Search Results for solution*


Biographies

  1. Finkel biography
    • One is his Mathematical Solution Book, and the other achievement, by far the most significant, is his founding of The American Mathematical Monthly.
    • Let us look first at his own description of how he came to publish the Mathematical Solution Book [Amer.
    • While teaching in a country school in Union County, Ohio, in 1887 I began the writing of my 'Mathematical Solution Book', designed to aid in improving the teaching of elementary mathematics in the rural schools, high schools and academies, and got it ready for publication the following year.
    • The Mathematical Solution Book has a title which takes up a large part of the page, namely A mathematical solution book containing systematic solutions of many of the most difficult problems.
    • Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions with Notes and Explanations by B F Finkel.
    • The Preface to Finkel's Solution Book is at THIS LINK .
    • The author is the well-known Professor of Mathematics and Physics at Drury College, and his aim is to provide systematic solutions of difficult questions in the earlier subjects.
    • He is very averse from "Short Cuts" and "Lightning Methods"; and insists that solutions should be written out step by step in logical order and the chain of reasoning made complete in every link.
    • At the same time, at my leisure, I was contributing problems and solutions to [various journals].
    • Most of our existing Journals deal almost exclusively with subjects beyond the reach of the average student or teacher of Mathematics or at least with subjects with which they are not familiar, and little, if any space, is devoted to the solution of problems.
    • Most of our existing Journals deal almost exclusively with subjects beyond the reach of the average student or teacher of Mathematics or at least with subjects with which they are not familiar, and little, if any space, is devoted to the solution of problems.
    • B F Finkel's Mathematical Solution Book .

  2. Kellogg Bruce biography
    • A major theme in Bruce's opus is the numerical solution of singular perturbation problems.
    • Another major theme of his research was the behaviour of solutions to partial differential equations near corners and interfaces.
    • An alternating direction iteration method is formulated, and convergence is proved, for the solution of certain systems of nonlinear equations.
    • 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.
    • Our object is to represent the solution to the problem as the sum of a finite number of special solutions, which may be considered as known, plus a remainder term whose regularity depends on the regularity of the data of the problem.
    • Kellogg's own summary to Discontinuous solutions of the linearized, steady state, compressible, viscous, Navier-Stokes equations (1988) is as follows:- .
    • Jump conditions across a possible curve of discontinuities of a solution of the linearised system are derived.
    • In a particular case, a discontinuous solution of the linearised system is constructed.
    • The singular perturbation causes boundary layers and interior layers in the solution, and the corners of the polygon cause corner singularities in the solution.
    • The paper considers pointwise bounds for derivatives of the solution that show the influence of these layers and corner singularities.
    • Its solution may have exponential and parabolic boundary layers, and corner singularities may also be present.
    • Sharpened pointwise bounds on the solution and its derivatives are derived.
    • We consider a one-dimensional convection-diffusion boundary value problem, whose solution contains a boundary layer at the outflow boundary, and construct a finite element method for its approximation.
    • It is shown that, measured in an e-weighted energy norm, the Galerkin finite element solution attains the same order of accuracy as the bilinear nodal interpolant.

  3. Kerr Roy biography
    • In 1963, Roy Kerr, a New Zealander, found a set of solutions of the equations of general relativity that described rotating black holes.
    • If the rotation is zero, the black hole is perfectly round and the solution is identical to the Schwarzschild solution.
    • it was conjectured that any rotating body that collapsed to form a black hole would eventually settle down to a stationary state described by the Kerr solution.
    • Finally, in 1973, David Robinson at Kings College, London, used Carter's and my results to show that the conjecture had been correct: such a black hole had indeed to be the Kerr solution.
    • 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.
    • 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.
    • Teddy Newman and Roger Penrose were working on a similar set of methods, but Teddy had come out with this as-yet unpublished theorem that basically 'proved' that my solution couldn't exist! Luckily, my neighbour, who was playing around with relativity, too, got hold of a preprint and I just scanned through it (I'm a lazy reader) and hit the crucial part which proved to me that my solution could exist! After that, I kept working like mad and found the solution in a few weeks.
    • Kip Thorne, in [Black Holes and Time Warps: Einstein\'s Outrageous Legacy (W W Norton & Co, 1995).',3)">3], recalls when Kerr announced his solution in a 10-minute presentation at the Texas Symposium on Relativistic Astrophysics in 1963:- .
    • 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.
    • Not only was the solution especially complex, lacking symmetry of previous solutions, but it became apparent that any stationary black hole can be described by Kerr's solution.

  4. 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.
    • Charles-Jean de La Vallee Poussin had asked in 1908: is it possible to approximate the ordinate of a polygonal line by means of a polynomial of degree n with error less than 1/n? Both de La Vallee Poussin and Bernstein made some progress in the following years and then the Belgium Academy of Science offered a prize for a solution.
    • Bernstein gave a complete solution in 1911, introducing what are now called the Bernstein polynomials and giving a constructive proof of Weierstrass's theorem (1885) that a continuous function on a finite subinterval of the real line can be uniformly approximated as closely as we wish by a polynomial.
    • 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.
    • 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).
    • He proved a special case of his own problem in Solution of a mathematical problem related to the theory of inheritance (1924).

  5. Khayyam biography
    • An approximate numerical solution was then found by interpolation in trigonometric tables.
    • Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by ruler and compass methods, a result which would not be proved for another 750 years.
    • 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 the science of algebra one encounters problems dependent on certain types of extremely difficult preliminary theorems, whose solution was unsuccessful for most of those who attempted it.
    • As for the Ancients, no work from them dealing with the subject has come down to us; perhaps after having looked for solutions and having examined them, they were unable to fathom their difficulties; or perhaps their investigations did not require such an examination; or finally, their works on this subject, if they existed, have not been translated into our language.
    • 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.
    • He did hope that "arithmetic solutions" might be found one day when he wrote (see for example [Dictionary of Scientific Biography (New York 1970-1990).
    • University of Georgia (Geometric solution of the cubic) .

  6. Mazya biography
    • As Maz'ya did not make this a secret, his fellow students all decided not to submit their solutions.
    • 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 book is in two parts, the first is on the higher-dimensional potential theory and the solution of the boundary problems for regions with irregular boundaries while the second part is on the space of functions whose derivatives are measures.
    • In collaboration with S A Nazarov and B A Plamenevskii, Maz'ya published Asymptotic behavior of solutions of elliptic boundary value problems under singular perturbations of the domain in 1981.
    • The book deals with the construction of asymptotic expansions of solutions of elliptic boundary value problems under singular perturbations of the domains (i.e.
    • blunted angles, cones or edges, small holes, narrow slits, etc.) A general approach is suggested, its main feature being systematic application of solutions to the so-called 'limit' problems.
    • Singularities of these solutions do not increase from one step to another.
    • The main term of an asymptotic expansion for a solution of the Dirichlet problem for the Laplacian in a three-dimensional domain with a narrow slit is obtained in the third chapter.
    • The fourth chapter deals with asymptotic expansions of solutions to a quasilinear equation of the second order.
    • 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).

  7. Bhaskara II biography
    • To give some examples before we examine his work in a little more detail we note that he knew that x2 = 9 had two solutions.
    • When p = 61 he found the solutions x = 226153980, y = 1776319049.
    • When p = 67 he found the solutions x = 5967, y = 48842.
    • Bhaskaracharya is finding integer solution to 195x = 221y + 65.
    • He obtains the solutions (x, y) = (6, 5) or (23, 20) or (40, 35) and so on.
    • Joy and happiness is indeed ever increasing in this world for those who have Lilavati clasped to their throats, decorated as the members are with neat reduction of fractions, multiplication and involution, pure and perfect as are the solutions, and tasteful as is the speech which is exemplified.
    • 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 problem is to find integer solutions to an equation of the form ax + by + cz = d.
    • Of course such problems do not have a unique solution as Bhaskaracharya is fully aware.
    • He finds one solution, which is the minimum, namely horses 85, camels 76, mules 31 and oxen 4.

  8. 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.
    • 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.
    • However, the fact that he was always satisfied with a rational solution and did not require a whole number is more sophisticated than we might realise today.
    • For example to find a square between 5/4 and 2 he multiplies both by 64, spots the square 100 between 80 and 128, so obtaining the solution 25/16 to the original problem.
    • In Book V he solves problems such as writing 13 as the sum of two square each greater than 6 (and he gives the solution 66049/10201 and 66564/10201).

  9. Ferro biography
    • Today we write the solutions to ax2 + bx + c = 0 as .
    • In del Ferro's time, although such solutions were known, they were not known in this form.
    • 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.
    • The subsequent developments in the story of the solution of the cubic, namely the contest in 1535 between Antonio Maria Fior (a student of del Ferro) and Tartaglia, then the involvement of Cardan, are told in detail in our biographies of Tartaglia and of Cardan.
    • As far as this biography of del Ferro is concerned we should stress that it was Cardan's discovery that del Ferro had been the first to solve the cubic and not Tartaglia which made him feel that he could honour his oath to Tartaglia not to divulge his method and still publish the solution in Ars Magna for there Cardan considered he is giving del Ferro's method, not that of Tartaglia.
    • 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.
    • Scipione Ferro of Bologna, almost thirty years ago, discovered the solution of the cube and things equal to a number [which in today's notation is the case x3+ mx = n], a really beautiful and admirable accomplishment.
    • The story that Fior was the only person to whom del Ferro divulged his solution is common in most histories of mathematics, yet it is false.
    • As we have seen above the solution was written down by del Ferro and certainly was known to Nave.
    • Pompeo Bolognetti, who lectured at the University of Bologna on mathematics from 1554 to 1568, also had access to the original solution by del Ferro as well as the solution as given by Cardan in Ars Magna which had been published by then.
    • 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.

  10. Ljunggren biography
    • The journal presented a collection of problems and each year the Crown Prince Olav Prize was given to the pupil who gave the best solutions to these problems.
    • He proved that there is only a finite number of solutions and that it is possible to determine an upper limit for this number; in the special case D = 2, D1 = 3 he showed that the only solution is x = 3, y = 2, z = 1.
    • where the left-hand side has no squared factor in x, has only a finite number of solutions.
    • However Mordell did not find the solution, nor was he able to find bounds on the finite number of solutions.
    • 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.
    • In the second of the two papers he proves that, under certain conditions on D, there are again at most two positive integer solutions.
    • where D + 1 is not a square has at most two solutions if n ≠ 2, and if there are two solutions, these will be determined by the fundamental unit of the domain Z[√D] and its second or fourth power.
    • [Ljunggren] was also a very gifted problem solver, contributing many original solutions to problems posed by his peers.

  11. Cherry biography
    • 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.
    • 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.
    • The author states that he has solved the problem of finding the exact solution of a two-dimensional uniform flow of a compressible perfect fluid about a cylinder.
    • The solution contains an infinite number of parameters which theoretically can be fixed to determine the shape of the 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.
    • The essential part of the paper is to overcome this difficulty by successfully continuing "analytically" the solutions into the region of higher speeds.
    • Also in 1949 he published Numerical solutions for transonic flow which:- .
    • presents the flow patterns past a cylinder, produced by superposition of a cosine-term solution and a sine-term solution to that generated from an incompressible flow past a cylinder without circulation.

  12. Choquet-Bruhat biography
    • 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.
    • Her research covers a very wide range of knowledge from the first mathematical proof for the existence of solutions of Einstein's relativistic theory of gravitation to the study of the conversion of electromagnetic waves into gravitational waves (or the reverse) in the vicinity of a black hole.
    • 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.
    • In 1963 she published Recueil de problemes de mathematiques a l'usage des physiciens which was translated into English as Problems and solutions in mathematical physics (1967).
    • It consists of problems and their solutions required for certificate examinations in the mathematical methods of physics.
    • 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.

  13. Lewy biography
    • In this paper criteria are given for determining conditions which guarantee the stability of numerical solutions of certain classes of differential equations.
    • 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.
    • 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).
    • The dock problem, written jointly with Friedrichs two years later, gives an explicit solution for the dock problem over a fluid of infinite depth.
    • The solution is given by the sum of two integrals of Laplace type taken over a complex path of integration.
    • 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.

  14. Bergman biography
    • This led him further to a general theory of integral operators that map arbitrary analytic functions into solutions of various partial differential equations.
    • It has been known for a century that the problem of finding the two-dimensional potential flow of an incompressible fluid can be solved by means of complex variables: To each analytic function of a complex variable corresponds a particular solution of the potential problem and vice versa.
    • 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.
    • Though all solutions obtained by Bers and Gelbart can be derived by Bergman's methods also, it must be expected that the new approach will prove very useful.
    • 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.
    • 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.
    • Results in the theory of one complex variable on such topics as analytic continuation, the residue theorem, Hadamard's theorems on the connection between the coefficients of the power series development of an analytic function and the character and location of the singularities and on Abelian integrals are used to give information concerning domains of regularity, series expansion, singularities and integral relations for the solutions.

  15. Baker Alan biography
    • where m is an integer and f is an irreducible homogeneous binary form of degree at least three, with integer coefficients, have at most finitely many solutions in integers.
    • 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.
    • Baker however went further and produced results which, at least in principle, could lead to a complete solution of this type of problem.
    • for any solution (x0, y0) of f (x, y) = m.
    • Of course this means that only a finite number of possibilities need to be considered so, at least in principle, one could determine the complete list of solutions by checking each of the finite number of possible solutions.
    • Hilbert himself remarked that he expected this problem to be harder than the solution of the Riemann conjecture.
    • Secondly, it shows that a direct solution of a deep problem develops itself quite naturally into a healthy theory and gets into early and fruitful contact with significant problems of mathematics.

  16. Lions biography
    • 70 (2) (1996), 125-135.',3)">3] is his work on "viscosity solutions" for nonlinear partial differential equations.
    • such nonlinear partial differential equation simply do not have smooth or even C1 solutions existing after short times.
    • The only option is therefore to search for some kind of "weak" solution.
    • Lions and Crandall at last broke open the problem by focusing attention on viscosity solutions, which are defined in terms of certain inequalities holding wherever the graph of the solution is touched on one side or the other by a smooth test function.
    • In 1989 Lions, in joint work with DiPerma, was the first to give a rigorous solution with arbitrary initial data.
    • 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.
    • And, amazingly enough, he was the first person to establish (in 2000) a result of existence and uniqueness of the solution of this type of problem.

  17. 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.
    • We also give sufficient conditions for a solution to stay in a weakly closed set." .
    • 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.
    • Some of the problems dealt with from a mathematical point of view involved obtaining asymptotic properties of the solutions, in the framework of semigroup theory of positive operators as well as the application of aggregation of variables methods to models formulated with two time scales.

  18. Al-Haytham biography
    • Huygens found a good solution which Vincenzo Riccati and then Saladini simplified and improved.
    • Ibn al-Haytham wrote a treatise Solution of doubts in which he gives his answers to these questions.
    • In Opuscula ibn al-Haytham considers the solution of a system of congruences.
    • Ibn al-Haytham gives two methods of solution:- .
    • The problem is indeterminate, that is it admits of many solutions.
    • Here ibn al-Haytham gives a general method of solution which, in the special case, gives the solution (7 - 1)! + 1.
    • Ibn al-Haytham's second method gives all the solutions to systems of congruences of the type stated (which of course is a special case of the Chinese Remainder Theorem).
    • Article: The Telegraph (The solution of Alhazen's problem) .

  19. Lax Peter biography
    • In 1957 he published an extremely important paper Asymptotic solutions of oscillating initial value problems where the beginnings of the theory of Fourier integral operators appears.
    • 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.
    • He constructed explicit solutions, identified classes of especially well-behaved systems, introduced an important notion of entropy, and, with Glimm, made a penetrating study of how solutions behave over a long period of time.
    • In addition, he introduced the widely used Lax-Friedrichs and Lax-Wendroff numerical schemes for computing solutions.
    • 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.
    • In the late 1960s a revolution occurred when Kruskal and co-workers discovered a new family of examples, which have "soliton" solutions: single-crested waves that maintain their shape as they travel.
    • 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".
    • Together with Phillips, Lax developed a broad theory of scattering and described the long-term behaviour of solutions (specifically, the decay of energy).
    • In 1970 Lax and Glimm published Decay of solutions of systems of nonlinear hyperbolic conservation laws, a difficult work which requires familiarity with earlier work of both authors.

  20. Harriot biography
    • he produced a practical numerical solution of the Mercator problem, most probably by the addition of secants ..
    • He gave a solution to Alhazen's problem which involved considering an equivalent problem, namely the problem of the maximum intercept formed between a circle and a diameter of a chord rotating about a point on a circle.
    • He came very close to a vector analysis solution of the problem of finding the velocity of the projectile and, certainly by 1607, he came to the conclusion that the path of the projectile was a tilted parabola.
    • 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.
    • For example, it does not discuss negative solutions.

  21. Besicovitch biography
    • The solutions submitted were carefully read and annotated by Besicovitch and the announcement "Perfect solutions of Problem 12 were sent in by M and N" spurred several young mathematicians on to develop their analytic powers.
    • When solving a problem most mathematicians need to make a commitment as to the nature of the solution long before the solution has been found, and this commitment interposes a psychological barrier to the consideration of other possibilities.
    • One of the achievements, with which he will always be associated, was his solution of the Kakeya problem on minimising areas.
    • Often he showed that the "obvious solution" to certain problems is false.
    • The solution generally accepted for this problem by around 1950 was that however the man moved, the lion first aimed to get onto the line joining the man to the centre of the arena (which it could always achieve) and then keeping on this radius however the man moved, it would end up catching the man.
    • He was more likely than anyone else to solve a problem which had seemed intractable, commonly the solution needed, by way of proof or counter-example, an ingenious and intricate construction.

  22. Menaechmus biography
    • Menaechmus's solution is described by Eutocius in his commentary to Archimedes' On the sphere and cylinder.
    • Of course we must emphasis that this in no way indicates the way that Menaechmus solved the problem but it does show in modern terms how the parabola and hyperbola enter into the solution to the problem.
    • Immediately following this solution, Eutocius gives a second solution.
    • ',1)">1], [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3] and [Greek Geometry from Thales to Euclid (Dublin-London, 1889), 153-179.',4)">4] all consider a problem associated with these solutions.
    • Plutarch says that Plato disapproved of Menaechmus's solution using mechanical devices which, he believed, debased the study of geometry which he regarded as the highest achievement of the human mind.
    • However, the solution described above which follows Eutocius does not seem to involve mechanical devices.
    • The solution proposed to this question in [Dictionary of Scientific Biography (New York 1970-1990).
    • What has come to be known as Plato's solution to the problem of duplicating the cube is widely recognised as not due to Plato since it involves a mechanical instrument.
    • it seems probable that someone who had Menaechmus's second solution before him worked to show how the same representation of the four straight lines could be got by a mechanical construction as an alternative to the use of conics.

  23. Mytropolsky biography
    • He further developed asymptotic methods and applied them to the solution of practical problems.
    • 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.
    • 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.

  24. Kantorovich biography
    • Kantorovich gave two lectures, "On conformal mappings of domains" and "On some methods of approximate solution of partial differential equations".
    • 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.
    • The trust's laboratory seemed unable to arrive at a satisfactory solution that could not be further improved upon.
    • The mathematical formulation of production problems of optimal planning was presented here for the first time and the effective methods of their solution and economic analysis were proposed.
    • Kantorovich introduced many new concepts into the study of mathematical programming such as giving necessary and sufficient optimality conditions on the base of supporting hyperplanes at the solution point in the production space, the concept of primal-dual methods, the interpretation in economics of multipliers, and the column-generation method used in linear programming.
    • 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.

  25. Rudolff biography
    • This is evident from old books on algebra, written many years ago, in which quantities are represented, not by characters, but by words written out in full, 'drachm', 'thing', 'substance', etc., and in the solution of each special example the statement was put, 'one thing', in such words as ponatur, una res, etc.
    • From this quote we see that he must have read the Latin Regensburg algebra of 1461 for in that work the solution to all problems begin with the words 'Pono quod lucrum sit una res'.
    • 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.
    • For several such problems concerned with "splitting the bill" (Zechenaufgaben) Rudolff supplied all the possible solutions.
    • Rudolff does not work out their solutions because, as he stated, he wanted to stimulate further algebraic research.
    • Rudolff was aware of the double root of the equation ax2 + b = cx and gave all the solutions to indeterminate first-degree equations.

  26. Faddeev biography
    • 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.
    • Until then it had been possible to prove only that there were non-trivial solutions for some A.
    • 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.
    • Two features are very characteristic of the mode of presentation: on the one hand the extensive use of geometrical considerations as a background for the true understanding of complicated situations which otherwise would remain obscure, and on the other hand, the care shown by the authors in inventing effective methods of solution, illustrated by actual application to numerical examples and to the construction of valuable tables.
    • 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).
    • This book consists of 983 problems (209 pages); hints to selected problems (37 pages), mostly the briefest possible; and 250 pages of solutions, ranging from mere answers to numerical problems to complete proofs ..

  27. Fermat biography
    • I have also found many sorts of analyses for diverse problems, numerical as well as geometrical, for the solution of which Viete's analysis could not have sufficed.
    • has no non-zero integer solutions for x, y and z when n > 2.
    • The second of the two problems, namely to find all solutions of Nx2 + 1 = y2 for N not a square, was however solved by Wallis and Brouncker and they developed continued fractions in their solution.
    • Brouncker produced rational solutions which led to arguments.
    • 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.

  28. 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.
    • By applying the Poincare-Lyapunov theory and the Poincare-Denjoy theory on trajectories on a torus, he examined the nature of the exact stationary solution in the vicinity of an approximate solution for a sufficiently small value of the parameter and proved theorems on the existence and stability of quasi-periodic solutions.
    • Moreover, general statements of methods for solving equations are illustrated by the explicit solution of examples.
    • In solving problems of this type it is exceedingly important to provide the degree of rigor required for the correct solution of these problems.

  29. Kublanovskaya biography
    • 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.
    • For all these cases, well-chosen numerical examples are analyzed and the solutions are tabulated.
    • Vera Kublanovskaya submitted her first QR summary 'Certain algorithms for the solution of the complete eigenvalue problem' on 5 July 1960, followed by two subsequent papers 'Some algorithms for the solution of the complete eigenvalue problem' (1961) and 'The solution of the eigenvalue problem for an arbitrary matrix' (1962) with details.
    • Kublanovskaya continued publishing significant papers on related topics (all written in Russian) such as: (with Vera Faddeeva) Computational methods for the solution of the general eigenvalue problem (1962); On a method of orthogonalizing a system of vectors (1964); Reduction of an arbitrary matrix to tridiagonal form (1964); A numerical scheme for the Jacobi process (1964); Some bounds for the eigenvalues of a positive definite matrix (1965); An algorithm for the calculation of eigenvalues of positive definite matrices (1965); On a certain process of supplementary orthogonalisation of vectors (1965); and A method for solving the complete problem of eigenvalues of a degenerate matrix (1966).
    • 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.

  30. Robinson Julia biography
    • Along with Martin Davis and Hilary Putman she gave a fundamental result which contributed to the solution to Hilbert's Tenth Problem, making what became known as the Robinson hypothesis.
    • She also did important work on that problem with Matijasevic after he gave the complete solution in 1970.
    • .,w) = 0 such that the sets of all values of x in all solutions of P = 0 is too complicated a set to be calculated by any method whatever.
    • .,w) = 0 has a solution for a given value of a, then we would have a method of calculating whether a belongs to the set S, and this is impossible.
    • As a result of her work at RAND she published An iterative method of solving a game in the Annals of Mathematics in 1951 in which she proved the convergence of an iterative process for approximating solutions for each player in a finite two-person zero-sum game.
    • 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?" .

  31. Matiyasevich biography
    • Does there exist an algorithm to determine whether a Diophantine equation has a solution in natural numbers? .
    • The fact that the second statement asks for a solution in natural numbers while the first asks for a solution in integers is not significant.
    • On the morning of 3 January 1970, I believed I had a solution of Hilbert's tenth problem, but by the end of that day I had discovered a flaw in my work.
    • This paper shows that every recursively enumerable relation is diophantine and so completes the solution of Hilbert's tenth problem in the negative sense.
    • Matiyasevich also published Solution of the tenth problem of Hilbert in Hungarian in 1970.
    • 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.
    • The first five lead to the negative solution of Hilbert's Tenth Problem; the remaining chapters are devoted to various applications of the method used by the author, which is, in a sense, more important than the solution itself: it has applications to Hilbert's eighth problem, decision problems in number theory, Diophantine complexity, decision problems in calculus, and Diophantine games.

  32. Marchenko biography
    • 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 these classes solutions are contained that can be obtained by the inverse problem method and by the methods of algebraic geometry, and also solutions that do not reduce to these methods.

  33. Serrin biography
    • A set of conditions is given for the solution of flow problems involving curved boundaries.
    • 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).
    • One of the fundamental questions which should be answered concerning any problem of applied mathematics is whether it is well set, that is, whether solutions actually exist and whether they are unique.
    • We shall be concerned here with the initial value problem for compressible fluid flow, and we shall study in particular the uniqueness of its solutions.
    • 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 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.

  34. Word problems
    • Here there are above all three fundamental problems whose solution is very difficult and which will not be possible without a penetrating study of the subject.
    • Each knotted space curve, in order to be completely understood, demands the solution of the three above problems in a special case.
    • He used his solution to show that right and left trefoils are distinct.
    • He published these results in 1927 and at the same time gave a simple rigorous proof of the solution of the word problem in a free group.
    • The solution to the word problem for these groups began with Dehn who stated the Freiheitssatz: .
    • In the following year Magnus published a paper containing a special case of the word problem for 1-relator groups, then in 1932 he published a complete proof of the solution of the word problem for this class of groups.
    • It required computability theory and developments in mathematical logic to even make the questions precise, but these areas were to not only provide explicit questions, they also provided solutions to the questions.
    • While sitting in the dentist's chair waiting for this unpleasant experience, inspiration struck and suddenly he saw the route to the solution.

  35. Cartan biography
    • This enabled Cartan to define what the general solution of an arbitrary differential system really is but he was not only interested in the general solution for he also studied singular solutions.
    • He did this by moving from a given system to a new associated system whose general solution gave the singular solutions to the original system.
    • He failed to show that all singular solutions were given by his technique, however, and this was not achieved until four years after his death.

  36. De Giorgi biography
    • as a child I had a special taste for puzzling out solutions to little problems, but I also had a certain passion for experimenting with little gadgets - experiments, if not of physics, of "pre-physics'".
    • 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.

  37. Word problems
    • Here there are above all three fundamental problems whose solution is very difficult and which will not be possible without a penetrating study of the subject.
    • Each knotted space curve, in order to be completely understood, demands the solution of the three above problems in a special case.
    • He used his solution to show that right and left trefoils are distinct.
    • He published these results in 1927 and at the same time gave a simple rigorous proof of the solution of the word problem in a free group.
    • The solution to the word problem for these groups began with Dehn who stated the Freiheitssatz: .
    • In the following year Magnus published a paper containing a special case of the word problem for 1-relator groups, then in 1932 he published a complete proof of the solution of the word problem for this class of groups.
    • It required computability theory and developments in mathematical logic to even make the questions precise, but these areas were to not only provide explicit questions, they also provided solutions to the questions.
    • While sitting in the dentist's chair waiting for this unpleasant experience, inspiration struck and suddenly he saw the route to the solution.

