendente dal parametro (1940) which examines the development of the Hilbert-Schmidt theory for a particular type of linear integral equation; and Observations on a theorem of Brouwer (1940) which gave an elementary proof of the equivalence of Brouwer's fixed point theorem and a special case of Kronecker's index theorem.
    • Antonio Avantaggiati describes Miranda's mathematical contributions in detail in [Methods of Functional Analysis and Theory of Elliptic Equations.
    • Proceedings of international Meeting dedicated to the memory of Professor Carlo Miranda (Naples, 1983).',1)">1] and divides these contributions into the following areas: (a) Integral equations, series expansions, summation methods; (b) Harmonic mappings, potential theory, holomorphic functions; (c) Calculus of variations, differential forms, elliptic systems; (d) Numerical analysis; (e) Propagation problems; (f) Differential geometry in the large; (g) General theory for elliptic equations; and (h) Functional transformations.
    • Jesus Hernandez writes in a review of [Methods of Functional Analysis and Theory of Elliptic Equations.
    • Several of these contributions are treated with some detail: results concerning normal families and approximation theory, the equivalence between Brouwer's fixed point theorem and a result for the zeroes of some systems of continuous functions, the Cauchy-Dirichlet problem for the propagation equation, the numerical integration of the Thomas-Fermi equation, integral equations (introducing the notions of pseudofunction and singular eigenvalue), problems of differential geometry "in the large", etc.
    • His work on partial differential equations, especially on first order linear systems in dimension greater than 2 (where conformal mapping cannot be used), is very interesting, and the same thing can be said about the integration of exterior differential forms of any degree (establishing for the first time in 1953 the algebraic-topological nature of the index for some elliptic problems).
    • The proof of the maximum modulus principle for elliptic equations of order 2m is also remarkable.
    • We point out his very special interest in two of the main tools of the "modern" theory of partial differential equations, namely a priori estimates and the application of functional analysis.
    • Some attention is paid to his important book on partial differential equations.
    • This book, Equazioni alle derivate parziali di tipo ellittico (Partial differential equations of elliptic type) (1955), was translated into Russian and published in 1957, and an English translation appeared in 1970.
    • This monograph is essentially a complete and thorough review of various methods that have been introduced in the mathematical literature to prove existence theorems for problems concerning second order partial differential equations of elliptic type, both linear and nonlinear.
    • With powerful synthesis and truly wonderful discerning exposition, the author succeeds in this goal by providing students of partial differential equations with a work of fundamental importance.
    • Following his sudden death, he was honoured by the Academy of Science Physics and Mathematics of Naples who set up an award in his name for young Italian analysts specialising in the study of elliptic equations.
    • An international conference on Methods of functional analysis and theory of elliptic equations was held in his honour in Naples 13-16 September 1982.

  11. Bergman biography
    • This was Erhard Schmidt who had been awarded his doctorate by the University of Gottingen for a thesis on integral equations written under Hilbert's supervision.
    • Bergman used the theory of integral equations as developed by Erhard Schmidt and David Hilbert [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • This led him further to a general theory of integral operators that map arbitrary analytic functions into solutions of various partial differential equations.
    • While at Brown University he participated in the Summer School in 1941 Advanced instruction and research in mechanics which resulted in the publications Partial Differential Equations and Fluid dynamics.
    • Several years ago Stefan Bergman discovered that essentially the same is true for a vast class of partial differential equations which includes the potential equation as the simplest case.
    • Bergman gave explicit formulae which allow a solution of a given differential equation to derive from an arbitrarily chosen analytic function (in some instances from a pair of real functions) and proved that all solutions can be derived in this way.
    • They consider a special type of differential equation, yet more general than the potential equation, and build up a system of solutions in close analogy to the procedure followed in the theory of analytic functions.
    • In 1953 Bergman and Schiffer published Kernel functions and elliptic differential equations in mathematical physics.
    • In this book the authors collect their researches of the last few years on elliptic partial differential equations.
    • The second part lays more stress on rigor, and treats fundamental solutions, reduction of boundary value problems to integral equations, orthonormal systems and kernel functions, eigenvalue problems associated with the kernels, variational theory of domain functions, comparison domains, basic existence theorems, and dependence of solutions on the boundary data or on the coefficients of the differential equation.
    • The presentation is in an easy flowing style, and the material should prove to be a most useful guide to those interested in the more advanced theory of linear elliptic partial differential equations.
    • Bergman published Integral operators in the theory of linear partial differential equations in 1961.
    • This treatise gives a summary of the author's numerous contributions from 1926 to 1961 to the theory of solutions of linear partial differential equations in two and three real variables by means of integral operators which usually involve analytic functions of one, or sometimes two, complex variables.
    • Awards are made every year or two in: 1) the theory of the kernel function and its applications in real and complex analysis; or 2) function-theoretic methods in the theory of partial differential equations of elliptic type with attention to Bergman's operator method.

  12. Sintsov biography
    • During this period he was being advised on research topics by Vasil'ev and, following his advice, he wrote his Master's Thesis The Theory of Connexes in Space in Connection with the Theory of First Order Partial Differential Equations.
    • Clebsch constructed the geometry of a ternary connex and applied it to the theory of ordinary differential equations.
    • Of course through his many years of research his interests varied but the main areas on which he worked were the theory of conics and applications of this geometrical theory to the solution of differential equations and, perhaps most important of all, the theory of nonholonomic differential geometry.
    • These were first published during the years 1927-1940 and include: A generalization of the Enneper-Beltrami formula to systems of integral curves of the Pfaffian equation Pdx + Qdy + Rdz = 0 (1927); Properties of a system of integral curves of Pfaff's equation, Extension of Gauss's theorem to the system of integral curves of the Pfaffian equation Pdx + Qdy + Rdz = 0 (1927); Gaussian curvature, and lines of curvature of the second kind (1928); The geometry of Mongian equations (1929); Curvature of the asymptotic lines (curves with principal tangents) for surfaces that are systems of integral curves of Pfaffian and Mongian equations and complexes (1929); On a property of the geodesic lines of the system of integral curves of Pfaff's equation (1936); Studies in the theory of Pfaffian manifolds (special manifolds of the first and second kind) (1940) and Studies in the theory of Pfaffian manifolds (1940).
    • There he studied the geometry of Monge equations and he introduced the important ideas of asymptotic line curvature of the first and second kind.
    • In 1903 he published two papers on the functional equation f (x, y) + f (y, z) = f (x, z), now called the 'Sintsov equation,' which are discussed by Detlef Gronau in [Notices of the South African Mathematical Society 31 (1) (2000), 1-8.',4)">4].
    • But before, it was Moritz Cantor who proposed these equations (there are two equations).
    • Cantor quotes these equations as examples of equations in three variables which can be solved by the method of differential calculus due to Niels Henrik Abel.

  13. Fredholm biography
    • As was always the case with all the deep mathematical results which Fredholm produced, this result was inspired by mathematical physics, in this case by the heat equation.
    • His 1898 doctoral dissertation involved a study of partial differential equations, the study of which was motivated by an equilibrium problem in elasticity.
    • He solved his operator equation in the particular cases which arise in the study of the physical problem in his thesis (and in the paper which appeared in 1900 based on that thesis) while the general case was solved by Fredholm somewhat later and not published until 1908.
    • Fredholm is best remembered for his work on integral equations and spectral theory.
    • Two years later in Stockholm a lecture about the 'principal solutions' of Roux and their connections with Volterra's equation led to a vivid discussion Finally, after a long silence Fredholm spoke and remarked in his usual slow drawl: in potential theory there is also such an equation.
    • In 1900 a preliminary report on his theory of Fredholm integral equations was published as Sur une nouvelle methode pour la resolution du probleme de Dirichlet.
    • Volterra had earlier studied some aspects of integral equations but before Fredholm little had been done.
    • Of course Riemann, Schwarz, Carl Neumann, and Poincare had all solved problems which now came under Fredholm's general case of an integral equation; this was an indication of how powerful his theory was.
    • Hilbert immediately saw the he importance of Fredholm's theory, and during the first quarter of the 20th century the theory of integral equations became a major research topic.
    • Fredholm published a fuller version of his theory of integral equations in Sur une classe d'equations fonctionelle which appeared in Acta Mathematica in 1903.
    • Hilbert extended Fredholm's work to include a complete eigenvalue theory for the Fredholm integral equation.
    • Fredholm's work on integral equations was met with great interest and boosted the morale and self-respect of Swedish mathematicians who so far had been working under the shadow of the continental cultural empires Germany and France.
    • Integral equations had now become a new mathematical tool not confined to symmetrical kernels.
    • Unlikely as it sounds, he built his first violin from half a coconut, while he also used his talents at building machines to make one to solve differential equations.
    • Fredholm received many honours for his mathematical contributions, including the V A Wallmarks Prize for the theory of differential equations in 1903, the Poncelet Prize from the French Academy of Sciences in 1908, and an honorary doctorate from the University of Leipzig in 1909.

  14. Marchenko biography
    • Marchenko's scientific interests generally centre around problems in mathematical analysis, the theory of differential equations, and mathematical physics.
    • Also in the 1950s he studied the asymptotic behaviour of the spectral measure and of the spectral function for the Sturm-Liouville equation.
    • He is well known for his original results in the spectral theory of differential equations, including the discovery of new methods for the study of the asymptotic behaviour of spectral functions and the convergence expansions in terms of eigenfunctions.
    • He also obtained fundamental results in the theory of inverse problems in spectral analysis for the Sturm-Liouville and more general equations.
    • With great success, Marchenko applied his methods to the Schrodinger equation.
    • Besides the basic results on the structure of the spectrum and the eigenfunction expansion of regular and singular Sturm-Liouville problems, it is in this domain that one-dimensional quantum scattering theory, inverse spectral problems and, of course, the surprising connections of the theory with nonlinear evolution equations first become related.
    • The periodic case of the Korteweg-de Vries equation was solved by Marchenko in 1972.
    • In addition to the important monographs mentioned above, other major texts written by Marchenko include Nonlinear equations and operator algebras (1986).
    • We systematically present a method for solving some physically important nonlinear equations that is based on the replacement of a given equation by an equation of the same form with respect to functions that take values in an arbitrary operator algebra.
    • The solution of an operator equation in the form of a travelling wave (a one-soliton solution) is elementary.
    • The solutions of the original equation are obtained from the one-soliton operator solutions by bordering them with special finite-dimensional projectors.
    • Arbitrariness in the choice of the operator algebra and the bordering projectors allows us to find broad classes of solutions of the Korteweg-de Vries, Kadomtsev-Petviashvili, nonlinear Schrodinger, sine-Gordon, Toda lattice, Langmuir and other equations.
    • In 1992 Marchenko's monograph Orthogonal functions of a discrete argument and their application in geophysics was published and in 2005, in collaboration with Evgeni Yakovlevich Khruslov, he wrote Homogenization of partial differential equations.

  15. Abel biography
    • While in his final year at school, however, Abel had begun working on the solution of quintic equations by radicals.
    • In 1823 Abel published papers on functional equations and integrals in a new scientific journal started up by Hansteen.
    • In Abel's third paper, Solutions of some problems by means of definite integrals he gave the first solution of an integral equation.
    • Abel began working again on quintic equations and, in 1824, he proved the impossibility of solving the general equation of the fifth degree in radicals.
    • Geometers have occupied themselves a great deal with the general solution of algebraic equations and several among them have sought to prove the impossibility.
    • The second of these explanations does seem the more likely, especially since Gauss had written in his thesis of 1801 that the algebraic solution of an equation was no better than devising a symbol for the root of the equation and then saying that the equation had a root equal to the symbol.
    • He had been working again on the algebraic solution of equations, with the aim of solving the problem of which equations were soluble by radicals (the problem which Galois solved a few years later).
    • Also after Abel's death unpublished work on the algebraic solution of equations was found.
    • If every three roots of an irreducible equation of prime degree are related to one another in such a way that one of them may be expressed rationally in terms of the other two, then the equation is soluble in radicals.
    • An extract from Abel's On the algebraic resolution of equations (1824) .

  16. Carleman biography
    • One reason was that many of his results, for instance the extension of Holmgren's uniqueness theorem, the analysis of the Schrodinger operator, and the existence theorem for Boltzmann's equation, were two decades ahead of their time and therefore not immediately appreciated.
    • As it is often the case with mathematicians who deal with differential or integral equations, Carleman carried a keen interest in the relationship between mathematics and applied sciences.
    • Before his professorship in Lund he published about thirty papers, the majority treating of the problems in the theory of integral equations, and the theory of real and complex functions, where he gave extraordinary evidence of originality, penetration and capacity to use various methods of analysis.
    • One of them is his fundamental contribution on singular integral equations and applications.
    • His first book Singular integral equations with real and symmetric kernel published in 1923 became fundamental.
    • Carleman is now remembered for remarkable results in integral equations (1923), quasi-analytic functions (1926), harmonic analysis (1944), trigonometric series (1918-23), approximation of functions (1922-27) and Boltzmann's equation (1944).
    • Names such as Carleman inequality, Carleman theorems (Denjoy-Carleman theorem on quasi-analytic classes of functions, Carleman theorem on conditions of well-definedness of moment problems, Carleman theorem on uniform approximation by entire functions, Carleman theorem on approximation of analytic functions by polynomials in the mean), Carleman singularity of orthogonal system, integral equation of Carleman type, Carleman operator, Carleman kernel, Carleman method of reducing an integral equation to a boundary value problem in the theory of analytic functions, Jensen-Carleman formula in complex analysis, Carleman continuum, Carleman linearization or Carleman embedding technique, Carleman polynomials, Carleman estimate in the unique continuation problem for solutions of partial differential equations and Carleman system in the kinetic theory of gas are well-known in mathematics (see [Encyclopaedia of Mathematics 2 (Kluwer 1988), 25-26.
    • In 1932 Carleman, following an idea of Poincare, showed that a finite dimensional system of nonlinear differential equations d u/dt = V(u), where Vk are polynomials in u, can be embedded in an infinite system of linear differential equations.
    • Results on unique continuation for solutions to partial differential equations are important in many areas of applied mathematics, in particular in control theory and inverse problems.
    • Carleman lectured at the Sorbonne in 1937 on Boltzmann's equation, which appears in the kinetic theory of gas, and published several papers on this subject.
    • Also his last book Mathematical problems of the kinetic theory of gas which deals with the mathematical aspects of the Boltzmann transport equation was published, after his death, in 1957 with some additional material submitted by L Carleson and O Frostman.

  17. Al-Khwarizmi biography
    • Having introduced the natural numbers, al-Khwarizmi introduces the main topic of this first section of his book, namely the solution of equations.
    • His equations are linear or quadratic and are composed of units, roots and squares.
    • He first reduces an equation (linear or quadratic) to one of six standard forms: .
    • Here "al-jabr" means "completion" and is the process of removing negative terms from an equation.
    • The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation.
    • Al-Khwarizmi then shows how to solve the six standard types of equations.
    • For example to solve the equation x2 + 10 x = 39 he writes [Muhammad ibn Musa Al-Khwarizmi : Algebra (London, 1831).',11)">11]:- .
    • The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned.
    • in his introductory section al-Khwarizmi uses geometrical figures to explain equations, which surely argues for a familiarity with Book II of Euclid's "Elements".
    • Al-Khwarizmi's concept of algebra can now be grasped with greater precision: it concerns the theory of linear and quadratic equations with a single unknown, and the elementary arithmetic of relative binomials and trinomials.
    • From its true emergence, algebra can be seen as a theory of equations solved by means of radicals, and of algebraic calculations on related expressions..
    • The final part of the book deals with the complicated Islamic rules for inheritance but require little from the earlier algebra beyond solving linear equations.
    • Al-Khwarizmi and quadratic equations .
    • History Topics: Quadratic, cubic and quartic equations .

  18. Euler biography
    • The core of his research program was now set in place: number theory; infinitary analysis including its emerging branches, differential equations and the calculus of variations; and rational mechanics.
    • Studies of number theory were vital to the foundations of calculus, and special functions and differential equations were essential to rational mechanics, which supplied concrete problems.
    • He introduced beta and gamma functions, and integrating factors for differential equations.
    • He discovered the Cauchy-Riemann equations in 1777, although d'Alembert had discovered them in 1752 while investigating hydrodynamics.
    • As well as investigating double integrals, Euler considered ordinary and partial differential equations in this work.
    • Problems in mathematical physics had led Euler to a wide study of differential equations.
    • He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others.
    • When considering vibrating membranes, Euler was led to the Bessel equation which he solved by introducing Bessel functions.
    • Euler here also begins developing the kinematics and dynamics of rigid bodies, introducing in part the differential equations for their motion.
    • He published a number of major pieces of work through the 1750s setting up the main formulae for the topic, the continuity equation, the Laplace velocity potential equation, and the Euler equations for the motion of an inviscid incompressible fluid.
    • However sublime are the researches on fluids which we owe to Messrs Bernoulli, Clairaut and d'Alembert, they flow so naturally from my two general formulae that one cannot sufficiently admire this accord of their profound meditations with the simplicity of the principles from which I have drawn my two equations ..
    • History Topics: Quadratic, cubic and quartic equations .
    • History Topics: Pell's equation .

  19. Gershgorin biography
    • The papers he published at this time are (all in Russian): Instrument for the integration of the Laplace equation (1925); On a method of integration of ordinary differential equations (1925); On the description of an instrument for the integration of the Laplace equation (1926); and On mechanisms for the construction of functions of a complex variable (1926).
    • He became Professor at the Institute of Mechanical Engineering in Leningrad in 1930, and from 1930 he worked in the Leningrad Mechanical Engineering Institute on algebra, theory of functions of complex variables, numerical methods and differential equations.
    • These were on the theory of elasticity, the theory of vibrations, the theory of mechanisms, methods of approximate numerical integration of differential equations and on other areas of mechanics and applied mathematics.
    • Gershgorin proposed an original and intricate mechanism for solving the Laplace equation, and he described such a device in detail in 'Instrument for the integration of the Laplace equation' (1925).
    • In 1929 Gershgorin published On electrical nets for approximate solution of the differential equation of Laplace (Russian) in which he gave a method for finding approximate solutions to partial differential equations by constructing a model based on networks of electrical components.
    • In the following year he published Fehlerabschatzung fur das Differenzverfahren zur Losung partieller Differentialgleichungen in which he made a careful analysis of the convergence of finite-difference approximation methods for solving the Laplace equation.
    • In this paper we present a method of conformal mapping of a given (finite or infinite) connected domain onto a disk, which is based on reducing the problem to a Fredholm integral equation.
    • L Lichtenstein [in 'Zur Theorie der konformen Abbildung: Konforme Abbildung nicht-analytischer, singularitatenfreier Flachenstucke auf ebene Gebiete' (1916)] had reduced that important problem to the solution of a Fredholm integral equation.
    • Independently of Lichtenstein, Gershgorin utilised Nystrom's method and reduced that conformal transformation problem to the same Fredholm integral equation.
    • Later, A M Banin solved the Lichtenstein-Gershgorin integral equation approximately, by reducing it to a finite system of linear differential equations.

  20. Bhaskara II biography
    • He reached an understanding of the number systems and solving equations which was not to be achieved in Europe for several centuries.
    • Bhaskaracharya studied Pell's equation px2 + 1 = y2 for p = 8, 11, 32, 61 and 67.
    • An example from Chapter 12 on the kuttaka method of solving indeterminate equations is the following:- .
    • The topics are: positive and negative numbers; zero; the unknown; surds; the kuttaka; indeterminate quadratic equations; simple equations; quadratic equations; equations with more than one unknown; quadratic equations with more than one unknown; operations with products of several unknowns; and the author and his work.
    • Equations leading to more than one solution are given by Bhaskaracharya:- .
    • The problem leads to a quadratic equation and Bhaskaracharya says that the two solutions, namely 16 and 48, are equally admissible.
    • The kuttaka method to solve indeterminate equations is applied to equations with three unknowns.
    • The problem is to find integer solutions to an equation of the form ax + by + cz = d.
    • Pell's equation .
    • History Topics: Pell's equation .

  21. Lorenz Edward biography
    • The paper A generalization of the Dirac equations appeared in the Proceeding of the National Academy of Sciences in 1941.
    • in 1948 after submitting the dissertation A Method of Applying the Hydrodynamic and Thermodynamic Equations to Atmospheric Models.
    • A generalized vorticity equation for a two-dimensional spherical earth is obtained by eliminating pressure from the equations of horizontal motion including friction.
    • The generalized vorticity equation is satisfied by formal infinite series representing the density and wind fields.
    • An approximate differential equation is presented, relating the change in speed of the zonal westerly winds to the contemporary zonal wind-speed and the meridional flow of absolute angular momentum.
    • This equation is tested statistically by means of values of the momentum flow and the zonal wind-speed, computed with the aid of the geostrophic-wind approximation, from pressure and height data extracted from analyzed northern-hemisphere maps.
    • He was using a computer to investigate models of the atmosphere which he had devised involving twelve differential equations.
    • Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow.
    • Solutions of these equations can be identified with trajectories in phase space.
    • The set of equations and the attractors described by this set of equations are now called the 'Lorenz equations' and 'Lorenz attractors', respectively.
    • Another account of aperiodic behaviour in ordinary differential equations, and difference equations, in which Lorenz describes how he arrived, starting from the description of convection in meteorology, at the Lorenz equations is contained in his paper On the prevalence of aperiodicity in simple systems delivered at the Biennial Seminar of the Canadian Mathematical Congress in Calgary, Canada, in 1978.

  22. Tartaglia biography
    • The first person known to have solved cubic equations algebraically was del Ferro but he told nobody of his achievement.
    • For mathematicians of this time there was more than one type of cubic equation and Fior had only been shown by del Ferro how to solve one type, namely 'unknowns and cubes equal to numbers' or (in modern notation) x3 + ax = b.
    • As negative numbers were not used this led to a number of other cases, even for equations without a square term.
    • In fact Tartaglia had also discovered how to solve one type of cubic equation since his friend Zuanne da Coi had set two problems which had led Tartaglia to a general solution of a different type from that which Fior could solve, namely 'squares and cubes equal to numbers' or (in modern notation) x3 + ax2 = b.
    • As public lecturer of mathematics at the Piatti Foundation in Milan, he was aware of the problem of solving cubic equations, but, until the contest, he had taken Pacioli at his word and assumed that, as Pacioli stated in the Suma published in 1494, solutions were impossible.
    • To Tartaglia's dismay, the governor was temporarily absent from Milan but Cardan attended to his guest's every need and soon the conversation turned to the problem of cubic equations.
    • Based on Tartaglia's formula, Cardan and Ferrari, his assistant, made remarkable progress finding proofs of all cases of the cubic and, even more impressively, solving the quartic equation.
    • Cardan and Ferrari travelled to Bologna in 1543 and learnt from della Nave that it had been del Ferro, not Tartaglia, who had been the first to solve the cubic equation.
    • In 1545 Cardan published Artis magnae sive de regulis algebraicis liber unus, or Ars magna as it is more commonly known, which contained solutions to both the cubic and quartic equations and all of the additional work he had completed on Tartaglia's formula.
    • For all the brilliance of his discovery of the solution to the cubic equation problem, Tartaglia was still a relatively poor mathematics teacher in Venice.
    • Ferrari clearly understood the cubic and quartic equations more thoroughly, and Tartaglia decided that he would leave Milan that night and thus leave the contest unresolved.
    • Fairly early in his career, before he became involved in the arguments about the cubic equation, he wrote Nova Scientia (1537) on the application of mathematics to artillery fire.
    • Quadratic, cubic and quartic equations .
    • History Topics: Quadratic, cubic and quartic equations .

  23. Wang Xiaotong biography
    • The important innovation which is incorporated in most of these problems is that they reduce to a cubic equation which Wang solves numerically.
    • We do not know of any earlier Chinese work on cubic equations.
    • Of course one has to understand that when we say that the text is concerned with cubic equations, we do not see expressions with x, x2 and x3 in them.
    • Rather the equations are expressed in words and Wang thinks in a geometrical way.
    • For example where we might say "Let the height be x" and then produce an equation in x, Wang writes:- .
    • He then goes on to set up a cubic equation for the height.
    • when he is about to set up a cubic equation for the depth.
    • In setting up cubic equations Wang Xiaotong utilised a rule which is the same as the "tian yuan".
    • Data given for the work done by the workers allows the volume to be calculated, and a cubic equation is arrived at for x.
    • To be able to solve this problem Wang has not only to be able to set up a cubic equation and solve it, but he also needs to know a formula for the volume of his dyke with trapezoidal ends and varying cross-section.
    • Wang calls a the unknown and finds a cubic equation in terms of a.
    • Writing x for the unknown a, we have the cubic equation .
    • Try to set up the necessary equations in these two cases in a similar way to our solution to Problem 15 above.
    • Not only did Wang's work influence later Chinese mathematicians, but it is said that it was his ideas on cubic equations which Fibonacci learnt, probably first transmitted into the Islamic/Arabic world, and then brought to Europe.

  24. Al-Tusi Sharaf biography
    • What is in this Treatise on equations by al-Tusi? Basically it is a treatise on cubic equations, but it does not follow the general development that came through al-Karaji's school of algebra.
    • it represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.
    • In the treatise equations of degree at most three are divided into 25 different types.
    • First al-Tusi discusses twelve types of equation of degree at most two.
    • He then looks at eight types of cubic equation which always have a positive solution, then five types which may have no positive solution.
    • We illustrate the method by showing how al-Tusi examined one of the five types of equation which under certain conditions has a solution, namely the equation x3 + a = bx, where a, b are positive.
    • Al-Tusi's first comment is that if t is a solution to this equation then t3 + a = bt and, since a > 0, t3 < bt so t < √b.
    • Thus the equation bx - x3 = a has a solution if a ≤ 2(b/3)3/2.
    • Then Al-Tusi deduces that the equation has a positive root if .
    • where D is the discriminant of the equation.
    • Al-Tusi then went on to give what we would essentially call the Ruffini-Horner method for approximating the root of the cubic equation.
    • Although this method had been used by earlier Arabic mathematicians to find approximations for the nth root of an integer, al-Tusi is the first that we know who applied the method to solve general equations of this type.

  25. Graffe biography
    • Graffe is best remembered for his "root-squaring" method of numerical solution of algebraic equations, developed to answer a prize question posed by the Berlin Academy of Sciences.
    • This was not his first numerical work on equations for he had published Beweis eines Satzes aus der Theorie der numerischen Gleichungen in Crelle's Journal in 1833.
    • Here, however, the prize decision would not have been made until the year 1838 and, on the other hand, the author flatters himself that his method for calculating the roots of numerical equations deserves to be considered even if other methods should lead more quickly to the result.
    • The Preface continues, explaining that he also presents previous attempts by other authors at giving methods to calculate the imaginary roots of an equation.
    • Algebraic equations are very often the subject of mathematical research, partly because of the remarkable relationships they offer and partly because of their versatile use.
    • Perhaps it is also the fact that for the general solution of equations that exceed the 4th degree, insuperable obstacles seem to stand in the way, which gives a peculiar charm to these investigations, which almost every mathematician is trying to use his powers to consider.
    • The process can be applied recursively obtaining equations whose roots are the fourth powers of the original roots, then the eighth powers etc.
    • The law by which the new equations are constructed is exceedingly simple.
    • If, for example, the coefficient of the fourth term of the given equation is c3, then the corresponding coefficient of the first transformed equation is c32 - 2c2c4 + 2c1c5 - 2c6.
    • justifying Graffe's principle and perfecting his method for finding the imaginary roots of an equation.
    • As presented by Graffe, the method is only applicable to the case where all the roots of the original equation are distinct but later improvements did away with this condition.
    • Lobachevsky, however, only seems to be thinking of the "root squaring" method as a way to calculate the largest root, not as a method for calculating all the roots of an equation.

  26. Morton biography
    • A discussion on numerical analysis of partial differential equations which he published in 1971 gives both a Culham and a University of Reading address for Morton.
    • Of all the partial differential equations which are solved numerically, those arising from fluid flow problems are certainly among the most important.
    • There is, however, a more pertinent reason for singling out this application area in discussing the numerical analysis of partial differential equations.
    • Each particular field has its own complicating difficulties and the equations necessary to describe physically interesting phenomena are often formidable.
    • It would therefore be inappropriate to work here with complete systems of realistic equations: instead it will be our aim to bring out some of the important points common to the whole class of fluid flow problems by using in each case the simplest model equations adequate for our purpose.
    • We should mention two important books he published: (with David F Mayers) Numerical Solution of Partial Differential Equations: an introduction (1994, 2nd edition 2005); and Numerical Solution of Convection-Diffusion Problems (1996).
    • This book is a solid introduction to numerical methods for partial differential equations.
    • The book includes parabolic, hyperbolic, and elliptic equations, each section starting with an analysis of the behaviour of solutions of the partial differential equations.
    • A very desirable feature of the book is that it goes beyond the usual investigation of the heat equation, the wave equation, and Poisson's equation.
    • Professor Keith William (Bill) Morton of the University of Oxford in recognition of his seminal contributions to the field of numerical analysis of partial differential equations and its applications and for services to his discipline.
    • The London Mathematical Society is proud to honour a mathematician who has changed the way we look at the numerical analysis of partial differential equations through his world-leading research results, his vision and his dynamic leadership qualities.
    • He gave the lecture Evolution Operators and Numerical Modelling of Hyperbolic Equations in the Mathematical Institute, Oxford.

  27. Lax Peter biography
    • He received his PhD in 1949, also from New York University, for his thesis Nonlinear System of Hyperbolic Partial Differential Equations in Two Independent Variables.
    • She was awarded a PhD in 1955 for her thesis Cauchy's Problem for a Partial Differential Equation with Real Multiple Characteristics.
    • for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.
    • The equations that arise in such fields as aerodynamics, meteorology and elasticity are nonlinear and much more complex: their solutions can develop singularities.
    • In the 1950s and 1960s, Lax laid the foundations for the modern theory of nonlinear equations of this type (hyperbolic systems).
    • Inspired by Richtmyer, Lax established with this theorem the conditions under which a numerical implementation gives a valid approximation to the solution of a differential equation.
    • Lax became fascinated by these mysterious solutions and found a unifying concept for understanding them, rewriting the equations in terms of what are now called "Lax pairs".
    • This phenomenon occurs not only for fluids, but also, for instance, in atomic physics (Schrodinger equation).
    • Their work also turned out to be important in fields of mathematics apparently very distant from differential equations, such as number theory.
    • In this monograph, written more than twenty years ago, we based our scattering theory on the wave equation rather than the Schrodinger equation.
    • Following up on a hint in Gelfand's address to the 1962 Stockholm International Congress, they showed that the Lax-Phillips scattering theory, applied to the wave equation appropriate to hyperbolic space, is a natural tool in the theory of automorphic functions.
    • Yet during the past five decades there has been an unprecedented outburst of new ideas about how to solve linear equations, carry out least square procedures, tackle systems of linear inequalities, and find eigenvalues of matrices.

  28. Mazya biography
    • Solution of Dirichlet's problem for an equation of elliptic type (Russian) was published in 1959 and Classes of domains and imbedding theorems for function spaces (Russian) in 1960.
    • He published the two papers Some estimates of solutions of second-order elliptic equations (Russian) and p-conductivity and theorems on imbedding certain functional spaces into a C-space (Russian) in 1961, and then four further papers in 1962, the year in which he was awarded his Candidate degree (equivalent to a doctorate) from Moscow State University.
    • The Dirichlet problem for an arbitrary order elliptic equation in a domain with a cut off tubular neighbourhood of a smooth closed submanifold is considered in the second chapter.
    • The fourth chapter deals with asymptotic expansions of solutions to a quasilinear equation of the second order.
    • In 1997 (with Vladimir Kozlov) Maz'ya published Theory of a higher-order Sturm-Liouville equation which Eastham summarises by writing that:- .
    • the authors have identified a special type of higher-order analogue of the hyperbolic Sturm-Liouville equation and they have developed a coherent theory based on the Green's function.
    • One year later, in 1999, Maz'ya, together with Vladimir Kozlov, published Differential equations with operator coefficients with applications to boundary value problems for partial differential equations.
    • All the proofs are complete and rely on undergraduate university courses on real and complex analysis and some basic facts of functional analysis and of the theory of partial differential equations.
    • For example we list a few recent works without detailing the co-authors: Spectral problems associated with corner singularities of solutions to elliptic equations (2000); Asymptotic theory of elliptic boundary value problems in singularly perturbed domains (2000); Spectral problems associated with corner singularities of solutions to elliptic equations (2001); and Linear water waves (2002).
    • In addition the American Mathematical Society published Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G Maz'ya's 70th Birthday in their Proceedings of Symposia in Pure Mathematics series.
    • in recognition of his contributions to the theory of differential equations.

  29. Diophantus biography
    • Diophantus, often known as the 'father of algebra', is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers.
    • The Arithmetica is a collection of 130 problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations.
    • The work considers the solution of many problems concerning linear and quadratic equations, but considers only positive rational solutions to these problems.
    • Equations which would lead to solutions which are negative or irrational square roots, Diophantus considers as useless.
    • To give one specific example, he calls the equation 4 = 4x + 20 'absurd' because it would lead to a meaningless answer.
    • In other words how could a problem lead to the solution -4 books? There is no evidence to suggest that Diophantus realised that a quadratic equation could have two solutions.
    • Diophantus looked at three types of quadratic equations ax2 + bx = c, ax2 = bx + c and ax2 + c = bx.
    • He solved problems such as pairs of simultaneous quadratic equations.
    • Diophantus would solve this by creating a single quadratic equation in x.
    • The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation..
    • We began this article with the remark that Diophantus is often regarded as the 'father of algebra' but there is no doubt that many of the methods for solving linear and quadratic equations go back to Babylonian mathematics.
    • History Topics: Pell's equation .

  30. Lopatynsky biography
    • His research interests then moved towards differential equations with his first paper on this topic Solution of the equation y ' = f (x, y) published in 1939, proving a general existence theorem.
    • He continued to undertake research on differential equations, but his interests now included differential operators.
    • The author studies linear (partial) differential equations from a formal algebraic point of view.
    • His treatment of this special case of the algebraic theory of algebraic differential equations yields a well-rounded ideal theory of linear differential operators; in many respects it differs essentially from the treatment due to Ritt (for instance, ideals and sums of integral manifolds are defined differently).
    • In 1945 Lopatynsky moved to Lvov where he was appointed to the chair of differential equations at Lvov University.
    • His seminar on differential equations at Lvov University attracted many mathematicians, both young men beginning their research activity and established researchers who found inspiration in the seminar.
    • Lopatynsky's research continued to impress as he continued to prove major results in the theory of systems of linear differential equations of the elliptic type.
    • In 1966 he became head of the partial differential equations Section of the Institute of Applied Mathematics and Mechanics of the Academy of Sciences of the Ukraine in Donetsk.
    • He was also appointed as Chairman of the Department of Differential Equations at Donetsk University.
    • Lopatynsky's contributions to the theory of differential equations are particularly important, with important contributions to the theory of linear and nonlinear partial differential equations.
    • He worked on the general theory of boundary value problems for linear systems of partial differential equations of elliptic type, finding general methods of solving boundary value problems.
    • he continued his studies of general boundary problems in differential equations using topological methods.
    • Recently he has obtained important results on solvability of the Cauchy problem for operator equations in Banach space and also on "almost everywhere" solvability of general linear and nonlinear boundary problems.
    • He also obtained some basic results in the solvability of the Cauchy problem for operator equations in Banach spaces.
    • In 1980 Lopatynsky published an important book Introduction to the Contemporary Theory of Partial Differential Equations.
    • This book makes the reader familiar with the basic notions and facts of algebra, topology, and functional analysis, and gives a general idea how to apply these notions to the theory of differential equations.
    • The book contains eight chapters: Sets; Basic algebraic notions; Algebraic equations; Topology; Differentiation and integration; Special linear spaces which are related to Euclidean spaces; Manifolds; and Elements of algebraic topology.
    • His next book, published in 1984 three years after his death, was entitled Ordinary differential equations.
    • We consider the basic methods of solving differential equations and methods of qualitative investigation of these solutions.
    • We emphasize the relation of the theory of differential equations to other areas of mathematics.
    • Finally let us mention that the Second International Conference for young mathematicians on Differential Equations and Applications held in November 2008 at the Donetsk National University was dedicated to Ya B Lopatinskii.
    • The conference is named after an outstanding mathematician, talented pedagogue and organiser, Academician of National Academy of Sciences of Ukraine Yaroslav Borisovich Lopatinskii who was a founder of both the Department of Differential Equations in Donetsk National University and the Department of Partial Differential Equations in the Institute of Applied Mathematics and Mechanics of National Academy of Sciences of Ukraine.
    • The conference is the continuation of the Conference on Differential Equations and Applications dedicated to the centenarian jubilee of Ya B Lopatinskii held in December 2006.

  31. Ruffini biography
    • On the other hand it gave him the chance to work on what was one of the most original of projects, namely to prove that the quintic equation cannot be solved by radicals.
    • To solve a polynomial equation by radicals meant finding a formula for its roots in terms of the coefficients so that the formula only involves the operations of addition, subtraction, multiplication, division and taking roots.
    • Quadratic equations (of degree 2) had been known to be soluble by radicals from the time of the Babylonians.
    • The cubic equation had been solved by radicals by del Ferro, Tartaglia and Cardan.
    • Certainly no mathematician has published such a claim and even Lagrange in his famous paper Reflections on the resolution of algebraic equations says he will return to the question of the solution of the quintic and, clearly, he still hoped to solve it by radicals.
    • In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals as the title shows: General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible.
    • The algebraic solution of general equations of degree greater than four is always impossible.
    • In writing this book, I had principally in mind to give a proof of the impossibility of solving equations of degree higher than four.
    • and recommend greatly the most important theorem which excludes the possibility of solving equations of degree greater than four.
    • your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgement, proves completely the impossibility of solving algebraically equations of higher than the fourth degree.

  32. Lewy biography
    • In this paper criteria are given for determining conditions which guarantee the stability of numerical solutions of certain classes of differential equations.
    • he published a series of fundamental papers on partial differential equations and the calculus of variations.
    • He solved completely the initial value problem for general non-linear hyperbolic equations in two independent variables.
    • On the basis of this, and using the daring idea of converting an elliptic equation into a hyperbolic one by penetrating into the complex domain, he developed a new proof of the analyticity of solutions of analytic elliptic equations in two independent variables, one which far exceeded the known proof in its elegance and simplicity.
    • He proved the well-posedness of the initial value problem for wave equations in what is now called Sobolev spaces two decades before these spaces became a common tool for specialists.
    • Nirenberg [D Kinderlehrer (ed.), Hans Lewy Selecta (Boston, MA, 2002).',6)">6] lists Lewy's mathematical papers under the following topics: (i) partial differential equations involving existence and regularity theory for elliptic and hyperbolic equations, geometric applications, approximation of solutions; (ii) existence and regularity of variational problems, free boundary problems, theory of minimal surfaces; (iii) partial differential equations connected with several complex variables; (iv) partial differential equations connected with water waves and fluid dynamics; (v) offbeat properties of solutions of partial differential equations.
    • Among the first papers he published after emigrating to the United States were A priori limitations for solutions of Monge-Ampere equations (two papers, the first in 1935, the second two years later), and On differential geometry in the large : Minkowski's problem (1938).
    • His paper An example of a smooth linear partial differential equation without solution (1957) gave a simple partial differential equation which has no solution, a result which had a substantial impact on the area.

  33. Ince biography
    • Ince's research was mainly on differential equations.
    • Emile Mathieu discovered the Mathieu functions, which are special cases of hypergeometric functions, in 1868 while solving the wave equation for an elliptical membrane moving through a fluid.
    • By the use of convergent infinite determinants and continued fractions, with asymptotic formulae for large values, he succeeded in making computations practicable and after eight years' devotion to this task he published in 1932 tables of eigenvalues for Mathieu's equation, and zeros of Mathieu functions.
    • These tables were useful not only in the problems originally envisaged but also in more recent investigations such as quantum-mechanical problems leading to Mathieu's equation.
    • Ince published a major text Ordinary Differential Equations (Longmans, Green and Co., London, 1926).
    • He contributed Integration of Ordinary Differential Equations and set out his aims in the Preface dated May 1939:- .
    • The object of this book is to provide in a compact form an account of the methods of integrating explicitly the commoner types of ordinary differential equation, and in particular those equations that arise from problems in geometry and applied mathematics.
    • With this qualification, it will be found to contain all the material needed by students in our Universities who do not specialize in differential equations, as well as by students of mathematical physics and technology.
    • A C Aitken and D E Rutherford wrote the Preface to the second edition of Integration of Ordinary Differential Equations of April 1943: .
    • The nucleus of an integral equation for one of the periodic Lame functions is expanded in series of products of the characteristic functions ..
    • One further paper, Simultaneous linear partial differential equations of the second order, was edited by Erdelyi after Ince's death and published in the Proceedings of the Royal Society of Edinburgh in 1942.

  34. Bateman biography
    • Bateman was awarded a Smith's prize in 1905 for an essay on differential equations.
    • Two further papers appeared in print in 1904, namely The solution of partial differential equations by means of definite integrals, and Certain definite integrals and expansions connected with the Legendre and Bessel functions.
    • It was during his visit to Gottingen that he learnt of work on integral equations being undertaken by Hilbert and his school.
    • One of these 1908 papers is his first publication on transformations of partial differential equations and their general solutions.
    • His 1908 paper was on the wave equation.
    • He is especially known for his work on special functions and partial differential equations.
    • In 1904 he extended Whittaker's solution of the potential and wave equation by definite integrals to more general partial differential equations.
    • Bateman was one of the first to apply Laplace transforms to integral equations in 1906.
    • In 1910 he solved systems of differential equations discovered by Rutherford which describe radio-active decay.
    • The finest contribution Bateman made to mathematics, however, was his work on transformations of partial differential equations, in particular his general solutions containing arbitrary functions.
    • In particular he applied his methods to equations resulting from electromagnetics, then later to those arising from hydrodynamics.
    • He wrote a number of texts that have been reprinted as classics: The mathematical analysis of electrical and optical wave-motion on the basis of Maxwell's equations (1915, reprinted 1955); Partial differential equations of mathematical physics (1932, reprinted 1944 and 1959); (written with H L Dryden and F D Murnaghan), Hydrodynamics, National Research Council, Washington, D.C.
    • (1932, reprinted 1956); and (written with A A Bennett and W E Milne), Numerical integration of differential equations (1933, reprinted 1956).
    • He only published five joint papers, one of those in 1924 being with Ehrenfest in which they looked at applications of partial differential equations to electromagnetic fields.

  35. Samoilenko biography
    • In 1963 he defended his candidate-degree thesis Application of Asymptotic Methods to the Investigation of Nonlinear Differential Equation with Irregular Right-Hand Side.
    • In 1974 Samoilenko became a professor and headed the Integral and Differential Equations section within the Department of Mechanics and Mathematics at the Kiev State University.
    • In 1987 Samoilenko was appointed head of the Department of Ordinary Differential Equations at the Institute of Mathematics of the Ukrainian Academy of Sciences in Kiev.
    • Samoilenko worked on both linear and nonlinear ordinary differential equations.
    • In the 1960s he studied nonlinear ordinary differential equations with impulsive action publishing papers such as Systems with pulses at given times (1967).
    • His work on boundary-value problems led to papers Numerical-analytic method for the investigation of systems of ordinary differential equations (2 parts both published in 1966) and many other innovative works.
    • His most original contribution was the numeric-analytic method for the study of periodic solutions of differential equations with periodic right hand side.
    • The latter is known as the method of successive changes of variables and its aim is to ensure the convergence of the iteration process in solving systems of nonlinear differential equations.
    • Their work continued over a long period and was written up in the important joint monograph Impulsive Differential Equations (Russian) in 1987.
    • In addition to the work mentioned above they worked jointly on the theory of multifrequency oscillation, then later on a system of evolutionary equations with periodic and conditional periodic coefficients.
    • This last work was done in collaboration with D Martyniuk and the three of them published, in 1984, the monograph Systems of Evolution Equations with Periodic and Quasiperiodic Coefficients (Russian) giving an excellent account of their results.
    • For example, with Mitropolskii and V L Kulik, he wrote Investigation of Dichotomy of Linear Systems of Differential Equations Using Lyapunov Functions (Russian) published in 1990.
    • In 1992 they published Numerical-analytic methods in the theory of boundary value problems for ordinary differential equations.
    • The book is devoted to the theory of generalized inverses of operators in a Banach space and its applications to linear and weakly nonlinear boundary-value problems for various classes of functional-differential equations, including systems of ordinary differential and difference equations, systems of differential equations with delay, systems with impulse action, and integro-differential systems.
    • A recent book by Samoilenko, written with Yu V Teplinskii, is Elements of the mathematical theory of evolution equations in Banach spaces (Ukranian) (2008).
    • The book Differential equations : Examples and problems (Russian) (1984) written with S A Krivosheya and N A Perestyuk contains the following authors' summary:- .
    • We give the solutions of typical problems in a course on ordinary differential equations.
    • The text is structured so as to develop practical skills in students for solving and investigating differential equations describing evolutionary processes in different fields of natural science.
    • He is on the Editorial Board of: Nonlinear Oscillations; the Ukrainian Mathematical Journal; Reports of the Ukrainian Academy of Sciences; the Bulletin of the Ukrainian Academy of Sciences; the Ukrainian Mathematical Bulletin; In the World of Mathematics; the Memoirs on Differential Equations and Mathematical Physics; the Miskolc Mathematical Notes; the Georgian Mathematical Journal; and the International Journal of Dynamical Systems and Differential Equations.

  36. Plemelj biography
    • Plemelj undertook research under von Escherich's supervision and in May 1898 was awarded his doctorate for a thesis on linear homogeneous differential equations with uniform periodical coefficients (uber lineare homogene Differentialgleichungen mit eindeutigen periodischen Koeffizienten).
    • An important mathematical event occurred while he was at Gottingen, for that was the year in which Holmgren lectured on Fredholm's theory of integral equations at Gottingen.
    • The contributions he made to integral equations and potential theory were brought together in a work he published in 1911 for which he was awarded the Prince Jablonowski Prize.
    • Riemann's problem, concerning the existence of a linear differential equation of the Fuchsian class with prescribed regular singular points and monodromy group, had been reduced to the solution of an integral equation by Hilbert in 1905.
    • Plemelj discovered equations relating to boundary values of holomorphic functions which are now called the "Plemelj formulae" and shortly after this was able to solve Riemann's problem in his paper Riemannian classes of functions with given monodromy group published in Monatshefte fur Mathematik und Physik in 1908.
    • The equations are today important in a number of different fields, including neutron transport theory where a singular integral equation is encountered.
    • Plemelj's methods for solving the Riemann's problem were further developed by Nikolai Ivanovich Mushelisvili into the theory of singular integral equations.
    • Within the theory of differential equations he worked mostly on equations of the Fuchs type and on Klein's theorems.
    • He used to hold a general course of mathematics and a three-year cycle of lectures on differential equations, the theory of analytic functions, and algebra including number theory.
    • They were The theory of analytic functions (1953), Differential and integral equations.

  37. Kerr Roy biography
    • He submitted his doctoral thesis Equations of Motion in General Relativity in 1958 and published the results of the thesis in three papers entitled The Lorentz-covariant approximation method in general relativity in Nuovo Cimento in 1959.
    • It is found that as well as the usual equations of motion and energy derived by Einstein, Infeld and Hoffman for the quasi-static approximation, there are three further equations, the equations of spin, which must be satisfied by the structural parameters of each particle.
    • These equations also appear as surface integral conditions in the quasi-static approximation.
    • It is only the differential equations satisfied by these that change in the higher orders.
    • In this first paper the equations of motion of a pole-dipole particle are calculated to the first approximation, and in the second paper this is continued to the second approximation.
    • In the third paper the method is applied to the combined Einstein-Maxwell equations.
    • In 1963, Roy Kerr, a New Zealander, found a set of solutions of the equations of general relativity that described rotating black holes.
    • In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein's equations of general relativity, discovered by the New Zealand mathematician, Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the universe.
    • Everybody who tried to solve the problem was going at it from the front, but I was trying to solve the equation from a different point of view - there were a number of new mathematical methods coming into relativity at the time and Josh [Goldberg] and I had had some success with these.
    • I was trying to look at the whole structure - the Bianchi identities, the Einstein equations and these Tetrads - to see how they fitted together and it all seemed to be pretty nice and it looked like lots of solutions were going to come out.
    • He, Papapetrou, had been trying for 30 years to find such a solution to Einstein's equation and had failed, as had other relativists.
    • In 1965, in collaboration with Alfred Schild who was a colleague at the University of Texas, Kerr published Some algebraically degenerate solutions of Einstein's gravitational field equations which introduced what are today known as Kerr-Schild spacetimes and the Kerr-Schild metric.
    • In the early 1960s Professor Kerr discovered a specific solution to Einstein's field equations which describes a structure now termed a Kerr black hole.

  38. Petryshyn biography
    • from Columbia University for his thesis Linear Transformations Between Hilbert Spaces and the Application of the Theory to Linear Partial Differential Equations.
    • In 1962, Direct and iterative methods for the solution of linear operator equations in Hilbert space was published which does much toward developing a unified point of view toward a number of important methods of solving linear equations.
    • In the same year, The generalized overrelaxation method for the approximate solution of operator equations in Hilbert space appeared and in the following year the two papers On a general iterative method for the approximate solution of linear operator equations and On the generalized overrelaxation method for operation equations.
    • His major results include the development of the theory of iterative and projective methods for the constructive solution of linear and nonlinear abstract and differential equations.
    • He has shown that the theory of A-proper type maps not only extends and unifies the classical theory of compact maps with some recent theories of condensing and monotone-accretive maps, but also provides a new approach to the constructive solution of nonlinear abstract and differential equations.
    • The theory has been applied to ordinary and partial differential equations.
    • Approximation-solvability of Nonlinear Functional and Differential Equations appeared in December 1992:- .
    • This outstanding reference/text develops an essentially constructive theory of solvability on linear and nonlinear abstract and differential equations involving A-proper operator equations in separable Banach spaces, treats the problem of existence of a solution for equations involving pseudo-A-proper and weakly-A-proper mappings, and illustrates their applications.
    • Facilitating the understanding of the solvability of equations in infinite dimensional Banach space through finite dimensional approximations, Approximation - solvability of Nonlinear Functional and Differential Equations: offers an important elementary introduction to the general theory of A-proper and pseudo-A-proper maps; develops the linear theory of A-proper maps; furnishes the best possible results for linear equations; establishes the existence of fixed points and eigenvalues for P-gamma-compact maps, including classical results; provides surjectivity theorems for pseudo-A-proper and weakly-A-proper mappings that unify and extend earlier results on monotone and accretive mappings; shows how Friedrichs' linear extension theory can be generalized to the extensions of densely defined nonlinear operators in a Hilbert space; presents the generalized topological degree theory for A-proper mappings; and applies abstract results to boundary value problems and to bifurcation and asymptotic bifurcation problems.
    • In 1995 his second monograph Generalized Topological Degree and Semilinear Equations appeared in print, published by Cambridge University Press.
    • In this monograph we develop the generalised degree theory for densely defined A-proper mappings, and then use it to study the solubility (sometimes constructive) and the structure of the solution set of [an] important class of semilinear abstract and differential equations ..
    • A-proper mappings arise naturally in the solution to an equation in infinite dimensional space via the finite dimensional approximation..
    • Using these tools, the author defines the generalised topological degree for densely defined A-proper mappings, gives applications to the solubility of an important class of semilinear abstract and differential equations, and discusses global bifurcation results.

  39. Volterra biography
    • In 1890 Volterra showed by means of his functional calculus that the theory of Hamilton and Jacobi for the integration of the differential equations of dynamics could be extended to other problems of mathematical physics.
    • During the years 1892 to 1894 Volterra published papers on partial differential equations, particularly the equation of cylindrical waves.
    • His most famous work was done on integral equations.
    • He began this study in 1884 and in 1896 he published papers on what is now called 'an integral equation of Volterra type'.
    • He continued to study functional analysi applications to integral equations producing a large number of papers on composition and permutable functions.
    • It was this principle which he applied to his celebrated researches on integral equations of Volterra's type.
    • He considered heuristically the integral equations as a limiting case of a system of linear algebraic equations and then checked his final formulae directly.
    • He studied the Verhulst equation and the logistic curve.
    • He also wrote on predator-prey equations.
    • In December 1938 he was affected by phlebitis: the use of his limbs was never recovered, but his intellectual energy was unaffected, and it was after this that his two last papers 'The general equations of biological strife in the case of historical actions' and 'Energia nei fenomeni elastici ereditarii' were published by the Edinburgh Mathematical Society and the Pontifical Academy of Sciences respectively.

  40. Dezin biography
    • The authors of [Differential Equations 44 (12) (2008), 1773-1775.',12)">12] write about Aleksei Alekseevich junior's extremely difficult upbringing:- .
    • He continued to undertake research at Moscow State University advised by Sergei Lvovich Sobolev and was awarded his candidate's degree (equivalent to a Ph.D.) in 1956 for his thesis Boundary Value Problems for Symmetric Systems of Partial Differential Equations.
    • He had begun writing research papers while still an undergraduate [Differential Equations 44 (12) (2008), 1773-1775.',12)">12]:- .
    • Before the award of his doctorate he had published several papers in Russian: On imbedding theorems and the problem of continuation of functions (1953); The second boundary problem for the polyharmonic equation in the space W2m (1954); Mixed problems for certain symmetric hyperbolic systems (1956); Concerning solvable extensions of the first order partial linear differential operators (1956); and Mixed problems for certain parabolic systems (1956).
    • papers concerned extension of functions, embedding theorems, and also an analysis of conditions for solubility of the second boundary-value problem for polyharmonic equations.
    • This was written in a period when he was publishing a particularly outstanding series of papers on invariant systems of first-order partial differential equations on smooth Riemannian manifolds.
    • These had the ultimate aim of trying to understand the structure of the Cauchy-Riemann equations in the plane.
    • These papers include Existence and uniqueness theorems for solutions of boundary problems for partial differential equations in function spaces (1959), Boundary value problems for invariant elliptic systems (1960), and Invariant elliptic systems of equations (1960).
    • An English translation with title Partial differential equations.
    • The author has intended this book to be an introduction to a general theory of boundary value problems for linear partial differential equations accessible to graduate students as well as researchers.
    • In simplest terms, this is a book about separation of variables in partial differential equations.
    • More accurately, the book may be considered an introduction to the use of spectral theory in solving initial- and two- point boundary value problems for ordinary differential equations with unbounded operator coefficients.
    • Dezin's next little book of 63 pages Equations, operators, spectra (1984) shows him to be a skilful and an innovative expositor.
    • Assume that you meet someone who knows only the elements of mathematics, for instance how to solve a system of two linear equations with two unknowns, but is eager to learn more, and you want to inform this person about the meaning of spectral theory of linear operators.
    • The potential reader is transported from very simple systems of linear equations to concepts as complex as that of linear space, invertible operator, eigenvalue and eigenvector, norm, adjoint, unitary and selfadjoint operator.
    • We describe special difference models of equations of mathematical physics, models of boundary value problems and objects of quantum mechanics.
    • He was married to Nataliya Borisovna and their home [Differential Equations 44 (12) (2008), 1773-1775.',12)">12]:- .

  41. Faddeev biography
    • Much of Faddeev's early work had been done in collaboration with Delone, particularly the highly significant results he obtained on Diophantine equations.
    • 3, 223-231.',2)">2] his early results on Diophantine equations are described:- .
    • Faddeev's very first results in Diophantine equations were remarkable.
    • He was able to extend significantly the class of equations of the third and fourth degree that admit a complete solution.
    • When he was studying, for example, the equation x3+ y3= A, Faddeev found estimates of the rank of the group of solutions that enabled him to solve the equation completely for all A ≤ 50.
    • For the equation x4+ Ay4= ±1 he proved that there is at most one non-trivial solution; this corresponds to the basic unit of a certain purely imaginary field of algebraic numbers of the fourth degree and exists only when the basic unit is trinomial.
    • This work included the results on Diophantine equations described in the above quotation and a wealth of other material.
    • Much of this work was done in collaboration with his wife Vera Nikolaevna Faddeeva but his first few papers on this topic are single authored: On certain sequences of polynomials which are useful for the construction of iteration methods for solving of systems of linear algebraic equations (1958), On over-relaxation in the solution of a system of linear equations (1958), and On the conditionality of matrices (1959).
    • The problems are grouped under seven heads: Complex Numbers, Determinants, Linear Equations, Matrices, Polynomials and Rational Functions of a single Indeterminate, Symmetric Functions, and Linear Algebra.

  42. Kantorovich biography
    • Kantorovich gave two lectures, "On conformal mappings of domains" and "On some methods of approximate solution of partial differential equations".
    • In 1936 he published On one class of functional equations (Russian) in which he applied semiordered spaces to numerical methods.
    • The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind.
    • Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations.
    • Consideration of semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form.
    • The authors are particularly concerned with applications of functional analysis to the theory of approximation and the theory of existence and uniqueness of solutions of differential and integral equations (both linear and non-linear).
    • These other areas include functional analysis and numerical analysis and within these topics he published papers on the theory of functions, the theory of complex variables, approximation theory in which he was particularly interested in using Bernstein polynomials, the calculus of variations, methods of finding approximate solutions to partial differential equations, and descriptive set theory.
    • The most interesting thing here was the paralleling of calculations for integrating the differential equation for the Bessel functions on these machines.

  43. Chazy biography
    • The research he undertook for his doctorate involved the study of differential equations, in particular looking at the methods used by Paul Painleve to solve differential equations that Henri Poincare and Emile Picard had failed to solve.
    • Chazy published several short papers while undertaking research, for instance Sur les equations differentielles dont l'integrale generale est uniforme et admet des singularites essentielles mobiles (1909), Sur les equations differentielles dont l'integrale generale possede une coupure essentielle mobile (1910) and Sur une equations differentielle du premier ordre et du premier degre (1911).
    • He was awarded his doctorate in 1911 for his thesis Sur les equations differentielles du troisieme ordre et d'ordre superieur dont l'integrale generale a ses points critiques fixes which he defended at the Sorbonne on 22 December 1910.
    • In his thesis he was able to extend results obtained by Painleve for differential equations of degree two to equations of degree three and higher.
    • The topic posed was: Improve the theory of differential algebraic equations of the second order and third order whose general integral is uniform.
    • Having done brilliant work on differential equations, Chazy's interests now turned towards the theory of relativity.
    • It discusses the principles of relativity, the equations of gravitation, the determination of ds2, Schwarzschild equations of motion, the n-body problem and finally cosmogonic hypotheses related to the ds2 of the universe.
    • (8) The ten differential equations of gravitation; .
    • (10) The Laplace equation and the Poisson equation.
    • Approximate equations of motion; .
    • Equations canoniques et variation des constantes.

  44. Aubin biography
    • His fundamental papers Metriques riemanniennes et courbure (1970), Equations differentielles non lineaires et probleme de Yamabe concernant la courbure scalaire (1976) and Meilleures constantes dans le theoreme d'inclusion de Sobolev et un theoreme de Fredholm non lineaire pour la transformation conforme de la courbure scalaire (1979) were fundamental in solving the Yamabe problem.
    • The analytic problem requires one to prove the existence of a solution of a highly nonlinear (complex Monge-Ampere) differential equation.
    • Aubin proved an important special case of the Calabi conjecture in Equations du type Monge-Ampere sur les varietes kahleriennes compactes.
    • To understand the scalar curvature equation on the sphere, he introduced the balancing condition on the conformal factors, and provided an improvement in the Sobolev inequality for such factors.
    • This became the basic tool of the compactness argument for a lot of subsequent work on this equation and later lead to the solution of prescribing curvature problem with no assumption on the symmetry of the curvature.
    • Monge-Ampere equations (1982), writes [Bull.
    • This book deals with certain nonlinear partial differential equations which arise from problems in global differential geometry.
    • This material is useful in other fields of mathematics such as partial differential equations, to name one.
    • Thierry Aubin was a very important mathematician whose work had great influence on the fields of Differential Geometry and Partial Differential Equations.
    • Aubin applied these methods in novel ways to fully nonlinear equations, and to delicate semilinear variational problems.
    • In the past fifty years there has been a surge of work on problems that involve the interplay of differential geometry and analysis, particularly partial differential equations.

  45. Tschirnhaus biography
    • He showed Collins and Wallis his methods for solving equations, but these turned out to be special cases of known results.
    • In it he discussed several mathematical questions including the solution of higher equations.
    • In his letter Leibniz also criticises Tschirnhaus's solution of algebraic equations.
    • Tschirnhaus worked on the solution of equations and the study of curves.
    • He discovered a transformation which, when applied to an equation of degree n, gave an equation of degree n with no term in xn-1 and xn-2.
    • We have indicated above that he had already discussed his methods for solving equations with Leibniz who had pointed out difficulties.
    • Nevertheless Tschirnhaus published his transformation in Acta Eruditorum in 1683 and, in this article, showed how it could be used to solve the general cubic equation.
    • However, his belief that the method would allow an equation of any degree to be solved is false as had already been pointed out to him by Leibniz.
    • History Topics: Quadratic, cubic and quartic equations .

  46. Jerrard biography
    • His most important work Mathematical Researches (1832-35) is on the theory of equations.
    • Viete and Cardan had shown how to transform an equation of degree n so that it had no term in xn-1.
    • These methods were, to a large extent, motivated by attempts to solve equations algebraically.
    • Abel and Ruffini showed this was impossible for general equations of degree greater than four.
    • In 1786 Bring reduced a general quintic to x5 + px + q = 0 while Jerrard generalised this to show that a transformation could be applied to an equation of degree n to remove the terms in xn-1, xn-2 and xn-3.
    • Hermite used Jerrard's result saying that it was the most important step in studying the quintic equation since Abel's results.
    • Jerrard wrote a further two volume work on the algebraic solution of equations An essay on the resolution of equations (1858).
    • Jerrard did not accept that the algebraic solution of the quintic equation was impossible.
    • In fact Jerrard had successfully shown that quintic equations could be solved but his error was to use methods which did not come under the precise definition of the 'method of radicals' which was required.

  47. Mordell biography
    • Rather remarkably, Mordell's future research interests were determined by these books, and his love of indeterminate equations came from this period.
    • For his Smith's Prize essay Mordell studied solutions of y2 = x3 + k, an equation which had been considered by Fermat.
    • Thue had already proved a result which, combined with Mordell's work showed that this equation had only finitely many solutions but Mordell only learned about Thue's work at a later date.
    • However he solved the equation for many values of k, giving complete solutions for some values.
    • Mordell was awarded the second Smith's Prize with his essay, and he went on to publish a long paper on this equation, now sometimes called Mordell's equation, in the Proceedings of the London Mathematical Society.
    • Mordell submitted his subsequent work on indeterminate equations of the third and fourth degree when he became a candidate for a Fellowship at St John's College, but he was not successful.
    • Indeterminate equations have never been very popular in England (except perhaps in the 17th and 18th centuries); though they have been the subject of many papers by most of the greatest mathematicians in the world: and hosts of lesser ones ..
    • marks the greatest advance in the theory of indeterminate equations of the 3rd and 4th degrees since the time of Fermat; and it is all the more remarkable that it can be proved by quite elementary methods.
    • He emphasised the fact that he was returning to Cambridge where he began his career by taking the equation y2 = x3 + k as the topic for his inaugural lecture to the Sadleirian Chair.

  48. Zhu Shijie biography
    • In dealing with simultaneous equations, Zhu certainly presented improvements, giving a method essentially equivalent to Gauss's pivotal condensation.
    • He treats polynomial algebra, and polynomial equations, by the "coefficient array method" or "method of the celestial unknown" which had been developed in northern China by the earlier thirteenth century Chinese mathematicians, but up till that time had not spread to southern China.
    • Zhu, however, wants to illustrate something more advanced than solving a quadratic equation.
    • Although we cannot be certain that Zhu's methods are exactly what we have presented here, he certainly arrived at the equation (2).
    • He has illustrated how to work with the four unknowns x, y, z, t and he can now illustrate how to solve a quartic equation.
    • It is phrased in terms of a right angled triangle, but the conditions are so artificial that he is really simply giving a system of equations.
    • The following problem in the Siyuan yujian is reduced by Zhu to a polynomial equation of degree 5 (see [First Australian Conference on the History of Mathematics (Clayton, 1980) (Clayton, 1981), 103-134.',7)">7] for a detailed solution as given by Zhu):- .
    • The Siyuan yujian also contains a transformation method for the numerical solution of equations which is applied to equations up to degree 14.
    • This is based on the method to solve polynomial equations which was rediscovered by Horner and Ruffini.

  49. Klein Oskar biography
    • He defended his doctorate in 1921 at Stockholm Hogskola and was opposed by Erik Ivar Fredholm the mathematical physicist best known for his work on integral equations and spectral theory.
    • In a paper in which he determined the atomic transition probabilities (prior to Dirac), he introduced the initial form of what would become known as the Klein-Gordon equation.
    • The Klein-Gordon equation was the first relativistic wave equation.
    • The equation can be written: .
    • It is interesting to note that this equation appeared exactly as it has been written in David Bohm's 1951 book Quantum Theory but was not called the Klein-Gordon equation.
    • However, Bethe and Jackiw's Intermediate Quantum Mechanics, originally written in 1964, does refer to the same equation as the Klein-Gordon equation.
    • Klein and Walter Gordon were thus eventually honoured with having the equation named after them, though it seems to have taken over a quarter of a century to receive the honour.
    • Oddly enough, Schrodinger himself privately developed a relativistic wave equation from his original wave equation, which, in reality, was not that difficult to do, and did so prior to Klein and Gordon, though he never published his results.
    • The trouble came when the equation did not result in the correct fine structure of the hydrogen atom and when Pauli introduced the concept of spin a year later (1927).
    • The equation turned out to be incompatible with spin and, as a result, is only useful for calculations involving spinless particles.
    • He and Jordan showed that one can quantize the non-relativistic Schrodinger equation and, in honour of this work, he was the recipient of yet another named mathematical tool, the Jordan-Klein matrices.
    • Despite the so-called Klein paradox, that being that the positron was not completely understood by physicists, he was able to convince physicists of the soundness of Dirac's relativistic wave equation.
    • Of the many he helped, one included Walter Gordon who would later join Klein in being the beneficiaries of the named equation we have just discussed.

  50. Nelson biography
    • The first of her two children, both daughters, was born shortly before she competed the work for her doctoral thesis The lattice of equational classes of commutative semigroups.
    • In addition to The lattice of equational classes of commutative semigroups referred to above, she published in 1971 the papers Embedding the dual of πm in the lattice of equational classes of commutative semigroups in the Proceedings of the American Mathematical Society and Embedding the dual of π∞ in the lattice of equational classes of semigroups in Algebra Universalis, both written jointly with Stanley Burris.
    • In the same year she published The lattice of equational classes of semigroups with zero in the Canadian Mathematical Bulletin.
    • Two which Nelson wrote jointly with Bernhard Banaschewski were On residual finiteness and finite embeddability and Equational compactness in equational classes of algebras both of which were published in Algebra Universalis.
    • In the following year she published Equational compactness in infinitary algebras again jointly with Bernhard Banaschewski.
    • W Taylor recently proved, among other results, that an equational class of finitary algebras contains enough equationally compact algebras if and only if the subdirectly irreducible algebras in the class constitute, up to isomorphism, a set.
    • This note provides a negative answer to the natural question whether the same equivalence holds for equational classes of infinitary algebras by exhibiting examples in which there are, up to isomorphism, only one subdirectly irreducible algebra in the class and no non-trivial equationally compact algebras at all.

  51. Fuchs biography
    • Fuchs worked on differential equations and the theory of functions.
    • Fuchs was a gifted analyst whose works form a bridge between the fundamental researches od Cauchy, Riemann, Abel, and Gauss and the modern theory of differential equations discovered by Poincare, Painleve, and Emile Picard.
    • In 1865 Fuchs studied nth order linear ordinary differential equations with complex functions as coefficients.
    • Fuchs enriched the theory of linear differential equations with fundamental results.
    • He discussed problems of the following kind: What conditions must be placed on the coefficients of a differential equation so that all solutions have prescribed proberties (e.g.
    • This led him (1865, 1866) to introduce an important class of linear differential equations (and systems) in the complex domain with analytic coeffivcients, a class which today bears hios name (Fuchaian equations, equations of the Fuchsian class).
    • He succeeded in characterising those differential equations the solutions of which have no essential singularity in the extended complex plane.
    • Fuchs later also studied non-linear fifferential equations and moveable singularities.
    • In a series of papers (1880-81) Fuchs studied functions obtained by inverting the integrals of solutions to a second-order linear differential equation in a manner generalising Jacobi's inversion problem.
    • Fuchs also investigated how to find the matrix connecting two systems of solutions of differential equations near two different points.

  52. Schmidt Harry biography
    • Another major publication during these years was his introduction to the theory of the wave equation Einfuhrung in die Theorie der Wellengleichung (1931).
    • While he was working at the laboratory Schmidt published, jointly with Kurt Schroder, a comprehensive report on the theory of laminar boundary layers deals with the basic conceptions and equations Laminare Grenzschichten.
    • The general equations of motion of hydrodynamics: .
    • The equation of continuity, the impulse theorem and the Navier-Stokes equations.
    • The equations of motion in orthogonal curvilinear coordinates.
    • Introduction of the boundary layer equation: .
    • A rigorous solution of the Navier-Stokes equations as an example.
    • The fundamental equations.
    • The equations of a steady two-dimensional motion with respect to the stream lines and their orthogonal trajectories.

  53. Waring biography
    • We shall comment further below on this important work, covering topics in the theory of equations, number theory and geometry.
    • Meditationes Algebraicae, covering the theory of equations and number theory, appeared in 1770 with an expanded version in 1782.
    • In Meditationes Algebraicae Waring proves that all rational symmetric functions of the roots of an equation can be expressed as rational functions of the coefficients.
    • He derived a method for expressing symmetric polynomials and he investigated the cyclotomic equation xn - 1 = 0.
    • The most significant aspect of Waring's treatment of this example is the symmetric relation between the roots of the quartic equation and its resolvent cubic.
    • k equations in k unknowns can be reduced to one equation with one unknown.
    • His result that the product of the degrees of the original equations is the degree of the single reduced equation is known as the Generalised Theorem of Bezout.

  54. Ljunggren biography
    • Papers such as Fermat's problem by Oystein Ore and On the indeterminate equation x2 - Dy2 = 1 by Trygve Nagell were in the issue which contained the problems that he solved to win his prize and, through studying these and other papers, he was already interested in number theory before beginning his university course.
    • Almost all of Ljunggren's research was on Diophantine equations.
    • For example in A note on simultaneous Pell equations (1941) Ljunggren studied the simultaneous Pell equations .
    • One of Ljunggren's main interests was Diophantine equations of degree 4.
    • In 1923 Mordell showed that the Diophantine equation .
    • In the paper Ljunggren found bounds for the number of integer solutions for some special equations of this type.
    • In the first of these he proves that the equation in question has at most two positive integer solutions and gives an example of D = 1785 which does indeed have two solutions, namely x = 13, y = 4 and x = 239, y = 1352.
    • He proved that the equation .

  55. Heaviside biography
    • Despite this hatred of rigour, Heaviside was able to greatly simplify Maxwell's 20 equations in 20 variables, replacing them by four equations in two variables.
    • Today we call these 'Maxwell's equations' forgetting that they are in fact 'Heaviside's equations'.
    • He introduced his operational calculus to enable him to solve the ordinary differential equations which came out of the theory of electrical circuits.
    • He replaced the differential operator d/dx by a variable p transforming a differential equation into an algebraic equation.
    • The solution of the algebraic equation could be transformed back using conversion tables to give the solution of the original differential equation.

  56. Viete biography
    • In 1593 Roomen had proposed a problem which involved solving an equation of degree 45.
    • (If I asked for a solution to ax = b nobody asks: "For which quantity do I solve the equation ?") .
    • Viete made many improvements in the theory of equations.
    • However, if we are to be strictly accurate we should say that he did not solve equations as such but rather he solved problems of proportionals which he states quite explicitly is the same thing as solving equations.
    • Viete therefore looked for solutions of equations such as A3 + B2A = B2Z where, using his convention, A was unknown and B and Z were knowns.
    • He presented methods for solving equations of second, third and fourth degree.
    • He knew the connection between the positive roots of equations and the coefficients of the different powers of the unknown quantity.
    • When Viete applied numerical methods to solve equations as he did in De numerosa potestatum he gave methods which were similar to those given by earlier Arabic mathematicians.
    • Although this seems to make Harriot's dependence on Viete clear, one would have to say that the two men give very similar approaches to solving equations algebraically, yet Harriot shows deeper understanding than does Viete.
    • History Topics: Quadratic, cubic and quartic equations .

  57. Kalman biography
    • Thus, such a system can be described by a finite number of simultaneous ordinary differential equations.
    • A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error.
    • The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics.
    • The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations.
    • The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field.
    • Properties of the variance equation are of great interest in the theory of adaptive systems.
    • Not only have they led to eminently useful developments, such as the Kalman-Bucy filter, but they have contributed to the rapid progress of systems theory, which today draws upon mathematics ranging from differential equations to algebraic geometry.
    • Among his many outstanding contributions were the formulation and study of most fundamental state-space notions (including controllability, observability, minimality, realizability from input/output data, matrix Riccati equations, linear-quadratic control, and the separation principle) that are today ubiquitous in control.

  58. Littlewood biography
    • In the late 1930's the Department of Scientific and Industrial Research tried to interest pure mathematicians in nonlinear differential equations which were important for radio engineers and scientists because they described the behaviour of electric circuits.
    • The impending war motivated this interest and in 1938 the Radio Research Board asked British pure mathematicians for help in dealing with certain types of nonlinear differential equations arising in radio engineering.
    • Littlewood, working jointly with Mary Cartwright, spent 20 years working on equations of this type such as van der Pol's equation.
    • Monthly 103 (10) (1996), 833-845.',16)">16], and in particular the work on van der Pol's equation is discussed in [Harmonic analysis and nonlinear differential equations, Riverside, CA, 1995, Contemp.
    • Van der Pol's experiments with nonlinear oscillators during the 1920s and 1930s stimulated mathematical interest in nonlinear differential equations arising in radio research.
    • Cartwright and Littlewood's analysis of the van der Pol equation and its generalizations led them to explore some interesting topological methods, including the development of a fixed-point theorem for continua invariant under a homeomorphism of the plane.
    • in recognition of his distinguished contributions to many branches of analysis, including Tauberian theory, the Riemann zeta-function, and non-linear differential equations.

  59. Qin Jiushao biography
    • There is a remarkable formula given in this Chapter which expresses the area of a figure as the root of an equation of degree 4.
    • Again equations of high degree appear, one problem involving the solution of the equation of degree 10.
    • Qin obtains the equation (really an equation of degree 5 in x2, where x2 is the diameter of the city):- .
    • Throughout the text, in addition to the tenth degree equation above, Qin also reduces the solution of certain problems to a cubic or quartic equation which he solves by the standard Chinese method (namely that which today is called the Ruffini-Horner method).
    • For example the following two equations .
    • Qin also solves linear simultaneous equations, in particular the system .

  60. Trudinger biography
    • in 1966 for his thesis Quasilinear Elliptical Partial Differential Equations in n Variables.
    • First there is the paper On the Dirichlet problem for quasilinear uniformly elliptic equations in n variables in which he extended previous work by his supervisor David Gilbarg, Olga Ladyzhenskaya and others on the solvability of the classical Dirichlet problem in bounded domains for certain second order quasilinear uniformly elliptic equations.
    • Secondly, in the paper The Dirichlet problem for nonuniformly elliptic equation he exploited the maximum principle to formulate general conditions for solvability of the Dirichlet problem for certain nonlinear elliptic equations.
    • In another 1967 paper On Harnack type inequalities and their application to quasilinear elliptic equations Trudinger examines weak solutions, subsolutions and supersolutions of certain quasilinear second order differential equations.
    • The book Elliptic partial differential equations of second order aimed to present (in the words of the authors):- .
    • the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process.
    • to the theory of quasilinear partial differential equations.
    • It had two new chapters one of which examined strong solutions of linear elliptic equations, and the other was on fully nonlinear elliptic equations.
    • The theory of nonlinear elliptic second order equations has continued to flourish during the last fifteen years and, in a brief epilogue to this volume, we signal some of the major advances.
    • In recent years, members of the programme have solved major open problems in curvature flow, affine geometry and optimal transportation, using techniques from nonlinear partial differential equations.