  38. 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.
    • 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.
    • 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).
    • 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.
    • The paper opens with a rapid expository review of the general relativity theory of gravitation, including discussion of kinematics, conservation laws, spherical symmetry, and the solutions of Schwarzschild and de Sitter in terms of comoving coordinates.

  39. 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.
    • Moreover, general statements of methods for solving equations are illustrated by the explicit solution of examples.
    • Before publishing this book with Bogolyubov, in 1931 Krylov had published the important monograph Les methodes de solution approchee des problemes de la physique mathematique.

  40. Schubert biography
    • Schubert is famed for his work on enumerative geometry which considers those parts of algebraic geometry that involves a finite number of solutions.
    • 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.

  41. 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.
    • 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).
    • Methods for exact solution published in 1974 was fundamental, and the ideas developed in it were later extended to dynamical systems, inverse scattering, and symplectic geometry.
    • 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.

  42. Halphen biography
    • The first result which brought him to the attention of mathematicians world-wide was his solution in 1873 of a problem of Chasles [Dictionary of Scientific Biography (New York 1970-1990).
    • Halphen's solution was ingenious ..
    • He defined the concepts of proper and improper solutions to an enumerative problem involving conics.
    • Then a particular number associated with a problem about conics has enumerative significance when it counts the number of proper solutions.
    • Halphen and Schubert engaged in a heated debate on whether an enumerative formula should be allowed to count degenerate solutions along with the nondegenerate solutions.

  43. Magiros biography
    • Magiros wrote other papers on stability such as The stability in the sense of Lyapunov, Poincare and Lagrange of some precessional phenomena (1970) and Remarks on stability concepts of solutions of dynamical systems (1974).
    • 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.

  44. 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.
    • Descartes presented a solution but it was rejected by Stampioen as not being complete.
    • He then posed two further public challenges under the alias of John Baptista of Antwerp involving the solution of cubics.
    • Having posed the problems as if by John Baptista of Antwerp, he then proceeded to give solutions to the problems under his own name, using his methods for finding the cube root of a + √b.
    • Stampioen rejected Waessenaer's solution which prompted Waessenaer to reply with a broadly based attack on the mathematics contained in Stampioen's Algebra or the New Method.
    • He published a topographical map in 1650 and, were it not for the fact that he is mentioned as being a member of a panel set up to adjudicate a proposed solution to the longitude problem in 1689, we might have wrongly supposed that he died shortly after 1650.

  45. Abel biography
    • While in his final year at school, however, Abel had begun working on the solution of quintic equations by radicals.
    • In Abel's third paper, Solutions of some problems by means of definite integrals he gave the first solution of an integral equation.
    • 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.

  46. Al-Quhi biography
    • Al-Quhi's solution to the problem is given in [Centaurus 38 (2-3) (1996), 140-207.',5)">5].
    • It is a classical style of solution using results from Euclid's Elements, Apollonius's Conics and Archimedes' On the sphere and cylinder.
    • If a solution exists, al-Quhi showed that it will have coordinates which lie on a particular rectangular hyperbola that he has constructed.
    • Next al-Quhi introduces the "cone of the surface" which, after many deductions, leads to showing that the solution has coordinates lying on a parabola.
    • 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.
    • Topics covered are quite varied, ranging from a discussion of what "known" means to solutions of specific problems such as the following Suppose we are given a circle and two intersecting straight lines l and m.

  47. Ferrari biography
    • They worked on problems set by Zuanne da Coi and eventually were able to extend solutions discovered in these special cases.
    • 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.
    • Cardan and Ferrari satisfied della Nave that they could solve the ubiquitous cosa and cube problem, and della Nave showed them in return the papers of the late del Ferro, proving that Tartaglia was not the first to discover the solution of the cubic.
    • Cardan published both the solution to the cubic and Ferrari's solution to the quartic in Ars Magna (1545) convinced that he could break his oath since Tartaglia was not the first to solve the cubic.

  48. Norlund biography
    • He studied the factorial series and interpolation series entering in their solutions, determining their region of convergence and by analytic prolongation and different summation methods he extended them in the complex plane, determining their singularities and their behaviour at infinity, also by use of their relations to continued fractions and asymptotic series.
    • 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 ..
    • Complete tables are given of the linear and quadratic relations which hold between the various solutions in every possible special case.
    • 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 ..
    • This coolness could be felt as an aloofness; maybe it was due to some sort of shyness, but on the other hand his words thus gained more importance and one could feel how he exerted himself to find the right solution to problems.

  49. Fantappie biography
    • The geometrical significance in abstract spaces of such notions as characteristic strips and singular solutions is given.
    • 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.
    • These solutions had been always rejected as "impossible", but suddenly they appeared "possible", and they explained a new category of phenomena which I later named "syntropic", totally different from the entropic ones, of the mechanical, physical and chemical laws, which obey only the principle of classical causation and the law of entropy.
    • Syntropic phenomena, which are instead represented by those strange solutions of the "anticipated potentials", should obey two opposite principles of finality (moved by a final cause placed in the future, and not by a cause which is placed in the past): differentiation and non-causable in a laboratory.
    • 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.

  50. Sluze biography
    • This work had been inspired by Mersenne who had informed them of Torricelli's solution of the problem of calculating the volume of the solid generated by revolving a hyperbola about the axis.
    • De Sluze had gained the experience necessary to solve such problems and sent his solution to Pascal who praised it highly.
    • 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.
    • They had both been stimulated to work on it by reading Isaac Barrow's Lectiones Opticae (1669) where he gave only a partial solution to the problem.
    • Huygens sent his results to Oldenburg in which he stated that he had found an elegant solution.
    • Some time later de Sluze told Oldenburg that he had found a good method of solution and was then sent details of Huygens' method by Oldenburg.

  51. Caccioppoli biography
    • For the linear case he considered a linear transformation acting on the vectors of a linear space (in which the solution is to be found).
    • If the image set entirely covers the second linear space then solutions exist independently on the given data.
    • the image set is a linear, closed subspace in the second linear space) then necessary and sufficient conditions are placed on the data set so that the problem has solutions.
    • 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.
    • Caccioppoli proved the analyticity of C2-class solutions.

  52. Lorenz Edward biography
    • The first few terms of a particular series solution are obtained explicitly.
    • Solutions of these equations can be identified with trajectories in phase space.
    • For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states.
    • Systems with bounded solutions are shown to possess bounded numerical solutions.

  53. Mahavira biography
    • He also discussed integer solutions of first degree indeterminate equation by a method called kuttaka.
    • The kuttaka (or the "pulveriser") method is based on the use of the Euclidean algorithm but the method of solution also resembles the continued fraction process of Euler given in 1764.
    • There is no unique solution but the smallest solution in positive integers is p = 15, x = 1, y = 3, z = 5.
    • Any solution in positive integers is a multiple of this solution as Mahavira claims.

  54. Pogorelov biography
    • It contains a number of new results on the setting of boundary-value problems, and on questions of uniqueness and regularity of generalized solutions.
    • The booklet contains a solution of Hilbert's well-known fourth problem concerning the determination of all realizations up to isomorphism of the system of axioms of classical geometries (Euclidean and non-Euclidean) supposing that the axioms of congruency are replaced by the axiom "triangle inequality".
    • Pogorelov's solution to Hilbert's Fourth Problem, which he presented to a meeting of the Kharkov Mathematical Society held at the Kharkov University, was described by I Kra as a "mathematical jewel" [Ukrainian Math.
    • In my article published in 1973 I have admitted some immodesty when I entitled it as "The Complete Solution of the Fourth Hilbert Problem".
    • In fact, it did not contain a complete solution of the fourth problem, because only the two-dimensional case was examined.
    • He was peerless in his skill in overcoming difficulties in the solution of hard mathematical problems.

  55. Fox Leslie biography
    • 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.
    • Another collaboration between Fox and Mayers led to Numerical solution of ordinary differential equations published in 1987, four years after Fox retired.
    • It considers the success achieved in the production of new techniques, machine-oriented techniques, error analysis, mathematical theorems and the solution of practical problems, and contrasts this with corresponding work in the field of linear algebra.

  56. Sokolov biography
    • 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.

  57. Ollerenshaw biography
    • Without fail, on waking in the morning, the details, the logical argument required or the facts that I needed to recall were clearly imprinted in my mind and, because of clarity, any required solution would often be clearly 'written' on the partition.
    • It was moreover a matter of geometry -- pure mathematics -- a nice problem that had a neat and successful solution.
    • Critical lattices relate to whole numbers in two or more dimensions and lead, by geometrical methods, to solutions concerned with 'close packing', for example, how best to stack tins in a cupboard or oranges in a box.
    • This became a good outlet for her papers which included the first general solution Rubik's cube, a solution to the twelve penny problem, and a solution to the nine prisoners problem.

  58. Tikhonov biography
    • 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.
    • However, his mathematical investigations are never confined to the solution of a given concrete problem, but serve as the starting point for stating a general mathematical problem that is a broad generalisation of the first problem.
    • 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.
    • Under his guidance many algorithms for the solution of various problems of electrodynamics, geophysics, plasma physics, gas dynamics, ..
    • He defined a class of regularisable ill-posed problems and introduced the concept of a regularising operator which was used in the solution of these problems.
    • Combining his computing skills with solving problems of this type, Tikhonov gave computer implementations of algorithms to compute the operators which he used in the solution of these problems.

  59. Morton biography
    • One may indeed expect a rapid expansion of this activity in the next few years as the solution of physically interesting two- and three-dimensional flows becomes a practical and economic proposition.
    • This is that from the inception of the subject scientists attempting the solution of fluid flow problems have continuously made outstanding contributions to the subject's development.
    • 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).
    • The book includes parabolic, hyperbolic, and elliptic equations, each section starting with an analysis of the behaviour of solutions of the partial differential equations.
    • The level of presentation is ideal for anyone with some knowledge of numerical analysis who wishes to learn about the solution of convection-diffusion problems.

  60. 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.
    • 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.
    • 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.
    • He established important theoretical results, including properties of existence and uniqueness of solutions of initial value problems, which are the basis of recent developments in methods for numerical simulation of gas in networks.

  61. Yang Hui biography
    • Firstly he explains the logic behind the problem, secondly he gives a numerical solution to the problem, and thirdly he shows how the method he has presented can be modified to solve similar problems.
    • 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.
    • Although Yang has presented a problem straight from the Nine Chapters his method of solution is quite different.
    • the additive method of multiplication and the subtractive method of division [relative to the] ten problems and their solutions.
    • Yang's solution is quoted in [The nine chapters on the mathematical art : Companion and commentary (Beijing, 1999).',8)">8]:- .
    • Then a modern solution would set up equations .

  62. Loyd biography
    • Click Solution White plays Ng4 as first move.
    • The subtle solutions are when Black plays Kf3 or Kh3 as first move.
    • Now if Black plays Kh3 or g2 then Qh7 for White mates, while if Black plays f3 then White mates with Rh8.')">HERE for the solution.
    • A prize of $1000, offered for the first correct solution to the problem, has never been claimed, although there are thousands of persons who say they have performed the required feat.
    • At first sight one would not expect there to be a unique solution to this problem, but Loyd was well aware that a logical argument would find the one and only one solution.

  63. Viete biography
    • (If I asked for a solution to ax = b nobody asks: "For which quantity do I solve the equation ?") .
    • The problem is that if we ask for a solution of x3 + x = 1 then we ask for the solution to a problem which does not make sense geometrically.
    • 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 gave geometrical solutions to doubling a cube and trisecting an angle in this book.

  64. Deligne biography
    • 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.
    • A solution of the three Weil conjectures was given by Deligne in 1974.
    • A solution to these problems required the development of a new kind of algebraic topology.
    • These conjectures were both exceptionally hard to settle (the best specialists, including A Grothendieck, had worked on them) and most interesting in view of the far-reaching consequences of their solution.

  65. Spencer Tony biography
    • As an example of his continued research activity, let us quote his own summary of his paper Exact solutions for a thick elastic plate with a thin elastic surface layer published in 2005:- .
    • 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.
    • It is shown that the interface tractions and in-plane stress discontinuities are determined only by the initial two-dimensional solution, without recourse to the three-dimensional elasticity theory.

  66. Gale biography
    • in 1949 for his thesis Solutions of Finite Two-Person Games.
    • Related to his thesis were the papers (with S Sherman) Solutions of finite two-person games, and two papers written jointly with H W Kuhn and A W Tucker, On symmetric games, and Reductions of game matrices.
    • The paper on symmetric games shows that if a zero-sum two-person game has an m × n pay-off matrix, then an optimal solution is immediately derivable from an optimal solution of a symmetric zero-sum two-person game with a square pay-off matrix of order m+n+1.
    • His contributions range from optimal assignment problems in a general setting to major contributions to mathematical economics such as his solution of the n-dimensional 'Ramsey Problem' and his important theory of optimal economic growth.

  67. 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.
    • He rightly deduced Emden's equation must have other solutions.
    • When Milne divulged his thoughts to Fowler, Ralph immediately developed a new solution for different values of n and all types of boundary solutions.
    • 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.

  68. Carmichael biography
    • He taught for three years at the Presbyterian College in Anniston and by 1909 he had around 170 publications in the American Mathematical Monthly, mostly problems and solutions to problems, as well as 13 papers in the Annals of Mathematics and the Bulletin of the American Mathematical Society.
    • in 1911 for his thesis Linear Difference Equations and their Analytic Solutions Linear Difference Equations and their Analytic Solutions.
    • 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.

  69. Douglas biography
    • It was during this period that he worked out a complete solution to the Plateau problem which had been posed by Lagrange in 1760 and then had been studied by leading mathematicians such as Riemann, Weierstrass and Schwarz.
    • Before Douglas's solution only special cases of the problem had been solved.
    • In a series of papers from 1927 onwards Douglas worked towards the complete solution: Extremals and transversality of the general calculus of variations problem of the first order in space (1927), The general geometry of paths (1927-28), and A method of numerical solution of the problem of Plateau (1927-28).
    • Douglas presented full details of his solution in Solution of the problem of Plateau in the Transactions of the American Mathematical Society in 1931.
    • After giving a complete solution to the Plateau Problem, Douglas went on to study generalisations of it.
    • In particular the award was for three papers all published in 1939: Green's function and the problem of Plateau and The most general form of the problem of Plateau published in the American Journal of Mathematics and Solution of the inverse problem of the calculus of variations published in the Proceedings of the National Academy of Sciences.
    • The third paper does not give the compete proof for the solution of the inverse problem of the calculus of variations but is an announcement of the result.

  70. Rado biography
    • In 1930 Rado published the work for which he is most famous, namely his solution to the Plateau Problem.
    • Plateau was a physicist who experimented with dipping thin wire frames into a soap solution and examining the soap film which was then stretched across the wire.
    • Garnier made a major breakthrough in 1928 followed soon after by independent solutions to the general problem by Douglas and by Rado.
    • His solution appeared 1930 in The problem of least area and the problem of Plateau published in Mathematische Zeitschrift.
    • Let us remark that the solution to the Plateau problem by both Douglas and by Rado did not exclude the possibility that the minimal surface contained a singularity.

  71. Keller Joseph biography
    • 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.
    • Two problems are inverses if the formulation of each involves all or part of the solution of the other.
    • The direct problem of the pair has been extensively studied or has a solution readily obtained by standard methods.
    • They can have no or only partly determined solutions even though the corresponding direct problems are well posed.

  72. Wu biography
    • 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.
    • 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.
    • We discuss results on the existence and uniqueness of solutions for given data, the regularity of solutions, singularity formation and the nature of the solutions after the singularity formation time.

  73. Margulis biography
    • Margulis's most spectacular achievement has been the complete solution of that problem and, in particular, the proof of the conjecture in question.
    • Margulis proved the full conjecture in 1986 and gives a beautiful survey of the work leading to this solution in [Fields Medallists Lectures (Singapore, 1997), 272-327.',3)">3].
    • One was the solution to a problem posed by Rusiewicz, about finitely additive measures on spheres and Euclidean spaces.
    • Though his work addresses deep unsolved problems, his solutions are housed in new conceptual and methodological frameworks of broad and enduring application.
    • Besides his celebrated results on super-rigidity and arithmeticity of irreducible lattices of higher rank semisimple Lie groups, and the solution of the Oppenheim conjecture on values of irrational indefinite quadratic forms at integral points, he has also initiated many other directions of research and solved a variety of famous open problems.

  74. Black Fischer biography
    • The solution was to demand a new level of mathematical technique.
    • The solution worked using a number of steps :- .
    • The famous Black-Scholes formula for the solution is:- .
    • He was in the forefront of recognising the importance of computer technology and good trading/engineering systems and thought that trying to model reality was more important than closed form analytical solutions.
    • ',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.

  75. Catalan biography
    • Liouville began publication of Journal de Mathematiques Pures et Appliquees in 1836 and a paper by Catalan, Solution d'un probleme de Probabilite relatif au jeu de rencontre, was published in the second volume in 1837.
    • The second of these contains the 'Catalan numbers' which appears in the solution of the problem of dissecting a polygon into triangles by means of non-intersecting diagonals.
    • 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.
    • He wrote: Elements de geometrie (1843); the two volume work Traite elementaire de geometrie descriptive (1850-52) which ran to 5 editions with the last appearing in 1881; Theoremes et problemes de geometrie elementaire (1852) which ran to 6 editions with the last appearing in 1879; Nouveau manuel des aspirants au baccalaureat es sciences (1852) which ran to 12 editions; Solutions des problemes de mathematique et de physique donnes a la Sorbonne dans les compositions du baccalaureat es sciences (1855-56); two volumes of Manuel des candidats a l'Ecole Polytechnique (1857-58); Notions d'astronomie (1860) which ran to 6 editions; Traite elementaire des series (1860); Histoire d'un concours (1865) with a second edition published in 1867; and Cours d'analyse de l'universite de Liege (1870) with a second edition published in 1880.

  76. 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:- .
    • The second subject covered in the book concerns the existence and uniqueness of positive periodic solutions for systems satisfying certain monotonicity assumptions.
    • The third part is concerned with the study of the connection between convexity and concavity of the displacement operator and the stability or instability of periodic solutions.
    • For example Approximate solution of operator equations (1969):- .

  77. 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.
    • Some of those problems were long standing, and their unexpected solutions caused wonder and surprise due to the originality and elegance of the method conceived by Gromov: famous instances are his proof of the old conjecture according to which a finitely generated group of polynomial growth has a nilpotent subgroup of finite index, or the beautiful construction (together with I Pyatetski-Shapiro) of non-arithmetic discrete groups of hyperbolic transformations in arbitrary dimension.
    • To summarise, Gromov has brought about not only solutions to famous and time-old problems, but also the bases of new fields of study for many scholars.
    • for his work in Riemannian geometry, which revolutionized the subject; his theory of pseudoholomorphic curves in symplectic manifolds; his solution of the problem of groups of polynomial growth; and his construction of the theory of hyperbolic groups.
    • The Abel committee says: "Mikhail Gromov is always in pursuit of new questions and is constantly thinking of new ideas for solutions to old problems.

  78. Lavrentev biography
    • Lavrent'ev gave excellent examples of practical application of theoretical solutions.
    • 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.
    • In his works written together with Mstislav Vsevolodovich Keldysh, important results obtained in the theory of conformal mappings were used for the solution of numerous applied problems.
    • The scientific work carried out at the Institute of Mathematics of the Ukrainian Academy of Sciences under the guidance of Lavrent'ev was aimed at the solution of not only fundamental theoretical problems but also numerous important applied problems.
    • A great merit of Lavrent'ev lies in the fact that he resolutely directed the theoretical investigations at the Institute of Mathematics of the Ukrainian Academy of Sciences towards the solution of the urgent problems of national economy.

  79. Kirkman biography
    • After Steiner asked his question, a solution was given by M Reiss in 1859.
    • The solution to the Fifteen Schoolgirls Problem is not particularly hard.
    • Cayley published a solution first, then Kirkman published his own solution, which of course he knew before asking the question.
    • Kirkman continued to study mathematics until his 89th year sending questions and solutions to the Educational Times up to a few months before his death.

  80. Bernoulli Johann biography
    • Five solutions were obtained, Jacob Bernoulli and Leibniz both solving the problem in addition to Johann Bernoulli.
    • The solution of the cycloid had not been found by Galileo who had earlier given an incorrect solution.
    • Johann's solution to this problem was less satisfactory than that of Jacob but, when Johann returned to the problem in 1718 having read a work by Taylor, he produced an elegant solution which was to form a foundation for the calculus of variations.

  81. Sundman biography
    • His methods were applicable more generally, however, and he went on to make a major breakthrough in the solution of the three-body problem.
    • The most famous contribution of Sundman was his solution of the three-body problem which he accomplished using analytic methods to prove the existence of an infinite series solution.
    • In fact the Academy was so impressed by his solution that, after receiving a report on his work from a committee headed by emile Picard, they decided to double the usual value of the prize in recognition of the brilliance of the work.
    • One of the reasons why the Academy was so impressed by Sundman's solution was that it was a problem to which Henri Poincare had devoted much effort.
    • The rigorous solution of the three-body problem is no further advanced today than it was in Lagrange's time, and one could say that it was clearly impossible.
    • In particular, their divergences reflect different views about the mathematical model of the three-body problem; and, finally, different conceptions about the idea of 'solution' of a physico-mathematical problem.
    • Sundman will always be known for his remarkable solution to the extremely difficult three-body problem, but he did other important work.

  82. Petryshyn biography
    • 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.
    • 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.
    • 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..

  83. Qin Jiushao biography
    • Again equations of high degree appear, one problem involving the solution of the equation of degree 10.
    • [Solution: x = 3, so diameter of city is x2= 9 li] .
    • 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).
    • Note the solution x = 1008 is not given] .
    • His next move, therefore, is to make various passes through the mk making replacements in such a way that eventually the new moduli are pairwise coprime but the solution to the original problem remains unchanged by the replacements.
    • He then solves each congruence and finally reassembles the answers to give the solution to the system of simultaneous congruences.
    • Shen discusses Qin's method of solution fully in [J.
    • How impressive is this work? Well suffice to say that Euler failed to provide a satisfactory solution to these problems and it was left to Gauss, Lebesgue and Stieltjes to rediscovered this method of solving systems of congruences.

  84. Stein biography
    • This theory led to important connections between harmonic analysis and probability theory, and facilitated the solution of numerous problems.
    • His explicit approximate solutions for the ∂-problems made it possible to prove sharp regularity results for solutions in strongly pseudoconvex domains.
    • Before Stein tells you his solution, the problems involved look utterly hopeless.

  85. Eutocius biography
    • the account of the solutions of the problem of the duplication of the cube, or the finding of two mean proportionals, by Plato, Heron, Philon, Apollonius, Diocles, Pappus, Sporus, Menaechmus, Archytas, Eratosthenes, Nicomedes; .
    • the fragment discovered by Eutocius himself containing the missing solution, promised by Archimedes in On the Sphere and Cylinder Book II.
    • 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.

  86. 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.
    • The author remarks, "there does not seem to appear in the literature any method for improving the accuracy of the characteristic values, except by cumbersome iteration." She then develops a method which corrects not only an approximate characteristic value, but also the coefficients in the series for the 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).

  87. Viviani biography
    • I was able to benefit from our intelligent conversations and his precious teachings and he was content that in the study of mathematics, which I had only recently begun, I could turn to his own voice for the solution of those doubts and difficulties that I often found through the natural weakness of my intellect.
    • The first of these contained solutions to twelve geometrical problems which had been published as challenges by Cristoforo Sadler.
    • It appears that Viviani found these easy and did not consider it worthwhile to publish his solutions but Prince Leopoldo, brother of the Grand Duke, had strongly encouraged him to do so.
    • His solution involved using the intersection of four right cylinders, the bases of which are tangent to the base of the hemisphere.

  88. Yamabe biography
    • Not only did Yamabe put the final touches to his solution to Hilbert's Fifth Problem in Princeton but while he was there his first child Kimiko was born and he submitted his doctoral dissertation to Osaka University and was awarded his doctorate.
    • In the following year Yamabe published A unique continuation theorem for solutions of a parabolic differential equation written jointly with Seizo Ito.
    • It was proved by P J Heawood that the four-colouring of a given "normal" map (having 2n vertices, 3n edges, and n + 2 faces or countries) is equivalent to the solution of a system of n + 2 congruences modulo 3 for 2n unknowns (each equal to ±1, and each occurring in three of the congruences).
    • The authors have coded this problem on the UNIVAC Scientific 1103 computer, which took 25 minutes to obtain the 146 solutions in a typical case with n = 18.

  89. Adian biography
    • The teacher asked everyone to solve only a couple of problems from each section, and he was immensely surprised when one of the students, Sergei Adian, handed him a thick notebook with complete solutions, drawings included, of all the problems from Rybkin's book! It is not surprising that the Education Department of Kirovabad submitted to Baku, the capital of the Azerbaijan Republic, a petition to send Sergei Adian to Moscow State University (MSU) to continue his education after completing his secondary school studies.
    • (Clearly, all continuous solutions of the equation are linear functions.) This result was not published at the time.
    • thesis, this new problem was more interesting, was mentioned in Kurosh's monograph, and was a difficult problem that had resisted solution by Novikov's methods.
    • Completing the project took intensive efforts from both collaborators in the course of eight years, and in 1968 their famous paper Infinite periodic groups appeared, containing a negative solution of the problem for all odd periods n > 4381, and hence for all multiples of those odd integers as well.

  90. Brahmagupta biography
    • Brahmagupta perhaps used the method of continued fractions to find the integral solution of an indeterminate equation of the type ax + c = by.
    • For example he solves 8x2 + 1 = y2 obtaining the solutions (x, y) = (1, 3), (6, 17), (35, 99), (204, 577), (1189, 3363), ..
    • For the equation 11x2 + 1 = y2 Brahmagupta obtained the solutions (x, y) = (3, 10), (161/5, 534/5), ..
    • He also solves 61x2 + 1 = y2 which is particularly elegant having x = 226153980, y = 1766319049 as its smallest solution.

  91. Bateman biography
    • 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.
    • One of these 1908 papers is his first publication on transformations of partial differential equations and their general solutions.
    • In 1904 he extended Whittaker's solution of the potential and wave equation by definite integrals to more general partial differential equations.
    • 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.

  92. Remez biography
    • 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.
    • All of Remez's work is characterised by great skill in applying the subtlest theoretical studies to finding a numerical solution to concrete problems.

  93. Aubin biography
    • The analytic problem requires one to prove the existence of a solution of a highly nonlinear (complex Monge-Ampere) differential equation.
    • Professor Aubin is widely known for his contribution to the solutions of the Calabi conjecture as well as the Yamabe problem.
    • 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.
    • The inclusion of a large number of interesting exercises (some of them with complete solutions) enhances the educational value of this book.

  94. MacDuffee biography
    • 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.

  95. 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.
    • He has no reservations about doing this, even though in the problems he subsequently treats he neglects possible negative solutions.
    • He then showed that, using his calculus of complex numbers, correct real solutions could be obtained from the Cardan-Tartaglia formula for the solution to a cubic even when the formula gave an expression involving the square roots of negative numbers.

  96. Taylor biography
    • For example Taylor wrote to Machin in 1712 providing a solution to a problem concerning Kepler's second law of planetary motion.
    • The paper gives a solution to the problem of the centre of oscillation of a body, and it resulted in a priority dispute with Johann Bernoulli.
    • 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.

  97. Riccati Vincenzo biography
    • 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.
    • Al-Haytham had a less than satisfactory solution to his own problem, Christiaan Huygens found a good solution which Vincenzo Riccati and Saladini simplified and improved.

  98. Tartaglia biography
    • 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.
    • 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.

  99. 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 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.
    • Next we present two extracts from reviews of Computer Solution of Linear Algebraic Systems.
    • 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.
    • The student should have cultivated and practiced the solution of mathematical problems new to him.

  100. De L'Hopital biography
    • L'Hopital had published a few brief mathematical notes, but in 1692, while Bernoulli was giving him lessons at Ouques, l'Hopital sent a solution of de Beaune's problem to Huygens.
    • Florimond de Beaune had asked for a curve for which the subtangent had a fixed length and Bernoulli had included the solution in the course he had given l'Hopital.
    • L'Hopital did not claim that the solution he sent Huygens was his own but Huygens made the reasonable assumption that it was.
    • Shortly after this l'Hopital published the solution under a pseudonym.
    • By the time Bernoulli saw the published solution he was back in Basel and, naturally enough, he was highly displeased.
    • It does appear, however, that he made few if any mathematical discoveries of his own and his solution of the brachystochrone problem was probably not his own.
    • The fact that this problem was solved independently by Newton, Leibniz and Jacob Bernoulli would put l'Hopital in very good company indeed if the solution was indeed due to him.

  101. Floer biography
    • Floer developed a new method for "counting" the solutions of maximum-minimum problems arising in geometry.
    • A certain quantity called the "index" traditionally used to classify solutions was infinite, and therefore unhelpful, in many important but apparently intractable problems.
    • Andreas realized that the difference between the indices of any two solutions could still be defined and could be used where the index itself was useless.
    • Combining this observation with detailed, careful analysis, and using work of many other mathematicians as well as his own, Andreas developed a theory that led to the solution of a number of outstanding problems.

  102. Krylov Aleksei biography
    • 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.
    • 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.
    • is to present simple methods of composition of the secular equation in the developed form, after which, its solution, i.e.
    • Krylov's practical interests were combined with a deep understanding of the ideas and methods of classical mathematics and mechanics of the seventeenth, eighteenth, and nineteenth centuries; and in the world of Newton, Euler, and Gauss, he found forgotten methods that were applicable to the solution of contemporary problems.

  103. Roomen biography
    • This should result in tables of sines, tangents and secants, and in a solution of the circle squaring problem, which for him meant the calculation of the proportion between the circumference and the diameter of a circle.
    • Section 11 would examine the many faulty or simply wrong solutions to the problem of squaring of the circle.
    • Viete's solution was published in 1595 and, at the end of his booklet, he proposed the Apollonian Problem of drawing a circle to touch three given circles.
    • Viete published a ruler and compass solution to the Apollonian Problem in 1600 which greatly impressed van Roomen [Descartes\'s Mathematical Thought (Springer, New York, 2003).',2)">2]:- .

  104. Haselgrove biography
    • Some of the applications of computers were immediately obvious; many well-formulated problems in science and engineering which required numerical solutions could benefit directly from faster computation.
    • This work involves the elucidation of classes of mathematical problems which are suitable for solution by a standardised approach.
    • The solution methods have to be studied analytically and tested on an extensive range of problems, to determine their applicability and limitations.
    • 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.

  105. Olech biography
    • He was not curious about the details, but he asked me to write out my solution, which of course I did.
    • 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).
    • 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.

  106. 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 (*).
    • Under these conditions (1) always has a solution for large values of m.
    • His solution of this case appeared in his paper On the representation of numbers in the form ax2 + by2 + cz2 + dt2 which was published in Acta Mathematica in 1926.

  107. Martin biography
    • Artemas Martin, a self-taught mathematician whose activity covered almost six decades, has been described as 'a unique example of what an inherent love of the solution of mathematical problems can do to a man even if he has not the advantages of advanced schooling'.
    • With his library there is a number of notebooks which contain his solutions to mathematical problems and well as lists of books that he was trying to purchase.
    • Martin published a very large number of problems and solutions to problems in a wide range of publications [American National Biography 14 (Oxford, 1999), 587.',3)">3]:- .
    • has that rare and happy faculty of presenting his solutions in the simplest mathematical language, so that those who have mastered the elements of the various branches of mathematics, are able to understand his reasoning.

  108. Colson biography
    • Notice that (a - b)3 + 3ab(a - b) = a3 - b3 so if a and b satisfy ab = m and a3 - b3 = 2n then a - b is a solution of x3 + 3mx = 2n.
    • 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.

  109. Ladyzhenskaya biography
    • 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.
    • 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.
    • 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.

  110. Sobolev biography
    • 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.
    • These ideas in the main concern generalised solutions of non-classical boundary value problems.

  111. Mikhlin biography
    • Dealing with the plane elasticity problem, he proposed two methods for its solution in multiply connected domains.
    • The second method is a certain generalisation of the classical Schwarz algorithm for the solution of the Dirichlet problem in a given domain by reducing it to simpler problems in smaller domains whose union is the original one.
    • He studied the error of the approximate solution for shells, similar to plane plates, and found out that this error is small for the so-called purely rotational state of stress.
    • When applied to the variational method, this notion enabled him to state necessary and sufficient conditions in order to minimise errors in the solution of the given problem when the error arising in the numerical construction of the algebraic system resulting from the application of the method itself is sufficiently small, no matter how large is the system's order.
    • Mikhlin also studied the finite element approximation in weighted Sobolev spaces related to the numerical solution of degenerate elliptic equations.
    • He found the optimal order of approximation for some methods of solution of variational inequalities.
    • Algorithm error: is the intrinsic error of the algorithm used for the solution of the approximating problem.

  112. Gregory biography
    • 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 .

  113. 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.
    • An original contribution is the axiomatic construction of the fundamentals of plane elasticity, the accuracy and generality of the mathematical procedures and some new numerical methods of solution.
    • His next important book, published in collaboration with Milan Prager and Emil Vitasek in 1964, was Numerical Solution of Differential Equations (Czech).

  114. Schoen biography
    • 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.
    • The solution of the Yamabe problem on compact manifolds, which Schoen discussed in this lecture, is one of his greatest achievements.
    • 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".

  115. 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.
    • In Book 4 al-Samawal also classifies problems into necessary problems, namely ones which can be solved; possible problems, namely ones where it is not known whether a solution can be found or not, and impossible problems which [The development of Arabic mathematics : between arithmetic and algebra (London, 1994).',3)">3]:- .
    • if one could assume the existence of their solution, this existence would lead to an absurdity.

  116. Nirenberg biography
    • 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.
    • 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).
    • 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.

  117. Kumano-Go biography
    • 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.
    • In two papers Kumano-Go also studied non-uniqueness of solutions of the Cauchy problem.
    • the construction of the fundamental solution of a first order hyperbolic system and the study of the wave front sets of solutions.

  118. Fichera biography
    • In 1949 he published the important paper Analisi esistenziale per le soluzioni dei problemi al contorno misti, relativi all'equazione e ai sistemi di equazioni del secondo ordine di tipo ellittico, autoaggiunti on uniqueness and existence of solutions of certain mixed boundary value problems.
    • 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.
    • concern the existence, uniqueness and regularity of solutions.

  119. Gleason biography
    • In his talk he sketched a possible approach to the solution of Hilbert's fifth problem, emphasizing the importance of one-parameter (local) subgroups in a locally Euclidean group G.
    • Then, in 1952, Gleason's paper Groups without small subgroups taken together with the results of Montgomery and Zippin, and Yamabe, gave a complete solution to Hilbert's problem.
    • Gleason won the Newcomb Cleveland Prize from the American Association for the Advancement of Science for his contribution to the solution of the problem.
    • In 1980 Gleason, together with R E Greenwood and L M Kelly, published The William Lowell Putnam Mathematical Competition which gave all the problems and their solutions from the beginning of the competition in 1938 up to 1964.

  120. 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.
    • A striking application is their solution of a conjecture of Mark Kac concerning large time asymptotics of a tubular neighbourhood of the Brownian motion path, the so-called 'Wiener sausage'.

  121. Dudeney biography
    • Click Solution Player one will fail if K tries to capture WP and H tries to capture BP.
    • It is a parity problem.')">HERE for the solution.
    • Other puzzles simply reduced to systems of linear equations if a mathematical solution was sought.
    • How could he have done it? There is no necessity to give measurements, for if the smaller piece (which is half a square) be made a little too large or small, it will not effect the method of solution.
    • Click Solution The problem can be solved with only two cuts, creating five pieces as shown.')">HERE for the solution.

  122. Al-Tusi Sharaf biography
    • 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.
    • We use, of course, modern notation to make the solution easy to understand, while al-Tusi would express all his mathematics in words.
    • 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.

  123. Trahtman biography
    • My first publications were [in semigroup theory], in particular, the solution of the Tarski problem ..
    • We present a solution of the road colouring problem.
    • Quite often it just requires a cunning new way to think about the problem and a solution drops out.
    • Unlike Fermat's Last Theorem which required lots of technical and sophisticated mathematical techniques to crack, the maths behind Trakhtman's solution is not complicated, it just required a clever new way to look for the solution.
    • Now he's working on a real algorithm to implement his solution.

  124. 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.
    • Pascal published a challenge under the name of Dettonville offering two prizes for solutions to these problems, and he lodged the prizes together with his own solutions with Carcavi.
    • He asked Carcavi and Roberval to judge the solutions submitted showing his respect for Carcavi's mathematical abilities.

  125. Keldysh Mstislav biography
    • The paper gives a very clear and concise exposition of various recent results concerning the solvability of Dirichlet's problem and also the stability of the solution when the boundary of the domain varies.
    • In Chapter I the author gives an exposition of the generalized solution of Dirichlet's problem in the sense of Wiener, and discusses the notions of regular and irregular points.
    • Chapter III is devoted to Wiener's criterion for regularity of a point, and to discussion of the behaviour of the solution at an irregular point.
    • Chapter IV contains an exposition of the harmonic measure and integral representation of the generalized solution.
    • Here he was able to use his experience in solving the flutter problem and his solution to shimmy, together with detailed instructions to engineers on how to overcome the problem, was described in Shimmy of the front gear of the three wheels undercart (1945).
    • He was named Hero of the Socialist Labour in 1956 for his solution to defence problems and received the Lenin Prize in the following year.

  126. Li Zhi biography
    • 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.
    • If we examine Li Zhi's solution closely we see a remarkable depth of understanding of equations.
    • Knowing that the solution cannot be a negative number (x = -16), Li Zhi works with the cubic factor and solves that to find the solution.
    • He gives the solution 20 pu which is the diameter of the pond.

  127. 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 .
    • They considered numerical methods which find an approximate solution on a grid of values of x and t, replacing ut(x, t) and uxx(x, t) by finite difference approximations.
    • Richardson's method yielded a numerical solution which was very easy to compute, but alas was numerically unstable and thus useless.
    • 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.

  128. Samoilenko biography
    • His most original contribution was the numeric-analytic method for the study of periodic solutions of differential equations with periodic right hand side.
    • They wrote Numerical-analytic methods for the study of periodic solutions (Russian) in 1976 and followed this with a new work on similar topics entitled Numerical-analytic methods for investigating the solutions of boundary value problems (Russian) in 1986.
    • In this monograph we present new promising directions in the development of numerical-analytic methods for studying the solutions of nonlinear boundary value problems in the case of a general form of boundary conditions, problems with controlling parameters, and also boundary value problems for impulse systems.
    • We give the solutions of typical problems in a course on ordinary differential equations.

  129. Dantzig George biography
    • Using hand-operated desk calculators, approximately 120 man-days were required to obtain a solution.
    • The particular problem solved was one which had been studied earlier by George Stigler (who later became a Nobel Laureate) who proposed a solution based on the substitution of certain foods by others which gave more nutrition per dollar.
    • He did not claim the solution to be the cheapest but gave his reasons for believing that the cost per annum could not be reduced by more than a few dollars.
    • Indeed, it turned out that Stigler's solution (expressed in 1945 dollars) was only 24 cents higher than the true minimum per year $39.69.
    • His work permits the solution of many previously intractable problems and has made linear programming into one of the most frequently used techniques of modern applied mathematics.

  130. Leray biography
    • This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.
    • He studied solutions of the initial value problem for three-dimensional Navier-Stokes equations.
    • He examined not only the existence and uniqueness of solutions but he showed that the solutions remained smooth for only a finite time after which turbulent solutions arise.

  131. Orszag biography
    • This book contains a wealth of solved problems and of techniques for approximating the exact solutions.
    • 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.
    • There are many examples and exercises, and a welcome feature is the large number of diagrams which compare the approximate solutions with the known solutions of some of the problems discussed.
    • It is not only the authors who hope for a similar book on approximate solutions of partial differential equations.

  132. Boruvka biography
    • To many people Boruvka is best known for his solution of the Minimal Spanning Tree problem which he published in 1926 in two papers On a certain minimal problem (Czech) and Contribution to the solution of a problem of economical construction of electrical networks (Czech).
    • He spent 1926-27 in Paris, where he lectured on his solution to the Minimal Spanning Tree problem, then returned to Masaryk University in Brno where he habilitated and was made a dozent in 1928.
    • And that is why I came with the idea that the solution of that problem was possible only in the following way that in the first period one would acquire some experience in the simplest cases and only in the second period, on the basis of the concepts introduced and experience obtained, one would go to the extension of those results to the most general case.
    • I would like to remember facts in my life that were essential not only for me personally, but chiefly for mathematics and for the future mathematical generation: Before every serious task I try to find carefully and dutifully how to fulfil it in the best way, and when I find a solution, I carry it out as best as I can according to my best sense and conscience and with all my might.

  133. Hall Marshall biography
    • This mathematical problem had been studied since about 1893, but the solution to the 92 by 92 matrix was unproven until 1961 because it required extensive computation.
    • Perhaps his best known result in group theory is his solution of the Burnside problem for groups of exponent 6.
    • He outlined his proof in Solution of the Burnside problem of exponent 6 (1957) and gave full details in the 22-page paper Solution of the Burnside problem of exponent six (1958).
    • John Thompson, then a graduate student at the University of Chicago, persuaded Saunders Mac Lane to invite me to talk on this subject [his solution of the Burnside problem for exponent 6].

  134. Fuchs biography
    • 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.
    • He succeeded in characterising those differential equations the solutions of which have no essential singularity in the extended complex plane.
    • 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.
    • In this interesting paper Gray also discusses the relationships between Fuchs' ideas and his mathematical tools, and illustrates how solutions of some problems led Fuchs to the study of further problems.

  135. 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 1980 Golub lectured on the numerical solution of large linear systems at a summer school in France.
    • 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.
    • 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.

  136. Mazur biography
    • We discussed problems proposed right there, often with no solution evident even after several hours of thinking.
    • We found a solution to a problem involving infinite dimensional vector spaces.
    • As with many of the problems in the Scottish Book the proposer would offer a prize for their solution.
    • Per Enflo showed in 1972 that the problem had a negative solution and, while in Warsaw lecturing on his solution, Mazur presented him with his prize, the live goose! .

  137. 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'.
    • In January of this year, Mr Friedmann submitted to me an extensive study of about 130 pages, in which he gave a quite satisfactory solution of the problem.
    • In reality it turns out that the solution given in it does not satisfy the field equations.

  138. Fatou biography
    • Although not giving a complete solution, Fatou's work also made a major contribution to finding a solution to the related question of whether conformal mapping of Jordan regions onto the open disc can be extended continuously to the boundary.
    • 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.

  139. 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).
    • J McCowan of University College at Dundee discussed this topic [waves] more fully and arrived at exact and complete solutions for certain cases.

  140. Appell biography
    • In 1878 he noted the physical significance of the imaginary period of elliptic functions in the solution of the pendulum which had been though to be purely a mathematical curiosity.
    • Appell submitted a solution which won second place.
    • his scientific work consists of a series of brilliant solutions of particular problems, some of the greatest difficulty.

  141. Perelman biography
    • He returned to St Petersburg at the end of April 2002 and, in July, put Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, the third instalment of his work, on the web.
    • In March 2010 the Clay Mathematics Institute announced that Perelman had met the conditions for the award of one million US dollars which they had offered for the solution of the Poincare Conjecture.
    • I consider that the American mathematician Hamilton's contribution to the solution of the problem is no less than mine.

  142. Dupre biography
    • Athanase Dupre has determined the surface tensions of solutions of soap by different methods.
    • A statistical method gives for one part of common soap in 5000 of water a surface tension about one-half as great as for pure water, but if the tension be measured on a jet close to the orifice, the value (for the same solution) is sensibly identical with that of pure water.
    • He explains these different values of the surface tension of the same solution as well as the great effect on the surface tension which a very small quantity of soap or other trifling impurity may produce, by the tendency of the soap or other substance to form a film on the surface of the liquid.

  143. Woods biography
    • This work led to two papers in 1950: Improvements to the accuracy of arithmetical solutions to certain two-dimensional field problems and The two-dimensional subsonic flow of an inviscid fluid about on aerofoil of arbitrary shape.
    • He now published a whole series of papers - the next two were: A new relaxation treatment of flow with axial symmetry (1951), and The numerical solution of two-dimensional fluid motion in the neighbourhood of stagnation points and sharp corners (1952).
    • 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.

  144. Sintsov biography
    • 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.
    • Sintsov gave in 1903 an elegant proof of its general real solution, which has the form f (x, y) = q(x) - q(y), where q is an arbitrary function in one variable.
    • in 1903) elementary simple proofs of its general real solutions.

  145. Sneddon biography
    • The book discusses applications of Fourier, Mellin, Laplace and Hankel transforms to the solution of problems in physics and engineering.
    • 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.
    • 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.

  146. Horner biography
    • 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.
    • It is also worth noting that he gave a solution to what has come to be known as the "butterfly problem" which appeared in The Gentleman's Diary for 1815 [Math.
    • The butterfly problem, whose name becomes clear on looking at the figure, has led to a wide range of interesting solutions.

  147. Pascal biography
    • Pascal published a challenge offering two prizes for solutions to these problems to Wren, Laloubere, Leibniz, Huygens, Wallis, Fermat and several other mathematicians.
    • Wallis and Laloubere entered the competition but Laloubere's solution was wrong and Wallis was also not successful.
    • Pascal published his own solutions to his challenge problems in the Letters to Carcavi.

  148. Fejer biography
    • Encouraged by Maksay, Fejer began submitting his solutions to the problems to Budapest [The Mathematical Intelligencer 15 (2) (1993), 13-26.',6)">6]:- .
    • Laszlo Racz, a secondary school teacher who led a problem study group in Budapest, often opened his session by saying, "Lipot Weiss has again sent in a beautiful solution." .
    • Poisson's integral provides a valid solution for Dirichlet's problem for the circle.

  149. Wald biography
    • Wald reported to the seminar on his work in econometrics, in particular he wrote a paper for the seminar on the existence of a solution to the competitive economic model.
    • He proved important results, perhaps the most significant being the existence of a solution to the competitive economic model which, as we noted above was written for Menger's seminar.
    • 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.

  150. Prthudakasvami biography
    • The solution of a first-degree indeterminate equation by a method called kuttaka (or "pulveriser") was given by Aryabhata I.
    • This method of finding integer solutions resembles the continued fraction process and can also be seen as a use of the Euclidean algorithm.
    • Brahmagupta seems to have used a method involving continued fractions to find integer solutions of an indeterminate equation of the type ax + c = by.

  151. 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).

  152. Kochin biography
    • The seminar participants also reviewed works containing solutions to specific problems related to the frontal model.
    • Soon Kochin's article was published, specifying Defant's solution.
    • He gave the solution to the problem of small amplitude waves on the surface of an uncompressed liquid in the paper Towards a Theory of Cauchy-Poisson Waves (Russian) in 1935.

  153. 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.
    • The existence of solutions with stronger hypothesis on f had been given earlier by Cauchy and then Lipschitz.
    • 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.

  154. 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.
    • Rather it presents a number of rules, some of which are far from easy, each given for the numerical solution of a geometric problem.
    • The Book of rare things in the art of calculation is concerned with solutions to indeterminate equations.

  155. 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.

  156. Bondi biography
    • The correct number of solutions to the problem is 332.
    • Bondi defines when two solutions are considered equivalent, then finds 123 distinct solution.

  157. 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.
    • 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.

  158. Euler biography
    • Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem.
    • 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.

  159. Gershgorin biography
    • 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.
    • 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.

  160. Gray Marion biography
    • Marion C Gray, A modification of Hallen's solution of the antenna problem, J.
    • 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).

  161. Ferrers biography
    • His first book was "Solutions of the Cambridge Senate House Problems, 1848 - 51".
    • Not at all - Ferrers contribution was the method of solution.
    • We see that there is a 1-1 correspondence between a partition and its conjugate and this 1-1 correspondence provides the solution to the problem stated by Sylvester (and stated by Adams in the Tripos paper of 1847).

  162. 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.

  163. Macaulay biography
    • Such problems have no complete solution, but Macaulay looks for structural properties of the set of solutions.
    • He also contributed a number of articles: Bolyai's science of absolute space (1900), On continued fractions (1900), Projective geometry (1906), On the axioms and postulates employed in the elementary plane constructions (1906), On a problem in mechanics and the number of its solutions (1906), and Some inequalities connected with a method of representing positive integers (1930).

  164. 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.
    • 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.

  165. Fibonacci biography
    • Three of these problems were solved by Fibonacci and he gives solutions in Flos which he sent to Frederick II.
    • 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.
    • Without explaining his methods, Fibonacci then gives the approximate solution in sexagesimal notation as 1.22.7.42.33.4.40 (this is written to base 60, so it is 1 + 22/60 + 7/602 + 42/603 + ..

  166. Schelp biography
    • Interaction with Paul Erdős started in 1972 as a result of a solution of an Erdős-Bondy problem on Ramsey numbers for cycles.
    • Enthusiasm in talking about it, in trying to solve a maths problem, in appreciating others' solutions, in taking part of the whole process of ups and downs culminating in a solution.

  167. Voevodsky biography
    • He continued to work on ideas coming from Grothendieck and in 1991 published Galois representations connected with hyperbolic curves (Russian) which gave partial solutions to conjectures of Grothendieck, made on nonabelian algebraic geometry, contained in his 1983 letter to Faltings and also in his unpublished 'Esquisse d'un programme' mentioned above.
    • In the present paper, the authors offer a very different solution to the problem of providing an algebraic formulation of singular cohomology with finite coefficients.
    • One consequence of Voevodsky's work, and one of his most celebrated achievements, is the solution of the Milnor Conjecture, which for three decades was the main outstanding problem in algebraic K-theory.

  168. Malgrange biography
    • The idea of finding an elementary solution for all differential operators with constant coefficients might seem a bit far fetched.
    • However, I had made the suggestion of the existence of such a solution using the theory of distributions.
    • 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.

  169. Romberg biography
    • I submitted the solution and received the following response: "The assignment was completely solved by the sender.
    • His only solution was to fly from Prague and so avoid the German checks.
    • It is also indicated how the method may reduce the labour for obtaining solutions of the analytical eigenvalue problem.

  170. 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.
    • Osgood showed that if f(x, y) is merely continuous there exists at least one solution ..

  171. Box biography
    • One feature of particular interest is practical discussion of genuinely nonlinear fitting problems and their solution with the help of tact and a special, publicly available, IBM-704 program.
    • The authors - all statistical practitioners themselves - take a fresh approach to statistics oriented toward the solution of problems in the physical, engineering, biological and social sciences.
    • The authors typically start with the statement of a problem faced by an experimenter, and then present one or more possible solutions, stating clearly the assumptions required for the validity of each.

  172. 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.

  173. 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.
    • We consider the basic methods of solving differential equations and methods of qualitative investigation of these solutions.
    • We illustrate the theoretical material with an analysis of the solution of many examples.

  174. Warschawski biography
    • He published The convergence of expansions resulting from a self-adjoint boundary problem in the Duke Mathematical Journal in 1940, jointly with A S Galbraith, which studied a problem of the Riesz-Fischer type concerning the expansion of a function with n derivatives in terms of the characteristic solutions of a self-adjoint boundary value problem of the second order.
    • 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.

  175. Picard Emile biography
    • He used methods of successive approximation to show the existence of solutions of ordinary differential equations solving the Cauchy problem for these differential equations.
    • Picard's solution was represented in the form of a convergent series.
    • He studied the transmission of electrical pulses along wires finding a beautiful solution to the problem.

  176. Seitz biography
    • Finkel writes [A mathematical solution book (1893), 440-441.',6)">6]:- .
    • The more difficult the question, the more determined was he to master it, and from [1872 to 1879], I never knew him to fail in the solution of any problem he undertook.
    • Seitz, although he died at the age of 37, contributed over 500 published problems and solutions in the Analyst, the Mathematical Visitor, the Mathematical Magazine, the School Visitor, and the Educational Times of London.

  177. Biot biography
    • Having discovered these laws he used them in analysis of saccharine solutions using an instrument called a polarimeter which he invented.
    • For this work on the polarisation of light passing through chemical solutions he was awarded the Rumford Medal of the Royal Society of London in 1840.
    • Arago supported the Daguerre photographic process with silver plates while Biot championed an approach with paper soaked in a silver solution as developed by Henry Fox Talbot.

  178. Chaplygin biography
    • He published a famous paper On gas streams in 1902 giving exact solutions to many cases of noncontinuous flow of a compressible gas.
    • Three decades later, however, Chaplygin's dissertation served as a starting point for many studies by aerodynamics specialists and provided the basis for the solution of problems of subsonic flows.
    • This postulate - the so-called Chaplygin-Zhukovsky postulate - gives a complete solution to the problem of the forces exerted by a stream on a body passing through it.

  179. Zelmanov biography
    • Let me explain the background to the restricted Burnside problem, the solution of which was the main reason for the award of the Medal, and also explain how Zelmanov, not a group theorist by training, came to solve one of the most fundamental questions in group theory.
    • This is equivalent to saying that a positive solution to the Restricted Burnside problem would show that there are only finitely many finite factor groups of B(d, n).
    • The General Burnside problem was shown to have a negative solution by Golod in 1964.
    • The greatest early contribution to the Restricted Burnside problem was by Hall and Higman in 1956 where they showed that, if the Schreier conjecture holds, then the Restricted Burnside problem has a positive solution if it could be proved for all prime powers n.

  180. Thompson John biography
    • The solution of Frobenius's conjecture was not done by simply pushing the existing techniques further than others had done; rather it was achieved by introducing many highly original ideas which were to lead to many developments in group theory.
    • To classify finite groups therefore reduces to two problems, namely the classification of finite simple groups and the solution of the extension problem, that is the problem of how to fit the building blocks together.
    • I like to say that I would like to see the solution of the problem of the finite simple groups and the part I expect Thompson's work to play in it.
    • His work on coding theory was to lay the foundation for the solution of a long standing problem, namely the fact that there is no finite plane of order 10.