  61. Arino biography
    • Solutions periodiques d'equations differentielles a argument retarde.
    • Oscillations autour d'un point stationnaire, conditions suffisantes de non-existence (1980); "Following a note by P Seguier the authors give some results on the non-existence of a nontrivial periodic solution to differential equations with delay, using mainly properties of monotonicity.
    • Stabilite d'un ensemble ferme pour une equation differentielle a argument retarde (1978); "Our aim is to establish a local existence result for a differential equation with delay in a reflexive Banach space, with the hypothesis of weak continuity in the second member.
    • Solutions oscillantes d'equations differentielles autonomes a retard (1978); "We show some results proving the existence, and specifying the behaviour, of solutions oscillating near a stationary point for some equations of the type x '(t) = L(xt) + N(xt) which have certain monotone and continuity properties.
    • Comportement des solutions d'equations differentielles a retard dans un espace ordonne (1980); "Using vectorial Ljapunov functionals, we give here some results related to the behaviour at infinity of solutions of a differential equation with delay in an ordered Banach space." .
    • Arino studied for a doctorate supervised by Maurice Gaultier and was awarded the degree in 1980 from the University of Bordeaux for his thesis Contributions a l'etude des comportements des solutions d'equations differentielles a retard par des methodes de monotonie et bifurcation.
    • As can be seen from this work his interest was mainly in differential equations, mainly with delay, but later he became primarily involved with applications of these ideas to biomathematics, particularly population dynamics.
    • His results in the field of delay differential equations stand out: oscillations, functional differential equations in infinite dimensional spaces, state-dependent delay differential equations.
    • In the light of this theory a cell equation involving unequal division is investigated in great detail.

  62. Khayyam biography
    • This problem in turn led Khayyam to solve the cubic equation x3 + 200x = 20x2 + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle.
    • Khayyam also wrote that he hoped to give a full description of the solution of cubic equations in a later work [Scripta Math.
    • Indeed Khayyam did produce such a work, the Treatise on Demonstration of Problems of Algebra which contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.
    • In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations.
    • Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of al-Khwarizmi).
    • However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations.
    • Another achievement in the algebra text is Khayyam's realisation that a cubic equation can have more than one solution.
    • He demonstrated the existence of equations having two solutions, but unfortunately he does not appear to have found that a cubic can have three solutions.
    • Khayyam's construction for solving a cubic equation .

  63. Sneddon biography
    • It is a major text containing around 550 pages and is mainly concerned with applications which involve the solution of ordinary differential equations, and boundary value and initial value problems for partial differential equations.
    • Sneddon's next text Elements of partial differential equations appeared the following year in 1957.
    • The aim of this book is to present the elements of the theory of partial differential equations in a form suitable for the use of students and research workers whose main interest in the subject lies in finding solutions of particular equations rather than in the general theory.
    • The applications of the methods are again the strength of the book which considers the use of partial differential equations in thermodynamics, stochastic processes, and birth and death processes for bacteria.
    • The book deals with, among other topics, Laplace's equation, mixed boundary value problems, the wave equation, and the heat equation.

  64. Schiffer biography
    • They wrote the monograph Kernel functions and elliptic differential equations in mathematical physics (1953) tying together their joint work.
    • In this book the authors collect their researches of the last few years on elliptic partial differential equations.
    • The second part lays more stress on rigour, and treats fundamental solutions, reduction of boundary value problems to integral equations, orthonormal systems and kernel functions, eigenvalue problems associated with the kernels, variational theory of domain functions, comparison domains, basic existence theorems, and dependence of solutions on the boundary data or on the coefficients of the differential equation.
    • The presentation is in an easy flowing style, and the material should prove to be a most useful guide to those interested in the more advanced theory of linear elliptic partial differential equations.
    • Although the method yields in all cases first-order differential equations for the analytic arcs bounding the extremal domains, these equations will contain - except in some of the simpler problems - accessory parameters which are not known a priori and which have to be determined by additional considerations using the special features of the problem on hand.
    • The paper concludes with a description of the author's method for obtaining a lower bound for the first eigenvalue of the Poincare-Fredholm integral equation in the case of a simply-connected domain bounded by an analytic curve.
    • He was keen to apply his complex analysis results to mathematical physics, particularly making important contributions to the partial differential equations of hydrodynamics.
    • All of the famous equations associated with that theory are derived here.

  65. Magenes biography
    • Sansone made important contributions to analysis, particularly with his studies of ordinary differential equations.
    • The paper considers the problem of the existence of solutions of the differential equation in the title which pass through a given point and are tangent to a given curve.
    • The first of these papers examines the values of λ for which the equation in the title, subject to certain boundary conditions, has a solution.
    • Stampacchia and I wanted to know and make known in Italy the results of the school of Laurent Schwartz on distributions and on partial differential equations.
    • Together they investigated inhomogeneous boundary problems for elliptic equations and inhomogeneous initial-boundary value problems for parabolic and hyperbolic evolution equations.
    • If there are still people who feel that the subject of partial differential equations is "dirty" mathematics, this work should refute them once and for all.
    • It is a work to be recommended to every serious student of partial differential equations and particularly to those who are fascinated by the manner in which modern functional analysis has aided and influenced their study.
    • In a remarkable series of papers, followed and made complete in a three-volume book in cooperation with J L Lions (Nonhomogeneous Boundary Value Problems and Applications), he set the foundations for the modern treatment of partial differential equations, and in particular the ones mostly used in applications.
    • After his death, students, friends and colleagues organised the conference 'Analysis and Numerics of Partial Differential Equations' in his memory at Pavia in November 2011.

  66. Sokolov biography
    • He also worked on functional equations and on such practical problems as the filtration of groundwater.
    • Other applications include On the determination of dynamic pull in shaft-lifting cables (Ukrainian) (1955) and On approximate solution of the basic equation of the dynamics of a hoisting cable (Ukrainian) (1955).
    • One of the topics which will always be associated with Sokolov's name is his method for finding approximate solutions to differential and integral equations.
    • Examples of his papers on this topic are On a method of approximate solution of linear integral and differential equations (Ukrainian) (1955), Sur la methode du moyennage des corrections fonctionnelles (Russian) (1957), Sur l'application de la methode des corrections fonctionnelles moyennes aux equations integrales non lineaires (Russian) (1957), On a method of approximate solution of systems of linear integral equations (Russian) (1961), On a method of approximate solution of systems of nonlinear integral equations with constant limits (Russian) (1963), and On sufficient tests for the convergence of the method of averaging of functional corrections (Russian) (1965).
    • This basic approach is developed by the author and applied to the approximate solution of Fredholm and Volterra-type integral equations of the second kind, to their nonlinear counterparts, to integral equations of mixed type, to linear and nonlinear one-dimensional boundary value problems, to initial-value problems in ordinary differential equations and to certain elliptic, hyperbolic and parabolic equations.
    • The first part of Sokolov's book discusses applications of his method to problems which can be modelled by linear integral equations with constant limits.
    • The next three parts look first at problems which can be modelled by nonlinear integral equations with constant limits and then extend the analysis to the situation where the upper limit is variable.
    • In the final part Sokolov examines applications of his method to integral equations of mixed type, then in a number of appendices he presents some generalisations of the method.

  67. Nirenberg biography
    • Also in 1953, the first year in which his publications appear, he published three further papers: A strong maximum principle for parabolic equations; A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic-hyperbolic type; and On nonlinear elliptic partial differential equations and Holder continuity.
    • Some other highlights are his research on the regularity of free boundary problems with [David] Kinderlehrer and [Joel] Spuck, existence of smooth solutions of equations of Monge-Ampere type with [Luis] Caffarelli and Spuck, and singular sets for the Navier-Stokes equations with Caffarelli and [Robert] Kohn.
    • His study of symmetric solutions of non-linear elliptic equations using moving plane methods with [Basilis] Gidas and [Wei Ming] Ni and later with [Henri] Berestycki, is an ingenious application of the maximum principle.
    • He is quoted as saying [Electronic Journal of Differential Equations, Conference 15 (2007), 221-228.',5)">5]:- .
    • This is used in Chapters III and IV in the discussion of bifurcation theory (the highlight being a complete proof of Rabinowitz' global bifurcation theorem) and the solution of nonlinear partial differential equations (the highlight being the global theorem of Landesman and Lazer).
    • Here is the first [Electronic Journal of Differential Equations, Conference 15 (2007), 221-228.',5)">5]:- .
    • The second quote related to an author's results he was describing [Electronic Journal of Differential Equations, Conference 15 (2007), 221-228.',5)">5]:- .
    • The nonlinear character of the equations is used in an essential way, indeed he obtains results because of the nonlinearity not despite it.
    • for his work in partial differential equations.
    • He was a plenary speaker at the International Congress of Mathematicians held in Stockholm in August 1962, giving the lecture Some Aspects of Linear and Nonlinear Partial Differential Equations.
    • He shared the Prize of 350000 Swedish crowns with Vladimir Igorevich Arnold for their achievements in the field of non-linear differential equations.
    • The equations are then called partial differential equations and again the most interesting ones are non-linear.
    • As an example from geometry one can mention the problem to find a surface with given curvature and from physics studies of the equations for viscose fluids and concerning existence of free streamlines.
    • The work of Louis Nirenberg has enormously influenced all areas of mathematics linked one way or another with partial differential equations: real and complex analysis, calculus of variations, differential geometry, continuum and fluid mechanics.
    • Caffarelli mentions Nirenberg's areas of interest in partial differential equations: Regularity and solvability of elliptic equations of order 2n; the Minkowski problem and fully nonlinear equations; the theory of higher regularity for free boundary problems; and symmetry properties of solutions to invariant equations.
    • His range of interest is very broad: differential equations, harmonic analysis, differential geometry, functional analysis, complex analysis, etc.

  68. Krylov Aleksei biography
    • There he was taught advanced mathematics by Aleksandr Nikolaevich Korkin, a student of Chebyshev, who was an expert in partial differential equations.
    • In 1904 he constructed a mechanical integrator to solve ordinary differential equations, being the first in Russia to make such an instrument.
    • He studied the acceleration of convergence of Fourier series in a paper in 1912, and studied the approximate solutions to differential equations in a paper published in 1917.
    • In 1931 he found a new method of solving the secular equation determining the frequency of vibrations in mechanical systems which is better than methods due to Lagrange, Laplace, Jacobi and Le Verrier.
    • This paper On the numerical solution of the equation by which, in technical matters, frequencies of small oscillations of material systems are determined deals with eigenvalue problems.
    • It is clear that, if for k = 2 and k = 3 it is easy to compose this [secular] equation, then for k = 4 the laying-out becomes cumbersome, and for values k more than 5 this is completely unrealisable in a direct way.
    • is to present simple methods of composition of the secular equation in the developed form, after which, its solution, i.e.
    • The first edition of On Some Differential Equations of Mathematical Physics Having Application to Technical Problems appeared in 1913, the second edition in 1932, and the fourth appeared in 1948 as part of Krylov's collected works.

  69. Brahmagupta biography
    • Brahmagupta developed some algebraic notation and presents methods to solve quardatic equations.
    • He presents methods to solve indeterminate equations of the form ax + c = by.
    • Brahmagupta perhaps used the method of continued fractions to find the integral solution of an indeterminate equation of the type ax + c = by.
    • Brahmagupta also solves quadratic indeterminate equations of the type ax2 + c = y2 and ax2 - c = y2.
    • For the equation 11x2 + 1 = y2 Brahmagupta obtained the solutions (x, y) = (3, 10), (161/5, 534/5), ..
    • Pell's equation .
    • History Topics: Quadratic, cubic and quartic equations .
    • History Topics: Pell's equation .

  70. Matiyasevich biography
    • While an undergraduate he had already published some important papers (all in Russian): Simple examples of unsolvable canonical calculi (1967), Simple examples of unsolvable associative calculi (1967), Arithmetic representations of powers (1968), A connection between systems of word and length equations and Hilbert's tenth problem (1968), and Two reductions of Hilbert's tenth problem (1968).
    • Devise a process according to which it can be determined by a finite number of operations whether a given polynomial equation with integer coefficients in any number of unknowns is solvable in rational integers.
    • Does there exist an algorithm to determine whether a Diophantine equation has a solution in natural numbers? .
    • In 1934 Thoralf Skolem showed that to solve Hilbert's Tenth Problem, it is sufficient to consider only Diophantine equations of total degree four.
    • It was connected with the special form of Pell's equation.
    • In this thesis, as well as giving a simplified proof that no algorithm exists to determine whether Diophantine equations have integer solutions, he gave a Diophantine representation of a wide class of natural number sequences produced by linear recurrence relations.
    • His paper Existential arithmetization of Diophantine equations (2009) continues work related to Hilbert's Tenth problem and, as Alexandra Shlapentokh writes:- .
    • continues his investigation of coding methods by introducing a coding scheme which, among other things, leads to the elimination of bounded quantifiers, arithmetization of Turing machines, and a much simplified construction of a universal Diophantine equation.

  71. Kato biography
    • II in 1950, and Note on Schwinger's variational method, On the existence of solutions of the helium wave equation, Upper and lower bounds of scattering phases and Fundamental properties of Hamiltonian operators of Schrodinger type in 1951.
    • The course covered, thoroughly but efficiently, most of the standard material from the theory of functions through partial differential equations.
    • In 1962 he introduced new powerful techniques for studying the partial differential equations of incompressible fluid mechanics, the Navier-Stokes equations.
    • In 1983 he discovered the "Kato smoothing" effect while studying the initial-value problem associated with the Korteweg-de Vries equation, which was originally introduced to model the propagation of shallow water waves.
    • Another contribution to this area was On the Korteweg-de Vries equation where, in Kato's words from the paper:- .
    • Existence, uniqueness, and continuous dependence on the initial data are proved for the local (in time) solution of the (generalized) Korteweg-de Vries equation on the real line ..
    • He lectured on Abstract differential equations and nonlinear mixed problems.

  72. MacDuffee biography
    • While at Princeton he published a number of papers including: On transformable systems and covariants of algebraic forms (1923), On covariants of linear algebras (1924), The nullity of a matrix relative to a field (1925) and On the complete independence of the functional equations of involution (1925).
    • This was the first of four classic books that MacDuffee wrote, the other three being An introduction to abstract algebra (1940), Vectors and matrices (1943) and Theory of equations (1954).
    • The course which I have been giving at Wisconsin for the last couple of years is still entitled the Theory of Equations, but might more properly be called the Theory of Polynomials.
    • This approach seems to unify the somewhat scattered topics in the theory of equations, and to give a deeper insight into the subject which is particularly valuable to those who go on in algebra and to those who contemplate teaching algebra.
    • After this bit of number theory it is easy to attack the problem of finding the integral solutions of an equation having integral coefficients, and the rational solutions of an equation having rational coefficients.
    • Let us consider for a moment the theorem that if an equation with integral coefficients has a rational solution, when this solution is expressed in lowest terms the numerator is a divisor of the constant term of the equation.

  73. Morawetz biography
    • In a series of three significant papers in the late 1950s, Cathleen Morawetz used functional analysis coupled with ingenious new estimates for an equation of mixed type, i.e.
    • with both elliptic and hyperbolic regions, to prove a striking new theorem for boundary value problems for partial differential equations.
    • During the 1970s she extended this work to examine other solutions to the wave equation.
    • She proved many important results relating to the non-linear wave equation.
    • for pioneering advances in partial differential equations and wave propagation resulting in applications to aerodynamics, acoustics and optics.
    • In addition to her deep contributions to partial differential equations, transonic flow, and other areas of applied mathematics, she provided guidance and inspiration to colleagues and students alike.
    • .for her deep and influential work in partial differential equations, most notably in the study of shock waves, transonic flow, scattering theory, and conformally invariant estimates for the wave equation.

  74. Cardan biography
    • There followed a period of intense mathematical study by Cardan who worked on solving cubic and quartic equations by radical over the next six years.
    • I have certainly grasped this rule, but when the cube of one-third of the coefficient of the unknown is greater in value than the square of one-half of the number, then, it appears, I cannot make it fit into the equation.
    • In 1540 Cardan resigned his mathematics post at the Piatti Foundation, the vacancy being filled by Cardan's assistant Ferrari who had brilliantly solved quartic equations by radicals.
    • In it he gave the methods of solution of the cubic and quartic equation.
    • In fact he had discovered in 1543 that Tartaglia was not the first to solve the cubic equation by radicals and therefore felt that he could publish despite his oath.
    • Solving a particular cubic equation, he writes:- .
    • Quadratic, cubic and quartic equations .
    • History Topics: Quadratic, cubic and quartic equations .

  75. Robinson Julia biography
    • Robinson was awarded a doctorate in 1948 and that same year started work on Hilbert's Tenth Problem: find an effective way to determine whether a Diophantine equation is soluble.
    • Hilbert in 1900 posed the problem of finding a method for solving Diophantine equations as the 10th problem on his famous list of 23 problems which he believed should be the major challenges for mathematical research this century.
    • Now you are going to ask how could he be sure? He couldn't check each possible method and maybe there were very involved methods that didn't seem to have anything to do with Diophantine equations but still worked.
    • The method of proof is based on the fact that there is a Diophantine equation say P(x,y,z,..
    • In 1971 at a conference in Bucharest Robinson gave a lecture Solving diophantine equations in which she set the agenda for continuing to study Diophantine equations following the negative solution to Hilbert's Tenth Problem problem.
    • Instead of asking whether a given Diophantine equation has a solution, ask "for what equations do known methods yield the answer?" .

  76. Choquet-Bruhat biography
    • She dedicated the book General Relativity and the Einstein Equations (2009) she wrote many years later to:- .
    • She was awarded her doctorate in 1951 for her thesis Theoreme d'existence pour certains systemes d'equations aux derivees partielles non lineaires.
    • Further papers followed: Theoreme d'existence pour les equations de la gravitation einsteinienne dans le cas non analytique (1950); Un theoreme d'existence sur les systemes d'equations aux derivees partielles quasi lineaires (1950); and Theoremes d'existence et d'unicite pour les equations de la theorie unitaire de Jordan-Thiry (1951).
    • One has to face difficult nonlinear partial differential equations in Einstein's theory of gravity.
    • Of very great significance is that Yvonne Bruhat's analysis enabled her to prove rigorously for the first time local-in-time existence and uniqueness of solutions of the Einstein equations.
    • Ondes Asymptotiques et Approchees pour des Systemes d'Equations aux Derivees Partielles non Lineaires, published in 1969, gives a method for constructing asymptotic and approximate wave solutions about a given solution for nonlinear systems of partial differential equations.
    • Global Solutions of the Problem of Constraints on a Closed Manifold, published in 1973, shows that the existence of global solutions of the constraint equations of general relativity on a closed manifold depend on subtle properties of the manifold.
    • Bruhat gave courses at the University of Paris to students taking the Master of Mathematics degree which prepared them for the practical use of distributions in the partial differential equations of theoretical physics.
    • They have succeeded in gathering in one volume the mathematical infrastructure of modern mathematical physics, which includes the theories of differentiable manifolds and global analysis, Riemannian and Kahlerian geometry, Lie groups, fibre bundles and their connections, characteristic classes and index theorems, distributions, and partial differential equations.
    • The original book already had a number of interesting applications, such as the Schrodinger equation, soap bubbles, electromagnetism, shocks, gravity, Hamiltonian systems, monopoles, spinors, degree theory applied to PDE, Wiener measure, etc.
    • Her latest book, General relativity and the Einstein equations was published in 2009.
    • As the title indicates, the emphasis is on the mathematical properties of the Einstein equations, in particular the local and global existence theorems of the initial value problem.
    • for their separate as well as joint work in proving the existence and uniqueness of solutions to Einstein's gravitational field equations so as to improve numerical solution procedures with relevance to realistic physical solutions.

  77. Lichtenstein biography
    • Die Losungen als Funktionen der Randwerte und der Parameter (On the theory of differential equations of the second order.
    • He did pioneering work in potential theory, integral equations, calculus of variations, differential equations and hydrodynamics.
    • [Lichtenstein] made important contributions to the theory of partial differential equations, and the calculus of variations.
    • The so-called "Schauder bounds" in the theory of elliptic differential equations can already be found quite precisely, for the two-dimensional case, in Lichtenstein's encyclopaedia articles.
    • His tracing back of a class of integro-differential equations to a system of integral equations was of far reaching importance.
    • By observing the ramification near any given equilibrium figure, which in contrast to Lyapunov's studies does not have to be an ellipsoid, he managed to advance a new integro-differential equation and to work out more clearly the basic ideas of ramification.
    • For the author, the fundamental problem of hydrodynamics is the integration of certain systems of partial differential equations with assigned boundary conditions, and he calls to his aid all the resources of modern mathematics.
    • It is not until we reach page 290 that the equations of motion are derived.
    • Since he presupposes on the part of his readers a familiarity with the theory of the Newtonian potential function and of integral equations, together with a good grasp of other branches of analysis, geometry, and celestial mechanics, the book is not easy to read, but the results are so significant that it should be carefully studied by everyone who is seriously interested in the mathematical treatment of the problem of figures of equilibrium of rotating fluid bodies.
    • He was glad to receive an invitation to teach for one trimester at the Jan Kazimierz University in Lwow [now the Ivan Franko University of Lviv] in 1930, where his lectures were on the theory of integral and integro-differential equations.

  78. Van der Pol biography
    • even in mathematics, his papers covered number theory, special functions, operational calculus and nonlinear differential equations.
    • Of course, to most mathematicians the name of van der Pol is associated with the differential equation which now bears his name.
    • This equation first appeared in his article On relaxation oscillation published in the Philosophical Magazine in 1926.
    • 35 (1960), 367-376.',4)">4], [Application of asymptotic methods in the theory of nonlinear differential equations (Russian) (Akad.
    • We explain the history of the development of the equation carrying his name, and also the origins of the method of finding the first approximation to the solution of this equation (the method of slowly varying coefficients).
    • Van der Pol did much to popularize his subject; he was an engaging lecturer, and often took the opportunity of bringing together phenomena over a wide field of science which could be elucidated by a single mathematical relation such as the equation for relaxation oscillations.
    • In fact van der Pol corresponded with Nikolai Mitrofanovich Krylov about the theory of nonlinear oscillations; a letter sent by van der Pol to Krylov is published in [Application of asymptotic methods in the theory of nonlinear differential equations (Russian) (Akad.

  79. Vladimirov biography
    • The numerical methods for solving the kinetic equation of neutron transfer in nuclear reactors which he presented in 1952 is now known as the 'Vladimirov method'.
    • In this thesis he presented his theoretical investigation of the numerical solution, using the method of characteristics, of the single-velocity transport equation for a multilayered sphere.
    • There he worked under Mikhail Alekseevich Lavrent'ev and he published the important paper On the application of the Monte Carlo methods for obtaining the lowest characteristic number and the corresponding eigenfunction for a linear integral equation in 1956.
    • Thus, he first proved the theorem on the uniqueness, existence, and smoothness of the solution of the single-velocity transport equation, established properties of the eigenvalues and eigenfunctions, and gave a new variational principle (the Vladimirov principle).
    • His thesis contained what is today known as the 'Vladimirov variational principle' which he applied to the one-velocity transport equation and derived the best boundary conditions in the method of spherical harmonics for convex regions.
    • In 1967 Vladimirov published the book The equations of mathematical physics (Russian) which was written at advanced undergraduate or beginning graduate level.
    • contains a comprehensive treatment of the standard boundary value problems for second order partial differential equations.
    • It is clear and well organised and contains much important material that is not presented in other introductory texts on partial differential equations.

  80. Schrodinger biography
    • In theoretical physics he studied analytical mechanics, applications of partial differential equations to dynamics, eigenvalue problems, Maxwell's equations and electromagnetic theory, optics, thermodynamics, and statistical mechanics.
    • In mathematics he was taught calculus and algebra by Franz Mertens, function theory, differential equations and mathematical statistics by Wilhelm Wirtinger (whom he found uninspiring as a lecturer).
    • One week later Schrodinger gave a seminar on de Broglie's work and a member of the audience, a student of Sommerfeld's, suggested that there should be a wave equation.
    • Within a few weeks Schrodinger had found his wave equation.
    • The solution of the natural boundary value problem of this differential equation in wave mechanics is completely equivalent to the solution of Heisenberg's algebraic problem.
    • I am simply fascinated by your [wave equation] theory and the wonderful new viewpoint it brings.
    • Note the wave equation!')">Schrodinger's grave at Alpbach in Austria .

  81. Galois biography
    • On 25 May and 1 June he submitted articles on the algebraic solution of equations to the Academie des Sciences.
    • Galois sent Cauchy further work on the theory of equations, but then learned from Bulletin de Ferussac of a posthumous article by Abel which overlapped with a part of his work.
    • Galois then took Cauchy's advice and submitted a new article On the condition that an equation be soluble by radicals in February 1830.
    • Galois was invited by Poisson to submit a third version of his memoir on equation to the Academy and he did so on 17 January.
    • .as correct as it is deep of this lovely problem: Given an irreducible equation of prime degree, decide whether or not it is soluble by radicals.
    • A page from Galois' Memoire sur les conditions de resolubilite des equationspar radicaus (published in his collected works in 1897) .

  82. Ferro biography
    • In one sense he is well known, for his role in solving cubic equations is explained in almost every general work on the history of mathematics ever written, and yet, surprisingly, his name remains relatively unknown.
    • The outstanding problem which del Ferro solved was to find a formula to solve a cubic equation similar to the formula which had been known since the time of the Babylonians for solving quadratic equations.
    • There has been much conjecture as to whether del Ferro came to work on the solution to cubic equations as a result of a visit which Pacioli made to Bologna.
    • It is not known whether the two discussed the algebraic solution of cubic equations, but certainly Pacioli had included this topic in his famous treatise the Summa which he had published seven years earlier.
    • Four years ago when Cardano was going to Florence and I accompanied him, we saw at Bologna Hannibal della Nave, a clever and humane man who showed us a little book in the hand of Scipione del Ferro, his father-in-law, written a long time ago, in which that discovery [solution of cubic equations] was elegantly and learnedly presented.
    • Dal Ferro's rule for the solution of cubic equations.
    • The manuscript gives a method of solution which is applied to the equation 3x3 + 18x = 60.
    • History Topics: Quadratic, cubic and quartic equations .

  83. Petrovsky biography
    • The 1928 paper deals with the Dirichlet problem for Laplace's equation and in the 1929 paper he solved a problem originally posed by Lebesgue.
    • Also in 1951 he was appointed as Head of the Department of Differential Equations at the University.
    • Petrovsky's main mathematical work was on the theory of partial differential equations, the topology of algebraic curves and surfaces, and probability.
    • Petrovsky also worked on the boundary value problem for the heat equation and this was applied to both probability theory and work of Kolmogorov.
    • Surveys 46 (6) (1991), 149-215.',32)">32] spoke about three of Petrovsky's papers on partial differential equations:- .
    • Apparently, Petrovskii was the first to use the Fourier transform to study higher-order equations with variable coefficients.
    • He published a number of important textbooks: Lectures on the Theory of Ordinary Differential Equations (1939) (based on courses of lectures he gave at the universities of Moscow and Saratov); Lektsii po teorii integralnykh uravneny (1948), translated into German as Vorlesungen uber die Theorie der Integralgleichungen (1953) (based on courses of lectures he gave at the universities of Moscow); Lektsii ob uravneniakh s chastnymi proizvodnymi (1948), translated into English as Lectures in Partial Differential Equations (1954) and into German as Vorlesungen uber partielle Differential gleichungen (1955) (based on courses of lectures he gave at the universities of Moscow); and Lectures on Partial Differential Equations (1950) (based on courses of lectures he gave at the universities of Moscow).

  84. Engel biography
    • With my investigations of differential equations which permit a finite continuous group, I've always had a vague idea of the analogy between substitution theory and transformation theory.
    • One can prove that certain problems in integration can be reduced to certain ancillary equations of particular order and with particular characteristics, while further reduction is impossible in general.
    • How far the analogy with the algebraic equations can be carried through, I can't say for the good reason that I have almost no knowledge of equation theory.
    • Lie had, for some time, thought of writing a larger work on transformation groups, but, without the impetus from the outside which he now was getting, it would have quite surely gone the way of the work on first-order partial differential equations which he had made plans to do in the eighteen-seventies.
    • Leaving Christiania in June 1885, Engel returned to Leipzig where he defended his Habilitation thesis Uber die Definitionsgleichungen der continuirlichen Transformationsgruppen (On the Defining Equations of the Continuous Transformation Groups) on 15 July.
    • On 14 October he gave his inaugural lecture entitled Anwendungen der Gruppentheorie auf Differentialgleichungen (Applications of Group Theory to Differential Equations) and became a lecturer there.
    • His first course of lectures was 'Theory of first-order partial differential equations' which he gave in the winter semester of 1885-86.
    • He also wrote on continuous groups and partial differential equations, translated works of Lobachevsky from Russian to German, wrote on discrete groups, Pfaffian equations and other topics.
    • If Lie had lived for a longer time and had summarised his ideas on partial differential equations of the first order in the form of a textbook, the book would be about the same as these lectures of his collaborator.

  85. Tamarkin biography
    • Tamarkin maintained his close friendship and academic collaboration with Friedmann and by 1908 the two were attending lectures by Steklov on partial differential equations.
    • I proposed to Tamarkin that he think about the asymptotic solution of differential equations (i.e.
    • and then submitted his thesis in 1917 on boundary value problems for linear differential equations.
    • It was published in English in Mathematische Zeitschrift in 1928 as Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions.
    • His research during this period continued on boundary value problems, but also included advances in mathematical physics, differential equations, and approximations.
    • Five papers were published in these journals in 1926 and 1927: On Laplace's integral equations; On Volterra's integro-functional equation; A new proof of Parseval's identity for trigonometric functions; On Fredholm's integral equations, whose kernels are analytic in a parameter; and The notion of the Green's function in the theory of integro-differential equations.
    • For example they published: On the summability of Fourier series (two papers), On a theorem of Hahn-Steinhaus, On a theorem of Paley and Wiener, On the theory of linear integral equations.
    • The problem of moments is the theory of an infinite system of integral equations under various hypotheses.

  86. Schoen biography
    • Not only did their work employ serious tools of geometric analysis, including partial differential equations and geometric measure theory, to resolve a question motivated by gravitational physics, but they also established a link between the positivity of the mass of an isolated gravitational system and the relationship between positive scalar curvature and topology, a topic of interest to a broad range of mathematicians.
    • In the early eighties, Schoen brought the Positive Mass Theorem to bear on the resolution of the famous Yamabe problem, providing more evidence to support the development of the mathematical theory of the constraint equations, and inspiring many others to do so.
    • At the Berkeley Congress he gave the lecture New Developments in the Theory of Geometric Partial Differential Equations.
    • The author surveys recent work on nonlinear elliptic partial differential equations which arise from geometric sources, concentrating especially on the Yamabe problem and the theory of harmonic mappings.
    • For the former, an outline is given of the recent solution of Yamabe's conjecture (that every metric on a compact manifold is pointwise conformally equivalent to one with constant scalar curvature), including the use of the positive mass theorem and a discussion of regularity of weak solutions of Yamabe's equation.
    • for his work on the application of partial differential equations to differential geometry, in particular his completion of the solution to the Yamabe Problem in "Conformal deformation of a Riemannian metric to constant scalar curvature".
    • Schoen, 40, continues his research in differential geometry, nonlinear partial differential equations and the calculus of variations.
    • The main topics are differential equations on a manifold and the relation between curvature and topology of a Riemannian manifold.
    • He serves on the editorial boards of: the Journal of Differential Geometry, Communications in Analysis and Geometry, Communications in Partial Differential Equations, Calculus of Variations and Partial Differential Equations, and Communications in Contemporary Mathematics.
    • His research has fundamentally shaped geometric analysis, and his results form many cornerstones within geometry, partial differential equations and general relativity.

  87. Mytropolsky biography
    • Mitropolskii has made major contributions to the theory of oscillations and nonlinear mechanics as well as the qualitative theory of differential equations.
    • Using a method of successive substitutes, he constructed a general solution for a system of nonlinear equations and studied its behaviour in the neighbourhood of the quasi-periodic solution.
    • the creation and mathematical justification of algorithms for constructing asymptotic expansions for non-linear differential equations describing non-stationary oscillatory processes; .
    • the investigation of systems of non-linear differential equations describing oscillatory processes in gyroscopic systems and strongly non-linear systems; .
    • the development of the averaging method for equations with slowly varying parameters, as well as for equations with non-differentiable and discontinuous right-hand sides, for equations with delayed argument, for equations with random perturbations, and for partial differential equations and equations in functional spaces; .
    • the development of the theory of reducibility in linear differential equations with quasi-periodic coefficients, and other equations.
    • This work was to lead to further advances by the Kiev school, in particular they applied asymptotic methods to partial and functional differential equations.
    • Asymptotic solutions of differential equations are worked out in great detail, the author always being willing to go the second mile with the reader in obtaining the inherently complicated formulas that arise.
    • We give various algorithms, schemes and rules for constructing approximate solutions of equations with small and large parameters, and obtain examples which in many cases graphically illustrate the effectiveness of the method of averaging and the breadth of its application to various problems which are, at first glance, very disparate.
    • Among the many co-authored works we mention Lectures on the application of asymptotic methods to the solution of partial differential equations (1968) co-authored with his former student Boris Illich Moseenkov, Lectures on the methods of integral manifolds (1968) co-authored with his former student Olga Borisovna Lykova, Lectures on the theory of oscillation of systems with lag (1969) co-authored with his former student Dmitrii Ivanovich Martynyuk, Asymptotic solutions of partial differential equations (1976) co-authored with his former student Boris Illich Moseenkov, Periodic and quasiperiodic oscillations of systems with lag (1979) also co-authored with D I Martynyuk, Mathematical justification of asymptotic methods of nonlinear mechanics (1983) co-authored with his former student Grigorii Petrovich Khoma, Group-theoretic approach in asymptotic methods of nonlinear mechanics (1988) co-authored with his former student Aleksey Konstantinovich Lopatin, and Asymptotic methods for investigating quasiwave equations of hyperbolic type (1991) co-authored with his former students G P Khoma and Miron Ivanovich Gromyak.
    • Some of these were Ukrainian journals such as the differential equations journal Differentsial'nye Uravneniya, while others were international journals such as the International Journal of Nonlinear Sciences and Numerical Simulations, the journal Nonlinear Analysis, the journal Nonlinear Dynamics, and the International Journal of Nonlinear Mechanics.
    • for his outstanding achievements in the theory of nonlinear differential equations and nonlinear oscillations.

  88. Yau biography
    • Yau was awarded a Fields Medal in 1982 for his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampere equations.
    • S-T Yau has done extremely deep and powerful work in differential geometry and partial differential equations.
    • The analytic problem is that of proving the existence of a solution of a highly nonlinear (complex Monge-Ampere ) differential equation.
    • .for his work in nonlinear partial differential equations, his contributions to the topology of differentiable manifolds, and for his work on the complex Monge-Ampere equation on compact complex manifolds.
    • As a result of Yau's work over the past twenty years, the role and understanding of basic partial differential equations in geometry has changed and expanded enormously within the field of mathematics.
    • His work has had, and will continue to have, a great impact on areas of mathematics and physics as diverse as topology, algebraic geometry, representation theory, and general relativity as well as differential geometry and partial differential equations.

  89. Lagrange biography
    • He solved the resulting system of n+1 differential equations, then let n tend to infinity to obtain the same functional solution as Euler had done.
    • In papers which were published in the third volume, Lagrange studied the integration of differential equations and made various applications to topics such as fluid mechanics (where he introduced the Lagrangian function).
    • Also contained are methods to solve systems of linear differential equations which used the characteristic value of a linear substitution for the first time.
    • In 1770 he also presented his important work Reflexions sur la resolution algebrique des equations which made a fundamental investigation of why equations of degrees up to 4 could be solved by radicals.
    • The paper is the first to consider the roots of an equation as abstract quantities rather than having numerical values.
    • The Mecanique analytique summarised all the work done in the field of mechanics since the time of Newton and is notable for its use of the theory of differential equations.
    • Extract of Lagrange's Reflexions sur la resolution algebrique des equations from his collected works (1869).
    • History Topics: Pell's equation .

  90. Rudolff biography
    • The first of these gives methods for solving linear and quadratic equations.
    • Earlier works on solving equations had presented the reader with 24 different cases but Rudolff reduces this to 8 cases.
    • In looking at the case of a quadratic of the form ax2 + b = cx he believed at first that there was only one solution to this equation which will solve the original problem, but he later recognised his error and realised that such equations have two solutions.
    • The second chapter is again concerned with solving equations and presents rules for solving them.
    • If there are more unknowns than equations, the problem is considered indeterminate.
    • Rudolff was aware of the double root of the equation ax2 + b = cx and gave all the solutions to indeterminate first-degree equations.

  91. Walsh biography
    • The equation of a curve transformed as above Mr Walsh calls its 'partial equation'.
    • Memoir on the Invention of Partial Equations; The Theory of Partial Functions; Irish Manufactures: A New Method of Tangents; An Introduction to the Geometry of the Sphere, Pyramid and Solid Angles; General Principles of the Theory of Sound; The Normal Diameter in Curves; The Problem of Double Tangency; The Geometric Base; The Theoretic Solution of Algebraic Equations of the Higher Orders.
    • Thus, in a page headed Cubic Equations, he writes the name of Cardan opposite to a well-known algebraic solution, that of Walsh opposite to the same result put under another and less convenient form, and below these he gives a formula headed For a Complete Cubic by Walsh only.
    • Discovered the general solution of numerical equations of the fifth degree at 114 Evergreen Street, at the Cross of Evergreen, Cork, at nine o'clock in the forenoon of July 7th, 1844; exactly twenty-two years after the invention of the Geometry of Partial Equations, and the expulsion of the differential calculus from Mathematical Science.
    • And the falsehood of the offspring of that method, namely, the no less celebrated doctrine of fluxions, differentials, limits, etc., the boast and glory of England, France and Germany, demonstrated by the great invention of the geometry of partial equations which has superseded them, at least in my hands, and indefinitely surpassed the old system in power.

  92. Tao biography
    • for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.
    • He combines sheer technical power, an other-worldly ingenuity for hitting upon new ideas, and a startlingly natural point of view that leaves other mathematicians wondering, " Why didn't anyone see that before?" At 31 years of age, Tao has written over eighty research papers, with over thirty collaborators, and his interests range over a wide swath of mathematics, including harmonic analysis, nonlinear partial differential equations, and combinatorics.
    • Another area in which Tao has worked is solving special cases of the equations of general relativity describing gravity.
    • Imposing cylindrical symmetry on the equations leads to the "wave maps" problem where, although it has yet to be solved, Tao's contributions have led to a great resurgence of interest since his ideas seem to have made a solution possible.
    • Another area where Tao has introduced novel ideas, giving the subject a whole new look, is the theory of the nonlinear Schrodinger equations.
    • These equations have considerable practical applications and again Tao's insights have shed considerable light on the behaviour of a particular Schrodinger equation.
    • Also in 2006, Tao published Nonlinear dispersive equations.
    • This monograph is a remarkable introduction to nonlinear dispersive evolution equations, in particular to their local and global well-posedness and scattering theory.
    • Tao's mathematical knowledge has an extraordinary combination of breadth and depth: he can write confidently and authoritatively on topics as diverse as partial differential equations, analytic number theory, the geometry of 3-manifolds, nonstandard analysis, group theory, model theory, quantum mechanics, probability, ergodic theory, combinatorics, harmonic analysis, image processing, functional analysis, and many others.

  93. Serrin biography
    • In the years 1948 to 1950, James Serrin had the unique opportunity of attending lecture courses on elliptic differential equations given by Professors Eberhard Hopf and David Gilbarg at Indiana University.
    • In addition to his work on hydrodynamics, he also began publishing very significant results on elliptic differential equations with papers such as On the Phragmen-Lindelof theorem for elliptic partial differential equations (1954), On the Harnack inequality for linear elliptic equations (1956) and (with David Gilbarg) On isolated singularities of solutions of second order linear elliptic equations (1956).
    • The period 1963-1964 highlights the full maturity of Jim Serrin's thought, in particular in mastering the art of well adapted test functions for studying general quasilinear equations.
    • It is during that period that his two articles ['Local behavior of solutions of quasi-linear equations' (1964) and 'Isolated singularities of solutions of quasi-linear equations' (1965)] on isolated singularities were published in 'Acta Mathematica'.
    • The climax of this work is his famous article 'The problem of Dirichlet for quasilinear differentials equations with many independent variables', published in London in 1969 in the 'Philosophical Transactions of the Royal Society'.
    • for his fundamental contributions to the theory of nonlinear partial differential equations, especially his work on existence and regularity theory for nonlinear elliptic equations, and applications of his work to the theory of minimal surfaces in higher dimensions.
    • The maximum principle enables us to obtain information about solutions of differential equations and inequalities without any explicit knowledge of the solutions themselves, and thus can be a valuable tool in scientific research.
    • The maximum principle moreover occurs in so many places and in such varied forms that anyone learning about it becomes acquainted with the classically important partial differential equations and, at the same time, discovers the reason for their importance.
    • Since its first applications in the study of the Laplace and linear elliptic operators in general, the maximum principle has found a wide range of applications in nonlinear partial differential equations and inequalities, including equations involving the celebrated p-Laplace operator.
    • He has served on the editorial boards of many major journals including the Archive for Rational Mechanics and Analysis, the Journal of Differential Equations, Communications in Partial Differential Equations, the Bulletin of the American Mathematical Society, Rendiconti Circolo Matematico di Palermo, Asymptotic Analysis, Differential and Integral Equations, Communications in Applied Analysis, and Advances in Differential Equations.

  94. Lions Jacques-Louis biography
    • Schwartz had made a big breakthrough in the understanding of partial differential equations which he saw should be completely recast in the context of distribution theory.
    • Lions was one of several students who Schwartz directed to take this new approach and his doctoral thesis developed what has become the standard variational theory of linear elliptic and evolution equations.
    • In one of these collaborations with Enrico Magenes, they were investigating inhomogeneous boundary problems for elliptic equations and inhomogeneous initial-boundary value problems for parabolic and hyperbolic evolution equations.
    • It is a work to be recommended to every serious student of partial differential equations and particularly to those who are fascinated by the manner in which modern functional analysis has aided and influenced their study.
    • The systems he had in mind are those described by linear and nonlinear partial differential equations.
    • He had already published a major work on control of systems Controle optimal de systemes gouvernes par des equations aux derivees partielles in 1968 which investigates deterministic optimisation problems involving partial differential equations.
    • One notable feature of this work is that Lions introduces an infinite dimensional version of the Riccati equation in it.
    • reports on methods of solving nonlinear boundary value problems for partial differential equations, on a theoretical and functional analysis basis.
    • Integral equations and numerical methods.

  95. Yang Hui biography
    • For example, if the problem reduced to the solution of a quadratic equation, then Yang would solve it numerically, then show how to solve a general quadratic equation numerically.
    • What Yang's method essentially reduces to is finding the determinant of the matrix of coefficients of the system of equations.
    • The topics covered by Yang include multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.
    • Then a modern solution would set up equations .
    • He is subtracting the second equation from the first: 300 - 100 coins, 21 - 1 Wenzhou oranges, 9 - 1 green oranges.
    • This is exactly what he is doing! Replace y in the above equation so that .

  96. Girard Albert biography
    • The first man who really has a place in the history of symmetric functions of roots of equations, a man who for clearness and grasp of material at hand in not only this topic but also in other phases of algebra could well hold his place a century later was Albert Girard ..
    • He gives an example of the equation (which we write in modern notation) .
    • then in an equation of any degree .
    • He was the first who spoke of the imaginary roots, and understood that every equation might have as many roots real and imaginary, and no more, as there are units in the index of the highest power.
    • He was the first who discovered the rules for summing the powers of the roots of any equation.
    • We should also mention his iterative approach to solving equations [Dictionary of Scientific Biography (New York 1970-1990).
    • With the aid of trigonometric tables Girard solved equations of the third degree having three real roots.

  97. Brioschi biography
    • Francesco Brioschi was an important mathematician in the European context owing to his contributions to the theory of algebraic equations and to the applications of mathematics to hydraulics.
    • One of his most important results was his application of elliptical modular functions to the solution of equations of the fifth degree in 1858.
    • Brioschi however later went on to solve sixth degree equations using similar techniques.
    • In 1888, Maschke proved that a particular sixth-degree equation could be solved by using hyperelliptic functions and Brioschi then showed that any sixth-degree algebraic equation could be reduced to Maschke's equation and therefore solved using hyperelliptic functions.
    • In mechanics Brioschi dealt with problems of statics, proving Mobius's results by analytic means; with the integration of equations in dynamics, according to Jacobi's method; with hydrostatics; and with hydrodynamics.

  98. Keller Joseph biography
    • First an inhomogeneous integral equation is derived for the sound field in an infinite medium containing a thin curved shell of different material.
    • By appropriate approximations, the solution of the integral equation is reduced to the evaluation of a surface integral.
    • Keller's first two single-author papers appeared in 1948: On the solution of the Boltzmann equation for rarefied gases; and The solitary wave and periodic waves in shallow water.
    • In particular, they present the definition of the integral given by E Nelson (1964), and show that the integral does satisfy the corresponding appropriate Schrodinger equation.
    • It also served as a starting point for development of the modern theory of linear partial differential equations .
    • Other areas in which he has contributed include singular perturbation theory, bifurcation studies in partial differential equations, nonlinear geometrical optics and acoustics, inverse scattering, effective equations for composite media, biophysics, biomechanics, carcinogenesis, optimal design, hydrodynamic surface waves, transport theory, and waves in random media.

  99. Whittaker biography
    • It also develops the theory of special functions and their related differential equations.
    • He studied these special functions as arising from the solution of differential equations derived from the hypergeometric equation.
    • His results in partial differential equations (described as 'most sensational' by Watson) included a general solution of the Laplace equation in three dimensions in a particular form and the solution of the wave equation.
    • He also worked on electromagnetic theory giving a general solution of Maxwell's equation, and it was through this topic that his interest in relativity arose.

  100. Fantappie biography
    • As an application the Cauchy theory of partial differential equations is considered.
    • A similar discussion of the notions of hyperbolic, parabolic and elliptic partial differential equations which pertain to the case of two independent variables also appears.
    • Suddenly I saw the possibility of interpreting a wide range of solutions (the anticipated potentials) of the wave equation which can be considered the fundamental law of the Universe.
    • For example in Deduzione autonoma dell'equazione generalizzata di Schrodinger, nella teoria di relativita finale (1955) Fantappie deduces the Klein-Gordon equation in quantum mechanics as a limit, as the radius of the universe tends to infinity, of a classical (non-quantized) equation in his extension of relativity based on a simple (pseudo-orthogonal) group having the Lorentz group as a type of limit.
    • Finally let us look briefly at some of the papers which Fantappie published in the last seven years of his life: Costruzione effettiva di prodotti funzionali relativisticamente invarianti (1949) constructs functional scalar products of two functions, as required in quantum mechanics, which are relativistically invariant; Caratterizzazione analitica delle grandezze della meccanica quantica (1952) gives conditions on an hermitian operator that he claims are necessary and sufficient for it to satisfy to represent a physically real observable; Determinazione di tutte le grandezze fisiche possibili in un universo quantico (1952) discusses aspects of group invariance of wave equations; Gli operatori funzionali vettoriali e tensoriali, covarianti rispetto a un gruppo qualunque (1953) discusses the role of operators and Lie groups in a quantum-mechanical universe; Deduzione della legge di gravitazione di Newton dalle proprieta del gruppo di Galilei (1955) shows that the inverse square law is a necessary consequence if certain specific assumptions are made; Les nouvelles methodes d'integration, en termes finis, des equations aux derivees partielles (1955) applies analytic functionals to find explicit solutions of partial differential equations; and Sur les methodes nouvelles d'integration des equations aux derivees partielles au moyen des fonctionnelles analytiques (1956) gives a new method for the solution of Cauchy's problem.

  101. Norlund biography
    • The thesis is the beginning of the penetrating study of difference equations that he accomplished in the following 15 years.
    • The problem in difference equations is to find general methods for determining a function when the size of its increase on intervals of a given length is known.
    • a long series of papers developing the theory of difference equations.
    • There he gave the lecture Sur les equations aux differences finies.
    • This is the first book to develop the theory of the difference calculus from the function-theoretic point of view and to include a significant part of the recent researches having to do with the analytic and asymptotic character of the solutions of linear difference equations.
    • This is The logarithmic solutions of the hypergeometric equation (1963) which was reviewed by L J Slater:- .
    • In this important paper the author discusses in a clear and detailed way the complete logarithmic solutions of the hypergeometric differential equation satisfied by the Gauss function ..
    • Tables are also given for the continuation formulae which hold between the logarithmic and other cases of Riemann's P-function, and the paper concludes with a very clear statement of the logarithmic solutions of the confluent hypergeometric equation satisfied by Kummer's function ..

  102. Tricomi biography
    • In this paper he studied the theory of partial differential equations of mixed type, in particular the equation .
    • now known as the 'Tricomi equation'.
    • The equation became important in describing an object moving at supersonic speed.
    • Of course there were no supersonic aircraft in 1923 but the equation was to play a major role in later studies of supersonic flight.
    • These papers cover a vast range of subjects including singular integrals, differential and integral equations, pseudodifferential operators, functional transforms, special functions, probability theory and its applications to number theory.
    • As well as having the 'Tricomi equation' named after him, there are also special functions called 'Tricomi functions'.

  103. Babuska biography
    • His first papers, all written in Czech, were Welding stresses and deformations (1952), Plane elasticity problem (1952), A contribution to the theoretical solution of welding stresses and some experimental results (1953), A contribution to one method of solution of the biharmonic problem (1954), Solution of the elastic problem of a half-plane loaded by a sequence of singular forces (1954), (with L Mejzlik) The stresses in a gravity dam on a soft bottom (1954), On plane biharmonic problems in regions with corners (1955), (with L Mejzlik) The method of finite differences for solving of problems of partial differential equations (1955), and Numerical solution of complete regular systems of linear algebraic equations and some applications in the theory of frameworks (1955).
    • Basically, the mathematical problem Babuska's group had to solve was to find a numerical solution to a nonlinear partial differential equation.
    • From the mathematical point of view it deals with the special method of solving a biharmonic equation for given boundary conditions.
    • His next important book, published in collaboration with Milan Prager and Emil Vitasek in 1964, was Numerical Solution of Differential Equations (Czech).
    • It was translated into English and published under the title Numerical processes in differential equations two years later.
    • Among the many other services to mathematics which Babuska has given, we mention the many journals which have benefited by his accepting a position on their editorial board: Communications in Applied Analysis; Communications in Numerical Methods in Engineering; Computer & Mathematics; Computer Methods in Applied Mechanics and Engineering; Computers and Structures; Communications in Applied Analysis; International Journal for Numerical Methods in Engineering; Modelling and Scientific Computing; Numerical Mathematics - A Journal of Chinese Universities; Numerical Methods for Partial Differential Equations; and Siberian Journal of Computer Mathematics.

  104. Wu biography
    • However, the full equations governing the motion of the waves are notoriously difficult to work with because of the free boundary and the inherent nonlinearity, which are non-standard and non-local.
    • Although many approximate treatments, such as linear theory and shallow-water theory as well as numerical computations, have been used to explain many important phenomena, it is certainly of importance to study the solutions of the equations which include the effects neglected by approximate models.
    • Her research interests centre on harmonic analysis and partial differential equations, in particular nonlinear equations from fluid mechanics.
    • The Ruth Lyttle Satter Prize in Mathematics is awarded to Sijue Wu for her work on a long-standing problem in the water wave equation, in particular for the results in her papers (1) "Well-posedness in Sovolev spaces of the full water wave problem in 2-D" (1997); and (2) "Well-posedness in Sobolev spaces of the full water wave problem in 3-D" (1999).
    • By applying tools from harmonic analysis (singular integrals and Clifford algebra), she proves that the Taylor sign condition always holds and that there exists a unique solution to the water wave equations for a finite time interval when the initial wave profile is a Jordan surface.
    • Recently, Wu's research has focused on nonlinear equations from fluid dynamics.
    • The Birkhoff-Rott equations provide a mathematical description of the evolution of a vortex sheet.
    • It is a longstanding open problem to determine a function space in which these equations are well-posed, or, alternatively, to describe the evolution past singularity formation; this is the problem addressed in the present paper.

  105. Rellich biography
    • He undertook research for his doctorate with Richard Courant as his advisor and, in 1929, he was awarded the degree for his thesis Verallgemeinerung der Riemannschen Integrationsmethode auf Differentialgleichungen n-ter Ordnung in zwei Veranderlichen (Generalization of Riemann's integration method on differential equations of n-th order in two variables).
    • In this dissertation he generalised the Riemann's integration method, namely the explicit representation of the solution of the initial value problem of a linear hyperbolic differential equation of second order, to the case of such equations any order.
    • During the years 1933-34 at Gottingen, Rellich has taught courses on Integral Equations and Spectral Theory (1933) and Partial Differential Equations (1934).
    • Another piece of work which brought him international recognition was his study of the Monge-Ampere differential equation of elliptic type which we already mentioned when we looked at his papers published in the period 1932-34.
    • He is also known for Rellich's theorem on entire solutions of differential equations which he proved in 1940.

  106. Dahlquist biography
    • BESK came into operation in December 1953 and Dahlquist used the machine to solve differential equations.
    • During this time Dahlquist wrote a number of papers such as The Monte Carlo-method (1954), Convergence and stability for a hyperbolic difference equation with analytic initial-values (1954), and Convergence and stability in the numerical integration of ordinary differential equations (1956).
    • He submitted his doctoral thesis Stability and error bounds in the numerical integration of ordinary differential equations to Stockholm University in 1958, defending it in a viva in December.
    • Dahlquist was to use this idea throughout his research in stiff differential equations.
    • In the same year of 1963 he published Stability questions for some numerical methods for ordinary differential equations, an expository paper on his fundamental results concerning stability of difference approximations for ordinary differential equations.
    • Awarded to a young scientist (normally under 45) for original contributions to fields associated with Germund Dahlquist, especially the numerical solution of differential equations and numerical methods for scientific computing.
    • He has created the fundamental concepts of stability, A-stability and the nonlinear G-stability for the numerical solution of ordinary differential equations.

  107. Bilimovic biography
    • He graduated with the gold medal in 1903 and was awarded a Master's degree (equivalent to a doctorate) in 1903 from Kiev University for his thesis Equations of motion for conservative systems and its application.
    • At this time Hilbert was undertaking research into integral equations, work which led him to develop functional analysis.
    • At the end of this analysis, using the energy integral, he eliminated time in the differential equations of motion.
    • Especially in rational mechanics, he was occupied by phenomenological principles, motion of the rigid body around fixed point, dynamics of elastic bodies and equations of motion.
    • Sur le mouvement d'un corps solide avec un corps supplementaire mobile (1939) examines the differential equations governing the motion of a system consisting of two rigid bodies A and B subject to the constraint that the relative motion of B is specified as a function of the motion of A.
    • A natural property of the differential equation of a conic section (Serbian) (1946) discusses some classical metric, intrinsic and projective relations for a conic section.
    • Pfaff's method in the geometrical optics (Serbian) (1946) derives the Hamiltonian equations in optics from Fermat's principle using vector notation.
    • Pfaff's expression and Pfaff's equations are closely related to other mathematical concepts, which occupy an important place in modern mathematics.
    • The relation between Pfaff's expression and differential equations in canonical form was established by George Prange and Ernst Schering.

  108. Mathisson biography
    • Mathisson studied general dynamical laws governing the motion of a particle, with possibly a spin or an angular momentum, in a gravitational or electromagnetic field, and developed a powerful method for passing from field equations to particle equations.
    • Mathisson proved that the variational equation can be solved when it has been defined so that the equations to be imposed upon the characteristic tensor will be compatible with the variations allowed in the fields.
    • He obtained the equations of motion for the angular momentum and for the centre of mass with arbitrary external forces.
    • Finally, he calculated the linear forces for the case of no electric moment, leading to the equations for linear motion.
    • M Mathisson, The variational equation of relativistic dynamics, Proc.

  109. Gray Marion biography
    • in 1926 for her thesis The theory of singular ordinary differential equations of the second order having offered physics as an allied subject.
    • In 1925 E T Whittaker communicated the paper The equation of conduction of heat by Marion C Gray to the Royal Society of Edinburgh.
    • Marion C Gray and S A Schelkunoff, The approximate solution of linear differential equations.
    • Various papers by Gray were read to the Society: The equation of telegraphy (which appeared in volume 42 of the Proceedings and she read to the meeting of the Society in November 1923), The equation of conduction of heat (which also appeared in volume 42 of the Proceedings), and On the equation of heat (which appeared as Particular solutions of the equation of conduction of heat in one dimension in volume 43 of the Proceedings).

  110. Laplace biography
    • His next paper for the Academy followed soon afterwards, and on 18 July 1770 he read a paper on difference equations.
    • This paper contained equations which Laplace stated were important in mechanics and physical astronomy.
    • Not only had he made major contributions to difference equations and differential equations but he had examined applications to mathematical astronomy and to the theory of probability, two major topics which he would work on throughout his life.
    • The main mathematical approach here is the setting up of differential equations and solving them to describe the resulting motions.
    • In the Mecanique Celeste Laplace's equation appears but although we now name this equation after Laplace, it was in fact known before the time of Laplace.

  111. Collatz biography
    • Collatz was awarded his doctorate in 1935 for his dissertation Das Differenzenverfahren mit hoherer Approximation fur lineare Differentialgleichunge (The finite difference method with higher approximation for linear differential equations).
    • Among his early papers are Genaherte Berechnung von Eigenwerten (1939) in which he considers various methods of approximating characteristic values, Das Hornersche Schema bei komplexen Wurzeln algebraischer Gleichungen (1940) in which he presents a more efficient way of using Horner's method to approximate the complex roots of an algebraic equation, and Schrittweise Naherungen bei Integralgleichungen und Eigenwertschranken (1940) in which inequalities between the eigenvalues of certain integral equations are studied.
    • Eigenwertprobleme und ihre numerische Behandlung (1945) contains three parts, the first containing a collection of practical applications which lead to boundary value problems for ordinary and partial differential equations.
    • This was followed by Numerische Behandlung von Differentialgleichungen (1951) which provides a comprehensive text on numerical methods for solving differential equations.
    • This small book gives a wealth of information on differential equations.
    • The book Aufgaben aus der Angewandten Mathematik (1972) (with J Albrecht) provides a collection of problems (with their solutions) on the solution of equations and systems of equations, interpolation, quadrature, approximation, and harmonic analysis.
    • Later texts by Collatz include Optimization problems (1975) and Differential equations (1986), the second of these being an English translation of an earlier German book.

  112. Goursat biography
    • He began teaching at the University of Paris in 1879, receiving his doctorate in 1881 from l'Ecole Normale Superieure for his thesis Sur l'equation differentialle lineaire qui admet pour integrale la serie hypergeometrique.
    • Goursat's papers on the theory of linear differential equations and their rational transformations, as well as his studies on hypergeometric series, Kummer's equation, and the reduction of abelian integrals form, in the words of Picard "a remarkable ensemble of works evolving naturally one from the other".
    • Goursat introduced the notion of orthogonal kernels and semiorthogonals in connection with Erik Fredholm's work on integral equations.
    • In 1891 Goursat wrote Lecons sur l'integration des equations aux derivees partielles du premier ordre.
    • Volume 2 explores functions of a complex variable and differential equations.
    • Volume 3 surveys variations of solutions and partial differential equations of the second order and integral equations and calculus of variations.

  113. Taylor biography
    • He gave an account of an experiment to discover the law of magnetic attraction (1715) and an improved method for approximating the roots of an equation by giving a new method for computing logarithms (1717).
    • It was, wrote Taylor, due to a comment that Machin made in Child's Coffeehouse when he had commented on using "Sir Isaac Newton's series" to solve Kepler's problem, and also using "Dr Halley's method of extracting roots" of polynomial equations.
    • Taylor initially derived the version which occurs as Proposition 11 as a generalisation of Halley's method of approximating roots of the Kepler equation, but soon discovered that it was a consequence of the Bernoulli series.
    • The second version occurs as Corollary 2 to Proposition 7 and was thought of as a method of expanding solutions of fluxional equations in infinite series.
    • These include singular solutions to differential equations, a change of variables formula, and a way of relating the derivative of a function to the derivative of the inverse function.
    • Taylor, in his studies of vibrating strings was not attempting to establish equations of motion, but was considering the oscillation of a flexible string in terms of the isochrony of the pendulum.
    • He tried to find the shape of the vibrating string and the length of the isochronous pendulum rather than to find its equations of motion.

  114. Black Fischer biography
    • The Black-Scholes-Merton partial differential equation for the price of a financial asset was derived in their famous paper [Journal of Political Economy, 81(3), 637-54.',15)">15], using Ito's Lemma, with the economics coming in by way of the observation, due to Merton R.
    • In 1973, Black published with Myron Scholes their famous paper entitled The Pricing of Options and Corporate Liabilities [Journal of Political Economy, 81(3), 637-54.',15)">15] which derived and solved the Black-Scholes-Merton differential equation thereby solving the stock-option pricing problem [Note 1] .
    • the model of the stock-price in continuous time was represented by a special type of differential equation (so called stochastic differential equation) which allowed for randomness in stock-price.
    • The differential equation for the stock-price, S(t), was:- .
    • When the above differential equation was integrated, it gave a stock-price distribution that was lognormal (i.e.
    • The famous paper The Pricing of Options and Corporate Liabilities [Journal of Political Economy, 81(3), 637-54.',15)">15] has two ways of deriving the relevant partial differential equation.
    • Putting a = μS(t) and b = σS(t) in Ito's Lemma and exploiting that there are no free lunches in a randomless/riskless portfolio, gave rise to the famous partial differential equation:- .
    • This equation is satisfied by the traded assets themselves, for example, f (S, t) = S(t) and by f (S, t) = A ert but these do not have the European call-option boundary conditions (at time T) that Black and Scholes were interested in (i.e.
    • By 1969, Black and Scholes had the above differential equation.
    • They tried to solve this partial differential equation with the European call-option boundary condition but could not solve it.
    • But Black and Scholes had noticed the curious absence (in the differential equation) of the investment return, μ, of the stock-price or any parameter representing the degree of preference, as to risk, on the part of option purchasers.
    • the return was the risk-free rate, irrespective of the purchaser's risk preferences, they found that Sprenkle's formula, with these adjustments, satisfied the partial differential equation.
    • It is clear that the unexpected aspect of the Black-Scholes-Merton differential equation was not at first accepted.
    • Inter alia, Bachelier, had shown in his thesis [The Random Character of Stock Market Prices, MIT Press, Cambridge, Massachusetts (contains the translation from French of Bachelier\'s doctoral thesis and contains Sprenkle\'s, 1961 paper).',88)">88] the close connection between random walks and the Fourier heat equation, something that was expanded on by Kac, in 1951, [Ito\'s stochastic calculus and probability theory, Tokyo, ix-xiv.
    • ',98)">98] and by Feynman [Review of Modern Physics, 20, 367-387.',89)">89], where it was shown that the solution of Fourier's equation could be expressed as the distribution function of a random variable arising of a large number of random walks each with n steps (and with each step size proportional to √(t/n)) and by letting n become very large (i.e.

  115. Fox Leslie biography
    • We should note that Fox was undertaking numerical work solving partial differential equations which arose in engineering problems but which could not be solved by analytic techniques.
    • For example he published Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations (1947), A short account of relaxation methods (1948), and The solution by relaxation methods of ordinary differential equations (1949).
    • This book was The numerical solution of two-point boundary problems in ordinary differential equations and it is a great tribute to his expository skills that it was reprinted by Dover Publications in 1990.
    • The book summarises at an elementary level the methods for numerical construction of the solutions of boundary-value problems which can be expressed in terms of ordinary differential equations of orders one to four.
    • With a practical yet rigorous approach, methods to investigate topics such as recurrence relations, zeros of polynomials, linear equations, eigenvalues and eigenvectors, approximations, interpolation, integration, and ordinary differential equations are described and analysed.
    • Another collaboration between Fox and Mayers led to Numerical solution of ordinary differential equations published in 1987, four years after Fox retired.
    • Though there are two main types of ordinary differential equations, those of initial value type and those of boundary value type, most books until quite recently have concentrated on the former, and again until recently the boundary value problem has had little literature.
    • Some numerical experiments with eigenvalue problems in ordinary differential equations (1960) considers methods for solving such equations using a computer.
    • Partial differential equations (1963) is summarised by Fox as follows:- .
    • This expository paper discusses the present state of our ability to solve partial differential equations.
    • Outstanding problems include a determination of the error of finite-difference approximations, the automatic machine production of finite-difference formulae in complicated regions, the smoothing of physical data, and the classification of equations for computing-machine library routines.
    • Some of his later papers examine numerical methods for factorising polynomials, for solving elliptic partial differential equations, and methods for treating singularities in boundary value problems.

  116. Chang biography
    • Chang's research interests include the study of certain geometric types of nonlinear partial differential equations.
    • The Ruth Lyttle Satter Prize is awarded to Sun-Yung Alice Chang for her deep contributions to the study of partial differential equations on Riemannian manifolds and in particular for her work on extremal problems in spectral geometry and the compactness of isospectral metrics within a fixed conformal class on a compact 3-manifold.
    • Yang and I have solved the partial differential equation of Gaussian/scalar curvatures on the sphere by studying the extremal functions for certain variation functionals.
    • The course was entitled "Geometric PDE" and described using analytic tools like that of partial differential equations to solve problems in geometry.
    • model differential equations like that of the Gaussian curvature equations on compact surfaces, the prescribing curvature equations and the evolution equations related to the curvature flows.

  117. Moser Jurgen biography
    • The difficulty that Moser had no money was overcome and he began to study the spectral theory of differential equations with Rellich as his advisor.
    • In 1955 several of Moser's papers were published including Singular perturbation of eigenvalue problems for linear differential equations of even order, and Nonexistence of integrals for canonical systems of differential equations.
    • Moser worked in ordinary differential equations, partial differential equations, spectral theory, celestial mechanics, and stability theory.
    • Next is Stable and random motions in dynamical systems (1973, reprinted 2001) which describes how stable behaviour and statistical behaviour take place together in analytic conservative systems of differential equations.
    • Here Moser examines inverse spectral theory for the one-dimensional Schrodinger equation with the aim, as he writes in the introduction, of showing that:- .
    • For his fundamental work on stability in Hamiltonian mechanics and his profound and influential contributions to nonlinear differential equations.

  118. Krylov Nikolai biography
    • He worked mainly on interpolation and numerical solutions to differential equations, where he obtained very effective formulas for the errors.
    • For example he published On the approximate solution of the integro-differential equations of mathematical physics (1926), and Approximation of periodic solutions of differential equations in French in 1929.
    • With his collaborator and former student N N Bogolyubov, he published On Rayleigh's principle in the theory of differential equations of mathematical physics and on Euler's method in calculus of variations (1927-8) and On the quasiperiodic solutions of the equations of the nonlinear mechanics.
    • Examples of physical systems are given which lead to the type of equation considered in the monograph.
    • Moreover, general statements of methods for solving equations are illustrated by the explicit solution of examples.
    • We present the fundamental results of the works of N M Krylov and N N Bogolyubov devoted to the establishment of effective error estimates for the Ritz method, the Bubnov-Galerkin method and the least squares method in connection with self-adjoint differential equations.
    • In 1939 Krylov and Bogolyubov published Sur les equations de Focker-Planck deduites dans la theorie des perturbations a l'aide d'une methode basee sur les proprietes spectrales de l'hamiltonien perturbateur (Application a la mecanique classique et a la mecanique quantique).

  119. Piaggio biography
    • His most famous work, An Elementary Treatise on Differential Equations, was published by G Bell & Sons in 1920.
    • Here list a few articles which Piaggio published in The Mathematical Gazette: Relativity rhymes with a mathematical commentary (January 1922); Geometry and relativity (July 1922); Mathematics for evening technical students (July 1924); Mathematical physics in university and school (October 1924); Probability and its applications (July 1931); Three Sadleirian professors: A R Forsyth, E W Hobson and G H Hardy (October 1931); Mathematics and psychology (February 1933); Lagrange's equation (May 1935); Fallacies concerning averages (December 1937); and The incompleteness of "complete" primitives of differential equations (February 1939).
    • In the Proceedings of the Glasgow Mathematical Association he published Exceptional integrals of a not completely integrable total differential equation (1953).
    • The usual theory of a single Pfaffian equation holds if the coefficients are of class C'.
    • He read papers to the Society such as Note on Linear Differential Equations with constant coefficients on 10 May 1912.
    • H T H Piaggio's Treatise on Differential Equations .

  120. Yamabe biography
    • This was a period when his mathematical interests began to move away from Lie groups to differential equations and differential geometry.
    • His next major research contribution was Kernel functions of diffusion equations.
    • This paper presents a new, elegant method for constructing Green's function G for the heat equation over any domain D in Euclidean space.
    • In the following year Yamabe published A unique continuation theorem for solutions of a parabolic differential equation written jointly with Seizo Ito.
    • The same theme was taken up in A unique continuation theorem of a diffusion equation which he published in 1959.
    • His second paper on Kernel functions of diffusion equations was published in 1959 and in the following year he published On a deformation of Riemannian structures on compact manifolds and Global stability criteria for differential systems.

  121. Sturm biography
    • One of Sturm's most famous papers Memoire sur la resolution des equations numeriques was published in 1829.
    • It considered the problem of determining the number of real roots of an equation on a given interval.
    • The 1829 paper was not the last of Sturm's work on this algebraic equations and in [Rev.
    • seeks to determine the mutual influence between A-L Cauchy's and Ch-F Sturm's research from 1829 to around 1840 on the roots of algebraic equations.
    • These were the years during which he published some important results on differential equations.
    • Sturm became interested in obtaining results on specific differential equations which occurred in Poisson's theory of heat.
    • Liouville was also working on differential equations derived from the theory of heat.
    • Papers of 1836-1837 by Sturm and Liouville on differential equations involved expansions of functions in series and is today well-known as the Sturm-Liouville problem, an eigenvalue problem in second order differential equations.

  122. Enskog biography
    • Enskog's thesis studied the Maxwell-Boltzmann equations.
    • Enskog began to work on this equation for his master's degree at Uppsala and made a remarkable prediction.
    • Hilbert published a new approach to the Maxwell-Boltzmann equations in 1912.
    • How to extend the Maxwell-Boltzmann equation to include collisions of more than two bodies was not clear.
    • Chapman, who was still working on the Maxwell-Boltzmann equations, immediately saw the importance of Enskog's methods and developed them further.
    • In 1917 Enskog published his Uppsala Dissertation, in which he perfected the determination of f from Boltzmann's equation.

  123. Magiros biography
    • This part includes papers on nonlinear differential equations, mathematical modelling of physical phenomena and linearization of nonlinear models.
    • Linearization by exact methods which presents various "exact" methods of linearizing problems in differential equations; On the linearization of nonlinear models of phenomena.
    • Linearization by approximate methods in which he points out that "approximate" linearizations may lose the whole qualitative behaviour of the original nonlinear equation; and Characteristic properties of linear and nonlinear systems in which he gives many examples, recalls the importance of identifying characteristic properties of solutions, such as the superposition property for linear systems, and the possibility of limit cycles and self-excited oscillations in nonlinear systems.
    • Two papers which Magiros published in 1977 are: Nonlinear differential equations with several general solutions in which he gives specific devices for finding solutions of some nonlinear ordinary differential equations; and The general solutions of nonlinear differential equations as functions of their arbitrary constants presenting some nonlinear differential equations for which, surprisingly, some superposition does occur, that is, there are families of solutions depending linearly on arbitrary constants.
    • covers a variety of topics from special functions and transforms to numerical methods for the solution of nonlinear differential equations and optimal control problems.

  124. Hermite biography
    • Also like Galois he was attracted by the problem of solving algebraic equations and one of the two papers attempted to show that the quintic cannot be solved in radicals.
    • The letters he exchanged with Jacobi show that Hermite had discovered some differential equations satisfied by theta-functions and he was using Fourier series to study them.
    • He had found general solutions to the equations in terms of theta-functions.
    • Although an algebraic equation of the fifth degree cannot be solved in radicals, a result which was proved by Ruffini and Abel, Hermite showed in 1858 that an algebraic equation of the fifth degree could be solved using elliptic functions.
    • Hermite is now best known for a number of mathematical entities that bear his name: Hermite polynomials, Hermite's differential equation, Hermite's formula of interpolation and Hermitian matrices.

  125. Ostrowski biography
    • One consequence of this association was his monograph Solution of equations and systems of equations which was published in 1960 and was the result of a series of lectures he had given at the National Bureau of Standards.
    • By 1973 the third edition of this monograph appeared, this time with a new title: Solution of equations in Euclidean and Banach spaces.
    • These are determinants, linear algebra, algebraic equations, multivariate algebra, formal algebra, number theory, geometry, topology, convergence, theory of real functions, differential equations, differential transformations, theory of complex functions, conformal mappings, numerical analysis and miscellany.
    • His work on aglebraic equations involved a study of the fundamental theorem of algebra, Galois theory, and estimating the roots of algebraic equations.
    • Other work of Ostrowski was on the Cauchy functional equation, the Fourier integral formula, Cauchy-Frullani integrals, and the Euler-Maclaurin formula.

  126. Richardson biography
    • He first developed his method of finite differences in order to solve differential equations which arose in his work for the National Peat Industries concerning the flow of water in peat.
    • a scheme of weather prediction which resembles the process by which the Nautical Almanac is produced in so far as it is founded upon the differential equations and not upon the partial recurrence of phenomena in their ensemble.
    • Making observations from weather stations would provide data which defined the initial conditions, then the equations could be solved with these initial conditions and a prediction of the weather could be made.
    • It was a remarkable piece of work but in a sense it was ahead of its time since the time taken for the necessary hand calculations in a pre-computer age took so long that, even with many people working to solve the equations, the solution would be found far too late to be useful to predict the weather.
    • However the way that Richardson modelled the causes of war was quite different, giving systems of differential equations which governed the interactions between countries caused by such things as attitudes and moods.
    • The equations are merely a description of what people would do if they did not stop and think.
    • He set up equations governing arms build-up by nations, taking into account factors such as the expense of an arms race, grievances between states, ambitions of states, etc.
    • Choosing different values for the various parameters in the equation he then tried to investigate when situations were stable and when they were unstable.

  127. Bring biography
    • This work describes Bring's contribution to the algebraic solution of equations.
    • Bring discovered an important transformation to simplify a quintic equation.
    • It enabled the general quintic equation to be reduced to one of the form .
    • By the time Jerrard discovered the transformation, Ruffini's work and Abel's work on the impossibility of solving the quintic and higher order equations had been published.
    • However, at the time of Bring's discovery, there was no hint that the quintic could not be solved by radicals and, although Bring does not claim that he discovered his transformation in an attempt to solve the quintic, it is likely that this is in fact why he was examining quintic equations.
    • History Topics: Quadratic, cubic and quartic equations .