  181. Yau biography
    • The analytic problem is that of proving the existence of a solution of a highly nonlinear (complex Monge-Ampere ) differential equation.
    • Yau's solution is classical in spirit, via a priori estimates.
    • However, there were still questions relating to whether Douglas's solution, which was known to be a smooth immersed surface, is actually embedded.
    • for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems.

  182. Zhang Qiujian biography
    • However, no reasons are given for the method of solution.
    • 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.
    • A modern solution sets A to have x coins, B to have y and C to have z.

  183. Whittaker biography
    • 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.

  184. Steinitz biography
    • He submitted a solution to a prize problem announced by the university and this won him not only 200 marks but also the right to submit his doctoral dissertation without payment of the usual fee.
    • Steinitz's solution of Konig's theorem twenty years before Konig is discussed in detail in [Acta.
    • If the reader is unaware of the solution, he will in places hardly be able to guess what is meant.
    • This does not indicate that M Levy has not also found a solution for the general case and it indicates even less that he was not able to find it.

  185. Mordell biography
    • 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.
    • At the time he wrote the essay Mordell believed that for some k there may be infinitely many solutions.
    • However he solved the equation for many values of k, giving complete solutions for some values.

  186. Piatetski-Shapiro biography
    • He had given a solution to a problem which had been posed by Raphael Salem.
    • Among his main achievements are: the solution of Salem's problem about the uniqueness of the expansion of a function into a trigonometric series; the example of a non symmetric homogeneous domain in dimension 4 answering Cartan's question, and the complete classification (with E Vinberg and G Gindikin) of all bounded homogeneous domains; the solution of Torelli's problem for K3 surfaces (with I Shafarevich); a solution of a special case of Selberg's conjecture on unipotent elements, which paved the way for important advances in the theory of discrete groups, and many important results in the theory of automorphic functions, e.g., the extension of the theory to the general context of semi-simple Lie groups (with I Gelfand), the general theory of arithmetic groups operating on bounded symmetric domains, the first 'converse theorem' for GL(3), the construction of L-functions for automorphic representations for all the classical groups (with S Rallis) and the proof of the existence of non arithmetic lattices in hyperbolic spaces of arbitrary large dimension (with M Gromov).

  187. Eudoxus biography
    • Eutocius wrote about Eudoxus's solution but it appears that he had in front of him a document which, although claiming to give Eudoxus's solution, must have been written by someone who had failed to understand it.
    • Tannery's ingenious suggestion was that Eudoxus had used the kampyle curve in his solution and, as a consequence, the curve is now known as the kampyle of Eudoxus.
    • Eudoxus was, I think, too original a mathematician to content himself with a mere adaptation of Archytas's method of solution.

  188. Erlang biography
    • During this time he kept up his interest in mathematics, and he received an award in 1904 for an essay on Huygens' solution of infinitesimal problems which he submitted to the University of Copenhagen.
    • Jensen persuaded Erlang to apply his skills to the solution of problems which arose from a study of waiting times for telephone calls.
    • In this paper he showed that if telephone calls were made at random they followed the Poisson distribution, and he gave a partial solution to the delay problem.
    • In 1917 he published Solution of some problems in the theory of probability of significance in automatic telephone exchanges in which he gave a formula for loss and waiting time which was soon used by telephone companies in many countries including the British Post Office.

  189. 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 .
    • They considered numerical methods which find an approximate solution on a grid of values of x and t, replacing ut(x, t) and uxx(x, t) by finite difference approximations.
    • Richardson's method yielded a numerical solution which was very easy to compute, but alas was numerically unstable and thus useless.
    • 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.

  190. Steiner biography
    • In his paper Several laws governing the division of planes and space, which also appeared in the first volume of Crelle's Journal, he considers the problem: What is the maximum number of parts into which a space can be divided by n planes? It is a beautiful problem and has the solution (n3 + 5n + 6)/6.
    • See [One Hundred Great Problems of Elementary Mathematics, Their History and Solution (Dover, 1965).',3)">3] for a solution.
    • The proof, essentially as given by Steiner, is reproduced in [One Hundred Great Problems of Elementary Mathematics, Their History and Solution (Dover, 1965).',3)">3].

  191. Picone biography
    • 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.

  192. Coulomb biography
    • A reason, perhaps, for the relative neglect of this portion of Coulomb's work was that he sought to demonstrate the use of variational calculus in formulating methods of approach to fundamental problems in structural mechanics rather than to give numerical solutions to specific problems.
    • his simple, elegant solution to the problem of torsion in cylinders and his use of the torsion balance in physical applications were important to numerous physicists in succeeding years.

  193. 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.
    • The Mathieu functions are solutions of the Mathieu equation which is .

  194. Tschirnhaus biography
    • 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.

  195. Moulton biography
    • He published On a class of particular solutions of the problem of four bodies in 1900 and A class of periodic solutions of the problem of three bodies with application to the lunar theory in 1906 both in the Transactions of the American Mathematical Society.
    • Other papers, this time in the Annals of Mathematics, include The straight line solutions of the problem of n bodies (1910) and The deviations of falling bodies (1913).

  196. Drinfeld biography
    • Although he only proved a special case of the Langlands conjecture, Drinfeld has introduced important new ideas in his solution and made a real breakthrough.
    • 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.

  197. Benedetti biography
    • In this work he discussed the general solution of all problems in Euclid's Elements, and other geometric problems, using only a compass of fixed opening.
    • [Benedetti] opens the section titled "De rationibus operationum perspectivae" with an impatient review of a perspective error committed by both Durer and Setlio, and he proceeds at once to prove the correct solution.
    • He must have been aware that Tartaglia considered that Cardan had stolen his solution of the cubic, although in this case Cardan fully acknowledged Tartaglia's contributions.

  198. Carleson biography
    • The citation emphasizes not only Carleson's fundamental scientific contributions, the best known of which perhaps are the proof of Luzin's conjecture on the convergence of Fourier series, the solutions of the corona problem and the interpolation problem for bounded analytic functions, the solution of the extension problem for quasiconformal mappings in higher dimensions, and the proof of the existence of 'strange attractors' in the Henon family of planar maps, but also his outstanding role as scientific leader and advisor.

  199. Chernikov biography
    • The practical importance of convenient algorithms for the solution of systems of linear inequalities and their connection with the theory of linear programming is well known.
    • the basis of this theory lies in the principle of boundary solutions; all its results are deduced from it by means of only a few finite methods..

  200. Machin biography
    • While Newton was planning for a third edition, he received two independent solutions of the problem of the motion of the nodes of the moon's orbit, one by John Machin, the other by Henry Pemberton.
    • One other publication by Machin is worth noting, namely The solution of Kepler's problem which was published in the Philosophical Proceedings of the Royal Society in 1738.

  201. Gelbart biography
    • In 1935 a solution to this problem by Abe Gelbart, Student, Central High School, Paterson, New Jersey was published.
    • 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.

  202. Manfredi biography
    • The difficulties in finding a solution were great for there were economic issues, technical issues, political issues and legal issues to overcome.
    • Pope Clement XI called the experts to a meeting in April 1718 but the proposed solution ran into technical objections.
    • This particular effort to find a solution came to an end in March 1721 when Clement XI died.

  203. Van Ceulen biography
    • Goudaan had posed a geometric problem which Van Ceulen solved but his solution was not accepted by Goudaan.
    • When Goudaan published his own solution to the problem, Van Ceulen realised that it was incorrect.
    • 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.

  204. Janovskaja biography
    • There then follows general comments concerning the theory of algorithms, and the mathematical concepts of proof, construction and solution.
    • 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.

  205. Hamming biography
    • In particular, Hamming investigated the Green's function and also the characteristic solutions for which he obtained asymptotic expressions.
    • Work in codes is related to packing problems and the error-correcting codes discovered by Hamming led to the solution of a packing problem for matrices over finite fields.

  206. 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.

  207. Fenyo biography
    • The structure of the solutions is then examined, including singular points and limit cycles, and the book concludes with an account of the elementary theory of non-linear oscillations.
    • The soundness of his basic culture, coupled with his innate curiosity, led Istvan Fenyo to seek the solution of problems in various areas of mathematics.

  208. 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.

  209. Cholesky biography
    • 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.
    • To solve Ax = b one now needs to solve LL'x = b so put y = L'x which gives Ly = b which is solved for y, then y = L'x is solved for x to obtain the solution.

  210. Kadets biography
    • The first steps towards a solution had been taken by Stanislaw Mazur, a student of Banach's, in 1929 and then by Stefan Kaczmarz, a colleague of Banach's, in 1932.
    • To the solution of the problem he applied arguments of approximation theory suggested by Bernstein's theorem on the recovery of a continuous function from its least deviations from polynomials.
    • He continued to work towards a complete solution of the Frechet-Banach problem and achieved this in 1966.

  211. Dynkin biography
    • a class of measure-valued Markov processes [which] can be used to give probabilistic solutions to certain nonlinear PDE's in a way which is analogous to the classical solution of the Dirichlet problem by means of Brownian motion.

  212. Walsh biography
    • 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.

  213. Motzkin biography
    • It was written as a partial solution to a problem which had been posed by Ostrowski and it gave Motzkin particular pleasure when he returned to the problem many years later and was able to give a complete solution.
    • The proof is very typical of Motzkin in that the Euclidean algorithm is given a new formulation, which at first seems to be leading away from the problem at hand, but is suddenly seen to be the decisive key to its solution.

  214. Girard Albert biography
    • He was the first who understood the use of negative roots in the solution of geometrical problems.
    • For those having only one root he indicated, beside Cardano's rules, an elegant method of numerical solution by means of trigonometric tables and iteration.
    • The negative solution is explained in geometry by moving backward, and the minus sign moves back when the + advances.

  215. Ajima biography
    • The first, the Gion shrine problem, he solved in an unpublished manuscript of 1774 entitled Kyoto Gion Dai Toujyutsu (The Solution to the Gion Shrine Problem).
    • Although his solution was unpublished, nevertheless Ajima became famous for his work on this problem.
    • Malfatti assumed that the solution would involve three circles, each of which is tangent to the other two.

  216. Truesdell biography
    • In its use of formal methods, its reliance on special kinematic hypotheses (membranes of revolution only) and its presentation of series solutions of a great many special problems, this work can be considered properly isolated from his other work.
    • Most important of all, he taught me that careful scholarship and the persistent search for insight and understanding are far more important than facile skill in the use of contemporary techniques for the solution of currently popular problems.

  217. Borok biography
    • 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 early 1960s Valentina worked on fundamental solutions and stability for partial differential equations well-posed in the sense of Petrovskii.
    • Her central results include the construction of maximal classes of uniqueness and well-posedness, Phragmen-Lindelof type theorems, and the study of asymptotic properties and stability of solutions of boundary-value problems in infinite layers.

  218. Simpson biography
    • It was his obvious mathematical skills demonstrated in these solutions which first brought his to the attention of other mathematicians of the day.
    • Simpson's attempt at an analytical solution is interpreted.

  219. Riccati biography
    • 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.

  220. Manfredi Gabriele biography
    • 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 difficulties in finding a solution were great for there were economic issues, technical issues, political issues and legal issues to overcome.

  221. Delone biography
    • Two features are very characteristic of the mode of presentation: on the one hand the extensive use of geometrical considerations as a background for the true understanding of complicated situations which otherwise would remain obscure, and on the other hand, the care shown by the authors in inventing effective methods of solution, illustrated by actual application to numerical examples and to the construction of valuable tables.
    • He also published a number of texts aimed at school pupils including (with O K Zhitomirski) Problems with solutions for a revision course in elementary mathematics (1928), (with O K Zhitomirski) Problems in geometry (1935), Analytical geometry I (1948), and (with D A Raikov) Analytical geometry II (1949).

  222. Moser Jurgen biography
    • First we mention Lectures on Hamiltonian systems (1968) which examines problems of the stability of solutions, the convergence of power series expansions, and integrals for Hamiltonian systems near a critical point.
    • The missing final two chapters would have been on KAM theory and unstable hyperbolic solutions.
    • for his contributions to the theory of Hamiltonian dynamical systems, especially his proof of the stability of periodic solutions of Hamiltonian systems having two degrees of freedom and his specific applications of the ideas in connection with this work.

  223. 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.
    • When Darboux proved the orthogonality of systems of homofocal ovals, he also showed that ovals provide a geometrical interpretation of the addition theorem and that they constitute the algebraic form of the integral solution.

  224. Zhu Shijie biography
    • Therefore Zhu does not necessarily give the simplest solution to a problem, but rather often introduces complications explicitly designed to illustrate how to handle more complicated situations.
    • 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.

  225. Cesari biography
    • The paper also contains applications of his general methods to more specialised problems, solutions to which had been found by Carl Jacobi and Richard von Mises.
    • Particularly noted for his study of the existence theorems for optimal solutions for both single- and multi-dimensional systems, he also contributed to the theory of necessary conditions and the analysis of Pareto problems.
    • He also investigated the existence of solutions to certain quasi-linear hyperbolic systems.

  226. Al-Karaji biography
    • Often he explicitely says that he is giving a solution in the style of Diophantus.
    • The solutions of quadratics are based explicitly on the Euclidean theorems ..

  227. Weise biography
    • Also mentioned are existence theorems as well as solutions by iteration, power series, and numerical methods.
    • Weise acted as supervisor of PhD students from a wide range of mathematical fields, a dozen of them went on to become professors, among them Wolfgang Gaschutz (finite groups), Wolfgang Haken (knot theory and the solution of the four-colour-problem), Wilhelm Klingenberg (differential geometry) and Jens Mennicke (topology).

  228. Schwarz biography
    • Weierstrass had shown that Dirichlet's solution to this was not rigorous, see [Rend.
    • 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.

  229. Kolosov biography
    • He passed his Master's examinations in 1893 and his Master's dissertation On certain modifications of Hamilton's principle and its application to the solution of problems of mechanics of solid bodies (Russian) (1903) contained his first really significant result.
    • Kolosov's thesis contained a formal solution of the plane problems of the theory of elasticity.
    • In addition to the important results we have mention above, we note that in 1907 Kolosov derived the solution for stresses around an elliptical hole.

  230. Schiffer biography
    • The 'Calculus of Variations' - formulating and solving problems in terms of a quantity to be maximized or minimized and analysing the properties of such extremal solutions - had already been and remains an established, highly developed, and highly effective area of mathematical analysis and its applications.
    • 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.

  231. Zolotarev biography
    • Zolotarev was able to give a much more effective solution.
    • They were able to give complete solutions in the case of four variables and of five variables.

  232. Archytas biography
    • Archytas solved the problem with a remarkable geometric solution (not of course a ruler and compass construction).
    • One interesting innovation which Archytas brought into his solution of finding two mean proportionals between two line segments was to introduce movement into geometry.
    • We know of Archytas's solution to the problem of duplicating the cube through the writings of Eutocius of Ascalon.

  233. Frenicle de Bessy biography
    • Frenicle de Bessy was such a computational juggernaut that whenever anyone would send him a numerical challenge, he would return awe-inspiring solutions in record time.
    • He solved many of the problems posed by Fermat but he did more than find numerical solutions for he also put forwards new ideas and posed further questions.
    • When Pierre de Fermat first began writing to de Bessy, he would challenge him with difficult number theory problems while giving no hint of their possible solution, which Frenicle found extremely frustrating, since he suspected that Fermat was teasing him.
    • We know that Frenicle found four solutions to the first of these problems on the day that he was given the problem, one of which is .
    • He found another six solutions the next day.
    • He gave solutions to both problems in Solutio duorm problematum circa numeros cubos et quadratos, quae tanquam insolubilia universis Europae mathematicis a clarissimo viro D Fermat sunt proposita (1657).
    • In the Solutio he gave a table of solutions for all values of m up to 150.
    • He also explained the way that he had discovered these solutions.
    • The other two manuscripts by Frenicle that were published in this volume were Methode pour trouver la solution des problemes par les exclusions and Abrege des combinaisons.
    • M de la Hire chose to put first the treatise by M Frenicle on 'Exclusions' because it gave a particular method which is used for the solution of problems, by means of which he easily resolved very difficult issues in number theory and algebra over which often there was little control, which led to it being admired by scholars with whom he had dealings, as can be seen in several places in their works.
    • We note that Frenicle's Methode pour trouver la solution des problemes par les exclusions presents ten rules which he suggests are useful in solving mathematical problems.
    • Rules are given to simplify problems and rules are given to make sure solutions are looked for in a systematic way so that nothing is missed.
    • In many ways these rules emphasise the point that we made earlier about Frenicle being primarily a remarkable calculator, for these rules give essentially an experimental approach to finding integer solutions to specific number theory problems.

  234. Schmidt F-K biography
    • Jean Dieudonne, writing in [The Mathematical Intelligencer 10 (1975), 7-21.',3)">3] about the solution of the Weil Conjectures, states that F-K Schmidt was one of the main contributors of essential ideas to the ultimate solution.
    • At the annual meeting of the Deutsche Mathematiker Vereinigung in September 1930 in Konigsberg, F K Schmidt gave a talk about [this] question and his solution.

  235. 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.
    • First published three years later, his Methods and theories for the solution of problems of geometrical construction appeared in various editions and languages.
    • Petersen never returned to cryptography; this seems to be another instance of a problem that must have occupied him intensely for a period, until he found a satisfactory solution and moved on to something else.

  236. Narayana biography
    • He then finds the solutions x = 6, y = 19 which give the approximation 19/6 = 3.1666666666666666667, which is correct to 2 decimal places.
    • Narayana then gives the solutions x = 228, y = 721 which give the approximation 721/228 = 3.1622807017543859649, correct to four places.
    • Finally Narayana gives the pair of solutions x = 8658, y = 227379 which give the approximation 227379/8658 = 3.1622776622776622777, correct to eight decimal places.

  237. Arnold biography
    • I spent a whole day thinking on this oldies, and the solution ..
    • The examining committee for the theis, which contained a solution to Hilbert's 13th problem, consisted of A G Vitushkin and L V Keldysh.
    • thesis contained a solution to Hilbert's 13th problem.

  238. Al-Kashi biography
    • In the richness of its contents and in the application of arithmetical and algebraic methods to the solution of various problems, including several geometric ones, and in the clarity and elegance of exposition, this voluminous textbook is one of the best in the whole of medieval literature; it attests to both the author's erudition and his pedagogical ability.
    • He was not the first to look at approximate solutions to this equation since al-Biruni had worked on it earlier.

  239. 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.
    • We are therefore compelled to give a solution adapted to Tables, the magnitudes of the retardation being set down in those Tables for velocities which are common in practice.
    • The amount of labour, however, in calculating all the quantities for a single component arc, even with the aid of copious tables, is so great that I was led to examine whether any thing could be done towards simplifying the solution and reducing the amount of calculation.

  240. 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.
    • A few years later Poincare extended Bruns' work to show that no solution to the three-body problem was possible given by algebraic expressions and integrals.

  241. Magenes biography
    • 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.

  242. Fontaine des Bertins biography
    • In 1732 Fontaine gave a solution to the brachistochrone problem, in 1734 he gave a solution of the tautochrone problem which was more general than that given by Huygens, Newton, Euler or Jacob Bernoulli, and in 1737 he gave a solution to an orthogonal trajectories problem.

  243. Montgomery biography
    • Iwasawa, himself a contributor to the solution of Hilbert's Fifth Problem, wrote in a review:- .
    • In the meantime, the theory of topological groups has made outstanding progress, culminating in the solution of Hilbert's fifth problem by Gleason and by the authors of the present book.
    • The next two chapters are devoted to the study of the structure of locally compact groups which leads to a solution of Hilbert's problem.

  244. Bers biography
    • 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.

  245. 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.
    • When d'Alembert attacked Clairaut's solution of the three-body problem as being too much based on observation and not, like his own work, based on theoretical results, Clairaut strongly attacked d'Alembert in the most bitter dispute of their lives.
    • The algebra book was an even more scholarly work and took the subject up to the solution of equations of degree four.

  246. 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.
    • 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 fact that he lost out because of errors in the way he submitted his solution for the prize was a major disappointment to Graffe.

  247. Ghetaldi biography
    • Also in 1607 Ghetaldi produced a pamphlet Variorum problematum collectio with 42 problems with solutions.
    • It is reasonable to ask: what is the most impressive ideas contained in Ghetaldi's work? Without doubt, it is his application of algebraic methods to the solution of problems in geometry.

  248. Vladimirov biography
    • The physicists passed mathematical assignments to the team in which Vladimirov was working and, prompted by these problems, he developed a new technique for the numerical solution of boundary value problems specifically designed for the type of problems which were encountered.
    • 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.
    • 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).

  249. Richardson biography
    • Having developed these methods by which he was able to obtain highly accurate solutions, it was a natural step to apply the same methods to solve the problems of the dynamics of the atmosphere which he encountered in his work for the Meteorological Office.
    • 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.

  250. Vallee Poussin biography
    • Vallee Poussin's first mathematical research was on analysis, in particular concentrating on integrals and solutions of differential equations.
    • In fact the solution of this major open problem was one of the major motivations for the development of complex analysis during the period from 1851 to 1896.

  251. Ford biography
    • 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).
    • He had gained a reputation as an excellent expositor and he wrote outstanding articles as well as contributing many mathematical problems and solutions.

  252. Zippin biography
    • He slowly returned to his joint research with Deane Montgomery, and together they made progress towards a solution to Hilbert's fifth problem [The honors class: Hilbert\'s problems and their solvers (A K Peters, 2002).',1)">1]:- .
    • It then follows immediately that every locally euclidean group is a Lie group, namely, the solution of Hilbert's fifth problem.
    • In the meantime, the theory of topological groups has made outstanding progress, culminating in the solution of Hilbert's fifth problem by Gleason and by the authors of the present book.

  253. La Faille biography
    • He determined the longitude by studying the phases of the moon and put his solution forward for the prize offered by Spain for a solution to the longitude problem.
    • De la Faille supported van Langren's solution to the longitude problem but no decision was reached about awarding the prize [Jesuit Science and the Republic of Letters (MIT Press, 2003), 339-340.',2)">2]:- .

  254. 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.
    • I had not expected that one could formulate the exact solution of the problem in such a simple way.
    • However, Schwarzschild himself makes clear that he believes that the theoretical solution is physically meaningless, so making it very clear that he did not believe in the physical reality of black holes.

  255. Robins biography
    • This was shown to Dr Pemberton, who, thence, conceiving a good opinion of the writer, for a further trial of his proficiency sent him some problems, of which the Doctor required elegant solutions, not those founded on algebraical calculations; adding an example of such a solution that the young geometer might the more readily comprehend his meaning.

  256. 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).

  257. Blichfeldt biography
    • This was not the immediate solution to all of Hans's problems for once in the United States he spent four years as a labourer working on farms and in sawmills.
    • 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.

  258. Varga biography
    • '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.
    • However, the title should prepare the reader for a modern treatment (thoroughly imbedded in functional analysis), for the subject not to be regarded merely as an end in itself (the numerical solution of elliptic and parabolic boundary value problems is the author's eventual target) and for theoretical depth (there are no details of explicit numerical procedures).
    • At the same time it shows that there are still a lot of unsolved mathematical problems, the solution of which may require deep mathematics, but probably also fast computers and highly-accurate numerical software.

  259. Atiyah biography
    • His first major contribution (in collaboration with F Hirzebruch) was the development of a new and powerful technique in topology (K-theory) which led to the solution of many outstanding difficult problems.
    • Subsequently (in collaboration with I M Singer) he established an important theorem dealing with the number of solutions of elliptic differential equations.

  260. 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).

  261. 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.
    • 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 ..

  262. Ree biography
    • Ree managed to solve the problem and sent his solution to Max Zorn.
    • When Zorn received Ree's solution he was impressed and sent it to the Bulletin of the American Mathematical Society.
    • One might imagine that Ree would be overjoyed to have his first paper published in a prestigious journal but he did not realise that the paper had been published for over five years after he sent his solution to Zorn.

  263. 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).
    • The utility of hyperbolic and elliptic arches, in the solution of various problems, and particularly in the business of computing fluents, has been shown by those eminent mathematicians, Maclaurin, Simpson and Landen; the last of whom has written a very ingenious paper on hyperbolic and elliptic arches, which was published in the first volume of his 'Mathematical Memoirs', in the year 1780.

  264. Delannoy biography
    • In all he seems to have published eleven articles but he also published many problems and solutions to problems set by others.
    • His first publication was Emploi de l'echiquier pour la solution de problemes arithmetiques (1886) followed by Sur la duree du jeu (1888).

  265. Ramsey biography
    • He accepted Russell's solution to remove the logical paradoxes of set theory arising from, for example, "the set of all sets which are not members of themselves".
    • Unsolved problems abound, and additional interesting open questions arise faster than solutions to the existing problems.

  266. Rolle biography
    • He made his solution known through publishing it in the Journal des scavans.
    • Rolle published another important work on solutions of indeterminate equations in 1699, Methode pour resoudre les equations indeterminees de l'algebre.

  267. Tarski biography
    • He had already made clear how pleased mathematicians should be that there is no solution to the general decision problem.
    • the solution of the decision problem in its most general form is negative.
    • Perhaps sometimes in their sleepless nights they thought with horror of the moment when some wicked metamathematician would find a positive solution, and design a machine which would enable us to solve any mathematical problem in a purely mechanical way..

  268. Ruffini biography
    • 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.

  269. Bianchi biography
    • As preparation for the general solution, the author applies the Lie-Killing methods for finding all three-dimensional spaces in which the motions of figures with given degrees of freedom are possible - this is enough to outline the goal and the general train of thought of Bianchi's work.
    • The importance of his results is known to every reader who is familiar with the awards of the Royal Jablonowski Society for 1901; their citation states that the strength of the methods and the elegance of the solutions need not be pointed out when we are talking about a paper whose author is Bianchi.

  270. Mason biography
    • The first problem which Hilbert suggested to him for a thesis topic was rapidly solved and he wrote up an elegant solution in two pages.
    • 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).

  271. 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).
    • The contemporary version of Hilbert's 21st problem is to find conditions implying a positive or negative solution to the Riemann-Hilbert problem.

  272. 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.

  273. 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.
    • He uses both algebraic methods of solution and geometric methods.
    • The solution had to be general and calculable at the same time and in a mathematical fashion, that is, geometrically founded.

  274. Abraham biography
    • It contains the complete solution of the general quadratic and is the first text in Europe to give such a solution.
    • 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.

  275. Efimov biography
    • The theorem which P S Aleksandrov refers to in this quote was one that Efimov became interested in while working for his doctorate but it was not until around 1950 that he began to concentrate all his efforts on obtaining a solution.
    • Only a few mathematicians have the determination shown by Efimov to spend over twelve years of their lives trying to solve a single problem but he did so and published his solution in 1963 in The impossibility in three-dimensional Euclidean space of a complete regular surface with a negative upper bound of the Gaussian curvature.
    • 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.