  128. Gregory biography
    • On the latter topic he had become interested in the problem of solving quintic equations algebraically and made some interesting discoveries on Diophantine problems.
    • However, we now summarise these and other contributions in the hope that, despite his reluctance to publish his methods, his remarkable contributions might indeed be more widely understood: Gregory anticipated Newton in discovering both the interpolation formula and the general binomial theorem as early as 1670; he discovered Taylor expansions more than 40 years before Taylor; he solved Kepler's famous problem of how to divide a semicircle by a straight line through a given point of the diameter in a given ratio (his method was to apply Taylor series to the general cycloid); he gives one of the earliest examples of a comparison test for convergence, essentially giving Cauchy's ratio test, together with an understanding of the remainder; he gave a definition of the integral which is essentially as general as that given by Riemann; his understanding of all solutions to a differential equation, including singular solutions, is impressive; he appears to be the first to attempt to prove that π and e are not the solution of algebraic equations; he knew how to express the sum of the nth powers of the roots of an algebraic equation in terms of the coefficients; and a remark in his last letter to Collins suggests that he had begun to realise that algebraic equations of degree greater than four could not be solved by radicals.
    • James Gregory's manuscripts on algebraic solutions of equations .

  129. Colson biography
    • In 1707 Colson published The universal resolution of cubic and biquadratic equations viz.
    • In the paper he gave a method to solve a cubic equation which was similar to that which had been discovered by several other mathematicians.
    • This is a quadratic equation in a3, so it can be solved for a3 using the usual formula for a quadratic.
    • Colson tested each of these 9 possible solutions to see if it satisfies the original equation, and was able to identify the three actual solutions.
    • This paper by Colson was the first to give all three solutions to a cubic equation.
    • In the final part of the paper he gives a method to solve cubic and quartic equations using geometric constructions of circles and parabolas.

  130. Picard Emile biography
    • Picard made his most important contributions in the fields of analysis, function theory, differential equations, and analytic geometry.
    • He used methods of successive approximation to show the existence of solutions of ordinary differential equations solving the Cauchy problem for these differential equations.
    • Starting in 1890, he extended properties of the Laplace equation to more general elliptic equations.
    • Picard also discovered a group, now called the Picard group, which acts as a group of transformations on a linear differential equation.

  131. Copson biography
    • Copson studied classical analysis, asymptotic expansions, differential and integral equations, and applications to problems in theoretical physics.
    • by Poisson's analytical solution of the equation of wave-motions.
    • .The analogue of Kirchhoff's formula, due to Volterra, is derived and an interesting account is given of a method, devised by Marcel Riesz and based on the theory of fractional integration, which provides a powerful method of solving initial value problems for equations like the wave equation.
    • In 1975 he published Partial differential equations which covers most of the classical techniques for first and second order linear partial differential equations, giving many examples and applications to physical problems.

  132. Nash biography
    • Meanwhile he went to Levinson to inquire about a differential equation that intervened and Levinson says it is a system of partial differential equations and if he could only [get] to the essentially simpler analog of a single ordinary differential equation it would be a damned good paper - and Nash had only the vaguest notions about the whole thing.
    • His research on the theory of real algebraic varieties, Riemannian geometry, parabolic and elliptic equations was, however, extremely deep and significant in the development of all these topics.
    • After this Nash worked on ideas that would appear in his paper Continuity of solutions of parabolic and elliptic equations which was published in the American Journal of Mathematics in 1958.
    • The outstanding results which Nash had obtained in the course of a few years put him into contention for a 1958 Fields' Medal but since his work on parabolic and elliptic equations was still unpublished when the Committee made their decisions he did not make it.

  133. D'Adhemar biography
    • The first part of the thesis, entitled Sur une classe d'equations aux derivees partielles du second ordre, du type hyperbolique, a 3 ou 4 variables independantes, dealt with the partial differential equation (1) with limit condition that arise in the two problems: the interior problem and the exterior problem.
    • For this problem, he dealt also with the equation (2).
    • It was in 1901 that Robert d'Adhemar's first publications were published, namely the mathematical papers Sur une integration par approximations successives and Sur une classe d'equations aux derives partielles du second ordre.
    • Although he had been writing philosophy book, he continued to publish mathematical papers such as Sur une equation aux derivees partielles du type hyperbolique (1905) and historical works such as Trois maitres: Ampere, Cauchy, Hermite (1905).
    • His first mathematics book Les equations aux derivees partielles a caracteristiques reelles was published in 1907.
    • Equations differentielles.
    • Equations integrales de M Fredholm et de M Volterra.
    • Equations aux derivees partielles du second ordre (1908) and L'equation de Fredholm et les problemes de Dirichlet et de Neumann (1909).
    • This book gave numerical methods for solving non-linear equations.
    • Equations integrales.
    • Equations differentielles et fonctionnnelles was published in 1912 with the second volume Fonctions synectiques, methodes des majorantes.
    • Equations aux derivees partielles du premier ordre.

  134. Fock biography
    • Schrodinger published his two fundamental papers on quantum theory in the spring of 1926 and Fock immediately started to develop the ideas and by the end of the year two of his own important papers on the Schrodinger equation had been published.
    • He became interested in the geometrization of the Dirac equation and he published an important paper in 1928 on Dirac's work on distributions.
    • the fundamental paper of 1935 in which the full symmetry structure of the hydrogen atom energy levels was shown to be given by the full Lorentz group; and the 1937 paper on the proper time parametrization of the Dirac equation, seminal for the later development of Schwinger's theory of field propagators and for the whole subject of parametrised field theories.
    • The reviewer feels that the author has made a major contribution to the understanding of gravitation theory, especially by his insistence on studying the solutions of the field equations and not merely the formal properties of the equations.
    • Some we have mentioned but now let us list a few: Fock space; Fock vacuum; the Fock method of quantisation; the Fock proper time method; the Hartree-Fock method; Fock symmetry; the Klein-Fock-Gordon equation; the Fock-Krylov theorem; and Dirac-Fock-Podolsky formalism.

  135. Sobolev biography
    • Sobolev became interested in differential equations, a topic which would dominate his research throughout his life, and even at this stage in his career he produced new results which he published.
    • published a number of profound papers in which he put forward a new method for the solution of an important class of partial differential equations.
    • Working with Smirnov, Sobolev studied functionally invariant solutions of the wave equation.
    • These methods allowed them to find closed form solutions to the wave equation describing the oscillations of an elastic medium.
    • By 1935 Sobolev was head of the Department of the Theory of Differential Equations at the Institute.
    • In 1958 Sobolev was part of the Soviet delegation to the International Mathematical Union, the delegation being led by Vinogradov, and Sobolev attended the International Congress at Edinburgh that year and gave an invited address on partial differential equations.

  136. Rolle biography
    • He published his most important work Traite d'algebre in 1690 on the theory of equations.
    • He also used it to solve Diophantine linear equations.
    • Let us see how this idea worked: If P(x) = 0 is a given polynomial equation with real roots a and b then he constructs a polynomial P'(x), which he called the 'first cascade,' so that P'(b) = (b - a)Q(b) where Q(x) is a polynomial of lower degree.
    • Some basic principles of the calculus and the theory of equations can definitely be traced to their origin as incidental propositions of the method.
    • It amplified the concepts of limits of roots of equations, provided the fundamentals from which Maclaurin derived his formula, began modern methods of series for determining roots, and discussed the relationship of imaginary roots in equations and their derivatives.
    • Rolle published another important work on solutions of indeterminate equations in 1699, Methode pour resoudre les equations indeterminees de l'algebre.

  137. Schmidt biography
    • His doctoral dissertation was entitled Entwickelung willkurlicher Funktionen nach Systemen vorgeschriebener and was a work on integral equations.
    • Schmidt's main interest was in integral equations and Hilbert space.
    • He took various ideas of Hilbert on integral equations and combined these into the concept of a Hilbert space around 1905.
    • Hilbert had studied integral equations with symmetric kernel in 1904.
    • He showed that in this case the integral equation had real eigenvalues, Hilbert's word, and the solutions corresponding to these eigenvalues he called eigenfunctions.
    • Schmidt published a two part paper on integral equations in 1907 in which he reproved Hilbert's results in a simpler fashion, and also with less restrictions.
    • In 1908 Schmidt published an important paper on infinitely many equations in infinitely many unknowns, introducing various geometric notations and terms which are still in use for describing spaces of functions and also in inner product spaces.

  138. Cherry biography
    • thesis Differential Equations Of Dynamics was written under guidance from Henry Baker and Ralph Fowler.
    • His first papers On the form of the solution of the equations of dynamics, On Poincare's theorem of 'the non-existence of uniform integrals of dynamical equations', and Note on the employment of angular variables in celestial mechanics were all published in 1924 and Some examples of trajectories defined by differential equations of a generalised dynamical type in the following year.
    • He undertook research on ordinary differential equations, particularly those arising from dynamics and celestial mechanics, for four years.
    • In 1937 he published Topological Properties of the Solutions of Ordinary Differential Equations and in 1947 he published the first part of Flow of a compressible fluid about a cylinder.
    • Since the author uses the solutions of Chaplygin, in the form of an infinite series of hypergeometric functions, of the linear second order partial differential equation in the hodograph variables of the potential function, this series diverges for values of the velocity whose speeds exceed the speed at infinity.

  139. Bhaskara I biography
    • He considers problems of indeterminate equations of the first degree and trigonometric formulae.
    • ',12)">12], [Ganita 23 (1) (1972), 57-79',13)">13] and [Ganita 23 (2) (1972), 41-50.',14)">14] Shukla discusses some features of Bhaskara's mathematics such as: numbers and symbolism, the classification of mathematics, the names and solution methods of equations of the first degree, quadratic equations, cubic equations and equations with more than one unknown, symbolic algebra, unusual and special terms in Bhaskara's work, weights and measures, the Euclidean algorithm method of solving linear indeterminate equations, examples given by Bhaskara I illustrating Aryabhata I's rules, certain tables for solving an equation occurring in astronomy, and reference made by Bhaskara I to the works of earlier Indian mathematicians.

  140. Bellman biography
    • His doctoral dissertation on the stability of differential equations was concerned with the behaviour of the solutions of real differential equations as the independent variable t tends to infinity.
    • Results from his dissertation appeared in the book Stability theory of differential equations which he published in 1953.
    • He went on to introduce Markovian decision problems in 1957 and in 1958 he published his first paper on stochastic control processes where he introduced what is today called the Bellman equation.
    • These include, in addition to those already mentioned: A Survey of the Theory of the Boundedness, Stability, and Asymptotic Behavior of Solutions of Linear and Nonlinear Differential and Difference Equations (1949); A survey of the mathematical theory of time-lag, retarded control, and hereditary processes (1954); Dynamic programming of continuous processes (1954); Dynamic programming (1957); Some aspects of the mathematical theory of control processes (1958); Introduction to matrix analysis (1960); A brief introduction to theta functions (1961); An introduction to inequalities (1961); Adaptive control processes: A guided tour (1961); Inequalities (1961); Applied dynamic programming (1962); Differential-difference equations (1963); Perturbation techniques in mathematics, physics, and engineering (1964); and Dynamic programming and modern control theory (1965).
    • After his death in 1984 his books continued to be published such as Partial differential equations (1985), Selective computation (1985), Methods in approximation (1986), and Wave propagation: An invariant imbedding approach (1986).

  141. Oleinik biography
    • Given Petrovsky's expertise in differential equations, the topology of algebraic curves and surfaces and mathematical physics, it is not difficult to see his influence on the direction that Oleinik's work would take.
    • In 1973 she became Head of the Department of Differential Equations at Moscow State.
    • The three chapters are: Basic mathematical aspects of the theory of elasticity; Homogenization of the equations of linear elasticity; Composites and perforated materials and Spectral problems in homogenization theory.
    • It is self-contained and the reader with background in partial differential equations and continuum mechanics can learn the homogenization techniques developed by Oleinik and her coauthors.
    • In 1996 Oleinik published Some asymptotic problems in the theory of partial differential equations.
    • Much of the book is devoted to the study of the asymptotic behaviour of solutions to nonlinear elliptic second-order equations.
    • Oleinik considers equations satisying Dirichlet boundary conditions and ones which satisfy Neumann boundary conditions.
    • Oleinik also studies the homogenization problem for linear elliptic equations in domains with the property that half is perforated and half contains no holes.
    • This book is a vast treasury of rigorous mathematical results about the Prandtl systems of partial differential equations in fluid dynamics.
    • The Prandtl equations were devised early in this century as a simpler replacement for the Navier-Stokes equations to describe viscous laminar fluid flows near boundaries to which the fluid adheres.
    • These replacement equations have a quite different mathematical character than the Navier-Stokes equations, and as one can easily see from this book, the theory goes along very different lines.

  142. Fowler biography
    • Early in his career, after receiving his degree, Fowler took to examining the behavior of the solutions to certain second-order differential equations.
    • In particular, he studied Emden's differential equation: .
    • Sir Arthur Eddington had originally shown that the equilibrium of gaseous stars could be found using the above equation with n = 3.
    • He rightly deduced Emden's equation must have other solutions.
    • The resulting general equation, which had considerable later influence on stellar astrophysics, was: .
    • These ions are closely packed leaving the free electrons to form a degenerate gas which Fowler described as "like a gigantic molecule in its lowest state." The equilibrium of the white dwarfs was later found to be described by a solution to Emden's equation as generalized by Fowler in the above equation with n = 3/2.

  143. Ito biography
    • Introducing the concept of regularisation, developed by Doob of the United States, I finally devised stochastic differential equations, after painstaking solitary endeavours.
    • He created the theory of stochastic differential equations, which describe motion due to random events.
    • Among them were On a stochastic integral equation (1946), On the stochastic integral (1948), Stochastic differential equations in a differentiable manifold (1950), Brownian motions in a Lie group (1950), and On stochastic differential equations (1951).
    • Stochastic differential equations, called "Ito Formula," are currently in wide use for describing phenomena of random fluctuations over time.
    • When I first set forth stochastic differential equations, however, my paper did not attract attention.

  144. Budan de Boislaurent biography
    • Budan is considered an amateur mathematician and he is best remembered for his discovery of a rule which gives necessary conditions for a polynomial equation to have n real roots between two given numbers.
    • In the early 19th century F D Budan and J B J Fourier presented two different (but equivalent) theorems which enable us to determine the maximum possible number of real roots that an equation has within a given interval.
    • Budan's rule was in a memoir sent to the Institute in 1803 but it was not made public until 1807 in Nouvelle methode pour la resolution des equations numerique d'un degre quelconque.
    • If an equation in x has n roots between zero and some positive number p, the transformed equation in (x - p) must have at least n fewer variations in sign than the original.
    • He quoted Lagrange to show that it would be useful to give the rules for solving numerical equations entirely by means of arithmetic, referring to algebra only if absolutely necessary.
    • Accordingly, the chief concern of Burdan's Nouvelle methode was to give the reader a mechanical process for calculating the coefficients of the transformed equation in (x - p).
    • Let us note that Charles-Francois Sturm in his famous paper Memoire sur la resolution des equations numeriques published in 1829 completely solved the problem of determining the number of real roots of an equation on a given interval.

  145. Bukreev biography
    • Ermakov was the first to show that some nonlinear differential equations of second order are simply related with linear differential equations of second order.
    • He visited Berlin where he heard lectures on the theory of hyperelliptic functions by Karl Weierstrass, lectures on the theory of Abelian functions and linear differential equations by Lazarus Fuchs, and lectures on the theory of numbers from Leopold Kronecker.
    • Bukreev determined the conditions for continuity of Fuchsian groups and constructed differential equations corresponding to each of the discontinuous groups.
    • Bukreev's work was broad and in addition to the areas of complex functions, differential equations, the theory and application of Fuchsian functions of rank zero, and geometry, he published papers on algebra such as On the composition of groups (1900).
    • He taught courses on analysis, differential and integral calculus and their applications to geometry, the theory of integration of differential equations, the theory of series, algebra, and other topics.
    • The geodesics as solutions of the Euler equation, Gauss curvature, geodesic curvature; II.

  146. Smithies biography
    • There he took courses by G H Hardy on Fourier analysis, John Whittaker on integral equations, and Ebenezer Cunningham on mechanics.
    • Smithies graduated in 1933 and began research on integral equations with Hardy at Cambridge.
    • He won the Rayleigh Prize in 1935 for an essay on differential equations of fractional order, and was awarded his doctorate for his thesis The Theory Of Linear Integral Equation which he submitted to the University of Cambridge in 1936.
    • Smithies early work was on integral equations and in 1958 his text Integral equations was published by Cambridge University Press in their Cambridge Tracts in Mathematics and Mathematical Physics Series.
    • the present work is intended as a successor to Maxime Bocher's tract, "An introduction to the study of integral equations" (University Press, Cambridge, 1909).

  147. Picone biography
    • In this period he developed research on ordinary differential equations and partial derivatives.
    • three different topics: (i) boundary value problems for second order linear ordinary differential equations, for which Picone developed his well-known "identity", and the subsequent extension of these results to second order linear partial differential equations of elliptic and parabolic types; (ii) partial differential equations of hyperbolic type (in two independent variables), for which Picone studied problems generalizing Goursat's problem; (iii) research on differential geometry in the direction set by L Bianchi, with particular attention to the characterization of the ds2 of a ruling and to W congruences.
    • While in Catania, he published two books: Teoria introduttiva delle equazioni differenziali ordinarie e calcolo delle variazioni (Introductory Theory of ordinary differential equations and calculus of variations) (1922) and Lezioni di Analisi infinitesimale (Lessons on Infinitesimal Analysis) (1923).
    • The work undertaken by the Institute included functional analysis, partial differentiation, integral equations, calculus of variations, special functions, probability theory, rational mechanics and mathematical physics.
    • Resulting from this were Picone's results on a priori bounds for the solutions of ordinary differential equations, as well as for those of linear partial differential equations of elliptic type and parabolic type for which the bound is obtained by means of the boundary data and the known terms; these results are contained in his well-known 'Notes on higher analysis' (Italian) a volume published in 1940 and which was, for its time, "truly avant-garde".
    • Gaetano Fichera highlights Picone's 1936 memoir which contains a characterization of a large class of linear partial differential equations whose solutions enjoy mean-value properties termed "integral properties" by Picone; using this theory Picone reconstructed M Nicolescu's theory of polyharmonic functions.
    • However, the works which led to the broadest and most important research are those based on the translation of boundary value problems for linear partial differential equations into systems of Fischer-Riesz integral equations; this method, whose object is the numerical calculation of the solutions, is similar to that of subsequent authors, who considered weak solutions of the same problems.
    • Some of his most important books which Picone published during his years in Rome are: Appunti di Analisi superiore (1940), which studies harmonic functions, Fourier, Laplace and Legendre series and the equations of mathematical physics; Lezioni di Analisi funzionale (1946), which concerns the calculus of variations; Teoria moderna dell'integrazione delle funzioni (1946), containing a detailed discussion of the r-dimensional Stieltjes integrals; (with Tullio Viola) Lezioni sulla teoria moderna dell'integrazione (1952), which is basically the previous work by Picone with three extra chapters by Viola; and (with Gaetano Fichera) Trattato di Analisi matematica (Vol 1, 1954, Vol 2, 1955), which puts into a treatise Picone's way of teaching calculus particularly slanted towards the applications studied at the Institute for Applied Calculus.

  148. Fenyo biography
    • The last topic in the book is the theory of non-linear ordinary differential equations, beginning with questions of existence consequences and stability.
    • The third volume published in 1980, although still presenting methods for engineers, is more involved with one of Fenyo's main research topics, namely integral equations.
    • The second section, which is over a quarter of the book, discusses linear integral equations.
    • Amongst the topics covered are Volterra integral equations and their relation with ordinary differential equations, Fredholm equations, self-conjugate and non-self-conjugate integral operators, and the associated eigenvalue theory.
    • The third section is on applications of integral equations.
    • Other books by Fenyo on integral equations are Integral equations - a book of problems (Hungarian) (1957), and the four volume work (written with H-W Stolle) Theorie und Praxis der linearen Integralgleichungen (1982, 1983, 1983, 1984).
    • These three volumes complete the encyclopaedic work (roughly 1700 pages) by Fenyo and Stolle on the theory and application of linear integral equations.
    • Their thesis is that the classical theory of linear integral equations produced many ideas for the later development of the theory of linear operators, and in turn functional analysis has helped the further development of integral equations.

  149. Tapia biography
    • in 1966 and then, in the following year submitted his thesis A Generalization of Newton's Method with an Application to the Euler-Lagrange Equation which led to the award of a Ph.D.
    • During this time he began to publish articles, the first one An application of a Newton-like method to the Euler-Lagrange equation in 1969 based on the work of his doctoral thesis.
    • In it Tapia considered the solution of the equation P(x) = 0, where P is a nonlinear mapping between Banach spaces.
    • He used Newton-like iterations to solve the generalized Euler-Lagrange equation of the calculus of variations.
    • It is also shown that this procedure can be applied to a class of two point boundary value problems containing the Euler-Lagrange equation for simple variational problems and most second order ordinary differential equations.

  150. Bogolyubov biography
    • He wrote his first scientific paper On the behavior of solutions of linear differential equations at infinity (Russian) in 1924.
    • The works of his first period, some of which were carried out by him jointly with his teacher N M Krylov, deal with direct methods of the calculus of variations, to the theory of nearly-periodic functions and approximate solutions of boundary-value differential equations.
    • Bogolyubov found methods to asymptotically integrate non-linear equations modelling oscillating systems [Russian Math.
    • Examples of physical systems are given which lead to the type of equation considered in the monograph.
    • Moreover, general statements of methods for solving equations are illustrated by the explicit solution of examples.
    • Bogolyubov himself completed a series of brilliant papers on the theory of stability of a plasma in a magnetic field and on the theory and applications of the kinetic equations, and he began his construction of axiomatic quantum field theory.

  151. Bers biography
    • Here Bers began work on the problem of removability of singularities of non-linear elliptic equations.
    • The nonparametric differential equation of minimal surfaces may be considered the most accessible significant example revealing typical qualities of solutions of non-linear partial differential equations.
    • The author sets as his goal the development of a function theory for solutions of linear, elliptic, second order partial differential equations in two independent variables (or systems of two first-order equations).
    • One of the chief stumbling blocks in such a task is the fact that the notion of derivative is a hereditary property for analytic functions while this is clearly not the case for solutions of general second order elliptic equations.

  152. Du Bois-Reymond biography
    • However, he continued to undertake research into applied mathematics and, as a consequence, became more and more involved with the theory of partial differential equations.
    • In this work he generalised Monge's idea of the characteristic of a partial differential equation from second order equations to third order equations.
    • Du Bois-Reymond's work is almost exclusively on calculus, in particular partial differential equations and functions of a real variable.
    • The standard technique to solve partial differential equations used Fourier series but Cauchy, Abel and Dirichlet had all pointed out problems associated with the convergence of the Fourier series of an arbitrary function.

  153. Bendixson biography
    • Bendixson also made interesting contributions to algebra when he investigated the classical problem of the algebraic solution of equations.
    • Abel had shown that the general equation of degree five could not be solved by radicals, while Galois had developed Galois theory which determined which equations could be solved by radicals.
    • Bendixson returned to Abel's original contribution and showed that Abel's methods could be extended to describe precisely which equations could be solved by radicals.
    • In examining periodic solutions of differential equations Bendixson used methods based on continued fractions.
    • The analysis problem which intrigued Bendixson more than all others was the investigation of integral curves to first order differential equations, in particular he was intrigued by the complicated behaviour of the integral curves in the neighbourhood of singular points.

  154. Steklov biography
    • For his Master's thesis Steklov worked on the equations of a solid body moving in an ideal non-viscous fluid.
    • There were four cases to be considered in integrating the equations which arose from this problem, and two of these cases had been solved by Clebsch in 1871.
    • Began lecturing at the University on the integration of partial differential equations.
    • His lecture course on the integration of partial differential equations was to third year students and the lectures went on until April 1909.
    • Finished my lectures on integration of equations.
    • In addition to the work for his master's thesis and his doctoral thesis referred to above, he reduced problems to boundary value problems of Dirichlet type where Laplace's equation must be solved on a surface.

  155. Kruskal Martin biography
    • An important paper on astronomy was Maximal extension of Schwarzschild's metric (1960) which showed that, using what are now called Kruskal coordinates, certain solutions of the equations of general relativity which are singular at the origin are not singular away from the origin, so allowing the study of black holes.
    • Kruskal's later work studied soliton equations, asymptotic analysis, and surreal numbers.
    • He was led to asymptotic analysis in his plasma physics studies and from there to solutions of Hamiltonian equations as in Asymptotic theory of Hamiltonian and other systems with all solutions nearly periodic (1962).
    • Kruskal's important paper (written jointly with Clifford S Gardner, John M Greene and Robert M Miura) Korteweg-de Vries equation and generalizations.
    • Before it, there was no general theory for the exact solution of any important class of nonlinear differential equations.
    • For his influence as a leader in nonlinear science for more than two decades as the principal architect of the theory of soliton solutions of nonlinear equations of evolution.

  156. Verhulst biography
    • He received his doctorate on 3 August 1825 after only three years study for his thesis De resolutione tum algebraica, tum lineari aequationum binominalium in which he studied the reduction of binomial equations.
    • Quetelet does not seem to have appreciated Verhulst's most important contribution, however, namely his work on the logistic equation and logistic function.
    • In the paper Verhulst argued against the model for population growth that Quetelet had proposed and instead proposed a model with a differential equation now known as the logistic equation.
    • He named the solution to the equation he had proposed in his 1838 paper the 'logistic function'.
    • In this last paper, Verhulst put forward some criticisms of his own model of population growth and this, together with Quetelet's criticisms in his obituary of Verhulst [Annuaire de l\'Academie royale des sciences de Belgique 16 (1850), 97-124.',13)">13], led to Verhulst's logistic equation being ignored for many years until the work of Raymond Pearl and Lowell Reed in 1920.

  157. Cimmino biography
    • Cimmino was only nineteen years old when he graduated with his thesis on approximate methods of solution for the heat equation in 2-dimensions, but he was appointed as an assistant to Picone who held the chair of analytical geometry at the University of Naples.
    • Some of Cimmino's most remarkable papers date to the period 1937-38 and concern the theory of partial differential equations of elliptic type.
    • Towards the end of that period, Professor Cimmino devised a numerical method for the approximate solution of systems of linear equations that he reminded me of in these days, following the recent publication by Dr Cesari ..
    • We have seen that Cimmino made contributions to partial differential equations of elliptic type and to computing approximate solutions to systems of linear equations.
    • It is also interesting to observe that Cimmino's impact on his main research area, the theory of partial differential equations, while non-negligible, has not been as great and as lasting as his work in numerical mathematics.

  158. Levinson biography
    • Norman decided to shift his field from gap and density theorems to non-linear differential equations, both ordinary and partial.
    • I recall our talking about this decision in 1940, and how difficult is was to move into this new field, and how hard Norman worked over a period of two or three years before he felt that he had enough mastery to obtain substantial results in this field; but this mastery he did achieve, and his outstanding contributions to non-linear differential equations were recognised officially in 1954 when the American Mathematical Society awarded Norman the Bocher Prize.
    • This was Theory of ordinary differential equations (written jointly with Earl Coddington) which [Norman Levinson : Selected papers of Norman Levinson (2 Vols.) (Boston, MA, 1998).',2)">2]:- .
    • The deep and original ideas of Norman Levinson have had a lasting impact on fields as diverse as differential and integral equations, harmonic, complex and stochastic analysis, and analytic number theory during more than half a century.
    • In other topics, Levinson provided the foundation for a rigorous theory of singularly perturbed differential equations.
    • He also made fundamental contributions to inverse scattering theory by showing the connection between scattering data and spectral data, thus relating the famous Gelfand-Levitan method to the inverse scattering problem for the Schrodinger equation.

  159. Uhlenbeck Karen biography
    • Uhlenbeck is a leading expert on partial differential equations and describes her mathematical interests as follows:- .
    • I work on partial differential equations which were originally derived from the need to describe things like electromagnetism, but have undergone a century of change in which they are used in a much more technical fashion to look at the shapes of space.
    • I did some very technical work in partial differential equations, made an unsuccessful pass at shock waves, worked in scale invariant variational problems, made a poor stab at three dimensional manifold topology, learned gauge field theory and then some about applications to four dimensional manifolds, and have recently been working n equations with algebraic infinite symmetries.
    • She described advances in geometry that have been achieved through the study of systems of nonlinear partial differential equations.
    • Among other things, she sketched some aspects of Simon Donaldson's work on the geometry of four-dimensional manifolds, instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.
    • She has also served on the editorial boards of many journals; a complete list to date is Journal of Differential Geometry (1979-81), Illinois Journal of Mathematics (1980-86), Communications in Partial Differential Equations (1983- ), Journal of the American Mathematical Society (1986-91), Ergebnisse der Mathematik (1987-90), Journal of Differential Geometry (1988-91), Journal of Mathematical Physics (1989- ), Houston Journal of Mathematics (1991- ), Journal of Knot Theory (1991- ), Calculus of Variations and Partial Differential Equations (1991- ), Communications in Analysis and Geometry (1992- ).
    • For her many pioneering contributions to global geometry that resulted in advances in mathematical physics and the theory of partial differential equations.
    • Karen Uhlenbeck is a distinguished mathematician of the highest international stature, specialising in differential geometry, non-linear partial differential equations and mathematical physics.

  160. Harriot biography
    • He introduced a simplified notation for algebra and his fundamental research on the theory of equations was far ahead of its time.
    • As an example of his abilities to solve equations, even when the roots are negative or imaginary, we reproduce his solution of an equation of degree 4.
    • As we have seen from the example above, Harriot did outstanding work on the solution of equations, recognising negative roots and complex roots in a way that makes his solutions look like a present day solution.
    • This is a major step forward in understanding which Harriot then carried forward to equations of higher degree.
    • History Topics: Quadratic, cubic and quartic equations .

  161. Naimark biography
    • With Krein, Naimark worked on applying Bezout's determinant to the problem of separating the roots of an algebraic equation.
    • The two collaborated in writing three papers on this topic: Uber eine Transformation der Bezoutiante, die zum Sturmschen Satze fuhrt (1933); On the application of Bezoutians to the separation of the roots of algebraic equations (1935); and The method of symmetric and hermitian forms in the theory of the separation of the roots of algebraic equations (1936).
    • Here he regularly gave courses in mathematical analysis, partial differential equations and functional analysis, he also supervised a group of post-graduate students and organized research seminars.
    • We have already noted that Naimark's first work was on the separation of roots of algebraic equations but, once he had established himself in Moscow, he worked on functional analysis and group representations.
    • For the pure mathematician it is a systematic treatment of the Lorentz groups in the classical tradition; for the theoretical physicist the long final chapter on invariant equations is of deep interest; for the social historian this Russian account of the theory of the Lorentz groups reveals the isolation of Russian mathematicians from the work of their Western colleagues, the same isolation manifest in Russian music and literature.

  162. Egorov biography
    • He was appointed an assistant lecturer at Moscow University on 27 January 1894, obtaining his Master's Degree in October 1899 for his thesis Second-order partial differential equations in two independent variables which he had defended on 22 September 1899.
    • Egorov also worked on integral equation publishing significant papers such as Sur quelques points de la theorie des equations integrates a limites fixes and Sur la theorie des equations integrates au noyau symmetrique which were both published in 1928.
    • In the following year he published lecture notes under the title Integration of differential equations.
    • At Moscow University, before his appointment as a professor, he taught courses on the synthetic theory of conics, number theory, the geometrical theory of partial differential equations, the theory of determinants, and the theory of binary forms.
    • After he was appointed as a professor, he taught courses on differential geometry, the integration of differential equations, integral equations, the calculus of variations, number theory, and the theory of surfaces.

  163. Tikhonov biography
    • His first-class achievements in topology and functional analysis, in the theory of ordinary and partial differential equations, in the mathematical problems of geophysics and electrodynamics, in computational mathematics and in mathematical physics are all widely known.
    • He defended his habilitation thesis in 1936 on Functional equations of Volterra type and their applications to mathematical physics.
    • The thesis applied an extension of Emile Picard's method of approximating the solution of a differential equation and gave applications to heat conduction, in particular cooling which obeys the law given by Josef Stefan and Boltzmann.
    • Thus, his research on the Earth's crust lead to investigations on well-posed Cauchy problems for parabolic equations and to the construction of a method for solving general functional equations of Volterra type.
    • However, in 1948 he began to study a new type of problem when he considered the behaviour of the solutions of systems of equations with a small parameter in the term with the highest derivative.

  164. Cramer biography
    • After giving the number of arbitrary constants in an equation of degree n as n2/2 + 3n/2, he deduces that an equation of degree n can be made to pass through n points.
    • Taking n = 5 he gives an example of finding the five constants involved in making an equation of degree 2 pass through 5 points.
    • This leads to 5 linear equations in 5 unknowns and he refers the reader to an appendix containing Cramer's rule for their solution.
    • He states a theorem by Maclaurin which says that an equation of degree n intersects an equation of degree m in nm points.

  165. Riesz Marcel biography
    • Marcel Riesz's interests ranged from functional analysis to partial differential equations, mathematical physics, number theory and algebra.
    • Riesz broadened his range of interests during the 1930 when he became interested in potential theory and in partial differential equations.
    • He was motivated by wave propagation and in particular Dirac's relativistic equation for the electron.
    • In 1949, Riesz published a 223 page paper L'integrale de Riemann-Liouville et le probleme de Cauchy in which he introduced a multiple integral of Riemann-Liouville type and showed how important this idea is in the theory of the wave equation.
    • In Problems related to characteristic surfaces Riesz extended these ideas to obtain the solution of the wave equation for a very general class of characteristic boundaries.

  166. Blanch biography
    • For the Mathieu equation y" + (a - 2 q cos 2x)y = 0, it is well known that certain values of a, described as characteristic values, lead to periodic solutions.
    • Among other papers that Blanch wrote before moving to Wright Patterson Air Force Base were: (with Roselyn Siegel) Table of modified Bernoulli polynomials (1950), On the numerical solution of equations involving differential operators with constant coefficients (1952), On the numerical solution of parabolic partial differential equations (1953) and (with Henry E Fettis) Subsonic oscillatory aerodynamic coefficients computed by the method of Reissner and Haskind (1953).
    • She continued to publish on Mathieu functions with D S Clemm after retiring, publishing the paper The double points of Mathieu's differential equation (1969) and the book Mathieu's equation for complex parameters.

  167. Lavrentev biography
    • For example, he applied variational properties of conformal mappings and reduced the important problem of flow around a wing to the solution of a singular integral equation of the first kind.
    • for his work on differential equations in 1982.
    • His outstanding scientific results in mathematics and its applications substantially affected the development of the theory of functions of complex variables, theory of differential equations, hydrodynamics, theory of motion of underground water, theory of long waves, dynamic stability, theory of cumulation, and many other fields of science and engineering.
    • In the 1940s he developed the theory of quasi-conformal mappings which gave a new geometrical approach to partial differential equations.
    • Other topics where he made substantial contributions were the theory of sets, the general theory of functions, and the theory of differential equations.

  168. Vandermonde biography
    • Vandermonde's four mathematical papers, with their dates of publication by the Academie des Sciences, were Memoire sur la resolution des equations (1771), Remarques sur des problemes de situation (1771), Memoire sur des irrationnelles de differents ordres avec une application au cercle (1772), and Memoire sur l'elimination (1772).
    • The first of these four papers presented a formula for the sum of the mth powers of the roots of an equation.
    • The paper also shows that if n is a prime less than 10 the equation xn - 1 = 0 can be solved in radicals.
    • Vandermonde's real and unrecognised claim to fame was lodged in his first paper, in which he approached the general problem of the solubility of algebraic equations through a study of functions invariant under permutations of the roots of the equation.
    • The reason for this strong claim by Muir is that, although mathematicians such as Leibniz had studied determinants earlier than Vandermonde, all earlier work had simply used the determinant as a tool to solve linear equations.

  169. Schlafli biography
    • For a given system of n equations of higher degree with n unknowns, I take a linear equation with undetermined coefficients a, b, c, ..
    • The work concluded with an examination of the class equation of third degree curves.
    • Other papers which he published investigate a variety of topics such as partial differential equations, the motion of a pendulum, the general quintic equation, elliptic modular functions, orthogonal systems of surfaces, Riemannian geometry, the general cubic surface, multiply periodic functions, and the conformal mapping of a polygon on a half-plane.

  170. D'Ocagne biography
    • Nomography consists in the construction of graduated graphic tables, nomograms, or charts, representing formulas or equations to be solved, the solutions of which were provided by inspection of the tables.
    • These papers by d'Ocagne included Nomographie; les Calculs usuels effectues au moyen des abaques (1891); Le Calcul simplifie par les procedes mecaniques et graphiques (1894); Sur la representation monographique des equations du second degre a trois variables (1896); Theorie des equations representables par trois systemes lineaires de points cotes (1897); and Application de la methode nomographique la plus generale, resultant de la superposition de deux plans, aux equations a trois et a quatre variables (1898).
    • If one makes a system of geometric elements (points or lines) correspond to each of the variables connected by a certain equation, the elements of each system being numbered in terms of the values of the corresponding variable, and if the relationship between the variables established by the equation may be translated geometrically into terms of a certain relation of position, easy to set up between the corresponding geometric elements, then the set of elements constitutes a chart of the equation considered.
    • This is the theory of charts, that is to say the graphical representation of mathematical laws defined by equations in any number of variables, which is understood today under the name Nomography.

  171. Heun biography
    • From 1886 to 1889 he lectured at the University of Munich on topics like: the theory of rational functions and their integrals, the theory of linear differential equations, introduction to the theory of linear substitutions and the general theory of differential equations.
    • The Heun equation is a second order linear differential equation of the Fuchsian type with four singular points.
    • It generalizes the hypergeometric differential equation which has three singular points, and is used today in mathematical physics, e.g.

  172. Jordan biography
    • The second part entitled Sur des periodes des fonctions inverses des integrales des differentielles algebriques was on integrals of the form ∫n u dz where u is a function satisfying an algebraic equation f (u, z) = 0.
    • Volumes 1 and 2 contain Jordan's papers on finite groups, Volume 3 contains his papers on linear and multilinear algebra and on the theory of numbers, while Volume 4 contains papers on the topology of polyhedra, differential equations, and mechanics.
    • He applied his work on classical groups to determine the structure of the Galois group of equations whose roots were chosen to be associated with certain geometrical configurations.
    • His work on group theory done between 1860 and 1870 was written up into a major text Traite des substitutions et des equations algebraique which he published in 1870.
    • The publication of Traite des substitutions et des equations algebraique did not mark the end of Jordan's contribution to group theory.
    • Generalising a result of Fuchs on linear differential equations, Jordan was led to study the finite subgroups of the general linear group of n × n matrices over the complex numbers.
    • Although given Jordan's work on matrices and the fact that the Jordan normal form is named after him, the Gauss-Jordan pivoting elimination method for solving the matrix equation Ax= b is not.

  173. Borok biography
    • Her undergraduate thesis on distribution theory and its applications to the theory of systems of linear partial differential equations was noted as outstanding and published in a top Russian journal.
    • In 1954, Valentina graduated from Kiev University and moved (following G E Shilov) to the graduate school at Moscow State University, where she received a PhD in 1957 for a thesis On Systems of Linear Partial Differential Equations with Constant Coefficients.
    • Her papers published in 1954-1959 contain a range of "inverse" theorems that allow partial differential equations to be characterized as parabolic or hyperbolic, by certain properties of their solutions.
    • In the same period she obtained formulae that made it possible to compute in simple algebraic terms the numerical parameters that determine classes of uniqueness and well-posedness of the Cauchy problem for systems of linear partial differential equations with constant coefficients.
    • In the early 1960s Valentina worked on fundamental solutions and stability for partial differential equations well-posed in the sense of Petrovskii.
    • Starting in the late 1960s, Valentina began a series of papers that lay the foundations for the theory of local and non-local boundary value problems in infinite layers for systems of partial differential equations.
    • In the early 1970s Valentina Borok founded a school on the general theory of partial differential equations in Kharkov.
    • The work of Valentina Borok and her school on boundary value problems in layers forms an important chapter in the general theory of partial differential equations.
    • Her other important contributions were in the area of difference, difference-differential, and functional-differential equations.
    • She also developed and published original lecture notes on a number of other core, as well as more specialized courses, in analysis and partial differential equations.

  174. Leray biography
    • This led to a collaboration between Leray and Schauder and their joint work led to a paper Topologie et equations fonctionelles published in the Annales scientifiques de l'Ecole normale Superieure.
    • This 1934 paper on topology and partial differential equations is of major importance:- .
    • This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.
    • He then returned to work on analysis, in particular studying differential equations arising from hydrodynamics.
    • He studied solutions of the initial value problem for three-dimensional Navier-Stokes equations.
    • He studied time dependent hyperbolic partial differential equations and also began to work on the Cauchy problem.
    • In particular he published a paper on the Cauchy problem for equations with variable coefficients in 1956.
    • He was able to generalise results in the theory of ordinary linear analytic differential equations to obtain similar results for partial differential equations.
    • In his hands, energy estimates for partial differential equations became combined with ideas from algebraic topology (such as fixed point theorems) in a highly original combination which cracked open the toughest problems.
    • Mathematician of penetration and originality, whose inventions revolutionized partial differential equations and algebraic topology.

  175. Bernstein Sergi biography
    • Bernstein returned to Paris and submitted his doctoral dissertation Sur la nature analytique des solutions des equations aux derivees partielles du second ordre to the Sorbonne in the spring of 1904.
    • This problem, posed by Hilbert at the International Congress of Mathematicians in Paris in 1900, was on analytic solutions of elliptic differential equations and asked for a proof that all solutions of regular analytical variational problems are analytic.
    • In 1906 he passed his Master's examination at St Petersburg but only with difficulty since Aleksandr Nikolayevich Korkin, who examined him on differential equations, expected him to use classical methods of solution (some sources say that Bernstein only passed the examination at the second attempt).
    • He moved to Kharkov in 1908 where he submitted a thesis Investigation and Solution of Elliptic Partial Differential Equations of Second Degree for yet another Master's degree.
    • As well as describing his approach to solving Hilbert's 19th Problem, it also solved Hilbert's 20th Problem on the analytic solution of Dirichlet's problem for a wide class of non-linear elliptic equations.
    • Mathematicians for a long time have confined themselves to the finite or algebraic integration of differential equations, but after the solution of many interesting problems the equations that can be solved by these methods have to all intents and purposes been exhausted, and one must either give up all further progress or abandon the formal point of view and start on a new analytic path.
    • The analytic trend in the theory of differential equations has only recently become established; and only seven years ago the late Professor Korkin in a conversation with me spoke scornfully of the "decadence" of Poincare's work.
    • As constructive function theory we want to call the direction of function theory which follows the aim to give the simplest and most pleasant basis for the quantitative investigation and calculation both of empirical and of all other functions occurring as solutions of naturally posed problems of mathematical analysis (for instance, as solutions of differential or functional equations).
    • The theory of probability is indebted to S N Bernstein for fundamental contributions on a number of topics; the axiomatic theory of probability, the foundations of normal correlation using limit theorems and the development of the general theory of correlation, the extension of the central limit theorem to sums of stochastically dependent variables, especially to heterogeneous Markov chains, and stochastic differential equations; the application of the theory of probability to biology and economics and applications of the methods of the theory of probability to the constructive theory of functions.

  176. Browder Felix biography
    • This area and partial differential equations have been my focus in the sixty years since, in particular nonlinear monotone operators from a Banach space to its dual.
    • He published The Dirichlet problem for linear elliptic equations of arbitrary even order with variable coefficients and The Dirichlet and vibration problems for linear elliptic differential equations of arbitrary order in the Proceedings of the National Academy of Sciences in 1952, and further papers in the same Proceedings in the following year.
    • He is in a field - a field of partial differential equations, which is a field in which the laws of radar, jet propulsion, atomic fission, all the basic laws of physics are expressed.
    • The subject had its origins in the study of nonlinear ordinary and partial differential equations, but it came to encompass a wider range of questions in all branches of analysis and in differential geometry, in theoretical physics, and in economics.
    • In the theory of linear elliptic partial differential equations, the work of Felix Browder and his school went well beyond the techniques first introduced by Russian analysts in establishing completeness theorems for the eigenfunctions of nonselfadjoint elliptic differential operators.
    • This theorem led to the proof of some deep existence theorems for nonlinear partial differential equations and began a massive development of monotone operator methods and their applications to partial differential equations.
    • He was their Colloquium Lecturer in 1973 giving the lectures Nonlinear functional analysis and its applications to nonlinear partial differential and integral equations.

  177. Olech biography
    • He defended his doctoral thesis On the asymptotic coincidence of sets filled up by integrals of two systems of ordinary differential equations on 26 April 1958.
    • Wazewski had there presented his ideas of applying the topological notion of a retract to the study of the solutions of differential equations and Lefschetz had seen the idea as being one of the most significant advances in the study of differential equations.
    • In particular he worked with Philip Hartman and they published the joint paper On global asymptotic stability of solutions of differential equations (1962).
    • During Olech's year at Princeton, Lawrence Markus lectured on problems of global stability of differential equations and Olech was able to generalise some of the results.
    • This thesis, On the global stability of an autonomous system on the plane, was published in the first volume of the journal Contributions to Differential Equations in 1963.
    • The most important results in Professor Czeslaw Olech's scientific work have been in the qualitative theory of differential equations and in control theory.
    • He obtained significant results for vector measures and their applications in the theory of differential equations and the theory of optimal control.
    • He also solved very important problems concerning autonomous systems on the plane with stable Jacobian matrix at each point of the plane and applied the Wazewski topological method in studying the asymptotic behaviour of solutions of differential equations.

  178. Sato biography
    • I then wanted to give some concrete example of it in the analysis of differential equations.
    • Sato explained the new theory of microlocal analysis in his lecture Regularity of hyperfunctions solutions of partial differential equations at the International Congress of Mathematicians at Nice in 1970, but the details appear in the 165 page paper by Sato, Kawai and Kashiwara Microfunctions and pseudo-differential equations in the proceedings of the Katata Conference held in 1971.
    • In this series we deal with these objects: (1) Deformation theory for linear differential equations (Riemann-Hilbert problem and its generalization to higher dimensions), (2) Quantum fields with critical strength (2-dimensional Ising model, etc.) and (3) Theory of Clifford group.
    • A rich theory for differential equations has been the result.
    • Hyperfunctions, together with integral Fourier operators, have become a major tool in linear partial differential equations.
    • Sato provided a unified geometric description of soliton equations in the context of tau functions and infinitedimensional Grassmann manifolds.
    • This was extended by his followers to other classes of equations, including self-dual Yang-Mills and Einstein equations.

  179. Pfaff biography
    • It investigates the use of some functional equations in order to calculate the differentials of logarithmic and trigonometrical functions as well as the binomial expansion and Taylor formula.
    • Pfaff did important work in analysis working on partial differential equations, special functions and the theory of series.
    • In the 1815 paper, which Pfaff submitted to the Berlin Academy on 11 May, he presented a transformation of a first-order partial differential equation into a differential system.
    • This theory of equations in total differentials is undoubtedly Pfaff's most significant contribution.
    • constituted the starting point of a basic theory of integration of partial differential equations which, through the work of Jacobi, Lie, and others, has developed into a modern Cartan calculus of extreme differential forms.

  180. Ferrari biography
    • Ferrari discovered the solution of the quartic equation in 1540 with a quite beautiful argument but it relied on the solution of cubic equations so could not be published before the solution of the cubic had been published.
    • Ferrari clearly understood the cubic and quartic equations more thoroughly than his opponent who decided that he would leave Milan that very night and thus leave the contest unresolved, so victory went to Ferrari.
    • Quadratic, cubic and quartic equations .
    • History Topics: Quadratic, cubic and quartic equations .

  181. Bachelier biography
    • ',4)">4] on Brownian Motion, in which Einstein derived the equation (the partial differential heat/diffusion equation of Fourier) governing Brownian motion and made an estimate for the size of molecules, Bachelier had worked out, for his Thesis, the distribution function for what is now known as the Wiener stochastic process (the stochastic process that underlies Brownian Motion) linking it mathematically with the diffusion equation.
    • In this course he may have drawn out the similarities between the diffusion of probability (the total probability of one being conserved) and the diffusion equation of Fourier (the total heat-energy being conserved).
    • Bachelier's work is remarkable for herein lie the theory of Brownian Motion (one of the most important mathematical discoveries of the 20th century), the connection between random walks and diffusion, diffusion of probability, curves lacking tangents (non-differentiable functions), the distribution of the Wiener process and of the maximum value attained in a given time by a Wiener process, the reflection principle, the pricing of options including barrier options, the Chapman-Kolmogorov equations in the continuous case, .

  182. Konigsberger biography
    • Much of Konigsberger's work on differential equations was influenced by the function theory developed by his friend Fuchs.
    • His work on differential equations was, however, also influenced by the applications which interested Bunsen, Kirchhoff and Helmholtz, with whom he was close friends in Heidelberg.
    • His approach to the differential equations of analytic mechanics showed novelty [Dictionary of Scientific Biography (New York 1970-1990).
    • Konigsberger was the first to treat not merely one differential equation, but an entire system of such equations in complex variables.

  183. Carmichael biography
    • in 1911 for his thesis Linear Difference Equations and their Analytic Solutions Linear Difference Equations and their Analytic Solutions.
    • by Indiana University for her thesis Transformations and Invariants Connected with Linear Homogeneous Difference Equations and Other Functional Equations in 1912.
    • Show that if the equation φ(x) = n has one solution it always has a second solution, n being given and x being the unknown.

  184. Korkin biography
    • He submitted On Determining Arbitrary Functions in Integrals of Linear Partial Differential Equations which he defended on 11 December 1860.
    • On the Paris visit he was particularly interested in Bertrand's lectures on partial differential equations and in Germany Kummer's lectures on quadratic forms fascinated him.
    • He defended his thesis On systems of first order partial differential equations and some questions on mechanics towards the end of 1867.
    • One of Korkin's major contributions was to the development of partial differential equations.
    • Initially Korkin was unimpressed with Zolotarev's investigation of an indeterminate equation of degree three which he presented in his Master' thesis.

  185. Dixon biography
    • Dixon's main area of research was in differential equations and he did early work on Fredholm integrals independently of Fredholm.
    • He worked both on ordinary differential equations and on partial differential equations studying abelian integrals, automorphic functions, and functional equations.
    • A spectacular generalisation of Dixon's beautiful identity is given by equation .31 on page 171 of [R L Graham, D E Knuth and O Patashnik, Concrete Mathematics (1989)] which must surely be the non plus ultra of the species.

  186. Prthudakasvami biography
    • Prthudakasvami is best known for his work on solving equations.
    • The solution of a first-degree indeterminate equation by a method called kuttaka (or "pulveriser") was given by Aryabhata I.
    • Brahmagupta seems to have used a method involving continued fractions to find integer solutions of an indeterminate equation of the type ax + c = by.
    • In this commentary Prthudakasvami writes the equation 10x + 8 = x2 + 1 as: .
    • The whole equation is therefore .

  187. Silva biography
    • He published his first paper in Portugaliae Mathematica in 1940, this being On the numerical resolution of algebraic equations (Portuguese).
    • In the following year he published Problems concerning rational functions of the roots of an algebraic equation (Portuguese) in the same journal.
    • Determine the equation whose roots are all sums of p roots; if there is a factor in the field, then this equation has a root in the field.
    • Assuming that it is possible to find all roots of an equation in a field, the preceding section furnishes a method of finding the coefficients of the factor, if it exists.

  188. McMullen biography
    • And I went to France and worked with Sullivan at Institut des Hautes Etudes Scientiques for a semester, and I met Steve Smale there who gave me this nice thesis problem on solving polynomial equations by iteration.
    • Once a proper understanding was achieved of which polynomial equations could be solved by radicals, there remained the problem of finding the roots of a polynomial equation by an iterative procedure for those for which no formula existed.
    • Newton had produced such a method and his iterative procedure generally converged for all quadratic polynomials and initial points, but this was not the case for polynomial equations of degree three.
    • He has made important contributions to various branches of the theory of dynamical systems, such as the algorithmic study of polynomial equations, the study of the distribution of the points of a lattice of a Lie group, hyperbolic geometry, holomorphic dynamics and the renormalization of maps of the interval.

  189. Varadhan biography
    • To Daniel Stroock and Srinivasa Varadhan for their four papers 'Diffusion processes with continuous coefficients I and II' (1969), 'On the support of diffusion processes with applications to the strong maximum principle (1970), Multidimensional diffusion processes (1979), in which they introduced the new concept of a martingale solution to a stochastic differential equation, enabling them to prove existence, uniqueness, and other important properties of solutions to equations which could not be treated before by purely analytic methods; their formulation has been widely used to prove convergence of various processes to diffusions.
    • In his landmark paper 'Asymptotic probabilities and differential equations' in 1966 and his surprising solution of the polaron problem of Euclidean quantum field theory in 1969, Varadhan began to shape a general theory of large deviations that was much more than a quantitative improvement of convergence rates.
    • Varadhan's book Lectures on diffusion problems and partial differential equations (1980) starts from Brownian motion and leads the students to stochastic differential equations and diffusion theory.

  190. Poisson biography
    • In his final year of study he wrote a paper on the theory of equations and Bezout's theorem, and this was of such quality that he was allowed to graduate in 1800 without taking the final examination.
    • During this period Poisson studied problems relating to ordinary differential equations and partial differential equations.
    • Poisson's name is attached to a wide variety of ideas, for example:- Poisson's integral, Poisson's equation in potential theory, Poisson brackets in differential equations, Poisson's ratio in elasticity, and Poisson's constant in electricity.

  191. Arbogast biography
    • The particular mathematical dispute which prompted the question set by the St Petersburg Academy in 1787, however, concerned the arbitrary functions which appeared when a differential equation was integrated.
    • d'Alembert claimed that these arbitrary functions were required to be continuous and must always be expressed in terms of algebraic or transcendental equations.
    • Euler argued that more general functions could be introduced when differential equations were integrated.
    • Do the arbitrary functions introduced when differential equations are integrated belong to any curves or surfaces either algebraic, transcendental, or mechanical, either discontinuous or produced by a simple movement of the hand? Or should they legitimately be applied only to continuous curves susceptible of being expressed by algebraic or transcendental equations? .

  192. Hudde biography
    • Hudde worked on maxima and minima and the theory of equations.
    • He gave an ingenious method to find multiple roots of an equation which is essentially the modern method of finding the highest common factor of a polynomial and its derivative.
    • If in an equation two roots are equal and if it be multiplied by any arithmetical progression, i.e.
    • the first term by the first term of the progression, the second by the second term of the progression, and so on: I say that the equation found by the sum of these products shall have a root in common with the original equation.

  193. Maschke biography
    • Maschke found working with Klein in his home in the evenings very rewarding and was fascinated with Klein's ideas on using group theory to solve algebraic equations.
    • Hermite, Kronecker and Brioschi had, in 1858, discovered how to solve the quintic equation by means of elliptic functions.
    • In 1888 Maschke proved that a particular sixth-degree equation could be solved by using hyperelliptic functions and Brioschi showed that any sixth-degree algebraic equation could be reduced to Maschke's equation and therefore solved in the same way.

  194. Spencer Tony biography
    • A particular strain energy function (Neo-Hookean) is chosen, and the condition for existence of an adjacent equilibrium position is obtained in the form of a transcendental equation, which is solved numerically for two loading conditions.
    • After introductory chapters on matrix algebra, vectors and Cartesian tensors, and an analysis of deformation and stress, the author examines the mathematical statements of the laws of conservation of mass, momentum and energy and the formulation of the mechanical constitutive equations for various classes of fluids and solids.
    • A procedure has been developed in previous papers for constructing exact solutions of the equations of linear elasticity in a plate (not necessarily thin) of inhomogeneous isotropic linearly elastic material in which the elastic moduli depend in any specified manner on a coordinate normal to the plane of the plate.
    • The essential idea is that any solution of the classical equations for a hypothetical thin plate or laminate (which are two-dimensional theories) generates, by straightforward substitutions, a solution of the three-dimensional elasticity equations for the inhomogeneous material.

  195. Mason biography
    • Mason's mathematical research interests lay in differential equations, the calculus of variations and electromagnetic theory.
    • He developed the relation between the algebra of matrices and integral equations as well as studying boundary value problems.
    • He published seven papers in the Transactions of the American Mathematical Society between 1904 and 1910: Green's theorem and Green's functions for certain systems of differential equations (1904), The doubly periodic solutions of Poisson's equation in two independent variables (1905), A problem of the calculus of variations in which the integrand is discontinuous (1906), On the boundary value problems of linear ordinary differential equations of second order (1906), The expansion of a function in terms of normal functions (1907); The properties of curves in space which minimize a definite integral (1908) and Fields of extremals in space (1910).

  196. Seki biography
    • Ten years later Leibniz, independently, used determinants to solve simultaneous equations although Seki's version was the more general.
    • He studied equations treating both positive and negative roots but had no concept of complex numbers.
    • In 1685, he solved the cubic equation 30 + 14x - 5x2 - x3 = 0 using the same method as Horner a hundred years later.
    • He discovered the Newton or Newton-Raphson method for solving equations and also had a version of the Newton interpolation formula.
    • Among other problems studied by Seki were Diophantine equations.

  197. Bocher biography
    • At Gottingen he also attended lecture courses by Klein on the potential function, on partial differential equations of mathematical physics and on non-euclidean geometry.
    • Bocher published around 100 papers on differential equations, series, and algebra.
    • Yet another exceptional service was rendered by his "Introduction to the Study of Integral Equations" ..
    • Special attention should be drawn also to his little known pamphlet on regular point of linear differential equations of the second order used for a number of years in connection with one of his courses of lectures.
    • When An introduction to the study of integral equations was reprinted in 1971 a reviewer wrote:- .
    • His final book was Lecons sur les methodes de Sturm dans la theorie des equations differentielles lineaires et leurs developpements modernes (1917) which was a record of lectures he gave in Paris in 1913-14 when he was Harvard Exchange Professor at the University of Paris.
    • He gave six lectures on Linear differential equations and their applications.
    • He was honoured with election to the National Academy of Sciences (United States) in 1909 and he served as president of the American Mathematical Society during 1909-1910 delivering his presidential address in Chicago on The published and unpublished works of Charles Sturm on algebraic and differential equations.
    • M Bocher: Integral equations .

  198. Forsythe biography
    • The books he wrote were: Bibliography of Russian Mathematics Books (1956); (with Wolfgang Wasow) Finite-Difference Methods for Partial Differential Equations (1967); and (with Cleve B Moler) Computer Solution of Linear Algebraic Systems (1967).
    • The author of [Finite-Difference Methods for Partial Differential Equations by George E Forsythe and Wolfgang R Wasow, Mathematics of Computation 16 (79) (1962), 379-380.',12)">12] puts Finite-Difference Methods for Partial Differential Equations into context:- .
    • The solution of partial differential equations by finite-difference methods constitutes one of the key areas in numerical analysis which have undergone rapid progress during the last decade.
    • As a result, the numerical solution of many types of partial differential equations has been made feasible.
    • The authors of this book have made an important contribution in this area, by assembling and presenting in one volume some of the best known techniques currently being used in the solution of partial differential equations by finite-difference methods.
    • The book is also praised by George Leo Watson in [Finite-Difference Methods for Partial Differential Equations by George E Forsythe and Wolfgang R Wasow, Biometrika 48 (3/4) (1961), 484.',15)">15]:- .
    • The aim of this monograph is to present, at the senior-graduate level, an up-to-date account of the methods presently in use for the solution of systems of linear equations.

  199. Golub biography
    • in 1959 for his thesis The Use of Chebyshev Matrix Polynomials in the Iterative Solution of Linear Equations Compared to the Method of Successive Overrelaxation which developed ideas in a paper by von Neumann.
    • In 1992 Golub, jointly with James M Ortega, published Scientific computing and differential equations.
    • A large part of scientific computing is concerned with the solution of differential equations, and thus differential equations are an appropriate focus for an introduction to scientific computing.
    • The need to solve differential equations was one of the original and primary motivations for the development of both analog and digital computers, and the numerical solution of such problems still requires a substantial fraction of all available computing time.
    • It is our goal in this book to introduce numerical methods for both ordinary and partial differential equations with concentration on ordinary differential equations, especially boundary value problems.
    • Although there are many existing packages for such problems, or at least for the main subproblems such as the solution of linear systems of equations, we believe that it is important for users of such packages to understand the underlying principles of the numerical methods.

  200. Levi Beppo biography
    • He had also studied the theory of integration, partial differential equations and the Dirichlet Principle, producing the famous "Beppo Levi theorem" and spaces now called "Beppo Levi spaces".
    • He wrote articles on logic, differential equations, complex variable, as well as on the border between analysis and physics.
    • He published Sistemas de ecuaciones analiticas en terminos finitos, diferenciales y en derivadas partiales (Systems of Analytic Equations: Equations in Finite Terms, Ordinary and Partial Differential Equations) (1944) as the first volume in the Monografias series.
    • The problems related to the resolution of equations, whether finite or differential, assume fundamentally different aspects according to whether we postulate only the existence of such properties of continuity and differentiability of the given functions and the unknowns as are strictly necessary if a particular problem is to have meaning, or admit additional hypotheses relative to the existence of a certain number of successive derivatives, or finally grant at once the existence of all derivatives.
    • This monograph is a clearly written exposition of the fundamental existence theorems for systems of analytic partial differential equations, together with necessary preliminary material on implicit functions and ordinary differential equations.

  201. Manfredi Gabriele biography
    • In the spring of 1706, Manfredi left Rome and returned to Bologna where he published his most famous work, De constructione aequationum differentialium primi gradus (Bologna 1707), the first monograph in the world dedicated to the study of differential equations [Dizionario Biografico degli Italiani 68 (2007).',4)">4]:- .
    • The work, in six sections, collected and presented in an orderly manner the results on first order differential equations scattered in the mathematical literature ..
    • He first studied equations with algebraic solutions, then those that lead to transcendental curves, then moved on to equations that are solved by means of substitution of variables.
    • The last section was a mixture of problems, some only proposed, such as integration of homogeneous equations.
    • He continued to produce works on differential equations, publishing Breve schediasma geometrico per la costruzione di una gran parte delle equazioni di primo grado, in the Giornale de' letterati d'Italia in 1714.
    • In this work he gave methods to integrate first order homogeneous differential equations.
    • Most of these memoirs concern the integration of ordinary differential equations.

  202. Stephansen biography
    • During the time that she was teaching Stephansen was working on her doctoral dissertation on partial differential equations.
    • Euler, d'Alembert and Lagrange had studied which second order partial differential equations which could be reduced to first order and this had been generalised by the Norwegian mathematician Alf Guldberg who, in 1900, had described all those third order partial differential equations which could be reduced to second order equations.
    • In her thesis, Stephansen generalised Guldberg's work and succeeded in describing all those fourth order partial differential equations which could be reduced to equations of the third order.
    • Stephansen published another paper in 1903 on differential equations, the idea for which came out of Hilbert's course of lectures that she attended.
    • She continued to undertake mathematical research and wrote two further papers, this time on difference equations, which were published in 1905 and 1906.

  203. Pell biography
    • Pell's equation y2 = ax2 + 1, where a is a non-square integer, was first studied by Brahmagupta and Bhaskara II.
    • It is often said that Euler mistakenly attributed Brouncker's work on this equation to Pell.
    • However the equation appears in a book by Rahn which was certainly written with Pell's help: some say entirely written by Pell.
    • Perhaps Euler knew what he was doing in naming the equation.
    • Pell's equation .
    • Pell's equation .
    • History Topics: Pell's equation .
    • Math Forum (Pell's equation) .

  204. Wilf biography
    • The title of the thesis was "The transmission of neutrons in multilayered slab geometry." It solved the transport equation in multilayered geometry by regarding each homogeneous layer as a little black box with prescribed inputs and outputs (which point of view was Jerry's hallmark), and it wired them together by representing each by a matrix.
    • Even before the award of his doctorate, Wilf had written a remarkable range of papers: (with M Kalos) Monte Carlo solves reactor problems (1957); An open formula for the numerical integration of first order differential equations (1957); An open formula for the numerical integration of first order differential equations.
    • After that normalization, the basic "WZ" equation F(n+1, k) - F(n, k) = G(n, k+1) - G(n, k) appeared in the room, and its self-dual symmetrical form was very compelling.

  205. Woodward biography
    • However, he also published several papers in the Bulletin of the American Mathematical Society such as: On the cubic equation defining the Laplacian envelope of the earth's atmosphere (1897), On the integration of a system of simultaneous linear differential equations (1897), On the differential equation defining the Laplacian distribution of density, pressure, and acceleration of gravity in the earth (1898), On the mutual gravitational attraction of two bodies whose mass distributions are symmetrical with respect to the same axis (1898), and An elementary method of integrating certain linear differential equations (1900).

  206. Korteweg biography
    • On this topic he published Sur la forme que prennent les equations du mouvement des fluids si l'on tient compte des forces capillaires causes par les variations de densite (On the form the equations of motions of fluids assume if account is taken of the capillary forces caused by density variations) in 1901.
    • He is remembered in particular for the Korteweg - de Vries equation on solitary waves, a courageous topic to attack since many mathematicians, including Stokes, were convinced such waves could not exist.
    • They found explicit, closed-form, travelling-wave solutions to the Korteweg - de Vries equation that decay rapidly.
    • We can mention, again showing Korteweg's versatility, his pure mathematics work on algebraic equations in papers such as Sur un theoreme remarquable, qui se rapporte a la theorie des equations algebriques a parametres reels, dont toutes les racines restent constamment reelles (1900).

  207. Klein biography
    • He owed some of his greatest successes to his development of Riemann's ideas and to the intimate alliance he forged between the later and the conception of invariant theory, of number theory and algebra, of group theory, and of multidimensional geometry and the theory of differential equations, especially in his own fields, elliptic modular functions and automorphic functions.
    • He showed it had equation x3y + y3z + z3x = 0 as a curve in projective space and its group of symmetries was PSL(2,7) of order 168.
    • Klein considered equations of degree greater than 4 and was particularly interested in using transcendental methods to solve the general equation of the fifth degree.

  208. Vinti biography
    • He remained at Palermo undertaking research on partial differential equations and when Baiada returned from three years studying in the United States in 1952, the two began a close collaboration.
    • The first of these papers gives an existence theorem for the equation zx = f (x, y, z, zy) using methods which had been developed by Baiada a couple of years earlier in solving a simpler equation.
    • The scientific interests of Calogero Vinti covered several areas of Mathematical Analysis, from Calculus of Variations to Differential Equations, from Approximation Theory to Real Analysis and Measure Theory.

  209. Kochina biography
    • An application of the theory of linear differential equations to some problems of ground-water motion published in 1940 is quite typical of many of her papers.
    • For example in 1948 she studied numerical solutions of a partial differential equation in On a nonlinear partial differential equation arising in the theory of filtration.
    • The papers in this book are divided into eight sections: Kinematics of atmospheric motions; Hydrodynamics; Applications of the analytical theory of linear differential equations in filtration theory; Steady flow in the presence of porous media; Unsteady motion of groundwater; Problems on oil filtration; Gas filtration through coal layers; and Filtration of liquids through porous media.

  210. Pascal Ernesto biography
    • He developed the intergraph, an instrument for the mechanical integration of differential equations.
    • A discussion is given of the theory of the integraph of Abdank-Abakanowicz with various improvements and modifications which the author has made in order to enable him to solve special types of first and second order differential equations, such as the general linear first order equation, the equation y" = (y3/x)1/2, etc.

  211. Crank biography
    • His main work was on the numerical solution of partial differential equations and, in particular, the solution of heat-conduction problems.
    • John Crank is best known for his joint work with Phyllis Nicolson on the heat equation, where a continuous solution u(x, t) is required which satisfies the second order partial differential equation .
    • Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level.

  212. Bevan-Baker biography
    • Huygens' geometrical construction, with its restriction that only one sheet of the envelope of the spherical wavelets is to be considered, is first justified in Chapter I by Poisson's analytical solution of the equation of wave-motions.
    • The analogue of Kirchhoff's formula, due to Volterra, is derived and an interesting account is given of a method, devised by Marcel Riesz and based on the theory of fractional integration, which provides a powerful method of solving initial value problems for equations like the wave equation.
    • A second edition of the book, which differed from the first by the addition of a new chapter on the application of the theory of integral equations to problems of diffraction theory by a plane screen, was published in 1950.

  213. Boussinesq biography
    • In his first derivation of the solitary wave, published in 1871 in the 'Comptes rendus', Boussinesq sought an approximate solution of Euler's equations that propagated at the constant speed c without deformation in a rectangular channel.
    • Lagrange had already tried this route and written the resulting series of differential equations, but had found their integration to exceed the possibilities of contemporary analysis unless nonlinear terms were dropped.
    • Mecanique 335 (2007), 479-495.',6)">6] Bois collects these under the following headings: The problem of static stresses in soils; Turbulent flows (first phase); Surface waves and Boussinesq's equation; The BBO equation and the 'historical term' of Basset-Boussinesq; and The method of potential, the 'Boussinesq problem' and the vibrations of bars.

  214. Bernoulli Daniel biography
    • The third part of Mathematical exercises was on the Riccati differential equation while the final part was on a geometry question concerning figures bounded by two arcs of a circle.
    • He was able to give the basic laws for the theory of gases and gave, although not in full detail, the equation of state discovered by Van der Waals a century later.
    • Daniel worked on mechanics and again used the principle of conservation of energy which gave an integral of Newton's basic equations.
    • It is especially unfortunate that he could not follow the rapid growth of mathematics that began with the introduction of partial differential equations into mathematical physics.

  215. Lipschitz biography
    • He carried out many important and fruitful investigations in number theory, in the theory of Bessel functions and of Fourier series, in ordinary and partial differential equations, and in analytical mechanics and potential theory.
    • Lipschitz's work on the Hamilton-Jacobi method for integrating the equations of motion of a general dynamical system led to important applications in celestial mechanics.
    • Lipschitz is remembered for the 'Lipschitz condition', an inequality that guarantees a unique solution to the differential equation y' = f (x, y).
    • Peano gave an existence theorem for this differential equation, giving conditions which guarantee at least one solution.

  216. Boruvka biography
    • He discussed these matters with Frantisek Vycichlo, a Prague mathematician, and their feeling was that differential equations would be a good direction to take his research team.
    • Boruvka had already written a paper on differential equations in 1934, but now he began to direct the research of Masaryk University towards that topic.
    • We discussed the matter thoroughly and arrived at the conclusion that it was essential to start pursuing the theory of differential equations which is immensely important as far as applications are concerned and which was much neglected before the war and in essence it was not at all developed.
    • In 1946 Boruvka became an ordinary professor at Masaryk University and in the following year he set up a Differential Equations Seminar.
    • The main aim of the seminar was to study global properties of linear differential equations of the nth order.
    • Boruvka's publications on this topic include Sur les integrales oscilatoires des equations differentielles lineaires du second ordre (1953), Remark on the use of Weyr's theory of matrices for the integration of systems of linear differential equations with constant coefficients (Czech) (1954), Uber eine Verallgemeinerung der Eindeutigkeitssatze fur Integrale der Differentialgleichung y' = f (x, y) (1956), and Sur la transformation des integrales des equations differentielles lineaires ordinaires du second ordre (1956).

  217. Schlesinger biography
    • He then studied mathematics and physics at the universities of Heidelberg and Berlin between 1896 and 1887, and he received a doctorate from the University of Berlin in 1887 for a thesis on differential equations entitled: Uber lineare homogene Differentialgleichungen vierter Ordnung, zwischen deren Integralen homogene Relationen hoheren als ersten Grades bestehen.
    • In this paper Schlesinger formulated the problem of isomonodromy deformations for a certain matrix Fuchsian equation.
    • Prove the existence of linear differential equations having a prescribed monodromic group.
    • The paper introduces what today are known as the Schlesinger transformations and Schlesinger equations which have an important role in differential geometry.

  218. Hertz Heinrich biography
    • There were several new factors in the equation which affected the issue such as, on the negative side, his unhappiness with the working environment of engineering firms, and on the positive side, his enjoyment of the mathematics he had learnt as part of his engineering studies.
    • However, it may have been a wise decision to delay beginning the work as S D'Agostino [Centaurus 36 (1) (1993), 46-82.',11)">11] suggests that Hertz's derivation of Maxwell's equations in 1884 formed an important part of the structural background to his studies on the propagation of electric waves which he now carried out.
    • He searched for a mechanical basis for electrodynamics starting from Maxwell's equations.
    • Maxwell's theory is Maxwell's system of equations.

  219. Zhang Qiujian biography
    • There are problems on extracting square and cube roots, problems on finding the solution to quadratic equations, problems on finding the sum of an arithmetic progression, and on solving systems of linear equations.
    • Zhang gives the solution by solving a quadratic equation, but his formulae are not particularly accurate.
    • In Chapter 3 problems which involve solving systems of equations occur.

  220. Feller biography
    • He transformed the relation between Markov processes and partial differential equations.
    • Other papers written by Feller while still at Brown University include: On the time distribution of so-called random events (1940), On the integral equation of renewal theory (1941), On A C Aitken's method of interpolation (1943), The fundamental limit theorems in probability (1945) and Note on the law of large numbers and "fair" games (1945).
    • outlines some new results and open problems concerning diffusion theory where we find an intimate interplay between differential equations and measure theory in function space.
    • It was also the first mathematics course I took at Princeton (a course in sophomore differential equations).

  221. Navier biography
    • Navier is remembered today, not as the famous builder of bridges for which he was known in his own day, but rather for the Navier-Stokes equations of fluid dynamics.
    • He gave the well known Navier-Stokes equations for an incompressible fluid in 1821 while in 1822 he gave equations for viscous fluids.
    • We should note, however, that Navier derived the Navier-Stokes equations despite not fully understanding the physics of the situation which he was modelling.
    • He did not understand about shear stress in a fluid, but rather he based his work on modifying Euler's equations to take into account forces between the molecules in the fluid.
    • The irony is that although Navier had no conception of shear stress and did not set out to obtain equations that would describe motion involving friction, he nevertheless arrived at the proper form for such equations.

  222. Schur biography
    • Third, he handled algebraic equations, sometimes proceeding to the evaluation of roots, and sometimes treating the so-called equation without affect, that is, with symmetric Galois groups.
    • He was also the first to give examples of equations with alternating Galois groups.
    • Sixth, in integral equations; .

  223. Siegel biography
    • Approximation of algebraic numbers by rationals and applications thereof to Diophantine equations.
    • These include his improvement of Thue's theorem, described above, given in his 1920 dissertation, and its application to certain polynomial Diophantine equations in two unknowns, proving an affine curve of genus at least 1 over a number field has only a finite number of integral points in 1929.
    • He had earlier than this in 1922, written papers on the functional equation of Dedekind's zeta functions of algebraic number fields and in 1921/23 made contributions to additive questions such as Waring type problems for algebraic number fields.
    • He examined Birkhoff's work on perturbation theory solutions for analytical Hamiltonian differential equations near an equilibrium point using formal power series.

  224. Arnold biography
    • Arnold has also made innumerable and fundamental contributions to the theory of differential equations, symplectic geometry, real algebraic geometry, the calculus of variations, hydrodynamics, and magneto- hydrodynamics.
    • The areas are Dynamical Systems, Differential Equations, Hydrodynamics, Magnetohydrodynamics, Classical and Celestial Mechanics, Geometry, Topology, Algebraic Geometry, Symplectic Geometry, and Singularity Theory.
    • He published Problemes ergodiques de la mecanique classique (with A Avez) (1967), Ordinary differential equations (Russian) (1971), Mathematical methods of classical mechanics (Russian) (1974), Supplementary chapters to the theory of ordinary differential equations (Russian) (1978), Singularity theory (1981), Singularities of differentiable mappings (Russian) (with A N Varchenko and S M Gusein-Zade) (1982), Catastrophe theory (1984), Huygens and Barrow, Newton and Hooke (Russian) (1989), Contact geometry and wave propagation (1989), Singularities of caustics and wave fronts (1990), The theory of singularities and its applications (1991), Topological invariants of plane curves and caustics (1994), Lectures on partial differential equations (Russian) (1997), Topological methods in hydrodynamics (with B A Khesin) (1998), and Arnold problems (Russian) (2000).
    • for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singularity theory.
    • In classical hydrodynamics the basic equations of an ideal fluid were derived by Euler in 1757 and major steps towards understanding them were taken by Helmholtz in 1858, and Kelvin in 1869.