  276. Watson biography
    • Although Watson was not interested in how best to model the situation, he was, however, very interested in using his expertise to determine mathematical solutions to the given model which others might then check against observations.
    • He obtained solutions to the problem in 1918 which showed conclusively that the model was not a satisfactory one.
    • Watson showed that if the layer was about 100 km above the Earth's surface and it had a certain conductivity, then indeed the solutions obtained closely matched observations.

  277. Saurin biography
    • His two papers on this topic both appeared in 1709, the first being Solutions et analyses de quelques probleme appartenants aux nouvelles methodes, and the second Solution generale du probleme ..

  278. Bendixson biography
    • Bendixson also made interesting contributions to algebra when he investigated the classical problem of the algebraic solution of equations.
    • In examining periodic solutions of differential equations Bendixson used methods based on continued fractions.

  279. Collatz biography
    • 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.

  280. Rellich biography
    • 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.
    • He is also known for Rellich's theorem on entire solutions of differential equations which he proved in 1940.

  281. Poisson biography
    • His approach to these problems was to use series expansions to derive approximate solutions.
    • Poisson submitted the first part of his solution to the Academy on 9 March entitled Sur la distribution de l'electricite a la surface des corps conducteurs.

  282. Jerrard biography
    • 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.
    • James Cockle was another mathematician who could not accept that Abel had proved the solution to be impossible, but Hamilton supported Abel and pointed out errors in Jerrard's work.

  283. 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.
    • Duarte also contributed to that part of mathematics and proposed problems and solutions to the American Mathematical Monthly for several years, and also to the journal Ciencia y Ingenieria (Science and Engineering) published in Merida.

  284. Tao biography
    • 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.
    • Covering number theory, algebra, analysis, Euclidean geometry, and analytic geometry, Solving mathematical problems includes numerous exercises and model solutions throughout.

  285. Kalman biography
    • The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics.
    • Analytic solutions are available in some cases.

  286. Steggall biography
    • It was while he was at Owens College, Manchester, that Steggall published London University Pure Mathematics Questions and Solutions which gave the University of London examination questions from 1877 to 1881 together with Steggall's solutions.
    • The generation of British mathematics to which Steggall belonged delighted in proposing and working out problems whose solutions might require the aid of any branch of pure or applied mathematics.

  287. Praeger biography
    • 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 Enumeration of rooted trees with a height distribution (1985) written jointly with P Schultz and N C Wormald, the authors used generating functions to find a new solution to the problem of determining the number of rooted trees whose vertices have a given height distribution.

  288. Dilworth biography
    • Indeed, it was the hope of many of the early researchers that lattice-theoretic methods would lead to the solution of some of the important problems in group theory.
    • Furthermore, these and other current problems are sufficiently difficult that imaginative and ingenious methods will be required in their solution.

  289. Kingman biography
    • This paper gave, in some sense, a complete solution to a problem which Kingman had been studying since his paper Markov transition probabilities.
    • Between this first paper and the complete solution in 1971, Kingman had published several further contributions building up to the final elegant result: Markov transition probabilities.

  290. Belanger biography
    • The revised paper was published as Essai sur la Solution Numerique de quelques Problemes Relatifs au Mouvement Permanent des Eaux Courantes (Essay on the numerical solution of some problems relating to the steady flow of water).

  291. Schlafli biography
    • are then values of the unknowns belonging to a single solution.
    • As in that theory, a 'group' of values of coordinates determines a point, so in this one a 'group' of given values of the n variables will determine a solution.

  292. Reinhardt biography
    • This was a step towards providing a solution to Hilbert's Eighteenth Problem for it solved the first part, showing that there are only finitely many essentially different space groups in n-dimensional Euclidean space.
    • It was in the last of these papers that Reinhardt completed the solution of Hilbert's Eighteenth Problem by finding a polyhedron which, although it is not the fundamental region of any space group, tiles 3-dimensional Euclidean space.

  293. Polya biography
    • What was the great novelty which made Polya and Szego's book of analysis problems so different? It was Polya's idea to classify the problems not by their subject, but rather by their method of solution.
    • its purpose is to discover the solution of the present problem.

  294. Artin biography
    • Artin himself proved that when O is the field of algebraic numbers, the subfield K of real algebraic numbers solves the problem and, moreover, it is the unique solution up to automorphisms of the field O.
    • Artin gave a complete solution in the paper Uber die Zerlegung definiter Funcktionen in Quadrate also published in 1927.

  295. 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.
    • During World War II, he remained at Bristol but did important work for the Ministry of Aircraft Production on optical systems for air reconnaissance and also for Mott's research group which was trying to provide rapid solutions to problems of a technical nature.

  296. Bolibrukh biography
    • Hilbert believed that the question had a positive solution and the problem appeared settled in 1908 when Josip Plemelj proved this by giving a reduction of the problem to a known result.
    • It was still believed that what was required was a correction in Plemelj's method to give the expected positive answer but Bolibrukh produced a major surprise when he proved in 1989 that certain prescribed conditions on the singularities led to a negative solution.

  297. Brocard biography
    • 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).
    • It is not even known whether there are only finitely many solutions.

  298. Haar biography
    • Each issue of the Kozepiskolai Matematikai Lapok contained a number of selected exercises from mathematics and shortly thereafter from physics, as well as solutions to the past months' problems and a list of those pupils who had sent in correct solutions.

  299. MacColl biography
    • He had published 'a pamphlet on ratios' in 1861 before moving to France, but once in Boulogne he became much more active in publishing, particularly in the 'Questions, Problems' section and in the 'Solutions' section of the Educational Times.
    • As to his books, his first was Algebraical Exercises and Problems with Elliptical Solutions (1870).

  300. Fergola biography
    • At this point (after the geometrical analysis, which is "an ontological principle of reduction"), we must proceed to the "geometrical composition" of the problem, that is, the construction of the solution, where the order of the analysis is reversed.
    • This last step is crucial: "the construction is the essential condition for the proper solution of a geometrical problem." Both geometrical analysis and composition must he accomplished according to the ancient criteria of elegance ..

  301. Al-Mahani biography
    • Therefore, this solution was declared impossible until the appearance of Ja'far al-Khazin who solved the equation with the help of conic sections.
    • It would be too easy to say that since al-Mahani has proposed a method of solution which he could not carry through then his work was of little value.

  302. Kalton biography
    • Invariably, in a few days he would have a solution for them.
    • Nigel solved it in forty-eight hours, and later this solution was used to answer a problem of Ramanujan.

  303. Poretsky biography
    • On 25 May 1886 Poretsky defended his thesis for a master's degree in astronomy, entitled "On the solution of some of the normal systems occurring in spherical astronomy, with an application to identify errors in the division of the Kazan Observatory meridian circle" which he had submitted to the Physical-Mathematical Faculty of Kazan University [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • He published major works on methods of solution of logical equations, and on the reverse mode of mathematical logic.

  304. Lions Jacques-Louis biography
    • By this we mean that approximate solutions are constructed by a reduction of the problem to a finite-dimensional one, and these are then shown to form a relatively compact family in a suitable topology, by means of a priori estimates and other evaluations.
    • These comprise methods in which the approximate solutions are regularized, or smoothed, before the passage to the limit is performed .

  305. Cardan biography
    • I have sent to enquire after the solution to various problems for which you have given me no answer, one of which concerns the cube equal to an unknown plus a number.
    • In it he gave the methods of solution of the cubic and quartic equation.

  306. Davis biography
    • Does there exist an algorithm to determine whether a Diophantine equation has a solution in natural numbers? .
    • 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.

  307. 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.
    • ',2)">2] then it refers to the optional number in a guessed solution and it is a feature which differs from the original method as presented by Bhaskara I.

  308. Heaviside biography
    • The solution of the algebraic equation could be transformed back using conversion tables to give the solution of the original differential equation.

  309. Menshov biography
    • He showed Luzin his solution to the problem that Luzin had just posed and before the end of 1914 the two had begun a firm mathematical friendship.
    • Menshov does not belong among the ranks of those mathematicians who undertake the solution of comparatively easy problems, or who continue the research of other authors on a course that has already been indicated.

  310. Kruskal Joseph biography
    • Roughly a year later, I had put a lot of work into this problem, but was still not close to a solution.
    • His work combined with mine finally led to a solution.

  311. Iyanaga biography
    • He studied this topic in T Yosiye's course and became interested in conditions under which dy/dx = f (x, y) has a unique solution.
    • conditions assure the uniqueness of solution.

  312. Linnik biography
    • In 1948-49 Linnik obtained results which contained, in principle, a complete solution to two central problems in the theory of the summation of variables forming a Markov chain.
    • Linnik substantially improved and developed the methods of his predecessors and gave an almost definitive solution of the problem for an inhomogeneous chain with an arbitrary finite number of events.

  313. Dezin biography
    • Already in his diploma work he developed a technique involving operators of averaging with variable radius, which even at present remains an effective tool in the theory of extension of functions and in the theory of boundary-value problems, in investigations of the problem of when weak and strong solutions coincide.
    • 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).

  314. 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.

  315. 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.
    • His different route to the solution, however, shows that he was looking for different methods than those of Euler, for whom Lagrange had the greatest respect.

  316. Graham biography
    • By the end of the semester, I finished the book and had a solution.
    • Every one of his collaborators knows that Ron will somehow find whatever hours or days are needed to come up with some substantial suggestion, and frequently with the crucial step towards the solution.

  317. Faddeeva biography
    • 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.
    • We mentioned above the 20 joint papers by Faddeeva and her husband, noting that some of the last few of these were: Natural norms in algebraic processes (1970), On the question of the solution of linear algebraic systems (1974), Parallel calculations in linear algebra (Part 1 in 1977, Part 2 in 1982), and A view of the development of numerical methods of linear algebra (1977).

  318. Shtokalo biography
    • 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).
    • 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.
    • Criteria of stability and unstability of their solutions was published being a translation of the Russian book which appeared one year earlier.
    • 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.

  319. Cafiero biography
    • He had begun published papers during this period, the first of these being Sull'approssimazione mediante poligonali degli integrali del sistema differenziale: y' = F(x, y), y(x0) = y0 (1947), which, under the assumption that F(x, y) is continuous in a rectangle, develops a method of polygonal approximation which produces every solution of the system specified in the title.
    • 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.

  320. Spence David biography
    • 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.
    • 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.

  321. Ringrose biography
    • The third and fourth volumes contain the solutions to the exercises.
    • R S Doran says the authors' solutions:- .

  322. Ibrahim biography
    • This is in contrast to The selected problems in which 41 difficult geometrical problems are solved, usually by analysis only, without a discussion of the number of solutions or conditions which make the solutions possible.

  323. Carleman biography
    • 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.
    • 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.

  324. 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.
    • The methods used are a combination of experimental observation, computation often on a very large scale, and analysis of the structure of the asymptotic form of the solution as the friction tends to zero.

  325. 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 formula is interpreted physically and the question of the uniqueness of the solution discussed.

  326. Titchmarsh biography
    • At Russell's first lecture the room was packed to the doors, and Russell said: "Ah, there's my clever pupil Mr Titchmarsh - he knows it all, he can go away." Russell dictated his lectures word for word and examples were handed out - and then, if necessary, solutions to examples.
    • Some of Titchmarsh's solutions replaced the official ones.

  327. Chuquet biography
    • In this work negative numbers, used as coefficients, exponents and solutions, appear for the first time.
    • The sections on equations cover quadratic equations where he discusses two solutions.

  328. Adamson biography
    • No solutions of the exercises, no proofs of the theorems are included in the first part of the book - this is a 'Workbook' and readers are invited to try their hand at solving the problems and proving the theorems for themselves.
    • The second part of the book contains complete solutions to all but the most utterly trivial exercises and complete proofs of the theorems.

  329. Lipschitz biography
    • 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.

  330. Kempe biography
    • Hence many problems - such as, for example, the trisection of an angle - which can readily be effected by employing other simple means, are said to have no geometrical solution, since they cannot be achieved by straight lines and circles only.
    • The first solution was found by a French army officer called Peaucellier and was brought to England by Professor Sylvester in a lecture at the Royal Institution in January 1874.

  331. Luzin biography
    • The study of effective sets that he embarked upon was pursued intensively for more than two decades and led to the solution of many important problems of set theory ..
    • The solution of the large problems that he undertook is distinguished by their subtlety, elegance, and simplicity of presentation.

  332. 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.
    • 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).

  333. Steinhaus biography
    • Steinhaus found a proportional but not envy free solution for n = 3.
    • An envy free solution to Steinhaus's problem for n = 3 was found in 1962 by John H Conway and, independently, by John Selfridge.

  334. Novikov Sergi biography
    • 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).

  335. Day biography
    • One of the first was in the paper A simple solution to the word problem for lattices (1970) where he gave a simple solution to the word problem in free lattices.

  336. 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.
    • The newfound interest in applicable mathematics took him in the 1960s, accompanied by a team of assistants, all over China to show workers of all kinds how to apply their reasoning faculty to the solution of shop-floor and everyday problems.

  337. D'Alembert biography
    • d'Alembert has tried to undermine [my solution to the vibrating strings problem] by various cavils, and that for the sole reason that he did not get it himself.
    • He wished to publish in our journal not a proof, but a bare statement that my solution is defective.

  338. Sturm biography
    • The first to give a complete solution was Cauchy but his method was cumbersome and impractical.
    • Sturm achieved fame with his paper which, using ideas of Fourier, gave a simple solution.

  339. Galois biography
    • On 25 May and 1 June he submitted articles on the algebraic solution of equations to the Academie des Sciences.
    • However the papers reached Liouville who, in September 1843, announced to the Academy that he had found in Galois' papers a concise solution .

  340. Rogosinski biography
    • Rogosinski's solution greatly pleased Landau and it was from this time that their close friendship developed.
    • Nevertheless, it is a very powerful tool for the solution of many questions in the theory of functions.

  341. 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).

  342. Minkowski biography
    • In 1881 the Academy of Sciences (Paris) announced that the Grand Prix for mathematical science to be awarded in 1883 would be for a solution to the problem of the number of representations of an integer as the sum of five squares.
    • Minkowski, although only eighteen years old at the time, reconstructed Eisenstein's theory of quadratic forms and produced a beautiful solution to the Grand Prix problem.

  343. Remak biography
    • [In order to answer questions about] (i) identification of an economic optimum, (ii) identification of an approximate economic solution, ..
    • There is, however, work in progress concerning the numerical solution of linear equations with several unknowns using electrical circuits.

  344. 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).
    • 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.

  345. Pisier biography
    • describes the development centred around the six problems formulated and discussed at the end of the Resume, and presents various results which led to their solutions.
    • For each of them, the solutions that were known (at the time of the writing of the book) are given with complete proofs and the necessary background.

  346. Brunelleschi biography
    • Huge engineering problems faced the placing of a dome on the octagonal Baptistry, and much argument had taken place on how to solve this and Brunelleschi set to work on finding an innovative solution.
    • He now combined his artistic skills, his mathematical skills, and his understanding of mechanical devices when he made a proposal to the wardens of works of the cathedral when they set up a competition in 1418 to find the best solution to the problem of designing and constructing the dome.

  347. Coble biography
    • 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).

  348. Dahlquist biography
    • 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.

  349. Cheng Dawei biography
    • Here is a modern solution.
    • How does Cheng Da Wei solve the problem? Basically he uses proportion supposing that the solution to the problem is that A has 10 sheep.

  350. 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).

  351. Federer biography
    • Federer and Morse wrote up the solution to the problem as the joint paper Some properties of measurable functions which was published in 1943.
    • It became dominant partly because of its successful application in the solution of the classical Plateau problem.

  352. Pacioli biography
    • XII (Wiesbaden, 1985), 237-246.',9)">9], although the solution he gave is incorrect.
    • 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.

  353. Trudinger biography
    • 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.
    • It had two new chapters one of which examined strong solutions of linear elliptic equations, and the other was on fully nonlinear elliptic equations.

  354. 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.

  355. Gray Andrew biography
    • that not only had he written out the solutions of those questions which had been answered incorrectly, but he had also written out complete solutions of all those which he had not attempted.

  356. Tinseau biography
    • Two papers were published in 1772 on infinitesimal geometry Solution de quelques problemes relatifs a la theorie des surfaces courbes et des lignes a double courbure and Sur quelques proptietes des solides renfermes par des surfaces composees des lignes droites.
    • He also wrote Solution de quelques questions d'astronomie on astronomy but it was never published.

  357. Malfatti biography
    • In 1802 he gave the first solution to the problem of describing in a triangle three circles that are mutually tangent, each of which touches two sides of the triangle, the so-called Malfatti problem.
    • His solution was published in a paper of 1803 on un problema stereotomica.

  358. Rennie biography
    • The JCMN is mainly concerned with mathematical problems, and their solution.
    • he offers a solution to the tomography problem (that is, to find out about the inside of a system through measurements on the outside) for electrical circuits with two-terminal linear components such as resistors.

  359. Szafraniec biography
    • 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).
    • Out of his many noticeable results, we mention a few: simplified forms (including the diagonal one) of the boundedness condition in the famous Szokefalvi-Nagy general dilation theory together with related integral representations of exponentially bounded operator-valued functions on abelian *-semigroups (unfortunately often attributed exclusively to a paper by Berg and Maserick, which appeared later), foundations of the theory of unbounded subnormal operators (together with Jan Stochel), new solutions to multidimensional real and complex moment problems (together with Jan Stochel), fresh look on interpolation theory, three term recurrence relations for orthogonal polynomials of several variables (together with Dariusz Cichon and Jan Stochel), and advances in the theory of quantum harmonic oscillators and canonical commutation relations.

  360. Schrodinger biography
    • 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.

  361. Foulis biography
    • Hyla wrote the solutions manual to Dave's first Calculus book, providing solutions to approximately 5000 problems, an impressive feat before the availability of graphing calculators.

  362. 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.
    • Under these conditions it is possible to show that the Schwarzschild field can be transformed into a solution of the problem.

  363. Petrovsky biography
    • the 1937 paper on the Cauchy problem for hyperbolic systems, the 1939 paper on the analyticity of solutions of elliptic systems, and the 1945 paper on lacunae for solutions of hyperbolic systems.

  364. 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.
    • In my opinion Wilkinson is single-handedly responsible for the creation of almost all of the current body of scientific knowledge about the computer solution of the problems of linear algebra.

  365. 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.

  366. Siegel biography
    • the Lagrangian solutions for the three-body problem.
    • He examined Birkhoff's work on perturbation theory solutions for analytical Hamiltonian differential equations near an equilibrium point using formal power series.

  367. 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.
    • 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.

  368. Wiener Christian biography
    • He also sought to simplify individual problems as much as possible and to find the easiest graphical solutions for them.
    • He was also interested in the problems and their solutions (such as shadow construction and brightness distribution), as well as in the development of the necessary geometric aids.

  369. Mansur biography
    • Abu Nasr Mansur's main achievements are his commentry on the Spherics of Menelaus, his role in the development of trigonometry from Ptolemy's calculation with chords towards the trigonometric functions used today, and his development of a set of tables which give easy numerical solutions to typical problems of spherical astronomy.
    • Menelaus's work formed the basis for Ptolemy's numerical solutions of spherical astronomy problems in the Almagest.

  370. Adleman biography
    • Altogether only about one fiftieth of a teaspoon of solution was used.
    • 542-545.',10)">10]) a DNA solution to another famous 'NP-complete' problem - the so-called "satisfaction" problem (SAT).

  371. Jungius biography
    • Empiricus remained verifiable through experience, Epistemonicus is grounded in principles and rules - as are the axioms of Euclid's geometry - and Heureticus reveals new methods for the solution of problems previously insoluble.
    • A similar replacement occurs when copper itself is introduced into a solution of silver in aqua fortis.

  372. Simon biography
    • The 1994 Bocher Prize is awarded to Leon Simon for his profound contributions towards understanding the structure of singular sets for solutions of variational problems.
    • These results left open basic questions about the structure of the set of singularities exhibited by the solutions of such variational problems.

  373. Hironaka biography
    • Hironaka gave a general solution of this problem in any dimension in 1964 in Resolution of singularities of an algebraic variety over a field of characteristic zero.
    • Hironaka talked about his solution in his lecture On resolution of singularities (characteristic zero) to the International Congress of Mathematicians in Stockholm in 1962.

  374. Von Neumann biography
    • If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me as soon as the lecture was over, with the complete solution in a few scribbles on a slip of paper.
    • Haar's construction of measure in groups provided the inspiration for his wonderful partial solution of Hilbert's fifth problem, in which he proved the possibility of introducing analytical parameters in compact groups.

  375. Garnir biography
    • 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.

  376. 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".

  377. Somov biography
    • In the theory of elliptical functions and their application to mechanics, he completed the solution of the problem concerning the rotation of a solid body around an immobile point in the Euler-Poinsot and Lagrange-Poisson examples.
    • The first in Russia to deal with the solution of kinematic problems, Somov included a chapter on this topic in his textbook on theoretical mechanics.

  378. 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 .
    • This has solution y = x + a(e-x/a - 1).

  379. Levin biography
    • Levin solved the problem and, a few years later, published his solution in the paper Generalization of a theorem of Holder (Russian) (1934).
    • 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.

  380. 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).
    • Two of Mendelsohn's papers An algorithmic solution for a word problem in group theory (1964) and (with Clark T Benson) A calculus for a certain class of word problems in groups (1966) were particularly important in launching this strand of my own research career - thank you Nathan! .

  381. Al-Khalili biography
    • The calculation of the direction of Mecca, as a function of terrestrial latitude and longitude, was one of the hardest of all problems of spherical trigonometry for which Islam required a solution.
    • One possible solution is that al-Khalili had computed more accurate auxiliary tables before calculating his tables for the direction of Mecca but these are now lost.

  382. 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:- .
    • Again in the case of torsion the presence of a couple about the axis suffices to give the distribution which leads to the solution.

  383. Sridhara biography
    • Often after stating a rule Sridhara gives one or more numerical examples, but he does not give solutions to these example nor does he even give answers in this work.
    • In [Ganita 1 (1950), 1-12.',7)">7] Shukla examines Sridhara's method for finding rational solutions of Nx2 ± 1 = y2, 1 - Nx2 = y2, Nx2 ± C = y2, and C - Nx2 = y2 which Sridhara gives in the Patiganita.

  384. Mascheroni biography
    • This encouraged him and he continued his work with two purposes in mind: to give a theoretical solution to the problem of constructions with compasses alone and to offer practical constructions that might be of help in making precision instruments.
    • From the solution to these problems he is able to prove theoretically that any construction which can be made using a ruler and compasses can be made with compasses alone.

  385. Banachiewicz biography
    • This, however, had been burnt down by the Germans on 15 September 1944, so after he took up his duties again after the war, Banachiewicz began to look for another solution.
    • 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]:- .

  386. 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.

  387. Rosenhain biography
    • Adolph Gopel independently solved the same problem but he did not submit his solution for the Paris Academy prize so basically they only received one solution to the problem.

  388. Bayes biography
    • The Essay, then, mainly, and perhaps justly, remembered for the solution of the problem posed by Bayes, should also be remembered for its contribution to pure mathematics.
    • This notebook contains a considerable amount of mathematical work, including discussions of probability, trigonometry, geometry, solution of equations, series, and differential calculus.

  389. Borelli biography
    • Daniele Spinola and Pietro Emmanuele both gave solutions to the problem and Borelli was asked to judge.
    • Although Borelli was critical of both solutions, he preferred that of Spinola and this led to an argument which became very heated.

  390. Meshchersky biography
    • This paper generalised the solution of the problem of the flow of a jet around a symmetric wedge obtained by D K Bobylev in 1881 to a nonsymmetric wedge.
    • Meshchersky obtained a complete solution for this more complex case of flow around a nonsymmetric wedge and the paper [Studies in the history of physics and mechanics (Moscow, 1988), 201-217.',3)">3] considers in detail the mathematical methods which he used, in particular comparing his methods to analogous ones of Western authors.

  391. Ince biography
    • Ince was the first to prove the proving the uniqueness of the Mathieu functions as periodic solutions.
    • It takes the existence of solutions for granted ..

  392. Kummer biography
    • In fact the prize of 3000 francs was offered for a solution to Fermat's Last Theorem but when no solution was forthcoming, even after extending the date, the Prize was given to Kummer even though he had not submitted an entry for the Prize.

  393. La Roche biography
    • Here is an example of one of his problems with solution which we have put into modern notation:- .
    • Solution: Let the number be x.

  394. Xiahou Yang biography
    • We have comments by Zhang Qiujian which criticise the accuracy of one of the solutions given in the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual).
    • The treatise contains three chapters in the usual style of problems and solutions.

  395. Novikov biography
    • Jointly with Adian he showed that the problem of the finiteness of periodic groups proposed by Burnside in 1902 had a negative solution.

  396. Oppenheim biography
    • The conjecture concerns Diophantine approximation and solutions of real quadratic forms which are not multiples of a rational form.

  397. Woodward biography
    • This problem was one requiring for its solution mathematical work of the highest order and, in addition, the experience of the engineer, so to shape his formulas that they could be applied directly by the computer.

  398. Freedman biography
    • The major innovation was the solution of the simply connected surgery problem by proving a homotopy theoretic condition suggested by Casson for embedding a 2-handle, i.e.

  399. Mohr Ernst biography
    • One of these 1951 papers looks at the numerical solution of the differential equation dy/dx = f (x, y).

  400. Al-Biruni biography
    • The contents of the work include the Arabic nomenclature of shade and shadows, strange phenomena involving shadows, gnomonics, the history of the tangent and secant functions, applications of the shadow functions to the astrolabe and to other instruments, shadow observations for the solution of various astronomical problems, and the shadow-determined times of Muslim prayers.

  401. Stokes biography
    • He pursued the usual school studies, and attracted the attention of the mathematical master by his solution of geometrical problems.

  402. Nagata biography
    • In fact Nagata announced his negative solution to Hilbert's 14th problem in his invited lecture On the fourteenth problem of Hilbert at the International Congress of Mathematicians held in Edinburgh, Scotland, in August 1958.

  403. Nash-Williams biography
    • This situation has, however, not deterred graph-theorists from studying the problem and obtaining some results which, although far from constituting a complete solution, are nevertheless interesting.

  404. Francais Jacques biography
    • He applied his methods to the famous problem of finding a sphere tangent to four given spheres, publishing a number of notes on the topic between 1808 and 1812, and giving a complete solution in the 1812 paper which appeared in Gergonne's Journal.

  405. Feigenbaum biography
    • The modernisation of cartography done to archival standards poses many problems, the solutions for which are strongly illuminated by the ideas and methods of nonlinear systems.

  406. Birnbaum biography
    • He showed me over and over again how mathematics could be used to look at a specific actuarial transaction, give it a general formulation, obtain a general solution, and make it part of an inventory of techniques to be used in similar situations in the future.

  407. Zeeman biography
    • You have to invent maths to get a solution to a problem but, in the process, I often discover a whole lot more which I didn't expect.

  408. 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.

  409. Delamain biography
    • After having largely explained the use of the 'Grammelogia' in the solution of a variety of questions in proportion, in interest, in annuities, in the extraction of roots, etc., the author concludes thus: "If there be composed three circles of equal thickness, A, B, and C, so that the inner edge of B, and the outer edge of A, be answerably graduated with logarithmic sines; and the outer edge of B, and the inner edge of C, with logarithms, and then, on the backside, be graduated the logarithmic tangents, and again the logarithmic sines opposite to the former graduations, it shall be fitted for the resolution of plain and spherical triangles.

  410. Feldman biography
    • Then follows a treatment of the methods of Gelfond and Schneider which led to the solution of Hilbert's seventh problem.

  411. Mazur Barry biography
    • Mazur had much earlier received the Cole prize for work which would prove important in the solution of Fermat's last Theorem.

  412. Mydorge biography
    • Mydorge left an unpublished manuscript Traite de geometrie of over 1000 geometric problems and their solutions.

  413. Lichtenstein biography
    • The solutions as functions of the boundary values and the parameter).

  414. Schauder biography
    • This degree is then used in an ingenious method to prove the existence of solutions to complicated partial differential equations.

  415. Girard Pierre biography
    • I thank you for the solution you sent me to the geometry problem..

  416. Mises biography
    • His Institute rapidly became a centre for research into areas such as probability, statistics, numerical solutions of differential equations, elasticity and aerodynamics.

  417. Wilder biography
    • He suggested Wilder write up the solution to the problem for his doctorate which indeed he did, becoming Moore's first Texas doctorate in 1923 with his dissertation Concerning Continuous Curves.

  418. 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.

  419. Gopel biography
    • finally, after ingenious calculations, obtained the result that the quotients of two theta functions are solutions of the Jacobian problem for p = 2.

  420. 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 ..

  421. Pascal Etienne biography
    • At the beginning of 1637 Fermat wrote his "Solution d'un probleme propose par M de Pascal".

  422. Broglie biography
    • He wrote at least twenty-five books including Ondes et mouvements (Waves and motions) (1926), La mecanique ondulatoire (Wave mechanics) (1928), Une tentative d'interpretation causale et non lineaire de la mecanique ondulatoire: la theorie de la double solution (1956), Introduction a la nouvelle theorie des particules de M Jean-Pierre Vigier et de ses collaborateurs (1961), Etude critique des bases de l'interpretation actuelle de la mecanique ondulatoire (1963).

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

  424. Plemelj biography
    • 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.

  425. Bliss Nathaniel biography
    • This was a period when clocks were proving to be the solution to the longitude problem and Harrison's clock H4 was being tested during his time in Greenwich.

  426. Picard Jean biography
    • Prodigious engineering efforts went into the solutions to this problem, and Picard's new levelling instruments with telescopic sights helped determine routes and avoided costly errors.

  427. Jensen biography
    • The theorem is important, but does not lead to a solution of the Riemann Hypothesis as Jensen had hoped.

  428. Mellin biography
    • He also extended his transform to several variables and applied it to the solution of partial differential equations.

  429. Anthemius biography
    • Heath [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2] gives Anthemius's solution:- .

  430. Vandiver biography
    • When he was eighteen years old he began to solve many of the number theory problems which were posed in the American Mathematical Monthly, regularly submitting solutions.

  431. Van der Waerden biography
    • These are solutions of the Burnside problem.

  432. Schmidt Harry biography
    • A rigorous solution of the Navier-Stokes equations as an example.

  433. Fefferman biography
    • 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.

  434. Dijkstra biography
    • she had a great agility in manipulating formulae and a wonderful gift for finding very elegant solutions.

  435. Epstein biography
    • Emigration would probably have been impossible in any case with the outbreak of war only a few weeks away, and Hitler's so-called final solution to the Jewish problem following shortly thereafter.

  436. Mitchell biography
    • He was instrumental in bringing the subject of spurious solutions to the fore.

  437. 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.

  438. Andreotti biography
    • Classical theorems on removable singularities and existence and uniqueness of solutions to the Cauchy problem are extended to some systems of partial differential operators through this technique.

  439. Lorentz George biography
    • Chapters nine, ten, and eleven concern entropy and Kolmogorov's solution of Hilbert's thirteenth problem.

  440. Diocles biography
    • The solution of this problem would, of course, have interesting consequences for the construction of a sundial.

  441. Thompson Abigail biography
    • Her paper "Thin position and the recognition problem for S3 " (1994), used the idea of thin position to reinterpret Rubenstein's solution to the recognition problem of the 3-sphere in a startling way.

  442. Mobius biography
    • His intuition, the problems he set himself, and the solutions that he found, all exhibit something extraordinarily ingenious, something original in an uncontrived way.

  443. Peierls biography
    • Surprises in theoretical physics are either theoretical results in disagreement with naive physical intuition, or simple solutions to apparently unmanageable problems.

  444. Jacobsthal biography
    • He also showed that it is possible to find a solution p = x2 + y2 where x and y can be expressed with simple sums over Legendre symbols.

  445. Wessel biography
    • He possesses a lot of theoretical knowledge of algebra, trigonometry and mathematical geometry, and as far as the last point is concerned, he has come up with some new and beautiful solutions to the most difficult problems in geographical surveying.

  446. Pringsheim biography
    • Before he left Germany to go to Zurich, Pringsheim gave to his friend Caratheodory a present of a very rare text from Jacob Bernoulli to his brother Johann Bernoulli containing the solution to the isoperimetric problem.

  447. Riesz Marcel biography
    • 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.

  448. Emerson biography
    • He returned to fluxions, publishing The Method of Increments herein the principles are demonstrated and the Practice thereof shown in the Solution of Problems in 1763.

  449. 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.

  450. 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.

  451. Hopf Eberhard biography
    • He held this post until 1947 by which time he had returned to the United States, where he presented the definitive solution of Hurewicz's problem.

  452. Pitiscus biography
    • The word 'trigonometry' is due to Pitiscus and first occurs in the title of his work Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus first published in Heidelberg in 1595 as the final section of A Scultetus's Sphaericorum libri tres methodice conscripti et utilibus scholiis expositi.

  453. 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.

  454. Cooper William biography
    • In the first part, Cooper and Henderson present the complete solution of a numerical example by the "simplex method" of G B Dantzig.

  455. Lehmer Derrick N biography
    • "Even a worm will turn," and now electricity and light, which have in the past gone to mathematics for solutions of their intricate problem, turn about and solve problems in mathematics which would require scores of years to complete.

  456. Smirnov biography
    • With Sobolev he devised a method for obtaining solutions on the propagation of waves in elastic media with plane boundaries.

  457. Schmidt biography
    • 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.

  458. Fuss biography
    • Most of Fuss's papers are solutions to problems posed by Euler on spherical geometry, trigonometry, series, differential geometry and differential equations.

  459. Tarry biography
    • He published a solution to the problem of finding the way out of a maze in 1895, a problem which had been of interest from classical times.

  460. Artin Michael biography
    • The point of the extension is that Artin's theorem on approximating formal power series solutions allows one to show that many moduli spaces are actually algebraic spaces and so can be studied by the methods of algebraic geometry.

  461. Le Cam biography
    • He made many contributions to the solution of practical problems such as studying stochastic models for rainfall, for the effects of radiation on living cells, sodium channel modelling and for cancer metastasis.

  462. 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).

  463. Bieberbach biography
    • There he worked out the details of his solution to the first part of Hilbert's eighteenth problem publishing them in two papers Uber die Bewegungsgruppen der Euklidischen Raume (1911, 1912).

  464. Morera biography
    • his results consisted of] solutions to complicated and difficult problems.

  465. Koch biography
    • 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.

  466. Thomae biography
    • He also discovered methods of solving difference equations giving solutions in the form of definite integrals.

  467. Birman biography
    • We had a course in Euclidean geometry, and every single night we would have telephone conversations and argue over the solutions to the geometry problems.

  468. Bezout biography
    • 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.

  469. Hippocrates biography
    • Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration.

  470. Skopin biography
    • At that time, a positive solution of the restricted Burnside problem was known only for a prime exponent.

  471. Bell John biography
    • The solution to the EPR problem that Einstein would have liked, rejecting (1) but retaining (2) was illegitimate.

  472. Bass biography
    • The solution of the congruence subgroup problem is presented as a pivotal event.

  473. Pfaff biography
    • In 1810 he contributed to the solution of a problem due to Gauss concerning the ellipse of greatest area which could be drawn inside a given quadrilateral.

  474. Lax Anneli biography
    • this book includes problems, solutions, historical notes, and bibliographical references that go beyond undergraduate enrichment.

  475. Schneider biography
    • The work on transcendence which had led Schneider to his solution of Hilbert's Seventh Problem led him to extend to a more general programme studying the transcendence of elliptic functions, modular functions and abelian functions.

  476. Severi biography
    • His most impressive work came before he went to Rome but, despite spending less time on mathematics, after this he still managed to produce work of the greatest importance like the solution of the Dirichlet problem and his development of the theory of rational equivalence.

  477. Lusztig biography
    • He was able to answer the question but they had asked F P Peterson, who was at the Massachusetts Institute of Technology, the same question a couple of months earlier and it was soon discovered that he had found a similar solution.

  478. 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.

  479. Lafforgue biography
    • This law allows one to describe, for any positive integer d, the primes p for which the congruence x2 = d mod p has a solution.

  480. Hill biography
    • His new idea on how to approach the solution to the three body problem involved solving the restricted problem.

  481. Lovasz biography
    • This book presents a nice survey of some recent developments towards the efficient solution of computational problems in areas like graph theory, number theory, and combinatorial optimization.

  482. Adams Edwin biography
    • The solution of two-dimensional electrical and hydrodynamical problems connected with a grating of rounded bars was obtained by H W Richmond beginning with a grating of bars of rectangular section.

  483. 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.

  484. Perron biography
    • One of the things he is best-known for is the Perron paradox which highlights the danger of assuming that a solution to a problem exists.

  485. Turing biography
    • He was criticised for his handwriting, struggled at English, and even in mathematics he was too interested with his own ideas to produce solutions to problems using the methods taught by his teachers.

  486. Kahler biography
    • His thesis advisor was Leon Lichtenstein but Kahler chose himself the topic for his doctoral dissertation Uber die Existenz von Gleichgewichtsfiguren, die sich aus gewissen Losungen des n-Korperproblems ableiten (On the existence of equilibrium figures that are derived from certain solutions of the n-body problem).

  487. Boscovich biography
    • His solution to this minimising problem took a geometric form.

  488. Szasz Domokos biography
    • The paper presents a partial solution of a classical open problem in mathematical physics, which is to prove rigorously the ergodicity of a system consisting of any number of identical hard balls in a box with periodic boundary conditions (i.e., on a torus).

  489. Cartwright biography
    • For something to do we went on and on at the thing with no earthly prospect of "results"; suddenly the whole vista of the dramatic fine structure of solutions stared us in the face.

  490. Tukey biography
    • the usefulness and limitation of mathematical statistics; the importance of having methods of statistical analysis that are robust to violations of the assumptions underlying their use; the need to amass experience of the behaviour of specific methods of analysis in order to provide guidance on their use; the importance of allowing the possibility of data's influencing the choice of method by which they are analysed; the need for statisticians to reject the role of 'guardian of proven truth', and to resist attempts to provide once-for-all solutions and tidy over-unifications of the subject; the iterative nature of data analysis; implications of the increasing power, availability and cheapness of computing facilities; the training of statisticians.

  491. Kodaira biography
    • Profusely illustrated and with plenty of examples, and problems (solutions to many of which are included), this book should be a stimulating text for advanced courses in complex analysis.

  492. 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.

  493. Al-Farisi biography
    • He noted the impossibility of giving an integer solution to the equation .

  494. Fine Nathan biography
    • Fine was also interested in problem solving and contributed both problems and solutions to problems to several different journals.

  495. Bessel-Hagen biography
    • Caratheodory thought Bessel-Hagen's disertation the first important advance in the theory of discontinuous solutions for problems in the calculus of variations since his own work in 1905.

  496. 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.

  497. 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.

  498. 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.

  499. Brisson biography
    • 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.

  500. Clifford biography
    • Without any diagram or symbolic aid he described the geometrical conditions on which the solution depended, and they seemed to stand out visibly in space.

  501. Hilbert biography
    • we hear within ourselves the constant cry: There is the problem, seek the solution.

  502. Baer biography
    • All of this will require careful planning, and there are some problems for which I cannot adequately envision solutions, and for this reason I am turning to you for advice.

  503. Bellavitis biography
    • In algebra he continued Ruffini's work on the numerical solution of algebraic equations and he also worked on number theory.

  504. Hall biography
    • He introduced the idea of a normal form which he used in the solution of the word problem for Lie rings and also for nilpotent groups.

  505. Piaggio biography
    • The author effectively remarks that this is not a necessary condition for the existence of a solution.

  506. Laurent Hermann biography
    • The last three volumes are devoted entirely to the solution and application of ordinary and partial differential equations.

  507. Donaldson biography
    • Using instantons, solutions to the equations of Yang-Mills gauge theory, he gained important insight into the structure of closed four-manifolds.

  508. Barclay biography
    • Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussion of new problems or new solutions, and comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.

  509. Gillman biography
    • This paper is concerned with the solution of an example of a zero-sum two-person game with essentially bounded measurable functions on (0, ) as pure strategies.

  510. Miranda biography
    • 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.

  511. Tapia biography
    • In it Tapia considered the solution of the equation P(x) = 0, where P is a nonlinear mapping between Banach spaces.

  512. 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.

  513. Sato biography
    • 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.

  514. Cooper biography
    • Other papers in which deal with applications include The uniqueness of the solution of the equation of heat conduction (1950).

  515. Sporus biography
    • His solution of the problem of duplicating the cube is similar to that of Diocles and in fact Pappus also followed a similar construction.

  516. Drach biography
    • of classifying the transcendental quantities satisfying the rational system verified by the solutions.

  517. Smullyan biography
    • Problems which had a unique solution, yet looked quite impossible.

  518. Friedrichs biography
    • This second paper gives an exact solution to the problem of the behaviour of a thin circular plate after buckling under a uniform edge force applied in the plane of the plate.

  519. Thue biography
    • I was fortunate enough to solve these, and I sent the solutions in to Copenhagen.
    • One day Holst came in and handed me a copy of the Danish mathematical periodical, where, to my pleasure I found a description of one of my solutions.
    • If f (x, y) is a homogeneous polynomial with integer coefficients, irreducible in the rationals and of degree > 2 and c is a non-zero integer then f (x, y) = c has only a finite number of integer solutions.

  520. Scorza biography
    • His first publication, which appear in print when he was only eighteen years old, gave the solution to two mathematical problems posed in the journal for secondary mathematics teaching.

  521. 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.

  522. Ozanam biography
    • He also wrote many works on mathematics, for example Methode generale pour tracer les cadrans (1673), La geometrie pratique du sr Boulenger (1684), Traite de la construction des equations pour la solution des problemes indeterminez (1687), Traite des lieux geometriques (1687), Traite des lignes du premier genre (1687), De l'usage du compas de proportion (1688).

  523. Dase biography
    • With small numbers, everybody that possesses any readiness in reckoning, sees the answer to such a question [the divisibility of a number] at once directly, for greater numbers with more or less trouble; this trouble grows in an increasing relation as the numbers grow, till even a practiced reckoner requires hours, yes days, for a single number; for still greater numbers, the solution by special calculation is entirely impractical.

  524. Valerio biography
    • (The authors mean this claim in the sense that Valerio was the first to systematize this ancient device, in the first three theorems of 'De centro'.) In applying these general theorems to the solution of a wide class of problems, Valerio thus advanced beyond his ancient Renaissance predecessors.

  525. Hobbes biography
    • it is clear that he hoped to assert preeminence in the learned world largely on the basis of the solution of the problem of squaring the circle.

  526. 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.

  527. Moser William biography
    • He edited (some with Ed Barbeau) the booklets containing the problems, solutions and results of the Canadian mathematical Olympiads from 1969 to 1978.

  528. Uhlenbeck Karen biography
    • 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.

  529. Baudhayana biography
    • The Sulbasutra of Baudhayana contains geometric solutions (but not algebraic ones) of a linear equation in a single unknown.

  530. Vajda biography
    • Of course, when these problems are cast in a Linear Programming form, the optimal solutions are integral, which results in the relevance of Linear Programming.

  531. 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.

  532. Bassi biography
    • In the end Benedict XIV went for a compromise solution by appointing Bassi as the twenty-fifth Benedettini but not giving her the same voting rights as the others.

  533. Sullivan biography
    • Beyond the solution of difficult outstanding problems, his work has generated important and active areas of research pursued by many mathematicians.

  534. Weldon biography
    • Biometrika will include (a) memoirs on variation, inheritance, and selection in animals and plants, based upon the examination of statistically large numbers of specimens (this will of course include statistical investigations of anthropometry); (b) those developments of statistical theory which are applicable to biological problems; (c) numerical tables and graphical solutions tending to reduce the labour of statistical arithmetic; (d) abstracts of memoirs, dealing with these subjects, which are published elsewhere; and (e) notes on current biometric work and unsolved problems.

  535. Hayes David biography
    • If a seemingly insuperable problem appeared, he would keep attempting a solution until all roadblocks were cleared and the way to further progress was open.

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

  537. Possel biography
    • announces the solution of several variants of the game of Nim with the following rules: Two players alternate in removing matches from p piles T1,..

  538. Christoffel biography
    • The procedure Christoffel employed in his solution of the equivalence problem is what Gregorio Ricci-Curbastro later called covariant differentiation, Christoffel also used the latter concept to define the basic Riemann-Christoffel curvature tensor.

  539. Jia Xian biography
    • Jia Xian is known to have written two mathematics books: Huangdi Jiuzhang Suanjing Xicao (The Yellow Emperor's detailed solutions to the Nine Chapters on the Mathematical Art), and Suanfa Xuegu Ji (A collection of ancient mathematical rules).

  540. Schubert Hans biography
    • Under certain conditions, he obtains solutions in terms of Bessel and Hankel functions and computes the resulting formulas for downwash.

  541. Saint-Venant biography
    • In the 1850s Saint-Venant derived solutions for the torsion of non-circular cylinders.

  542. Crelle biography
    • The solution was simple, even if it required a change in policy, and that was to have a second journal for more practical mathematics and this he moved to a second journal which he started in 1829, the Journal fur die Baukunst.

  543. 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.

  544. Bukreev biography
    • The geodesics as solutions of the Euler equation, Gauss curvature, geodesic curvature; II.

  545. Julia biography
    • The present book contains a small number of carefully chosen problems, each problem followed by one or more complete solutions.

  546. Miller Kelly biography
    • But he realized early that the Negro college student of that period and in the years immediately ahead needed to be awakened to a realization of the problems of the race and an interest in their solution.

  547. Routledge biography
    • Hall gave his students old examination papers to solve and then criticised the solutions they produced.

  548. Bombieri biography
    • He has repeatedly demonstrated an ability to quickly master essentials of a complicated new field, to select important problems which are accessible, and to apply intense energy and insight to their solution, making liberal use of deep results of other mathematicians in widely differing areas.

  549. Airy biography
    • He had earlier published a paper On a peculiar Defect in the Eye on this problem for which he was the first to provide a practical solution.

  550. 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.

  551. Beaugrand biography
    • In 1634 he was appointed to the committee which was set up by Cardinal Richelieu to evaluate Jean-Baptiste Morin's solution to the longitude problem by measuring absolute time from the position of the Moon relative to the stars.

  552. Post biography
    • Post showed that the word problem for semigroups was recursively insoluble in 1947, giving the solution to a problem which had been posed by Thue in 1914.

  553. Rademacher biography
    • A large part of the recent developments were initiated by the author's solution of the problem of unrestricted partitions and many of the results are due either to him or to his direct inspiration.

  554. Lexell biography
    • It involves the solution of polygons given certain sides and angles between them, their mensuration, division by diagonals, circumscribing polygons around circles and inscribing polygons in circles.

  555. Wren biography
    • Of course Charles II was not having an observatory built to push forward scientific research, rather he wanted a solution to the longitude problem which would give England a huge advantage over its competitors as a sea-faring nation.

  556. 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.

  557. 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.

  558. Lerch biography
    • 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).

  559. Slater biography
    • He wrote papers on this topic such as: (with R H Fowler) Collision numbers in solutions (1938), The rates of unimolecular reactions in gases (1939), Aspects of a theory of unimolecular reaction rates (1948), and Gaseous unimolecular reactions: theory of the effects of pressure and of vibrational degeneracy (1953).

  560. Stifel biography
    • 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.

  561. Plancherel biography
    • He also contributed to the solutions to variational problems via Ritz' method and to ergodic theory.

  562. Gegenbauer biography
    • The Gegenbauer polynomials are solutions to the Gegenbauer differential equation and are generalizations of the associated Legendre polynomials.

  563. 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.

  564. Watson Henry biography
    • The only solution received did not please Galton ..

  565. Ehrenfest biography
    • He corresponded with Klein who told him that what was required was a survey, not a complete solution of all the problems of the subject by Ehrenfest himself.

  566. Amsler biography
    • That was the problem of the attraction of an ellipsoid, which was first studied in depth by Ivory whose solution was later generalised by Poisson.

  567. Malcev biography
    • mathematical work is distinguished by the abundance of new ideas and the creation of new mathematical trends on the one hand, and the solution of a number of classical problems on the other hand.

  568. Mahler biography
    • I almost immediately posed him the following problem: An integer is called powerful if p | m implies p2 | m; are there infinitely many consecutive powerful numbers? Mahler immediately answered: Trivially, yes! x2 - 8 y2 = 1 has infinitely many solutions.

  569. Aryabhata I biography
    • 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.

  570. Morawetz biography
    • During the 1970s she extended this work to examine other solutions to the wave equation.

  571. Hermite biography
    • He had found general solutions to the equations in terms of theta-functions.

  572. Beurling biography
    • Quite possibly the finest feat of cryptoanalysis performed during the Second World War was Arne Beurling's solution of the secret of the Geheimschreiber.

  573. Rouche biography
    • 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.

  574. De Beaune biography
    • According to Beaugrand, the first of these problems - which in the present state of textual study appears to concern itself only with the determination of the tangent to an analytically defined curve - interested Debeaune "in a design touching on dioptrics." As to the second of these problems, the one that has been particularly identified with Debeaune and that ushered in what was called at the end of the seventeenth century the "inverse of tangents" i.e., the determination of a curve from a property of its tangent - Debeaune told Mersenne on 5 March 1639 that he sought a solution with only one precise aim: to prove that the isochronism of string vibrations and of pendulum oscillations was independent of the amplitude.

  575. Euclid biography
    • Euclid's geometric solution of a quadratic equation .

  576. Hopf biography
    • The boldness of the questions deserves as much admiration as the surprising results of the solutions.

  577. 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.

  578. Stallings biography
    • This paper is an ingenious solution of the conjecture for dimension n ≥ 7.

  579. Tacquet biography
    • Tacquet rejected all notions that solids are composed of planes, planes of lines, and so on, except as heuristic devices for finding solutions.

  580. Peterson biography
    • 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.

  581. Wilkins biography
    • He looked at the problem of different languages in different parts of the world and looked for a solution to this curse which hindered learning.

  582. Browder Felix biography
    • In nonlinear functional analysis the introduction of monotone and, later, accretive operator theory led to the solution of problems that had heretofore been out of reach.

  583. Plato biography
    • He remained in Syracuse for part of 360 BC but did not achieve a political solution to the rivalry.

  584. Lie biography
    • 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.

  585. De Moivre biography
    • Dupont looks at this problem, and Todhunter's solution, in [Atti Accad.

  586. Bott biography
    • Synge then came up with the perfect solution - why not move straight to a doctorate? Richard Duffin became Bott's supervisor and the first problem they solved was one which Bott suggested himself.

  587. Adler biography
    • His theoretical solution involved giving specific constructions, such as bisecting a circular arc, using only a compass.

  588. Kirillov biography
    • This text, based on courses and seminars at Moscow University, consists of three main parts: expository text, problems and hints for solution.

  589. Bronowski biography
    • In 1933 he published a solution of the classical functional Waring problem, to determine the minimal n such that a general degree d polynomial f can be expressed as a sum of dth powers of n linear forms, but his argument was incomplete.

  590. Prodi biography
    • 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.

  591. Kneser biography
    • But above all, the decisive advances towards the solution of the so-called Mayer Problem, recently introduced to the calculus of variations, are due to Kneser.

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

  593. 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.

  594. Eisenstein biography
    • When he was about ten years old his parents tried to find a solution to his continual health problems by sending him to Cauer Academy in Charlottenburg, a district of Berlin which was not incorporated into the city until 1920.

  595. 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.

  596. 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.

  597. Lobachevsky biography
    • 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.

  598. Mauchly biography
    • For his pioneering contributions to automatic computing by participating in the design and construction of the ENIAC, the world's first all-electronic computer, and of the BINAC and the UNIVAC, and for his pioneering efforts in the application of electronic computers to the solution of scientific and business problems.

  599. 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, ..

  600. Jordanus biography
    • The solution illustrates the use of letters by Jordanus:- .

  601. Ehrenfest-Afanassjewa biography
    • They corresponded with Klein who told them that what was required was a survey, not a complete solution of all the problems of the subject by the Ehrenfests themselves.

  602. Netto biography
    • Despite this, Netto's "proof" was widely accepted as providing a solution to the dimension problem until Jurgens' criticism in 1899 of Netto's proof.

  603. Salem biography
    • Much of this progress is due to [Salem] and people influenced by his ideas, and acquaintance with his work seems to be a prerequisite for those who would like to contribute to the solution of the problem.

  604. Wang Xiaotong biography
    • Try to set up the necessary equations in these two cases in a similar way to our solution to Problem 15 above.

  605. Van der Pol biography
    • 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).

  606. Haret biography
    • Taking also into account commensurabilities, and using generalized Fourier series (which generate quasiperiodic solutions), Poincare proved the divergence of these series, which means instability, confirming in this way Haret's result.

  607. Sripati biography
    • 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 .

  608. Macdonald biography
    • He corrected his 1903 solution to the problem of a perfectly conducting sphere embedded in an infinite homogeneous dielectric in 1904 after a subtle error was pointed out by Poincare.

  609. Maurolico biography
    • Demonstratio algebrae, which is an elementary text looking at quadratic equations and problems whose solution reduces to solving a quadratic; .

  610. Somerville biography
    • In this correspondence they discussed the mathematical problems set in the Mathematical Repository and in 1811 Mary received a silver medal for her solution to one of these problems.

  611. Dodgson biography
    • As early as 1894 Dodgson used truth tables for the solution of specific logic problems.

  612. Helly biography
    • He taught in a Gymnasium, gave private tuition, and wrote solution manuals for a series of standard textbooks.

  613. Kaczmarz biography
    • 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.

  614. Morin Jean-Baptiste biography
    • His solution, proposed in 1634, was based on measuring absolute time by the position of the Moon relative to the stars.

  615. Saunderson biography
    • The final book presents the solution of cubic and quartic equations.

  616. Gowers biography
    • Dr Gowers' achievements include the following: a solution to the notorious Banach hyperplane problem (to find a Banach space which is not isomorphic to any hyperplane), a counterexample to the Banach space Schroder-Bernstein theorem, a proof that if all closed infinite-dimensional subspaces of a Banach space are isomorphic then it is a Hilbert space, and an example of a Banach space such that every bounded operator is a Fredholm operator.