  225. Hamming biography
    • His doctoral dissertation Some Problems in the Boundary Value Theory of Linear Differential Equation was supervised by Waldemar Trjitzinsky (1901-1973).
    • His interests were in analysis, particularly measure theory, integration and differential equations.
    • Hamming did, however, develop interests in ideas that were quite far removed from his study of differential equations [Amer.
    • Hamming also worked on numerical analysis, integrating differential equations, and the Hamming spectral window which is much used in computation for smoothing data before Fourier analysing it.

  226. Stampacchia biography
    • For three years he produced outstanding examination results in a wide range of courses such as Tutorial Sessions in Analysis and in Geometry, Calculus of Variations, Theory of Functions, and Ordinary Differential Equations.
    • His thesis was concerned with an adaptation of an approximation procedure for Volterra integral equations due to Tonelli to boundary value problems for systems of ordinary differential equations.
    • From the time Stampacchia took up his appointment in Naples, his research output was impressive consisting mainly of papers on differential equations and the calculus of variations.
    • The years that Stampacchia spent in Pisa and Naples characterize the formation of his personality as an analyst: he was a passionate specialist in calculus of variations and in the theory of partial differential equations, a practitioner and an inspirer of research works of considerable depth and originality of thought.
    • His 326 page text Equations elliptiques du second ordre a coefficients discontinus was published in 1966, then in 1967 he was elected President of the Italian Mathematical Union (Unione Matematica Italiana).
    • On the one hand, variational inequalities have stimulated new and deep results dealing with nonlinear partial differential equations.

  227. Anosov biography
    • His work was supervised by Pontryagin and during this period Anosov published a number of papers including: On stability of equilibrium states of relay systems (Russian) (1959); Averaging in systems of ordinary differential equations with rapidly oscillating solutions (Russian) (1960); and Limit cycles of systems of differential equations with small parameters in the highest derivatives (Russian) (1960).
    • Anosov defended his thesis on averaging in systems of ordinary differential equations at Moscow University in 1961.
    • Hilbert's 21st problem (the Riemann-Hilbert problem) belongs to the theory of linear systems of ordinary differential equations in the complex domain: does there exist a Fuchsian system having these singularities and monodromy? Hilbert was convinced that such a system always exists.
    • He continues to work (2009) at the Department of Differential Equations in the Steklov Mathematical Institute and at present he is Head of Department and serves on the Academic Council of the Institute.
    • He is also Chairman of the Dissertation Council covering the areas of Differential equations, Mathematical physics, and Theoretical physics.
    • He also has been involved with the International Congresses of Mathematicians in other capacities, being a member of the panel to decide the scientific programme in the section "Dynamical systems and differential equations" for three of the Congresses, and on two of these occasions he chaired the panel.

  228. Leimanis biography
    • Immediately he was on his travels again, this time going to Paris where he spent a year at the Henri Poincare Institute undertaking research on differential equations and celestial mechanics.
    • Leimanis continued to publish and, when he was approaching 80, the paper On integration of the differential equation of central motion appeared.
    • Assuming that the force acting on a particle is of the form f(r)g(q), the theory of infinitesimal transformations is applied to determine the forms of f(r) and g(q) for which the differential equation of central motion is integrable by quadratures or reducible to a first-order differential equation.

  229. Guo Shoujing biography
    • We should now look at the rather remarkable work which Guo did on spherical trigonometry and solving equations.
    • To solve this equation Guo used a numerical method similar to Horner's method.
    • The equation has two real roots, the smaller being the solution to the problem while the other, being numerically larger than the length of the arc, was rightly discarded by Guo.
    • Two of the coefficients of the equation, namely the constant term and the coefficient of x2, involve the length a of the arc, so require a value to be chosen for π.

  230. Pincherle biography
    • His research mainly concerned functional equations and functional analysis.
    • From about 1890, Pincherle published several papers in which he used the axiomatic approach with differential and integral equations.
    • even though he was the author of the article on functional equations and operators in the French version of the "Encyclopedie des mathematique pures et appliquees" (1912), in which he gave a very detailed historical account and referred to his own work, Pincherle's work itself did not have much influence.
    • the 1888 paper (in Italian) of S Pincherle on the 'Generalized Hypergeometric Functions' led him to introduce the afterwards named Mellin-Barnes integral to represent the solution of a generalized hypergeometric differential equation investigated by Goursat in 1883.

  231. Gelfond biography
    • Gelfond developed basic techniques in the study of transcendental numbers, that is numbers that are not the solution of an algebraic equation with rational coefficients.
    • He also contributed to the study of differential and integral equations and to the history of mathematics.
    • This book is very much in the spirit of the modern Russian school concerned with the so-called constructive theory of functions, approximative methods for the solution of differential equations, and so forth.
    • Also in 1952 Gelfond published the low level Solving equations in integers which was translated into English in 1960.

  232. Spence David biography
    • Obtaining equations under special conditions, Spence found numerical results for lift, pitching moment, and jet shape, which he compared with experimental results obtained from a wind tunnel.
    • By similarity considerations, the displacements are expressed in terms of the solution of a pair of nonlinear ordinary differential equations satisfying two-point boundary conditions.
    • We consider boundary value problems for the biharmonic equation in the open rectangle x > 0, -1 < y < 1, with homogeneous boundary conditions on the free edges y = ±1, and data on the end x = 0 of a type arising both in elasticity and in Stokes flow of a viscous fluid, in which either two stresses or two displacements are prescribed.
    • For such 'noncanonical' data, coefficients in the eigenfunction expansion can be found only from the solution of infinite sets of linear equations, for which a variety of methods of formulation have been proposed.

  233. De Vries biography
    • Gustav de Vries's name is well known to mathematicians because of the work of his doctoral dissertation which contained the Korteweg-de Vries equation.
    • On 1 December 1894 de Vries had an oral examination on his thesis Bijdrage tot de kennis der lange golven which contained the famous Korteweg-de Vries equation.
    • They found explicit, closed-form, travelling-wave solutions to the Korteweg - de Vries equation that decay rapidly.
    • However, both Korteweg and de Vries seem to have completely missed the fact that the equation, now called the Korteweg-de Vries equation, had already appeared in the work of Joseph Valentin Boussinesq.
    • In fact the equation appears as a footnote in Boussinesq's 680-page treatise Essai sur la theorie des eaux courantes (1885), but this should not in any way diminish the importance of de Vries's contribution.
    • to commemorate the centennial of the equation by and named after Korteweg and de Vries.

  234. Henrici Peter biography
    • His next contribution Bergmans Integraloperator erster Art und Riemannsche Funktion (1952) is an elegant study of the representation of solutions of an elliptic partial differential equation in terms of analytic functions.
    • His first book Discrete variable methods in ordinary differential equations, published by John Wiley & Sons in 1962, quickly won international acclaim and became a classic standard text on the topic.
    • This book contains a comprehensive and up-to-date treatment of methods for the numerical integration of ordinary differential equations, especially those associated with initial-value problems.
    • There is no doubt that this book is a valuable contribution to numerical analysis, and it will certainly have an important influence on future developments in the numerical integration of ordinary differential equations.

  235. Aronhold biography
    • Certain linear partial differential equations which he came across in his work are characteristic of invariant theory and are named after him.
    • Aronhold established his theory in general and does not derive any specific equations.
    • His efforts to obtain equations independent of substitution coefficients led to linear partial differential equations of the first order, which have linear coefficients.
    • These equations, which are characteristic of the theory of invariants, are known as 'Aronhold's differential equations'.
    • Aronhold explicitly established the required fourth degree equations and formulated a theorem on plane curves of the fourth order.

  236. Hilbert biography
    • Hilbert's work in integral equations in about 1909 led directly to 20th-century research in functional analysis (the branch of mathematics in which functions are studied collectively).
    • Making use of his results on integral equations, Hilbert contributed to the development of mathematical physics by his important memoirs on kinetic gas theory and the theory of radiations.
    • Many have claimed that in 1915 Hilbert discovered the correct field equations for general relativity before Einstein but never claimed priority.
    • In this paper the authors show convincingly that Hilbert submitted his article on 20 November 1915, five days before Einstein submitted his article containing the correct field equations.
    • Einstein's article appeared on 2 December 1915 but the proofs of Hilbert's paper (dated 6 December 1915) do not contain the field equations.
    • In the printed version of his paper, Hilbert added a reference to Einstein's conclusive paper and a concession to the latter's priority: "The differential equations of gravitation that result are, as it seems to me, in agreement with the magnificent theory of general relativity established by Einstein in his later papers".
    • Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations.

  237. Peschl biography
    • this work lies on the common boundary between differential geometry, function theory (of one and several variables) and partial differential equations.
    • in the Schwarz lemma; the essence is the relation between the standard hyperbolic metric in the unit disc and the Beltrami equation, to which particular differential invariants are associated.
    • This situation has been generalized to different types of metrics, to equations of higher order and to more than one variable.
    • Partielle Differentialgleichungen erster Ordnung (1973) provides an elementary introduction to first order partial differential equations while Differential-geometrie (1973) provides a clear, elementary and concisely presented introduction to local differential geometry in Euclidean and Riemannian spaces.

  238. Saunderson biography
    • The chapters on algebra introduce the idea of an equation and how real life problems can be reduced to equations.
    • The reader is shown how to solve quadratic equations, there other topics such as magic squares are studied.
    • The final book presents the solution of cubic and quartic equations.

  239. Garnir biography
    • In Sur la theorie de la lumiere de M L de Broglie (1945) he compared the approach by Kemmer to the theory of the meson to de Broglie's modifications of Maxwell's equations.
    • His fields of research were broad, including algebra and mathematical analysis, in particular the very active field of functional analysis, and the still booming area of partial differential equations, especially boundary value problems.
    • Another of his interests was in the theory of boundary value problems for partial differential equations.
    • In particular, he studied Green's functions as solutions to boundary value problems for the wave and diffusion equations.
    • In the later part of his career, Garnir became interested in the propagation of singularities of solutions of boundary value problems for evolution partial differential equations.
    • In recent years the use of such tools as operators in Hilbert and Banach spaces, the theory of distributions and other methods of functional analysis has become commonplace in investigations of problems in partial differential equations.
    • Thus from its very beginning it was a Centre of international cooperation in the broad realms of functional analysis and partial differential equations.

  240. Gelbart biography
    • derive the equation of the locus of its centre.
    • The basic idea was to construct a theory similar to complex function theory for the solutions of a system of generalized Cauchy-Riemann equations arising in the mechanics of continua.
    • They published two joint papers on S-monogenic functions, namely On a class of differential equations in mechanics of continua (1943) and On a class of functions defined by partial differential equations (1944).

  241. Cooper biography
    • He then played a major role in the British mathematical scene serving on the Mathematics Panel of the University Grants Committee in 1971-75 and other committees, as well as organising a major conference on differential equations at Chelsea College.
    • His research was on a wide range of different but related topics: operator theory, transform theory, thermodynamics, functional analysis and differential equations.
    • Using the exact equations of elasticity, and a Fourier transform integrating technique, conclusions are obtained as to (1) the velocities of propagation which can be obtained and in particular their upper bounds; (2) the dispersive nature of the waves, both longitudinal and transverse; (3) the velocity at which elastic energy can be expected to be transported.
    • Other papers in which deal with applications include The uniqueness of the solution of the equation of heat conduction (1950).

  242. Brouncker biography
    • Brouncker gave a method of solving the diophantine equation .
    • See our article Pell's equation for more details.
    • It is believed that Euler made an error in naming the equation 'Pell's equation', and that he was intending to acknowledge the outstanding contribution made by Brouncker.
    • It is interesting to think that if Euler had not made this error then Brouncker, instead of being relatively unknown as a mathematician, would be universally known through 'Brouncker's equation'.
    • Pell's equation .
    • History Topics: Pell's equation .

  243. Day biography
    • Within nine months he had completed his Master's degree and submitted a Master's thesis On modular equational classes.
    • Some of Day's early papers are: Injectives in non-distributive equational classes of lattices are trivial (1970), A note on the congruence extension property (1971), Injectivity in equational classes of algebras (1972), Splitting algebras and a weak notion of projectivity (1973), Filter monads, continuous lattices and closure systems (1975), and Splitting lattices generate all lattices (1975).
    • I came to work in the morning, wrote the equations down, and tried to manipulate them.

  244. Riccati Vincenzo biography
    • Vincenzo continued his father's work on integration and differential equations but Giorgio Bagni notes differences in their approach in [Riv.
    • Vincenzo's favorite field of research is analysis, in particular setting out the analytical treatment of mechanical problems, conducted by solving differential equations, properly constructed.
    • Vincenzo gave a collection of methods to solve certain specific types of differential equations in his memoir De usu motus tractorii in constructione Aequationum Differentialium Commentarius (1752).
    • Here he notes new approaches by Euler and these influence him in finding a new method for solving differential equations.
    • Vincenzo Riccati, somehow, put an end to this trend by showing that one could construct in a simple continuous way all transcendental curves from the differential equations that define them.
    • It probably came too late, at the end of the period of construction of the curves, when geometry has given way to algebra, and when series became the tool of choice to represent the solutions of differential equations.
    • Vincenzo studied hyperbolic functions and used them to obtain solutions of cubic equations.

  245. Luzin biography
    • Many of these mathematicians turned to other topics such as topology, differential equations, and functions of a complex variable.
    • In 1931 Luzin himself turned to a new area when he began to study differential equations and their application to geometry and to control theory.
    • Finikov had derived differential equations that determine all principal on a given surface, and Byushgens had obtained differential equations that determine surfaces which have a given linear element and admit a bending on a principal base.
    • However, the question of solubility of these equations, in general, remained unclear.
    • no example was found in which the equations ..
    • up to 1938, when Luzin, by means of a subtle analysis of these equations, established that the existence of a principal base is rather rare.

  246. Petersen biography
    • The interest he had shown in ruler and compass constructions when he was at school had continued to influence his research topic and his doctoral thesis was entitled On equations which can be solved by square roots, with application to the solution of problems by ruler and compass.
    • If the equation of degree 2n can be solved by square roots, one of the roots can be expressed by n such different square roots, where each can appear several times.
    • His research was on a wide variety of topics from algebra and number theory to geometry, analysis, differential equations and mechanics.
    • He published The theory of algebraic equations in 1877 which was written in a concise style, treating as many topics as possible without using Galois theory.

  247. Gilbarg biography
    • His was work there took him into new areas of mathematics and involved fluid dynamics and nonlinear partial differential equations.
    • Except for a paper relating to his thesis which was published in the Duke Mathematical Journal in 1942, all his remaining mathematical publications were in the areas of fluid dynamics and nonlinear partial differential equations.
    • For many mathematicians, Gilbarg is best known for his remarkable book Elliptic Partial Differential Equations of Second Order written in collaboration with Neil Trudinger and published in 1977.
    • I could never have imagined forty years ago when my book with David Gilbarg on elliptic partial differential equations was first published that it would get such recognition.
    • the systematic development of the general theory of second order quasilinear elliptic equations and of the linear theory required in the process.
    • Aside from his fluid dynamics work, undoubtedly [Gilbarg's] best-known contribution to the mathematical literature is the monograph "Elliptic Partial Differential Equations of Second Order," co-authored with his former Stanford PhD student Neil S Trudinger.
    • At the recent commemorative conference in the Stanford Mathematics Department, James Serrin described the Gilbarg-Trudinger text as being "on the bookshelf of everyone working in partial differential equations, a monumental work which is one of the great lasting achievements of analysis." .

  248. Carmeli biography
    • Among his publications at this time are The motion of a particle of finite mass in an external gravitational field (1964), Has the geodesic postulate any significance for a finite mass? (1964), Semigenerally covariant equations of motion.
    • Derivation (1965), Semigenerally covariant equations of motion.
    • The significance of the "tail" and the relation to other equations of motion (1965), Motion of a charge in a gravitational field (1965), The equations of motion of slowly moving particles in the general theory of relativity (1965), and Equations of motion without infinite self-action terms in general relativity (1965).
    • During his time in this post he published papers such as Group analysis of Maxwell's equations (1969), Infinite-dimensional representations of the Lorentz group (1970), and SL(2, C) symmetry of the gravitational field dynamical variables (1970).
    • students, and E Leibowitz) Gauge fields : Classification and equations of motion (1989):- .

  249. Davis biography
    • Devise a process according to which it can be determined by a finite number of operations whether a given polynomial equation with integer coefficients in any number of unknowns is solvable in rational integers.
    • Does there exist an algorithm to determine whether a Diophantine equation has a solution in natural numbers? .
    • During these years when Davis was moving around spending time at various institutions, he published a number of important papers such as Arithmetical problems and recursively enumerable predicates (1953), The definition of universal Turing machine (1957), (with Hilary Putnam) Reductions of Hilbert's tenth problem (1958) and (with Hilary Putnam and Julia Robinson) The decision problem for exponential diophantine equations (1961).
    • is a completely self-contained exposition of the proof that there is no algorithm for determining whether an arbitrary Diophantine polynomial equation with integer coefficients has an integer solution.

  250. Levi-Civita biography
    • the main mathematical and physical questions discussed by Einstein and Levi-Civita in their 1915 - 1917 correspondence: the variational formulation of the gravitational field equations and their covariance properties, and the definition of the gravitational energy and the existence of gravitational waves.
    • Its major achievements are two: a derivation of the equations of motion of n point masses, free from the subtle errors besetting most of the standard treatments; and a careful discussion of the possible contributions, in the Einsteinian approximation, of the finite size and internal constitution of the bodies involved.
    • He also wrote on the theory of systems of ordinary and partial differential equations.
    • In [Italian mathematics between the two world wars (Pitagora, Bologna, 1987), 125-141.',18)">18] the authors argue that Levi-Civita was interested in the theory of stability and qualitative analysis of ordinary differential equations for three reasons: his interest in geometry and geometric models; his interest in classical mechanics and celestial mechanics, in particular, the three-body problem; and his interest in stability of movement in the domain of analytic mechanics.
    • Their results include the conception of the localized induction approximation for the induced velocity of thin vortex filaments, the derivation of the intrinsic equations of motion, the asymptotic potential theory applied to vortex tubes, the derivation of stationary solutions in the shape of helical vortices and loop-generated vortex configurations, and the stability analysis of circular vortex filaments.
    • In 1933 Levi-Civita contributed to Dirac's equations of quantum theory.

  251. Lemaitre biography
    • Einstein was at the conference and he spoke to Lemaitre in Brussels telling him that the ideas in his 1927 paper had been presented by Friedmann in 1922, but he also said that although he thought Lemaitre's solutions of the equations of general relativity were mathematically correct, they presented a solution which was not feasible physically.
    • In 1942 he published L'iteration rationnelle in which he discussed Gauss's method of successive approximations applied to a system of two equations in two unknowns to determine the orbit of a planet from three observations.
    • Lemaitre then applied these ideas to accelerate the orthodox process of iteration, taking the Picard iterative solution of first order differential equations as an example.
    • He applied the same techniques in another paper published in the same year, namely Integration d'une equation differentielle par iteration rationnelle.
    • In Sur un cas limite du probleme de Stormer (1945) he studied trajectories of an electron in the neighborhood of lines of force of a magnetic dipole field, then returned to his study of numerical solutions to first order differential equations in Interpolation dans la methode de Runge-Kutta (1947).
    • Is it possible to account for the existence of more or less permanent concentrations of galaxies in which no single galaxy remains long in the same place? The two-fold purpose of the paper is to delineate the underlying mechanical model and to write down the fundamental equations of the problem.
    • It is shown how these equations can be applied toward the solution of the well-known problem of uniform distribution in a homogeneous, expanding universe.

  252. Bolibrukh biography
    • This was a useful activity connected with the applied orientation of the Moscow Institute of Physics and Technology and requiring a certain qualification, but it had nothing to do with linear differential equations in the complex domain.
    • Does there exist a Fuchsian system of linear ordinary differential equations in the complex domain having prescribed singularities and monodromy group? .
    • He was an invited lecturer at the International Congress of Mathematicians held in Zurich in 1994 giving the lecture The Riemann-Hilbert problem and Fuchsian differential equations on the Riemann sphere.
    • In those years he gave the specialized lecture course on the analytic theory of differential equations, vector bundles and the Riemann-Hilbert problem.
    • He prepared his lectures thoroughly, and one of his courses was published as the book Fuchsian differential equations and holomorphic bundles, which is a good example of his style.
    • He agreed, and we joined the group working on differential equations, and later, together with him, joined the newly formed dynamical systems group.

  253. Belanger biography
    • The application of the momentum principle to the hydraulic jump is commonly called the Belanger equation, but few know that his original treatise was focused on the study of gradually varied open channel flows (Belanger 1828).
    • The originality of Belanger's 1828 essay was the successful development of the backwater equation for steady, one-dimensional gradually-varied flows in an open channel, together with the introduction of the step method, distance calculated from depth, and the concept of critical flow conditions.
    • Belanger gave some specific examples in the paper to show the applicability of his equation.
    • Belanger provided a stepwise integration of this equation in the simple case of the horizontal aqueduct that had been built recently to bring the waters of the Ourcq River into Paris.
    • This 1841 work contained the hydraulic jump equation now known as the Belanger equation.

  254. Skorokhod biography
    • He was awarded his doctorate in 1962 for his thesis Stochastic differential equations and limit theorems for random processes.
    • They have ranged over almost all the fundamental areas of these theories, and Skorokhod's contribution to the development of subjects such as limit theorems for random processes, stochastic differential equations, and probability distributions in infinite dimensional spaces can scarcely be exaggerated.
    • The book is primarily devoted to the development of certain probabilistic methods in the field of stochastic differential equations and limit theorems for Markov processes.
    • 42 (9) (1990), 1157-1170.',11)">11] Korolyuk and Portenko survey Skorokhod's work on limit theorems for random processes, stochastic differential equations and the theory of Markov processes.
    • Limit theorems for random processes and stochastic differential equations are the areas of probability theory in which, over 35 years ago, Skorokhod started his scientific career and contributed much to their far-reaching advances.
    • III (1975), Controlled random processes (1977); and Stochastic differential equations and their applications (1982).

  255. Kolchin biography
    • Other papers around this time were Algebraic matric groups (1946) and The Picard-Vessiot theory of homogeneous linear ordinary differential equations (1946).
    • His deep and abiding interest has always been in the application of the powerful and clarifying techniques of algebra to problems in the theory of differential equations.
    • Following the tradition set by Joseph Fels Ritt (1893 - 1951), the founding father of differential algebra, his desire has been to remove the algebraic aspects of differential equations from analysis.
    • It is intended that such a theory bear to algebraic groups the same relation that the theory of differential equations bears to the theory of algebraic equations.
    • Algebraic groups can be viewed as groups in the category of algebraic varieties, where the latter are taken to be locally given as sets of simultaneous solutions of algebraic equations.

  256. Lie biography
    • Although not on the permanent staff, Sylow taught a course, substituting for Broch, in which he explained Abel's and Galois' work on algebraic equations.
    • Lie had started examining partial differential equations, hoping that he could find a theory which was analogous to the Galois theory of equations.
    • the theory of differential equations is the most important discipline in modern mathematics.
    • He examined his contact transformations considering how they affected a process due to Jacobi of generating further solutions of differential equations from a given one.
    • It was during the winter of 1873-74 that Lie began to develop systematically what became his theory of continuous transformation groups, later called Lie groups leaving behind his original intention of examining partial differential equations.

  257. Haack biography
    • However, around 1936, he began to work on problems in gas dynamics and differential equations, collaborating with his wife on these topics.
    • As well as these books on geometry, he also continued his work on gas dynamics; for example in 1958 he published the paper (published jointly with his doctoral student Gerhard Bruhn) Ein Charakteristikenverfahren fur dreidimensionale instationare Gasstromungen in which he derived the characteristic equations for the unsteady three-dimensional motion of inviscid perfect gas.
    • At the Technische Hochschule of Berlin, Haach lectured on systems of two linear partial differential equations of first order in two independent variables, in particular, elliptic systems.
    • This book combines the study of partial and Pfaff differential equations.
    • The point of view is to consider partial differential equations in the framework of Pfaff equations.

  258. Atkinson biography
    • A new phase of his work began when he began to study eigenfunction expansions both for difference equations and differential equations.
    • We shall present the theory of certain recurrence relations in the spirit of the theory of boundary problems for differential equations.
    • Second, we shall present the theory of boundary problems for certain ordinary differential equations, emphasizing cases in which the coefficients may be discontinuous, or may have singularities of delta function type.
    • Finally, we give some account of theories which unify the topics of differential and difference equations, relying mainly on the method of replacement by integral equations.

  259. Ford biography
    • Ford read the paper On the Roots of a Derivative of a Rational Function to the meeting of the Society on Friday 14 May 1915, the paper On the Oscillation Functions derived from a Discontinuous Function to the meeting on 11 June 1915, and the paper A method of solving algebraic equations to the meeting on 12 January 1917.
    • Following his contributions to the war effort, Ford joined the faculty at the Rice Institution, Houston, Texas and while there he published papers such as On the closeness of approach of complex rational fractions to a complex irrational number (1925), The Solution of Equations by the Method of Successive Approximations (1925), On motions which satisfy Kepler's first and second laws (1927/28), and The limit points of a group (1929).
    • Two significant books published by Ford are Automorphic Functions (1929) and Differential Equations (1933, second edition 1955).
    • See reviews at THIS LINK for Differential Equations and THIS LINK for Automorphic Functions.
    • Some of the papers are related to the fields of Ford's major interests: complex functions, interpolation, differential equations, and numerical analysis.
    • L R Ford - Differential Equations .

  260. Hopf Eberhard biography
    • Another important contribution from this period was the Wiener-Hopf equations, which he developed in collaboration with Norbert Wiener from the Massachusetts Institute of Technology.
    • By 1960, a discrete version of these equations was being extensively used in electrical engineering and geophysics, their use continuing until the present day.
    • Other work which he undertook during this period was on stellar atmospheres and on elliptic partial differential equations.
    • An example of this was the dropping of Hopf's name from the discrete version of the so called Wiener-Hopf equations, which are currently referred to as "Wiener filter".
    • His interests and principal achievements were in the fields of partial and ordinarydifferential equations, calculus of variations, ergodic theory, topological dynamics, integral equations, differential geometry, complex function theory and functional analysis.

  261. Infeld biography
    • During this time he wrote six joint papers with Max Born - examples of their papers in the Proceedings of the Royal Society are Foundations of the new Field Theory (1934) and On the Quantization of the New Field Equations (1935).
    • For example he published (jointly with A Einstein and B Hoffmann) The gravitational equations and the problem of motion (1938) and a second part, jointly with Einstein, two years later.
    • the gravitational field equations, satisfied in regions free of matter, imply the vanishing of certain surface integrals taken over 2-dimensional surfaces enclosing spatial regions containing the particles responsible for the field.
    • In obtaining this result the authors made use of a normalizing condition restricting the choice of coordinates; with it they were able to show that the vanishing of the surface integrals led to equations of motion for the particles.
    • Other papers he published around this time include (with P R Wallace, one of his doctoral students) The equations of motion in electrodynamics (1940), On the Theory of Brownian Motion (1940), On a new treatment of some eigenvalue problems (1941), A generalization of the factorization method for solving eigenvalue problems (1942), and Clocks, rigid rods and relativity theory (1943).
    • The problem is a substantial one, because the field equations are non-linear, and because they are connected by differential identities.

  262. Malgrange biography
    • Malgrange was awarded his doctorate in 1955 from the Universite Henri Poincare at Nancy for his thesis Existence et approximation des solutions des equations aux derivees partielles et des equations de convolution.
    • These were the two-part paper Equations aux derivees partielles a coeficients constants (1953, 1954), and the papers Sur quelque proprietes des equations de convolution (1954) and Formes harmonique sur un espace de Riemann a ds2 analytique (1955).
    • We have mentioned his important work on linear partial differential equations above, but he has made numerous other very significant contributions to differential geometry, non-linear differential equations, and singularities of functions and mappings.
    • In particular, he has studied hypoelliptic operators, ideals of differentiable functions, the classification of differential equations with regular singular points, and the algebraic theory of partial differential equations with variable coefficients.
    • In 1991 Malgrange published Equations differentielles a coefficients polynomiaux.
    • The modern algebraic theory of differential systems, or "theory of D-modules'', brings us new relations between two mathematical areas traditionally far apart: the theory of systems of linear partial differential equations and algebraic geometry.
    • This book documents the development of the theory of linear differential equations with irregular singularities in the interaction between Deligne, Malgrange and Ramis.

  263. Krasovskii biography
    • Barbashin organized a seminar on qualitative methods in the theory of differential equations gathering round him an excellent team working on mathematics and mechanics.
    • Barbashin influenced Krasovskii to work on the stability theory of motion and Krasovskii published his first papers Theorems on the stability of motions governed by a system of two equations and (with Evgenii Alekseevich Barbashin) The stability of motion as a whole in 1952.
    • Applications of Lyapunov's second method to differential systems and equations with delay and published in 1963.
    • At the Gorkii Ural State University he set up a school of control theory and differential equations.
    • The author is himself a distinguished contributor to the theory of optimal control on the basis of his previous valuable results in the qualitative theory of differential equations and Lyapunov stability theory.
    • The problems are formulated in terms of linear and quasilinear ordinary differential equations.

  264. Halphen biography
    • He was led to extend results due to Max Noether which, in turn, had him examine projective transformations which fix certain differential equations.
    • A characterisation of such invariant differential equations appeared in Halphen's doctoral dissertation On differential invariants which he presented in 1878.
    • Halphen made major contributions to linear differential equations and algebraic space curves.
    • He examined problems in the areas of systems of lines, classification of space curves, enumerative geometry of plane conics, singular points of plane curves, projective geometry and differential equations, elliptic functions, and assorted questions in analysis.
    • For example, in 1880 he won the Grand Prix of the Academie des Sciences for his work on linear differential equations.
    • Other work such as that on linear differential equations was overtaken by Lie group methods.

  265. Kublanovskaya biography
    • Using analytical computational devices, this group was solving systems of linear algebraic equations.
    • the method is applied to particular problems such as, for instance, solution of systems of linear equations, determination of eigenvalues and eigenvectors of a matrix, integration of differential equations by series, solution of Dirichlet problem by finite differences, solution of integral equations, etc.
    • The paper is concerned with finding, without the use of the Gaussian transformation, the normal generalized (in the sense of the least-squares method) solution for a system of linear algebraic equations with a rectangular matrix.
    • We have only been able to give a brief indication of the many papers which she had published - MathSciNet lists 134 publications in the areas of Commutative rings and algebras, Functions of a complex variable, History and biography, Linear and multilinear algebra, Numerical analysis, Ordinary differential equations and Systems theory.

  266. Caccioppoli biography
    • After 1930 Caccioppoli devoted himself to the study of differential equations and he provided existence theorems for both linear and non-linear problems.
    • His idea was to use a topological- functional approach to the study of differential equations.
    • Carrying on in this way Caccioppoli, in 1931, extended in some cases Brouwer's fixed point theorem, and applied his results to existence problems of both partial differential equations and ordinary differential equations.
    • In the period between 1933 and 1938 Caccioppoli applied his method to elliptic equations, providing the a priori upper bound for their solutions, in a more general way than Bernstein did for the two-dimensional case.
    • In 1935 he dealt with the question introduced in 1900 by Hilbert during the International Congress of Mathematicians, namely whether or not the solutions of analytical elliptic equations are analytic.

  267. Stepanov biography
    • He returned to Moscow in 1915 and, much influenced by Egorov and Luzin, he worked on periodic functions and differential equations.
    • In the qualitative theory of differential equations he worked on the general theory of dynamical systems studied by G D Birkhoff.
    • Besides writing articles on the study of almost periodic trajectories and on a generalisation of Birkhoff's ergodic theorem (which found an important application in statistical physics), Stepanov organised a seminar on the qualitative methods of the theory of differential equations (1932) that proved of great importance for the creation of the Soviet scientific school in this field.
    • A graduate-level text Qualitative Theory of Differential Equations by Stepanov and his student Viktor V Nemytskii became a classic, the 1960 edition being reprinted in 1989.
    • It considers existence and continuity theorems, integral curves of a system of two differential equations, systems of n-differential equations, general theory of dynamical systems, systems with an integral invariant, and many related topics.

  268. Szafraniec biography
    • Wazewski had built an important seminar at the Jagiellonian University which was mainly devoted to the study of differential equations.
    • He was famed for his topological approach to the study of differential equations, and had obtained remarkable results applying Borsuk's theory of retracts.
    • He was awarded a Master's degree (equivalent to a Ph.D.) in 1968 for a thesis on the theory of differential equations.
    • The papers he wrote while he was undertaking research included: On a certain sequence of ordinary differential equations (1963); (with Andrzej Lasota) Sur les solutions periodiques d'une equation differentielle ordinaire d'ordre n (1966) and (with Andrzej Lasota) Application of the differential equations with distributional coefficients to the optimal control theory (1968).
    • At the 1997 workshop 'Special functions and differential equations' held at Madras in India, he gave the talk The quantum harmonic oscillator in L2(R) in which he introduced the Hilbert-space model of the quantum harmonic oscillator couple of the creation and annihilation operators, he obtained some new interrelations between these operators.

  269. Griffiths Lois biography
    • She received the degree in 1927 after submitting her dissertation Certain quaternary quadratic forms and diophantine equations by generalized quaternion algebras.
    • In 1945 Griffiths produced a typewritten set of notes Determinants and systems of linear equations.
    • She expanded the notes into a book which was published by John Wiley and Sons as Introduction to the theory of equations in 1945.
    • slowly through the proofs of the important general theorems in the elementary theory of algebraic equations.
    • III (1945); and A note on linear homogeneous Diophantine equations (1946).
    • She attended the American Mathematical Society meeting in Chicago in April 1947 and delivered the lecture Linear homogeneous diophantine equations on the afternoon of Friday 26 April.

  270. Kneser biography
    • After writing this thesis on algebraic functions and equations, he then worked on space curves.
    • Adolf Kneser's early work was on algebraic functions and equations.
    • One of these areas is that of linear differential equations; in particular he worked on the Sturm-Liouville problem and integral equations in general.
    • He wrote an important text on integral equations.
    • the first to introduce Hilbert's new methods into analysis in his textbook on integral equations.

  271. Capelli biography
    • In this course Battaglini followed the approach given by Camille Jordan in his Traite des substitutions et des equations algebraique which he published in 1870.
    • Capelli had proved the theorem, known today as the Rouche-Capelli theorem, which gives conditions for the existence of the solution of a system of linear equations.
    • In 1879 Frobenius defined the rank of a system of equations to be the maximal order of a nonzero minor.
    • In 1886 Capelli and Garbieri in Corso di analisi algebrica showed that a system of equations having rank k is equivalent to a triangular system with exactly k nonzero diagonal terms.
    • This approach is very important in effective methods for solving systems of linear equations ..
    • They also showed that a system of equations is consistent if and only if the rank of the array of its coefficients is the same as the rank of the array augmented by the row of constant terms.

  272. Baker Alan biography
    • This was awarded for his work on Diophantine equations.
    • [Diophantine equations], carrying a history of more than one thousand years, was, until the early years of this century, little more than a collection of isolated problems subjected to ingenious ad hoc methods.
    • It was A Thue who made the breakthrough to general results by proving in 1909 that all Diophantine equations of the form .
    • Turan goes on to say that Carl Siegel and Klaus Roth generalised the classes of Diophantine equations for which these conclusions would hold and even bounded the number of solutions.
    • He proved that for equations of the type f (x, y) = m described above there was a bound B which depended only on m and the integer coefficients of f with .

  273. Ramanathan biography
    • Another fascinating paper is Ramanujan's modular equations (1990).
    • The present author commences with a very informative historical survey of modular equations.
    • Of course, in a paper of only 18 pages in length, the author can only discuss a small portion of Ramanujan's modular equations and he concentrates therefore on equations of composite degree.
    • He gives some proofs, shows connections to previous work, and offers insights into how Ramanujan may have discovered some of his equations.

  274. Clairaut biography
    • The following year Clairaut studied the differential equations now known as 'Clairaut's differential equations' and gave a singular solution in addition to the general integral of the equations.
    • In 1739 and 1740 he published further work on the integral calculus, proving the existence of integrating factors for solving first order differential equations (a topic which also interested Johann Bernoulli, Reyneau and Euler).
    • The algebra book was an even more scholarly work and took the subject up to the solution of equations of degree four.

  275. Lichnerowicz biography
    • His thesis was published under this title, and also under the title Sur certains problemes globaux relatifs au systeme des equations d'Einstein.
    • Chapter I (Axiomatique de la theorie de la gravitation) [gives] relevant results on the initial value problem associated with the field equations of general relativity; most important for the sequel are those which deal with the continuation of an "interior field," in a region containing matter, across a boundary into an "exterior field" in regions free of matter.
    • The first part treats linear equations, determinants and matrices, Hermitian forms, characteristic roots and resolvents, and tensor algebra.
    • The second surveys the theory of exterior differential forms, the general form of Stokes's theorem and its specialization to two and three dimensions, orthogonal series, Fourier integrals, bounded linear operators in Hilbert space, and the classical theory of integral equations for L2integrable kernels.
    • except during his lectures when he would fill the blackboard with equations in his dense handwriting, equations almost always comprising many tensorial indices.

  276. Saint-Venant biography
    • Perhaps his most remarkable work was that which he published in 1843 in which he gave the correct derivation of the Navier-Stokes equations.
    • Seven years after Navier's death, Saint-Venant re-derived Navier's equations for a viscous flow, considering the internal viscous stresses, and eschewing completely Navier's molecular approach.
    • Why his name never became associated with those equations is a mystery.
    • We should remark that Stokes, like Saint-Venant, correctly derived the Navier-Stokes equations but he published the results two years after Saint-Venant.
    • In 1871 he derived the equations for non-steady flow in open channels.

  277. Warga biography
    • He wrote a book which has become a classic, namely Optimal Control of Differential and Functional Equations (1972).
    • Along with the unilateral constraints on the state variable, the side conditions extensively investigated in the book are ordinary differential equations with or without delays, functional integral equations and functional equations in general Banach spaces.
    • Several weeks later however, I received a phone call from Jack asking me to help him find two articles on differential equations.

  278. Ghizzetti biography
    • In addition to derivation of the basic properties of this transform, applications are indicated to ordinary linear differential equations with constant coefficients as well as systems of such equations.
    • In the next three parts, the author applies the basic material to the solution of some of the ordinary and partial differential equations in electrotechnics.
    • In part two, differential equations arising in lumped circuit phenomena are handled.
    • After beginning work in Rome, his research interests moved towards the theory of moments and the theory of partial differential equations.

  279. Al-Samawal biography
    • In Book 2 of al-Bahir al-Samawal describes the theory of quadratic equations but, rather surprisingly, he gave geometric solutions to these equations despite algebraic methods having been fully described by al-Khwarizmi, al-Karaji, and others.
    • Al-Samawal also described the solution of indeterminate equations such as finding x so that a xn is a square, and finding x so that axn + bxn-1 is a square.
    • The final book of al-Bahir contains an interesting example of a problem in combinatorics, namely to find ten unknowns given the 210 equations which give their sums taken 6 at a time.
    • Of course such a system of 210 equations need not be consistent and al-Samawal gave the 504 conditions which are necessary for the system to be consistent.

  280. Gordan biography
    • Moving to Konigsberg, Gordan studied under Jacobi, then he moved to Berlin where he began to become interested in problems concerning algebraic equations.
    • In the year 1874-75 when Gordan and Klein were together at Erlangen they undertook a joint research project examining groups of substitutions of algebraic equations.
    • They investigated the relationship between PSL(2,5) and equations of degree five.
    • Later Gordan went on to examine the relation between the group PSL(2,7) and equations of degree seven, then he studied the relation of the group A6 to equations of degree six.

  281. Hille biography
    • Kirsti Hille wrote the article [Integral Equations Operator Theory 4 (3) (1981), 304-306.',5)">5] after the death of her husband.
    • Hille's main work was on integral equations, differential equations, special functions, Dirichlet series and Fourier series.
    • Among Hille's other texts were Analytic function theory Vol 1 (1959), Vol 2 (1964); Analysis Vol 1 (1964), Vol 2 (1966); Lectures on ordinary differential equations (1969); Methods in classical and functional analysis (1972); and Ordinary differential equations in the complex domain (1976).

  282. Murnaghan biography
    • Harry Bateman had been appointed there in 1912 and his interests in partial differential equations fitted perfectly with Murnaghan's interests at the time.
    • Arriving at Johns Hopkins University in Baltimore, Murnaghan began doctoral studies working on differential equations which arose in the study of radio-active decay.
    • Of course this meant that he was deeply involved in solving differential equations, and indeed he also wrote papers on this topic.
    • It covers topics such as: vectors and matrices; Fourier series; boundary value problems; Legendre and Bessel functions; integral equations; the calculus of variations and dynamics; and the operational calculus.
    • The first of these is a short book of less that 100 pages written for engineers and scientists, while the second consists of 19 lectures on such topics as: the Fourier integral; the Laplace integral transformation; the differential equations of Laguerre and Bessel; properties of special functions; asymptotic series for an error function, and for certain Bessel functions.

  283. Church biography
    • Early contributions included the papers On irredundant sets of postulates (1925), On the form of differential equations of a system of paths (1926), and Alternatives to Zermelo's assumption (1927).
    • For example he published Remarks on the elementary theory of differential equations as area of research in 1965 and A generalization of Laplace's transformation in 1966.
    • The first examines ideas and results in the elementary theory of ordinary and partial differential equations which Church feels may encourage further investigation of the topic.
    • The paper includes a discussion of a generalization the Laplace transform which he extends to non-linear partial differential equations.
    • This generalization of the Laplace transform is the topic of study of the second paper, again using the method to obtain solutions of second-order partial differential equations.

  284. Remez biography
    • He gave courses at these institutions on analysis, differential equations and differential geometry while undertaking research for his doctorate.
    • from Kiev State University in 1929 with his thesis Methods of Numerical Integration of Differential Equations with an Estimate of Exact Limits of Allowable Errors.
    • Remez generalised Chebyshev-Markov characterisation theory and used it to obtain approximate solutions of differential equations.
    • He also worked on approximate solutions of differential equations and the history of mathematics.
    • The book has two parts: Part I - Properties of the solution of the general Chebyshev problem; Part II - Finite systems of inconsistent equations and the method of nets in Chebyshev approximation.

  285. Wantzel biography
    • Wantzel is famed for his work on solving equations by radicals.
    • In 1845 Wantzel, continuing his researches into equations, gave a new proof of the impossibility of solving all algebraic equations by radicals.
    • In meditating on the researches of these two mathematicians, and with the aid of principles we established in an earlier paper, we have arrived at a form of proof which appears so strict as to remove all doubt on this important part of the theory of equations.
    • It was he who first gave the integration of differential equations of the elastic curve.

  286. Reizins biography
    • He graduated in 1948 with distinction and became a member of the Department of Mathematical Analysis at the University while he undertook research on differential equations under Arvids Lusis.
    • However, he continued to undertake research in mathematics and in 1951 his first paper The behaviour of the integral curves of a system of three differential equations in the neighbourhood of a singular point was published by the Latvian Academy of Sciences.
    • thesis he studied the qualitative behaviour of homogeneous differential equations and obtained results that were highly regarded by specialists.
    • Of the many other important contributions made by Reizins we should mention in particular his work on Pfaff's equations and his contributions to the history of mathematics.
    • Other important historical papers include Mathematics in University of Latvia 1919-1969 (1975, joint with E Riekstins) and From the History of the General Theory of Ordinary Differential Equations (1977).

  287. Mineur biography
    • While still at High School, preparing for the second part of his degree, Henri Mineur had developed a passion for functional equations.
    • There are remarks such as "the superiority of functional equations over differential equations is that they allow one to define discontinuous functions." .
    • He was awarded his doctorate in 1924 for his thesis Discontinuous solutions of a class of functional equations in which he established an addition theorem for Fuchsian functions.
    • It gives one of the very few up-to-date discussions of the subject, which is not merely intended for the layman or the general public, but treats the entire problem concisely, "from the ground up," using directly the equations of Einstein, de Sitter, Lemaitre, and others, and discusses the properties of the various types of space resulting from each.

  288. Mitchell biography
    • He worked on an idea of Ron's of incorporating higher derivatives into methods for Ordinary Differential Equations, apparently one of the few times Ron strayed away from PDE's to ODE's.
    • He was mainly interested at that time in finite difference methods for both ordinary and partial differential equations.
    • He was joined in March 1964 by Donald Kershaw, whose main interests were differential and integral equations, and some students, including Alistair Watson.
    • In this talk Olec Zienkiewicz described instabilities they had experienced in converting their successful finite element codes for structural problems into codes for solving the Navier-Stokes and related equations in fluid dynamics.
    • Some of the problems arose from Mathematical Biology, on which "Mano" Manoranjan did much of his PhD work, but Ron was also interested in solitons, particularly those arising from the Korteweg-de Vries and Schrodinger equations.

  289. Van Vleck biography
    • Almost all Van Vleck's research papers were in the fields of function theory and differential equations.
    • For example he published On the determination of a series of Sturm's functions by the calculation of a single determinant (1899), On linear criteria for the determination of the radius of convergence of a power series (1900), On the convergence of continued fractions with complex elements (1901), A determination of the number of real and imaginary roots of the hypergeometric series (1902), On an extension of the 1894 memoir of Stieltjes (1903), and On the extension of a theorem of Poincare for difference-equations (1912).
    • Of the American Mathematical Society sometime president, and editor of the Transactions; always wise counsellor and leader; creative mathematician and successful investigator in the theory of functions, and in the theories of differential and difference equations and of functional equations; for these eminent services in mathematics, and especially for your important researches concerning functional equations and analytic continued fractions.

  290. Deligne biography
    • He also worked closely with Jean-Pierre Serre, leading to important results on the l-adic representations attached to modular forms, and the conjectural functional equations of L-functions.
    • Andre Weil gave for the first time a theory of varieties defined by equations with coefficients in an arbitrary field, in his Foundations of Algebraic Geometry (1946).
    • Weil's work on polynomial equations led to questions on what properties of a geometric object can be determined purely algebraically.
    • Weil's work related questions about integer solutions to polynomial equations to questions in algebraic geometry.
    • He conjectured results about the number of solutions to polynomial equations over the integers using intuition on how algebraic topology should apply in this novel situation.

  291. McClintock biography
    • One paper treats difference equations as differential equations of infinite order and others look at quintic equations which are soluble algebraically.
    • This led he to restate difference equations as differential equations of infinite order.

  292. Povzner biography
    • At this stage in his career, however, his interests changed from algebra to analysis and his first publication in his new area of research was Sur les equations du type de Sturm-Liouville et les fonctions "positives" (1944).
    • This did not prevent him continuing to undertake research and he published On equations of the Sturm-Liouville type on a semi-axis (1946) and a joint paper with Boris Moiseevich Levitan in the same year entitled Differential equations of the Sturm-Liouville type on the semi-axis and Plancherel's theorem.
    • He also worked on partial differential equations which describe non-stationary processes.
    • Systems of ordinary differential equations.
    • Partial differential equations.

  293. Heinonen biography
    • Three were written jointly with Tero Kilpelainen: A-superharmonic functions and supersolutions of degenerate elliptic equations; Polar sets for supersolutions of degenerate elliptic equations; and On the Wiener criterion and quasilinear obstacle problems.
    • The others were the single author publications Boundary accessibility and elliptic harmonic measures and Asymptotic paths for subsolutions of quasilinear elliptic equations, and the paper On quasiconformal rigidity in plane and space written with K Astala.
    • Heinonen published two important books: (with Olli Martio and Tero Kilpelainen) Nonlinear Potential Theory of Degenerate Elliptic Equations (1993); and Lectures on Analysis on Metric Spaces (2001).
    • This excellent book is the first monograph dealing with a potential theory of second-order quasilinear elliptic equations of [a certain] type ..

  294. Krasnosel'skii biography
    • For example Positive solutions of operator equations (1962) which studied the existence, uniqueness, and properties of positive solutions of linear and non-linear equations in a partially ordered Banach space, Vector fields in the plane (1963) which the angular variation of a plane vector field relative to a curve, and Displacement operators along trajectories of differential equations (1966) which is described by C Olech as follows:- .
    • For example Approximate solution of operator equations (1969):- .
    • is devoted to the investigation of approximate methods of solving operator equations.

  295. Fichera biography
    • Gaetano Fichera was at the heart of the important developments connecting physics (mostly elasticity), functional analysis and the theory of partial differential equations and inequalities which took place after WWII, many of them in Italy where the mathematical study of the mechanics of continuous media was a well-established tradition.
    • In pure mathematics Gaetano Fichera achieved considerable results in the following fields: mixed boundary value problems of elliptic equations; generalized potential of a simple layer; second order elliptic-parabolic equations; well posed problems; weak solutions; semicontinuity of quasi-regular integrals of the calculus of variations; two-sided approximation of the eigenvalues of a certain type of positive operators and computation of their multiplicity; uniform approximation of a complex function f(z); extension and generalization of the theory for potentials of simple and double layer; specification of the necessary and sufficient conditions for the passage to the limit under integral sign for an arbitrary set; analytic functions of several complex variables; solution of the Dirichlet problem for a holomorphic function in a bounded domain with a connected boundary, without the strong conditions assumed by Francesco Severi in a former study; construction of a general abstract axiomatic theory of differential forms; convergence proof of an approximating method in numerical analysis and explicit bounds for the error.
    • Throughout the book the author gives special attention to methods and results having applications in the theory of partial differential equations.
    • This contrasts with much current work on differential equations where "error bounds" commonly involve unspecified constants.

  296. Cesari biography
    • During this period he studied surfaces given by parametric equations, in particular the Lebesgue area of such a surface.
    • Three years later, in 1959, Cesari published the monograph Asymptotic behavior and stability problems in ordinary differential equations.
    • In fact much of this appears in his book Optimization - theory and applications: Problems with ordinary differential equations published in 1983.
    • In the last twenty years, much of his attention was devoted to the study of questions arising in nonlinear analysis and its applications to differential equations.
    • We promised to return to his book Optimization - theory and applications: Problems with ordinary differential equations published three years after he retired.

  297. Schauder biography
    • While Schauder was in Paris he collaborated with J Leray and their joint work led to a paper Topologie et equations fonctionelles published in the Annales scientifiques de l'Ecole normale Superieure.
    • This 1934 paper on topology and partial differential equations is of major importance [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.
    • His last work was to generalise results of Courant, Friedrichs and Lewy on hyperbolic partial differential equations.
    • In particular, Schauder's formulation of a fixed point theorem originated a new, extremely fruitful method in the theory of differential equations, known as Schauder's method ..
    • Schauder's fixed point theorem and his skillful use of function space techniques to analyse elliptic and hyperbolic partial differential equations are contributions of lasting quality.

  298. Calderon biography
    • Calderon, on the other hand, with his background as an engineer, saw that such operators held an important key to understanding the theory of partial differential equations.
    • In 1958 Calderon published one of his most important results on uniqueness in the Cauchy problem for partial differential equations.
    • for his ground-breaking work on singular integral operators leading to their application to important problems in partial differential equations, including his proof of uniqueness in the Cauchy problem, the Atiyah-Singer index theorem, and the propagation of singularities in nonlinear equations..
    • Calderon's techniques have been absorbed as standard tools of harmonic analysis and are now propagating into nonlinear analysis, partial differential equations, complex analysis, and even signal processing and numerical analysis.

  299. Spence biography
    • Spence published his last work "Outlines of a theory of algebraical equations: deduced from the principles of Harriot, and extended to the fluxional or differential calculus" in 1814.
    • It is clear that Spence must have read Lagrange's 1770 paper Reflexions sur la resolution algebrique des equations for he tries to make a systematic approach to solving equations of degree 2, 3 and 4 using symmetrical functions of the roots as does Lagrange.
    • This makes it sound as if Spence follows Lagrange rather closely but this is certainly not the case for he gives his own approach to solving these equations.
    • Of course he has no success when he tries to generalise his approach to solving fifth degree equations.
    • Of these tracts, the first only was intended by the author to meet the public eye in its present shape, though a few copies of another of them, demominated 'Outlines of a Theory of Algebraical Equations', had been printed and distributed among the author's friends.

  300. Pierpont biography
    • Two series of lectures were given, one by Maxime Bocher on Linear Differential Equations, and their Application and the other by Pierpont on Galois's Theory of Equations.
    • He also wrote some good historical articles on the theory of equations such as Lagrange's place in the theory of substitutions (1894), and Early history of Galois' theory of equations (1897).
    • On the other hand the author, having in mind the needs of the students of applied mathematics, has dwelt at some length on the theory of linear differential equations, especially as regards the functions of Legendre, Laplace, Bessel, and Lame.

  301. Wazewski biography
    • Wazewski made important contributions to the theory of ordinary differential equations, partial differential equations, control theory and the theory of analytic spaces.
    • was to bring him fame and lead to the development of a new school of differential equations.
    • he succeeded in applying with amazing effect the topological notion of retract (introduced by K Borsuk) to the study of the solutions of differential equations.
    • Lefschetz considered his method of retracts one of the most important achievements in the theory of differential equations since the war.

  302. John biography
    • He applied this in his study of general properties of linear partial differential equations, convex geometry and the mathematical theory of water waves.
    • It was in this period that John introduced the space of functions of bounded mean oscillations which plays a fundamental role in harmonic analysis and nonlinear elliptic equations.
    • He retired in 1981 but at this time his work was concentrating on the theory of nonlinear wave equations.
    • For anyone interested in the analysis of partial differential equations, the work of Fritz John is especially rewarding.
    • He wrote by now classical papers in convexity, ill-posed problems, the numerical treatment of partial differential equations, quasi-isometry and blow-up in nonlinear wave propagation.

  303. Al-Quhi biography
    • The geometric problems that al-Quhi studied usually led to quadratic or cubic equations.
    • One, which requires the solution of a quadratic equation, had been found by Abu Kamil in the ninth century.
    • The other, which requires the solution of a quartic equation, is the one presented by al-Quhi.

  304. Koch biography
    • Von Koch's first results were on infinitely many linear equations in infinitely many unknowns.
    • In 1891 he wrote the first of two papers on applications of infinite determinants to solving systems of differential equations with analytic coefficients.
    • Yet this work can be said to be the first step on the long road which eventually led to functional analysis, since it provided Fredholm with the key for the solution of his integral equation.

  305. Hill biography
    • Examples of papers he published in the Annals of Mathematics include: On the lunar inequalities produced by the motion of the ecliptic (1884), Coplanar motion of two planets, one having a zero mass (1887), On differential equations with periodic integrals (1887) (these differential equations are now called Hill's differential equation), On the interior constitution of the earth as respects density (1888), The secular perturbations of two planets moving in the same plane; with application to Jupiter and Saturn (1890), On intermediate orbits (1893), Literal expression for the motion of the Moon's perigee (1894) and Application of Chebyshev's principle in the projection of maps (1908).

  306. Nicolson biography
    • Phyllis Nicolson is best known for her joint work with John Crank on the heat equation, where a continuous solution u(x, t) is required which satisfies the second order partial differential equation .
    • Crank and Nicolson's method, which is numerically stable, requires the solution of a very simple system of linear equations (a tridiagonal system) at each time level.

  307. Pairman biography
    • Her thesis advisor was George Birkhoff and after submitting her thesis Expansion Theorems for Solution of a Fredholm's Linear Homogeneous Integral Equation of the Second Kind with Kernel of Special Non-Symmetric Type she was awarded a Ph.D.
    • Pairman joined the Edinburgh Mathematical Society in January 1917, and read the paper On a difference equation due to Stirling to the meeting of the Society on 11 January 1918, and the paper A new form of the remainder in Newton's interpolation formula to the next meeting of the Society on 8 February.
    • In 1927 she published a joint mathematics paper On a class of integral equations with discontinuous kernels with Rudolph E Langer, who was a friend who graduated from Harvard in a June 1922 ceremony as Eleanor Pairman and Bancroft Brown and, like Eleanor, had George Birkhoff as his thesis advisor.

  308. Coble biography
    • His interests in research relate to finite geometries and the group theory related to them, and to Cremona transformations related to the Galois theory of equations.
    • His early papers, written while he was at Johns Hopkins University, include: On the relation between the three-parameter groups of a cubic space curve and a quadric surface (1906); An application of the form-problems associated with certain Cremona groups to the solution of equations of higher degree (1908); An application of Moore's cross-ratio group to the solution of the sextic equation (1911); An application of finite geometry to the characteristic theory of the odd and even theta functions (1913); and Point sets and allied Cremona groups (1915).

  309. Apery biography
    • However, in the 1950s he became interested in number theory and worked on diophantine equations.
    • In particular he studied the diophantine equation .
    • Two short papers, both entitled Sur une equation diophantienne, are devoted to a study of this equation.

  310. Clausius biography
    • The basic equation set up by Clausius was therefore dQ = dU + dW where dQ was the increment in the heat, dU was the change in energy of the body, and dW was the change in external work done.
    • The Clausius-Clapeyron equation appears which expresses the relation between the pressure and temperature at which two phases of a substance are in equilibrium.
    • Clausius deliberately made choices in setting up the equations so that they were:- .

  311. Roach biography
    • His early papers were: On the approximate solution of elliptic, self adjoint boundary value problems (1967); Fundamental solutions and surface distributions (1968); Approximate Green's functions and the solution of related integral equations (1970); and (jointly with Robert A Adams) An intrinsic approach to radiation conditions (1972).
    • As indicated by the title, this book is intended to give a self-contained and systematic introduction to the theory of Green's functions and the general ideas involved in their application to boundary value problems associated with ordinary and partial differential equations.
    • The required preliminaries do not exceed an elementary knowledge of the initial-boundary value problem for the wave equation.

  312. Schubert Hans biography
    • Uber eine lineare Integrodifferentialgleichung mit Zusatzkern (1950) looked at certain aerodynamical problems which lead to integrodifferential equations.
    • At Halle Schubert taught a variety of different courses such as differential and integral calculus, partial differential equation, and integral equations.

  313. Ferrand biography
    • Pierre Lelong's name is attached to several mathematical concepts, for example the Poincare-Lelong equation, the Lelong-Demailly numbers, Lelong's problem, and the Lelong-Skoda transform.
    • Volume III covered multivariable integral calculus, further topics in functions of a complex variable, Fourier series and ordinary differential equations.
    • The fourth and final volume was published in 1974 and covered ordinary differential equations, multivariable integral calculus and holomorphic functions.

  314. Schouten biography
    • He produced 180 papers and 6 books on tensor analysis, applying tensor analysis to Lie groups, general relativity, unified field theory, and differential equations.
    • This is a complete exposition of the classical theory of the Pfaffian equation and of the present results in the theory of systems of Pfaffian equations, to which the authors themselves have substantially contributed.

  315. Cafiero biography
    • His next paper was Un'osservazione sulla continuita rispetto ai valori iniziali degli integrali dell'equazione: y' = f (x, y) (1947), which proves that any group of conditions sufficient to assure the existence and uniqueness, with respect to the initial values, of the integral of the equation y' = f (x, y) is also sufficient to assure the continuous dependence of the solution on the initial values.
    • In each of the years 1948, 1949 and 1950, Cafiero published three papers most of which studied ordinary differential equations.
    • We have already seen that Cafiero made contributions to the theory of ordinary differential equations and to the theory of measure and integration.

  316. Goldstine biography
    • Most of the paper is taken up with the more difficult problem of determining such conditions when the class of admissible points is (1940) required to satisfy an equation of an abstract functional character.
    • There followed A generalized Pell equation.
    • In particular the School used a Bush analyser, designed by Vannevar Bush, specifically to integrate systems of ordinary differential equations.

  317. Stifel biography
    • Also in this book he solves cubic and quartic equations using methods from Cardan.
    • In particular, he solves the quartic equation .
    • One of the advances in Stifel's notes is an early attempt to use negative numbers to reduce the solution of a quadratic equation to a single case.

  318. Fibonacci biography
    • Indeed, although mainly a book about the use of Arab numerals, which became known as algorism, simultaneous linear equations are also studied in this work.
    • Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction.
    • And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.

  319. Fiedler biography
    • He published his thesis in three parts (1954, 1955, 1956) but these were not his first publications, having already published Solution of a problem of Professor E Čech (1952), On certain matrices and the equation for the parameters of singular points of a rational curve (1952), and (with L Granat) Rational curve with the maximum number of real nodal points (1954).
    • Examples of papers he published on these topics are: Numerical solution of algebraic equations which have roots with almost the same modulus (1956); Numerical solution of algebraic equations by the Bernoulli-Whittaker method (1957); On some properties of Hermitian matrices (1957); (with Jiri Sedlacek) On W-bases of directed graphs (1958); and (with Josej Bily and Frantisek Nozieka) Die Graphentheorie in Anwendung auf das Transportproblem (1958).

  320. Sylvester biography
    • He was a very active researcher and by the time he resigned the chair of natural philosophy in 1841 he had published fifteen papers on fluid dynamics and algebraic equations.
    • In 1851 he discovered the discriminant of a cubic equation and first used the name 'discriminant' for such expressions of quadratic equations and those of higher order.

  321. Bouquet biography
    • Bouquet and Briot developed Cauchy's work on the existence of integrals of a differential equation.
    • For example Etude des fonctions d'une variable imaginaire (Study of functions with one imaginary variable); Recherches sur les proprietes des fonctions definies par des equations differentielles (Research on the properties of functions defined by differential equations); and Memoire sur l'integration des equations differentielles au moyen des fonctions elliptiques (Memoir on the integration of differential equations by means of elliptic functions).

  322. Hermann biography
    • He lectured on mechanics in November 1708 and in December of that year he wrote to Grandi giving him a detailed explanation of how to use Leibniz's calculus to deduce the differential equation of the logarithm function.
    • Hermann discussed such topics as finding the radius of curvature and normals to plane curves; the division of an angle or an arc of a circumference into n parts, by the use of an infinite series; orthogonal trajectories for a given family of curves, by the use of differential equations; and the use of polar coordinates in the analysis of plane curves other than spirals.
    • In his work on curves in space, Hermann discusses the spherical epicycloid; the problem of finding the shortest distance between two points on a given surface; and the equations and properties of various surfaces from the point of view of analytic geometry of three dimensions.

  323. Sripati biography
    • His work on equations in this chapter contains the rule for solving a quadratic equation and, more impressively, he gives the identity: .
    • Other mathematics included in Sripati's work includes, in particular, rules for the solution of simultaneous indeterminate equations of the first degree that are similar to those given by Brahmagupta .

  324. Spitzer biography
    • A rapid treatment of the Boltzmann equation, in an appendix, brings us in Chapter 2 to the transport equation for a fluid.
    • This is joined with Maxwell's equations, and the simple limits of high and low magnetic fields are briefly considered.

  325. Plucker biography
    • The characteristic features of Plucker's analytic geometry were already present in this work, namely, the elegant operations with algebraic symbols occurring in the equations of conic sections and their pencils.
    • This work also contains the celebrated 'Plucker equations' relating the order and class of a curve.
    • In this way of specifying coordinates, a point has a linear equation, namely that of all lines through the point while a line has a pair of numbers namely the x and y coordinates of where it cuts the axes.

  326. Cowling biography
    • The equations of Boltzmann and Maxwell are then developed, Enskog's generalization of Maxwell's equation of transfer being given.
    • In the important chapter on the non-uniform state for a simple gas, use is made of Enskog's method of solving the integral equation and of Burnett's calculation of certain quantities A and B with the aid of Sonine's polynomials.

  327. Peano biography
    • In 1886 Peano proved that if f (x, y) is continuous then the first order differential equation dy/dx = f (x, y) has a solution.
    • Four years later Peano showed that the solutions were not unique, giving as an example the differential equation dy/dx = 3y2/3 , with y(0) = 0.
    • The following year he discovered, and published, a method for solving systems of linear differential equations using successive approximations.

  328. Voronoy biography
    • He was awarded a Master's Degree in 1894 for a dissertation on the algebraic integers associated with the roots of an irreducible cubic equation.
    • In the essay I am now presenting, results from the general theory of algebraic integers are applied to the particular case of numbers depending on the root of an irreducible equation x3 = rx + s.
    • In our exposition the resolution of these questions is based on a detailed study of the solutions of third-degree equations relative to a prime and a composite modulus.

  329. Franklin biography
    • While still an undergraduate, Franklin was taking part in published discussions in The American Mathematical Monthly, his contribution to the Discussion: Relating to the Real Locus Defined by the Equation xy = yx appearing in March 1917.
    • However, he is best known for textbooks he published on calculus, differential equations, complex variable and Fourier series.
    • In particular he wrote Differential equations for electrical engineers (1933), Treatise on advanced calculus (1940), The four color problem (1941), Methods of advanced calculus (1944), Fourier methods (1949), Differential and integral calculus (1953), Functions of a complex variable (1958) and Compact calculus (1963).

  330. Wallis biography
    • He also discovered methods of solving equations of degree four which were similar to those which Harriot had found but Wallis claimed that he made the discoveries himself, not being aware of Harriot's contributions until later.
    • He also criticises Descartes' Rule of Signs stating, quite correctly, that the rule which determines the number of positive and the number of negative roots by inspection, is only valid if all the roots of the equation are real.
    • History Topics: Pell's equation .

  331. Dyson biography
    • The first, written in 1941 (published in 1944) is A proof that every equation has a root.
    • there are so many proofs of the theorem that every equation has a root that it seems almost criminal to produce another.
    • The historical account of the breakdown in communications between mathematicians and physicists and of the lack of interest in Maxwell's equations constitutes an indictment of the mathematical community.

  332. Baiada biography
    • The first of these papers gives an existence theorem for the equation zx = f (x, y, z, zy) using methods which had been developed by Baiada a couple of years earlier in solving a simpler equation.
    • We have mentioned some of Baiada's publications above but we note that his output totals 60 scientific publications on a wide range of different fields in analysis: ordinary and partial differential equations, Fourier series and the series expansion of orthonormal functions, topology, real analysis, functional analysis, calculus of variations, measure and integration, optimisation, and the theory of functions.

  333. Mayer Adolph biography
    • In the following years he taught Differential and Integral Calculus, Theory of Definite Integrals, Some chapters from mechanics and the calculus of variations, Higher Algebra, Differential Equation of Mechanics and the Calculus of Variations, Analytic Geometry, and many more courses of a similar type.
    • Mayer worked on differential equations, the calculus of variations and mechanics.
    • His work on the integration of partial differential equation and a search to determine maxima and minima using variational methods brought him close to the investigations which Lie was carrying out around the same time.

  334. Jeffery biography
    • He did one years teacher training in 1911 but he was already undertaking research and his first paper On a form of the solution of Laplace's equation suitable for problems relating to two spheres was read to the Royal Society in 1912.
    • He made effective use of Whittaker's general solution to Laplace's equation which Whittaker found in 1903.
    • Jeffery also worked on general relativity and produced exact solutions to Einstein's field equations in certain special cases.

  335. Redei biography
    • In 1953 L Redei published his famous article "Die 2-Ringklassen-gruppe des quadratischen Zahlkorpers und die Theorie des Pell-schen Gleichung", after many years of investigation of Pell's equation.
    • He gave a unified theory for the structure of class groups of real quadratic number fields and conditions for solvability of Pell's equation and other indeterminate equations.

  336. Betti biography
    • His early work is in the area of equations and algebra.
    • In 1854 Betti showed that the quintic equation could be solved in terms of integrals resulting in elliptic functions.
    • Although Jordan, in his Traite des substitutions et des equations algebriques (1870) credits Betti with having filled the gaps in Galois' arguments and with having been the first to establish the sequence of Galois' theorems rigorously, the fact is that Betti's work contains substantial obscurities and errors.

  337. Wilkins Ernest biography
    • In 1944 four of his papers appeared: On the growth of solutions of linear differential equations; Definitely self-conjugate adjoint integral equations; Multiple integral problems in parametric form in the calculus of variations; and A note on skewness and kurtosis.
    • In the following year he published The differential difference equation for epidemics in the Bulletin of Mathematical Biophysics.

  338. Tietz biography
    • Erich Hecke, although he was mortally ill, lectured on Linear Differential Equations.
    • The 1953 paper Die Kinematik des starren Korpers was a joint work with Rudolf Iglisch in which the authors derive Euler's equation in kinematics by means of vector algebra.
    • Differential equations; 8.

  339. Arf biography
    • At that time, I was thinking about making a list of the algebraic equations or Galois algebraic equations that could be solved.
    • Arf presented a paper On a generalization of Green's formula and its application to the Cauchy problem for a hyperbolic equation to the volume Studies in mathematics and mechanics presented to Richard von Mises in 1954.

  340. Bombelli biography
    • It is unclear exactly how Bombelli learnt of the leading mathematical works of the day, but of course he lived in the right part of Italy to be involved in the major events surrounding the solution of cubic and quartic equations.
    • Scipione del Ferro, the first to solve the cubic equation was the professor at Bologna, Bombelli's home town, but del Ferro died the year that Bombelli was born.
    • History Topics: Quadratic, cubic and quartic equations .

  341. Vessiot biography
    • In 1892 he submitted his doctoral dissertation Sur l'integration des equations differentielles lineaires.
    • In this he studied Lie groups of linear transformations, in particular considering the action of these Lie groups on the independent solutions of a differential equation.
    • He published his thesis in the Annales Scientifiques de l'Ecole Normale Superieure in 1892 and over the following few years published papers such as Sur une classe d'equations differentielles (1893), Sur une methode de transformation et sur la reduction des singularites d'une courbe algebrique (1894), Sur les systemes d'equations differentielles du premier ordre qui ont des systemes fondamentaux d'integrales (1894), and Sur quelques equations differentielles ordinaires du second ordre (1895).
    • As we mentioned above, Vessiot applied continuous groups to the study of differential equations.
    • He extended results of Jules Joseph Drach (1902) and Elie Cartan (1907) and also extended Fredholm integrals to partial differential equations.

  342. Mathieu Emile biography
    • From his late twenties his main efforts were devoted to the then unfashionable continuation of the great French tradition of mathematical physics, and he extended in sophistication the formation and solution of partial differential equations for a wide range of physical problems.
    • He discovered these functions, which are special cases of hypergeometric functions, while solving the wave equation for an elliptical membrane moving through a fluid.
    • The Mathieu functions are solutions of the Mathieu equation which is .

  343. Evans biography
    • His doctoral dissertation Volterra's integral equation of the second kind with discontinuous kernel was published in the Transactions of the American Mathematical Society in two parts in 1910 and 1911.
    • His work dealt with potential theory, functional analysis, integral equations and the problem of minimal surfaces, the Plateau Problem.
    • Among the important texts he wrote were Functional equations and their applications (1918), The logarithmic potential (1927), and Mathematical Introduction to economics (1930).

  344. Scholtz biography
    • They considered projective geometrical questions about conic sections and they transformed these questions into algebraic equations, where the determinant came into play.
    • In Six points lying on a conic section, and the theorem hexagrammum mysticum (1877) and Sechs Punkte eines Kegelschnittes (1878) he proved Pascal's theorem in Steiner's generality, by reducing it to an equation involving certain determinants.
    • Scholtz' later papers appeared in the Nouvelle Annales de Mathematique, for example Resolution de l'equation du troisieme degre (1881), and in the Yearbook of the Grammar School in Iglo, see A remark on light interference (1886).

  345. Osgood biography
    • Osgood's main work was on the convergence of sequences of continuous functions, solutions of differential equations, the calculus of variations and space filling curves.
    • In 1898 Osgood published an important paper on the solutions of the differential equation dy/dx = f(x, y) satisfying the prescribed initial conditions y(a) = b.
    • Some papers over the next few years included: Sufficient conditions in the calculus of variations (1900), On a fundamental property of a minimum in the calculus of variations and the proof of a theorem of Weierstrass's (1901), A Jordan curve of positive area (1903), On Cantor's theorem concerning the coefficients of a convergent trigonometric series, with generalizations (1909), On the gyroscope (1922), and On normal forms of differential equations (1925).

  346. Recorde biography
    • The book was the Second Part of Arithmetic, The Grounde of Artes being the first, covering the extraction of roots, the theory of equations and arithmetic with surds.
    • In his study of quadratic equations, Recorde does not allow solutions which are negative, but he does allow negative coefficients.
    • He makes good use of the sum and product of the roots stressing that for the equation .

  347. Descartes biography
    • Harriot's work on equations, however, may indeed have influenced Descartes who always claimed, clearly falsely, that nothing in his work was influenced by the work of others.
    • Descartes' geometric solution of a quadratic equation .
    • History Topics: Quadratic etc equations .

  348. Kurschak biography
    • Another topic which Kurschak investigated was the differential equations of the calculus of variations.
    • He proved invariance of the differential equations he was considering under contact transformations.
    • a second-order differential expressions to provide the equation belonging to the variation of a multiple integral.

  349. Von Staudt biography
    • Von Staudt also gave a nice geometric solution to quadratic equations.
    • We are given the quadratic equation x2 - gx + h = 0 which we wish to solve geometrically.
    • Then a and b are the roots of the given quadratic equation.

  350. Friedmann biography
    • In his last year at the University he was working on an essay on the subject I assigned: 'Find all orthogonal substitutions such that the Laplace equation, transformed for the new variables, admits particular solutions in the form of a product of two functions, one of which depends only on one, and the other on the other two variables'.
    • Also in this letter he asked Steklov's advice on integrating equations he had obtained from theoretically modelling bombs dropping.
    • In reality it turns out that the solution given in it does not satisfy the field equations.

  351. Oppenheim biography
    • Even at this stage it was number theory which appealed to him and he began solving Diophantine equations.
    • He continued with his early interest in Diophantine equations and looked to apply methods coming from ergodic theory.
    • Examples of his papers are Rational approximations to irrationals (1941), On the representation of real numbers by products of rational numbers (1953), On indefinite binary quadratic forms (1954), On the Diophantine equation x3+ y3+ z3= x + y + z (1966), The irrationality of certain infinite products (1968), Representations of real numbers by series of reciprocals of odd integers (1971) and The prisoner's walk: an exercise in number theory (1984).

  352. Sridhara biography
    • We give details below of Sridhara's rule for solving quadratic equations as given by Bhaskara II.
    • Sridhara was one of the first mathematicians to give a rule to solve a quadratic equation.
    • Multiply both sides of the equation by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the coefficient of the unknown; then take the square root.

  353. Bernoulli Nicolaus(I) biography
    • There he worked on geometry and differential equations.
    • Other problems he worked on involved differential equations.
    • He also made significant contributions in studying the Riccati equation.

  354. Mohr Ernst biography
    • From this time on his work was on applied mathematics, mainly fluid dynamics and differential equations, but he also published the occasional paper on polynomials.
    • One of these 1951 papers looks at the numerical solution of the differential equation dy/dx = f (x, y).
    • Hermann Weyl, in two pioneering papers, described a class of differential equations in the limit point case where complete determination of the continuous spectrum is possible.

  355. Stokes biography
    • After he had deduced the correct equations of motion Stokes discovered that again he was not the first to obtain the equations since Navier, Poisson and Saint-Venant had already considered the problem.
    • The work also discussed the equilibrium and motion of elastic solids and Stokes used a continuity argument to justify the same equation of motion for elastic solids as for viscous fluids.

  356. Feigenbaum biography
    • In 1973 it had been conjectured that the behaviour of the logistic equation was the same in a qualitative sense for all g(x) which have a maximum value and decrease monotonically on either side of this maximum.
    • The remarkable result obtained by Feigenbaum was to show that not only was the behaviour qualitatively similar but there was a very precise mathematical result which held for all such logistic equations.
    • Feigenbaum did not actually work with the precise logistic equation which May studied and in fact his work was independent of that by May.

  357. Legendre biography
    • In "Elements" Legendre gave a simple proof that π is irrational, as well as the first proof that π2 is irrational, and conjectured that π is not the root of any algebraic equation of finite degree with rational coefficients.
    • I have thought that what there was better to do in the problem of comets was to start out from the immediate data of observation, and to use all means to simplify as much as possible the formulas and the equations which serve to determine the elements of the orbit.
    • His method involved three observations taken at equal intervals and he assumed that the comet followed a parabolic path so that he ended up with more equations than there were unknowns.

  358. Stampioen biography
    • In 1633 he challenged Descartes to a public competition by giving him a geometric problem whose solution involved the solution of a quartic equation.
    • In fact Stampioen's criticism was fair for although Descartes had taken the geometric problem and derived the correct quartic equation, he left the problem there without solving the quartic.
    • The problem which Stampioen was interested in came as a consequence of using the Cardan-Tartaglia formula to solve cubic equations.

  359. Drinfeld biography
    • He discussed the concepts of quantum groups and quantization, and also talked about Poisson groups, Lie bi-algebras and the classical Yang-Baxter equation.
    • The interactions between mathematics and mathematical physics studied by Atiyah led to the introduction of instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.

  360. Horner biography
    • Horner is largely remembered only for the method, Horner's method, of solving algebraic equations ascribed to him by Augustus De Morgan and others.
    • This discussion is somewhat moot because the method was anticipated in 19th century Europe by Paolo Ruffini (it won him the gold medal offered by the Italian Mathematical Society for Science who sought improved methods for numerical solutions to equations), but had, in any case, been considered by Zhu Shijie in China in the thirteenth century.
    • Horner made other mathematical contributions, however, publishing a series of papers on transforming and solving algebraic equations, and he also applied similar techniques to functional equations.

  361. Ingarden biography
    • For example his early work includes Equations of motion and field equations in five-dimensional unified relativity theory (Russian) (1953) in which he:- .
    • The gravitational equations of the general relativity theory involving the energy-momentum density tensor of matter are generalized to five dimensions.
    • Other papers by Ingarden around this time include: A generalization of the Young-Rubinowicz principle in the theory of diffraction (1955); On a new type of relativistically invariant linear local field equations (Russian) (1956); On the geometrically absolute optical representation in the electron microscope (1957); and Composite variational problems (1959) [Rep.

  362. Liouville biography
    • This he did in October of 1830 but even at this stage he had written a number of papers which he had submitted to the Paris Academy on electrodynamics, partial differential equations and the theory of heat.
    • His work on boundary value problems on differential equations is remembered because of what is called today Sturm-Liouville theory which is used in solving integral equations.
    • Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.

  363. Zaanen biography
    • his study of the theory of linear integral equations.
    • Measure and integral, Banach and Hilbert space, linear integral equations (1953) which contained much of his own research as well as material from a lecture course by N G de Bruijn.
    • Its main feature is the emphasis laid on integral equations and especially on those with symmetrizable kernel, a domain of research in which we owe to the author many personal results.
    • Measure and integral, Banach and Hilbert space, linear integral equations (1953), we have already mentioned above.

  364. Boltzmann biography
    • The equations of Newtonian mechanics are reversible in time and Poincare proved that if a mechanical system is in a given state it will return infinitely often to a state arbitrarily close to the given one.
    • The actual irreversibility of natural phenomena thus proves the existence of processes that cannot be described by mechanical equations, and with this the verdict on scientific materialism is settled.
    • Boltzmann continued to defend his belief in atomic structure and in a 1905 publication Populare Schriften he tried to explain how the physical world could be described by differential equations which represented the macroscopic view without representing the underlying atomic structure.
    • May I be excused for saying with banality that the forest hides the trees for those who think that they disengage themselves from atomistics by the consideration of differential equations.

  365. Edge biography
    • The equation of the scroll of tangents of the common curve of two quadrics is due to Cayley in 1850.
    • Salmon, in his famous text, gave an equation in covariant form.
    • Edge gave a procedure for finding this equation in 1979.
    • Bring's curve was first studied in Klein's 1884 book in connection with the transformation to reduce the general quintic equation to the form x5 + ex + f = 0.

  366. Curry biography
    • He was given a topic in the theory of differential equations by George Birkhoff but he began reading books on logic which seemed to him far more interesting that his research topic.
    • Curry now made his final change in direction and decided to give up his doctoral studies on differential equations and to write a doctoral dissertation on logic.
    • this advantage, of course, implies a restriction on the scope of the treatment, because it is limited to the rational aspects such as arise from ordinary linear differential equations with constant coefficients.
    • For the more general cases of partial differential equations, fractional operators, etc., the theory of integral transforms is doubtless unavoidable.

  367. Jacobi biography
    • By the time Jacobi left school he had read advanced mathematics texts such as Euler's Introductio in analysin infinitorum and had been undertaking research on his own attempting to solve quintic equations by radicals.
    • Kummer had made advances beyond what Jacobi had achieved on third-order differential equations and Jacobi wrote to his brother Moritz in 1836 describing how Kummer had managed to solve problems which had defeated him.
    • Jacobi carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics.

  368. Von Neumann biography
    • He was notorious for dashing out equations on a small portion of the available blackboard and erasing expressions before students could copy them.
    • It was then that he became aware of the mysteries underlying the subject of non-linear partial differential equations.
    • His work, from the beginnings of the Second World War, concerns a study of the equations of hydrodynamics and the theory of shocks.
    • The phenomena described by these non-linear equations are baffling analytically and defy even qualitative insight by present methods.

  369. Shtokalo biography
    • His research on the theory of functions of a complex variable and its applications included work on conformal mappings, differential equations, and variational statistics.
    • However, during the second half of the 1930s he worked almost exclusively on differential equations.
    • Shtokalo continued to undertake research on differential equations while in Ufa, but the direction of his research was at this time influenced by the joint work of Bogolyubov with Nikolai Mitrofanovich Krylov in which they developed a theory of non-linear oscillations; they called their topic 'non-linear mechanics'.
    • or the habilitation) on Asymptotic and Symbolic-Analytic Methods in the Solution of Certain Classes of Linear Differential Equations with Variable Coefficients.
    • In 1945 several papers appeared including: Methode asymptotique pour la solution de certaines classes d'equations differentielles lineaires a coefficients variables; Generalisation de la formule fondamentale de la methode symbolique pour le cas des equations differentielles a coefficients variables; Criteria for stability and instability of the solutions of linear differential equations with quasiperiodic coefficients (Ukrainian, Russian), Linear differential equations of the n th order with quasiperiodic coefficients (Ukrainian, Russian), Systems of linear differential equations with quasiperiodic coefficients (Ukrainian, Russian), and Generalized Gibbs formula for the case of linear differential equations with variable coefficients (Ukrainian, Russian).
    • Shtokalo worked mainly in the areas of differential equations, operational calculus and the history of mathematics.
    • After 1945 he became particularly interested in the qualitative and stability theory of solutions of systems of linear ordinary differential equations in the Lyapunov sense and in the 1940s and 1950 published a series of articles and three monographs in these areas.
    • Shtokalo's work had a particular impact on linear ordinary differential equations with almost periodic and quasi-periodic solutions.
    • He extended the applications of the operational method to linear ordinary differential equations with variable coefficients.
    • In the following year his 128-page monograph Operational methods and their development in the theory of linear differential equations with variable coefficients (Russian) was published.
    • In 1961 the English text Linear differential equations with variable coefficients.
    • It differs from other books in this domain by the consideration of solutions of ordinary differential equations with variable coefficients, based on papers of the author and of K G Valeev.
    • Teacher in primary, intermediate, and advanced schools, scholar, organiser and director of scientific endeavours, researcher in the area of the theory of differential equations, in the domain of operational calculus, in the area of the history of mathematics, organiser of the publication of works which are classics of indigenous science - such is a far from complete enumeration of the contributions of Iosif Zakharovich Shtokalo to the development of domestic science and culture.

  370. Kumano-Go biography
    • His supervisor was M Nagumo who supervised his work on the singular perturbation of second order partial differential equations.
    • During these years Kumano-Go published a series of papers which studied the local and global uniqueness of the solutions of the Cauchy problem for partial differential equations.
    • This is Partial differential equations which was again written in Japanese and was published in 1978.
    • This is a textbook which in addition to studying partial differential equations provides an introduction to pseudo-differential operators.

  371. Brodetsky biography
    • The 5th International Congress for Applied Mechanics was held at Cambridge, Massachusetts, in 1938 and Brodetsky delivered a paper on the equations of motion of an airplane.
    • Sections: Equations of motion, coefficients of statical and dynamical stability k, t; first approximation.
    • Sections: Equations of motion, first approximation.
    • Sections: Symmetrical aeroplane: equations of motion, first approximation.

  372. Peterson biography
    • The dissertation contains a derivation of two equations equivalent to those of Mainardi and Codazzi, and in it Peterson outlined a proof of the fundamental theorem of surface theory.
    • His main work is in differential geometry but he obtained an honorary doctorate for his work on partial differential equations.
    • This was in 1879 from the Novorossiiskii University of Odessa in recognition for his outstanding contributions to the theory of characteristics of partial differential equations [Dictionary of Scientific Biography (New York 1970-1990).
    • by means of a uniform general method, he deduced nearly all the devices known at that time for finding general solutions of different classes of equations.
    • These included: Sur l'integration des equations aux derivees partielles; Sur la deformation des surfaces du second ordre; Sur les courbes tracees sur les surfaces; and Sur les relations et les affinites entre les surfaces courbes.

  373. Drach biography
    • Drach viewed Emile Picard's application, in 1887, of Galois theory to linear differential equations as a model of perfection and he tried to extend Galois theory to differential equations in general, building on the work of Lie and Vessiot in addition to that of Emile Picard.
    • Other papers by Drach include three published in 1908: Sur les systemes completement orthogonaux de l'espace euclidien a n dimensions; Recherches sur certaines deformations remarquables a reseau conjugue persistant; and Sur le probleme logique de l'integration des equations differentielles.
    • After the war ended he published his geometric approach to such problems in L'equation differentielle de la balistique exterieure et son integration par quadratures (1920).
    • Drach's results can be compared with the modern treatment of the same class of equations.
    • Another example of his work is Sur la theorie des corps plastiques et l'equation d'Airy-Tresca which he published in 1946.
    • Around the same time Drach published two papers on partial differential equations: Sur les equations aux derivees partielles du premier ordre dont les caracteristiques sont lignes asymptotiques des surfaces integrales (1947); and Sur des equations aux derivees partielles du premier et du second ordre dont les caracteristiques sont lignes asymptotiques des surfaces integrales (1948).

  374. Faedo biography
    • We have already seen that he made contributions to a wide variety of areas such as the calculus of variations, the theory of linear ordinary differential equations, the theory of partial differential equations, measure theory, the Laplace transform for functions of several variables, questions relating to existence for linear equations in Banach spaces, and foundational problems such as his work on Zermelo's principle in infinite-dimensional function spaces.
    • This paper contains a description and analysis of a method for solving time-dependent partial differential equations which is today known as the Faedo-Galerkin method [Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia.',5)">5]:- .

  375. Hesse biography
    • the special forms of linear equation and of planar equation that Hesse used in these books are called Hesse's normal form of the linear equation and of the planar equation in all modern textbooks on the discipline.

  376. Cockle biography
    • Most of his work, however, was in pure mathematics where he studied algebra, the theory of equations, and differential equations.
    • Describing Cockle's contributions to differential equations Harley explained that his [Proc.
    • mode of dealing with the theory of differential equations was marked by originality and independence of mind.

  377. Chaplygin biography
    • His paper On the motion of a heavy body of revolution in a horizontal plane (1897) was the first to present the general equation of motion of a nonholonomic system.
    • This equation is a generalisation of Lagrange's equation.
    • He had developed methods of approximation for solving differential equation and he presented his first results on that topic to the Moscow Mathematical Society in 1905.

  378. Abu Kamil biography
    • The Book on algebra by Abu Kamil is in three parts: (i) On the solution of quadratic equations, (ii) On applications of algebra to the regular pentagon and decagon, and (iii) On Diophantine equations and problems of recreational mathematics.
    • The Book of rare things in the art of calculation is concerned with solutions to indeterminate equations.
    • Sesiano in [Centaurus 21 (2) (1977), 89-105.',11)">11] discusses Abu Kamil's work on indeterminate equations and he argues that his methods are very interesting for three reasons.

  379. Ritt biography
    • Ritt resigned his position at the Naval Observatory and began working for his doctorate which was awarded in 1917 for his thesis On a general class of linear homogeneous differential equations of infinite order with constant coefficients.
    • Ritt had begun a new major research topic in the 1930s when he began to create a theory of ordinary and partial differential equations.
    • The first book was Differential equations from an algebraic standpoint (1932) and the second, a very major revision and extension of the first, was Differential Algebra (1950).
    • In the last three years of his life Ritt began a deep study of the applications of Lie theory to homogeneous differential equations.

  380. Fine Henry biography
    • Two further paper On the functions defined by differential equations with an extension of the Puiseux polygon construction to these equations, and Singular solutions of ordinary differential equations appeared in 1889 and 1890 respectively.
    • He gave his retiring address as president on An unpublished theorem of Kronecker respecting numerical equations.

  381. Dandelin biography
    • He gave a method of approximating the roots of an algebraic equation, now named the Dandelin-Graffe method, and published this in Recherches sur la resolution des equations numeriques (1826).
    • Dandelin then considers the possibility of accelerating both processes by applying them to the equation whose roots are the squares of those of the original.
    • The method he proposes for doing this is to form the product f (x) f (-x), where f (x) = 0 is the original equation (note that this is not quite the way one does it now).
    • The first is that if the zeros of a polynomial are widely separated into one group of very large modulus, and one of very small modulus, then the equation which remains when final terms are dropped is approximately satisfied by the large zeros.

  382. Stein biography
    • He was among the first to appreciate the interplay among partial differential equations, classical Fourier analysis, several complex variables and representation theory.
    • For more than a century there has been a significant and fruitful interaction between Fourier analysis, complex function theory, partial differential equations, real analysis, as well as ideas from other disciplines such as geometry and analytic number theory, etc.
    • Stein's fusion of complex analysis, partial differential equations, analysis on nilpotent Lie groups, and Euclidean harmonic analysis has deeply influenced countless mathematicians.
    • Stein is one of the foremost experts in harmonic analysis in the world, and he has made stellar contributions to this field as well as related fields such as the theory of several complex variables and partial differential equations.

  383. Penrose biography
    • In this paper Penrose defined a generalized inverse X of a complex rectangular (or possibly square and singular) matrix A to be the unique solution to the equations AXA = A, XAX = X, (AX)T = AX, (XA)T = XA.
    • He used this generalized inverse for problems such as solving systems of matrix equations, and finding a new type of spectral decomposition.
    • In the following year Penrose published On best approximation solutions of linear matrix equations which used the generalized inverse of a matrix to find the best approximate solution X to AX = B where A is rectangular and non-square or square and singular.
    • His development of Twistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics.

  384. Fourier biography
    • Having left St Benoit in 1789, he visited Paris and read a paper on algebraic equations at the Academie Royale des Sciences.
    • The second objection was made by Biot against Fourier's derivation of the equations of transfer of heat.
    • the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.
    • If they had illuminated this branch of physics by important and general views and had greatly perfected the analysis of partial differential equations, if they had established a principal element of the theory of heat by fine experiments ..

  385. Malfatti biography
    • Malfatti wrote an important work on equations of the fifth degree.
    • His papers dealt with many subjects from probability to mechanics and he participated in the debate around Ruffini's attempt to prove the impossibility of solving (in the meaning of that period) equations of higher degree than four.
    • These include: Problems and methods of mathematical analysis in the work of Gianfrancesco Malfatti, Contributions of Gianfrancesco Malfatti to combinatorial analysis and to the theory of finite difference equations, The work of Malfatti in the realm of mechanics, The geometrical research of Gianfrancesco Malfatti, Gianfrancesco Malfatti and the theory of algebraic equations, and Gianfrancesco Malfatti and the support problem.

  386. Wheeler biography
    • The records show that Anna studied Algebra and Trigonometry in 1899-1900, Modern Geometry, the Theory of Equations, and Solid Analytical Geometry in 1900-1901, Calculus, Analytical mechanics and Plane Analytical Geometry in 1901-02, and the Theory of Substitutions and Potential, Partial Differential Equations and Fourier Series, and Differential Equations in 1902-03.
    • After winning a scholarship to study for her master's degree at the University of Iowa, she was awarded the degree for a thesis The extension of Galois theory to linear differential equations in 1904.
    • After returning to the United States in August 1907, where her husband was by now Dean of Engineering in South Dakota, she taught courses in the theory of functions and differential equations.
    • in 1910 for her thesis Biorthogonal Systems of Functions with Applications to the Theory of Integral Equations being the one written originally at Gottingen.
    • She took various courses at Chicago: General analysis; Periodic Orbits; Theory of Numbers; Integral Equations; Modern Analysis applied to celestial Mechanics; and Theory of Algebraic Numbers.
    • Anna Pell had published two papers in the Bulletin of the American Mathematical Society in 1909-10, namely On an integral equation with an adjoined condition and Existence theorems for certain unsymmetric kernels.
    • While at this College she published Non-homogeneous linear equations in infinitely many unknowns (1914) and (with Ruth L Gordon) The modified remainders obtained in finding the highest common factor of two polynomials (1916).
    • Under his guidance she worked on integral equations studying infinite dimensional linear spaces.
    • As an example of one of the papers she wrote while on the Faculty at Bryn Mawr, we mention Linear Ordinary Self-Adjoint Differential Equations of the Second Order (1927).
    • In 1914 Lichtenstein made connection, without the intermediary of the theory of integral equations, between the theory of linear differential systems of the second order and the theory of linear equations in infinitely many unknowns.

  387. Floquet biography
    • Floquet submitted his doctoral thesis Sur la theorie des equations differentielles lineaires (On the theory of linear differential equations) to the Faculty of Science in Paris on 8 April 1879.
    • For example he published three papers with the title Sur les equations differentielles lineaires a coefficients periodiques (On linear differential equations with periodic coefficients), two in 1881 and the third in 1883.
    • In 1884 he published Addition a un memoire sur les equations differentielles lineaires (Addition to a memoir on linear differential equations) and Sur les equations differentielles lineaires a coefficients doublement periodiques (On linear differential equations with double periodic coefficients).
    • For example he published Sur une classe d'equations differentielles lineaires non homogenes (1887), Sur une propriete de la surface xyz = l3 (1888), Sur le mouvement d'un fil dans un plan fixe (1889), Sur l'equation de Lame (1895), Sur le mouvement d'un point ou d'un fil glissant sur un plan horizontal fixe lorsqu'on tient compte de la rotation de la terre et du frottement (1898), Sur les equations intrinseques du mouvement d'un fil et sur le calcul de sa tension (1901), and L'astronome Messier (1902).

  388. Hadamard biography
    • The topic proposed for the prize had been one on geodesics and Hadamard's work in studying the trajectories of point masses on a surface led to certain non-linear differential equations whose solution also gave properties of geodesics.
    • Matrices whose determinants satisfied equality in the relation are today called Hadamard matrices and are important in the theory of integral equations, coding theory and other areas.
    • In particular he worked on the partial differential equations of mathematical physics producing results of outstanding importance.
    • He continued to produce books and papers of the highest quality, publishing perhaps his most famous text Lectures on Cauchy's problem in linear partial differential equations in 1922.

  389. Prodi biography
    • He graduated from the University of Parma on 24 November 1948 having submitted his thesis on problems of stability in the theory of differential equations.
    • He published papers on differential equations during this period which were directly related to his thesis: Un'osservazione sugl'integrali dell'equazione y" + A(x)y = 0 nel caso A(x)→ +∞ per x→ ∞ (1950); Nuovi criteri di stabilita per l'equazione y" + A(x)y = 0.
    • It was while he was in Trieste that Prodi produced the most famous of his research results when he proved important uniqueness theorems for two-dimensional Navier-Stokes equations.
    • This book was in the line of the work of Renato Caccioppoli and like the pioneering monograph 'Problemi di esistenza in analisi funzionale' of Carlo Miranda [1949], it put the emphasis upon the application of global implicit function theorems to the existence and multiplicity of solutions of nonlinear elliptic partial differential equations.

  390. Bernoulli Jacob biography
    • In May 1690 in a paper published in Acta Eruditorum, Jacob Bernoulli showed that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation.
    • After finding the differential equation, Bernoulli then solved it by what we now call separation of variables.
    • In 1696 Bernoulli solved the equation, now called "the Bernoulli equation", .

  391. Levy Hyman biography
    • Levy's main work was in numerical methods, numerical solution of differential equations, finite difference equations and statistics.
    • Among other mathematical works he published were Numerical Studies in Differential Equations (1934), Elements of the Theory of Probability (1936), and Finite Difference Equations (1958).

  392. Simon biography
    • in 1971 for his thesis Interior Gradient Bounds for Non-Uniformly Elliptic Equations.
    • supervisor James H Michael) Sobolev and mean-value inequalities on generalized submanifolds of Rn (1973); Global estimates of Holder continuity for a class of divergence-form elliptic equations (1974); (with Richard M Schoen and Shing-Tung Yau) Curvature estimates for minimal hypersurfaces (1975); Interior gradient bounds for non-uniformly elliptic equations (1976); and Remarks on curvature estimates for minimal hypersurfaces (1976).
    • This development began with his 1983 paper "Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems." The first stage of his work on general singular sets is principally described in "Cylindrical tangent cones and the singular set of minimal submanifolds" (1993), and the remaining work appears in his paper "Rectifiability of the singular set of energy minimizing maps" (1995).

  393. Brisson biography
    • His favourite field of study was the theory of partial differential equations.
    • The main idea in these reports was the application of the functional calculus, through symbols, to the solution of certain kinds of linear differential equations and of linear equations with finite differences.
    • The 1823 report was the object of lively discussion in 1825 before the Academy and was approved of by Cauchy, who, although he had some reservations about the validity of some of the symbols used and the equations obtained, emphasized the elegance of the method and the importance of the objects to which they were applied.

  394. Kluvanek biography
    • The book, covered Differential and Integral calculus, Analytic geometry, Differential equations and Complex variable [S Tkacik, J Guncaga, P Valihora and M Gerec (eds.), Igor Kluvanek: Prispevky zo seminara venovaneho nedozitym 75.
    • This is a monograph on the geometry of the range of a vector measure and applications to control systems governed by partial differential equations.
    • Moreover, it opens the new area of applications of vector measures to control systems governed by partial differential equations.
    • The subject matter of the monograph under review is motivated by the fact that in describing superpositions of evolution processes one often encounters serious problems in solving the corresponding evolution equations.

  395. Ricci-Curbastro biography
    • In 1875 Ricci-Curbastro was awarded a doctorate for his thesis On Fuchs's research concerning linear differential equations.
    • Three of these articles appeared in Nuovo Cimento in 1877 and, in the same year, an article appeared in Giornale di matematiche di Battaglini which Dini had asked him to write on Lagrange's problem on a system of linear differential equations.
    • Ricci-Curbastro's early work was in mathematical physics, particularly on the laws of electric circuits and differential equations.
    • In the paper, applications are given by Ricci-Curbastro and Levi-Civita to the classification of the quadratic forms of differentials and there are other analytic applications; they give applications to geometry including the theory of surfaces and groups of motions; and mechanical applications including dynamics and solutions to Lagrange's equations.

  396. Kronecker biography
    • students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy.
    • The topics on which he lectured were very much related to his research: number theory, the theory of equations, the theory of determinants, and the theory of integrals.
    • We have already indicated that Kronecker's primary contributions were in the theory of equations and higher algebra, with his major contributions in elliptic functions, the theory of algebraic equations, and the theory of algebraic numbers.

  397. Monge biography
    • The four memoirs that Monge submitted to the Academie were on a generalisation of the calculus of variations, infinitesimal geometry, the theory of partial differential equations, and combinatorics.
    • Over the next few years he submitted a series of important papers to the Academie on partial differential equations which he studied from a geometrical point of view.
    • .the composition of nitrous acid, the generation of curved surfaces, finite difference equations, partial differential equations (1785); double refraction and the structure of Iceland spar, the composition of iron, steel, and cast iron, and the action of electricity sparks on carbon dioxide gas (1786); capillary phenomena (1787); and the causes of certain meteorological phenomena (1788); and a study in physiological optics (1789).

  398. Lexell biography
    • Lexell made a detailed investigation of exact equations differential equations.
    • In addition Lexell developed a theory of integrating factors for differential equations at the same time as Euler but, although it has often been thought that he learnt of the technique from Euler, the author of [Istor.-Mat.
    • Lexell did work in analysis on topics other than differential equations, for example he suggested a classification of elliptic integrals and he worked on the Lagrange series.

  399. Birkhoff biography
    • Birkhoff read Poincare's works on differential equations and celestial mechanics and he learnt more, and was more strongly influenced in the direction his research was taking, by Poincare than from his supervisor.
    • The doctoral thesis which Birkhoff submitted was entitled Asymptotic Properties of Certain Ordinary Differential Equations with Applications to Boundary Value and Expansion Problems and it led to the award of his Ph.D.
    • Birkhoff's work on linear differential equations, difference equations and the generalised Riemann problem mostly all arose from the basis he laid in his thesis.

  400. Zylinski biography
    • Can you tell me what is the Schrodinger equation?" I said: "I don't know," and then he said: "All right.
    • The Schrodinger equation is the fundamental equation of (non-relativistic) quantum mechanics, which was not part of the syllabus, and I am quite sure that Żyliński himself didn't know the Schrodinger equation.

  401. Szekeres biography
    • The reference to chaos theory here refers in particular to his interest in Feigenbaum's functional equation.
    • Another unusual quality was George's interest in computational and "experimental" mathematics, which he maintained until his last paper on Abel's equation.
    • In the mid-90s, I worked with him on Feigenbaum's functional equation.
    • We wrote programs to solve several cases of this equation, and I was very impressed by this 80 year-old who knew more about how to actually get computers to do real mathematics than many of my younger colleagues.

  402. Pisot biography
    • He then saw that he could get better and better approximations from the equation x2 - 2y2 = 1.
    • There he looks at diophantine equations, the Goldbach conjecture, Roth's theorem, transcendental numbers, the distribution of primes, and p-adic analysis.

  403. Daniell biography
    • Among the courses that Hilbert was giving at this time were Partial Differential Equations, Mathematical Foundations of Physics, and Theory of the Electron.
    • presents one of the earliest mathematical treatments of continuous time Markov processes, including the Chapman-Kolmogorov equation (ten years before Kolmogorov) and a short treatment of the Wiener process (two years before Wiener).

  404. Borchardt biography
    • Borchardt's doctoral work, on non-linear differential equations, was supervised by Jacobi and submitted in 1843.
    • Borchardt also generalised results of Kummer on equations determining the secular disturbances of the planets.
    • In several further papers Borchardt applied the theory of determinants to algebraic equations, mostly in connection with symmetric functions, the theory of elimination, and interpolation.

  405. Dudeney biography
    • Other puzzles simply reduced to systems of linear equations if a mathematical solution was sought.
    • Problem 11 from the same book reduces to a quadratic equation:- .

  406. Ramanujan biography
    • Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic.
    • He devoloped relations between elliptic modular equations in 1910.
    • Ramanujan worked out the Riemann series, the elliptic integrals, hypergeometric series and functional equations of the zeta function.

  407. D'Alembert biography
    • He was a pioneer in the study of partial differential equations and he pioneered their use in physics.
    • The article contains the first appearance of the wave equation in print but again suffers from the defect that he used mathematically pleasing simplifications of certain boundary conditions which led to results which were at odds with observation.

  408. Lifshitz biography
    • The requirements were: ability to evaluate any indefinite integral (in terms of elementary functions) and to solve any ordinary differential equation of the standard type, knowledge of vector analysis and tensor algebra as well as of principles of the theory of functions of complex variable (theory of residues, Laplace method).
    • This 1974 prize was awarded jointly to Lifshitz, V A Belinskii and I M Khalatnikov for their work on the singularities of cosmological solutions of the gravitational equations which was presented in sixteen papers between 1961 and 1985.

  409. Kirchhoff biography
    • Kirchhoff considered an electrical network consisting of circuits joined at nodes of the network and gave laws which reduce the calculation of the currents in each loop to the solution of algebraic equations.
    • An early form of the theory had been developed by Germain and Poisson but it was Navier who gave the correct differential equation a few years later.

  410. Cohen biography
    • In addition to his work on set theory, Cohen worked on differential equation and harmonic analysis.
    • Like many great mathematicians, his mathematical interests and contributions were very broad, ranging from mathematical analysis and differential equations to mathematical logic and number theory.

  411. Runge biography
    • Runge then worked on a procedure for the numerical solution of algebraic equations in which the roots were expressed as infinite series of rational functions of the coefficients.
    • There were three standard methods for the numerical solution of such equations, namely by Newton, Bernoulli and Graffe, and the method found by Runge had all three of the standard methods as special cases.
    • He worked out many numerical and graphical methods, gave numerical solutions of differential equations, etc.

  412. Eckert Wallace biography
    • The first is the development of the theory or the solution of the differential equations of motion expressing the coordinates of the moon as explicit functions of time.
    • In order to bring the Tables within even their present length, various parts of the basic equations were curtailed whenever permissible in the light of observational requirements (as then visualised).
    • Eckert therefore decided not to recompute new tables but to compute the ephemerides directly from Brown's equations.

  413. Bellavitis biography
    • enables us to express by means of formulae the results of geometric constructions, to represent geometric propositions by means of equations, and to replace a logical argument by the transformation of equations.
    • In algebra he continued Ruffini's work on the numerical solution of algebraic equations and he also worked on number theory.

  414. Milne-Thomson biography
    • In 1948 he published Applications of elliptic functions to wind tunnel interference while in 1957 he wrote a review paper A general solution of the equations of hydrodynamics which M G Scherberg reviews as follows:- .
    • For example he wrote Consistency equations for the stresses in isotropic elastic and plastic materials (1942), and Stress in an infinite half-plane (1947).
    • He gave two lectures in Madrid in 1951 on the elements of finite elasticity theory, the first lecture covering the topics of deformation tensors, stress, equations of motion, and energy.

  415. Rouche biography
    • In 1875 Rouche published a two page paper Sur la discussion des equations du premier degre in volume 81 of Comptes Rendus of the Academie des Sciences.
    • This short paper contained his result on solving systems of linear equations.
    • This is the well-known criterion which says that a system of linear equations has a solution if and only if the rank of the matrix of the associated homogeneous system is equal to the rank of the augumented matrix of the system.

  416. Bobillier biography
    • The second and the third Books, deal with the solution of problems, and the equations which derive from them; the latter, with certain algebraic methods which enable numerical calculations to be shortened.
    • He first set up a problem in the form of an equation in a particular case, simple enough so that the analytic geometry of his time could deal with it.

  417. Herglotz biography
    • In this last paper Herglotz solved Abel's integral equation which results from the inversion of measured seismic travel times into a velocity-depth function.
    • There are two sections, one of five chapters on classical theory of the mechanics of continua based on Hamilton's principle and another of four chapters on partial differential equations.

  418. Cole biography
    • Cole returned to Harvard and wrote a thesis A Contribution to the Theory of the General Equation of the Sixth Degree which, as the title indicates, studied equations of degree 6.

  419. Killing biography
    • Lie algebras were introduced by Lie in about 1870 in his work on differential equations.
    • Finally, before we leave our discussion of Killing's work, it is worth noting that he introduced the term 'characteristic equation' of a matrix.

  420. Brocard biography
    • The text consists of brief descriptive paragraphs, with diagrams and equations of these curves.
    • In 1876, Brocard asked if the only solutions to the equation n! + 1 = m2, in positive integers (n, m), are (4, 5), (5, 11), (7, 71).

  421. Moisil biography
    • While working there he wrote the paper On a class of systems of equations with partial derivatives from mathematical physics.
    • Before reading this work Moisil had worked on differential equations, the theory of functions and mechanics.
    • Among Moisil's other books we mention: Associated matrices of systems of partial differential equations.

  422. Gauss biography
    • In the first volume he discussed differential equations, conic sections and elliptic orbits, while in the second volume, the main part of the work, he showed how to estimate and then to refine the estimation of a planet's orbit.
    • Gauss used the Laplace equation to aid him with his calculations, and ended up specifying a location for the magnetic South pole.

  423. Poincare biography
    • His thesis was on differential equations and the examiners were somewhat critical of the work.
    • The idea was to come in an indirect way from the work of his doctoral thesis on differential equations.
    • He can be said to have been the originator of algebraic topology and, in 1901, he claimed that his researches in many different areas such as differential equations and multiple integrals had all led him to topology.

  424. Straus biography
    • An approximate solution of the field equations for empty space is obtained and the gravitational potentials thus determined are required to piece together continuously with the known gravitational potentials for a pressure free, spatially constant density of matter.
    • This presented a new derivation of the field equations which was necessary since the derivation in Einstein's single authored paper published in the previous year was based on an error.
    • Algebraic equations satisfied by roots of natural numbers.

  425. Zygmund biography
    • Among other topics, he worked on summability of numerical series, summability of general orthogonal series, trigonometric integrals, sets of uniqueness, summability of Fourier series, differentiability of functions, smooth functions, approximation theory, absolutely convergent Fourier series, multipliers and translation invariant operators, conjugate series and Taylor series, lacunary trigonometric series, series of independent random variables, random trigonometric series, the Littlewood-Paley, Luzin and Marcinkiewicz functions, boundary values of analytic and harmonic functions, singular integrals, partial differential equations and interpolation operators.
    • Their famous joint papers over the next few years on singular integrals and partial differential equations, the most significant of which appeared in 1952, have had a major impact on modern analysis.
    • For outstanding contributions to Fourier analysis and its applications to partial differential equations and other branches of analysis, and for his creation and leadership of the strongest school of analytical research in the contemporary mathematical world.

  426. Novikov Sergi biography
    • We should mention especially Sergei's uncle Mstislav Keldysh who made major contributions to complex function theory, differential equations and applications to aerodynamics.
    • These include a systematic study of finite-gap solutions of two-dimensional integrable systems, formulation of the equivalence of the classification of algebraic-geometric solutions of the KP equation with the conformal classification of Riemann surfaces, and work (with Krichever) on "almost commuting" operators that appear in string theory and matrix models ("Krichever-Novikov algebras", now widely used in physics).

  427. Atiyah biography
    • Subsequently (in collaboration with I M Singer) he established an important theorem dealing with the number of solutions of elliptic differential equations.
    • Beyond these linear problems, gauge theories involved deep and interesting nonlinear differential equations.
    • In particular, the Yang-Mills equations have turned out to be particularly fruitful for mathematicians.

  428. Denjoy biography
    • These great mathematicians gave Denjoy a strong background in complex function theory, continued fractions and differential equations and set him on the road to his great discoveries.
    • In 1934 he wrote that his greatest achievements had been the integration of derivatives, the computation of the coefficients of a converging trigonometric series, a theorem on quasi-analytic functions, and differential equations on a torus.
    • Choquet, very fairly, suggests that Denjoy's work on differential equations on a torus, not nearly so highly rated by Denjoy himself, is one of his most influential pieces of work and has [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .

  429. Hurewicz biography
    • He gave a series of lectures at Brown University in 1943 and these were published in mimeographed form by Brown University as Ordinary differential equations in the real domain with emphasis on geometric method.
    • Lectures on ordinary differential equations was a reprinting, with minor revisions, of the mimeographed notes of his Brown University lectures.
    • This textbook is a beautiful introduction to ordinary differential equations which again reflects the clarity of his thinking and the quality of his writing.

  430. Pfeiffer biography
    • When he returned to Kiev he was made head of the Department of Ordinary Differential Equations at the University.
    • Pfeiffer's first research was on the problem of solving equations by radicals, and he next looked at algebraic geometry.
    • Georgii Vasil'evich Pfeiffer (1872-1946), professor at Kiev University, is known as a specialist in the field of integration of differential equations and systems of partial differential equations.
    • As this quote indicates, however, his most important contributions involve work on partial differential equations following on from the methods developed by Lie and Lagrange.
    • some work of the Ukrainian mathematician G V Pfeiffer , showing how to find integrals of a general system of partial differential equations by using sequential complete systems instead of passing to Jacobian systems.
    • Pfeiffer also constructed all the infinitesimal operators of a system of equations.
    • For example, he published three papers in 1946: La reception et l'integration par la methode speciale des equations, systemes d'equations semi-Jacobiens, des equations, systemes d'equations semi-Jacobiens generalises aux derivees partielles du premier ordre de plusieurs fonctions inconnues; Sur les equations, systemes d'equations semi-jacobiens, semi-jacobiens generalises aux derivees partielles de premier ordre a plusieurs fonctions inconnues; and Sur les equations, systemes d'equations semi-mixtes aux derivees partielles du premier ordre a plusieurs fonctions inconnues.
    • As the title indicates, the paper is concerned with a special method for solving semi-Jacobian systems of partial differential equations.
    • Such a system arises from a linear partial differential equation of the first order containing one or more parameters whose elimination leads to the Jacobian system of equations which are nonlinear.
    • The author sketches two rules for integration of systems of "semi-mixed'' partial differential equations of the first order with several unknown functions.
    • The author sketches two ways of applying his method of integration [first published in 1923] to systems of generalized semi-Jacobian equations of the first order with several unknown functions.
    • He also published books such as his 346-page 1937 book Integration of differential equations.
    • He also used his language skills to translate books such as the 1891 French text by Edouard Goursat Lecons sur l'integration des equations aux derivees partielles du premier ordre which appeared in Ukrainian in 1940.

  431. Carcavi biography
    • (2) The equation x3 + y3 = z3 has no solutions in integers.
    • (3) The equation y2 + 2 = x3 admits no solutions in integers except x = 3, y = 5.
    • (4) The equation y2 + 4 = x3 admits no solutions in integers except x = 2, y = 2 and x = 5, y = 11.

  432. Lobachevsky biography
    • Despite this heavy administrative load, Lobachevsky continued to teach a variety of different topics such as mechanics, hydrodynamics, integration, differential equations, the calculus of variations, and mathematical physics.
    • In 1834 Lobachevsky found a method for the approximation of the roots of algebraic equations.
    • This method of numerical solution of algebraic equations, developed independently by Graffe to answer a prize question of the Berlin Academy, is today a particularly suitable method for using computers to solve such problems.

  433. Ortega biography
    • In the second part of the book, devoted mostly to geometry, Ortega gives a method of extracting square roots very accurately using Pell's equation, which is surprising since a general solution to Pell's equation does not appear to have been found before Fermat over 100 years later.
    • Pell's equation .

  434. Goldberg biography
    • Several other mathematicians at the university were equally important for Goldberg's development, including Boris Vladimirovich Gnedenko, Yaroslav Borisovich Lopatynsky, who held the chair of differential equations, and Lev Israelevich Volkovyskii who had been a student of Mikhail Alekseevich Lavrent'ev and was working on complex analysis, particularly quasiconformal mappings and Riemann surfaces.
    • Goldberg was asked to give the opening memorial plenary lecture on memorial meeting in honour of Shlomo Strelitz at the 'Conference of Differential Equations and Complex Analysis' at the University of Haifa in December 2000.
    • He gave the lecture On the growth of entire solutions of algebraic differential equations which was published in 2005.

  435. Todd John biography
    • Solution of differential equations by recurrence relations (1950); Experiments on the inversion of a 16 × 16 matrix (1953); Experiments in the solution of differential equations by Monte Carlo methods (1954); The condition of the finite segments of the Hilbert matrix (1954); Motivation for working in numerical analysis (1954); and A direct approach to the problem of stability in the numerical solution of partial differential equations (1956).

  436. Wu Wen-Tsun biography
    • He based his method on the idea of a characteristic set which had been introduced by Joseph Ritt in his algebraic and algorithmic approach to differential equations.
    • It is in Chapter 4 that Wu explains how to translate geometrical problems into polynomial equations.
    • In 2000 Wu published Mathematics mechanization : Mechanical geometry theorem-proving, mechanical geometry problem-solving and polynomial equations-solving.

  437. Holder biography
    • Klein's lectures on Galois theory at Gottingen had interested Holder who began to study the Galois theory of equations and from there he was led to study composition series of groups.
    • Holder was one of the first to give a rigorous account of the famous classical case where a splitting field is not a radical extension: the irreducible cubic equation over the rationals with three real roots, where it is nonetheless necessary to adjoin complex roots of unity.

  438. Euclid biography
    • Euclid's geometric solution of a quadratic equation .
    • History Topics: Quadratic, cubic and quartic equations .

  439. Riemann biography
    • In October he set to work on his lectures on partial differential equations.
    • Riemann studied the convergence of the series representation of the zeta function and found a functional equation for the zeta function.

  440. Markov biography
    • He wrote his first mathematics paper while at the Gymnasium but his results on integration of linear differential equations which were presented in the paper were not new.
    • Markov graduated in 1878 having won the gold medal for submitting the best essay for the prize topic set by the faculty in that year - On the integration of differential equations by means of continued fractions.
    • During his lectures he did not bother about the order of equations on the blackboard, nor about his personal appearance.

  441. Andrews biography
    • Andrews had published three papers by the time he had completed his thesis work: An asymptotic expression for the number of solutions of a general class of Diophantine equations (1961); A lower bound for the volume of strictly convex bodies with many boundary lattice points (1963); and On estimates in number theory (1963).
    • This last paper, in the American Mathematical Monthly, gave a method for finding an upper bound for the number of solutions of a Diophantine equation of the form y = f (x).

  442. Vashchenko biography
    • After his studies in Kazan, he returned to Kiev and he taught at Kiev Cadet School from 1855 to 1862, receiving his Master's Degree in 1862 for a dissertation on the operational method and its application to solving linear differential equations.
    • In particular he worked on the theory of linear differential equations, the theory of probability (see [A N Bogolyubov (ed.), On the history of the mathematical sciences 167 \'Naukova Dumka\' (Kiev, 1984), 36-39.',3)">3]) and non-euclidean geometry.
    • He published The Symbolic Calculus and its Application to the Integration of Linear Differential Equations in 1862.

  443. Schwartz biography
    • This theory provides a convenient formalism for many common situations in theoretical and applied analysis, but its greatest significance may be in connection with partial differential equations, particularly those of hyperbolic type, where its adaptability to local problems gives it an advantage over Hilbert space (and other primarily global) techniques.
    • This has led to extensive studies of topological vector spaces beyond the familiar categories of Hilbert and Banach spaces, studies that, in turn, have provided useful new insights in some areas of analysis proper, such as partial differential equations or functions of several complex variables.
    • I think every reader of his cited paper, like myself, will have left a considerable amount of pleasant excitement, on seeing the wonderful harmony of the whole structure of the calculus to which the theory leads and on understanding how essential an advance its application may mean to many parts of higher analysis, such as spectral theory, potential theory, and indeed the whole theory of linear partial differential equations ..

  444. Paoli biography
    • His research was on analytic geometry, calculus, partial derivatives, and differential equations.
    • Although most mathematicians ignored Paolo Ruffini's proof of the impossibility of solving equations of degree greater than four by the method of radicals, Paoli read Ruffini's proof and wrote to him in 1799:- .
    • and recommend greatly the most important theorem which excludes the possibility of solving equations of degree greater than four.

  445. Lehmer Derrick biography
    • The chapter headings are: Lucas' functions; Tests for primality; Continued fractions; Bernoulli numbers and polynomials; Diophantine equations; Numerical functions; Matrices; Power residues; Analytic number theory; Partitions; Modular forms; Cyclotomy; Combinatorics; Sieves; Equation solving; Computing techniques; and Miscellaneous.

  446. Darboux biography
    • Darboux generalised results of Kummer giving a system defined by a single equation with many interesting properties.
    • This integral was introduced in a paper on differential equations of the second order which he wrote in 1870.

  447. Kodaira biography
    • These include applications of Hilbert space methods to differential equations which was an important topic in his early work and was largely the result of influence by Weyl.
    • In 1979 he published the five volume Introduction to analysis in Japanese covering real numbers, functions, differentiation, integration, infinite series, functions of several variables, curves and surfaces, Fourier series, Fourier transforms, ordinary differential equations, and distributions.
    • In mathematics and science it is a familiar occurrence to have objects, such as systems of equations, depending on parameters.

  448. Abraham biography
    • Rather strangely, however, 1145 was also the year that al-Khwarizmi's algebra book was translated by Robert of Chester so Abraham bar Hiyya's work was rapidly joined by a second text giving the complete solution to the general quadratic equation.
    • History Topics: Quadratic, cubic and quartic equations .

  449. Juel biography
    • Many readers must have felt that if all that projective geometry could tell us of a problem involving a cubic equation was that it has at least one solution, and not more than three, then projective geometry had not by any means justified its claims to replace the ordinary algebraic kind.
    • The main topics covered are problems leading to cubic and quartic equations in one variable, reduced to finding the intersections of two conics, of which in the first case one intersection is known; the two-dimensional chain, and its relations; anti-collineations; the projective metric, Euclidean and non-Euclidean; the quadratic transformation, the rational plane cubic, and the general plane cubic.

  450. Wright Sewall biography
    • Another paper by Wright which shows his mathematical approach to the subject is The differential equation of the distribution of gene frequencies which he published in 1945.
    • He derives differential equations which are satisfied by the probability density function of the distribution of gene frequencies under certain conditions.

  451. Levi Eugenio biography
    • This important work studied partial differential equations of order 2n, linear in two variables fully elliptic in a certain region of the plane.
    • Since the problem could not be solved in general, he examined special cases starting with linear equations in two variables and then extending the ideas to the non-linear case.
    • However, he also wrote on issues relating to: differential geometry, Lie groups, partial differential equations and the minimum principle.

  452. Al-Kashi biography
    • He also considered the equation associated with the problem of trisecting an angle, namely a cubic equation.
    • He was not the first to look at approximate solutions to this equation since al-Biruni had worked on it earlier.

  453. Milne Archibald biography
    • He read papers at meetings of the Society such as Notes on the equation of the parabolic cylinder on Friday 9 January 1914, The Conformal Representation of the Quotient of two Bessel Functions on 24 January 1916, and Note on the Peano-Baker method of solving linear differential equations on 11 February 1916.

  454. Sylow biography
    • Finding Abel's papers on the solvability of algebraic equations by radicals more interesting, Sylow was led from them (by the professor in applied mathematics, Carl Bjerknes) to Galois.
    • Although at first Sylow found reading Abel's papers a difficult task, soon he found that Abel had achieved a far deeper understanding of the theory of equations than his published papers indicated.
    • made myself acquainted with newer works, particularly in the theory of equations.
    • he worked instead in the library, studying number theory and the theory of equations.
    • It is interesting to note that no lectures in algebra or the theory of equations are mentioned from his stay either in Paris or in Berlin.
    • In his lectures Sylow explained Abel's and Galois's work on algebraic equations.
    • 15 (1) (1988), 40-52.',11)">11] Winfried Scharlau describes how Sylow was led to his discovery by his study of Galois' work, in particular of Galois' criterion for the solvability of equations of prime degree.

  455. Valiron biography
    • Among the papers that Valiron published in the years following World War I, we mention: Les theoremes generaux de M Borel dans la theorie des fonctions entieres (1920); Recherches sur le theoreme de M Picard (1921); Recherches sur le theoreme de M Picard dans la theorie des fonctions entieres (1922); Sur les fonctions entieres verifiant une classe d'equations differentielles (1923); Sur l'abscisse de convergence des series de Dirichlet (1924); Sur les surfaces qui admettent un plan tangent en chaque point (1926); and Sur la distribution des valeurs des fonctions meromorphes (1926).
    • The second volume, published in 1945, was entitled Equations Fonctionnelles; Applications.
    • It covers ordinary differential equations, partial differential equations, and algebraic equations of two variables.

  456. Cauchy biography
    • He did important work on differential equations and applications to mathematical physics.
    • Numerous terms in mathematics bear Cauchy's name:- the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations and Cauchy sequences.

  457. Arzela biography
    • He also published Sviluppo, in serie ordinate secondo le potenze decrescenti della variabile, di n funzioni algebriche definite da altrettante equazioni a coefficienti determinati (Expansion of n algebraic functions, defined by equally many equations with fixed coefficients, in series arranged according to decreasing powers of the variable) in 1873.
    • He had also worked on the theory of equations publishing the paper Sopra la teoria dell' eliminazione algebraica (On the theory of algebraic elimination) in 1877.
    • In the year long course of lectures Arzela proved that equations of degree greater than four cannot be solved by radicals.

  458. Painleve biography
    • He worked on differential equations, particularly studying their singular points, and on mechanics.
    • His interest in mechanics was a natural one since this subject provided a natural setting for applications of the results which he had proved for differential equations.
    • He solved, using Painleve functions, differential equations which Poincare and Emile Picard had failed to solve, showing, as Hadamard wrote, that:- .

  459. Dionysodorus biography
    • The Dionysodorus we are interested in here is the mathematician Dionysodorus who Eutocius states solved the problem of the cubic equation using the intersection of a parabola and a hyperbola.
    • Strabo distinguishes this Dionysodorus from Dionysodorus of Amisene and it is now thought that the Dionysodorus referred to by Pliny is not the mathematician who solved the problem of the cubic equation.
    • Shortly after Cronert published details of the fragments of papyri relating to Dionysodorus which had been found at Herculaneum, Schmidt published a commentary on the material in which he argued convincingly that the Dionysodorus who solved the cubic equation using the intersection of a parabola and a hyperbola was the Dionysodorus of Caunus referred to in the Herculaneum papyrus.

  460. Sitter biography
    • He found solutions to Einstein's field equations in the absence of matter.
    • This is a particularly simple solution of the field equations of general relativity for an expanding universe.
    • He is not a cold, dispassionate juggler of Greek letters, a balancer of equations, but rather an artist in whom wild flights of the imagination are restrained by the formalism of a symbolic language and the evidence of observation.

  461. Sonin biography
    • He continued working on his doctorate, essentially equivalent to the German habilitation, and after submitting a thesis on partial differential equations of the second order to the Moscow University he was awarded the degree in 1874.
    • He has a sequence of polynomials named after him - the Sonin polynomials Tnm(x) satisfy the differential equation .

  462. Rahn biography
    • The book, written in German, contains an example of Pell's equation.
    • Pell's equation .
    • History Topics: Pell's equation .

  463. Haselgrove biography
    • The problem of stellar evolution is expressed, mathematically, by a set of non-linear partial differential equations describing the variation of density and temperature as a function of time and of distance from the star centre.
    • In The solution of non-linear equations and of differential equations with two-point boundary conditions (1961) Haselgrove suggests general iterative techniques, based on an n-dimensional extension of the Newton-Raphson process.

  464. Golab biography
    • Professor Golab dealt with different fields of mathematics such as geometry, topology, algebra, analysis, logic, functional and differential equations, the theory of numerical methods and various applications of mathematics.
    • may be considered as the father figure of the Polish school of functional equations.
    • All Polish mathematicians working in the theory of functional equations are - directly or indirectly - pupils of Professor Golab.

  465. Meissel biography
    • Meissel's mathematical interests covered the following fields: number theory (in particular, properties of prime numbers), theta functions, elliptic functions, spherical trigonometry, hydrodynamics, ordinary differential equations, asymptotic expansions, and Bessel functions.
    • a forerunner (in the theory of Bessel functions, in connection with Emden's equation etc.).

  466. Fermi biography
    • He showed great talents, especially in mathematics and by the time he left elementary school at the age of ten he was puzzling out how the equation x2 + y2 = r2 represented a circle.
    • In his essay Fermi derived the system of partial differential equations for a vibrating rod, then used Fourier analysis to solve them.

  467. Apollonius biography
    • He gives propositions determining the centre of curvature which lead immediately to the Cartesian equation of the evolute.
    • Included in it are a series of propositions which, though worked out by the purest geometrical methods, actually lead immediately to the determination of the evolute of each of the three conics; that is to say, the Cartesian equations of the evolutes can be easily deduced from the results obtained by Apollonius.

  468. Ulugh Beg biography
    • This excellent book records the main achievements which include the following: methods for giving accurate approximate solutions of cubic equations; work with the binomial theorem; Ulugh Beg's accurate tables of sines and tangents correct to eight decimal places; formulae of spherical trigonometry; and of particular importance, Ulugh Beg's Catalogue of the stars, the first comprehensive stellar catalogue since that of Ptolemy.
    • The calculation is built on an accurate determination of sin 1° which Ulugh Beg solved by showing it to be the solution of a cubic equation which he then solved by numerical methods.

  469. Frobenius biography
    • On the algebraic solution of equations, whose coefficients are rational functions of one variable.
    • The theory of linear differential equations.
    • In his work in group theory, Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups.

  470. Cholesky biography
    • To solve the condition equations in the method of least squares, Cholesky invented a very ingenious computational procedure which immediately proved extremely useful: it is now know as the Method of Cholesky and we describe it below.
    • After his death one of his fellow officers, Commandant Benoit, published Cholesky's method of computing solutions to the normal equations for some least squares data fitting problems in Note sur une methode de resolution des equations normales provenant de l'application de la methode des moindres carres a un systeme d'equations lineaires en nombre inferieure a celui des inconnues.
    • Application de la methode a la resolution d'un systeme defini d'equations lineaires (Procede du Commandant Cholesky), published in the Bulletin geodesique in 1924.
    • The beauty of the method is that it is trivial to solve equations of the type Mx = b when M is a triangular matrix.

  471. Joachimsthal biography
    • This work, which studied the integrability of differential equations with more than two variables, gave a complete answer to a question that had been considered by Lagrange.
    • Joachimsthal introduced a notation that can be used to write down the equations of tangents and polars of plane and projective conics.
    • The various notations introduced by Joachimsthal in the area of second order equations and conic sections have an influence that has extended far beyond these areas, for example into the important work of Frank Morley.

  472. Vallee Poussin biography
    • Vallee Poussin's first mathematical research was on analysis, in particular concentrating on integrals and solutions of differential equations.
    • One of his first papers in 1892 on differential equations was awarded a prize by the Belgium Academy.
    • Volume 2 covered multiple integrals, differential equations, and differential geometry.

  473. Gutzmer biography
    • He submitted his theses Uber gewisse partielle Differentialgleichungen hoherer Ordnung (On certain partial differential equations of higher order) to the University of Halle-Wittenberg and was awarded his doctorate on 13 January 1893.
    • He submitted his thesis Zur Theorie der adjungierten Differentialgleichungen (On the theory of adjoint differential equations) to the University of Halle-Wittenberg on 23 April 1896 and worked there as a privatdozent until March 1899.
    • Among the advanced courses he taught we list: ordinary differential equations, analytic mechanics, calculus of variations, number theory, higher algebra, function theory and the theory of algebraic curves.

  474. Zaremba biography
    • Much of Zaremba's research work was in partial differential equations and potential theory.
    • He studied elliptic equations and in particular contributed to the Dirichlet principle.
    • And as for my speciality, why, how could I forget the splendid results in the domain of mixed boundary problems and of harmonic functions, as well as of hyperbolic equations, research by means of which he opened a new path along which contemporary knowledge will proceed in the near future.

  475. Germain biography
    • Lagrange, who was one of the judges in the contest, corrected the errors in Germain's calculations and came up with an equation that he believed might describe Chladni's patterns.
    • She demonstrated that Lagrange's equation did yield Chladni's patterns in several cases, but could not give a satisfactory derivation of Lagrange's equation from physical principles.

  476. Fox biography
    • He submitted two further papers in June 1925, The Expression of Hypergeometric Series in Terms of Similar Series, and Some Further Contributions to the Theory of Null Series and Their Connexion with Null Integrals to the same Proceedings; both were published in 1927 as was his next paper A Generalization of an Integral Equation Due to Bateman which he submitted in 1926.
    • Fox's main contributions were on hypergeometric functions, integral transforms, integral equations, the theory of statistical distributions, and the mathematics of navigation.

  477. Smith Karen biography
    • A parabola, defined by the polynomial equation y = x2, is a familiar example of an algebraic variety.
    • In general, algebraic varieties are defined by many equations in many unknowns, and can be quite complicated.

  478. Thymaridas biography
    • Thymaridas also gave methods for solving simultaneous linear equations which became known as the 'flower of Thymaridas'.
    • For the n equations in n unknowns .
    • He also shows how certain other types of equations can be put into this form.

  479. Donaldson biography
    • Moreover the methods are new and extremely subtle, using difficult nonlinear partial differential equations.
    • His methods have been described as extremely subtle, using difficult nonlinear partial differential equations.
    • Using instantons, solutions to the equations of Yang-Mills gauge theory, he gained important insight into the structure of closed four-manifolds.

  480. Andreotti biography
    • Perhaps his most famous results are his proof of the theorem of Leonida Tonelli (1958), his proof of the duality of Picard and Albanese varieties of algebraic surfaces, his work with A L Mayer on the Schottky problem (1967), and the Andreotti-Vesentini separation theorem which appeared in their joint 1965 paper Carleman estimates for the Laplace-Beltrami equation on complex manifolds.
    • The author extends various classical results of the theory of Cauchy-Riemann equations to general complexes of linear partial differential operators.

  481. Ayyangar biography
    • His papers include: Ancient Hindu Mathematics (1921); The Hindu sine Table (1923-24); The mathematics of Aryabhata (1926); The Hindu Arabic numerals (2 parts) (1928,1929); Bhaskara and samclishta kuttaka (1929-30); New light on Bhaskara's chakravala or cyclic method of solving indeterminate equations of the second degree in two variables (1929-30); New proofs of old theorems - Apollonius and Brahmagupta (1920-30); Astronomy - past and present (1930); Some glimpses of ancient Hindu mathematics (1933); Fourteen calendars (1937); A new continued fraction (1937-38); The Bhakshali manuscript (1939); Theory of the nearest square continued fraction (2 parts) (1940, 1941); Peeps into India's mathematical past (1945); and Remarks on Bhaskara's approximation to the sine of an angle (1950).
    • He read the paper On the Sexi-Sectional Equation at a meeting of the Society on Friday 7 November 1924.

  482. Cantor biography
    • the numbers which are roots of polynomial equations with integer coefficients, were countable.
    • A transcendental number is an irrational number that is not a root of any polynomial equation with integer coefficients.

  483. Laurent Hermann biography
    • He wrote 30 books and a fair number of papers on infinite series, equations, differential equations and geometry.
    • The last three volumes are devoted entirely to the solution and application of ordinary and partial differential equations.

  484. Klugel biography
    • The reason for this is that analysis expresses the connection of the quantities by equations, and that it uses the general properties of the equations, as well as the rules for connecting them, to give the value of each quantity by those belonging together with it, or to develop their relations.
    • While the synthetic method avails itself of such propositions which state an equality, it does not use algebraic equations.

  485. Keldysh Mstislav biography
    • Mstislav Keldysh, was also a very talented mathematician in the theory of functions of a complex variable and in differential equations.
    • Examples of papers he published during this period illustrate the work he undertook at the Steklov Institute: Sur l'approximation en moyenne par polynomes des fonctions d'une variable complexe (1945), Sur l'interpolation des fonctions entieres (1947), On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations (Russian) (1951), and On a Tauberian theorem (Russian) (1951).
    • For instance a paper he published in 1951 on boundary value-problems for elliptic equations that degenerate on the boundary, that attracted much attention from Russian and foreign mathematicians had its origins in work he was doing in aerodynamics.

  486. Sokhotsky biography
    • His doctoral dissertation On definite integrals and functions with applications to expansion of series was an early investigation of the theory of singular integral equations.
    • One of the first to approach problems of the theory of singular integral equations, Sokhotsky in this work considered important boundary properties of integrals of the type of Cauchy and, essentially, arrived at the so-called formulas of I Plemelj (1908).
    • His work is important in the development of the theory of functions, in particular having applications in the theory of hypergeometric series and differential equations.

  487. Scheffers biography
    • In 1891, he published Vorlesungenuber Differentialgleichungen mit bekannten infinitesimalen Transformationen (Lectures on Differential Equations with Known Infinitesimal Transformations) and, two years later Vorlesungenuber continuierliche Gruppen mit geometrischen und anderen Anwendungen (Lectures on Continuous Groups with Geometric and Other Applications).
    • A new chapter on implicit functions, with a thorough discussion of the functional determinant and of the independence of functions and of equations has been inserted.
    • He is not therefore concerned with the equations of the projections, and though a certain amount of mathematics is necessary for his purpose, yet he concerns himself as far as possible with the geometrical constructions of the parallels and meridians, rather than with the trigonometrical calculations of their positions and sizes.

  488. Wedderburn biography
    • He began mathematical research while still an undergraduate and his first paper, On the isoclinal lines of a differential equation of the first order was published in the Proceedings of The Royal Society of Edinburgh in 1903.
    • Two other papers which he published in the same year in publications of the Royal Society of Edinburgh were on the scalar functions of a vector and on an application of quaternions to differential equations.

  489. Ostrogradski biography
    • This is an important work in the theory of partial differential equations and was reprinted in Crelle's Journal in 1836 and an English translation was made by Todhunter and published in 1861.
    • His important work on ordinary differential equations considered methods of solution of non-linear equations which involved power series expansions in a parameter alpha.
    • In addition to his important contributions to partial differential equations, he made significant advances to the theory of elasticity and to algebra publishing over 80 reports and giving lectures.

  490. Mei Wending biography
    • Mei's first mathematical work was the Fangcheng lun (On simultaneous linear equations) which he wrote in 1672.
    • Indeed, Western missionaries who went to China in the 16th and 17th centuries did not mention simultaneous linear equations because the subject was then only in its infancy in the West.
    • Mei Wending clearly wished to demonstrate the superiority of early Chinese mathematics over the methods Western scholars had brought to China, and at least in this case, the example of simultaneous linear equations was an excellent one to stress.

  491. Levin biography
    • Morduhai-Boltovskoi had Levin undertaking research from his second year of study, proposing a problem to him about generalising the functional equation for the Euler G-function.
    • He was already publishing papers and, in addition to the one we mentioned above, he had published The arithmetic properties of holomorphic functions (1933), The intersection of algebraic curves (1934), Entire functions of irregular growth (1936), and The growth of the Sturm-Liouville integral equation (1936).
    • Of his results on the spectral theory of differential operators we shall mention only the construction, dating from the 50's, of the operator "attached to infinity" of the transformation for the Schrodinger equation, which played an important part in the solution of the inverse problem in the theory of scattering.

  492. Lame biography
    • He used them to transform Laplace's equation into ellipsoidal coordinates and so separate the variables and solve the resulting equation.
    • This happened with curvilinear coordinates for he was led to study the equation .

  493. Privalov biography
    • Of the mathematicians, Konstantin Alekseevich Andreev was best known for his work on geometry and was Dean of the Faculty during Privalov's undergraduate years, Dimitri Fedorovich Egorov was a leading researcher in differential geometry and integral equations, Leonid Kuzmich Lakhtin was interested in analysis and probability, and Boleslav Kornelievich Mlodzeevskii had been the first to give lectures at Moscow University on set theory and the theory of functions.
    • He graduated from the University of Moscow in 1913 after being examined on his paper The reducibility problem in the theory of linear differential equations.
    • Later textbook were: Fourier series (1930); Course of differential calculus (1934); Course of integral calculus (1934); Integral equations (1935); Foundation of the analysis of infinitesimals, textbook for self-education (1935); and Elements of the theory of elliptic functions (1939).

  494. Tauber biography
    • Of lesser importance is Tauber's work on differential equations and the gamma function, but let us give the title of one of his papers on this latter topic, namely uber die unvollstandigen Gammafunktionen (1906).
    • In particular his papers Uber die Hypothekenversicherung (1897) and Gutachten fur die sechste internationale Tagung der Versicherungswissenschaften (1909) contain his formulation of the Tisiko equation.

  495. Wiman biography
    • One of the topics that Wiman studied was the solubility of algebraic equations.
    • For example, in Uber die metacyklischen Gleichungen von Primzahlgrad published in the volume of Acta Mathematica dedicated to the memory of Niels Abel, he studied the Galois group of soluble equations of prime degree.
    • For example, Wiman-Valiron theory, Wiman-Valiron discs, Wiman theorems for quasiregular mappings, Wiman surfaces, the Wiman inequality, Wiman's sextic, the Wiman-Valiron method for difference equations, the conjecture of Wiman, and the Wiman bound.

  496. Goldbach biography
    • He also studied equations and worked out in his correspondence with Euler how to provide a quick test for whether an algebraic equation has a rational root.

  497. Sundman biography
    • To regularize the singularity of the differential equations of motion, in the 1912 paper mentioned above, Sundman introduced a new independent variable which regularizes the motion within a band of finite breadth.
    • It is a matter of construction a machine for solving systems of second order differential equations.
    • Although designed to calculate astronomical perturbations, the machine essentially functions as an integrator for differential equations and could be used for a large number of other problems.

  498. Baudhayana biography
    • The Sulbasutra of Baudhayana contains geometric solutions (but not algebraic ones) of a linear equation in a single unknown.
    • Quadratic equations of the forms ax2 = c and ax2 + bx = c appear.

  499. Barnes biography
    • His early work was concerned with various aspects of the gamma function, including generalisations of this function given by the so-called Barnes G-function, which satisfies the equation .
    • He also considered second-order linear difference equations connected with the .

  500. Bolza biography
    • While at Clark, Bolza published the important paper On the theory of substitution groups and its application to algebraic equations in the American Journal of Mathematics.
    • Immediately after his return to Germany Bolza continued teaching and research, in particular on function theory, integral equations and the calculus of variations.
    • Bolza returned to Chicago for part of 1913 giving lecturers during the summer on function theory and integral equations.

  501. Kaczmarz biography
    • On 13 October 1924, Kaczmarz was awarded his doctorate for his thesis The relationships between certain functional and differential equations.
    • He published results from his thesis in his first paper Sur l'equation fonctionnelle f (x) + f (x + y) = φ(y) f (x + y/2) which appeared in Fundamenta Mathematicae in 1924.
    • From 1923 to 1939 Kaczmarz taught many university level courses at Lwow such as: Analytical Geometry, Higher Analysis, Integral Equations, Algebraic Curves, Trigonometric Series, Non-Euclidean Geometry and the Theory of Groups, and Differential Geometry.
    • There is Kaczmarz's algorithm for the approximate solution for systems of linear equations which appears in his paper Angenaherte Auflosung von Systemen linearer Gleichungen published in the Bulletin International de l'Academie Polonaise des Sciences et des Lettres in 1937.

  502. Chebyshev biography
    • Chebyshev submitted a paper on The calculation of roots of equations in which he solved the equation y = f (x) by using a series expansion for the inverse function of f.

  503. Kline biography
    • His research publications during his first years as director of the Division of Electromagnetic Research, now in applied areas, included: Some Bessel equations and their application to guide and cavity theory (1948); A Bessel function expansion (1950); An asymptotic solution of Maxwell's equations (1950); and An asymptotic solution of linear second-order hyperbolic differential equations (1952).

  504. Maxwell biography
    • Maxwell showed that a few relatively simple mathematical equations could express the behaviour of electric and magnetic fields and their interrelation.
    • The four partial differential equations, now known as Maxwell's equations, first appeared in fully developed form in Electricity and Magnetism (1873).

  505. Mahavira biography
    • He also discussed integer solutions of first degree indeterminate equation by a method called kuttaka.
    • An example of a problem given in the Ganita Sara Samgraha which leads to indeterminate linear equations is the following: .

  506. Mollweide biography
    • The second piece of work to which Mollweide's name is attached today is the Mollweide equations which are sometimes called Mollweide's formulas.
    • One of the more puzzling aspects is why these equations should have become known as the Mollweide equations since in the 1808 paper in which they appear Mollweide refers the book by Antonio Cagnoli (1743-1816) Traite de Trigonometrie Rectiligne et Spherique, Contenant des Methodes et des Formules Nouvelles, avec des Applications a la Plupart des Problemes de l'astronomie (1786) which contains the formulas.

  507. Boersma biography
    • The discussion centres around the use of integral representation theory to reduce such problems to Fredholm integral equations which are suitable for the study of low frequency oscillations.
    • The results of the airfoil analysis are infinite systems of linear equations, from which numerical results can be obtained by truncation.
    • Complex Function Theory, Applied Analysis and Partial Differential Equations which provided the interesting combination of mathematical theory applied to physics problems.

  508. Codazzi biography
    • The formulas give two relations between the first and second quadratic forms over a surface together with an equation, already found by Gauss, which gives necessary and sufficient conditions for the existence of a surface which admits two given quadratic forms.
    • 6 (2) (1979), 137-163.',3)">3] shows that, in 1853, Karl M Peterson, then a student of Minding at the University of Dorpat (now named Tartu), submitted a dissertation containing a derivation of two equations equivalent to those of Mainardi and Codazzi and outlining a proof of the fundamental theorem of surface theory.

  509. Koopmans biography
    • He showed that the desired result is obtainable by the straightforward solution of a system of equations involving the costs of the materials at their sources and the costs of shipping them by alternative routes.
    • He also devised a general mathematical model of the problem that led to the necessary equations.
    • In Identification problems in economic model construction (1949) he used matrix methods to study structural equations within a linear economic model.

  510. Ferrers biography
    • They range over such subjects as quadriplanar co-ordinates, Lagrange's equations and hydrodynamics.
    • In 1853 Sylvester published On Mr Cayley's impromptu demonstration of the rule for determining at sight the degree of any symmetrical function of the roots of an equation expressed in terms of the coefficients in the Philosophical Magazine.

  511. Upton biography
    • Whatever he did and worked on was executed in a purely mathematical manner and any Wrangler at Cambridge would have been delighted to see him juggle with integral and differential equations with a dexterity that was surprising.
    • He drew the shape of the bulb exactly on paper, and got the equation of its lines with which he was going to calculate its contents, when Mr Edison again appeared and asked him what it was.

  512. Narayana biography
    • He did this by using an indeterminate equation of the second order, Nx2 + 1 = y2, where N is the number whose square root is to be calculated.
    • If x and y are a pair of roots of this equation with x < y then √N is approximately equal to y/x.
    • History Topics: Pell's equation .

  513. Sommerfeld biography
    • In this thesis he studied the representation of arbitrary functions by the eigenfunctions of partial differential equations and other given sets of functions.
    • His work on this topic contains important theory of partial differential equations.
    • He lectured on a wide range of topics, giving lectures on probability and also on the partial differential equations of physics.

  514. Boole biography
    • Boole had begun to correspond with De Morgan in 1842 and when in the following year he wrote a paper On a general method of analysis applying algebraic methods to the solution of differential equations he sent it to De Morgan for comments.
    • Boole also worked on differential equations, the influential Treatise on Differential Equations appeared in 1859, the calculus of finite differences, Treatise on the Calculus of Finite Differences (1860), and general methods in probability.

  515. Bloch biography
    • While other patients constantly requested that they be given their freedom, he was perfectly happy to study his equations and keep his correspondence up to date.
    • Bloch worked on a large range of mathematical topics; for example, function theory, geometry, number theory, algebraic equations and kinematics.
    • He published articles such as: Sur les integrales de Fresnel (1919), Memoire d'analyse diophantienne lineaire (1922), Les proprietes diametrales des coniques deduites de la definition focale (1924), Les theoremes de M Valiron sur les fonctions entieres et la theorie de l'uniformisation (1925), Les fonctions holomorphes et meromorphes dans le cercle-unite (1926), Le probleme de la cubique lacunaire (1927), and Racines multiples des systemes de m equations a m inconnues (1927).
    • He wrote two papers in collaboration with Polya, namely On the roots of certain algebraic equations (1932), and Abschatzung des Betrages einer Determinante (1933).

  516. Varga biography
    • I took the chance and joined the staff at Bettis in June, 1954, immediately after getting my Ph.D., and it was a truly exciting experience for me! The problems to be solved on the computers were two- and three-dimensional multigroup diffusion equations, used to design nuclear reactors for submarines, and aircraft carriers, for example, as well as large land-based electric power generators.
    • 'Matrix Iterative Analysis' belongs in the personal library of every numerical analyst interested in either the practical or theoretical aspects of the numerical solution of partial differential equations.
    • The iterative procedures for solving finite-difference approximations to second-order partial differential equations of elliptic and parabolic types have been described in many places.

  517. Praeger biography
    • Bernhard Neumann suggested a problem to her which she solved and so published her first paper Note on a functional equation while still an undergraduate.
    • Praeger had studied the functional equation x(n+1) - x(n) = x2(n), where x2(n) = x(x(n)) and x is an integer-valued function of the integer variable n, and found a three-parameter family of solutions.
    • In fact Praeger wrote one joint paper with her husband, Note on primitive permutation groups and a Diophantine equation, which was published by the journal Discrete Mathematics in 1980.

  518. Raphson biography
    • His election to that Society was on the strength of his book Analysis aequationum universalis which was published in 1690 contained the Newton method for approximating the roots of an equation.
    • Newton-Raphson method of solving equations .

  519. Mikusinski biography
    • thesis Sur un probleme d'interpolation pour les integrales des equations differentielles lineaires and, after defending it, was awarded the degree on 25 July 1945.
    • The Mikusinski operational calculus was successfully used in ordinary differential equations, integral equations, partial differential equations and in the theory of special functions.

  520. Simpson biography
    • By way of compensation, however, the Newton-Raphson method for solving the equation f (x) = 0 is, in its present form, due to Simpson.
    • Newton described an algebraic process for solving polynomial equations which Raphson later improved.

  521. Poretsky biography
    • the theoretical portion of [Poretsky's thesis] dealt with reducing the number of unknowns and equations for certain systems of cyclic equations that occur in practical astronomy.
    • He published major works on methods of solution of logical equations, and on the reverse mode of mathematical logic.

  522. Christoffel biography
    • Christoffel published papers on function theory including conformal mappings, geometry and tensor analysis, Riemann's o-function, the theory of invariants, orthogonal polynomials and continued fractions, differential equations and potential theory, light, and shock waves.
    • How does one compare someone who worked solely in one area with another who contributed to many areas? Again how does one compare someone who worked on differential equations with a geometer? Despite the obvious difficulties, and minor differences of opinion, it is still surprising how much agreement there is on such a ranking.
    • It is difficult to compare a differential geometer with a function theorist, or those working on ordinary and partial differential equations with numerical analysts.

  523. Clebsch biography
    • Pure mathematics became Clebsch's main research topic when he began to study the calculus of variations and partial differential equations.
    • Clebsch, by taking as his starting-point an algebraic curve defined by its equation, made the theory more accessible to the mathematicians of his time, and added a more concrete interest to it by the geometrical theorems that he deduced from the theory of Abelian functions.

  524. Antonelli biography
    • However the war had ended before the machine came into service but it was still used for the numerical solution of differential equations as intended.
    • Petzinger, in [Wall Street Journal (November 1996).',3)">3], describes the way that McNulty used ENIAC to solve differential equations after the construction of the machine was complete in February 1946:- .
    • The first task was breaking down complex differential equations into the smallest possible steps.

  525. Friedrichs biography
    • He collaborated with Lewy on linear hyperbolic partial differential equations and they wrote a joint paper in 1927, and another joint paper, with Courant and Lewy, considered the stability of difference schemes for partial differential equations.
    • He was now interested in operator on Hilbert spaces and applied these tools to initial value problem for hyperbolic equations.

  526. Titchmarsh biography
    • Other topics to which he made major contributions included entire functions of a complex variable and, working with Hardy, integral equations.
    • From 1939 Titchmarsh concentrated on the theory of series expansions of eigenfunctions of differential equations, work which helped to resolve problems in quantum mechanics.
    • His work on this topic occupied him for the last 25 years of his life and he published much of it in Eigenfunction Expansions Associated with Second-Order Differential Equations (1946, 1958).

  527. De Giorgi biography
    • In 1955 De Giorgi gave an important example which showed nonuniqueness for solutions of the Cauchy problem for partial differential equations of parabolic type whose coefficents satisfy certain regularity conditions.
    • In the following year he proved what has become known as "De Giorgi's Theorem" concerning the Holder continuity of solutions of elliptic partial differential equations of second order.
    • The authors of this paper are all students of De Giorgi and they describe his contributions to geometric measure theory, the solution of Hilbert's XIXth problem in any dimension, the solution of the n-dimensional Plateau problem, the solution of the n-dimensional Bernstein problem, some results on partial differential equations in Gevrey spaces, convergence problems for functionals and operators, free boundary problems, semicontinuity and relaxation problems, minimum problems with free discontinuity set, and motion by mean curvature.