  617. Cohen Wim biography
    • It concerned fundamental research in direct relation to practical engineering problems: his elegant and deep mathematical solution of the problem of stresses and displacements in helicoidal shells and ship propeller blades has proved of great importance in ship engineering building.

  618. Alberti biography
    • It was again Alberti who found the solution that remained influential up to our own days.

  619. Levy Hyman biography
    • Levy's main work was in numerical methods, numerical solution of differential equations, finite difference equations and statistics.

  620. Hipparchus biography
    • it seems highly probable that Hipparchus was the first to construct a table of chords and thus provide a general solution for trigonometrical problems.

  621. Gemma Frisius biography
    • It is worth noting that although there were many methods of finding longitude proposed in the 250 years following Gemma Frisius's work, ultimately the methods he proposed were to become the solution to finding the longitude at sea.

  622. Ricci Giovanni biography
    • Hilbert himself remarked that he expected this Seventh Problem to be harder than the solution of the Riemann conjecture.

  623. Flugge-Lotz biography
    • Her contributions have spanned a lifetime during which she demonstrate, in a field dominated by men, the value and quality of a woman's intuitive approach in searching for and discovering solutions to complex engineering problems.

  624. Goursat biography
    • Volume 3 surveys variations of solutions and partial differential equations of the second order and integral equations and calculus of variations.

  625. Caramuel biography
    • He published the philosophy work Rationalis et realis philosophia in 1642 and, in the following year, published his theological work Theologia moralis ad prima atque clarissima principia reducta which sought solutions to theological problems through applying mathematical rules.

  626. Fresnel biography
    • The mathematical difficulties were formidable, and a solution was to require many months of effort.

  627. Faber biography
    • Only in the 1980s was Faber's idea seen to be an important ingredient for the efficient solution of partial differential equations.

  628. Ghizzetti biography
    • 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.

  629. 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.

  630. Begle biography
    • Throughout, Ed dictated no solutions but strove to harmonise the opinions of all.

  631. Voronoy biography
    • 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.

  632. 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.

  633. Iacob biography
    • The author's more than twenty years' research on boundary-value problems for plane harmonic functions is reflected in a thorough account of the potential-theoretical background for the solution of plane incompressible-flow and linearized compressible-flow problems.

  634. Levi-Civita biography
    • 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.

  635. 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.

  636. Lissajous biography
    • This apparatus which is particularly remarkable because it represents the solution to a difficult problem, has, unfortunately, no chance of being sold in quantity, since the number of pictures need is very considerable: not less than 32 pictures are required.

  637. Kaplansky biography
    • He completed the solution of Kurosh's problem on algebraic algebras of bounded degree, where Jacobson had made a decisive reduction, and considered numerous questions in the area of Banach algebras, always from the algebraist's viewpoint.

  638. Franklin biography
    • In A Step-Polygon of a Denumerable Infinity of Sides which Bounds No Finite Area (1933), written in collaboration with Jesse Douglas, the authors gave an explicit construction for functions which had been shown to exist by Jesse Douglas in his solution of Plateau's problem in his paper published in 1931 (for which he received a Fields Medal).

  639. Torricelli biography
    • Around 1640, Torricelli devised a geometrical solution to a problem, allegedly first formulated in the early 1600s by Fermat: 'given three points in a plane, find a fourth point such that the sum of its distances to the three given points is as small as possible'.

  640. Hoyle biography
    • Fred believed that, as a general rule, solutions to major unsolved problems had to be sought by exploring radical hypotheses, whilst at the same time not deviating from well-attested scientific tools and methods.

  641. Whiston biography
    • Having been one of the main enthusiasts for the Longitude Act, Whiston now proposed a number of methods of finding the longitude at sea; there was a lot of money for a good solution but he never succeeded.

  642. Nash biography
    • 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.

  643. Goldstein biography
    • He studied numerical solutions to steady-flow laminar boundary-layer equations in 1930.

  644. 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.

  645. Goldberg biography
    • He gave the lecture On the growth of entire solutions of algebraic differential equations which was published in 2005.

  646. Geocze biography
    • No difficulties at home, no noise from his children, no horrors of the trenches or the thunder of guns were able to distract Zoard Geocze from concentrating on the solution of his favourite problem and making efforts to widen and deepen our knowledge of the subject.

  647. 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]:- .

  648. Vinogradov biography
    • His methods reached their height in Some theorems concerning the theory of prime numbers written in 1937 which provides a partial solution to the Goldbach conjecture.

  649. Budan de Boislaurent biography
    • Fashions in mathematics change and solving a problem which Lagrange had deemed important does not guarantee your solution will achieve fame.

  650. Cramer biography
    • This leads to 5 linear equations in 5 unknowns and he refers the reader to an appendix containing Cramer's rule for their solution.

  651. Smith biography
    • Hermite asked Smith if he would cooperate in trying not to make the Academy look foolish, and simply submit a solution to the Grand Prix question.

  652. Chazy biography
    • In the same year, Chazy published a paper on the three-body problem, Sur les solutions isosceles du Probleme des Trois Corps, an area he had begun to study in 1919 and for which he has become famous.

  653. 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.

  654. De Vries biography
    • They found explicit, closed-form, travelling-wave solutions to the Korteweg - de Vries equation that decay rapidly.

  655. 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.

  656. Ahmes biography
    • The Verso has 87 problems on the four operations, solution of equations, progressions, volumes of granaries, the two-thirds rule etc.

  657. 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.

  658. Orlicz biography
    • Working in Lvov Orlicz participated in the famous meetings at the Scottish Cafe (Kawiarnia Szkocka) where Stefan Banach, Hugo Steinhaus, Stanislaw Ulam, Stanislaw Mazur, Marek Kac, Juliusz Schauder, Stefan Kaczmarz and many others talked about mathematical problems and looked for their solutions.

  659. Wallace biography
    • He published two books after he retired, A Geometrical Treatise on the Conic Sections with an Appendix Containing Formulae for their Quadrature (1838) and Geometrical Theorems and Analytical Formulae with their application to the Solution of Certain Geodetical Problems and an Appendix (1839).

  660. Hawking biography
    • In 2005 Hawking published Information loss in black holes in which he proposed a solution to the information loss paradox.

  661. Mercator Nicolaus biography
    • A known theoretical solution was to devise a clock which would keep accurate time at sea.

  662. Lansberge biography
    • In the solution of spherical triangles Van Lansberge employs a device similar to that of Maurice Bressieu in his 'Metrices astronomicae' (Paris, 1581), the marking of the given parts of a triangle by two strokes.

  663. Oleinik biography
    • Much of the book is devoted to the study of the asymptotic behaviour of solutions to nonlinear elliptic second-order equations.

  664. Menabrea biography
    • He had always worked for a compromise solution to the Italian unification question and up to 1859 believed that a compromise between the Vatican and the state was possible.

  665. Huygens biography
    • At this time Huygens patented his design of pendulum clock with the solution of the longitude problem in mind.

  666. Lindelof biography
    • Lindelof's first work in 1890 was on the existence of solutions for differential equations.

  667. Boutroux biography
    • There he lectured at the College de France on functions which are the solutions of first order differential equations.

  668. 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.

  669. Mineur biography
    • 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.

  670. Taussky-Todd biography
    • who provided significant contributions to solutions of problems associated with applications of computers.

  671. Hollerith biography
    • Hollerith realised that cards would provide a better solution.

  672. Kubilius biography
    • He published three papers (all in Russian) during his time as a research student: On the application of I M Vinogradov's method to the solution of a problem of the metric theory of numbers (1949); The distribution of Gaussian primes in the sectors and contours (1950); and On the decomposition of prime numbers as the sum of two squares (1951).

  673. Camus biography
    • treatment of toothed wheels and their use in clocks, studies of the raising of water from wells by buckets and pumps, an evaluation of an alleged solution to the problem of perpetual motion, and works on devices and standards of measurement.

  674. Turan biography
    • Turan mentioned these problems and told me that they were not only interesting in themselves but their positive solution would have many applications.

  675. Korteweg biography
    • They found explicit, closed-form, travelling-wave solutions to the Korteweg - de Vries equation that decay rapidly.

  676. 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.

  677. Kolchin biography
    • 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.

  678. Povzner biography
    • He published The representation of smallest degree isomorphic to a given Abelian group (1937), written to give his partial solution to a problem stated by Otto Yulevich Schmidt in his book The abstract theory of groups, namely given an abstract group, find a permutation representation of least degree.

  679. Egorov biography
    • In this paper, in addition to the independent, very elegant and simple solution of the problem proposed, the originality and logical rigour of the exposition of the basic general geometrical principles deserve special mention, as does also the very successful working out of many details.

  680. 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.

  681. Redei biography
    • 32-33 (1990), 199-211.',1)">1] examines the work which led up to the solution of the problem by Redei in 1953:- .

  682. Banach biography
    • We discussed problems proposed right there, often with no solution evident even after several hours of thinking.

  683. Banneker biography
    • The surviving manuscript journal contains mathematical puzzles and their solutions.

  684. Razmadze biography
    • He also did important work on discontinuous solutions.

  685. 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).

  686. Schwerdtfeger biography
    • Admirable features of the book include a list of examples for solution, footnotes with historical notes, references to the original literature and to extensions of the topics treated, a geometric framework for the algebraic results, and a summary of the principal problems considered ..

  687. Nicomedes biography
    • As indicated in this quote Pappus also wrote about Nicomedes, in particular he wrote about his solution to the problem of trisecting an angle (see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]):- .

  688. 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.

  689. 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.

  690. 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).

  691. Al-Jayyani biography
    • The work, which is published together with a Spanish translation and a commentary in [La trigonometria europea en el siglo XI : Estudio de la obra de Ibn Mu\'ad, \'el Kitab mayhulat\' (Barcelona, 1979).',3)">3], contains formulae for right-handed triangles, the general law of sines, and the solution of a spherical triangle by means of the polar triangle.

  692. Philon biography
    • The solution is effectively produced by the intersection of a circle and a rectangular hyperbola.

  693. Gergonne biography
    • Gergonne provided an elegant solution to the Problem of Apollonius in 1816.

  694. Lifshitz biography
    • 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.

  695. Chrystal biography
    • The result is that algebra, as we teach it, is neither an art nor a science, but an ill-farrago of rules; whose object is the solution of examination problems.

  696. Cohen biography
    • These results present the long-awaited solutions of the most outstanding open problems of axiomatic set theory and should be rated as the most important advance in the study of axiomatic set theory since the publication of Godel's 1940 monograph 'The consistency of the continuum hypothesis' (1940).

  697. Bari biography
    • Already this first piece of work by Nina Bari testified to her great mathematical talent, since it included the solution of several very difficult problems in the theory of trigonometric series that had lately been engaging the attention of many outstanding mathematicians.

  698. Richmond biography
    • It is true that the scope of these methods is restricted, but there is compensation in the fact that when geometry is successful in solving a problem the solution is almost invariably both simple and beautiful.

  699. Kluvanek biography
    • Indeed, to express the required solutions in integral form one may have to integrate with respect to a vector-valued measure of infinite total variation.

  700. Zeno of Elea biography
    • Again Zeno has presented a deep problem which, despite centuries of efforts to resolve it, still seems to lack a truly satisfactory solution.

  701. Seki biography
    • For example, in 1683, he considered integer solutions of ax - by = 1 where a, b are integers.

  702. Wazewski biography
    • 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.

  703. Plateau biography
    • He later used a solution of soapy water and glycerine and dipped wire contours into it, noting that the surfaces formed were minimal surfaces.

  704. Cox Elbert biography
    • In 1925 Cox was awarded his doctorate for his thesis Polynomial solutions of difference equations.

  705. Leonardo biography
    • Leonardo studied Euclid and Pacioli's Suma and began his own geometry research, sometimes giving mechanical solutions.

  706. Poincare biography
    • His results applied only to restricted classes of functions and Poincare wanted to generalise these results but, as a route towards this, he looked for a class functions where solutions did not exist.

  707. Capelli biography
    • 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.

  708. Bhaskara I biography
    • ',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.

  709. Wilks biography
    • In 1947 he was awarded the Presidential Certificate of Merit for his contributions to anti-submarine warfare and the solution of convoy problems.

  710. Pincherle biography
    • 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.

  711. Fricke biography
    • 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.

  712. Guo Shoujing biography
    • 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.

  713. Christiansen biography
    • Yet his great integrity led him to be intolerant of injustice, of those who were rude, self-seeking, inefficient and not disposed to think, and of those who peddled simple solutions to complex problems.

  714. Wrinch biography
    • After discussions they decided (correctly it turned out) that computers would not give an immediate solution because of the difficult problem of determining phases.

  715. 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.

  716. Polozii biography
    • Original results in the theory of functions of a complex variable were obtained in the 1950s and 1960s by G Polozii of Kiev, who introduced a new notion of p-analytic functions, defined the notion of derivative and integral for these functions, developed their calculus, obtained a generalised Cauchy formula, and devised a new approximation method for solution of problems in elasticity and filtration.

  717. Brioschi biography
    • One of his most important results was his application of elliptical modular functions to the solution of equations of the fifth degree in 1858.

  718. 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.

  719. Warga biography
    • This is a scholarly, clear presentation of that aspect of optimal control theory which deals with the existence of usual, generalized (relaxed) and approximate optimal solutions, and necessary conditions for optimality.

  720. Menelaus biography
    • Another Arab reference to Menelaus suggests that his Elements of Geometry contained Archytas's solution of the problem of duplicating the cube.

  721. Boggio biography
    • In the first of these, Boggio obtained a solution for the problem of an elastic membrane, displaced in its own plane with known displacements on the boundary.

  722. Adams Frank biography
    • in recognition of his solution of several outstanding problems of algebraic topology and of the methods he invented for this purpose which have proved of prime importance in the theory of that subject.

  723. Third biography
    • In the Mathematical Questions and Solutions section of the Educational Times of 1902 he published The perpendicular from the isogonal conjugate of any point on the Euler line of a triangle to the trilinear polar of the point passes through the orthocentre.

  724. Zhukovsky biography
    • Those Joukowski aerofoils were actually used on some aircraft, and today these techniques provide a mathematically rigorous reference solution to which modern approaches to aerofoil design can be compared for validation.

  725. Simplicius biography
    • Damascius had written Problems and Solutions about the First Principles which develops the Neoplatonist philosophy as expounded by Proclus.

  726. Lehmer Emma biography
    • To perform the operation with pencil and paper one must start with the million or so numbers among which the solution is known to lie.

  727. Chudakov biography
    • Chudakov made a substantial contribution to the solution when he proved that all, except possibly a finite number, of even integers greater than 2 can be represented as the sum of two primes.

  728. Nygaard biography
    • In 1952 he published On the solution of integral equations by Monte-Carlo methods as a Norwegian Defence Research Establishment Report.

  729. Lehmer Derrick biography
    • He was a pioneer in the application of mechanical methods, including digital computers, to the solution of problems in number theory and he talked about some of the methods used to factorise numbers including: factor tables, trial division, Legendre's method, factor stencils, the continued fraction method, Fermat's method, methods based on quadratic forms, and Shanks' method.

  730. Kovacs biography
    • It is an open-book competition and the competitors have ten days in which to produce solutions.

  731. Darboux biography
    • brilliant are his reductions of various geometrical problems to a common analytic basis, and their solution and development from a common point of view.

  732. Gromoll biography
    • provided one of the cornerstones of the Poincare Conjecture solution.

  733. Higman biography
    • This result plays a vital part in Zelmanov's positive solution to the restricted Burnside problem in the early 1990s.

  734. Thomason biography
    • For example in a 1983 paper he found a partial solution of Grothendieck's absolute cohomological purity conjecture.

  735. Bonferroni biography
    • More than anything else, however, I was struck by his personal style and the simplifying solutions to the very complex procedures which he proposed.

  736. Kolmogorov biography
    • The time of their graduate studies remains for all of Kolmogorov's students an unforgettable period in their lives, full of high scientific and cultural strivings, outbursts of scientific progress and a dedication of all one's powers to the solutions of the problems of science.

  737. Wielandt biography
    • I am indebted to that time for valuable discoveries: on the one hand the applicability of abstract tools to the solution of concrete problems, on the other hand, the - for a pure mathematician - unexpected difficulty and unaccustomed responsibility of numerical evaluation.

  738. Chatelet Albert biography
    • Clairin, who applied group theory to the solution of differential equations, had published Cours de mathematiques generales (1910).

  739. Scheffe biography
    • 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.

  740. Lob biography
    • Examples of papers by Lob in the Journal of Symbolic Logic in the 1950s are Concatenation as basis for a complete system of arithmetic (1953), Solution of a problem of Leon Henkin (1955), and Formal systems of constructive mathematics (1956).

  741. Vessiot biography
    • 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.

  742. Cauchy biography
    • 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.

  743. Rudolph biography
    • Although we all tried very hard to solve it; Dan beat us all with a ingenious solution.

  744. Householder biography
    • For his impact and influence on computer science in general and particularly for his contributions to the methods and techniques for obtaining numerical solutions to very large problems through the use of digital computers, and for his many publications, including books, which have provided guidance and help to workers in the field of numerical analysis, and for his contributions to professional activities and societies as committee member, paper referee, conference organiser, and society President.

  745. Frenet biography
    • It contains problems with full solutions and often historical remarks.

  746. Lehto biography
    • It led in due time to a simple solution of the geometric problem of moduli, and there are encouraging signs of a fruitful theory in several dimensions.

  747. Rahn biography
    • in the solutions, and in the arithmetic too, I make use of a completely new method, which has not been used by any writer on algebra in a published work, that I first learned from an eminent and very learned person to whom I should very gladly acknowledge indebtedness and humble respect, had he permitted.

  748. Simson biography
    • a proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions.

  749. Bartel biography
    • Throughout the book care is taken to present the theoretical foundations on which are based the solutions of the problems considered.

  750. Bachmann biography
    • Bachmann surveyed the attempts that had been made over nearly 300 years attempting to give a positive or a negative solution to Fermat's Last Theorem in Das Fermatproblem in Seiner Bisherigen Entwicklung (1919).

  751. Scholtz biography
    • He also submitted solutions to problems that had been posed in this journal.

  752. Copson biography
    • by Poisson's analytical solution of the equation of wave-motions.

  753. Karlin biography
    • Karlin published papers such as Solutions of discrete, two-person games; Polynomial games; and Games with continuous, convex pay-off all in 1950.

  754. Humbert Georges biography
    • He thus enriched analysis and gave the complete solution of the two great questions of the transformation of hyperelliptic functions and of their complex multiplication.

  755. Frobenius biography
    • On the algebraic solution of equations, whose coefficients are rational functions of one variable.

  756. Church biography
    • 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.

  757. 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).

  758. Ulam biography
    • While Ulam was at Los Alamos, he developed the 'Monte-Carlo method' which searched for solutions to mathematical problems using a statistical sampling method with random numbers.

  759. Recorde biography
    • In his study of quadratic equations, Recorde does not allow solutions which are negative, but he does allow negative coefficients.

  760. Shelah biography
    • The solution appears in the second edition of his book 'Classification Theory and the Number of Non-isomorphic Models' (1990).

  761. Ceva Giovanni biography
    • The first mathematical problem he attacked was the classic problem of squaring the circle and he produced several incorrect solutions to it.

  762. Bremermann biography
    • During this time he continued to produce high quality results in complex analysis continuing to push the results of his doctoral dissertation towards a general solution to the Levi problem.

  763. Thymaridas biography
    • then Thymaridas gives the solution .

  764. Durell biography
    • All chapters conclude with a series of exercises, with solutions at the end of the book.

  765. Hudde biography
    • Hudde proposed a "mechanical" solution, which was not mathematically exact.

  766. Gronwall biography
    • Only a mathematician with Gronwall's gift for analysis and most uncommon grasp of the literature of chemistry and physics could have contributed the elegant solution which he gave.

  767. Bollobas biography
    • All come with solutions, many with hints, and most with illustrations.

  768. 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.

  769. De Bruijn biography
    • Also in 1943, in addition to his doctoral thesis, he published On the absolute convergence of Dirichlet series, On the number of solutions of the system ..

  770. Kochina biography
    • 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.

  771. Descartes biography
    • Descartes' geometric solution of a quadratic equation .

  772. Magnitsky biography
    • Geometry and trigonometry were not abstract entities, but solution methods for navigational problems, just as contemporary English and American texts relied on examples of commercial transactions to induce students to do their calculations.

  773. McClintock biography
    • He published A simplified solution of the cubic in 1900 in the Annals of Mathematics.

  774. Gaschutz biography
    • This is an example that shows how minor variations of the initial conditions can influence the solutions of an equation considerably.

  775. Fraser biography
    • Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussion of new problems or new solutions, and comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.

  776. Courant biography
    • The first application as a numerical method, however, was given by Courant in 1943 in his solution of a torsion problem.

  777. Roth Klaus biography
    • Speaking of Roth's solution to this problem of approximating algebraic numbers Davenport said [2]:- .

  778. Schramm biography
    • This work led to the solution of many problems by him, many together with his collaborators Greg Lawler now at Cornell University, and Wendelin Werner in Strasbourg, France, as well as by many other mathematical researchers.

  779. Ostrogradski biography
    • They include a special case of Green's theorem, a general development (the first such, according to Yushkevich) of the method of separation of variables, and the first solution of the problem of heat diffusion in a triangular prism.
    • His important work on ordinary differential equations considered methods of solution of non-linear equations which involved power series expansions in a parameter alpha.

  780. Fock biography
    • 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.

  781. 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.

  782. Sun Zi biography
    • In fact the solution given, although in a special case, gives exactly the modern method.

  783. Rudin biography
    • Her 1952 paper A primitive dispersion set of the plane provided a positive solution to an unsolved problem contained in R L Wilder's book Topology of manifolds (1949).

  784. 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.

  785. Conway biography
    • To really understand and prove everything in the book, not to mention to attempt solutions of the many questions inspired on every page of the book, will engage many people for many years.

  786. Akhiezer biography
    • His most outstanding work consisted of deep approximation results in the constructive function theory, including the solution of the problem of Zolotarev.

  787. Verhulst biography
    • He named the solution to the equation he had proposed in his 1838 paper the 'logistic function'.

  788. Fredholm biography
    • 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.

  789. 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.

  790. Eratosthenes biography
    • Eratosthenes erected a column at Alexandria with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [A History of Greek Mathematics (2 vols.) (Oxford, 1921).',4)">4]:- .

  791. Freitag biography
    • She has also contributed prodigiously to the Elementary Problems and Solutions Section of the Fibonacci Quarterly and published many papers in that journal.

  792. Ricci-Curbastro biography
    • 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.

  793. Bring biography
    • This work describes Bring's contribution to the algebraic solution of equations.

  794. 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.

  795. Tamarkin biography
    • I proposed to Tamarkin that he think about the asymptotic solution of differential equations (i.e.

  796. Barsotti biography
    • Thus it is not surprising that Chevalley's solution of the problem has no evident link with the methods that, according to the suggestions of the classical geometers, should have been used in order to define the intersection multiplicity; rather, it is linked to the analytical approach, and it is therefore a strictly "local" theory, thus having the advantage of providing an intersection multiplicity also for algebroid varieties.

  797. Hertz Heinrich biography
    • A prize had been announced by the Philosophy Faculty for the solution of an experimental problem concerning electrical inertia and Hertz was very keen to enter.

  798. 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.

  799. Roberval biography
    • He therefore solved this problem before Torricelli who found a solution after 1644.

  800. Greenhill biography
    • The chief characteristics of Greenhill's work were a desire for concrete realisation of abstract theories and the direction of investigation to the solution of definite problems.

  801. Taylor Geoffrey biography
    • Taylor continued his research after the end of the War, taking the opportunity to complete some more thorough investigations into problems where previously the pressure of finding solutions had prevented him from taking his study further.

  802. Von Staudt biography
    • Von Staudt also gave a nice geometric solution to quadratic equations.

  803. 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.

  804. Lhuilier biography
    • Lhuilier also corrected Euler's solution of the Konigsberg bridge problem.

  805. Sidler biography
    • In his paper Die Schale Vivianis (1901) he showed an elementary solution to the problem and new results relating to Viviani's curve.


History Topics

  1. Pell's equation
    • where n is a given integer and we are looking for integer solutions (x, y).
    • 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' .
    • 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.
    • He also noted that, using a similar argument to what we have just given, if x = a, y = b is a solution of nx2 + k = y2 then applying the method of composition to (a, b) and (a, b) gave (2ab, b2 + na2) as a solution of nx2 + k2 = y2 and so, dividing through by k2, gives .
    • as a solution of Pell's equation nx2 + 1 = y2.
    • Well if k = 2 then, since (a, b) is a solution of nx2 + k = y2 we have na2 = b2 - 2.
    • 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.
    • Often he could find trial solutions which worked for one of these values of k and so in many cases he was able to give solutions.
    • as another solution.
    • Among the examples Brahmagupta gives himself is a solution of Pell's equation .
    • and applies his method to find the solution .
    • We can now generate a sequence of solutions (x,y): .
    • 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.
    • We can assume that a and b are coprime, for otherwise we could divide each by their gcd and get a "closer" solution with smaller k.
    • is a solution to .
    • With such a choice of m he therefore has integer solutions .
    • 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.
    • as a solution to the 'Pell type equation' nx2 - 4 = y2.
    • as the smallest solution to 61x2 + 1 = y2.
    • 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.
    • However, when one writes down a proof it should become clear that the algorithm switching to Brahmagupta's method is never necessary (although can reach the solution more quickly).
    • 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 .
    • Finally Narayana applies Brahmagupta's method to this last equation to get the solution .
    • We await these solutions, which, if England or Belgic or Celtic Gaul do not produce, then Narbonese Gaul will.
    • 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 .
    • Brouncker discovered a method of solution which is essentially the same as the method of continued fractions which was later developed rigorously by Lagrange.
    • 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.
    • Brouncker found the smallest solutions, using his method, which is .
    • We should note that by this time several mathematicians had claimed that Pell's equation nx2 + 1 = y2 had solutions for any n.
    • Wallis, describing Brouncker's method, had made that claim, as had Fermat when commenting on the solutions proposed to his challenge.
    • In fact Fermat claimed, correctly of course, that for any n Pell's equation had infinitely many solutions.
    • This established rigorously the fact that for every n Pell's equation had infinitely many solutions.
    • The solution depends on the continued fraction expansion of √n.
    • will be the smallest solution to Pell's equation .
    • To find the infinite series of solutions take the powers of 170 + 39√19.
    • will give a second solution to the equation.
    • as the next solution.
    • 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 .

  2. Braids arithmetic
    • May 12th 1868." In addition to the collection of arithmetic problems with solutions written by William Braid, he has written a Scots Proverb at the end of most of the problems.
    • The Voluntary Exercises in Arithmetic are mostly elementary and the solutions for those near the beginning are not worth recording.
    • However, we give an occasional example of a solution to illustrate William Braid's methods.
    • The examples dealing with repeating decimals fall into this category and we give more details of Braid's solutions (including his errors) when giving these.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.