  528. Kontsevich biography
    • A striking consequence is that it satisfies an infinite integrable hierarchy of Korteweg-de Vries equations completed by a so-called "string equation".

  529. Tarski biography
    • In 1968 Tarski wrote another famous paper Equational logic and equational theories of algebras in which he presented a survey of the metamathematics of equational logic as it then existed as well as giving some new results and some open problems.

  530. Julia biography
    • Volume 3 contains four parts: (i) Functional equations and conformal mapping; (ii) Conformal mapping; (iii) General lectures; and (iv) Isolated works in analysis on Implicit function defined by the vanishing of an active function, and on certain series.
    • Volume 4 is again in four parts: (i) Functional calculus and integral equations; (ii) Quasianalyticity; (iii) Various techniques of analysis; and (iv) Works concerning Hilbert space.
    • The applications to the theory of matrices and equations, which are largely implicit, in certain of the more abstract treatments, are elaborated here with a wealth of detail which renders them unusually accessible to the student.

  531. Montroll biography
    • He used both his expertise in chemistry and mathematics in his thesis Applications of the characteristic value theory of integral equations in which he applied integral equations to the study of imperfect gases.
    • In the proceeding of the conference Nonlinear equations in abstract spaces held in 1977 at the University of Texas he published On some mathematical models of social phenomena in which he examined models for population growth and statistical models of other social phenomena.

  532. Pogorelov biography
    • He has solved a number of key problems in geometry in the large, in the foundations of geometry, and in the theory of the Monge-Ampere equations, and he also has obtained remarkable results in the geometric theory of stability of thin elastic shells.
    • On Monge-Ampere equations of elliptic type (1960).
    • This book presents a systematic exposition of a number of publications of A D Aleksandrov and his students, dealing with Monge-Ampere equations of elliptic type.

  533. Wilczynski biography
    • By that time he had published over a dozen papers in astronomy, but his interests moved towards differential equations which arose in his study of the dynamics of astronomical objects.
    • From there his interests became pure mathematical interests in differential equations.
    • But Wilczynski was the first ever to appreciate, demonstrate and exploit the utility of completely integrable systems of linear homogeneous differential equations for projective differential geometry.

  534. FitzGerald biography
    • This was Electricity and Magnetism by Maxwell which, for the first time, contained the four partial differential equations, now known as Maxwell's equations.
    • His first work On the equations of equilibrium of an elastic surface filled in cases of a problem studied by Lagrange.

  535. Bell John biography
    • We may say that when the state-vector is α+ or α- respectively, sz is equal to /2 and -/2 respectively, but, if one restricts oneself to the Schrodinger equation, sx and sy just do not have values.
    • It is clear that we could try to recover realism and determinism if we allowed the view that the Schrodinger equation, and the wave-function or state-vector, might not contain all the information that is available about the system.
    • Nevertheless it would appear natural that the possibility of supplementing the Schrodinger equation with hidden variables would have been taken seriously.

  536. Hoyle biography
    • In 1945 he published On the integration of the equations determining the structure of a star which discussed the most advantageous way of integrating the equations of stellar equilibrium.

  537. Graham biography
    • It took a knowledge of differential equations to solve it.
    • He was a plenary speaker at the British Mathematical Colloquium in East Anglia in 1990 when he gave the lecture Arithmetic progressions: from Hilbert to Shelah and he was again a plenary speaker in 2009 in Galway when he gave the lecture The combinatorics of solving linear equations.

  538. Segre Beniamino biography
    • By 1931 when he was appointed to the Chair of Geometry at the University of Bologna he already had 40 publications in algebraic geometry, differential geometry, topology and differential equations.
    • This was a period during which he made exceptional research contributions on algebraic geometry but his interests also broadened, stimulated by discussions with Mordell and Kurt Mahler, to diophantine equations and the arithmetic of algebraic varieties.

  539. Scorza biography
    • Now, Cassel profoundly observes that there is nothing in that event that is essentially characteristic of free competition, and therefore the reasoning by which this conclusion is drawn from the presence of the equations of maximum satisfaction in the system that determine the equilibrium, is nothing other than gross sophistry.
    • The mathematician, who does not possess the general theory of what is called 'corpi numerici' knows, through algebra, the theory of equations; through number theory, the theory of congruencies with respect to a prime modulus; through the treatises on algebraic numbers, the theory of congruencies with respect to a prime ideal; three theories of which, even if he has spotted some analogy, he does not see the innermost bounds.

  540. Gronwall biography
    • In 1898, at the age 21, he was the author of ten mathematical papers and received his doctor's degree at Uppsala University for the thesis On system of linear total differential equations particularly with 2n-periodic coefficients.
    • Gronwall's work contains classical analysis (Fourier series, Gibbs phenomenon, summability theory, Laplace and Legendre series), differential and integral equations, analytic number theory (transcendental numbers, divisor function, L-function of Dirichlet), complex function theory (Dirichlet L-series, conformal mappings, univalent functions), differential geometry, mathematical physics (problems of elasticity, ballistics, induction, potential theory, kinetic theory of gases, optics), nomography, atomic physics (wave mechanics of hydrogen and helium atom, lattice theory of crystals) and physical chemistry where he is especially known as a very important contributor.

  541. Faddeeva biography
    • This is a textbook on numerical methods for solving finite systems of linear equations, inverting matrices, and calculating the eigenvalues of finite matrices, all with desk calculators.
    • The second chapter deals with numerical methods for the solution of systems of linear equations and the inversion of matrices, and the third with methods for computing characteristic roots and vectors of a matrix.

  542. Orlicz biography
    • In recent decades those spaces have been used in analysis, constructive theory of functions, differential equations, integral equations, probability, mathematical statistics, etc.

  543. Plancherel biography
    • He applied his results in the theory of hyperbolic and parabolic partial differential equations.
    • In algebra Plancherel obtained results on quadratic forms and their applications, to the solvability of systems of equations with infinitely many variables and to the theory of commutative Hilbert algebras (theorem of Plancherel-Godement).

  544. Gateaux biography
    • He recalled that Volterra introduced this notion to study problems including an hereditary phenomenon, but also that it was used by others (Jacques Hadamard and Paul Levy) to study some problems of mathematical physics - such as the equilibrium problem of fitted elastic plates - finding a solution to equations with functional derivatives, or, in other words, by calculating a relation between this functional and its derivative.
    • Though I am still mobilized, I work on lectures I should read at the College de France on the functions of lines and the equations with functional derivatives and at this occasion I would like to develop several chapters of the theory.

  545. Stevin biography
    • In the latter Stevin presented a unified treatment for solving quadratic equations and a method for finding approximate solutions to algebraic equations of all degrees.

  546. Lukacs biography
    • In 1942 Lukacs had made an important contribution to mathematical statistics by introducing, for the first time, the method of differential equations in characteristic function theory.
    • Other topics to which Lukacs made major contributions include characterisations of distributions, stability of characterisation results and functional equations.

  547. Laurent Pierre biography
    • Find the limiting equations that must be joined to the indefinite equations in order to determine completely the maxima and minima of multiple integrals.

  548. Backlund biography
    • All the candidates gave lectures and, because he was not already on the staff of a university, Bjorling was asked to defend one of his recently published papers on roots of algebraic equations.
    • It was in this new area of differential equations that Backlund produced his more notable results, namely on what are today called Backlund transformations.

  549. Wilkinson biography
    • He began to put his greatest efforts into the numerical solution of hyperbolic partial differential equations, using finite difference methods and the method of characteristics.
    • He worked on numerical methods for solving systems of linear equations and eigenvalue problems.

  550. Crighton biography
    • He gave a mathematical model in which the problem reduce to solving two singular integral equations with Cauchy-type kernels, and with variable coefficients.
    • Solving the equations he showed that he boundary converts the energy stored in the turbulent boundary layer into the sound waves which generate noise.

  551. Sharkovsky biography
    • He had already published a number of high quality papers (all in Russian), such as: Necessary and sufficient conditions for convergence of one-dimensional iterative processes (1960), Rapidly converging iterative processes (1961), Solutions of a class of functional equations (1961), and The reducibility of a continuous function of a real variable and the structure of the stationary points of the corresponding iteration process (1961).
    • He was made head of the Department of Differential Equations at the Institute of Mathematics at the Ukrainian branch of the USSR Academy of Sciences in 1974.
    • He also works in the theory of functional and functional differential equations, and the study of difference equations and their application.
    • The book, written with G P Pelyakh, Introduction to the theory of functional equations (Russian) was published in 1974.
    • In 1986, in collaboration with Yu L Maistrenko and E Yu Romanenko, Sharkovsky published the Russian monograph Difference equations and their applications.
    • The book contains four parts: (I) One-dimensional dynamical systems; (II) Difference equations with continuous time ; (III) Differential-difference equations ; and (IV) Boundary-value problems for hyperbolic systems of partial differential equations.
    • The aim of the present book is to acquaint the reader with some recently discovered and (at first sight) unusual properties of solutions to nonlinear difference equations.
    • These properties enable us to use difference equations in order to model complicated oscillating processes (this can often be done in those cases when it is difficult to apply ordinary differential equations).
    • Difference equations are also a useful tool in synergetics - an emerging science concerned with the study of ordered structures.
    • The application of these equations opens up new approaches in solving one of the central problems of modern science - the problem of turbulence.
    • This book is especially interesting for specialists in differential equations applying their results to some practical problems in the natural sciences and technology.

  552. Epstein biography
    • However Mr Christoffel only sought to study the behaviour of the integrals at infinity, and gave the important equation (7) in §2 without proof.
    • He proved a functional equation, the analytic continuation and the Kronecker limit formula for these functions.

  553. Fermat biography
    • Fermat posed further problems, namely that the sum of two cubes cannot be a cube (a special case of Fermat's Last Theorem which may indicate that by this time Fermat realised that his proof of the general result was incorrect), that there are exactly two integer solutions of x2 + 4 = y3 and that the equation x2 + 2 = y3 has only one integer solution.
    • History Topics: Pell's equation .

  554. Minding biography
    • Minding also worked on differential equations, algebraic functions, continued fractions and analytic mechanics.
    • In differential equations he used integrating factor methods.

  555. Al-Karaji biography
    • He does not treat equations above the second degree except for ones which can easily be reduced to at most second degree equations followed by the extraction of roots.

  556. Gelfand biography
    • Another important area of his work is that on differential equations where he worked on the inverse Sturm-Liouville problem.
    • He worked on computational mathematics, developing general methods for solving the equations of mathematical physics by numerical means.

  557. Hedrick biography
    • He was awarded a doctorate by Gottingen in February 1901 for a dissertation, supervised by Hilbert, Uber den analytischen Charakter der Losungen von Differentialgleichungen (On the analytic character of solutions of differential equations).
    • This strengthen his interests in differential equations, the calculus of variations, and functions of a real variable which he would work on for the rest of his life.

  558. Pacioli biography
    • During this time Pacioli worked with Scipione del Ferro and there has been much conjecture as to whether the two discussed the algebraic solution of cubic equations.
    • History Topics: Quadratic, cubic and quartic equations .

  559. Knopp biography
    • Volume 1 covers numbers, functions, limits, analytic geometry, algebra, set theory; volume 2 covers differential calculus, infinite series, elements of differential geometry and of function theory; and volume 3 covers integral calculus and its applications, function theory, differential equations.
    • Friedrich Losch added a fourth volume in 1980 to cover more modern material: set theory, Lebesgue measure and integral, topological spaces, vector spaces, functional analysis, integral equations.

  560. Adian biography
    • In his first work as a student in 1950, he proved that the graph of a function f (x) of a real variable satisfying the functional equation f (x + y) = f (x) + f (y) and having discontinuities is dense in the plane.
    • (Clearly, all continuous solutions of the equation are linear functions.) This result was not published at the time.

  561. Butzer biography
    • Moreover, it has already become indispensable in classical approximation theory, in the study of the initial and boundary behaviour of solutions of partial differential equations and in the theory of singular integrals, because of the new results obtained by the authors in these areas.
    • Special as this approach may seem, it not only embraces many of the topics of the classical theory but also leads to significant new results, e.g., on summation processes of Fourier series, conjugate functions, fractional integration and differentiation, limiting behaviour of solutions of partial differential equations, and saturation theory.

  562. Kloosterman biography
    • Kloosterman was examining the number of solutions in integers xn, to the equation .
    • He had managed to find, provided s ≥ 5 and the an satisfy suitable congruence conditions, an asymptotic formula for the number of solution to the equation (*).

  563. Patodi biography
    • His doctoral thesis, Heat equation and the index of elliptic operators, was supervised by M S Narasimhan and S Ramanan and the degree was awarded by the University of Bombay in 1971.
    • An analytic approach, via the heat equation yields easily a formula for the index of an elliptic operator on a compact manifold: but, the formula involves an integrand containing too many derivatives of the symbol, while from the Atiyah-Singer index theorem one would expect only two derivatives to figure.

  564. Kovalevskaya biography
    • The three papers were on Partial differential equations, Abelian integrals and Saturn's Rings.
    • The first of these three articles was still a valuable paper however, because it contained an exposition of Weierstrass's theory for integrating certain partial differential equations.

  565. Alexiewicz biography
    • In fact, because of the disruption caused by the war, his habilitation thesis was published before his doctoral thesis, but he had already a number of earlier publications such as: (with W Orlicz) Remarque sur l'equation fonctionelle f (x + y) = f (x) + f (y) (1945); Linear operations among bounded measurable functions I and II (1946); On Hausdorff classes (1947); On multiplication of infinite series (1948); and Linear functionals on Denjoy-integrable functions (1948).
    • Scalar and vector measurable functions; sequences of linear operators; the Denjoy integral; differentiation of vector functions; differential equations and equations with vector functions; two norm spaces and two norm algebras and their applications in summability theory; analytic functions; and applications of functional analysis to classical problems of mathematical analysis.

  566. Zorawski biography
    • He spent time in Leipzig where he studied continuous groups of transformations now called Lie groups, and Gottingen where he studied differential equations.
    • After returning to Krakow, Zorawski continued to teach courses on analytical and synthetic geometry, differential geometry, the formal theory of the differential equations, the theory of the forms, and the theory of the Lie groups.

  567. Bottasso biography
    • Torino, 1912) Bottasso underlined the analogy between vector homography and integral equations, and used vector homography to solve integral equations.

  568. Orszag biography
    • The title of this volume is somewhat misleading in that the subjects discussed are approximate analytic solutions of ordinary differential and difference equations, and no other topics are considered.
    • It is not only the authors who hope for a similar book on approximate solutions of partial differential equations.

  569. Wronski biography
    • A piece of work which he had undertaken during this period resulted in a publication Resolution generale des equations de tous degres in 1812 claiming to show that every equation had an algebraic solution.
    • For good measure, it contains a summary of the "general solution of the fifth degree equation".

  570. Wald biography
    • seasonal corrections to time series, approximate formulas for economic index numbers, indifference surfaces, the existence and uniqueness of solutions of extended forms of the Walrasian system of equations of production, the Cournot duopoly problem, and finally, in his much used work written with Mann (1943), stochastic difference equations.

  571. Fefferman biography
    • Fefferman's work on partial differential equations, Fourier analysis, in particular convergence, multipliers, divergence, singular integrals and Hardy spaces earned him a Fields Medal at the International Congress of Mathematicians at Helsinki in 1978.
    • Professor Charles Fefferman's contributions and ideas have had an impact on the development of modern analysis, differential equations, mathematical physics and geometry, with his most recent work including his sharp (computable) solution of the Whitney extension problem.

  572. Martin Lajos biography
    • Making physically and technically untenable simplifications, he put down the equation he had figured out, and although the contraptions constructed on its basis all failed at the testing, martin deemed it justified to publish his theoretical findings.
    • In a reply in a highly ironical tome [Application of the differential coefficient for solving the equation of the propeller surface (Hungarian) (1877)] he called Szily's variational method an unnecessary and erroneous extravagance.

  573. Morera biography
    • Morera studied the fundamental problems which arise in dynamics with particular regard to the use of the Pfaff method applied to Jacobian systems of partial differential equations and to the problem of Lie transformations of the canonical equations of motion.

  574. Hormander biography
    • After Marcel Riesz retired in 1952 Hormander began working on the theory of partial differential equations.
    • In 1962 the International Congress was held in Stockholm and Hormander, as well as being heavily involved in the organisation, received a Fields Medal for his work on partial differential equations.

  575. Stiefel biography
    • They feel that from the point of view of the applications to stability and vibrational questions in mechanics the variational approach is the most suitable one (as compared with the approach by differential or integral equations).
    • Whenever possible, we derive the basic differential equations or at least we interpret them.

  576. Eells biography
    • In it the whole geometry or topology of the spaces involved play a role, rather than just the equations describing the behaviour or motion in small areas.
    • This he did with "Global Analysis" in 1971-72, "Geometry of the Laplace Operator" in 1976-77, and "Partial Differential Equations in Differential Geometry", in 1989-90.

  577. Watson biography
    • Watson worked on a wide variety of topics, all within the area of complex variable theory, such as difference equations, differential equations, number theory and special functions.

  578. Woods biography
    • Woods own description of his 1953 paper The relaxation treatment of singular points in Poisson's equation states:- .
    • If F is harmonic or is a solution to Poisson's equation, it may have singular points in the field or on the boundary at which it (a) has finite values, but has infinite derivatives, (b) has logarithmic infinities, or (c) has simple discontinuities.

  579. Rado Ferenc biography
    • Treated are: Nomograms for equations with two variables, with three variables (6 types), order and class of nomograms, nomograms with several variables, projective and homographic transformation of nomograms, classification of nomograms.
    • He published papers such as the following written in Romanian: Two theorems concerning the separation of variables in nomography (1955); On rhomboidal nomograms (1956); The best projective transformation of the scales of alignment nomograms (1957); and Functional equations in connection with nomography (1958).

  580. Forsyth biography
    • Famous texts which Forsyth published before his 1893 work Theory of Functions of a complex variable , are A treatise on differential equations (1885), and Theory of differential equations published in six volumes between 1890 and 1906.

  581. Leibniz biography
    • Another major mathematical work by Leibniz was his work on determinants which arose from his developing methods to solve systems of linear equations.
    • History Topics: Quadratic, cubic and quartic equations .

  582. Toeplitz biography
    • When he arrived there Hilbert was completing his theory of integral equations.
    • A major joint project with Hellinger to write a major encyclopaedia article on integral equations, which they worked on for many years, was completed during this time and appeared in print in 1927.

  583. Shannon biography
    • At the Massachusetts Institute of Technology he also worked on the differential analyser, an early type of mechanical computer developed by Vannevar Bush for obtaining numerical solutions to ordinary differential equations.
    • The most important results [mostly given in the form of theorems with proofs] deal with conditions under which functions of one or more variables can be generated, and conditions under which ordinary differential equations can be solved.

  584. Aryabhata I biography
    • It also contains continued fractions, quadratic equations, sums of power series and a table of sines.
    • This work is the first we are aware of which examines integer solutions to equations of the form by = ax + c and by = ax - c, where a, b, c are integers.

  585. Lefschetz biography
    • He tackled problems related to dissipative nonlinear ordinary differential equations but did not take the usual approach of using linear theory to tackle nonlinear differential equations.

  586. Jonquieres biography
    • He also worked on algebra, in particular the theory of equations, and, in the latter part of his life, on the theory of numbers where he examined Diophantine equations and the distribution of primes.

  587. Aepinus biography
    • During this period he undertook research in several different areas of mathematics including algebraic equations, solving partial differential equations, and on negative numbers.

  588. Tonelli biography
    • The fourth, and final, volume Argomenti vari, published in 1963, contains papers on trigonometric series, ordinary differential equations and integral equations (all published in or after 1924-25), and some miscellaneous work (from 1909 onwards), including Tonelli's biography of Salvatore Pincherle.

  589. Gohberg biography
    • In addition to Gohberg's outstanding work in analysis and in particular in operator theory and matrix methods, he founded the major international journal Integral equations and operator theory in the late 1980s.
    • is devoted to the publication of current research in integral equations, operator theory and related topics with emphasis on the linear aspects of the theory.

  590. Dehn biography
    • I attended his course in 'Non-linear Partial Differential Equations'.
    • I attended his course in Non-linear Partial Differential Equations.

  591. Al-Mahani biography
    • However, he was led to an equation involving cubes, squares and numbers which he failed to solve after giving it lengthy meditation.
    • Therefore, this solution was declared impossible until the appearance of Ja'far al-Khazin who solved the equation with the help of conic sections.

  592. Grave biography
    • He obtained his masters degree in 1889 (equivalent to a Ph.D.) for his thesis On the Integration of Partial Differential Equations of the First Order (Russian) and, in the autumn of that year, began teaching at the University of St Petersburg.
    • In particular he worked on Galois theory, ideals and equations of the fifth degree.

  593. Morishima biography
    • This paper On the Diophantine equation xp + yp = czp published in the Proceedings of the American Mathematical Society was, as all of Morishima's work, subject to the criticism that he did not give full enough explanations.
    • The authors obtain elegant criteria generalising the classical Wieferich and Mirimanoff criteria for the first case of Fermat's equation.

  594. Fubini biography
    • In this area he worked on differential equations, analytic functions and functions of several complex variables.
    • Another analysis topic he studied was non-linear integral equations.

  595. Cipolla biography
    • The work includes the theory of abstract groups, the theory of groups of substitutions, and Galois's theory of algebraic equations.
    • Also with Vincenzo Amato (1881-1963), another of his students in Catania who studied the properties of those algebraic equations whose Galois group was the fundamental subgroup of the whole group, he wrote secondary school texts such as Algebra elementare : per il ginnasio superiore e per le classi 3 e 4 dell'istituto magistrale inferiore (1926), and Aritmetica prattica per le scuole industriale.

  596. Eutocius biography
    • 4, of the auxiliary problem amounting to the solution by means of conics of the cubic equation (a - x) x2 = b c2.
    • the solutions (a) by Diocles of the original problem of II.4 without bringing in the cubic, (b) by Dionysodorus of the auxiliary cubic equation.

  597. Maddison biography
    • When she first reached Bryn Mawr College, Maddison continued to work on this topic but later, advised by Scott, she began to work on singular solutions of differential equations.
    • in 1896 for her thesis On Singular Solutions of Differential Equations of the First Order in Two Variables and the Geometrical Properties of Certain Invariants and Covariants of Their Complete Primitives and in the same year appointed as Reader in Mathematics at Bryn Mawr.

  598. Branges biography
    • The classical ingredients of the proof, the Loewner differential equation and the inequalities conjectured by Robertson and Milin, as well as the Askey-Gasper inequalities from the theory of special functions, are clearly described in the volume 'The Bieberbach Conjecture' (published by the American Mathematical Society).
    • The key was to find norms for which the necessary inequalities could be propagated by the Loewner equation.

  599. Lerch biography
    • He also studied elliptic functions and integral equations.
    • He is remembered today for his solution of integral equations in operator calculus and for the 'Lerch formula' for the derivative of Kummer's trigonometric expansion for log G(v).

  600. Laguerre biography
    • Laguerre studied approximation methods and is best remembered for the special functions the Laguerre polynomials which are solutions of the Laguerre differential equations.
    • This memoir of Laguerre is significant not only because of the discovery of the Laguerre equations and polynomials and their properties, but also because it contains one of the earliest infinite continued fractions which was known to be convergent.

  601. Boutroux biography
    • There he lectured at the College de France on functions which are the solutions of first order differential equations.
    • He worked on multiform functions and also continued Painleve's work on singularities of differential equations.

  602. Bjerknes Vilhelm biography
    • Vilhelm Bjerknes and his associates at Bergen succeeded in devising equations relating the measurable components of weather, but their complexity precluded the rapid solutions needed for forecasting.
    • The next step forward in the mathematical approach was due to Richardson in 1922 when he reduced the complicated equations produced by Bjerknes's Bergen School to long series of simple arithmetic operations.

  603. Bourgain biography
    • Bourgain's work touches on several central topics of mathematical analysis: the geometry of Banach spaces, convexity in high dimensions, harmonic analysis, ergodic theory, and finally, nonlinear partial differential equations from mathematical physics.
    • ',2)">2] contains a survey relating to Bourgain's work on nonlinear partial differential equations from mathematical physics, including later results than was covered in the articles describing his work up to the award of the Fields Medal.

  604. Girard Pierre biography
    • one may learn to find the equation for some solid as one finds the equation for a curved plane.

  605. Coulson biography
    • In almost every case the fundamental problem is the same, since it consists in solving the standard equation of wave motion; the various applications differ chiefly in the conditions imposed upon these solutions.
    • Let us indicate the contents of the 156 page book: The equation of wave motion; Waves on strings; Waves in membranes; Longitudinal waves in bars and springs; Waves in liquids; Sound waves; Electric waves; General considerations.

  606. Mauchly biography
    • In particular the School used a Bush analyser, designed by Vannevar Bush specifically to integrate systems of ordinary differential equations.
    • Von Neumann was working on this project and became involved with the ENIAC computer and used it to solve systems of partial differential equations which were crucial in the work on atomic weapons at Los Almos.

  607. Thiele biography
    • One of his most important contributions to actuarial science was a differential equation for the net premium reserve Vt at time t for a life insurance, namely .
    • Although, as we have said, this differential equation is Thiele's most significant contribution to actuarial science, he never published the result.

  608. Ehrenfest biography
    • In 1917 and 1920 Ehrenfest published papers investigating the problem of the extent to which the three-dimensional nature of physical space is determined by the structure of basic physical equations or is reflected in these basic equations.

  609. Pontryagin biography
    • He began to study applied mathematics problems, in particular studying differential equations and control theory.
    • Another book by Pontryagin Ordinary differential equations appeared in English translation, also in 1962.

  610. Libri biography
    • A little-known consequence of these disputes is that Liouville made his famous announcement of Evariste Galois's important work on the theory of equations in response to an attack by Libri in 1843.
    • However he made many contributions to number theory and to the theory of equations.

  611. Vitali biography
    • His thesis, Le equazioni di Appell del 2o ordine e le loro equazioni integrali (Appell's 2nd order equations and their integral equations) was published in 1902 in a shortened version and as a full version in the following year.

  612. Haar biography
    • He examined the standard systems of orthonormal trigonometric functions and also orthonormal systems related to Sturm-Liouville differential equations.
    • After the work of his thesis, which we gave some details of above, he went on to study partial differential equations with applications to elasticity theory.

  613. Nekrasov biography
    • He also investigated mathematical questions which were related to these applications, in particular writing important works on non-linear integral equations.
    • In the same year another important work on the applications of integral equations to aerodynamics was published.

  614. Remak biography
    • these equations are very awkward to handle mathematically.
    • There is, however, work in progress concerning the numerical solution of linear equations with several unknowns using electrical circuits.

  615. Grosswald biography
    • For the second edition of the text published in 1984, Grosswald had added material on L-functions and primes in arithmetic progressions, the arithmetic of number fields, and Diophantine equations.
    • In Bessel polynomials Grosswald studies: the relationship between Bessel functions and Bessel polynomials, differential equations and differential recurrence relations satisfied by the generalized Bessel polynomials, recurrence relations satisfied by the generalized Bessel polynomials, orthogonality properties of the generalized Bessel polynomials and the corresponding moment problem, relations of the generalized Bessel polynomials and the classical orthogonal polynomials, generating functions, Rodrigues-type formulas, the generalized Bessel polynomials and continued fractions, the zeros of the Bessel polynomials, algebraic properties of the Bessel polynomials, and the Galois group of the Bessel polynomials.

  616. Einstein biography
    • This seemed to contradict classical electromagnetic theory, based on Maxwell's equations and the laws of thermodynamics which assumed that electromagnetic energy consisted of waves which could contain any small amount of energy.
    • In fact Hilbert submitted for publication, a week before Einstein completed his work, a paper which contains the correct field equations of general relativity.

  617. Levy Paul biography
    • Not only did Levy contribute to probability and functional analysis but he also worked on partial differential equations and series.
    • He undertook a large-scale work on generalised differential equations in functional derivatives.

  618. Radon biography
    • In mathematics he took lecture courses by Hans Hahn (one on Theoretical arithmetic and one on the Foundations of geometry), Wilhelm Wirtinger (Ordinary differential equations) and Franz Mertens (one on Algebra and one on Number theory) among others.
    • While creating a theory of absolutely additive set functions which, heretofore, has barely been investigated, the author succeeds with the development of a theory that contains the theory of integral equations, linear and bilinear forms in infinitely many variables, as a special case.

  619. Cauer biography
    • Interested in using computers to solve systems of linear equations, he contacted Richard Courant at Gottingen and Vannevar Bush who was developing mechanical computers at the Massachusetts Institute of Technology.
    • Back in Gottingen he wanted to build a calculating machine to solve linear equations but, despite having progressed well with the project, this was a time when funding was impossible due to the Depression so it could not proceed.

  620. Chuquet biography
    • The sections on equations cover quadratic equations where he discusses two solutions.

  621. Beltrami biography
    • Some of Beltrami's last work was on a mechanical interpretation of Maxwell's equations.
    • (dated December, 1888) is devoted to the mechanical interpretation of Maxwell's equations.

  622. Kutta biography
    • It contains the now famous Runge-Kutta method for solving ordinary differential equations.
    • The former contains the Runge-Kutta method for solving ordinary differential equations while the latter contains the Zhukovsky- Kutta (Joukowski -Kutta) theorem giving the lift on an aerofoil.

  623. Fontaine des Bertins biography
    • His papers are rather confused, and ignorant of the work of others, but do contain some very original ideas in the calculus of variations, differential equations and the theory of equations.

  624. Bohl biography
    • He graduated in 1887 with a degree in mathematics having won a Gold Medal for an essay he wrote on The Theory of Invariants of Linear Differential Equations in 1886.
    • Bohl's doctoral dissertation applied topological methods to systems of differential equations.

  625. Wangerin biography
    • He taught many courses at the University of Halle including: linear partial differential equations; calculus of variations; theory of elliptical functions; synthetic geometry; hydrostatics and capillarity theory; theory of space curves and surfaces; analytic mechanics; potential theory and spherical harmonics; celestial mechanics; the theory of the map projections; hydrodynamics; and the partial differential equations of mathematical physics.

  626. Newson biography
    • She spoke to Klein while he was in the United States and he tested her to see if her understanding of mathematics, in particular of differential equations, was good enough to profit from doctoral studies.
    • She completed her thesis Uber den Hermiteschen Fall der Lameschen Differentialgleichungen (On the Hermite case of the Lame differential equations) in the summer of 1896 and was examined in July 1896.

  627. Cartwright biography
    • The Radio Research Board of the Department of Scientific and Industrial Research produced a memorandum regarding certain differential equations which came out of modelling radio and radar work.
    • They began to collaborate studying the equations.

  628. Dirichlet biography
    • In 1852 he studied the problem of a sphere placed in an incompressible fluid, in the course of this investigation becoming the first person to integrate the hydrodynamic equations exactly.
    • These series had been used previously by Fourier in solving differential equations.

  629. Taussky-Todd biography
    • While in Gottingen Taussky also edited Artin's lectures in class field theory (1932), assisted Emmy Noether in her class field theory and Courant with his differential equations course.
    • For the first time I realised the beauty of research on differential equations - something that my former boss, Professor Courant, had not been able to instil in me.

  630. Kramp biography
    • He published his memoir on double refraction Sur la double refraction de la chaux carbonatee in 1811 and, in 1820, he published Equations des nombres which contains a new approximate method to solve numerical equations.

  631. Scheffe biography
    • He wrote a doctoral thesis on differential equations and was awarded his PhD in 1935.
    • Scheffe's doctoral dissertation The Asymptotic Solutions of Certain Linear Differential Equations in Which the Coefficient of the Parameter May Have a Zero was supervised by Rudolph E Langer.

  632. Skolem biography
    • It was entitled Einige Satze uber ganzzahlige Losungen gewisser Gleichungen und Ungleichungen, and was on integral solutions of certain algebraic equations and inequalities.
    • Skolem was remarkably productive publishing around 180 papers on topics such as Diophantine equations, mathematical logic, group theory, lattice theory and set theory.

  633. Francais Francois biography
    • Francois worked on partial differential equations and his memoir of 1795 on this topic was developed further and presented to the Academie des Sciences in 1797.
    • Lacroix praised Francais' work and described it as making a major contribution to the study of partial differential equations; however, it was not published.

  634. Bortolotti biography
    • In the 1940 paper on Babylonian mathematics, Bortolotti gives a summary of problems published by Neugebauer but argues that the fact that large series of examples for quadratic equations are made up from the same roots demonstrates that this pair of roots has an 'arcane mystic property'.
    • It is also wrong to deny the existence of approximations to irrational square roots, to assume a geometrical basis of the quadratic equations or to deny the existence of texts of this type in the Hellenistic period.

  635. McCowan biography
    • A regular attendee at meetings of the Edinburgh Mathematical Society, he presented the papers: On a representation of elliptic integrals by curvilinear arcs (12 June 1891); On the solution of non-linear partial differential equations of the second order (13 May 1892); and Note on the solution of partial differential equations by the method of reciprocation (11 November 1892).

  636. Nikodym biography
    • the Radon-Nikodym theorem and derivative, the Nikodym convergence theorem, the Nikodym-Grothendieck boundedness theorem), in functional analysis (the Radon-Nikodym property of a Banach space, the Frechet-Nikodym metric space, a Nikodym set), projections onto convex sets with applications to Dirichlet problem, generalized solutions of differential equations, descriptive set theory and the foundations of quantum mechanics.
    • Some of his other books were: Introduction to differential calculus, (Warsaw, 1936) (jointly with his wife), Theory of tensors with applications to geometry and mathematical physics, I, (Warsaw, 1938), Differential Equations, (Poznan, 1949).

  637. Quillen biography
    • for a thesis on partial differential equations in 1964 entitled Formal Properties of Over-Determined Systems of Linear Partial Differential Equations.

  638. Cramer Harald biography
    • One interesting paper by Cramer over this period which we should note is one he published in 1920 discussing prime number solutions x, y to the equation ax + by = c, where a, b, c are fixed integers.
    • Note that if a = b = 1 then the question of whether this equation has a solution for all c is Goldbach's conjecture, while if a = 1, b = -1, c = 2, then the question about prime solutions to x = y + 2 is the twin prime conjecture.

  639. Mansion biography
    • The Royal Belgium Academy of Science proposed for its prize competition for 1871 the task "to summarise and simplify the theory of partial differential equations of the first two orders".
    • This was a vast and difficult undertaking and Mansion decided to enter but to restrict himself to the theory of first order partial differential equations.
    • Mansion submitted the 289-page memoir Memoire sur la theorie des equations aux derivees partielles du premier ordre which was judged the winning entry.

  640. MacRobert biography
    • Its special features are an emphasis on geometrical methods, extensive discussion of special functions and second-order differential equations, and a profusion of illustrative examples.
    • It is a very useful text-book on special functions, and an introduction to their application to partial differential equations of mathematical physics.

  641. Federer biography
    • He was building on work by Lamberto Cesari who had studied surfaces given by parametric equations, in particular the Lebesgue area of such a surface, while at Pisa from 1938 to 1942.
    • It has depth and beauty of its own, but its greatest worth should be in its effect on other areas of mathematics, e.g., differential geometry, differential topology, partial differential equations, algebraic geometry, potential theory.

  642. Mercer biography
    • Mercer was a mathematical analyst of originality and skill; he made noteworthy advances in the theory of integral equations, and especially in the theory of the expansion of arbitrary functions in series of orthogonal functions.
    • Mercer's theorem about the uniform convergence of eigenfunction expansions for kernels of operators appears in his 1909 paper Functions of positive and negative types and their connection with the theory of integral equations published in the Philosophical Transactions.

  643. Bombieri biography
    • The award was made for his major contributions to the study of the prime numbers, to the study of univalent functions and the local Bieberbach conjecture, to the theory of functions of several complex variables, and to the theory of partial differential equations and minimal surfaces.
    • He has significantly influenced number theory, algebraic geometry, partial differential equations, several complex variables, and the theory of finite groups.

  644. Householder biography
    • He started publishing on this new topic with Some numerical methods for solving systems of linear equations which appeared in 1950.
    • In a remarkable series of papers he effectively classified the algorithms for solving linear equations and computing eigensystems, showing that in many cases essentially the same algorithm had been presented in a large variety of superficially quite different algorithms.

  645. Duarte biography
    • He published papers on the general solution of a diophantine equation of the third degree x3 + y3 + z3 - 3xyz = v3, simplified Kummer's criterion and gave a simple proof of the impossibility of solving the Fermat equation x3 + y3 + z3 = 0 in nonzero integers.

  646. Thue biography
    • His contributions to the theory of Diophantine equations are discussed in [Normat 38 (4) (1990), 153-159; 192.',3)">3].
    • In fact Thue wrote 35 papers on number theory, mostly on the theory of Diophantine equations, and these are reproduced in [A Thue, Selected mathematical papers of Axel Thue, Introduction by Carl Ludwig Siegel (Universitetsforlaget, Oslo, 1977).',2)">2].

  647. Klug biography
    • All the way up to the solving of fourth order algebraic equations.
    • And at the time of Euler, it was not known that the equations above the fourth order cannot be solved.

  648. Stoilow biography
    • Stoilow's thesis advisor was Emile Picard, and in 1914 he submitted his doctoral thesis Sur une classe de fonctions de deux variables definies par les equations lineaires aux derivees partielles.
    • He did, however, publish his first paper in 1914, namely Sur les integrales des equations lineaires aux derivees partielles a deux variables independantes.
    • He published two further, namely Sur les fonctions quadruplement periodiques (1915) and Sur l'integration des equations lineaires aux derivees partielles et la methode des approximations successives (1916), before publishing his doctoral thesis in 1916.
    • He published three further papers in 1919 including Sur les singularites mobiles des integrales des equations lineaires aux derivees partielles et sur leur integrale generale, and two further papers in 1920.
    • Before he took up his first university appointment in 1919, Stoilow concentrated on the theory of partial differential equations in the complex domain.

  649. Reeb biography
    • Equations differentielles, written jointly with Robert Campbell, was a 78-page booklet published in 1964.
    • It presented the theory of ordinary differential equations in a form which would prove useful to physicists and engineers.
    • He explained the geometric approach to the theory of differential equations which he had adopted and indicated that it followed the approach begun by Henri Poincare, Paul Painleve and Elie Cartan.
    • For example, at the Fourth International Colloquium on Differential Geometry at Santiago de Compostela in 1978 he gave the talk Equations differentielles et analyse non classique which surveyed results on the perturbation of dynamical systems obtained using methods of nonstandard analysis.

  650. Milne biography
    • Milne combined the two approaches and came up with an integral equation of great mathematical interest which is now known as Milne's integral equation.

  651. Cartan biography
    • Cartan worked on continuous groups, Lie algebras, differential equations and geometry.
    • By 1904 Cartan was writing papers on differential equations and in many ways this work is his most impressive.

  652. Fricke biography
    • The present volume is a very attractive exposition of the modern theory of equations of degrees 5, 6, 7.
    • Fricke's long experience with the latter subject made it easy for him to give a simple authoritative exposition of those portions of it which suffice for the transcendental solutions of equations of low degrees.

  653. Hutton biography
    • The first volume looks at topics such as: arithmetic including discussion of square and cube roots, arithmetical and geometrical progressions, compound interest, double position and permutations and combinations; logarithms; algebra including the study of quadratic equations and the Cardan-Tartaglia method for cubic equations; geometry which follows the approach in Euclid's Elements; surveying; and conic sections.

  654. Gegenbauer biography
    • The Gegenbauer polynomials are solutions to the Gegenbauer differential equation and are generalizations of the associated Legendre polynomials.
    • However, the name of Gegenbauer occurs in many other places, such as Gegenbauer functions, Gegenbauer transforms, Gegenbauer series, Fourier-Gegenbauer sums, Gauss-Gegenbauer quadrature, Gegenbauer's integral inequalities, Gegenbauer's partial differential operators, the Gegenbauer equation, Gegenbauer approximation, Gegenbauer weight functions, the Gegenbauer oscillator, and the Gegenbauer addition theorem published in 1875.

  655. Ajima biography
    • Tsuda Nobuhisa solved the problem with an equation of degree 1024.
    • Ajima's remarkable achievement was to reduce this to an equation of degree 10.

  656. Banach biography
    • However, an exception was made to allow him to submit On Operations on Abstract Sets and their Application to Integral Equations.
    • The theory generalised the contributions made by Volterra, Fredholm and Hilbert on integral equations.

  657. Hurwitz biography
    • He worked on how to derive class number relations from modular equations.
    • Further topics studied by Hurwitz include complex function theory, the roots of Bessel functions, and difference equations.

  658. Truesdell biography
    • He took courses from Bateman on the partial differential equations of mathematical physics in 1940-41, and in 1941-42 on methods of mathematical physics (where for the entire year he and C-C Lin were the only students), aerodynamics of compressible fluids, and potential theory.
    • In subject matter, in particular their coverage of continuum mechanics and physics, and their use of the theory of partial differential equations and special functions, the courses formed a solid foundation for the young researcher.

  659. Schubert biography
    • Algebraically, the solution of the problems of enumerative geometry amounts to finding the number of solutions for certain systems of algebraic equations with finitely many solutions.
    • Since the direct algebraic solution of the problems is possible only in the simplest cases, mathematicians sought to transform the system of equations, by continuous variation of the constants involved, into a system for which the number of solutions could be determined more easily.

  660. Barsotti biography
    • He moved to this in Theta functions in positive characteristic (1979) and, staying with the theme of theta functions, published Differential equations of theta functions (1983) and Theta functions and differential equations (1985).

  661. Smoluchowski biography
    • He taught a variety of courses: potential theory, mechanics, electricity, optics, thermodynamics, kinetic theory of gases, differential equations, and mathematical physics.
    • Smoluchowski made many contributions to physics and mathematics, particularly to the theory of Brownian motion, stochastic processes and related problems, of which the most important are the 'Smoluchowski equations' bearing his name.

  662. Caratheodory biography
    • He added important results to the relationship between first order partial differential equations and the calculus of variations.
    • Caratheodory wrote many fine books including Lectures on Real Functions (1918), Conformal representation (1932), Calculus of Variations and Partial Differential Equations (1935), Geometric Optics (1937), Real functions Vol.

  663. Duhamel biography
    • He published articles such as Sur les equations generales de la propagation de la chaleur dans les corps solides dont la conductibilite n'est pas la meme dans tous les sens (1832) and Sur la methode generale relative au mouvement de la chaleur dans les corps solides plonges dans des milieux dont la temperature varie avec le temps (1833) in the Journal of the Ecole Polytechnique.
    • Duhamel worked on partial differential equations and applied his methods to the theory of heat, to rational mechanics, and to acoustics.
    • 'Duhamel's principle' in partial differential equations arose from his contributions to the distribution of heat in a solid with a variable boundary temperature.

  664. Smeal biography
    • Among Smeals' publications are (with Ernest Frederick John Love) The psychrometric formula (1911), (with S Brodetsky) On Graeffe's method for complex roots of algebraic equations (1924) and The equations of the gravitational field in orthogonal coordinates (1926).

  665. Subbotin biography
    • He had already published two papers prior to submitting his Master's thesis, one was On the determination of singular points of analytic functions while the second was published in France and was on singular points of certain differential equations.
    • Later he worked in celestial mechanics producing new methods of calculating orbits from three observations based on solving the Euler-Lambert equations.

  666. Roomen biography
    • Van Roomen had proposed a problem which involved solving an equation of degree 45 in Ideae mathematicae (1593).
    • The formal object, however, is the equality (aequalitas) of quantities, since only those problems, in which some equation is either explicitly given or can be deduced from the data of the problem, are analytic.

  667. Warschawski biography
    • The first was a single author paper On the solution of the Lichtenstein-Gershgorin integral equation in conformal mapping.
    • Theory while the second, On the solution of the Lichtenstein-Gershgorin integral equation in conformal mapping.

  668. Loewy biography
    • Loewy worked on linear groups, the algebraic theory of differential equations and actuarial mathematics.
    • He also published papers (in German) in the Transactions of the American Mathematical Society such as: On the reducibility of real groups of linear homogeneous substitutions (1903); On group theory, with applications to the theory of linear homogeneous differential equations (1904); and On completely reducible groups that belong to a group of linear homogeneous substitutions (1905).

  669. Nunes biography
    • The book is in three parts, the first part dealing with equations of the first and second degree and the third part dealing with equations of the third degree.

  670. Bott biography
    • We mentioned Smale above, and the second was Daniel Quillen who wrote his thesis Formal Properties of Over-Determined Systems Of Linear Partial Differential Equations at Harvard.
    • The main themes of the papers included in [Volume 4] are the geometry and topology of the Yang-Mills equations and the rigidity phenomena of vector bundles.

  671. Schmidt F-K biography
    • Heffter, an expert on differential equations, complex analysis and analytic geometry, had been appointed to Freiburg in 1911 having previously been a full professor at RWTH Aachen and Kiel.
    • Galois theory and algebraic equations; 6.

  672. Gromov biography
    • He did, however, contribute the text of his lecture A topological technique for the construction of solutions of differential equations and inequalities which was published in the Conference Proceedings in 1971.
    • It has been emphasised above that he tends to look at all questions from the geometric side: he translates them in ad hoc geometric terms, and uses his extraordinary geometric intuition to investigate them thoroughly; it should be added that he is also able to treat, in the same way, questions coming from the most diverse branches of mathematics: algebra, analysis, differential equations, probability theory, theoretical physics, etc.

  673. Lagny biography
    • In about 1690 he developed a method of giving approximate solutions of algebraic equations and, in 1694, Halley published a twelve page paper in the Philosophical Transactions of the Royal Society giving his method of solving polynomial equations by successive approximation which is essentially the same as that given by Lagny a few years earlier.

  674. Livsic biography
    • He was particularly interested in the courses in complex variable, integral equations and differential equations.

  675. La Hire biography
    • He began with their focal definitions and applied Cartesian analytic geometry t the study of equations and the solution of indeterminate problems; he also displayed the Cartesian method for solving certain types of equations by intersections of curves.

  676. Lorentz George biography
    • Kamke was writing a book on differential equations.
    • I wrote some 20 papers: joint papers with Kamke and Knopp, papers related to differential equations, papers on summability, on Fourier series, and papers where rearrangements play a role.

  677. Offord biography
    • It was during these last three years at Cambridge that he worked with J E Littlewood on the topic for which he is best known today, and they published a series of important joint papers beginning with On the number of real roots of a random algebraic equation in 1938.
    • In the following year they published a second paper with this title and in it they give estimates of the expected number of real roots for an equation of degree n when the coefficients are identically distributed random variables.

  678. De Beaune biography
    • Every trace of the work was lost until 1963, when it was rediscovered among manuscripts in the Roberval Archive at the Academie des Sciences in Paris, and thus it appears for the first time in the present critical edition (the book [The early theory of equations: on their nature and constitution (Golden Hind Press, Fairfield, CT, 1986).',2)">2]).
    • The first from Costabel [The early theory of equations: on their nature and constitution (Golden Hind Press, Fairfield, CT, 1986).',2)">2]:- .

  679. Bernoulli Johann biography
    • Integration to Bernoulli was simply viewed as the inverse operation to differentiation and with this approach he had great success in integrating differential equations.
    • He summed series, and discovered addition theorems for trigonometric and hyperbolic functions using the differential equations they satisfy.

  680. Somov biography
    • During this time he wrote his first mathematical work on algebraic equations Theory of determinate algebraic equations of higher degree which was published in 1838 [Dictionary of Scientific Biography (New York 1970-1990).',1)">1].

  681. Berge biography
    • He then applied this symbolic calculus to combinatorial analysis, Bernoulli numbers, difference equations, differential equations and summability factors.

  682. Wintner biography
    • Wintner published on analysis, number theory, differential equations and probability (with several joint papers with Norbert Wiener).
    • A study of certain astronomical equations led Wintner to consider almost periodic functions.

  683. Mises biography
    • His Institute rapidly became a centre for research into areas such as probability, statistics, numerical solutions of differential equations, elasticity and aerodynamics.
    • He classified his own work, not long before his death, into eight areas: practical analysis, integral and differential equations, mechanics, hydrodynamics and aerodynamics, constructive geometry, probability calculus, statistics and philosophy.

  684. Durell biography
    • Contents include the properties of the triangle and the quadrilateral; equations, sub-multiple angles, and inverse functions; hyperbolic, logarithmic, and exponential functions; and expansions in power-series.
    • Further topics encompass the special hyperbolic functions; projection and finite series; complex numbers; de Moivre's theorem and its applications; one- and many-valued functions of a complex variable; and roots of equations.

  685. Pick biography
    • However more than half of his papers were on functions of a complex variable, differential equations, and differential geometry.

  686. Voevodsky biography
    • And then we apply in a new situation, in this case in the situation of algebraic equations which is purely algebraic.

  687. Plessner biography
    • Then in session 1921/22 he studied in Berlin where von Mises lectured on differential and integral equations, Bieberbach on differential geometry and Schur on algebra.

  688. Dickson biography
    • Dickson: Theory of Equations .

  689. Vijayanandi biography
    • This system led to much work on integer solutions of equations and their application to astronomy.

  690. Vilant biography
    • Those parts for which some originality may be claimed are: (a) a method for finding the cube root of binomials of form R ± √S, where S may be positive or negative, and (b) a method for finding rational and whole-number solutions of indeterminate problems involving linear, quadratic and cubic equations.

  691. Catalan biography
    • In the same journal, he published two papers in 1838: Note sur un Probleme de combinaisons, and Note sur une Equation aux differences finies.
    • Four papers by Catalan are published in Volume 4 in 1839: Note sur la Theorie des Nombres; Solution nouvelle de cette question: Un polygone etant donne, de combien de manieres peut-on le partager en triangles au moyen de diagonales?; Addition a la Note sur une Equation aux differences finies; and Memoire sur la reduction d'une classe d'integrales multiples.
    • Two consecutive whole numbers, other than 8 and 9, cannot be consecutive powers; otherwise said, the equation xm - yn = 1 in which the unknowns are positive integers only admits a single solution.

  692. Whyburn biography
    • His research was mostly on second order ordinary differential equations, see [3] for details.

  693. Hartree biography
    • However Niels Bohr gave a lecture course in Cambridge in 1921 and Hartree was much influenced, working on applications of numerical methods for integrating differential equations to calculate atomic wave functions.

  694. Pars biography
    • He based his treatment on the theorem of Lagrange that he called the fundamental equation, which he proceeded to translate into six different forms, each exploited in appropriate contexts.

  695. Riesz biography
    • He built on ideas introduced by Frechet in his dissertation, using Frechet's ideas of distance to provide a link between Lebesgue's work on real functions and the area of integral equations developed by Hilbert and his student Schmidt.

  696. Weierstrass biography
    • ., from the differential equation defining this function, was the first mathematical task I set myself; and its fortunate solution made me determined to devote myself wholly to mathematics; I made this decision in my seventh semester ..

  697. Dantzig George biography
    • It was a system with nine equations in seventy-seven unknowns.

  698. Bartholin biography
    • The problem is the first example of an inverse tangent problem which in modern notation results in requiring the solution to the differential equation .

  699. Ferrel biography
    • However this assumption is not realistic, but the realistic assumption that the friction is proportional to the square of the velocity produced non-linear equations which were much more difficult to treat.

  700. Bondi biography
    • The exact relativistic form of the equation of hydrostatic support by an isotropic pressure is found in an especially convenient form.

  701. Mendelsohn biography
    • He wrote papers on a wide variety of combinatorial problems, for example: Symbolic solution of card matching problems (1946), Applications of combinatorial formulae to generalizations of Wilson's theorem (1949), Representations of positive real numbers by infinite sequences of integers (1952), A problem in combinatorial analysis (1953), The asymptotic series for a certain class of permutation problems (1956), and Some elementary properties of ill conditioned matrices and linear equations (1956).

  702. Kummer biography
    • He extended Gauss's work on hypergeometric series, giving developments that are useful in the theory of differential equations.

  703. Rees David biography
    • [He] was never happier than when sitting in front of the television scribbling down algebraic equations to find a solution to some mathematical challenge he had set himself.

  704. Jia Xian biography
    • He generalised a method of finding square roots and cube roots to finding nth roots, for n > 3, and then extended the method to solving polynomial equations of arbitrary degree.

  705. Dionis biography
    • Dionis du Sejour also worked on the theory of equations, not attaining the depth of results of Bezout or Lagrange.

  706. Todhunter biography
    • He also wrote some more elementary texts, for example Algebra (1858), Trigonometry (1859), Theory of Equations (1861), Euclid (1862), Mechanics (1867) and Mensuration (1869).

  707. Gopel biography
    • Gopel linked four more of these quadratics through a homogeneous fourth degree relation, later named the 'Gopel relation' which coincides with the equation of the Kummer surface.

  708. Konig Denes biography
    • It includes not only popular puzzling arithmetic problems including riddles which usually can be solved with linear equations, but also more remarkable works which were printed in foreign countries as mentioned above, and which lead us into the wonderful world of the numbers.

  709. Velez-Rodriguez biography
    • Her doctoral work consisted of studying differential equations which arose in the study of astronomical orbits.

  710. Crelle biography
    • Crelle realised the importance of Abel's work and published several articles by him in this first volume, including his proof of the insolubility of the quintic equation by radicals.

  711. Robbins biography
    • Robbins' paper with his student Sutton Monro on Stochastic Approximation provided an analogue of an iterative method due to Isaac Newton for finding the root of a function, even when the function's equation is unknown and the evaluation of the function involves experimental error.

  712. Moulton biography
    • His books include An Introduction to Celestial Mechanics (1902), An introduction to astronomy (1906), Descriptive astronomy (1912), Periodic orbits (1920) The Nature of the World and Man (1926), Differential equations (1930), Astronomy (1931), and Consider the Heavens (1935).

  713. Blades biography
    • For example, he communicated On Spheroidal Harmonics and Allied Functions, by Mr G B Jeffery to the meeting on Friday 11 June 1915 and Transformations of Axes for Whittaker's Solution of Laplace's Equation, by Dr G B Jeffery to the meeting on Friday 9 March 1917.

  714. Newton biography
    • [Newton] brought me the other day some papers, wherein he set down methods of calculating the dimensions of magnitudes like that of Mr Mercator concerning the hyperbola, but very general; as also of resolving equations; which I suppose will please you; and I shall send you them by the next.

  715. Fergola biography
    • He did learn some mathematics from Giuseppe Marzucco (1713-1800) but this was only up to quadratic equations.

  716. Schwarzschild biography
    • Schwarzschild's relativity papers give the first exact solution of Einstein's general gravitational equations, giving an understanding of the geometry of space near a point mass.

  717. Apastamba biography
    • The general linear equation was solved in the Apastamba's Sulbasutra.

  718. Tate biography
    • In his thesis, which has become a classic, he proved the functional equation for Hecke's L-series by a novel method involving Fourier analysis on idele groups.

  719. Vranceanu biography
    • Other topics he studied include the absolute differential calculus of congruences, analytical mechanics, partial differential equations of the second order, non-holonomic unitary theory, conformal connection spaces, metrics in spherical and projective spaces, Lie groups, global differential geometry, discrete groups of affine connection spaces, locally Euclidean connection spaces, Riemannian spaces of constant connection, differentiable varieties, embedding of lens spaces into Euclidean space, tangent vectors of spheres and exotic spheres, the equivalence method, non-linear connection spaces, and the geometry of mechanical systems.

  720. Frenicle de Bessy biography
    • History Topics: Pell's equation .

  721. Herstein biography
    • Among the methods and problems discussed in some detail are a derivation of the Slutsky equation via the calculus, a problem in Welfare Economics treated by the theory of convex sets, matrix theory as applied to international trade, and a game-theoretical approach to the personnel assignment problem.

  722. Rosanes biography
    • he scribbled equations which his students never quite saw because as he wrote he hid them with his body and as he moved along he rubbed them out with his sponge.

  723. Davenport biography
    • At the most advanced level he wrote a monograph Analytic methods for Diophantine equations and Diophantine inequalities (1962) which includes many of his contributions extending the Hardy-Littlewood method.

  724. Airy biography
    • This text was one of eleven books which Airy published, some of the others being Trigonometry (1825), Gravitation (1834), and Partial differential equations (1866).

  725. Williams biography
    • In 1925 Cox was awarded his doctorate by Cornell University for his thesis Polynomial solutions of difference equations.

  726. Niven biography
    • In the present state of our knowledge of the resistance of the air to shot, the problem of integrating the equations of motion of the shot and of plotting-out a representation of the curve described by it is peculiar, because, according to the best experiments we possess, the law of the retardation cannot be expressed by a single exact formula which is available for the solution.

  727. Butters biography
    • He also contributed to the mathematical work of the Society, For example at the meeting of the Society on Friday 11 January 1889, J Watt Butters discussed the solution of an algebraic equation.

  728. Post biography
    • thesis was on mathematical logic, and we shall discuss it further in a moment, but first let us note that Post wrote a second paper as a postgraduate, which was published before his first paper, and this was a short work on the functional equation of the gamma function.

  729. Ollerenshaw biography
    • Before falling asleep, I 'drew' with my finger any relevant geometrical figure or algebraic equation on the partitioning of the dormitory cubicle that formed a bedside wall.

  730. Delone biography
    • He studied Tschirnhaus's inverse problem, producing methods to determine whether or not two given cubic equations determine the same field.

  731. Mathews biography
    • Mathews also wrote Algebraic equations (1907) which is a clear exposition of Galois theory, and Projective geometry (1914).

  732. Schwartz Jacob biography
    • The present volume I under review contains the topological theory of spaces and operators, and the spectral theory of "arbitrary" operators and some applications; the second volume will contain the spectral theory of completely reducible operators and further applications, e.g., applications to differential operators and partial differential equations.

  733. Cosserat Francois biography
    • The most practical results concerning elasticity were the introduction of the systematic use of the movable trihedral and the proposal and resolution, before Fredholm's studies, of the functional equations of the sphere and ellipsoid.

  734. Macaulay biography
    • What ideas were there then in this work? The main theme underlying the book is the problem of solving equations of systems of polynomials in several variables.

  735. Nevanlinna biography
    • For example he wrote the paper Calculus of variation and partial differential equations (1967).

  736. Farkas biography
    • In 1881 Gyula Farkas published a paper on Farkas Bolyai's iterative solution to the trinomial equation, making a careful study of the convergence of the algorithm.

  737. Kneser Hellmuth biography
    • For example he produced a beautiful solution to the functional equation f ( f (x) ) = ex which he published in 1950, and the deep understanding he achieved of the strange properties of manifolds without a countable basis of neighbourhoods between 1958 and 1964.

  738. Li Rui biography
    • In 1813 while working on his edition of the lost work by Yang Hui, Li Rui wrote the Kai Fang Shuo (Theory of Equations of Higher Degree).

  739. Blichfeldt biography
    • Some of the many topics that he covered were diophantine approximations, orders of linear homogeneous groups, theory of geometry of numbers, approximate solutions of the integers of a set of linear equations, low-velocity fire angle, finite collineation groups, and characteristic roots.

  740. Cosserat biography
    • The most practical results concerning elasticity were the introduction of the systematic use of the movable trihedral and the proposal and resolution, before Fredholm's studies, of the functional equations of the sphere and ellipsoid.

  741. Babbage biography
    • He wrote two major papers on functional equations in 1815 and 1816.

  742. Liu Hui biography
    • In Chapter 8 he looks at simultaneous linear equations and computes with both positive and negative numbers.

  743. Stewartson biography
    • Keith Stewartson's abiding passion in mathematical research lay in the solution of the equations governing the motion of liquids and gases, and in the comparison of his theoretical predictions with experiment and observation.

  744. Romberg biography
    • In this work they investigated the hydrodynamical equations for an ideal incompressible fluid on a rotating sphere which is subjected to the influence of tidal forces.

  745. Burkhardt biography
    • Other topics on which Burkhardt published papers included groups, differential equations, differential geometry and mathematical physics.

  746. Bethe biography
    • Oddly, though, he left the neutrino out of the proton-proton reaction equation.

  747. Uhlenbeck biography
    • He extended Boltzmann's equation to dense gasses and wrote two important papers on Brownian motion.

  748. Newman biography
    • The first was an early inroad on Hilbert's Fifth Problem, in which he proved that abelian continuous groups do not have arbitrarily small subgroups, the second was a simplified proof of a difficult fixed point theorem of Cartwright and Littlewood arising in the study of differential equations.

  749. Nielsen biography
    • He turned to number theory and studied Bernoulli numbers in Traite elementaire des nombres de Bernoulli (Gauthier-Villars, Paris, 1923) and Fermat's equation writing good textbooks on these topics.

  750. Linfoot biography
    • In fact despite still being an undergraduate, Linfoot was already undertaking research and published his first paper The domains of convergence of Kummer's solutions to the Riemann P-equation in 1926.

  751. Strassen biography
    • Using this new matrix multiplication routine, Strassen was able to show that Gaussian elimination (an efficient algorithm for solving systems of linear equations) is not an optimal solution.

  752. Prager biography
    • The numerical methods aspect also shows the hand of a master and covers all the material that is usually given in a one term course, including ordinary differential equations, in a reasonably rigorous and at the same time practical manner.

  753. Watson Henry biography
    • In addition to these books he wrote on Lagrange's method and Monge's method for solving partial differential equations and, jointly with Galton, he wrote On the probability of extinction of families.

  754. Young Andrew biography
    • He read the paper On the quasi-periodic solutions of Mathieu's differential equation to the Society at its meeting on Friday 13 February 1914.

  755. Wiltheiss biography
    • His doctoral studies on systems of hyperelliptic differential equations were supervised by Weierstrass and he submitted his thesis Die Umkehrung einer Gruppe von Systemen allgemeiner hyperelliptischer Differentialgleichungen to the University of Berlin.

  756. Kempe biography
    • Kempe was taught mathematics by Cayley and graduated in 1872 with distinction in mathematics and in the same year he published his first mathematical paper A general method of solving equations of the nth degree by mechanical means.

  757. Cohn biography
    • His second book Linear equations was published in 1958 and another book Solid geometry was published in 1961.

  758. Gram biography
    • Gram later published this work in the Journal fur Mathematik and it proved to be of fundamental importance in the development of the theory of integral equations.

  759. Cremona biography
    • The geometric method is principally a use of terms or descriptive relations instead of equations.

  760. Schroeter biography
    • He attended courses by Dirichlet on number theory and on differential equations which influenced Schroter's teaching for the whole of his career but it was Steiner who was a major influence on Schroter's research.

  761. Trail biography
    • His widely used Elements of Algebra, which he published for his students in 1770, ranged from first principles to equations of all orders and included applications to problem solving, physics and geometry.

  762. Clapeyron biography
    • The Clapeyron relation, a differential equation which determines the heat of vaporisation of a liquid, is named after him.

  763. Whiteside biography
    • 1 - 14, especially 10 where he uses a proto-Lagrangian analysis to deduce a third-order defining differential equation).

  764. Dirac biography
    • Also in 1928 he found a connection between relativity and quantum mechanics, his famous spin-1/2 Dirac equation.

  765. Beurling biography
    • Beurling worked on the theory of generalized functions, differential equations, harmonic analysis, Dirichlet series and potential theory.

  766. Chowla biography
    • He wrote on additive number theory (lattice points, partitions, Waring's problem), analysis, Bernoulli numbers, class invariants, definite integrals, elliptic integrals, infinite series, the Weierstrass approximation theorem), analytic number theory (Dirichlet L-functions, primes, Riemann and Epstein zeta functions), binary quadratic forms and class numbers, combinatorial problems (block designs, difference sets, Latin squares), Diophantine equations and Diophantine approximation, elementary number theory (arithmetic functions, continued fractions, and Ramanujan's tau function), and exponential and character sums (Gauss sums, Kloosterman sums, trigonometric sums).

  767. Hua biography
    • Hua wrote several papers with H S Vandiver on the solution of equations in finite fields and with I Reiner on automorphisms of classical groups.

  768. Fields biography
    • After the award of the degree in 1887 for his thesis Symbolic Finite Solutions, and Solutions by Definite Integrals of the Equation (dn/dxn)y - (xm)y = 0, he remained teaching at Johns Hopkins for a further two years.

  769. Dinghas biography
    • His work is in many areas of mathematics including differential equations, functions of a complex variable, functions of several complex variables, measure theory and differential geometry.

  770. Blaschke biography
    • In Leipzig he became a close friend of Gustav Herglotz who was interested in partial differential equations, function theory and differential geometry, and succeeded Runge in Gottingen 10 years later.

  771. Suschkevich biography
    • Examples of courses he attended at this stage in his career are: Determinants, by Frobenius in the summer of 1908; Algebra I and Algebra II by Frobenius in 1908-09; Ordinary differential equations, by Schur in the summer of 1909; Chebyshev's Theorem, by Frobenius in November 1909; Bernoulli numbers by Frobenius in the summer of 1910, and Matrices, by Frobenius in the winter of 1910-11.

  772. Banachiewicz biography
    • Two years later he again defended a thesis at Dorpat, this time on the Gauss equation, and was promoted to assistant professor.
    • For example: An outline of the Cracovian algorithms of the method of least squares (1942); On the accuracy of least squares solution (1945); Sur la resolution des equations normales de la methode des moindres carres (1948); Sur l'interpolation dans le cas des intervalles inegaux (1949); A general least squares interpolation formula (1949); Les cracoviens et quelques-unes de leurs applications en geodesie (1949); On the general least squares interpolation formula (1950); and Resolution d'un systeme d'equations lineaires algebriques par division (written much earlier by only published in 1951 due to World War II) [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .

  773. Sidler biography
    • He attended lectures by J Bertrand (analysis), M Chasles (geometry), H Faye (astronomy), G Lame (mathematical physics), U J Le Verrier (popular astronomy), J Liouville (differential equations) and V Puiseux (celestial mechanics).

  774. Jyesthadeva biography
    • Other mathematical results presented by Jyesthadeva include topics studied by earlier Indian mathematicians such as integer solutions of systems of first degree equation solved by the kuttaka method, and rules of finding the sines and the cosines of the sum and difference of two angles.

  775. Conway Arthur biography
    • The first of these papers extended work of Love on electromagnetic waves in an isotropic medium while the next two concerned equations of classical electromagnetic field theory.

  776. Blackwell biography
    • The most interesting thing I remember from calculus was Newton's method for solving equations.

  777. Takagi biography
    • Hilbert had left this topic immediately after writing the Zahlbericht and by the time Takagi reached Gottingen he was engaged in studying the foundations of geometry and then integral equations.

  778. Walsh Joseph biography
    • The topics he taught, rotating them from year to year, included calculus, algebra, mechanics, differential equations, complex variable, probability, number theory, potential theory, approximation theory, and function theory.

  779. Li Shanlan biography
    • The "tian yuan" or "coefficient array method" or "method of the celestial unknown" of setting up equations, which Li learnt from Li Zhi's famous text, had a huge influence on him and he began to push these algebraic techniques forward solving a whole variety of new problems.

  780. Pade biography
    • It deals with the development into a continued fraction of the generating function of a sequence satisfying a difference equation.

  781. Kemeny biography
    • There were many long calculations, deriving one formula from another to solve a differential equation.

  782. Ampere biography
    • In mathematics he worked on partial differential equations, producing a classification which he presented to the Institut in 1814.

  783. Granville biography
    • from Yale, Granville spent a postdoctoral year at the New York University Institute of Mathematics working on differential equations with Fritz John.

  784. Van Ceulen biography
    • In 1595 the two men competed in the solution of a forty-fifth degree equation proposed by van Roomen in his 'Ideae mathematicae' (1593) and recognised its relation to the expression of sin 45A in terms of sin A.

  785. Flato biography
    • The second is the cohomological study of nonlinear representations of covariance groups of nonlinear partial differential equations which leads to important mathematical developments with nontrivial physical consequences ..

  786. Kalicki biography
    • Kalicki worked on logical matrices and equational logic and published 13 papers on these topics from 1948 until his death five years later.

  787. Halley biography
    • Halley's other activities included studying archaeology, geophysics, the history of astronomy, and the solution of polynomial equations.

  788. Luke biography
    • His work on these topics led him to require much information on special functions and he was led to develop tables of special functions and to use numerical techniques to solve equations.

  789. Le Paige biography
    • Some of these papers were on topics he had worked on before he settled on geometry as his main interest, for example there are papers on continued fractions, differential equations, the difference calculus, and Bernoulli numbers.

  790. Kac biography
    • in the summer of 1930 I became obsessed with the problem of solving cubic equations.

  791. Jitomirskaya biography
    • He travelled all night to see me, only to have to wait for another three hours since I didn't want to miss a lecture on differential equations by Vladimir Igorevich Arnold.

  792. Wang Yuan biography
    • However he did write a number of books such as: (with Hua Loo Keng) Applications of number theory to numerical analysis (1978); Goldbach Conjecture (1984); (with Hua Loo Keng) Popularising mathematical methods in the People's Republic of China (1989); Diophantine equations and inequalities in algebraic number fields (1991); (with Fang Kai-Tai) Number theoretic methods in statistics (1994); Hua Loo Keng (1995); and (with Fong Yuen) Calculus (1997).

  793. Eisenstein biography
    • While in Ireland in 1843 Eisenstein met Hamilton in Dublin, a city he would have dearly liked to have settled in, and Hamilton gave him a copy of a paper that he had written on Abel's work on the impossibility of solving quintic equations.

  794. Lindemann biography
    • He is famed for his proof that π is transcendental, that is, π is not the root of any algebraic equation with rational coefficients.

  795. Brink biography
    • Finally we give some examples of Brink's papers: A new integral test for the convergence and divergence of infinite series (1918); A new sequence of integral tests for the convergence and divergence of infinite series (1919); The May Meeting of the Minnesota Section (1927); Recent Publications: Reviews: Studies in the History of Statistical Method - With Special Reference to Certain Education Problems (1929); The May Meeting of the Minnesota Section (1930); A Simplified Integral Test for the Convergence of Infinite Series (1931); Recent Publications: Reviews: Differential Equations (1932); The Annual Meeting of the Minnesota Section (1937); and College Mathematics During Reconstruction (1944).

  796. Green biography
    • The formula connecting surface and volume integrals, now known as Green's theorem, was introduced in the work, as was "Green's function" the concept now extensively used in the solution of partial differential equations.

  797. Kostrikin biography
    • The meaning of an algebraic concept can be of a number-theoretic or geometric nature, and frequently its roots lie in computational aspects of mathematics and in the solution of equations.

  798. Mayer Tobias biography
    • with respect to the inequalities of motions, from that famous theory of the great Newton, which that eminent mathematician Eulerus first elegantly reduced to general analytic equations.

  799. Pythagoras biography
    • For example they solved equations such as a (a - x) = x2 by geometrical means.

  800. Erdelyi biography
    • He also worked on asymptotic analysis, fractional integration and singular partial differential equations.

  801. Poinsot biography
    • In addition Poinsot worked on number theory and on this topic he studied Diophantine equations, how to express numbers as the difference of two squares and primitive roots.

  802. Wigner biography
    • epoch-making work on how symmetry is implemented in quantum mechanics, the determination of all the irreducible unitary representations of the Poincare group, and his work with Bargmann on realizing those irreducible unitary representations as the Hilbert spaces of solutions of relativistic wave equations, ..

  803. Lacroix biography
    • Not only did Monge use his influence to obtain this position for Lacroix, but he also acted as his mathematical advisor, recommending that he undertake research on partial differential equations and the calculus of variations.

  804. Seitz biography
    • Find the equation to the locus of the centres of all the circles that can be incribed in a given semi-ellipse.

  805. Jordanus biography
    • In De numeris datis Jordanus gives results on solving quadratic equations similar to those given by al-Khwarizmi except general forms are given rather than the numerical examples of the earlier text.

  806. Good biography
    • Quickly realising that the calculation would never terminate, he found rational approximations from Pell's equation x2 - 2y2 = 1, discovering this method for himself.

  807. Gillman biography
    • Next he took courses on Integral Calculus, Matrix Theory, and Differential Equations.

  808. Schroder biography
    • Schroder's concept of solving a relational equation was a precursor of Skolem functions, and he inspired Lowenheim's formulation and proof of the famous theorem that every sentence with an infinite model has a countable model, the first real theorem of modern logic.

  809. Weinstein biography
    • In examining singular partial differential equations he introduced a new branch of potential theory and applied the results to many different situations including flow about a wedge, flow around lenses and flow around spindles.

  810. Griffiths biography
    • Though the papers selected cover a broad range of topics in complex analysis, algebraic geometry and differential equations ..

  811. Peres biography
    • Peres' work on analysis and mechanics was always influenced by Volterra, extending results of Volterra's on integral equations.

  812. Servois biography
    • Servois worked in projective geometry, functional equations and complex numbers.

  813. Taylor Mary biography
    • There she carried out research on her specialist topics of the magneto-ionic theory of radio wave propagation and also in differential equation, particularly their applications to physics.

  814. Hecke biography
    • Schoeneberg describes Hecke's contributions to a number of topics which he lists as follows: Hilbert modular functions, Dedekind zeta functions, arithmetical notions and methods, elliptic modular forms of level N, algebraic functions, Dirichlet series with functional equation, Hecke-operators Tn, and physics where he made contributions to the kinetic theory of gases.

  815. Hellins biography
    • Hellins published many papers; the following were all in the Philosophical Transactions of the Royal Society: A new method of finding the equal roots of an equation by division (1782); Dr Halley's method of computing the quadrature of the circle improved; being a transformation of his series for that purpose, to others which converge by the powers of 60 (1794); Mr Jones' computation of the hyperbolic logarithm of 10 compared (1796); A method of computing the value of a slowly converging series, of which all the terms are affirmative (1798); An improved solution of a problem in physical astronomy, by which swiftly converging series are obtained, which are useful in computing the perturbations of the motions of the Earth, Mars, and Venus, by their mutual attraction (1798); A second appendix to the improved solution of a problem in physical astronomy (1800); and On the rectification of the conic sections (1802).

  816. Maurolico biography
    • Demonstratio algebrae, which is an elementary text looking at quadratic equations and problems whose solution reduces to solving a quadratic; .

  817. Manin biography
    • He has written papers on: algebraic geometry including ones on the Mordell conjecture for function fields and a joint paper with V Iskovskikh on the counter-example to the Luroth problem; number theory including ones about torsion points on elliptic curves, p-adic modular forms, and on rational points on Fano varieties; and differential equations and mathematical physics including ones on string theory and quantum groups.

  818. Cunningham biography
    • He wrote on linear differential equations, prompted by Pearson's work and other work related to statistics.

  819. Bernoulli Johann(III) biography
    • In the field of mathematics he worked on probability, recurring decimals and the theory of equations.

  820. Urysohn biography
    • At this stage Urysohn was interested in analysis, in particular integral equations, and this was the topic of his habilitation.

  821. Lanczos biography
    • He worked on relativity theory and after writing his dissertation Relation of Maxwell's Aether Equations to Functional Theory he sent a copy to Einstein.

  822. Galerkin biography
    • His visits around European construction sites ended around 1914 but his academic work then turned to the area for which he is today best known, namely the method of approximate integration of differential equations known as the Galerkin method.

  823. Petzval biography
    • He was influenced by the work of Liouville and wrote both a long paper and a two volume treatise on the Laplace transform and its application to ordinary linear differential equations.

  824. Peirce Benjamin biography
    • For example An Elementary Treatise on Plane Trigonometry (1835), First Part of an Elementary Treatise on Spherical Trigonometry (1836), An Elementary Treatise on Sound (1836), An Elementary Treatise on Algebra : To which are added Exponential Equations and Logarithms (1837), An Elementary Treatise on Plane and Solid Geometry (1837), An Elementary Treatise on Plane and Spherical Trigonometry (1840), and An Elementary Treatise on Curves, Functions, and Forces Vol 1 (1841), Vol 2 (1846).

  825. Zuse biography
    • It was the first operational program-controlled calculating machine and was used by the German aircraft industry to solve systems of simultaneous equations and other mathematical systems which were produced by the problems of dealing with the vibration of airframes put under stress.

  826. Bianchi biography
    • Bianchi partial differential equations play an important role.

  827. Widman biography
    • Widman used Cossist notation, as was usual at that time, discussing 24 different types of equations.

  828. Picken biography
    • He read papers to the Society such as A Proof of the Addition Theorem in Trigonometry to the meeting on Friday 9 December 1904, On a Direct Method of Obtaining the Foci and Directrices from the General Equation of the Second Degree to the meeting on 9 June 1905, On Simson Line and Related Theory: and An Exercise in Geometric Generality (communicated by A W Young) on 8 May 1914.

  829. Moore Eliakim biography
    • Other topics he worked on include algebraic geometry, number theory and integral equations.

  830. Flugge-Lotz biography
    • Here she applied her mathematical skills in solving differential equations to solve an important problem on the distribution of lift on wings (one that Prandtl himself had been unable to solve).

  831. Wolf Frantisek biography
    • Consideration of the Schrodinger equation leads to perturbation problems for partial differential operators, where the change may occur in the coefficients of the operator or in boundary conditions.

  832. Aryabhata II biography
    • In Mahasiddhanta Aryabhata II gives in about twenty verses detailed rules to solve the indeterminate equation: by = ax + c.

  833. Schmetterer biography
    • was mostly concerned with differential equations in the field of aerodynamics.

  834. Northcott biography
    • In particular he was taught to solve simultaneous equations and prove elementary theorems in Euclidean geometry which gave him a love of mathematics at this early stage in his education.

  835. Faber biography
    • Only in the 1980s was Faber's idea seen to be an important ingredient for the efficient solution of partial differential equations.

  836. Chisini biography
    • Indeed, Enriques immediately recognised his talent, led him to obtain a degree in mathematics in 1912, and engaged him as assistant and coauthor in the writing of the treatise Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche (Lessons on the geometric theory of equations and algebraic functions).

  837. Maupertuis biography
    • By 1731 he had written his first paper on astronomy and another on differential equations, and was rapidly developing a reputation as an all round mathematician and scientist.

  838. Casorati biography
    • Differential equations were of great interest to him and his research in this area was undertaken with the aim of making the existing theories more accurate and more complete.

  839. Roberts biography
    • In theory of numbers he was interested in the Pellian equation and similar problems.

  840. Karsten biography
    • He wrote an important article in 1768 Von den Logarithmen vermeinter Grossen in which he discussed logarithms of negative and imaginary numbers, giving a geometric interpretation of logarithms of complex numbers as hyperbolic sectors, based on the similarity of the equations of the circle and of the equilateral hyperbola.

  841. Burckhardt biography
    • After this the instructor should teach algebra up to and including equations of the second degree.

  842. Noether Max biography
    • This result showed that given two algebraic curves f (x, y) = 0, g(x, y) = 0 which intersect in a finite number of isolated points, then the equation of an algebraic curve which passes through all those points of intersection can be expressed in the form af + bg = 0, where a and b are polynomials in x and y, is and only if certain conditions are satisfied.

  843. Polya biography
    • He also worked on conformal mappings and potential theory, and he was led to study boundary value problems for partial differential equations and the theory of various functionals connected with them.

  844. Feuerbach biography
    • one day he appeared in class with a drawn sword and threatened to cut off the head of every student in the class who could not solve the equations he had written on the blackboard.

  845. Osipovsky biography
    • His most famous work was the three-volume handbook A Course of Mathematics (1801-1823) which covered function theory, differential equations, and the calculus of variations.

  846. Aiken biography
    • These plans were made for a very specific purpose, for Aiken's research had led to a system of differential equations which had no exact solution and which could only be solved using numerical techniques.

  847. Hecht biography
    • His later texts covered topics such as quadratic and cubic equations, differential and integral calculus, and arithmetic and geometry.

  848. Fagnano Giulio biography
    • Fagnano suggested new methods of solving equations of degree 2, 3 and 4.

  849. Schmid biography
    • Following his move to Berlin, the direction of Schmid's research changed somewhat and he moved away from algebraic number theory, becoming interested in topics in algebraic geometry and Lame differential equations.

  850. Chernoff biography
    • He took courses by Bers, Feller, Loewner, Tamarkin, and others, and wrote a Master's dissertation Complex Solutions of Partial Differential Equations under Bergman's supervision.

  851. Fatou biography
    • Using existance theorems for the solutions to differential equations, Fatou was able to prove rigorously certian results on planetary orbits which Gauss had suggested by only verified with an intuitive argument.

  852. Hay biography
    • Until high school, I was not particularly mathematically inclined - indeed, I was much better at verbal subjects; combinatorial aspects of numbers and equations have never been my strong point ..

  853. McShane biography
    • McShane is famous for his work in the calculus of variations, Moore-Smith theory of limits, the theory of the integral, stochastic differential equations, and ballistics.

  854. Young Lai-Sang biography
    • It is generally regarded as a study of the iteration of maps, of time evolution of differential equations, and of group actions on manifolds.

  855. Stone biography
    • His doctorate was awarded in 1926 for a thesis entitled Ordinary Linear Homogeneous Differential Equations of Order n and the Related Expansion Problems.

  856. Kochin biography
    • Kochin also edited the works of Ivan Aleksandrovich Lappo-Danilevskii (1896-1931), who was an expert on applying matrix theory to differential equations, and of Aleksandr Mikhailovich Lyapunov.

  857. Abraham Max biography
    • He loved his absolute aether, his field equations, his rigid electron just as a youth loves his first flame, whose memory no later experience can extinguish.

  858. Loewner biography
    • We should also note that Loewner's proof uses the Loewner differential equation which has been studied extensively since he introduced it, and was used by de Branges in his celebrated proof of the Bieberbach conjecture.

  859. Goldstein biography
    • He studied numerical solutions to steady-flow laminar boundary-layer equations in 1930.

  860. Elliott biography
    • All ageing mathematicians should be particularly pleased to learn that a second piece of work by Elliott, which was again of major importance, was his contribution to the theory of integral equations which he made after he retired.

  861. Robertson biography
    • Around this time he built on de Sitter's solution of the equations of general relativity in an empty universe and developed what are now called Robertson-Walker spaces [Biographical Memoirs National Academy of Sciences 51 (1980), 343-361.',2)">2]:- .

  862. Fine Nathan biography
    • Fine writes from the viewpoint of a number theorist, and his slim volume is rich with examples and results from the theory of partitions, the study of Ramanujan's mock theta functions, and modular equations.

  863. Rutherford biography
    • Rutherford's papers in the 1940s included On the relations between the numbers of standard tableaux, On the matrix representation of complex symbols, On substitutional equations, Some continuant determinants arising in physics and chemistry, On commuting matrices and commutative algebras; these being published either by the Edinburgh Mathematical Society or by the Royal Society of Edinburgh.

  864. Polkinghorne biography
    • Also in 1955 he published Temporally ordered graphs and bound state equations and On the classification of fundamental particles.

  865. Al-Farisi biography
    • He noted the impossibility of giving an integer solution to the equation .

  866. Jevons biography
    • The 'logical piano', a machine designed by Jevons and constructed by a Salford watchmaker, had 21 keys for operations in equational logic.

  867. Fano biography
    • Early studies deal with line geometry and linear differential equations with algebraic coefficients ..

  868. Fuchs Klaus biography
    • He also published On the Invariance of Quantized Field Equations (1938/39), On the Stability of Nuclei Against -Emission (1939) and (with Born) On Fluctuations in Electromagnetic Radiation (1939).

  869. Bolyai Farkas biography
    • For example he gave iterative procedures to solve equations which he then proved convergent by showing them to be monotonically increasing and bounded above.

  870. Kahler biography
    • He submitted his habilitation thesis on the integrals of algebraic differential equations to Hamburg in 1930 and became a Privatdozent.

  871. Hensel biography
    • He showed, at least for quadratic forms, that an equation has a rational solution if and only if it has a solution in the p-adic numbers for each prime p and a solution in the reals.

  872. Delsarte biography
    • He worked during that year at the private mansion of the Foundation, undertaking research for his doctoral thesis and also working on his first two papers Sur les rotations dans l'espace fonctionnel and E de certaines equations integrales qui generalisent celles de Fredholm which were published by the Academy of Science.
    • At Nancy he developed a new course on differential equations in the academic year 1933-34 and in the following year, also at Nancy, he gave a course on Riemann spaces and relativity.
    • He published a series of papers on this topic in 1934-35: Les fonctions moyenne-periodiques (1934); Application de la theorie des fonctions moyenne-periodiques a la resolution de certaines equations integrales (1934); Application de la theorie des fonctions moyenne-periodiques a la resolution des equations de Fredholm-Norlund (1935); and Les fonctions moyenne-periodiques (1935).

  873. Perron biography
    • However he also worked on differential equations, matrices and other topics in algebra, continued fractions, geometry and number theory.

  874. Schoenberg biography
    • Schoenberg is noted worldwide for his realisation of the importance of spline functions for general mathematical analysis and in approximation theory, their key relevance in numerical procedures for solving differential equations with initial and/or boundary conditions, and their role in the solution of a whole host of variational problems.

  875. Dougall biography
    • Examples of papers he read at meeting of the Society are Elementary Proof of the Collinearity of the Mid Points of the Diagonals of a Complete Quadrilateral on Friday 12 February 1897; Methods of Solution of the Equations of Elasticity on 10 December 1897; and Notes on Spherical Harmonics on 12 December 1913.

  876. Dedekind biography
    • He attended courses by Dirichlet on the theory of numbers, on potential theory, on definite integrals, and on partial differential equations.

  877. Freundlich biography
    • His occasional inability to comprehend these ideas had the salutary effect of making Einstein seek to simplify their mathematical formulation, for if one of Felix Klein's pupils could not make sense of his equations who could? Through his intimate contact with Einstein, Freundlich was the first to become thoroughly acquainted with the fundamental principles of the new gravitational theory and, as Einstein himself remarks in the foreword of Freundlich's book, he was particularly well qualified as its exponent because he had been the first to attempt to put it to the test.

  878. Routh biography
    • In fact the impact of this prize winning work was very significant since Thomson and Tait rewrote for the second edition of their text Natural philosophy treatise the part dealing with equations of motion using Routh's developments.

  879. Cajori biography
    • Before looking at his main work on the history of mathematics, let us first note that he did write some textbooks which were not historical texts such as An introduction to the modern theory of equations (1904) and Elementary algebra: First year course (1915).

  880. Ahmes biography
    • The Verso has 87 problems on the four operations, solution of equations, progressions, volumes of granaries, the two-thirds rule etc.

  881. Walfisz biography
    • Walfisz published further monographs: Pell's equation (Russian) (1952); Lattice points in many-dimensional spheres (Russian) (1960); and Weylsche Exponentialsummen in der neueren Zahlentheorie (1963).

  882. Appell biography
    • He then wrote on algebraic functions, differential equations and complex analysis.

  883. Machin biography
    • Machin had explained to Taylor in Child's Coffeehouse how to use Newton's series to solve Kepler's problem and also how Halley's method finds roots of polynomial equations.

  884. Bugaev biography
    • He wrote on algebraic integrals of certain differential equations.

  885. Lonie biography
    • Topics taught in Lonie's Mathematics Department in 1879 are: Mathematics - Euclid, Elementary Modern Geometry and Conic Sections, Plane and Spherical Trigonometry, with practice, Elementary Algebra with Higher Equations, Mensuration, and Mechanics; Physics - after Balfour Stewart and Modern Views of Natural Forces including Energy, Sound, Heat, Light; Geography - Modern Geography.

  886. Chernikov biography
    • There are properties such as solubility, a concept which goes back to Galois and attempts to classify which polynomial equations could be solved by radicals, which make perfect sense for infinite groups.

  887. Rado biography
    • His first paper On the roots of algebraic equations was published in 1921 and in the following year he published his first paper on conformal mappings.

  888. Magnus biography
    • In addition to research in group theory and special functions, he worked on problems in mathematical physics, including electromagnetic theory and applications of the wave equation.

  889. Eckert John biography
    • Von Neumann was working on this project and became involved with the ENIAC computer and used it to solve systems of partial differential equations which were crucial in the work on atomic weapons at Los Almos.

  890. Valyi biography
    • His doctoral dissertation was on the theory of the propeller which led to his developing a theory of partial differential equations of the second order.

  891. Sankara biography
    • It is a text which covers the standard mathematical methods of Aryabhata I such as the solution of the indeterminate equation by = ax ± c (a, b, c integers) in integers which is then applied to astronomical problems.

  892. Marcinkiewicz biography
    • For in the field of real variable Marcinkiewicz had exceptionally strong intuition and technique, and the results he obtained in the theory of conjugate functions, had they been extended to functions of several variables might have given (as we see clearly now) a strong push to the theory of partial differential equations.

  893. Moser Leo biography
    • These include On the sum of digits of powers (1947), Some equations involving Euler's totient (1949), Linked rods and continued fractions (1949), On the danger of induction (1949) and A theorem on the distribution of primes (1949).

  894. Pople biography
    • He read about the differential and integral calculus, teaching himself how to solve differential equations.

  895. Rees biography
    • In 1931 Rees graduated with her doctorate for a thesis entitled Division algebras associated with an equation whose group has four generators.

  896. Davidov biography
    • As well as his work on the equilibrium of a floating body, Davidov also worked on partial differential equations, elliptic functions and the application of probability to statistics.

  897. Reye biography
    • While undertaking research at Gottingen, he had attended inspiring lectures on partial differential equations by Bernhard Riemann.

  898. Lyapunov biography
    • (4) (1993), 3-47.',8)">8] include: stability, particularly the stability of critical points; the construction and the application of the Lyapunov function; stability of functional- differential equations; the second Lyapunov method; and the method of the Lyapunov vector function in stability theory and nonlinear analysis.

  899. Weil biography
    • At this time he was particularly fascinated by solving Diophantine equations.

  900. Al-Biruni biography
    • These include: theoretical and practical arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, Archimedes' theorems, trisection of the angle and other problems which cannot be solved with ruler and compass alone, conic sections, stereometry, stereographic projection, trigonometry, the sine theorem in the plane, and solving spherical triangles.

  901. Perelman biography
    • A possible approach to attacking the Poincare Conjecture had been developed by Richard Hamilton who had introduced a significant idea in 1982 when he began to study a particular equation he called the Ricci flow.

  902. Menabrea biography
    • The principle of Menabrea states that the elastic energy of a body in perfect elastic equilibrium is a minimum with respect to any possible system of stress-variation compatible with the equations of the statics of continua in addition to the boundary conditions.

  903. Huygens biography
    • History Topics: Quadratic, cubic and quartic equations .

  904. Lindelof biography
    • Lindelof's first work in 1890 was on the existence of solutions for differential equations.

  905. Iyanaga biography
    • Although it was this latter topic which would eventually attract Iyanaga, at this stage of his undergraduate career he was attracted to differential equations.

  906. Spanier biography
    • Interestingly, one of Spanier's theories, now called Alexander-Spanier homology, is currently being applied to analyse differential equations - a return to Poincare's original use of algebraic topology.

  907. Stirling biography
    • In the minutes of a meeting of the Royal Society of London on 4 April 1717, when Brook Taylor lectured on extracting roots of equations and on logarithms, it is recorded:- .

  908. Krein biography
    • During this time he worked on topics such as Banach spaces, the moment problem, integral equations and matrices, and on spectral theory for linear operators.

  909. Linnik biography
    • The systems of diophantine equations studied by these methods and the flows of lattice points introduced by these methods are closely related to the behaviour of the ideal classes of the corresponding algebraic fields.

  910. Hamel biography
    • He wrote papers on differential equations, and on fluid dynamics, for example on the critical velocity of a fluid, that is the velocity at which the flow changes from laminar to turbulent.

  911. Turan biography
    • Their importance first of all is that they lead to interesting deep problems of a completely new type; they have quite unexpectedly surprising consequences in many branches of mathematics - differential equations, numerical algebra, and various branches of function theory.

  912. Morin Ugo biography
    • Several of the mathematicians at Padua were very active opponents of Fascism, including Eugenio Curiel (1912-1945), who had been appointed assistant professor of rational mechanics in February 1934, Ernesto Laura (1879-1949), the professor of rational mechanics from 1922 and director of the mathematics seminar, and Giuseppe Zwirner (1904-1979), who worked on ordinary differential equations and was very active in the anti-Fascist Giustizia e Liberta movement.

  913. Potts biography
    • In addition to research on Ising-type models in mathematical physics and on road traffic analysis, Potts contributed to three other areas of research: operations research, especially networks; difference equations; and robotics.

  914. Mackey biography
    • The new material in the present book is concentrated in the last 50 pages and it centres around lattice models in statistical mechanics, PDEs in hydrodynamics, Kac-Moody Lie algebras, and the Korteweg-de Vries equation.

  915. Lorgna biography
    • We indicate below the titles of some of his works but let us record here that, among the pure mathematical topics he worked on, was geometry, convergence of series and algebraic equations.

  916. Aida biography
    • Aida explained the use of algebraic expressions and the construction of equations.

  917. Nalli biography
    • In the year 1919 Nalli started to work on issues related to the theory of linear integral equations and the study of integral operators.

  918. Weingarten biography
    • In this work he reduced the problem of finding all surfaces isometric to a given surface to the problem of determining all solutions to a partial differential equation of the Monge-Ampere type.

  919. Seifert biography
    • Seifert, still able to do mathematical research, worked on differential equations and wrote a series of papers on the topic through the war years.

  920. Pollaczek biography
    • In my theory the task of carrying out these integrations is reduced to the problem of resolving one or several systems of s linear non-homogeneous integro-functional equations of a new kind.

  921. Stackel biography
    • In 1891, Stackel's habilitation thesis, Integration of Hamiltonian-Jacobian differential equations by means of separation of variables, was accepted by the University of Halle, near Leipzig, and he took up a lectureship there.

  922. Slutsky biography
    • While at the Kiev Institute of Commerce, Slutsky gave the fundamental equation of value theory to economics.

  923. Lusztig biography
    • The thesis was published under the same title in the Journal of Differential Equations in 1972.

  924. Foster biography
    • Equational characterization of factorization in Mathematische Annalen.

  925. Conforto biography
    • a compilation of formulas and tables of coefficients relevant to a generalisation of the Clapeyron three-moment equation to the case of a continuous beam with piecewise linear variation of bending stiffness, supported at a finite number of points and subject to a uniform transverse loading and to an axial thrust.

  926. Wiener Norbert biography
    • In 1914 he went to Gottingen to study differential equations under Hilbert, and also attended a group theory course by Edmund Landau.

  927. Van Kampen biography
    • Wintner had worked at Johns Hopkins since 1930, the year before van Kampen arrived, and his interests were in almost-periodic functions and differential equations.

  928. Burchnall biography
    • In both the joint papers and his single author papers he wrote on differential equations, hypergeometric functions and Bessel functions.

  929. Krawtchouk biography
    • He wrote papers on differential and integral equations, studying both their theory and applications.

  930. Faltings biography
    • My main interests are arithmetic geometry (diophantine equations, Shimura-varieties), p-adic cohomology (relation crystalline to etale, p-adic Hodge theory), and vector bundles on curves (Verlinde-formula, loop-groups, theta-divisors).

  931. Lax Anneli biography
    • The title was On Cauchy's Problem for Partial Differential Equations with Multiple Characteristics, and it was published in Communications on Pure and Applied Mathematics in 1956.

  932. Adrain biography
    • Adrain's first papers in the Mathematical Correspondent concerned the steering of a ship and Diophantine algebra (the study of rational solutions to polynomial equations).

  933. Bochner biography
    • He also published papers on the gamma function, the zeta function and partial differential equations.

  934. Grandi biography
    • Grandi also applied the term "clelies" to the curves determined by certain trigonometric equations involving the sine function .

  935. Castelli biography
    • He remarks with wry amusement on the gay times had by the many knights and gentlefolk in the cardinal's entourage, while he devoted himself instead to the solution of hundreds of equations.

  936. Wilton biography
    • Papers Wilton published during this period include: On plane waves of sound (1913); On the highest wave in deep water (1913); On deep water waves (1914); Figures of equilibrium of rotating fluid under the restriction that the figure is to be a surface of revolution (1914); On the potential and force function of an electrified spherical bowl (1914-15); On ripples (1915); On the solution of certain problems of two-dimensional physics (1915); A pseudo-sphere whose equation is expressible in terms of elliptic functions (1915); and A formula in zonal harmonics (1916-17).

  937. Hippocrates biography
    • Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration.

  938. Sperner biography
    • The first chapter deals with affine space and linear equations.

  939. Hellinger biography
    • With Toeplitz he wrote a monumental survey of the literature on integral equations up to 1923 for Klein's Enzyklopadie der Mathematischen Wissenschaften.

  940. Savage biography
    • His grades began to improve: C in analytic geometry; B in calculus; B in differential equations; A in Raymond Wilder's foundations of mathematics; and A in Raymond Wilder's point set topology course.

  941. Bromwich biography
    • T J I'A Bromwich's method for solving the source-free Maxwell equations for electromagnetic waves.

  942. Lasker biography
    • He did not neglect his mathematics however, and in 1893 he lectured on differential equations at Tulane University in New Orleans.

  943. Zolotarev biography
    • He then continued his studies at the Faculty of Physics and Mathematics investigating an indeterminate equation of degree three.

  944. Chrystal biography
    • Chrystal's mathematical publications cover many topics including non-euclidean geometry, line geometry, determinants, conics, optics, differential equations, and partitions of numbers.

  945. Rao biography
    • Putting chance to work (1989), (with D N Shanbhag) Choquet-Deny type functional equations with applications to stochastic models (1994), (with H Toutenburg) Linear models.

  946. Trahtman biography
    • At Bar-Ilan University, Trahtman taught courses in discrete mathematics, theory of sets, algebra, analytical geometry, mathematical logic, finite automata, formal languages, rings and modules, and differential equations.

  947. Kaluza biography
    • Kaluza is remembered for this in Kaluza-Klein (named after the mathematician Oskar Klein) field theory, which involved field equations in five-dimensional space.

  948. Archimedes biography
    • History Topics: Pell's equation .

  949. Segre Corrado biography
    • Among other important work which Segre produced was an extension of ideas of Darboux on surfaces defined by certain differential equations.

  950. Castillon biography
    • He also studied conic sections, cubic equations and artillery problems.
    • Among his later mathematical publications we note: Memoire sur la regle de Cardan, et sur les equations cubique, avec quelques remarques sur les equations en general (1783) and two memoirs in 1790 and 1791 entitled Examen philosophique de quelque principes de l'algebre.

  951. Efimov biography
    • Thus, although applications (e.g., to rigid body dynamics and elasticity theory) are mentioned and the usual matrix theory is covered (including, e.g., reduction to the Jordan canonical form), there is none of the standard material on the solution of systems of linear equations.

  952. Bruns biography
    • He worked on the three-body problem showing that the series solutions of the Lagrange equations can change between convergent to divergent for small perturbations of the constants on which the coefficients of the time depend.

  953. Birman biography
    • The particular problems that are suggested by braids have led me to knot theory, to operator algebras, to mapping class groups, to singularity theory, to contact topology, to complexity theory and even to ordinary differential equations and chaos.

  954. Thomae biography
    • He also discovered methods of solving difference equations giving solutions in the form of definite integrals.

  955. Cox Elbert biography
    • In 1925 Cox was awarded his doctorate for his thesis Polynomial solutions of difference equations.

  956. Kelland biography
    • He wrote analytical papers on General Differentiation (1839), and Differential Equations (1853), and gave a geometrical Theory of Parallels outlining a version of non-Euclidean geometry.

  957. Ghetaldi biography
    • We now think of Descartes as founding the application of algebra to geometry, and although Ghetaldi never quite managed to achieve this breakthrough (nowhere in his work are there algebraic equations for geometric objects) nevertheless he came very close.

  958. Bernstein Felix biography
    • His range of interests were remarkable and he worked on convex functions, isoperimetric problems, the Laplace transform, number theory (including Fermat's Last Theorem), differential equations and the mathematical theory of genetics.

  959. Fomin biography
    • He worked with a number of collaborators from 1973 on the writing of a monograph on measure theory and differential equations.

  960. Kellogg biography
    • In 1908 he published three papers, namely Potential functions on the boundary of their regions of definition and Double distributions and the Dirichlet problem, both in the Transactions of the American Mathematical Society, and A necessary condition that all the roots of an algebraic equation be real in the Annals of Mathematics.

  961. Morse biography
    • Morse theory is important in the field of global analysis which is the study of ordinary and partial differential equations from a global or topological point of view.

  962. Tinseau biography
    • Tinseau wrote on the theory of surfaces, working out the equation of a tangent plane at a point on a surface, and he generalised Pythagoras's theorem proving that the square of a plane area is equal to the sum of the squares of the projections of the area onto mutually perpendicular planes.

  963. Rothblum biography
    • You could do and did many things very well, but nobody, like nobody, could push nasty equations around like you could ..

  964. Helmholtz biography
    • Helmholtz attempted to give a mechanical foundation to thermodynamics, and he also tried to derive Maxwell's electromagnetic field equations from the least action principle.

  965. Francais Jacques biography
    • It was a work on the integration of first order partial differential equations, but the memoir had been lost so there are few details as to its precise contents.

  966. Baker biography
    • Its contents are as follows: Euclid's theory of parallel lines; Propositions of incidence; The symbolic representation and Pappus' theorem; Theorems proved from the propositions of incidence; The fundamental hypothesis; The symbols of the real points of a line; Involution and harmonic ranges; Related ranges and pencils; Conics; Assignment of two absolute points, properties of circles; The parabola; The rectangular hyperbola; Theorems on conics; Length and distance; Equation of conic and line.

  967. Schlomilch biography
    • Conics are treated at first individually and in detail so as to bring out particular geometrical properties; then, starting with the equation of the second degree, they are discussed and reduced to their simplest forms and incidentally, as it were, we are introduced to the general properties of the curves of the second and higher orders.

  968. Fuss biography
    • Most of Fuss's papers are solutions to problems posed by Euler on spherical geometry, trigonometry, series, differential geometry and differential equations.

  969. Wright biography
    • He [was] interested in many different strands of analysis, being one of the first to work on difference-differential equations.

  970. Smirnov biography
    • Smirnov was a very active member of this circle, for example lecturing on the theory of algebraic equations, particularly the work of Goursat and Appell.

  971. Suetuna biography
    • In particular he read Hardy and Littlewood's paper The approximate functional equation in the theory of the zeta function with applications to the divisor problems of Dirichlet and Piltz which appeared in the Proceedings of the London Mathematical Society.

  972. Rudin Walter biography
    • The first is that the choice of topics serves as a superior introduction into much of what is current in analysis, in particular to the branches of harmonic analysis, partial differential equations, several complex variables, and Banach algebras.

  973. Ostrovskii biography
    • The authors describe the applications to the interpolation by entire functions, to entire and meromorphic solutions of ordinary differential equations, to the Riemann boundary problem with an infinite index and to the arithmetic of the convolution semigroup of probability distributions.

  974. Peirce B O biography
    • master of the methods dealing with the partial differential equations of mathematical physics.

  975. Lowenheim biography
    • Lowenheim analysed and improved upon the customary methods of solving equations in the calculus of classes or domains (that is, set theory in its Peirce-Schroder [Charles Peirce and Ernst Schroder] setting) and proved what is now known as Lowenheim's general development theorem for functions of functions.

  976. Chen biography
    • An outstanding and original mathematician, Chen's work falls naturally into three periods: his early work on group theory and links in the three sphere; his subsequent work on formal differential equations, which gradually developed into his most powerful and important work; and his work on iterated integrals and homotopy theory, which occupied him for the last twenty years of his life.

  977. Grauert biography
    • Much space is occupied by the treatment of systems of linear equations (the Gaussian algorithm), the theory of determinants and the theory of eigenvalues of linear mappings in 'Euclidean' vector spaces (transformation of principal axes).

  978. Feldman biography
    • The last part of the book describes Alan Baker's work on linear forms in the logarithms of algebraic numbers and its applications to Diophantine equations and to the determination of imaginary quadratic fields with class number 1 or 2.

  979. Ivory biography
    • Ivory wrote several articles for encyclopaedias, including the influential Equations in Encyclopaedia Britannica.

  980. Bayes biography
    • This notebook contains a considerable amount of mathematical work, including discussions of probability, trigonometry, geometry, solution of equations, series, and differential calculus.

  981. Khinchin biography
    • It was first published in 1943 and the eight lectures it contains are: Continuum; Limits; Functions; Series; Derivative; Integral; Series expansions of functions; and Differential equations.

  982. Zhukovsky biography
    • Today it is known as the Kutta-Joukowski theorem, since Kutta pointed out that the equation also appears in his 1902 dissertation.

  983. Ward Seth biography
    • Arithmetic and geometry are sincerely and profoundly taught, analytical algebra, the solution and application of equations, containing the whole mystery of both those sciences, being faithfully expounded in the Schools by the Professor of Geometry, and in several Colleges by particular tutors.

  984. Adams biography
    • He began this work in 1851 when elected as President of the Royal Astronomical Society and he presented a paper to the Royal Society in 1853 in which he showed that Laplace had omitted terms from his equations which were not negligible.

  985. Nygaard biography
    • In 1952 he published On the solution of integral equations by Monte-Carlo methods as a Norwegian Defence Research Establishment Report.

  986. Meders biography
    • Adolf Kneser, who had been taught by Kronecker and written a thesis on algebraic functions and equations, was the professor at Dorpat.

  987. Langlands biography
    • my only active encounter with partial differential equations, a subject to which I had always hoped to return but in a different vein.

  988. Atwood biography
    • Atwood also published on equations for the use of Hadley's quadrant.

  989. Tinbergen biography
    • In his contribution to the debate Tinbergen projected a 'quantitative stylising of the Dutch economy' to isolate the important factors and their effects by means of a set of definitions and equations.

  990. Stueckelberg biography
    • Occasionally all three of us would gather around the table while Kramers wrote out a few equations, illustrating how they fitted together to explain some atomic property or other.

  991. Lang biography
    • Your famous theorem in Diophantine equations earned you the distinguished Cole Prize of the American Mathematical Society.

  992. Krull biography
    • his earlier studies, but also dealt with other fields of mathematics: group theory, calculus of variations, differential equations, Hilbert spaces.

  993. Roy biography
    • This work led to partial differential equations which could only be solved numerically, but at this time the Applied Mathematics Department had no computing facilities.

  994. Wielandt biography
    • It was a matter of estimating eigenvalues of non-self-adjoint differential equations and matrices.

  995. Chatelet Albert biography
    • Clairin, who applied group theory to the solution of differential equations, had published Cours de mathematiques generales (1910).

  996. Aleksandrov Aleksandr biography
    • These first three works were all as a result of his mathematical work with Delone but also in 1934 he published two physics papers on quantum mechanics On the calculation of the energy of a bivalent atom by Fok's method and Remark on the commutation rule in Schrodinger's equation.

  997. Coriolis biography
    • It is not the ideas of 'work' for which Coriolis is best remembered, however, rather it is for the Coriolis force which appears in the paper Sur les equations du mouvement relatif des systemes de corps (1835).
    • He showed that the laws of motion could be used in a rotating frame of reference if an extra force called the Coriolis acceleration is added to the equations of motion.

  998. Campbell biography
    • In a paper published two years later "On the Theory of Simultaneous Partial Differential Equations" he develops a system of formulas by which it may be determined whether such a system is or is not integrable.

  999. Kothe biography
    • In the following years I had the pleasure to attend his inspiring lectures on "Hilbert Space Theory", "Partial Differential Equations", "Game Theory" and especially about the field of his main interest "Topological Vector Spaces".

  1000. Birnbaum biography
    • After arriving in Gottingen, Edmund Landau became his advisor, and he attended several lecture courses: differential equations given by Courant; calculus of variations given by Courant; power series given by Landau; higher geometry given by Herglotz; probability calculus given by Bernays; analysis of infinitely many variables given by Wegner; and attended the mathematical seminar directed by Courant and Herglotz.

  1001. Bernoulli Nicolaus(II) biography
    • Nicolaus worked on curves, differential equations and probability.

  1002. Lehto biography
    • Originally not much more than a curiosity, its notions began to pervade the theory of elliptic differential equations in two variables (Lavrent'ev, Morrey), and Teichmuller saw it not merely as an efficient tool in geometric function theory, but as a gateway to new problems of unmistakably classical flavour.

  1003. Rudio biography
    • He reduced this problem to the problem of solving a differential equation.

  1004. Bachmann biography
    • The book gives in very convenient form the chief results of 284 years of struggle with the problem of proving the possibility or impossibility in integers of the equation xn + yn = zn for values of n greater than 2.

  1005. Sperry biography
    • Wilczynski had begun his research career as a mathematical astronomer but his study of the dynamics of astronomical objects had turned his interests towards differential equations and then to projective differential geometry and ruler surfaces.

  1006. Griffiths Brian biography
    • We give the titles of a few of his mathematical education article which give an overview of his interests in that topic: Pure mathematicians as teachers of applied mathematicians (1968); Mathematics Education today (1975); Successes and failures of mathematical curricula in the past two decades (1980); Simplification and complexity in mathematics education (1983); The implicit function theorem: technique versus understanding (1984); A critical analysis of university examinations in mathematics (1984); Cubic equations, or where did the examination question come from? (1994); The British Experience of Teaching Geometry since 1900 (1998); and The Divine Proportion, matrices and Fibonacci numbers (2008).

  1007. Zeeman biography
    • I suppose I am particularly fond of having unknotted spheres in 5-dimensions, of spinning lovely examples of knots in 4-dimensions, of proving Poincare's Conjecture in 5-dimensions, of showing that special relativity can be based solely on the notion of causality, and of classifying dynamical systems by using the Focke-Plank equation.

  1008. Karlin biography
    • These ideas play a basic role in problems involving convexity, moment spaces, orthogonal polynomials, Chebyshev systems, the oscillation properties of linear differential equations, and the theory of approximation.

  1009. Eddington biography
    • It was Dirac's 1928 paper on the wave equation of the electron which had first set Eddington on the path of seeking ways to unify quantum mechanics and general relativity.

  1010. Kaluznin biography
    • The school provided a solid background in mathematics, including topics in the foundations of analysis, differential equations and complex variables.

  1011. Mobius biography
    • He avoided the army and completed his Habilitation thesis on Trigonometrical equations.

  1012. Mertens biography
    • Mertens is perhaps best known for his determination of the sign of Gauss sums, his work on the irreducibility of the cyclotomic equation, and the hypothesis which bears his name.

  1013. Bliss biography
    • They were An existence theorem for a differential equation of the second order, with an application to the calculus of variations and Sufficient condition for a minimum with respect to one-sided variations.

  1014. Carre biography
    • Between 1701 and 1705, Carre published over a dozen papers on a variety of mathematical and physical subjects: Methode pour la rectification des lignes courbes par les tangentes (1701); Solution du probleme propose aux Geometres dans les memoires de Trevoux, des mois de Septembre et d'Octobre (1701); Reflexions ajoutees par M Carre a la Table des Equations (1701); Observation sur la cause de la refraction de la lumiere (1702); Pourquoi les marees vont toujours en augmentant depuis Brest jusqu'a Saint-Malo, et en diminuant le long des cotes de Normandie (1702); Nombre et noms des instruments de musique (1702); Observations sur la vinaigre qui fait rouler de petites pierres sur un plan incline (1703); Observation sur la rectification des caustiques par reflexions formees par le cercle, la cycloide ordinaire, et la parabole, et de leurs developpees, avec la mesure des espaces qu'elle renferment (1703); Methode pour la rectification des courbes (1704); Observation sur ce qui produit le son (1704); Examen d'une courbe formee par le moyen du cercle (1705); Experiences physiques sur la refraction des balles de mousquet dans l'eau, et sur la resistance de ce fluide (1705); and Probleme d'hydrodynamique sur la proportion des tuyaux pour avoir une quantite d'eau determinee (1705).

  1015. Esclangon biography
    • Esclangon elaborated a theory for these functions, studied their differentiation and integration, and examined the differential equations which allow them as coefficients.

  1016. Divinsky biography
    • You use a lot of differential equations and what is called the maximum likelihood function.

  1017. Stormer biography
    • Poincare had, in the same year, solved the differential equations resulting from the motion of a charged particle in the field of a single pole.

  1018. Wussing biography
    • His main thesis, ably defended and well documented, is that the roots of the abstract notion of group do not lie, as frequently assumed, only in the theory of algebraic equations, but that they are also to be found in the geometry and the theory of numbers of the end of the 18th and the first half of the 19th centuries.

  1019. Borel biography
    • In [Enseignement mathematique 11 (1965), 1-95.',8)">8] Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics.

  1020. Frege biography
    • He lectured on all branches of mathematics, in particular analytic geometry, calculus, differential equations, and mechanics, although his mathematical publications outside the field of logic are few.

  1021. Birkhoff Garrett biography
    • He attended a course on potential theory given by Oliver Kellogg which gave him a good understanding of differential equations.

  1022. Feynman biography
    • Feynman seemed to possess a frightening ease with the substance behind the equations, like Einstein at the same age, like the Soviet physicist Lev Landau - but few others.

  1023. Merrifield biography
    • This was on the strength of some excellent mathematical papers on the calculation of elliptic functions, the first of which was The geometry of the elliptic equation which he published in 1858.

  1024. Weyl biography
    • This thesis investigated singular integral equations, looking in depth at Fourier integral theorems.

  1025. Rota biography
    • The topics were wide-ranging: differential equations, ergodic theory, nonstandard analysis, probability, and of course, combinatorics.

  1026. Eckmann biography
    • Peter Hilton, who had been a personal friend of Eckmann's for many years spoke in detail of Eckmann's research in topology: continuous solutions of systems of linear equations, a group-theoretical proof of the Hurwitz-Radon theorem, complexes with operators, spaces with means, simple homotopy type.

  1027. Xu Guangqi biography
    • The brilliant "tian yuan" or "coefficient array method" or "method of the celestial unknown" for solving equations which had been expounded with such skill by Li Zhi in the 13th century was no longer understood in China.

  1028. Cassels biography
    • After further papers on Diophantine equations and Diophantine approximation he wrote a series of five papers on Some metrical theorems in Diophantine approximation.

  1029. De Bruijn biography
    • He began publishing papers on combinatorics relevant to his work during this period such as The problem of optimum antenna current distribution (1946), A combinatorial problem (1946), On the zeros of a polynomial and of its derivative (1946), and A note on van der Pol's equation (1946) [Applied Logic Series 28 (Kluwer Academic Publishers, Dordrecht, 2003).',1)">1]:- .

  1030. Lamb biography
    • In a famous paper in the Proceedings of the London Mathematical Society he showed how Rayleigh's results on the vibrations of thin plates fitted with the general equations of the theory.

  1031. Russell Scott biography
    • This is now recognised as a fundamental ingredient in the theory of 'solitons', applicable to a wide class of nonlinear partial differential equations.

  1032. Ascoli biography
    • Between 1926 and 1930 he published twelve important works on partial differential equations: these include Sul problema di Dirichlet nei campi sferici e ipersferici (1927); Sulle singolarita isolate delle funzioni armoniche (1928); Sull'unicita della soluzione nel problema di Dirichlet (1928); and Sull'equazione di Laplace dello spazio iperbolico (1929).

  1033. Gaschutz biography
    • This is an example that shows how minor variations of the initial conditions can influence the solutions of an equation considerably.

  1034. Duhem biography
    • It is to this reading, to these exchanges of views, that I owe the greater part of my later works, almost all of which deal with the calculus of variations, the theory of Hugoniot, hyperbolic partial differential equations, Huygens' principle.

  1035. Hahn biography
    • These include a report on integral equation he wrote in 1911, his modification of Hellinger's theory of invariants of quadratic forms, in which he dispensed with the use of the Hellinger integral, and his work on duality in Banach spaces, culminating with his proof of the Hahn-Banach theorem in 1927.

  1036. Schramm biography
    • Schramm was recognized for his development of stochastic Loewner equations and for his contributions to the geometry of Brownian curves in the plane.

  1037. Bour biography
    • Bour continued his studies at the Ecole des Mines in Paris and worked on a major paper Sur l'integration des equations differentielles de la mecanique analytique which was read before the Academie des Sciences on 5 March 1855 and published in the Journal de mathematiques pures et appliquees.
    • in line with the analogous studies of Bonnet and Codazzi, contained several theorems on ruled surfaces and minimal surfaces; but in its printed version this work does not include the test for the integration of the problem's equations in the case of surfaces of revolution, which had enabled Bour to surpass the other competitors for the Academy's grand prize.

  1038. Turnbull biography
    • Turnbull's major beautifully written works include The Theory of Determinants, Matrices, and Invariants (1928), The Great Mathematicians (1929), Theory of Equations (1939), The Mathematical Discoveries of Newton (1945), and An Introduction to the Theory of Canonical Matrices (1945), which was jointly written with Aitken.

  1039. Neumann Carl biography
    • In 1890 Emile Picard used Neumann's results to develop his method of successive approximation which he used to give existence proofs for the solutions of partial differential equations.

  1040. Dini biography
    • In this last work he devoted a chapter to integral equations in which he presented many of his own innovative ideas.

  1041. Burgess biography
    • And Concurrency of lines joining vertices of a triangle to opposite vertices of triangles on its sides; determinants connected with the periodic solutions of Mathieu's equation.

  1042. Van der Waerden biography
    • In Galois theory he showed the asymptotic result that almost all integral algebraic equations have the full symmetric group as Galois group.

  1043. Janovskaja biography
    • An analysis is given for the problem of finding geometric solutions for algebraic equations of degree higher than two by locating points of intersection of conic sections with other curves.

  1044. Akhiezer biography
    • Examples of his papers from his time in Kiev are: On polynomials deviating least from zero (Russian) (1930), On the extremal properties of certain fractional functions (Russian) (1930), On a minimum problem in the theory of functions, and on the number of roots of an algebraic equation which lie inside the unit circle (Russian) (1931), and Uber einige Funktionen, welche in zwei gegebenen Intervallen am wenigsten von Null abweichen (3 parts, 1932-1933).

  1045. Sluze biography
    • This work was on geometrical construction in which he discussed the cubature of various solids and the solutions to third and fourth degree equations which he obtained geometrically using the intersection of any conic section with a circle.

  1046. Weise biography
    • In years soon after the war Weise published a small but concise book Gewohnliche Differentialgleichungen (1948) in which he discusses Legendre, Bessel, and Sturm-Liouville equations.

  1047. Schwarz biography
    • An idea from this work, in which he constructed a function using successive approximations, led Emile Picard to his existence proof for solutions of differential equations.

  1048. Coulomb biography
    • From examination of many physical parameters, he developed a series of two-term equations, the first term a constant and the second term varying with time, normal force, velocity, or other parameters.

  1049. Heine biography
    • Before arriving at Halle, Heine published on partial differential equations and during his first few years teaching at Halle he wrote papers on the theory of heat, summation of series, continued fractions and elliptic functions.

  1050. Mellin biography
    • He also extended his transform to several variables and applied it to the solution of partial differential equations.

  1051. Stieltjes biography
    • Stieltjes also contributed to ordinary and partial differential equations, the gamma function, interpolation, and elliptic functions.

  1052. Berwick biography
    • Berwick also gave, in 1915, necessary and sufficient conditions for a quintic equation to be soluble by radicals.

  1053. Bouvard biography
    • Using all the data at his disposal, Bouvard produced a system of 77 equations but was unable to find a possible orbit for the planet from them.

  1054. Stark biography
    • In 1948 he published On a functional equation and, in the following year, the paper On a ratio test of Frink in which he gives an extension of Raabe's test for convergence.

  1055. Helly biography
    • His thesis was on Fredholm equations.


History Topics

  1. Weather forecasting
    • 3.3 Primitive Equations .
    • Moreover, the primitive equations describing atmospheric processes, which are used in forecasting models, are presented and explained as well.
    • The finite element method, which is another method for numerically solving partial differential equations, is described briefly.
    • Furthermore, I have assumed that the reader is familiar with differential equations, differentiation of functions of several variables, Fourier series and Gaussian elimination.
    • However, as the article should provide only an overview of the mathematical methods used in current forecasting models, I have chosen to include only simple equations and explain some mathematical symbols in order to make understanding the methods easier.
    • Also, a number of mathematical concepts are explained in words rather than with equations.
    • In the early 20th century, scientists, in particular Vilhelm Bjerknes and Lewis Fry Richardson, pioneered numerical weather forecasting, which is based on applying physical laws to the atmosphere and solving mathematical equations associated to these laws.
    • We must apply the equations of theoretical physics not to ideal cases only, but to the actual existing atmospheric conditions as they are revealed by modern observations.
    • Bjerknes' equations were very complicated and not very practical for predicting the weather as they required immense computational power, a fact of which he was aware himself.
    • Nevertheless, he firmly believed that one day, meteorology would be a proper science and weather forecasts based on solving mathematical equations would be feasible: .
    • The first attempt to use mathematics in order to predict the weather was made by the British mathematician Lewis Fry Richardson (1881-1953), who simplified Bjerknes' equations so that solving them became more feasible.
    • He also worked for the National Peat Industries for some time, and in order to solve differential equations modelling the flow of water in peat, he invented his method for finite differences, which produces highly accurate results.
    • Basically, this method allows finding approximate solutions to differential equations.
    • A differential equation with a smooth variable is converted into a function (or an approximation thereof) that relates the changes of the variable and given steps in time and/or space, meaning that the changes are calculated at discrete points rather than at infinitely many points.
    • Then the derivatives in the differential equation are replaced by finite difference approximations (this method will be explained in more detail in section 4.1).
    • So in the place of the differential equation you get many equations which can be solved using arithmetic.
    • He remodelled the fundamental equations describing atmospheric processes such that it was possible to solve them numerically.
    • By dividing the surface of the Earth into thousands of grid squares, and the atmosphere into several horizontal layers, he obtained a large number of grid boxes, connected to one another by mathematical equations.
    • In terms of numerical weather prediction, this equation is important as it facilitated extended five-day forecasts [Stormwatchers.
    • The prerequisite for computer-generated forecasts was to simplify the full primitive equations that govern the atmosphere (they will be discussed in section 3.3), as the early computers were unable to deal with all the equations included in Richardson's model.
    • In 1948, Charney developed the quasi-geostrophic approximation, which reduces several equations of atmospheric motions to only two equations in two unknown variables [Atmospheric Dynamics (Chapter 1: Basics, Chapter 5: Balance of Forces in Synoptic Scale Flow, Chapter 13: Quasi-Geostrophic Equations) (University of Edinburgh, 2005) ',14)">14, chapter 13].
    • These equations are much easier to solve and could be handled by the early computers.
    • Furthermore, this approximation filters out all but the slow long-wave motions that are important in meteorology, so that you do not have to solve the primitive equations for acoustic and gravity waves as Richardson did 30 years earlier.
    • Although the computers were fed with simplified equations only, the limited computer power demanded a barotropic (i.e.
    • In 1963, a six-layer model based on the primitive equations was used for producing a forecast.
    • Since then, as computer power increased, the models have constantly been refined (meaning that more layers, a finer grid, more equations, topography and landscape characteristics were included) [Meteorology for Scientists and Engineers, second edition (Pacific Grove CA, 2000)',4)">4, p.
    • Lorenz reasoned that the dynamical equations that describe the atmosphere are exceedingly sensitive to initial conditions.
    • Dynamical equations are deterministic; meaning that given initial conditions, they determine how the process they describe will evolve in the future.
    • But before looking at current forecasting models, let us look at the primitive equations that form the basis of every such model.
    • 3.3) nnnnnPrimitive Equations .
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    • Scientists treat the Earth's atmosphere as if it were a fluid on a rotating sphere in order to describe large-scale atmospheric processes using the fundamental laws of thermodynamics and hydrodynamics, also called the primitive equations.
    • Essentially, they are the equations of motion, one for each of the three wind directions, the continuity equation, describing the conservation of mass, the ideal gas law, and the first law of thermodynamics, describing the conservation of energy.
    • There is also an equation for determining the humidity, which is not always included in the set of the primitive equations (and is not treated here).
    • Here, only the very basic versions of the primitive equations are described.
    • The equations of motion are based on Newton's second law : force equals the product of mass and acceleration.
    • Thus, the equation of motion can be rewritten as .
    • Then the equations of motion in spherical coordinates are: .
    • The basic principle underlying the continuity equation is the conservation of mass.
    • The continuity equation is used to determine the air density.
    • In the Eulerian reference frame, the continuity equation is .
    • Substituting the continuity equation by the above filter condition is called anelastic approximation [A Description of the Nonhydrostatic Regional Model LM.
    • The pressure in an air parcel is found using the equation of state, which relates pressure, temperature and density.
    • The equation describing the change of temperature T with respect to time t is: .
    • The other processes in the equation are adiabatic, meaning that there is no heat transfer.
    • The primitive equations were first used in a weather forecast by Lewis Fry Richardson.
    • Jule Charney and his colleagues had simplified them so that the early computers could handle them, but nowadays meteorologists have gone back to use all of Richardson's equations.
    • All current weather forecasting models are based on the primitive equations -- or versions thereof -- but each model uses different approximations and assumptions, resulting in slightly different outcomes.
    • Also, the models include equations accounting for the effects of small-scale processes such as convection, radiation, turbulence and the effects of mountains that cannot be represented explicitly by the forecasting models, as their resolution is not high enough.
    • Some models, such as the regional model COSMO developed by the German weather service Deutscher Wetterdienst (DWD), have therefore abandoned this assumption and are based on non-hydrostatic thermodynamic equations (similar to the equations used in fluid mechanics).
    • smaller grid spacing), but as a result, the primitive equations are much more complex and computationally more demanding as vertical wind components are included in the model [http://www.dwd.de',29)">29].
    • Again, the primitive equations have to be re-written in terms of ζ [A Description of the Nonhydrostatic Regional Model LM.
    • But let us now look at the methods employed by meteorologists to solve these equations.
    • Similarly, Richardson's finite difference method is not the only method for solving the primitive equations anymore; its strongest "opponent" is the so-called spectral method, which will be described in section 4.2.
    • Before the primitive equations can be solved, they have to be discretized with respect to space and time.
    • The atmosphere is then divided into a number of layers, resulting in a three-dimensional grid, in which the primitive equations can be solved for each grid point.
    • The regional model of the DWD has a resolution of 7 km, and the local model has a resolution of up to 2.8 km (the Met Office's models have coarser resolution as the Met Office does not work with non-hydrostatic equations).
    • The great advantage of the triangular grid is that the primitive equations can be solved in air parcels close to the poles without any problems, as opposed to the rectangular grid, where the longitudes approach each other, resulting in erroneous computations.
    • Now, the primitive equations have to be re-written in finite difference form.
    • The implicit scheme, on the other hand, is absolutely stable, but it results in a system of simultaneous equations, so is more difficult to solve [Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ',17)">17, p.
    • When the primitive equations are expressed in terms of finite differences, the equations soon become very long and take some computational effort to solve.
    • This means that the primitive equations are subdivided into forcing terms fψ referring to slowly varying modes and source terms sψ directly related to the fast-moving sound waves: .
    • representing a set of equations that can be solved using Gaussian elimination [A Description of the Nonhydrostatic Regional Model LM.
    • For equations including acoustically active terms, i.e.
    • If you re-write the primitive equations using finite differences, you get, "after considerable algebra" [A Description of the Nonhydrostatic Regional Model LM.
    • 67], a linear tridiagonal system of simultaneous equations which can be written in the general form .
    • The equation system can be solved for using a solving method based on Gaussian elimination and back-substitution.
    • In a nutshell, the derivatives in the primitive equations can be approximated by finite differences, such that the equations can be transformed into a linear equation system.
    • It takes modern supercomputers at the leading weather services quite a while to solve all these equations, so it is astonishing that Richardson managed to produce a numerical weather forecast at all, even if it was for a limited area.
    • However, the application of this method to the primitive equations was crucial to the development of numerical weather forecasting, as it was the only mathematical method that could simplify partial differential equations needed for forecasting for several decades.
    • A further disadvantage of the finite difference method, other than the great number of equations you have to solve, is that it does not reveal anything about the behaviour of the variables between the individual grid points.
    • One of the advantages of the spectral method is that the primitive equations can be solved in terms of global functions rather than in terms of approximations at specific points as in the finite difference method.
    • The partial differential equations are represented in terms of spherical harmonics, which are truncated at a total wave number of 799.
    • When this series is substituted into an equation of the form Lψ = f (x), where L is a differential operator, you get a so-called residual function: .
    • The residual function is zero when the solution of the equation above is exact, therefore the series coefficients an should be chosen such that the residual function is minimised, i.e.
    • A simple example that can be solved in terms of a Fourier series illustrates the idea of the spectral method: One of the processes described by the primitive equations is advection (which is the transport of for instance heat in the atmosphere), and the non-linear advection equation is given by .
    • Having chosen appropriate boundary conditions, the equation can be expanded in terms of a finite Fourier series: .
    • The advection equation then is: .
    • As each of the terms on the left-hand side of the equation has been truncated at a different wave number, there will always be a residual function.
    • There are several methods which convert differential equations to discrete problems, for example the least-square method or the Galerkin method, and which can be used in order to choose the time derivative such that the residual function is as close to zero as possible [Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ',17)">17, p.
    • It is difficult to calculate the non-linear terms of a differential equation in the context of the spectral method, but you can get around this problem by using a so-called transform method.
    • Using a transform method requires three steps, which will be shown for the non-linear term in the advection equation above: .
    • Still, using transform methods is necessary in order to solve differential equations in spectral space.
    • Spherical harmonics Ynm(λ, φ) are the angular part of the solution to Laplace's equation.
    • At the poles, the solutions to differential equations become infinitely differentiable; therefore the poles are usually excluded from the spectral space, which actually simplifies the method [Chebyshev and Fourier Spectral Methods, second edition (Mineola NY, 2000) ',5)">5, p.
    • A third technique for finding approximate solutions to partial differential equations and hence to the primitive equations is the finite element method.
    • The domain for which the partial differential equations have to be solved is divided into a number of subdomains, and a different polynomial is used to approximate the solution for each subdomain.
    • These approximations are then incorporated into the primitive equations.
    • One of the easiest ways to increase the quality of weather predictions is to increase the orders of the numerical approximations to partial differential equations.

  2. Quadratic etc equations
    • Quadratic, cubic and quartic equations .
    • It is often claimed that the Babylonians (about 400 BC) were the first to solve quadratic equations.
    • This is an over simplification, for the Babylonians had no notion of 'equation'.
    • What they did develop was an algorithmic approach to solving problems which, in our terminology, would give rise to a quadratic equation.
    • In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation.
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    • Euclid had no notion of equation, coefficients etc.
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    • He has six chapters each devoted to a different type of equation, the equations being made up of three types of quantities namely: roots, squares of roots and numbers i.e.
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    • Al-Khwarizmi gives the rule for solving each type of equation, essentially the familiar quadratic formula given for a numerical example in each case, and then a proof for each example which is a geometrical completing the square.
    • Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book Liber embadorum published in 1145 which is the first book published in Europe to give the complete solution of the quadratic equation.
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    • Pacioli does not discuss cubic equations but does discuss quartics.
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    • Dal Ferro is credited with solving cubic equations algebraically but the picture is somewhat more complicated.
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    • We believe that dal Ferro could only solve cubic equation of the form x3 + mx = n.
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    • However, without the Hindu's knowledge of negative numbers, dal Ferro would not have been able to use his solution of the one case to solve all cubic equations.
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    • Remarkably, dal Ferro solved this cubic equation around 1515 but kept his work a complete secret until just before his death, in 1526, when he revealed his method to his student Antonio Fior.
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    • Soon rumours started to circulate in Bologna that the cubic equation had been solved.
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    • Nicolo of Brescia, known as Tartaglia meaning 'the stammerer', prompted by the rumours managed to solve equations of the form x3 + mx2 = n and made no secret of his discovery.
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    • Cardan invited Tartaglia to visit him and, after much persuasion, made him divulge the secret of his solution of the cubic equation.
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    • This is a quadratic equation in a3, so solve for a3 using the usual formula for a quadratic.
    • Cardan knew that you could not take the square root of a negative number yet he also knew that x = 4 was a solution to the equation.
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    • After Tartaglia had shown Cardan how to solve cubics, Cardan encouraged his own student, Lodovico Ferrari, to examine quartic equations.
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    • Cardan published all 20 cases of quartic equations in Ars Magna.
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    • Rewrite this last equation as .
    • Solve this quadratic and we have the required solution to the quartic equation.
    • In the years after Cardan's Ars Magna many mathematicians contributed to the solution of cubic and quartic equations.
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    • In it he made many contributions to the understanding of cubic equations.
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    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Quadratic_etc_equations.html .

  3. Pell's equation
    • Pell's equation .
    • We will discuss below whether Pell's equation is properly named.
    • By this we mean simply: did Pell contribute at all to the study of Pell's equation? There is no doubt that the equation had been studied in depth for hundreds of years before Pell was born.
    • First let us say what Pell's equation is.
    • We are talking about the indeterminate quadratic equation .
    • Now, although it is fair to say that Brahmagupta was the first to study this equation, it is equally possible to see that earlier authors had studied problems related to Pell's equation.
    • To mention some briefly: Diophantus examines problems related to Pell's equation and we can reduce Archimedes' "cattle problem" to solving Pell's equation although there is no evidence that Archimedes made this connection.
    • In other words, if (a, b) and (c, d) are solutions to Pell's equation then so are .
    • This fundamentally important fact generalises easily to give Brahmagupta's lemma, namely that if (a, b) and (c, d) are integer solutions of 'Pell type equations' of the form .
    • are both integer solutions of the 'Pell type equation' .
    • In fact this method of composition allowed Brahmagupta to make a number of fundamental discoveries regarding Pell's equation.
    • One property that he deduced was that if (a, b) satisfies Pell's equation so does (2ab, b2 + na2).
    • Now of course the method of composition can be applied again to (a, b) and (2ab, b2+ na2) to get another solution and Brahmagupta immediately saw that from one solution of Pell's equation he could generate many solutions.
    • as a solution of Pell's equation nx2 + 1 = y2.
    • and this is an integer solution to Pell's equation.
    • If k = -2 then essentially the same argument works while if k = 4 or k = -4 then a more complicated method, still based on the method of composition, shows that integer solutions to Pell's equation can be found.
    • So Brahmagupta was able to show that if he could find (a, b) which "nearly" satisfied Pell's equation in the sense that na2 + k = b2 where k = 1, -1, 2, -2, 4, or -4 then he could find one, and therefore many, integer solutions to Pell's equation.
    • For example, if we attempt to solve 23x2 + 1 = y2 we see that a = 1, b = 5 satisfies 23a2 + 2 = b2 so, by the above argument, x = 5, y = 24 satisfies Pell's equation.
    • Among the examples Brahmagupta gives himself is a solution of Pell's equation .
    • He discovered the cyclic method, called chakravala by the Indians, which was an algorithm to produce a solution to Pell's equation nx2 + 1 = y2 starting off from any "close" pair (a, b) with na2 + k = b2.
    • The method relies on a simple observation, namely that, for any m, (1, m) satisfies the 'Pell type equation' .
    • to the 'Pell type equation' nx2 + (m2 - n)/k = y2 where (m2 - n)/k is also an integer.
    • If (m2 - n)/k is one of 1, -1, 2, -2, 4, -4 then we can apply Brahmagupta's method to find a solution to Pell's equation nx2 + 1 = y2.
    • If (m2 - n)/k is not one of these values then repeat the process starting this time with the solution x = (am + b)/k, y = (bm + na)/k to the 'Pell type equation' nx2 + (m2 - n)/k = y2 in exactly the same way as we applied the process to na2 + k = b2.
    • This happens when an equation of the form nx2 + t = y2 is reached where t is one of 1, -1, 2, -2, 4, -4.
    • as a solution to the 'Pell type equation' nx2 - 4 = y2.
    • But this is an equation which Brahmagupta's method solves giving .
    • Secondly the algorithm always reaches a solution of Pell's equation after a finite number of steps without stopping when an equation of the type nx2 + k = y2 where k = -1, 2, -2, 4, or -4 is reached and then applying Brahmagupta's method.
    • The next contribution to Pell's equation was made by Narayana who, in the 14th Century, wrote a commentary on Bhaskara II's Bijaganita.
    • Take m = 7 to get the equation .
    • Take m = 11 to get the equation .
    • Now Narayana applies Brahmagupta's method, in the form we gave above for equations with k = 2, to obtain the solutions .
    • His next example is a solution of Pell's equation .
    • which leads successively, by applying the cyclic method, to the equations .
    • Finally Narayana applies Brahmagupta's method to this last equation to get the solution .
    • Narbonese Gaul, of course, was the area around Toulouse where Fermat lived! One of Fermat's challenge problems was the same example of Pell's equation which had been studied by Bhaskara II 500 years earlier, namely to find solutions to .
    • Frenicle de Bessy tabulated the solutions of Pell's equation for all n up to 150, although this was never published and his efforts have been lost.
    • He challenged Brouncker who was claiming to be able to solve any example of Pell's equation to solve .
    • In 1658 Rahn published an algebra book which contained an example of Pell's equation.
    • This book was written with help from Pell and it is the only known connection between Pell and the equation which has been named after him.
    • Wallis published Treatise on Algebra in 1685 and Chapter 98 of that work is devoted to giving methods to solve Pell's equation based on the exchange of letters he had published in Commercium epistolicum in 1658.
    • We should note that by this time several mathematicians had claimed that Pell's equation nx2 + 1 = y2 had solutions for any n.
    • In fact Fermat claimed, correctly of course, that for any n Pell's equation had infinitely many solutions.
    • He was, of course, aware of the work of Brouncker on Pell's equation as presented by Wallis, but he was totally unaware of the contributions of the Indian mathematicians.
    • He gave the basis for the continued fractions approach to solving Pell's equation which was put into a polished form by Lagrange in 1766.
    • The other major contribution of Euler was in naming the equation "Pell's equation" and it is generally believed that he gave it that name because he confused Brouncker and Pell, thinking that the major contributions which Wallis had reported on as due to Brouncker were in fact the work of Pell.
    • Lagrange published his Additions to Euler's Elements of algebra in 1771 and this contains his rigorous version of Euler's continued fraction approach to Pell's equation.
    • This established rigorously the fact that for every n Pell's equation had infinitely many solutions.
    • will be the smallest solution to Pell's equation .
    • will give a second solution to the equation.
    • Here are the first few powers of (170 + 39√19), starting with its square, which gives the first few solutions to the equation 19x2 + 1 = y2 .
    • Although the continued fraction approach to solving Pell's equation is a very nice one for small values on n, the difficulty of the method has been analysed to see if it is the most efficient for large n.
    • The continued fraction method is not a polynomial time algorithm, and indeed it is now known that no polynomial time algorithm exists for solving Pell's equation.

  4. Matrices and determinants
    • It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations.
    • The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive.
    • He sets up the coefficients of the system of three linear equations in three unknowns as a table on a 'counting board'.
    • Our late 20th Century methods would have us write the linear equations as the rows of the matrix rather than the columns but of course the method is identical.
    • Cardan, in Ars Magna (1545), gives a rule for solving a system of two linear equations which he calls regula de modo and which [Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 766-774.',7)" onmouseover="window.status='Click to see reference';return true">7] calls mother of rules ! This rule gives what essentially is Cramer's rule for solving a 2 cross 2 system although Cardan does not make the final step.
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    • For example de Witt in Elements of curves, published as a part of the commentaries on the 1660 Latin version of Descartes' Geometrie , showed how a transformation of the axes reduces a given equation for a conic to canonical form.
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    • Using his 'determinants' Seki was able to find determinants of 2 cross 2, 3 cross 3, 4 cross 4 and 5 cross 5 matrices and applied them to solving equations but not systems of linear equations.
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    • He explained that the system of equations .
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    • two characters, the first marking in which equation it occurs, the second marking which letter it belongs to.
    • As well as studying coefficient systems of equations which led him to determinants, Leibniz also studied coefficient systems of quadratic forms which led naturally towards matrix theory.
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    • It arose out of a desire to find the equation of a plane curve passing through a number of given points.
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    • Cramer does go on to explain precisely how one calculates these terms as products of certain coefficients in the equations and how one determines the sign.
    • In 1772 Laplace claimed that the methods introduced by Cramer and Bezout were impractical and, in a paper where he studied the orbits of the inner planets, he discussed the solution of systems of linear equations without actually calculating it, by using determinants.
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    • Using observations of Pallas taken between 1803 and 1809, Gauss obtained a system of six linear equations in six unknowns.
    • Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix.
    • Cauchy also introduced the idea of similar matrices (but not the term) and showed that if two matrices are similar they have the same characteristic equation.
    • Jacques Sturm gave a generalisation of the eigenvalue problem in the context of solving systems of ordinary differential equations.
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    • In fact the concept of an eigenvalue appeared 80 years earlier, again in work on systems of linear differential equations, by D'Alembert studying the motion of a string with masses attached to it at various points.
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    • An algorithm for calculation can be based on this, it consists of applying the usual rules for the operations of multiplication, division, and exponentiation to symbolic equations between linear systems, correct symbolic equations are always obtained, the sole consideration being that the order of the factors may not be altered.
    • Cayley also proved that, in the case of 2 cross 2 matrices, that a matrix satisfies its own characteristic equation.
    • That a matrix satisfies its own characteristic equation is called the Cayley-Hamilton theorem so its reasonable to ask what it has to do with Hamilton.
    • In 1870 the Jordan canonical form appeared in Treatise on substitutions and algebraic equations by Jordan.
    • Frobenius also proved the general result that a matrix satisfies its characteristic equation.
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  5. Babylonian mathematics
    • In this article we now examine some algebra which the Babylonians developed, particularly problems which led to equations and their solution.
    • Now these could be used to solve equations.
    • For example they constructed tables for n3 + n2 then, with the aid of these tables, certain cubic equations could be solved.
    • For example, consider the equation .
    • Nevertheless the Babylonians could handle numerical examples of such equations by using rules which indicate that they did have the concept of a typical problem of a given type and a typical method to solve it.
    • For example in the above case they would (in our notation) multiply the equation by a2 and divide it by b3 to get .
    • Putting y = ax/b this gives the equation .
    • Again a table would have been looked up to solve the linear equation ax = b.
    • To solve a quadratic equation the Babylonians essentially used the standard formula.
    • They considered two types of quadratic equation, namely .
    • For example problems which led the Babylonians to equations of this type often concerned the area of a rectangle.
    • For example if the area is given and the amount by which the length exceeds the breadth is given, then the breadth satisfies a quadratic equation and then they would apply the first version of the formula above.
    • The equation .
    • Notice that the scribe has effectively solved an equation of the type x2 + bx = c by using x = √[(b/2)2 + c] - (b/2).
    • 40 (1956), 185-192.',10)">10] Berriman gives 13 typical examples of problems leading to quadratic equations taken from Old Babylonian tablets.
    • If problems involving the area of rectangles lead to quadratic equations, then problems involving the volume of rectangular excavation (a "cellar") lead to cubic equations.
    • The clay tablet BM 85200+ containing 36 problems of this type, is the earliest known attempt to set up and solve cubic equations.

  6. Quadratic etc equations references
    • References for: Quadratic, cubic and quartic equations .
    • A R Amir-Moez, Khayyam, al-Biruni, Gauss, Archimedes, and quartic equations, Texas J.
    • A E Berriman, The Babylonian quadratic equation, Math.
    • Isoperimetric problems and the origin of the quadratic equations, Isis 32 (1940), 101-115.
    • J P Hogendijk, Sharaf al-Din al-Tusi on the number of positive roots of cubic equations, Historia Mathematica 16 (1) (1989), 69-85.
    • B Hughes, The earliest correct algebraic solutions of cubic equations, Vita mathematica (Washington, DC, 1996), 107-112.
    • C Romo Santos, Cardano's 'Ars magna' and the solutions of cubic and quartic equations (Spanish), Rev.
    • P Schultz,Tartaglia, Archimedes and cubic equations, Austral.
    • G S Smirnova, Geometric solution of the cubic equations in Raffaele Bombelli's 'Algebra' (Russian), Istor.
    • P D Yardley, Graphical solution of the cubic equation developed from the work of Omar Khayyam, Bull.
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Quadratic_etc_equations.html .

  7. Quadratic etc equations references
    • References for: Quadratic, cubic and quartic equations .
    • A R Amir-Moez, Khayyam, al-Biruni, Gauss, Archimedes, and quartic equations, Texas J.
    • A E Berriman, The Babylonian quadratic equation, Math.
    • Isoperimetric problems and the origin of the quadratic equations, Isis 32 (1940), 101-115.
    • J P Hogendijk, Sharaf al-Din al-Tusi on the number of positive roots of cubic equations, Historia Mathematica 16 (1) (1989), 69-85.
    • B Hughes, The earliest correct algebraic solutions of cubic equations, Vita mathematica (Washington, DC, 1996), 107-112.
    • C Romo Santos, Cardano's 'Ars magna' and the solutions of cubic and quartic equations (Spanish), Rev.
    • P Schultz,Tartaglia, Archimedes and cubic equations, Austral.
    • G S Smirnova, Geometric solution of the cubic equations in Raffaele Bombelli's 'Algebra' (Russian), Istor.
    • P D Yardley, Graphical solution of the cubic equation developed from the work of Omar Khayyam, Bull.
    • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Quadratic_etc_equations.html] .

  8. Group theory
    • the theory of algebraic equations at the end of the 18th Century leading to the study of permutations.
    • (3) Permutations were first studied by Lagrange in his 1770 paper on the theory of algebraic equations.
      Go directly to this paragraph
    • Lagrange's main object was to find out why cubic and quartic equations could be solved algebraically.
      Go directly to this paragraph
    • In studying the cubic, for example, Lagrange assumes the roots of a given cubic equation are x', x'' and x'''.
      Go directly to this paragraph
    • The first person to claim that equations of degree 5 could not be solved algebraically was Ruffini.
      Go directly to this paragraph
    • In 1799 he published a work whose purpose was to demonstrate the insolubility of the general quintic equation.
      Go directly to this paragraph
    • In a paper of 1802 he shows that the group of permutations associated with an irreducible equation is transitive taking his understanding well beyond that of Lagrange.
      Go directly to this paragraph
    • His first paper on the subject was in 1815 but at this stage Cauchy is motivated by permutations of roots of equations.
      Go directly to this paragraph
    • Galois in 1831 was the first to really understand that the algebraic solution of an equation was related to the structure of a group le groupe of permutations related to the equation.
      Go directly to this paragraph
    • Betti began in 1851 publishing work relating permutation theory and the theory of equations.
      Go directly to this paragraph
    • In fact Betti was the first to prove that Galois' group associated with an equation was in fact a group of permutations in the modern sense.
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  9. Fund theorem of algebra
    • Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers.
    • Early studies of equations by al-Khwarizmi (c 800) only allowed positive real roots and the FTA was not relevant.
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    • This discovery was made in the course of studying a formula which gave the roots of a cubic equation.
      Go directly to this paragraph
    • The formula when applied to the equation x3 = 15x + 4 gave an answer involving √-121 yet Cardan knew that the equation had x = 4 as a solution.
      Go directly to this paragraph
    • Descartes in 1637 says that one can 'imagine' for every equation of degree n, n roots but these imagined roots do not correspond to any real quantity.
      Go directly to this paragraph
    • Viete gave equations of degree n with n roots but the first claim that there are always n solutions was made by a Flemish mathematician Albert Girard in 1629 in L'invention en algebre .
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    • They believed that a polynomial equation of degree n must have n roots, the problem was, they believed, to show that these roots were of the form a + bi, a, b real.
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    • His proof in Recherches sur les racines imaginaires des equations is based on decomposing a monic polynomial of degree 2n into the product of two monic polynomials of degree m = 2n-1.
    • Lagrange used his knowledge of permutations of roots to fill all the gaps in Euler's proof except that he was still assuming that the polynomial equation of degree n must have n roots of some kind so he could work with them and deduce properties, like eventually that they had the form a + bi, a, b real.
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    • if one carries out operations with these impossible roots, as though they really existed, and says for example, the sum of all roots of the equation xm+axm-1 + bxm-2 + .
    • Lagrange's 1808 2nd Edition of his treatise on equations makes no mention of Gauss's new proof or criticisms.
      Go directly to this paragraph
    • In 1849 (on the 50th anniversary of his first proof!) Gauss produced the first proof that a polynomial equation of degree n with complex coefficients has n complex roots.
    • Euler gave the most algebraic of the proofs of the existence of the roots of an equation, the one which is based on the proposition that every real equation of odd degree has a real root.

  10. Pell's equation references
    • References for: Pell's equation .
    • A A Antropov, Two methods for the solution of Pell's equation in the work of J Wallis (Russian), Istor.
    • A A Antropov, Wallis' method of "approximations" as applied to the solution of the equation x - ny = 1 in integers (Russian), Istor.-Mat.
    • S P Arya, On the Brahmagupta- Bhaskara equation, Math.
    • C Baltus, Continued fractions and the Pell equations : The work of Euler and Lagrange, Comm.
    • I-Kh I Gerasim, On the genesis of Redei's theory of the equation x -Dy = -1 (Russian), Istor.-Mat.
    • T N Sinha, Pell's equation: its history and significance, Math.
    • B L van der Waerden, Pell's equation in the mathematics of the Greeks and Indians (Russian), Uspehi Mat.

  11. Pell's equation references
    • References for: Pell's equation .
    • A A Antropov, Two methods for the solution of Pell's equation in the work of J Wallis (Russian), Istor.
    • A A Antropov, Wallis' method of "approximations" as applied to the solution of the equation x - ny = 1 in integers (Russian), Istor.-Mat.
    • S P Arya, On the Brahmagupta- Bhaskara equation, Math.
    • C Baltus, Continued fractions and the Pell equations : The work of Euler and Lagrange, Comm.
    • I-Kh I Gerasim, On the genesis of Redei's theory of the equation x -Dy = -1 (Russian), Istor.-Mat.
    • T N Sinha, Pell's equation: its history and significance, Math.
    • B L van der Waerden, Pell's equation in the mathematics of the Greeks and Indians (Russian), Uspehi Mat.

  12. Alcuin's book
    • Alcuin, of course, doesn't use equations.
    • From the second equation we see that z must be divisible by 5 so write z = 5t.
    • The first equation gives t < 10 but, subtracting the equations gives -x + 9t = 80 and so t ≥ 9 (since 9t > 80).
    • Substitute 2x = y into the first equation to get x = 4 so y = 8.
    • Subtract the first equation from the second to obtain 5x + 3y = 20.
    • Subtract the first equation from the second to obtain 5x + 3y = 30.
    • Subtract the first equation from the second to obtain 5x + 3y = 90.
    • Subtract the first equation from the second to obtain 5x + 3y = 100.
    • Thus x is divisible by 23 and, by the second equation, cannot be as large as 46.
    • Thus x is divisible by 19 and, by the second equation, cannot be larger than 19.
    • Multiply the second equation by 4 and subtract twice the first equation from it to obtain 6x - z = 24.
    • But the second equation shows that x can't be as big as 6 (for then y and z would have to be 0 or negative) Hence x = 5, y = 1 and z = 6.

  13. Debating topics
    • What is an equation? .
    • What about x in 0x = 1? Is this an equation? .
    • Can we solve equations without having "unknowns"? .
    • How did the Chinese represent equations? .
    • Does the equation ax = b always have a solution? Do quadratic, cubic and quartic equations always have solutions? .
    • What does it mean to say that equations of degree 5 cannot be solved? .
    • Why did the ancient Chinese not worry about solving cubic, quartic, quintic equations? .
    • The equation x2 + 1 = 0 has no real number solution.
    • Do we need to introduce negative numbers to get solutions of such equations? .

  14. Bakhshali manuscript
    • Equations are given with a large dot representing the unknown.
    • Here is an example of an equation as it appears in the Bakhshali manuscript.
    • Equation .
    • 21 (1) (1986), 51-61.',9)">9] and some of these lead to indeterminate equations.
    • These problems can all be reduced to solving a linear equation with one unknown or to a system of n linear equations in n unknowns.
    • If we use modern methods we would solve the system of three equation for x1, x2, x3 in terms of k to obtain .

  15. General relativity
    • Poisson used the gravitational potential approach to give an equation which, unlike Newton's, could be solved under rather general conditions.
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    • Einstein also realised that the gravitational field equations were bound to be non-linear and the equivalence principle appeared to only hold locally.
    • How much they learnt from each other is hard to measure but the fact that they both discovered the same final form of the gravitational field equations within days of each other must indicate that their exchange of ideas was helpful.
    • Of course Einstein's 18 November paper still does not have the correct field equations but this did not affect the particular calculation regarding Mercury.
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    • On 25 November Einstein submitted his paper The field equations of gravitation which give the correct field equations for general relativity.
    • Five days before Einstein submitted his 25 November paper Hilbert had submitted a paper The foundations of physics which also contained the correct field equations for gravitation.
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    • Immediately after Einstein's 1915 paper giving the correct field equations, Karl Schwarzschild found in 1916 a mathematical solution to the equations which corresponds to the gravitational field of a massive compact object.
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  16. Set theory
    • However Cantor examines the set of algebraic real numbers, that is the set of all real roots of equations of the form .
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    • For an equation of the above form define its index to be .
    • There is only one equation of index 2, namely x = 0.
    • There are 3 equations of index 3, namely .
    • For each index there are only finitely many equations and so only finitely many roots.
    • The first person to explicitly note that he was using such an axiom seems to have been Peano in 1890 in dealing with an existence proof for solutions to a system of differential equations.
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  17. Chinese overview
    • He solved cubic equations by extending an algorithm for finding cube roots.
    • He improved methods for finding square and cube roots, and extended the method to the numerical solution of polynomial equations computing powers of sums using binomial coefficients constructed with Pascal's triangle.
    • The treatise contains remarkable work on the Chinese remainder theorem, gives an equation whose coefficients are variables and, among other results, Heron's formula for the area of a triangle.
    • Equations up to degree ten are solved using the Ruffini-Horner method.
    • It contains the "tian yuan" or "coefficient array method" or "method of the celestial unknown" which was a method to work with polynomial equations.
    • He described multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.
    • He produced the Shou shi li (Works and Days Calendar), worked on spherical trigonometry, and solved equations using the Ruffini-Horner numerical method.

  18. Indian mathematics
    • The main topics of Jaina mathematics in around 150 BC were: the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.
    • He introduced trigonometry in order to make his astronomical calculations, based on the Greek epicycle theory, and he solved with integer solutions indeterminate equations which arose in astronomical theories.
    • Far from it for he made other major contributions in to the understanding of integer solutions to indeterminate equations and to interpolation formulas invented to aid the computation of sine tables.
    • In addition to this device, they sometimes also used the theory of quadratic equations, or applied the method of successive approximations.
    • Brahmagupta is probably the earliest astronomer to have employed the theory of quadratic equations and the method of successive approximations to solving problems in spherical astronomy.
    • This period saw developments in sine tables, solving equations, algebraic notation, quadratics, indeterminate equations, and improvements to the number systems.
    • Citrabhanu was a sixteenth century mathematicians from Kerala who gave integer solutions to twenty-one types of systems of two algebraic equations.
    • These types are all the possible pairs of equations of the following seven forms: .

  19. Doubling the cube
    • To see, using modern mathematics, why this works we note that the cylindrical surface has equation .
    • the toroidal surface has equation .
    • and the conical surface has equation .
    • The first comes from the rectangular hyperbola and the parabola which are the first two equations in our list.
    • For his second solution Menaechmus uses the intersection of the two parabolas y2 = bx and x2 = ay which are the second and third equations in our list.

  20. Golden ratio
    • With the development of algebra by the Arabs one might expect to find the quadratic equation (or a related one) to that which we have given above.
    • Al-Khwarizmi does indeed give several problems on dividing a line of length 10 into two parts and one of these does find a quadratic equation for the length of the smaller part of the line of length 10 divided in the golden ratio.
    • Abu Kamil gives similar equations which arise from dividing a line of length 10 in various ways.
    • Cardan, Bombelli and others included problems in their texts on finding the golden ratio using quadratic equations.

  21. Real numbers 2
    • Up to this time there was no proof that numbers existed that were not the roots of polynomial equations with rational coefficients.
    • Clearly √2 is the root of a polynomial equation with rational coefficients, namely x2 = 2, and it is easy to see that all roots of rational numbers arise as solutions of such equations.
    • A number is called transcendental if it is not the root of a polynomial equation with rational coefficients.

  22. function concept
    • Thus began the long controversy about the nature of functions to be allowed in the initial conditions and in the integrals of partial differential equations, which continued to appear in an ever increasing number in the theory of elasticity, hydrodynamics, aerodynamics, and differential geometry.
    • In this work Condorcet distinguished three types of functions: explicit functions, implicit functions given only by unsolved equations, and functions which are defined from physical considerations such as being the solution to a diffferential equation.
    • One indicates this correspondence by the equation y = f(x).

  23. Topology history
    • The problem arose from studying a polynomial equation f(w, z) = 0 and considering how the roots vary as w and z vary.
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    • Riemann introduced Riemann surfaces, determined by the function f(w, z), so that the function w(z) defined by the equation f(w, z) = 0 is single valued on the surfaces.
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    • Schmidt in 1907 examined the notion of convergence in sequence spaces, extending methods which Hilbert had used in his work on integral equations to generalise the idea of a Fourier series.
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    • Poincare developed many of his topological methods while studying ordinary differential equations which arose from a study of certain astronomy problems.
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  24. Babylonian Pythagoras
    • ',17)">17], claims that the tablet is connected with the solution of quadratic equations and has nothing to do with Pythagorean triples:- .
    • Now this to us is quite an easy exercise in solving equations.
    • We would substitute y = 0.75/x into the second equation to obtain a quadratic in x2 which is easily solved.
    • This however is not the method of solution given by the Babylonians and really that is not surprising since it rests heavily on our algebraic understanding of equations.

  25. Arabic mathematics
    • This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.
    • Omar Khayyam (born 1048) gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.
      Go directly to this paragraph
    • Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work [Arch.
      Go directly to this paragraph
    • He wrote a treatise on cubic equations, which [The development of Arabic mathematics : between arithmetic and algebra (London, 1994).',11)">11]:- .
      Go directly to this paragraph
    • represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.
    • Omar Khayyam combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means.
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  26. Babylonian mathematics references
    • A E Berriman, The Babylonian quadratic equation, Math.
    • An old Babylonian catalogue text with equations for squares and circles, J.
    • Isoperimetric problems and the origin of the quadratic equations, Isis 32 (1940), 101-115.
    • Quadratic equations among the Babylonians (Russian), Molotov.

  27. Babylonian mathematics references
    • A E Berriman, The Babylonian quadratic equation, Math.
    • An old Babylonian catalogue text with equations for squares and circles, J.
    • Isoperimetric problems and the origin of the quadratic equations, Isis 32 (1940), 101-115.
    • Quadratic equations among the Babylonians (Russian), Molotov.

  28. History overview
    • Systems of linear equations were studied in the context of solving number problems.
    • Quadratic equations were also studied and these examples led to a type of numerical algebra.
    • Major progress in mathematics in Europe began again at the beginning of the 16th Century with Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic equations.
      Go directly to this paragraph
    • The 19th Century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations.
      Go directly to this paragraph
    • Lie's work on differential equations led to the study of topological groups and differential topology.
      Go directly to this paragraph
    • The study of integral equations was driven by the study of electrostatics and potential theory.
      Go directly to this paragraph
    • There is no reason why anyone should introduce negative numbers just to be solutions of equations such as x + 3 = 0.

  29. Planetary motion
    • This is Law I: the equation of the elliptic path with respect to the origin at one focus: see Kepler's Planetary Laws: Section 6.
    • We consider the equation of a circle with origin at some eccentric point: as an illustration we may take the circle CQD, centre B, shown in the figure, where A is to be regarded as the origin or pole; just for our present purpose, we set AB = ae to represent the 'polar distance' alone (since the focal distance for a circle is zero).
    • Since both the focal distance and the polar distance are measured from the centre B of the ellipse, it is only when these two distances coincide (aε = ae), uniquely, that we obtain the simplest possible equation -- as expressed in (5).
    • [For a less precise version of equation (10) - simply that the transradial motion is proportional (inverse-linearly) to the distance - see Kepler's Planetary Laws: Section 10.] .
    • We return to equation (5), the formula for the radius vector: .
    • Now we return to equation (9), which stated the area-time law (in kinematical terms) for one complete circuit: .

  30. Abstract groups
    • The first version of Galois' important paper on the algebraic solution of equations was submitted to the Paris Academie des Sciences in 1829.
    • Galois was invited by Poisson to submit a third version of his memoir on equations to the Academie and he did so on 17 January 1831.
    • Although Galois had proved the results in general, the paper only considered equations of prime degree.
    • It was unclear to him how Galois' results classified which equations were soluble by radicals.
    • Although Galois had used groups extensively throughout his paper on equations, he had not given a definition.
    • This was reinforced when Jordan published his major group theory text Traite des substitutions et des equations algebraique in 1870.

  31. Egyptian Papyri
    • This is called the "red auxiliary" equation since the scribe wrote this equation in red ink.
    • Now the answer to the red auxiliary equation is 4 so the original equation had solution twice × (twice × 1/15).
    • Another example of solving an equation is Problem 24 which asks: .

  32. Babylonian mathematics references
    • A E Berriman, The Babylonian quadratic equation, Math.
    • Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations, Historia Mathematica 8 (1981), 277-318.
    • II : An old Babylonian catalogue text with equations for squares and circles, J.

  33. Babylonian and Egyptian references
    • A E Berriman, The Babylonian quadratic equation, Math.
    • Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations, Historia Mathematica 8 (1981), 277-318.
    • II : An old Babylonian catalogue text with equations for squares and circles, J.

  34. Real numbers 1
    • The Arabic mathematicians went further with constructible magnitudes for they used geometric methods to solve cubic equations which meant that they could construct magnitudes whose ratio to a unit length involved cube roots.
    • For example Omar Khayyam showed how to solve all cubic equations by geometric methods.
    • Fibonacci, using skills learnt from the Arabs, solved a cubic equation showing that its root was not formed from rationals and square roots of rationals as Euclid's magnitudes were.

  35. Brachistochrone problem
    • (where g is the acceleration due to gravity) and substituting for v gives the equation of the curve as .
    • The cycloid x(t) = h(t - sin t), y(t) = h(1 - cos t) satisfies this equation.
    • He found what has now come to be known as the Euler-Lagrange differential equation for a function of the maximising or minimising function and its derivative.
    • In this way δz expresses a difference of z which is different from dz, but which, however, will satisfy the same rules; such that where we have for any equation dz = m dx, we can equally have δz = mδx, and likewise in other cases.

  36. Nine chapters
    • Essentially linear equations are solved by making two guesses at the solution, then computing the correct answer from the two errors.
    • Here 18 problems which reduce to solving systems of simultaneous linear equations are given.
    • The problems involve up to six equations in six unknowns and the only difference with the modern method is that the coefficients are placed in columns rather than rows.
    • Quadratic equations are considered for the first time in Chapter 9, are solved by an analogue of division using ideas from geometry, in fact from the Chinese square-root algorithm, rather than from algebra.

  37. Squaring the circle
    • His proof essentially attempted to prove that π was transcendental, that is not the root of a rational polynomial equation.
    • However, others such as Huygens, believed that π was algebraic, that is that it is the root of a rational polynomial equation.
    • The final solution to the problem of whether the circle could be squared using ruler and compass methods came in 1880 when Lindemann proved that π was transcendental, that is it is not the root of any polynomial equation with rational coefficients.

  38. Jaina mathematics
    • the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.

  39. Fractal Geometry
    • the first iteration of the set is the whole set, scaled down by a factor of r1) satisfies the following two equations [Introduction to Fractals and Chaos (London, 1995).',2)">2]: .
    • These equations, however, do not appear in Hausdorff's paper, as they relate directly to fractals (and calculating the dimension of a fractal), which were ideas that would have been unknown to Hausdorff.
    • Still, from these two equations, it is easy to see how one can obtain a dimension that is not a whole number, as [Introduction to Fractals and Chaos (London, 1995).',2)">2] .

  40. Bourbaki 1
    • A large number of subcommittees were formed, given the size of the group, and these were to cover the following topics: algebra, analytic functions, integration theory, differential equations, existence theorems for differential equations, partial differential equations, differentials and differential forms, calculus of variations, special functions, geometry, Fourier series, and representations of functions.

  41. Special relativity
    • and showed that certain equations were invariant under these transformations.
      Go directly to this paragraph
    • These transformations, with a different scale factor, are now known as the Lorentz equations and the group of Lorentz transformations gives the geometry of special relativity.
      Go directly to this paragraph
    • Also in 1908 Minkowski published an important paper on relativity, presenting the Maxwell-Lorentz equations in tensor form.
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  42. Orbits
    • Euler developed methods of integrating linear differential equations in 1739 and made known Cotes' work on trigonometric functions.
      Go directly to this paragraph
    • The other line was to produce a sophisticated theory to transform and integrate the equations of motion.
    • Liouville made a number of very important mathematical discoveries while working on the theory of perturbations including the discovery of Liouville's theorem "when a bounded domain in phase space evolves according to Hamilton's equations its volume is conserved".
      Go directly to this paragraph

  43. Tartaglia versus Cardan
    • Tartaglia to Cardano (August 1539): Master Girolamo, I have received a letter of yours, in which you write that you understand the rule; but that when the cube of one-third of the coefficient of the unknown is greater in value than the square of one-half of the number you cannot resolve the equation by following the rule, and therefore you request me to give you the solution of this equation "One cube equal to nine unknowns plus ten".
    • And as to resolving you the equation you have sent, I must say that I am very sorry that I have given you already so much as I have done, for I have been informed, by person worthy of faith, that you are about to publish another algebraic work, and that you have gone boasting through Milan of having discovered some new rules in Algebra.

  44. The number e
    • Of course from the equation x = at, we deduce that t = log x where the log is to base a, but this involves a much later way of thinking.
    • Gentlemen, we have not the slightest idea what this equation means, but we may be sure that it means something very important.

  45. Cosmology
    • However, Einstein realised he could introduce a arbitrary constant into his mathematical equations, which could balance the gravitational force and keep the galaxies apart.
    • The Russian mathematician and meteorologist Friedmann had realised in 1917 that Einstein equations could describe an expanding universe.
      Go directly to this paragraph

  46. Longitude2
    • This variation between clock time and sundial time is known as the Equation of Time or the Equation of Natural Days and had been known to the Greeks and Arabs many centuries earlier (although of course the reason for the variation was not then understood).

  47. Abstract linear spaces
    • His systemes lineaires is a table of coefficients of a system of linear equations denoted by a single upper-case letter and Laguerre defines addition, subtraction and multiplication of of these linear sysyems.
      Go directly to this paragraph
    • We suppose that for any real number m the notation ma has a meaning such that the preceeding equations are valid.

  48. Classical light
    • The four partial differential equations, now known as Maxwell's equations, which completely describe the classical electromagnetic theory appeared in fully developed form in Maxwell's paper Electricity and Magnetism (1873).

  49. Bourbaki 2
    • This chapter covers ordinary first order differential equations and systems of such equations, again presented with considerably more generality than previous texts.

  50. references
    • (1951), On stochastic differential equations, Memoirs, American Mathematical Society, 4, 1-51.
    • (1951), On some connections between probability theory and differential and integral equations, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, 189-215, University of California Press.

  51. Modern light
    • He gave his famous four partial differential equations, now known as Maxwell's equations, which completely describe classical electromagnetic theory.

  52. Babylonian Pythagoras references
    • An old Babylonian catalogue text with equations for squares and circles, J.
    • J Friberg, Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples and the Babylonian triangle parameter equations, Historia Math.

  53. Quantum mechanics history

  54. Babylonian Pythagoras references
    • An old Babylonian catalogue text with equations for squares and circles, J.
    • J Friberg, Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean triples and the Babylonian triangle parameter equations, Historia Math.

  55. Weather forecasting references
    • R S Harwood, Atmospheric Dynamics (Chapter 1: Basics, Chapter 5: Balance of Forces in Synoptic Scale Flow, Chapter 13: Quasi-Geostrophic Equations) (University of Edinburgh, 2005) .

  56. General relativity references
    • J Norton, Einstein's discovery of the field equations of general relativity : some milestones, Proceedings of the fourth Marcel Grossmann meeting on general relativity (Amsterdam-New York, 1986), 1837-1848.

  57. Indian mathematics references
    • R Lal and R Prasad, Integral solutions of the equation Nx^2+1 = y^2 in ancient Indian mathematics (cakravala or the cyclic method), Ganita Bharati 15 (1-4) (1993), 41-54.

  58. Nine chapters references
    • K Chemla, Different concepts of equations in 'The nine chapters on mathematical procedures' and in the commentary on it by Liu Hui (3rd century), Historia Sci.

  59. Babylonian numerals references
    • Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations, Historia Mathematica 8 (1981), 277-318.

  60. Weather forecasting references
    • R S Harwood, Atmospheric Dynamics (Chapter 1: Basics, Chapter 5: Balance of Forces in Synoptic Scale Flow, Chapter 13: Quasi-Geostrophic Equations) (University of Edinburgh, 2005) .

  61. General relativity references
    • J Norton, Einstein's discovery of the field equations of general relativity : some milestones, Proceedings of the fourth Marcel Grossmann meeting on general relativity (Amsterdam-New York, 1986), 1837-1848.

  62. Indian mathematics references
    • R Lal and R Prasad, Integral solutions of the equation Nx^2+1 = y^2 in ancient Indian mathematics (cakravala or the cyclic method), Ganita Bharati 15 (1-4) (1993), 41-54.

  63. Nine chapters references
    • K Chemla, Different concepts of equations in 'The nine chapters on mathematical procedures' and in the commentary on it by Liu Hui (3rd century), Historia Sci.

  64. Tait's scrapbook
    • Maxwell became dp/dt since there is an equation in the Treatise on Natural Philosophy which reads .

  65. Measurement
    • Egyptian papyri, for example, contain methods for solving equations which arise from problems about weights and measures.

  66. Knots and physics
    • Maxwell also gave equations in three dimensions which represented knotted curves.

  67. Ring Theory
    • The equation xn+ yn= zn has no solution for positive integers x, y, z when n > 2.

  68. Indian Sulbasutras
    • If we had taken this into account we would have obtained the equation .

  69. Newton's bucket
    • In 1918 Joseph Lense and Hans Thirring obtained approximate solutions of the equations of general relativity for rotating bodies.

  70. Fair book insert
    • First calculates the other angle as 47°24 then uses the same type of proportional equations as before to calculate the sides: .

  71. Arabic numerals
    • a person from India presented himself before the Caliph al-Mansur in the year [ 776 AD] who was well versed in the siddhanta method of calculation related to the movment of the heavenly bodies, and having ways of calculating equations based on the half-chord [essentially the sine] calculated in half-degrees ..

  72. Fair book
    • He then squares the equation obtaining a quadratic in x2 which he solves by completing the square.

  73. Gravitation
    • There is the physics of the situation which involves finding the equations which govern bodies acted on by gravity, and for this a proper understanding of the laws of mechanics is required.

  74. Chinese numerals
    • This was a notation for an equation and Li Zhi gives the earliest source of the method, although it must have been invented before his time.

  75. Pi history
    • Shortly after Shanks' calculation it was shown by Lindemann that π is transcendental, that is, π is not the solution of any polynomial equation with integer coefficients.
      Go directly to this paragraph

  76. Egyptian mathematics
    • Some problems ask for the solution of an equation.

  77. Zero
    • Cardan solved cubic and quartic equations without using zero.

  78. Physical world
    • Although Newton had made a clear distinction between a mathematical theory and a physical reality, Berkeley argued that he had fallen into his own trap for he spoke of forces as physical entities, where Berkeley believed that they were nothing other than terms in the equations set up by Newton.

  79. EMS History
    • (Professor of Mathematics in the University of Edinburgh), on The Solution of Algebraic and Transcendental Equations in the Mathematical Laboratory.

  80. Babylonian numerals references
    • Plimpton 322, Pythagorean triples, and the Babylonian triangle parameter equations, Historia Mathematica 8 (1981), 277-318.


Famous Curves

  1. Newtons
    • Cartesian equation: .
    • In this classification of cubics, Newton gives four classes of equation.
    • The third class of equations is the one given above which Newton divides into five species.
    • In the third Case the Equation was yy = ax3 + bxx + cx + d and defines a Parabola whose Legs diverge from one another, and run out infinitely contrary ways.
    • The five types depend on the roots of the cubic in x on the right hand side of the equation.

  2. Cardioid
    • Cartesian equation: .
    • Polar equation: .
    • We can easily give parametric equations for the cardioid, namely .
    • (i) The curve with Cartesian equation: y = 0.75 x2/3 plusminus √(1 - x2).
    • (ii) The curve with Polar equation: r = sin2(π/8 - θ/4).

  3. Circle
    • Cartesian equation: .
    • Polar equation: .
    • Apollonius, in about 240 BC, showed effectively that the bipolar equation r = kr' represents a system of coaxial circles as k varies.
    • In terms of bipolar equations mr2 + nr'2 = c2 represents a circle whose centre divides the line segment between the two fixed points of the system in the ratio n to m.

  4. Trident
    • Cartesian equation: .
    • Newton was the first to undertake such a systematic study of cubic equations and he classified them into 72 different cases.
    • Cartes (Descartes) constructed equations of six dimensions.

  5. Right
    • Cartesian equation: .
    • Polar equation: .
    • The general strophoid has equation .
    • The particular case of a right strophoid in where a = π/2 and the equation, in cartesians and polars, is that given above.

  6. Catenary
    • Cartesian equation: .
    • Its equation was obtained by Leibniz, Huygens and Johann Bernoulli in 1691.
    • They were responding to a challenge put out by Jacob Bernoulli to find the equation of the 'chain-curve'.

  7. Lemniscate
    • Cartesian equation: .
    • Polar equation: .
    • The bipolar equation of the lemniscate is rr' = a2/2.

  8. Curve definitions
    • Algebraic curve : A curve whose cartesian equation can be expressed in terms of powers of x and y together with the operations of addition, subtraction, multiplication and division.
    • A curve may be defined by an equation, called the bipolar equation, connecting r and r'.

  9. Serpentine
    • Parametric Cartesian equation: .
    • The first of these is equations of the form .

  10. Trisectrix
    • Cartesian equation: .
    • Polar equation: .
    • Another form of the equation is r = a sec(θ/3) where the origin is inside the loop and the crossing point is on the negative x-axis.

  11. Double
    • Cartesian equation: .
    • Polar equation: .

  12. Devils
    • Cartesian equation: .
    • Polar equation (Special case): .

  13. Kappa
    • Cartesian equation: .
    • Polar equation: r = a cot(θ) .

  14. Folium
    • Cartesian equation: .
    • Polar equation: .

  15. Cissoid
    • Cartesian equation: .
    • Polar equation: .

  16. Eight
    • Cartesian equation: .
    • Polar equation: .

  17. Cartesian
    • Cartesian equation: .
    • The Cartesian Oval has bipolar equation r + mr' =a.

  18. Limacon
    • Cartesian equation: .
    • Polar equation: .

  19. Pearls
    • Cartesian equation: .
    • The curves with the equation given above, where n, p and m are integers, were studied by de Sluze between 1657 and 1698.

  20. Rhodonea
    • Polar equation: .
    • It has polar equation r = a sin(2θ) and cartesian form (x2+ y2) 3 = 4 a2x2y2.

  21. Cayleys
    • Cartesian equation: .
    • Polar equation: .

  22. Conchoidsl
    • Cartesian equation: .
    • Polar equation: .

  23. Kampyle
    • Cartesian equation: .
    • Polar equation: .

  24. Quadratrix
    • Cartesian equation: .
    • Polar equation: .

  25. Conchoid
    • Cartesian equation: .
    • Polar equation: .

  26. Trifolium
    • Cartesian equation: .
    • Polar equation: .

  27. Foliumd
    • Cartesian equation: .
    • The equation of the tangent at the point with t = p is .

  28. Cassinian
    • Cartesian equation: .
    • They are defined by the bipolar equation rr' = k2.

  29. Hyperbola
    • Cartesian equation: .
    • The evolute of the hyperbola with equation given above is the Lame curve .

  30. Lame
    • Cartesian equation: .
    • In 1818 Lame discussed the curves with equation given above.

  31. Ellipse
    • Cartesian equation: .
    • The evolute of the ellipse with equation given above is the Lame curve.

  32. Astroid
    • Cartesian equation: .
    • The equation of this tangent T is .

  33. Straight
    • Cartesian equation: .

  34. Watts
    • Polar equation: .

  35. Epicycloid
    • Parametric Cartesian equation: .

  36. Hypotrochoid
    • Parametric Cartesian equation: .

  37. Spiric
    • Cartesian equation: .

  38. Lissajous
    • Parametric Cartesian equation: .

  39. Sinusoidal
    • Polar equation: .

  40. Freeths
    • Polar equation: .

  41. Nephroid
    • Parametric Cartesian equation: .

  42. Hypocycloid
    • Parametric Cartesian equation: .

  43. Pursuit
    • Cartesian equation: .

  44. Durers
    • Cartesian equation: .

  45. Neiles
    • Cartesian equation: .

  46. Plateau
    • Parametric Cartesian equation: .

  47. Epitrochoid
    • Parametric Cartesian equation: .

  48. Tricuspoid
    • Cartesian equation: .

  49. Hyperbolic
    • Polar equation: .

  50. Lituus
    • Polar equation: .

  51. Involute
    • Parametric Cartesian equation: .

  52. Talbots
    • Parametric Cartesian equation: .

  53. Fermats
    • Polar equation: .

  54. Frequency
    • Cartesian equation: .

  55. Tractrix
    • Parametric Cartesian equation: .

  56. Parabola
    • Cartesian equation: .

  57. Tschirnhaus
    • Cartesian equation: 3a y2 = x(x-a)2 .

  58. Witch
    • Cartesian equation: .

  59. Spiral
    • Polar equation: .

  60. Cycloid
    • Parametric Cartesian equation: .

  61. Pearshaped
    • Cartesian equation: .

  62. Bicorn
    • Cartesian equation: .

  63. Cochleoid
    • Polar equation: .

  64. Equiangular
    • Polar equation: .


Societies etc

  1. International Congress Speaker
    • Paul Painleve, Le probleme moderne de l'integration des equations differentielles.
    • Andrew Russell Forsyth, On the Present Condition of Partial Differential Equations of the Second Order as Regards Formal Integration.
    • Niels Erik Norlund, Sur les equations aux differences finies.
    • Jean Marie Le Roux, Considerations sur une equation aux derivees partielles de la physique mathematique.
    • Torsten Carleman, Sur la theorie des equations integrales lineaires et ses applications.
    • Edward Charles Titchmarsh, Eigenfunction Problems Arising from Differential Equations.
    • Kosaku Yosida, Semigroup Theory and the Integration Problem of Diffusion Equations.
    • Lars Garding, Some Trends and Problems in Linear Partial Differential Equations.
    • Peter Henrici, Problems of Stability and Error Propagation in the Numerical Integration of Ordinary Differential Equations.
    • Louis Nirenberg, Some Aspects of Linear and Nonlinear Partial Differential Equations.
    • Enrico Bombieri, Variational Problems and Elliptic Equations.
    • Heinz-Otto Kreiss, Initial Boundary Value Problems for Hyperbolic Partial Differential Equations.
    • Shing-Tung Yau, The Role of Partial Differential Equations in Differential Geometry.
    • Peter David Lax, Problems Solved and Unsolved Concerning Linear and Non-Linear Partial Differential Equations.
    • Richard Melvin Schoen, New Developments in the Theory of Geometric Partial Differential Equations.
    • Arnold Schonhage, Equation Solving in Terms of Computational Complexity.
    • Andrei Andreevich Bolibrukh The Riemann-Hilbert problem and Fuchsian differential equations on the Riemann sphere.
    • Jean Bourgain, Harmonic Analysis and Nonlinear Partial Differential Equations.
    • Pierre-Louis Lions, On Some Recent Methods for Nonlinear Partial Differential Equations.
    • Luis Caffarelli, Nonlinear Elliptic Theory and the Monge-Ampere Equation.
    • Sun-Yung Alice Chang and Paul Chien-Ping Yang, Non-linear Partial Differential Equations in Conformal Geometry.
    • Jean-Michel Coron, On the Controllability of Nonlinear Partial Differential Equations .
    • Carlos Kenig, The Global Behavior of Solutions to Critical Non-linear Dispersive Equations .
    • Shige Peng, Backward Stochastic Differential Equations, Nonlinear Expectations and Their Applications .

  2. AMS Steele Prize
    • for three fundamental papers: "On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables", "An example of a smooth linear partial differential equation without solution", and "On hulls of holomorphy".
    • for three papers of fundamental and lasting importance "Abzweigung einer periodischen Losung von einer stationaren Losung eines Differential systems", "A mathematical example displaying features of turbulence", and "The partial differential equation ut + uux = uxx".
    • for his paper "Uniqueness in the Cauchy Problem for Partial Differential Equation".
    • for his numerous and fundamental contributions to the theory and applications of linear and nonlinear partial differential equations and functional analysis, for his leadership in the development of computational and applied mathematics, and for his extraordinary impact as a teacher.
    • for his numerous basic contributions to linear and nonlinear partial differential equations and their application to complex analysis and differential geometry.
    • for two seminal papers "Viscosity solutions of Hamilton-Jacobi equations" (joint with P-L Lions), and "Generation of semi-groups of nonlinear transformations on general Banach spaces" (joint with T M Liggett).
    • for the "Evans-Krylov theorem" as first established in the papers Lawrence C Evans "Classical solutions of fully nonlinear convex, second order elliptic equations", and N V Krylov "Boundedly inhomogeneous elliptic and parabolic equations".
    • for their paper "Korteweg de Vries equation and generalizations.
    • for his book "Elliptic Partial Differential Equations of Second Order", written with the late David Gilbarg.
    • one of the world's greatest mathematicians studying nonlinear partial differential equations.

  3. Young Mathematician prize
    • for work on the stability of solutions of operator Hamiltonian equations with periodic coefficients.
    • for a study of well-defined characteristic problems for an ultra-hyperbolic equation.
    • for works on the theory of stochastic differential equations and Banach geometry.
    • for finite-gap and isodromic solutions of equations of nonlinear Schrodinger type.
    • for a study of some algebraic structures related to the Young-Baxter equation.
    • for work in representation theory for solutions of the Young-Baxter equation.
    • for convolution equations and an asymptotic holomorphic function.
    • for works on the theory of perturbed linear systems of ordinary differential equations.

  4. European Mathematical Society Prize
    • In a different direction, his results on partial differential equations, in particular on flame propagation and combustion, are very significant.
    • has greatly contibuted to the asymptotic analysis of Euler and Navier-Stokes equations with large Coriolis force.
    • The simplest case (when the equations are set on the unit cube with periodic boundary conditions) has been solved by Grenier around 1995.
    • Grenier obtained both positive and negative important results on the problem of convergence of the Navier-Stokes equations to the Euler equations in a domain with solid boundary conditions.
    • He also justified the hydrostatic limit of the Euler equations in a two dimensional infinitesimally thin strip.
    • Grenier gave a very elegant proof of convergence for the semi-classical limit of the nonlinear Schrodinger equations (before appearance of shocks).
    • In the technically demanding proof the travelling waves are constructed as solutions of a functional equation, applying centre manifold theory in an infinite dimensional space.
    • was the first to make a systematic and impressive asymptotic analysis for the case of large parameters in Theory of Ginzburg-Landau equation.

  5. Clay Award
    • for his discovery of the Ricci Flow Equation and its development into one of the most powerful tools of geometric analysis.
    • for his ground-breaking work in analysis, notably his optimal restriction theorems in Fourier analysis, his work on the wave map equation (the hyperbolic analogue of the harmonic map equation), his global existence theorems for KdV type equations, as well as significant work in quite distant areas of mathematics, such as his solution with Allen Knutson of Horn's conjecture, a fundamental problem about hermitian matrices that goes back to questions posed by Hermann Weyl in 1912 .
    • This conjecture posits an essentially geometric necessary and suffcient condition, "Psi", for a pseudo-differential operator of principal type to be locally solvable, i.e., for the equation Pu = f to have local solutions given a finite number of conditions on F Dencker's work provides a full mathematical understanding of the surprising discovery by Hans Lewy in 1957 that there exist a linear partial differential operator - a one-term, third-order perturbation of the Cauchy-Riemann operator - which is not local solvable in this sense.
    • Taubes' affirmative solution of the Weinstein conjecture for any 3-dimensional contact manifold is based on a novel application of the Seiberg-Witten equations to the problem.

  6. AMS/SIAM Birkhoff Prize
    • for his outstanding work in partial differential equations, in numerical analysis, and, particularly, in nonlinear elasticity theory; the latter work has led to his study of quasi-isometric mappings as well as functions of bounded mean oscillation, which have had impact in other areas of analysis.
    • for his fundamental contributions to the theory of nonlinear partial differential equations, especially his work on existence and regularity theory for nonlinear elliptic equations, and applications of his work to the theory of minimal surfaces in higher dimensions.
    • for his important contributions to partial differential equations, to the mathematical analysis of problems of transonic flow and airfoil design by the method of complexification, and to the development and application of scientific computing to problems of fluid dynamics and plasma physics.
    • for her deep and influential work in partial differential equations, most notably in the study of shock waves, transonic flow, scattering theory, and conformally invariant estimates for the wave equation.
    • for his leadership, originality, depth, and breadth of work in dynamical systems, differential equations, mathematical biology, shock wave theory, and general relativity.

  7. AMS Bcher Prize
    • for his contributions to the theory of linear, nonlinear, ordinary, and partial differential equations contained in his recent papers.
    • for his work in partial differential equations.
    • in recognition of his fundamental work on the theory of singular integrals and partial differential equations, and in particular for his paper "Cauchy integrals on Lipschitz curves and related operators".
    • for his deep and fundamental work in nonlinear partial differential equations, in particular his work on free boundary problems, vortex theory and regularity theory.
    • for his work on the application of partial differential equations to differential geometry, in particular his completion of the solution to the Yamabe Problem in "Conformal deformation of a Riemannian metric to constant scalar curvature".
    • for his contributions to nonlinear hyperbolic equations .
    • for his fundamental paper "On global existence and scattering for the wave maps equations".
    • for his recent fundamental breakthrough on the problem of critical regularity in Sobolev spaces of the wave maps equations "Global regularity of wave maps I.
    • for his fundamental contributions to our understanding of the Ginzburg-Landau equations with a small parameter.
    • for his fundamental work in the analysis of nonlinear dispersive equations.
    • for his important contributions to harmonic analysis, partial differential equations, and nonlinear dispersive PDE.

  8. BMC 2007
    • McKean, H Camassa-Holm: a 1-dimensional caricature of Euler's equation .
    • Qian, Z On strong solutions of the 3D-Navier-Stokes equations .
    • Vassiliev, D Teleparallelism: difficult word but simple way of reinterpreting the Dirac equation .
    • Erdogan, M BDispersive estimates for Shrodinger equations with large magnetic potentials .
    • Vargas, ABilinear restriction theorems and applications to dispersive equations .

  9. Wolf Prize
    • for pioneering work on the development and application of topological methods to the study of differential equations.
    • for initiating many, now classic and essential, developments in partial differential equations.
    • for fundamental work in modern analysis, in particular, the application of pseudo-differential and Fourier integral operators to linear partial differential equations.
    • for his groundbreaking work on singular integral operators and their application to important problems in partial differential equations.
    • for his innovating ideas and fundamental achievements in partial differential equations and calculus of variations.
    • for his revolutionary contributions to global Riemmanian and symplectic geometry, algebraic topology, geometric group theory and the theory of partial differential equations.
    • for his fundamental work on stability in Hamiltonian mechanics and his profound and influential contributions to nonlinear differential equations.
    • for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singularity theory.
    • for his creation of "algebraic analysis", including hyperfunction and microfunction theory, holonomic quantum field theory, and a unified theory of soliton equations.

  10. Lagrange Prize
    • To give only a few examples one should mention first the systematic use of functional analysis and weak solutions for solving elliptic and parabolic differential equations, both theoretically and numerically, further the various methods he developed for solving nonlinear problems and his profound studies on control problems for systems governed by partial differential equations, optimal control first and controllability later with the introduction of the now standard Hilbert Uniqueness Method.
    • In a remarkable series of papers, followed and made complete in a three-volume book in cooperation with J L Lions (Nonhomogeneous Boundary Value Problems and Applications), he set the foundations for the modern treatment of partial differential equations, and in particular the ones mostly used in applications.
    • It also served as a starting point for development of the modern theory of linear partial differential equations.
    • He has also made important and often seminal contributions to many other fields, including singular perturbation theory, bifurcation studies in partial differential equations, nonlinear geometrical optics and acoustics, inverse scattering, effective equations for composite media, biophysics, biomechanics, carcinogenesis, optimal design, hydrodynamic surface waves, transport theory and waves in random media.

  11. AMS Wiener Prize
    • for his pioneering work in the area of dynamic programming, and for his related work on control, stability, and differential-delay equations.
    • for his broad contributions to applied mathematics, in particular, for his work on numerical and theoretical aspects of partial differential equations and on scattering theory.
    • for his contributions to applied mathematics in the areas of supersonic aerodynamics, plasma physics and hydromagnetics, and especially for his contributions to the truly remarkable development of inverse scattering theory for the solution of nonlinear partial differential equations.
    • for his outstanding contribution of original and non-perturbative mathematical methods in statistical mechanics by means of which he was able to solve several long open important problems concerning critical phenomena, phase transitions, and quantum field theory; and to Jerrold E Marsden for his outstanding contributions to the study of differential equations in mechanics: he proved the existence of chaos in specific classical differential equations; his work on the momentum map, from abstract foundations to detailed applications, has had great impact.

  12. BMC 1991
    • Brezis, HUniform estimates and blow-up for a nonlinear partial differential equation .
    • Goldie, C M Renewal theory for measures satisfying convolution equations .
    • Ward, R S Completely-solvable partial differential equations and Lie algebras .

  13. AMS Veblen Prize
    • for his work in nonlinear partial differential equations, his contributions to the topology of differentiable manifolds, and for his work on the complex Monge-Ampere equation on compact complex manifolds.
    • for his continuing study of the Ricci flow and related parabolic equations for a Riemannian metric, and to Gang Tian for his contributions to geometric analysis.

  14. AMS Satter Prize
    • for her deep contributions to the study of partial differential equations on Riemannian manifolds and in particular for her work on extremal problems in spectral geometry and the compactness of isospectral metrics within a fixed conformal class on a compact 3-manifold.
    • for her work on a long-standing problem in the water wave equation.
    • for her fundamental work on the hydrodynamic limits of the Boltzmann equation in the kinetic theory of gases.

  15. BMC 1997
    • Lions, P On compressible Euler and Navier-Stokes equations .
    • Special session: Partial differential equationsn Organiser: E B Davies .
    • Brezis, HRecent developments on the Ginzburg Landau equation in 2D .

  16. MAA Chauvenet Prize
    • Differential Equations: Linearity vs.
    • On Local Solvability of Linear Partial Differential Equations, Bull.
    • Ramanujan, Modular Equations, and Approximations to Pi, or, How to Compute One Billion Digits of Pi, Amer.
    • Niels Hendrik Abel and equations of the fifth degree, Amer.

  17. BMC 2001
    • Bournaveas, N Low regularity solutions of the Klein-Gordon-Dirac equations .
    • Special session: Partial differential equationsn .
    • Shirikyan, AErgodicity for the randomly forced equations of mathematical physics .

  18. BMC 2001

  19. Ukrainian Academy of Sciences
    • Research priorities in the institute include: theory of differential equations, mathematics of physics, statistical theory, theory of functions, topology, algebra, the dynamics of special mechanical systems, computer programming and the institute develops various fields of mathematics for the natural sciences and technology.
    • Research emphasis includes: nonlinear problems of mathematical physics having free boundaries, theory of the structure of differential equations, applied hydraulics, metal welding, rock stress, and automated planning and control systems for industrial enterprises.
    • The institute studies functional analysis, fundamental and applied problems of algebra, solid state mechanics and mathematical physics, including the theory of differential and integral equations and matrix polynomials.

  20. BMC 1971
    • Edmunds, D EQuasi-linear partial differential equations .

  21. BMC 2006
    • Mihailescu, P Cyclotomic norm equations and Catalan's Conjecture .
    • Dafermos, M The red-shift effect and decay rates for the wave equation of a Schwarzschild black hole exterior .

  22. Ukrainian Academy of Sciences
    • Research priorities in the institute include: theory of differential equations, mathematics of physics, statistical theory, theory of functions, topology, algebra, the dynamics of special mechanical systems, computer programming and the institute develops various fields of mathematics for the natural sciences and technology.
    • Research emphasis includes: nonlinear problems of mathematical physics having free boundaries, theory of the structure of differential equations, applied hydraulics, metal welding, rock stress, and automated planning and control systems for industrial enterprises.
    • The institute studies functional analysis, fundamental and applied problems of algebra, solid state mechanics and mathematical physics, including the theory of differential and integral equations and matrix polynomials.

  23. NAS Award in Applied Mathematics
    • for his penetrating, variegated, and fundamental contributions to mathematical theory and its applications to problems in functional analysis, numerical analysis, linear and non linear partial differential equations, wave propagation, and scattering theory.
    • for his extraordinary insight and invaluable contributions to the analysis and application of partial differential equations, especially to supersonic flow, combustion, vortex motion, and turbulent diffusion.
    • for his seminal contribution to the understanding of differential and difference equations and for his many outstanding contributions to numerical analysis, fluid dynamics, and meteorology.

  24. Maxwell Prize
    • for discovering the particle-like behaviour of solitary waves, which he named 'solitons'; for introducing the inverse scattering transform method of solving the initial-value problem for the KdV equation; and for many other contributions to applied mathematics.
    • for his contributions to algorithm-oriented numerical analysis which are fundamental and range from highly nonlinear algebraic systems through large-scale ordinary and partial differential equations to Markov chains.

  25. BMC 1951
    • Cartwright, M LNon-linear differential equations .
    • Friedlander, F GNon-linear differential equations .
    • Reuter, G E HNon-linear differential equations .

  26. BMC 1982
    • Yau, S T Non-linear equations in differential geometry .
    • Lloyd, N GClosed orbits of differential equations .

  27. Fermat Prize
    • for his impressive contributions to the Calculus of Variations and Geometric Measure Theory, and their link with partial differential equations.
    • for his contributions to the fine analysis of planar Brownian motions, his invention of the Brownian snake and its applications to the study of non-linear partial differential equations.

  28. BMC 2005
    • Khovanskii, A The solvability and unsolvability of equations in finite terms .
    • Special session: Differential Equationsn Organiser: E Shargorodsky .

  29. BMC 1986
    • Roquette, P Diophantine equations for algebraic integers .
    • Howie, J Equations over groups .

  30. BMC 1983
    • Quillen, D G Infinite determinants over algebraic curves arising from problems in geometry, differential equations and number theory .
    • Cook, R JSimultaneous additive equations .

  31. BMC 1964
    • Lighthill, M JAsymptotic properties of Fourier integrals and of solutions of partial differential equations .
    • Smithies, FHomotopy invariants of elliptic equations .

  32. Mathematics 2005
    • Differential Equations .

  33. BMC 1980
    • McLeod, J BConnection problems for ordinary differential equations .

  34. BMC 2008
    • Romito, MSome recent results concerning the 3D Navier-Stokes equations driven by a random force.

  35. BMC 1973
    • Birch, B JDifferential equations over finite fields .

  36. Minutes for 1982
    • Prof Everitt had agreed to bear in mind future BMC date when organizing Dundee Differential Equations Conferences.

  37. BMC 1992
    • MacCallum, M A H Algebraic algorithms for solving ordinary differential equations .

  38. Minutes for 2000
    • Four principal speakers have accepted invitations, and there will be Special Sessions in Modular Forms and Partial Differential Equations.

  39. BMC 1956
    • de Rham, G WElementary solutions of certain differential equations .

  40. BMC 1978
    • Wilkinson, J HLinear differential equations and canonical forms for matrix pencils .

  41. BMC 1996
    • Nakajima, HMonopoles and Nahm's equation .

  42. BMC 2004
    • Wood, A The influence of G C Stokes on the modern asymptotic theory of differential equations .

  43. BMC 1963
    • McLeod, J BNew results in eigenfunction theory of differential equations .

  44. BMC 1987
    • Elworthy, K D Geometric aspects of stochastic differential equations .

  45. Minutes for 2000
    • There will be special sessions on modular forms, organised by A J Baker and S Delbourgo, and on partial differential equations, organised by S Kuksin and S Merkulov.

  46. BMC 1988
    • Sparrow, C T Chaotic differential equations .

  47. BMC 1999
    • Bruce, J W Curves, surfaces and differential equations .

  48. BMC 1981
    • Schmidt, W Diophantine equations and inequalities .

  49. Rolf Schock Prize
    • for his important contributions to the theory of nonlinear partial differential equations.

  50. Czech Academy of Sciences
    • The Institute is concerned mainly with mathematical analysis (differential equations, numerical analysis, functional analysis, theory of functions, mathematical physics), probability theory and mathematical statistics, mathematical logic, theoretical computer science and graph theory, numerical algebra, topology (general and algebraic) and theory of teaching mathematics.

  51. German Society for Applied Mathematics and Mechanics
    • These are at present: Efficient Numerical Methods for Partial Differential Equations, Computer Arithmetic and Scientific Computation, Inverse Problems: Analysis and Numerical Methods, Applied Stochastic Analysis and Optimization, Material-Theory, Mathematical Analysis of Nonlinear Phenomena, Dynamics and Control Theory, Scientific Computing, Experimental Mechanics, Didactics in Mechanics, Analysis of Microstructure, Applied and Numerical Algebra, and Multiple Field Problems in Solid Mechanics.

  52. Abel Prize
    • for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.

  53. LMS Presidential Addresses
    • Partial Differential Equations: some Criticisms and Suggestions.

  54. Sylvester Medal
    • for his fundamental work in arithmetic geometry and his many contributions to the theory of ordinary differential equations.

  55. Balaguer Prize
    • Parabolic Quasilinear Equations Minimizing Linear Growth Functionals .

  56. SIAM W T and Idalia Reid Prize in Mathematics
    • The Society for Industrial and Applied Mathematics awards the W T and Idalia Reid Prize in Mathematics for work related to differential equations or control theory.

  57. Turin Mathematical Society
    • This paper contained equations which Laplace stated were important in mechanics and physical astronomy.

  58. Dahlquist Prize
    • The prize, established in 1995, is awarded to a young scientist (normally under 45) for original contributions to fields associated with Germund Dahlquist, especially the numerical solution of differential equations and numerical methods for scientific computing.

  59. Latvian Academy of Sciences
    • It was composed partly from the Institute of Physics of the Latvian Academy of Sciences and partly from the Departments of Differential Equations and General Mathematics of the Faculty of Physics and Mathematics of the University of Latvia.

  60. Paris Academy of Sciences
    • In 1880 Halphen won the Grand Prix for his work on linear differential equations.

  61. Academy of Sciences of Belarus
    • It has Divisions of: algebra; number theory; control processes theory; differential equations; mathematical cybernetics; mathematical theory of systems; non-linear analysis; numerical methods of mathematical physics; numerical modelling; parallel computational processes; and stochastic analysis.

  62. Collatz Prize
    • for his highly original and profound contributions to applied mathematics, calculus of variations and nonlinear partial differential equations, the mechanics of continua, and mathematical material sciences.

  63. BMC 1998
    • Uhlenbeck, K K Geometric partial differential equations: a comparison between elliptic and hyperbolic theory .


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  77. References for Faedo
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  105. References for Ladyzhenskaya
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  106. References for Galerkin
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  108. References for Banachiewicz
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  109. References for Riccati Vincenzo
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  114. References for Waring
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  115. References for Luzin
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  116. References for Aryabhata I
    • K S Shukla, Use of hypotenuse in the computation of the equation of the centre under the epicyclic theory in the school of Aryabhata I, Indian J.

  117. References for Laplace
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  118. References for De Beaune
    • R Schmidt and E Black (trans.), Francois Viete, Albert Girard, Florimond de Beaune, The early theory of equations: on their nature and constitution (Golden Hind Press, Fairfield, CT, 1986).

  119. References for Sylvester
    • P Holgate, Waring and Sylvester on random algebraic equations, Biometrika 73 (1) (1986), 228-231.

  120. References for Lie
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  123. References for Eisenstein
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  126. References for Griffiths Lois
    • T A Brown, Review: Introduction to the Theory of Equations by Lois Wilfred Griffiths, The Mathematical Gazette 33 (303) (1949), 57-58.

  127. References for Van der Pol
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    • S P Arya, On the Brahmagupta- Bhaskara equation, Math.

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    • M N Saltykow, L'Oeuvre de Jacobi dans le domaine des equations aux derivees partielles du premier ordre, Bull.

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  132. References for Al-Karaji
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    • J Sesiano, Le traitement des equations indeterminees dans le Badi fi al-Hisab d'Abu Bakr Al-Karaji, Arch.

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  134. References for Lefschetz
    • J Adem, A sketch of Solomon Lefschetz's life in Mexico (Spanish), Differential equations, Math.

  135. References for Bhaskara II
    • S P Arya, On the Brahmagupta- Bhaskara equation, Math.

  136. References for Levi-Civita
    • C Cattani and M De Maria, Einstein's path toward the generally covariant formulation of gravitational field equations: the contribution of Tullio Levi-Civita, in Proceedings of the fourth Marcel Grossmann meeting on general relativity, Part A, B, Rome, 1985 (North-Holland, Amsterdam, 1986), 1805-1826.
    • L Dell'Aglio and G Israel, La theorie de la stabilite et l'analyse qualitative des equations differentielles ordinaires dans les mathematiques italiennes : le point de vue de Tullio Levi-Civita, in Cahiers du Seminaire d'Histoire des Mathematiques 10 (Univ.

  137. References for Fermat
    • A P Kauchikas, Double equations in the work of Diophantus and of P Fermat (Russian), Istor.-Mat.

  138. References for Sokolov
    • V P Filcakova, A sketch of the life and scientific activity of Ju D Sokolov (on his seventy-fifth birthday) (Russian), in Approximate and qualitative methods of the theory of differential and integral equations, Izdanie Inst.

  139. References for Tikhonov
    • A B Vasileva and V M Volosov, The work of Tikhonov and his pupils on ordinary differential equations containing a small parameter, Russian Math.

  140. References for Aryabhata
    • K S Shukla, Use of hypotenuse in the computation of the equation of the centre under the epicyclic theory in the school of Aryabhata I, Indian J.

  141. References for Chazy
    • S Chakravarty and M J Ablowitz, Parameterizations of the Chazy equation, Stud.

  142. References for Mytropolshy
    • The studies of Yu A Mitropolskii on the field of the theory of nonlinear oscillations and the theory of nonlinear differential equations (Russian), in Problems of the asymptotic theory of nonlinear oscillations 275 "Naukova Dumka" (Kiev, 1977), 7-14.

  143. References for Sankara
    • P K Majumdar, A rationale of Bhatta Govinda's method for solving the equation ax - c = by and a comparative study of the determination of 'Mati' as given by Bhaskara I and Bhatta Govinda, Indian J.

  144. References for Steklov
    • A I Demcisin and V S Sologub, V A Steklov's studies in the theory of boundary value problems for ordinary differential equations (Ukrainian), Narisi Istor.

  145. References for Oleinik
    • V A Ilin, L D Kudryavtsev, Yu S Osipov and S I Pokhozhaev, Olga Arsenievna Oleinik (on the occasion of her seventieth birthday), Differential Equations 31 (7) (1995), 1035-1055 .

  146. References for Lorgna
    • E Badolati, Kepler's equation and the Neapolitan mathematical school (Italian), Rend.

  147. References for Boole
    • L de Ledesma, and L M Laita, George Boole : From differential equations to mathematical logic, in Proceedings of the Mathematical Meeting in Honor of A Dou (Madrid, 1989), 341-351.

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    • N Ya Tsyganova, Gauss's principle of least force in the research of P Stackel (Russian), Differential equations and applied problems (Tula, 1985), 68-74.

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    • M A Szalek, Pauli versus the Maxwell equations and the Biot-Savart law, Phys.

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    • I-Kh I Gerasim, On the genesis of Redei's theory of the equation x2 - Dy2 = -1 (Russian), Istor.-Mat.

  151. References for Bachelier
    • K Ito, On stochastic differential equations, Memoirs, American Mathematical Society, 4 (1951) 1-51.

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    • Ja O Matviisin, Charles Eugene Delaunay (1816-1872) : an outline of his life and scientific activity (Ukrainian), in Projective-iterative methods for the solution of differential and integral equations (Ukrainian) (Kiev, 1974), 116-130.

  153. References for Al-Biruni
    • A R Amir-Moez, Khayyam, al-Biruni, Gauss, Archimedes, and quartic equations, Texas J.

  154. References for Bers
    • L Nirenberg, Louis Lipman Bers and partial differential equations, in Lipa's legacy, New York, 1995 (Providence, RI, 1997), 455-461.

  155. References for Maschke
    • G Zappa, History of the solution of fifth- and sixth-degree equations, with an emphasis on the contributions of Francesco Brioschi (Italian), Rend.


Additional material

  1. James Gregory's manuscripts
    • James Gregory's manuscripts on algebraic solutions of equations .
    • The following is a version of Herbert Turnbull's description of his discovery of James Gregory's attempts at finding an algebraic method to solve polynomial equations.
    • Gregory's work gave new methods for solving cubic and quartic equations [often called biquadratic equations] which he hoped would generalise to allow quintic equations to be solved.
    • A perusal of the letters which passed between Gregory and Collins will show that one of the final subjects to be discussed was the theory of equations.
    • Ever since the famous discoveries of solutions for cubic and biquadratic equations by Scipio Ferro, Tartaglia and Cardan, of the Italian school, mathematicians of all countries had attempted a generalization, and particularly addressed themselves to find an algebraic solution of the quintic equation.
    • It was not, indeed, until the beginning of the nineteenth century that the matter was settled, when Abel demonstrated the impossibility of such a solution, in general, for the quintic and higher equations.
    • These three manuscripts give a new way of solving the cubic and biquadratic equations, together with a remarkable attempt upon the quintic.
    • "I have now abundantly satisfied myself in these things I was searching after in the analytics, which are all about reduction and solution of equations.
    • A week or two later Tschirnhausen, then a young man of twenty-four years, arrived in London, to spend the summer in a search for the solution of all equations up to the eighth dimension.
    • Evidently James Gregory believed that he could reduce a quintic to a quartic, and, more generally, an n-ic to a lower equation.
    • Had Dary, an energetic and skilled computer, embarked upon this enterprise, a highly interesting development in the theory of equations would have followed.
    • Cubic equations: .
    • From this is made the following equation .
    • If through the first equation a is removed both from the second and the third, the second equation becomes -q2 + 6z2 = b2, and the third .
    • From these two equations and the fourth are found three values .
    • From these emerge the following two equations without b, .
    • and hence there results the following equation having no unknown quantity except z, .
    • Quartic equations: .
    • Let x = z + v; hence the following equation .
    • Quintic equations: .
    • From these is made the following equation .
    • Let this equation be multiplied by the following .
    • The resulting equation has 20 dimensions.
    • If all its terms are separately equated to zero, except those of dimension 20, 15, 10, 5, 0, equations are formed, 16 in number.
    • But there are 17 unknown quantities (namely v, z, a, b, etc.); and since the resulting 16 equations and the one given make 17, it is clear that with the help of these operations all the unknown quantities can be removed except any one namely v.
    • If this is done, a value of v will be given by a biquadratic equation having a quintic root: and so the value of v will be given with the help of a biquadratic equation, all whose terms are explicit, and a pure quintic.
    • Therefore x = z - v is given since both z and v are made known: and hence it is clear in what manner the quintic equation can be reduced to a biquadratic, if only someone may be found who would not recoil from such labour.
    • [Gregory actually writes out this equation in full.
    • This gives him four equations for z, a, b, c in terms of the known q and r, from which he eliminates a, b, c in succession, and obtains the relation .
    • In the St Andrews manuscript there is an error in sign in the last equation for b2 and in the next for c3, but the final result is correct.
    • Unfortunately the elimination processes would have yielded at least one irreducible equation of degree 6, had the attempted work been completed: and thus the hoped-for solution - by lowering the degree from 5 to 4 - would have had to be abandoned.
    • It may also be remarked that, to complete the theory of the cubic and biquadratic, it would be necessary to show that the values of z and v, so found, would yield roots of the original equation, and not merely those of the arbitrary factor y(v).
    • (2)n Why is the degree in z of the final eliminant lower than that of x in the original equation? .
    • At any rate he readily gets the equation .
    • Thus the problem which apparently led to a biquadratic equation is, in fact, soluble in quadratics.
    • In this way we see that whenever a biquadratic equation lacks its second and fourth term, but contains one unknown quantity to one dimension [y] in the third and to two [y2] in the last term, it is possible to eliminate the unknown which occurs to four dimensions [z4], and thereby to resolve the said biquadratic into two quadratics, where the unknown quantity is confined to two dimensions.

  2. Dickson: 'Theory of Equations
    • Dickson: Theory of Equations .
    • L E Dickson published his book Elementary theory of equations in 1953.
    • ELEMENTARY THEORY of EQUATIONS .
    • For instance, one may be sure that a given cubic equation has only the one real root seen in the graph, if the bend points lie on the same side of the x-axis.
    • Emphasis is here placed upon Newton's method of solving numerical equations, both from the graphical and the numerical standpoint.
    • One of several advantages (well recognized in Europe) of Newton's method over Horner's is that it applies as well to non-algebraic as to algebraic equations.
    • In this elementary book, the author has of course omitted the difficult Galois theory of algebraic equations (certain texts on which are very erroneous) and has merely illustrated the subject of invariants by a few examples.
    • It is surprising that the theorems of Descartes, Budan, and Sturm, on the real roots of an equation, are often stated inaccurately.
    • An easy introduction to determinants and their application to the solution of systems of linear equations is afforded by Chapter XI, which is independent of the earlier chapters.
    • Elementary Theory of Equations.
    • This book occupies a middle ground in difficulty, being too advanced for the average freshman, but still of an elementary character, suitable for a second course in the theory of equations.
    • It is such a book as may be read with profit by any one who wants an exact statement and rigorous proof of the elementary theorems - not involving group-theory or invariants - concerning algebraic equations; a work of value to all teachers of algebra, whether elementary or advanced.
    • This chapter should be read by everyone who thinks that complex numbers are "imaginary" and that we gain nothing by their use except to make certain equations have roots.
    • The theorem that an integral root of an equation with integral coefficients divides the constant term might well be supplemented by the similar theorem that if an equation with integral coefficients has a fractional root a/b, a must divide the constant term and b the coefficient of the highest power of x.
    • The use of elementary calculus allows a clear treatment and a complete solution of the problem, "given an equation to locate its real roots," while the methods of Chap.
    • Besides Horner's well-known method for the numerical computation of roots, Newton's is given and emphasized as one that is effective for non-algebraic as well as for algebraic equations; and Graffe's little known but very ingenious scheme of solution by forming equations whose roots are powers of the roots of the given equation, and Lagrange's solution by continued fractions are also explained.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Dickson_equations.html .

  3. ELOGIUM OF EULER
    • After having provided the steps to the roots of algebraic equations, and their general solvability, numerous new theories and some ingenious and insightful views, Mr.
    • Cotes had already provided the way in which to represent the roots of certain algebraic equations by sine and cosine.
    • He is responsible for the general solution of linear equations which are so varied and useful as well as the first of all formulas for approximations.
    • There we are able to observe by the propitious method of substitutions or by using an already known method to solve obstinate equations or by reductions to first differentials of first order equations and then by considering the integrals' forms; he deduced the differential equations conditions to which they may be satisfied, sometimes by the thorough examination of the factors which provides for a complete differential and other times lead him to conclude the formulation of a general class of integral equations.
    • There is a particularity that he noticed in an equation which provided him with the opportunity to separate the indiscriminant which appeared confusing, otherwise if in an equation, where they are separate slips through the known methods, it is by mixing the indiscriminant that he was able to recognize the integral.
    • Euler, one will soon discover what made him presume that the operation would produce the given effect that envisaged, and if one examines the form of one of his best methods, he expects the factors of a second degree equation and one will note that he has stopped at one of those which is particular to this order of equations.
    • Euler arrived at the conclusion that differential equations are susceptible to particular solutions which are not included in general solutions in which Mr.
    • This occurs when one is seeking particular integrals for a certain determined value of the unknowns which are contained within the equation.
    • No one has made a more extended and better use of the methods which provide for the approximate value of a determined quantity through the use of differential equations for which one has obtained the first values.
    • He has also provided a direct method to immediately deduce the same equation with a value very close to the real one, through which the powers are removed from their difference and can be discarded; a method by which the approximations used by the mathematicians can not be extended though the equations for which the observations or the particular considerations do not provide the first value for these known methods.
    • Euler has plumbed the nature of differential equations, the fountainhead of difficulties which oppose integration and the way in which to elucidate or conquer them.
    • This was done by searching for the sums or the expression of their general terms and to those of the roots or determinant equations, by which to obtain with a simple calculation the approximate value of the products or the indefinite sums of certain numbers.
    • He knew how to best present it, through the deep understanding of the theory which provided him the way in which to solve a great number of these equations, to distinguish the forms of the orders of integrals, for the different number of variables, to reduce these equations when they attain a certain form and become ordinary integrations, to provide for a way in which to remember these forms, through substitutions after which they vanish; in one word, to discover within the nature of these partial differential equations most of these singular properties which render the general theory so difficult and thorny, qualities which are nearly inseparable in Geometry where the degree of difficulty is so often the measurement of interest that one takes in a question and the honor that one attaches to a discovery.
    • Equally, for the developments of general equations of curves of the second and third degrees, in other words the development of any random order which are similar to the generating curve; a remarkable equation due to its extreme simplicity.
    • These two works contain a link to the application of partial differential equations to geometrical problems that may be employed in many applications which Mr.
    • The solution to the problem that is sought for the motion of an object which is launched into space and is attracted towards two points has become famous by Euler's ability of make the necessary substitutions thought a reduction to quadratic equations so that their complexity and form might have made them appear to be insoluble.
    • The vibrating string problem and all those that belong to the theory of sound or the laws of oscillations in air had been subjected to analysis by these new methods which in turn enriched the calculus of partial differentials equations.
    • In such a way that in the course of his work there sometimes appeared a unique method to integrate a differential equation or sometimes a remark concerning a question in Analysis or Mechanics lead him to a solution to a very complicated differential equation which did not lend itself to direct methods.

  4. Euler Elogium.html.html
    • After having provided the steps to the roots of algebraic equations, and their general solvability, numerous new theories and some ingenious and insightful views, Mr.
    • Cotes had already provided the way in which to represent the roots of certain algebraic equations by sine and cosine.
    • He is responsible for the general solution of linear equations which are so varied and useful as well as the first of all formulas for approximations.
    • There we are able to observe by the propitious method of substitutions or by using an already known method to solve obstinate equations or by reductions to first differentials of first order equations and then by considering the integrals' forms; he deduced the differential equations conditions to which they may be satisfied, sometimes by the thorough examination of the factors which provides for a complete differential and other times lead him to conclude the formulation of a general class of integral equations.
    • There is a particularity that he noticed in an equation which provided him with the opportunity to separate the indiscriminant which appeared confusing, otherwise if in an equation, where they are separate slips through the known methods, it is by mixing the indiscriminant that he was able to recognize the integral.
    • Euler, one will soon discover what made him presume that the operation would produce the given effect that envisaged, and if one examines the form of one of his best methods, he expects the factors of a second degree equation and one will note that he has stopped at one of those which is particular to this order of equations.
    • Euler arrived at the conclusion that differential equations are susceptible to particular solutions which are not included in general solutions in which Mr.
    • This occurs when one is seeking particular integrals for a certain determined value of the unknowns which are contained within the equation.
    • No one has made a more extended and better use of the methods which provide for the approximate value of a determined quantity through the use of differential equations for which one has obtained the first values.
    • He has also provided a direct method to immediately deduce the same equation with a value very close to the real one, through which the powers are removed from their difference and can be discarded; a method by which the approximations used by the mathematicians can not be extended though the equations for which the observations or the particular considerations do not provide the first value for these known methods.
    • Euler has plumbed the nature of differential equations, the fountainhead of difficulties which oppose integration and the way in which to elucidate or conquer them.
    • This was done by searching for the sums or the expression of their general terms and to those of the roots or determinant equations, by which to obtain with a simple calculation the approximate value of the products or the indefinite sums of certain numbers.
    • He knew how to best present it, through the deep understanding of the theory which provided him the way in which to solve a great number of these equations, to distinguish the forms of the orders of integrals, for the different number of variables, to reduce these equations when they attain a certain form and become ordinary integrations, to provide for a way in which to remember these forms, through substitutions after which they vanish; in one word, to discover within the nature of these partial differential equations most of these singular properties which render the general theory so difficult and thorny, qualities which are nearly inseparable in Geometry where the degree of difficulty is so often the measurement of interest that one takes in a question and the honor that one attaches to a discovery.
    • Equally, for the developments of general equations of curves of the second and third degrees, in other words the development of any random order which are similar to the generating curve; a remarkable equation due to its extreme simplicity.
    • These two works contain a link to the application of partial differential equations to geometrical problems that may be employed in many applications which Mr.
    • The solution to the problem that is sought for the motion of an object which is launched into space and is attracted towards two points has become famous by Euler's ability of make the necessary substitutions thought a reduction to quadratic equations so that their complexity and form might have made them appear to be insoluble.
    • The vibrating string problem and all those that belong to the theory of sound or the laws of oscillations in air had been subjected to analysis by these new methods which in turn enriched the calculus of partial differentials equations.
    • In such a way that in the course of his work there sometimes appeared a unique method to integrate a differential equation or sometimes a remark concerning a question in Analysis or Mechanics lead him to a solution to a very complicated differential equation which did not lend itself to direct methods.

  5. M Bcher: 'Integral equations
    • M Bocher: Integral equations .
    • An introduction to the study of integral equations by Maxime Bocher was No 10 in the series and published in 1909.
    • AN INTRODUCTION TO THE STUDY OF INTEGRAL EQUATIONS .
    • In this tract I have tried to present the main portions of the theory of integral equations in a readable and, at the same time, accurate form, following roughly the lines of historical development.
    • AN INTRODUCTION TO THE STUDY OF INTEGRAL EQUATIONS .
    • The theory and applications of integral equations, or, as it is often called, of the inversion of definite integrals, have come suddenly into prominence and have held during the last half dozen years a central place in the attention of mathematicians.
    • By an integral equation [a term first suggested by du Bois-Reymond in 1888] is understood an equation in which the unknown function occurs under one or more signs of definite integration.
    • Mathematicians have so far devoted their attention mainly to two peculiarly simple types of integral equations, - the linear equations of the first and second kinds, - and we shall not in this tract attempt to go beyond these cases.
    • We shall also restrict ourselves to equations in which only simple (as distinguished from multiple) integrals occur.
    • In this respect integral equations are in striking contrast to the closely related differential equations, where the passage from ordinary to partial differential equations is attended with very serious complications.
    • The theory of integral equations may be regarded as dating back at least as far as the discovery by Fourier of the theorem concerning integrals which bears his name; for, though this was not the point of view of Fourier, this theorem may be regarded as a statement of the solution of a certain integral equation of the first kind.
    • Abel and Liouville, however, and after them others began the treatment of special integral equations in a perfectly conscious way, and many of them perceived clearly what an important place the theory was destined to fill.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Bocher_integral_equations.html .

  6. L R Ford - Differential Equations
    • Lester R Ford: Differential Equations .
    • Lester R Ford's book Differential Equations was published by the McGraw-Hill Book Company in 1933.
    • This text on differential equations definitely breaks away from the traditional form of introduction to the subject.
    • Many elementary texts on differential equations leave the student bewildered by the mass of special methods that are used for solving different types of equations.
    • As a result, the student acquires some skill and technique in handling the classical differential equations that occur in physics and elsewhere but he finds himself in a strange country when he takes a more advanced course that contains rigorous discussions.
    • Chapter VIII on linear equations contains the Gramian, the Wronskian and linear dependence, as well as symbolic methods.
    • The next chapter on certain classical equations gives a good introduction to the hypergeometric, the Legendre, and the Bessel differential equations.
    • In Chapter X on partial differential equations of the first order the distinction between complete and general solutions is well brought out, also the geometrical interpretations of solutions are emphasized.
    • In the preface, the author states: "The partial differential equation of the second order is a vexatious problem for the writer of an elementary text.
    • Notice the use of the word except instead of unless at the bottom of page 115; also the sentence on page 173 that reads "A symbolic method in which the integrations are hitched abreast instead of tandem avoids this difficulty"; also such section headings as "Aids to Good Guessing" and "On the Making of Rules." Moreover it is well to note how the special cases are grouped under general methods with the comment that an energetic and intelligent student could keep on forever discovering new rules for integrating particular types of equations.
    • In every way this is a very good text on differential equations.
    • A second edition of Differential Equations was published in 1955.
    • It is unusual to find Clairaut's equation and simple examples of solution in series in the first chapter of a text-book on differential equations, but the idea is a good one.
    • Subsequent chapters cover special methods for equations of first order, linear equations of any order with a brief account of the use of the Laplace transform, solution in series of the hypergeometric, Legendre's and Bessel's equations, approximate numerical solutions, and two chapters on partial differential equations.
    • General solutions of simple types of partial differential equations are obtained before separation of variables is used to solve problems of vibration and the Laplace equation in two dimensions.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Ford_Diff_Equations.html .

  7. Henry Baker addresses the British Association in 1913, Part 2
    • One of the earliest conscious applications of the notion was in the problem of solving algebraic equations by means of equations of lower order.
    • All equation of the fourth order can be solved by means of a cubic equation, because there exists a rational function of the four roots which takes only three values when the roots are exchanged in all possible ways.
    • Following out this suggestion for an equation of any order, we are led to consider, taking any particular rational function of its roots, what is the group of interchanges among them which leaves this function unaltered in value.
    • On these lines a complete theory of equations which are soluble algebraically can be given.
    • Then I mention the historical fact that the problem of ascertaining when that well-known linear differential equation called the hypergeometric equation has all its solutions expressible in finite terms as algebraic functions, was first solved in connection with a group of similar kind.
    • For any linear differential equation it is of primary importance to consider the group of interchanges of its solutions when the independent variable, starting from an arbitrary point, makes all possible excursions, returning to its initial value.
    • And it is in connection with this consideration that one justification arises for the view that the equation can be solved by expressing both the independent and dependent variables as single-valued functions of another variable.
    • Theory of Functions of Complex Variables: Differential Equations.
    • And then there are the problems of the theory of differential equations.
    • But our whole physical outlook is based on the belief that the problems of Nature are expressible by differential equations; and our knowledge of even the possibilities of the solutions of differential equations consists largely, save for some special types, of that kind of ignorance which, in the nature of the case, can form no idea of its own extent.
    • There are subjects whose whole content is an excuse for a desired solution of a differential equation; there are infinitely laborious methods of arithmetical computation held in high repute of which the same must be said.
    • And yet I stand here to-day to plead with you for tolerance of those who feel that the prosecution of the theoretic studies, which alone can alter this, is a justifiable aim in life! Our hope and belief is that over this vast domain of differential equations the theory of functions shall one day rule, as already it largely does, for example, over linear differential equations.

  8. St Andrews Mathematics Examinations
    • Solve the following equations .
    • Mark in a diagram the positions of the points (-1, 2), (-2, 1), and find the equation to the line joining these two points.
    • Find the equation to a circle of radius a passing through the origin, and having its centre situated on the axis of x.
    • Draw the figure represented by the following equations, and obtain the equation to the straight line drawn from the intersection of the first two perpendicular to the third:- .
    • If α and β be the roots of the equation ax2 + bx + c = 0, prove α + β = -b/a and αβ = c/a.
    • What does the equation to any locus give? What do m and c stand for in y = mx + c? Find the equation to the straight line through (5, -7), (i) parallel to 3x + 5y = 8 ; and (ii) perpendicular to the same line.
    • Prove that (2, 3) is the centre of one of the circles touching the lines 4x + 3y = 7, 5x + 12y = 20, and 3x + 4y = 8, and find the equation to the circle.
    • Find the equation to the tangent to the circle x2 + y2 = 10 at the point (-3, 1).
    • Define an ellipse, and from your definition deduce its equation in the form:- .
    • Find the conditions that the equation .
    • Solve either of the equations- .
    • Prove that in an equation with real coefficients, imaginary roots occur in pairs.
    • Prove that in any equation the number of positive roots cannot exceed the number of changes in the signs of the coefficients.
    • Prove that the equation x7 - 3x4 - 4x2 + x - 1 = 0 cannot have more than three real roots.
    • If α = 0, β = 0, γ = 0, be the equations to the sides of a triangle ABC opposite the angles A, B, C, prove that α sin A - β sin B is the equation to the straight line bisecting AB from C.
    • Find the equation to the tangent to the hyperbola in terms of the tangent of the angle which it makes with the axis of x.

  9. Goursat: 'Cours d'analyse mathmatique
    • Contents: Theorie des fonctions analytiques; Equations differentielles; Equations aux derivees partielles du premier ordre .
    • Chapter XVIII - Equations differentielles.
    • Formation des equations differentielles.
    • - Equations du premier ordre.
    • - Equations d'ordre superieur.
    • Chapter XX - Equations differentielles lineaires.
    • - Etude de quelques equations particulieres.
    • Equations a coefficients periodiques.
    • - Systemes d'equations lineaires.
    • Chapter XXI - Equations differentielles non lineaires.
    • - Etude de quelques equations du premier ordre.
    • Chapter XXII - Equations aux derivees partielles du premier ordre.
    • Equations lineaires du premier ordre.
    • - Equations aux differentielles totales.
    • - Equations du premier ordre a trois variables.
    • - Equations simultanees.
    • - Generalites sur les equations d'ordre superieur.
    • Contents: Integrales infiniment voisines; Equations aux derivees partielles du second ordre; Equations integrales; Calcul des variations .
    • Equations aux variations.
    • Chapter XXIV - Equations de Monge-Ampere.
    • Classification des equations lineaires.
    • Chapter XXV - Equations lineaires a n variables.
    • Classification des equations a n variables.
    • Chapter XXVI - Equations lineaires du type hyperbolique.
    • Etude de quelques problemes relatifs a l'equation s = f (x, y).
    • - Equations a plus de deux variables.
    • Chapter XXVII - Equations lineaires du type elliptique.
    • - Equation generale du type elliptique.
    • Chapter XXIX - Equation de la chaleur.
    • Chapter XXX - Resolution des equations integrales par approximations successives.
    • Equations integrales lineaires a limites variables.
    • - Equations integrales lineaires a limites fixes.
    • Chapter XXXI - L'equation de Fredholm.
    • Chapter XXXIII - Applications des equations integrales.
    • Applications aux equations differentielles.
    • - Applications aux equations aux derivees partielles.

  10. Collins and Gregory discuss Tschirnhaus
    • he affirms he can give general new methods for quadratures of curvilinear figures and straightening of curves, has much amplified the doctrine of constructions, and lastly a new method for the roots of all equations, whereby Hudde's reductions and breaking of equations are rendered useless, of which new method he gave me only one specimen in an easy case of a biquadratic equation ..
    • I received this from him on Friday last, and then proposed this equation to be solved by his new method: .
    • This is an equation as cannot be broke by Descartes' cubical mallet, for instead of being reduced to a cubic equation it comes to an impossible quadratic.
    • By mine of the 3rd instant I gave you some account of a new method for finding the roots of equations etc invented by Mr Tschirnhaus, a gent of Saxony, who I told you was just upon departing for Paris; and, presuming you have that letter, I proceed.
    • I received lately tw