  3. Alcuin's book
    • Alcuin gives solutions in the book but often these give an answer with a verification that it is correct rather than a method of proof.
    • We shall usually give both a modern approach to solving the problem as well as a comment on Alcuin's solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • It would appear from the surviving manuscripts that Alcuin didn't give a solution to this puzzle.
    • Solution.
    • Again it would appear that Alcuin didn't give a solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • This is Alcuin's solution, although of course he writes it out in words rather than the symbols we have used.
    • See if you can find a solution with only 9 crossings.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Alcuin's solution appears simply wrong unless the statement of the puzzle has been corrupted..
    • Solution.
    • Solution.
    • Solution.
    • Of course, as well as the incorrect formula for the area of the field, this solution has another problem, namely that you can't fit that many houses into the field.
    • Solution.
    • This problem is similar to the previous one and Alcuin's solution has all the same errors in it.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • These are the only 5 possible solutions with x, y, z all positive.
    • Solution.
    • These are the only 6 possible solutions with x, y, z all positive.
    • Solution.
    • The original problem that Alcuin is basing this puzzle on might be based on Islamic law or Alcuin may be assuming the reader will base the solution on Roman law.
    • However, his solution assumes that the each will receive the average of the two possibilities, namely: .
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Alcuin gives a good solution to this puzzle.
    • Solution.
    • Of course the problem has no solution since the sum of three odd numbers can never be even.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • Solution.
    • This is precisely Alcuin's solution.
    • Solution.
    • Solution.
    • Solution.
    • Alcuin gives a solution in which the camel makes three trips to a point 20 leagues from the start and moves the grain to here at a cost of 60 measures.
    • Solution.

  4. Doubling the cube
    • We next consider the solution proposed by Archytas.
    • This is a quite beautiful solution showing quite outstanding innovation by Archytas.
    • The solution by Archytas is the most remarkable of all, especially when his date is considered (first half of the fourth century BC), because it is not a plane construction but a bold construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution..
    • This is the solution of Archytas, reported by Eudemus: .
    • Through the writings of Eutocius, we know that Eudoxus also gave a solution to the problem of doubling the cube.
    • His solution is lost, however, since the version which Eutocius had in front of him was rather trivially incorrect and he therefore did not reproduce it.
    • Nobody believes that Eudoxus had an elementary error in his solution (he was far too good a mathematician for that) so the error must have been an error introduced when his solution was copied by someone who did not understand it properly.
    • Paul Tannery suggested that Eudoxus's solution was a two-dimensional version of the one given by Archytas which we have just described, in effect the solution obtained by projecting Archytas's construction onto a plane.
    • too original a mathematician to content himself with a mere adaptation of Archytas's method of solution.
    • Menaechmus's solution to finding two mean proportionals is described by Eutocius in his commentary to Archimedes' On the sphere and cylinder.
    • Menaechmus gave two solutions.
    • Of course we must emphasis again that this in no way indicates the way that Menaechmus solved the problem but it does show in modern terms how the parabola and hyperbola enter into the solution to the problem.
    • 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.
    • One of the great puzzles concerning the solution of the problem of doubling the cube is that there is a mechanical solution known as Plato's machine.
    • Now it seems highly unlikely that Plato would give a mechanical solution, particularly given his views on such solutions.
    • One theory is that Plato invented the mechanical solution to show how easy it is to devise such solutions, but the more widely held theory is that Plato's machine was invented by one of his followers at the Academy.
    • He erected a column at Alexandria dedicated to King Ptolemy with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [A history of Greek mathematics I (Oxford, 1931).',2)">2]:- .
    • Other solutions to the problem were by Philon and Heron who both gave similar methods.
    • Their solution is effectively produced by the intersection of a circle and a rectangular hyperbola.
    • Nicomedes, who was highly critical of Eratosthenes mechanical solution, gave a construction which used the conchoid curve which he also used to solve the problem of trisection of an angle.
    • Although these many different methods were invented to double the cube and remarkable mathematical discoveries were made in the attempts, the ancient Greeks were never going to find the solution that they really sought, namely one which could be made with a ruler and compass construction.

  5. Brachistochrone problem
    • Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument.
    • If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.
    • Perhaps we are reading too much into Johann Bernoulli's references to Pascal and Fermat, but it interesting to note that Pascal's most famous challenge concerned the cycloid, which Johann Bernoulli knew at this stage to be the solution to the brachistochrone problem, and his method of solving the problem used ideas due to Fermat.
    • Leibniz persuaded Johann Bernoulli to allow a longer time for solutions to be produced than the six months he had originally intended so that foreign mathematicians would also have a chance to solve the problem.
    • Five solutions were obtained, Newton, Jacob Bernoulli, Leibniz and de L'Hopital solving the problem in addition to Johann Bernoulli.
    • Newton sent his solution to Charles Montague, the Earl of Halifax, who was an innovative finance minister and the founder of the Bank of England.
    • He was President of the Royal Society during the years 1695 to 1698 so it was natural that Newton send him his solution to the brachistochrone problem.
    • The Royal Society published Newton's solution anonymously in the Philosophical Transactions of the Royal Society in January 1697.
    • His solution was explained to Montague as follows:- .
    • Solution.
    • The May 1697 publication of Acta Eruditorum contained Leibniz's solution to the brachistochrone problem on page 205, Johann Bernoulli's solution on pages 206 to 211, Jacob Bernoulli's solution on pages 211 to 214, and a Latin translation of Newton's solution on page 223.
    • The solution by de L'Hopital was not published until 1988 when, nearly 300 years later, Jeanne Peiffer presented it as Appendix 1 in [Die Gesammelten Werke der Mathematiker und Physiker der Familie Bernoulli (Basel, 1988).',1)" onmouseover="window.status='Click to see reference';return true">1].
    • Johann Bernoulli's solution divides the plane into strips and he assumes that the particle follows a straight line in each strip.
    • Huygens had shown in 1659, prompted by Pascal's challenge about the cycloid, that the cycloid is the solution to the tautochrone problem, namely that of finding the curve for which the time taken by a particle sliding down the curve under uniform gravity to its lowest point is independent of its starting point.
    • Johann Bernoulli ended his solution of the brachistochrone problem with these words:- .
    • Despite the friendly words with which Johann Bernoulli described his brother Jacob Bernoulli's solution to the brachistochrone problem (see above), a serious argument erupted between the brothers after the May 1697 publication of Acta Eruditorum.
    • Euler, however, commented that his geometrical approach to these problems was not ideal and it only gave necessary conditions that a solution has to satisfy.
    • The question of the existence of a solution was not solved by Euler's contribution.
    • The solution was found by considering special cases, and it was only some time later, in research isoperimetric curves, that the great mathematician of whom we speak and his famous brother Jacob Bernoulli gave some general rules for solving several other problems of the same type.

  6. Orbits
    • However Newton later said that an exact solution for three bodies .
      Go directly to this paragraph
    • Even in this form the problem does not lead to exact solutions.
      Go directly to this paragraph
    • Euler, however, found a particular solution with all three bodies in a straight line.
      Go directly to this paragraph
    • Lagrange submitted Essai sur le probleme des trois corps in which he showed that Euler's restricted three body solution held for the general three body problem.
      Go directly to this paragraph
    • He also found another solution where the three bodies were at the vertices of an equilateral triangle.
      Go directly to this paragraph
    • Lagrange considers his solutions do not apply to the solar system but we now know the both the Earth and Jupiter have asteroids sharing their orbits in the equilateral triangle solution configuration discovered by Lagrange.
      Go directly to this paragraph
    • Laplace's work of 1787, that of Adams of 1854 and later Delaunay's work described below eventually provided solutions.
      Go directly to this paragraph
    • Bessel also proposed this solution to the problem but died before completing his calculations.
      Go directly to this paragraph
    • He treated it as a restricted three body problem and used transformations to produce infinite series solutions for the longitude, latitude and parallax for the Moon.
      Go directly to this paragraph
    • He discussed convergence and uniform convergence of the series solutions discussed by earlier mathematicians and proved them not to be uniformly convergent.
      Go directly to this paragraph
    • He conjectured that there are infinitely many periodic solutions of the restricted problem, the conjecture being later proved by Birkhoff.
      Go directly to this paragraph

  7. Bakhshali manuscript
    • The Bakhshali manuscript is a handbook of rules and illustrative examples together with their solutions.
    • The solution to the example is then given and finally a proof is set out.
    • To illustrate we give the following indeterminate problem which, of course, does not have a unique solution:- .
    • The solution, translated into modern notation, proceeds as follows.
    • We seek integer solutions x1, x2, x3 and k (where x1 is the price of an asava, x2 is the price of a haya, and x3 is the price of a horse) satisfying .
    • For integer solutions k - (x1 + x2 + x3) must be a multiple of the lcm of 4, 6 and 7.
    • This is the indeterminate nature of the problem and taking different multiples of the lcm will lead to different solutions.
    • This is not the minimum integer solution which would be k = 131.
    • so we obtain integer solutions by taking k = 131 which is the smallest solution.
    • This solution is not given in the Bakhshali manuscript but the author of the manuscript would have obtained this had he taken k - (x1 + x2 + x3) = lcm(4, 6, 7) = 84.
    • Here is another equalisation problem taken from the manuscript which has a unique solution:- .

  8. Trisecting an angle
    • Later, however, they trisected the angle by means of the conics, using in the solution the verging described below ..
    • Now one of the reasons why the problem of trisecting an angle seems to have attracted less in the way of reported solutions by the best ancient Greek mathematicians is that the construction above, although not possible with an unmarked straight edge and compass, is nevertheless easy to carry out in practice.
    • A mechanical type of solution is easily found.
    • So as a practical problem there was little left to do although the Greeks still were not satisfied in general with mechanical solutions from a purely mathematical point of view they did not find them.
    • There is another mechanical solution given by Archimedes.
    • Pappus gives two solutions which both involve the drawing of a hyperbola.
    • The passage from Pappus from which this solution is taken is remarkable as being one of three passages in Greek mathematical works still extant ..
    • These constructions described by Pappus show how the Greeks 'improved' their solutions to the problem of trisecting an angle.
    • From a mechanical solution they had progressed to a solution involving conic sections.
    • They could never progress to plane solutions since we know that such are impossible.

  9. Weather forecasting
    • Basically, this method allows finding approximate solutions to differential equations.
    • All the different methods need to be stable, in other words, it has to be guaranteed that the numerical solution does not diverge from the true solution as the time span for which the forecast is made increases.
    • 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.
    • 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.
    • As a result, the solutions of many problems are very accurate.
    • 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.

  10. Tartaglia versus Cardan
    • Tartaglia: I cannot do that, because as soon as he shall have one of the said cases with its solution, his excellency will at once understand the rule discovered by me, with which many other rules may perhaps be found, based on the same material.
    • I know them by the two last, because a similar one to the seventh he sent me two years ago, and I made him confess that he did not understand the same, and a similar one to that last (which induces an operation of the square and cube equal to a number) I gave him out of courtesy solved, not a year ago, and for that solution I found a rule specially bearing upon such problems.
    • I send you two questions with their solutions, but the solutions shall be separate from the questions, and the messenger will take them with him; and if you cannot solve the questions he will place the solutions in your hand.
    • In addition to this, be pleased to send me the propositions offered by you to Master Antonio Maria Fior, and if you will not send me the solutions, keep them by you, they are not so very precious.
    • And if it should please you, in receiving the solutions of my said questions - should you yourself be unable to solve them, after you have satisfied yourself that my first six questions are different in kind - to send me the solution of any one of them, rather for friendship's sake, a for a test of your great skill, than for any other purpose, you will do me a very singular pleasure.
    • 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".

  11. Squaring the circle
    • This is really asking whether squaring the circle is a 'plane' problem in the terminology of Pappus given above (we shall often refer to a 'plane solution' rather than use the more cumbersome 'solutions using ruler and compass").
    • The ancient Greeks, however, did not restrict themselves to attempting to find a plane solution (which we now know to be impossible), but rather developed a great variety of methods using various curves invented specially for the purpose, or devised constructions based on some mechanical method.
    • Oenopides is thought by Heath to be the person who required a plane solution to geometry problems.
    • It only led to a greater flood of amateur solutions to the problem of squaring the circle and in 1775 the Paris Academie des Sciences passed a resolution which meant that no further attempted solutions submitted to them would be examined.
    • A few years later the Royal Society in London also banned consideration of any further 'proofs' of squaring the circle as large numbers of amateur mathematicians tried to achieve fame by presenting the Society with a solution.
    • 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.

  12. Fair book
    • Although no 'common rule' is specified, the solution tells us that the rule gives the tonnage as x .
    • Although no words appear in the solution, we see that the tonnage, found from log tables, is 651.5.
    • Here Walker produces a strange method of solution.
    • The answer is correct but an extremely lengthy and difficult method to obtain the solution.
    • It is worth noting that some (but not all) of the solutions have a tick to the left of the question.
    • The solution is carried out with two applications of Heron's formula to the triangles ABD and BCD.
    • No solution given.
    • No solution is given.
    • The cask is drawn but no calculations or solution are given.
    • No solution is worked out.
    • At this point the solution terminates and no answer is given.
    • The solution to this question again terminates and no answer is given.

  13. Quadratic etc equations
    • 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.
      Go directly to this paragraph
    • 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.
      Go directly to this paragraph
    • Cardan invited Tartaglia to visit him and, after much persuasion, made him divulge the secret of his solution of the cubic equation.
      Go directly to this paragraph
    • Here, in modern notation, is Cardan's solution of x3 + mx = n.
      Go directly to this paragraph
    • so if a and b satisfy 3ab = m and a3 - b3 = n then a - b is a solution of x3 + mx = n.
    • Then x = a - b is the solution to the cubic.
    • 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.
      Go directly to this paragraph
    • Here, again in modern notation, is Ferrari's solution of the case: x4 + px2 + qx + r = 0.
      Go directly to this paragraph
    • 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.
      Go directly to this paragraph
    • One of the most elementary to us, yet showing a marked improvement in understanding, was the observation that if x = b, x = c, x = d are solutions of a cubic then the cubic is .
      Go directly to this paragraph

  14. Fair book insert
    • The problems on this page all give solutions in square links.
    • The first solution given to this problem is incorrect but then the correct solution is given using the rule area = (a + b)h/2 where a, b are the lengths of the parallel sides and h is the perpendicular distance between them.
    • [We have corrected this problem which is incorrectly gives the diagonal as AB.] The solution just computes the areas of the two triangles ABC and ADC.
    • The method of solution here is to solve triangles, one after the other, using b = a sin B/sin A, all done with six-figure logs.
    • It is the first time that a method of solution is given, all the previous problems being solved by use of a rule (there is no evidence that the rule is thought of in terms of a formula, rather it seems to be thought of as a procedure which has been learnt by heart).
    • The x here is my invention - it is not used in the solution where he has calculated that 36 goes in the space where I have the x.
    • Various errors are made in the solution.
    • There then follows a problem which is not stated but the solution clearly indicates that the problem was to solve a right angled triangle where the two shorter sides are 500 and 480.
    • Solution given using methods as above is correct hypotenuse is 693.1 and the angles are 43°50 and 46°10.

  15. Mathematical games
    • In fact Lucas (the inventor of the Towers of Hanoi) gives a pretty solution to Cardan's Ring Puzzle using binary arithmetic.
      Go directly to this paragraph
    • Tartaglia, who with Cardan jointly discovered the algebraic solution of the cubic, was another famous inventor of mathematical recreations.
      Go directly to this paragraph
    • He invented many arithmetical problems, and contributed to problems with weighing masses with the smallest number of weights and Ferry Boat type problems which now have solutions using graph theory.
      Go directly to this paragraph
    • Ozanam and Montucla quote the solutions of both De Moivre and Montmort.
      Go directly to this paragraph
    • There is a unique solution (up to symmetry) to the 6 × 6 problem and the puzzle, in the form of a wooden board with 36 holes into which pins were placed, was sold on the streets of London for one penny.
      Go directly to this paragraph
    • Solutions for n = 9, 15, 27 were given in 1850 and much work was done on the problem thereafter.
      Go directly to this paragraph
    • This problem was shown by computer to have exactly 65 solutions in 1958.

  16. Word problems
    • Here there are above all three fundamental problems whose solution is very difficult and which will not be possible without a penetrating study of the subject.
    • Each knotted space curve, in order to be completely understood, demands the solution of the three above problems in a special case.
    • He used his solution to show that right and left trefoils are distinct.
    • He published these results in 1927 and at the same time gave a simple rigorous proof of the solution of the word problem in a free group.
    • The solution to the word problem for these groups began with Dehn who stated the Freiheitssatz: .
    • In the following year Magnus published a paper containing a special case of the word problem for 1-relator groups, then in 1932 he published a complete proof of the solution of the word problem for this class of groups.
    • It required computability theory and developments in mathematical logic to even make the questions precise, but these areas were to not only provide explicit questions, they also provided solutions to the questions.
    • While sitting in the dentist's chair waiting for this unpleasant experience, inspiration struck and suddenly he saw the route to the solution.

  17. Longitude2
    • Hooke, therefore, like almost all scientists of that time was a biased judge of longitude solutions since he hoped to solve the problem himself.
    • Jonas Moore, although keen to see a solution of the longitude problem, seems to have seen his role as making it possible for others to solve it rather than himself.
    • More and more pressure was mounting for a solution to the longitude problem as the continuing failure to solve it was costing England vast sums of money.
    • Everyone believed that mathematicians and astronomers would provide the solution but it is not to be.
    • every hour, on the dot, immerse the bandage in a solution of the powder of sympathy and the dog on shipboard would yelp the hour.
    • James Bradley, who had succeeded Halley as Astronomer Royal in 1742, and Tobias Mayer were convinced that the lunar distance method would lead to the solution of the longitude problem.
      Go directly to this paragraph

  18. Debating topics
    • Does the equation ax = b always have a solution? Do quadratic, cubic and quartic equations always have solutions? .
    • The equation x2 + 1 = 0 has no real number solution.
    • Let i be a symbol representing its solution.
    • Do we need to introduce negative numbers to get solutions of such equations? .

  19. function concept
    • It was a concept whose introduction was particularly well timed as far as Johann Bernoulli was concerned for he was looking at problems in the calculus of variations where functions occur as solutions.
    • In 1746 d'Alembert published a solution to the problem of a vibrating stretched string.
    • The solution, of course, depended on the initial form of the string and d'Alembert insisted in his solution that the function which described the initial velocities of the each point of the string had to be E-continuous, that is expressed by a single analytic expression.
    • 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.

  20. Kepler's Planetary Laws
    • Kepler's principal aim was to find a solution that would satisfy observations - and in that respect he possessed the outlook of a modern scientist.
    • And because it was expressed geometrically, the solution would potentially be exact - the closed orbit of a single planet in a plane round the fixed Sun.
    • This is the process that was described (in Section 4) as idealization because it ensured an exact solution (of the one-body problem) which was uniquely simple.
    • The solutions reached in each case are in some senses provisional, but they are certainly vital steps on the way to the presentday solution.

  21. Quadratic etc equations references
    • 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.
    • 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.

  22. The Scottish Book
    • From items at the end of this collection, it will be seen that some Russian mathematicians must have visited the town; they left several problems (and prizes for their solutions).
    • Many of the problems have since found their solution, some in the form of published papers.
    • (I know of some of my own problems, solutions to which were published in periodicals, among them, e.g.
    • I should be grateful if the recipients of this collection were willing to point out errors, supply information about solution to problems, or indicate developments contained in recent literature in topics connected with the subjects discussed in the problems.

  23. Quadratic etc equations references
    • 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.
    • 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.

  24. Egyptian Papyri
    • The method of solution was to "get rid of" the fractions by multiplying through.
    • Now the answer to the red auxiliary equation is 4 so the original equation had solution twice × (twice × 1/15).
    • Doubling this gives 1/5 + 1/15 which is the required solution to Problem 21.
    • Finally Ahmes checks his solution, or proves his answer is correct.
    • As a final look at the Rhind papyrus let us give the solution to Problem 50.
    • What is its area? Here is the solution as given by Ahmes.
    • Notice that the solution is equivalent to taking π = 4(8/9)2 = 3.1605.

  25. Fund theorem of algebra
    • 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
    • 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 .
      Go directly to this paragraph
    • However he does not assert that solutions are of the form a + bi, a, b real, so allows the possibility that solutions come from a larger number field than C.
      Go directly to this paragraph

  26. Longitude1
    • A great solution if one were able to determine where land was relative to the line .
    • The method is theoretically correct but Werner had not solved the longitude problem since the cross-staff could not make accurate enough measurements, and more seriously there was no mathematical theory of the Moons orbit (and even when Newton gave his theory of gravitation 150 years later the Moon's motion, a three body problem, was beyond solution).
      Go directly to this paragraph
    • The position of the Spice Islands was in dispute and Spain sought a solution to these costly problems.
      Go directly to this paragraph
    • The fact that no solution to this problem had been found was costing countries vast sums of money.
    • A solution had to be found, so countries began to adopt the standard method, namely to offer money, prizes, pensions, wealth beyond belief to mathematicians and astronomers who could give a method to find the longitude at sea.
    • The Academie Royale was desperate to examine every chance for a solution and money was no problem.
      Go directly to this paragraph
    • The members of the Academie Royale des Sciences made observations of the Moon over the years 1667 to 1669 which convinced them that the mathematics of the position of the Moon was too difficult to make it useful as a solution to the longitude problem.
      Go directly to this paragraph

  27. Babylonian mathematics
    • In this article we now examine some algebra which the Babylonians developed, particularly problems which led to equations and their solution.
    • When a solution was found for y then x was found by x = by/a.
    • The solution given by the scribe is to compute 0; 40 times 0; 40 to get 0; 26, 40.
    • The form that their solutions took was, respectively .

  28. Quantum mechanics history
    • In 1896 Wilhelm Wien proposed a solution to the Kirchhoff challenge.
      Go directly to this paragraph
    • However although his solution matches experimental observations closely for small values of the wavelength, it was shown to break down in the far infrared by Rubens and Kurlbaum.
      Go directly to this paragraph
    • The year 1926 saw the complete solution of the derivation of Planck's law after 26 years.
      Go directly to this paragraph
    • Dirac, in 1928, gave the first solution of the problem of expressing quantum theory in a form which was invariant under the Lorentz group of transformations of special relativity.
      Go directly to this paragraph
    • However Niels Bohr had the final triumph, for the next day he had the solution.
      Go directly to this paragraph

  29. Planetary motion
    • Nowadays astronomers accept that planetary motion has to be treated dynamically, as a many-body problem, for which there is bound to be no exact solution.
    • The astronomical solution to the one-body problem consists of the two laws: .
    • This composite solution represents what is in fact the earliest instance of a planetary orbit: it will be succinctly referred to in what follows as 'the Sun-focused ellipse'.
    • We shall now prove that, subject to its obvious external limitations, this unique solution is of universal applicability as a self-contained piece of mathematics.
    • Moreover, the kinematical solution is qualitatively different from any later, dynamical one in that it possesses exact geometrical representation - while the adoption of the auxiliary angle as variable ensures that the treatment turns out to be the simplest possible.

  30. Greek sources II
    • Although Archimedes promises a solution later in his text, it does not appear.
      Go directly to this paragraph
    • Eutocius quotes from a solution by Diocles of this problem.
      Go directly to this paragraph
    • A number of different solutions have been proposed but this is leading us away from the question of dating which we are discussing in this article.

  31. Neptune and Pluto
    • Strangely Airy, who now knew that both Adams and Le Verrier had come to almost identical solutions to the same problem, did not tell either of them about the other, nor did he tell Le Verrier of his plans to begin a search.
      Go directly to this paragraph
    • His first solution had depended on assuming a distance for the "new planet" of twice that of Uranus from the Sun.
      Go directly to this paragraph
    • He was unhappy with this arbitrary part of his solution and he had redone the mathematical analysis finding a better estimate of the distance of the "new planet" by testing different distances against the observed perturbations of Uranus.
      Go directly to this paragraph

  32. Topology history
    • In 1736 Euler published a paper on the solution of the Konigsberg bridge problem entitled Solutio problematis ad geometriam situs pertinentis which translates into English as The solution of a problem relating to the geometry of position.
      Go directly to this paragraph
    • involved looking at the totality of all solutions rather than at particular trajectories as had been the case earlier.
      Go directly to this paragraph

  33. General relativity
    • Many possible solutions were proposed, Venus was 10% heavier than was thought, there was another planet inside Mercury's orbit, the sun was more oblate than observed, Mercury had a moon and, really the only one not ruled out by experiment, that Newton's inverse square law was incorrect.
      Go directly to this paragraph
    • 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.
      Go directly to this paragraph
    • At the time this was purely theoretical work but, of course, work on neutron stars, pulsars and black holes relied entirely on Schwarzschild's solutions and has made this part of the most important work going on in astronomy today.
      Go directly to this paragraph

  34. 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

  35. Classical time
    • During the 16th century the solution of problems relating to time became of utmost importance because of its relation to finding the longitude.
    • In an age of exploration on a world scale, determining position became a crucial problem and much effort was put into its solution.
    • Theoretically this provided a solution to the longitude problem, but in practice observing the eclipses of Jupiter's moons from the deck of a ship was essentially impossible.
    • Several large prizes were offered for a solution to the problem of determining longitude and Galileo tried the persuade the Spanish Court in 1616 that he could determine absolute time using Jupiter's moons and, after failing to convince them, tried to persuade Holland of his method when they offered a large prize in 1636.

  36. The four colour theorem
    • De Morgan kept asking if anyone could find a solution to Guthrie's problem and several mathematicians worked on it.
      Go directly to this paragraph
    • However the final ideas necessary for the solution of the Four Colour Conjecture had been introduced before these last two results.
      Go directly to this paragraph
    • The year 1976 saw a complete solution to the Four Colour Conjecture when it was to become the Four Colour Theorem for the second, and last, time.
      Go directly to this paragraph

  37. Water-clocks
    • The solution to the first problem was to have the klepsydra supplied from a large reservoir of water kept at a constant level.
    • Vitruvius gives several solutions, the simplest being as follows [Vitruvius : ten books on architecture / translation [from the Latin] by Ingrid D Rowland ; commentary and illustrations by Thomas Noble Howe ; with additional commentary by Ingrid D Rowland and Michael J Dewar, ed.

  38. Matrices and determinants
    • from which the solution can be found for the third type of corn, then for the second, then the first by back substitution.
    • had a solution because .
    • 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.
      Go directly to this paragraph

  39. History overview
    • 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
    • There is no reason why anyone should introduce negative numbers just to be solutions of equations such as x + 3 = 0.

  40. Chinese overview
    • Perhaps it is most famous for presenting the 'Hundred fowls problem' which is an indeterminate problem with three non-trivial solutions.
    • 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.

  41. Fermat's last theorem

  42. Indian mathematics
    • 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.
    • Citrabhanu was a sixteenth century mathematicians from Kerala who gave integer solutions to twenty-one types of systems of two algebraic equations.

  43. EMS History
    • Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussion of new problems or new solutions, and comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.
    • (Professor of Mathematics in the University of Edinburgh), on The Solution of Algebraic and Transcendental Equations in the Mathematical Laboratory.

  44. Forgery 1
    • If this is so then he had not lost the ability to produce mathematics of the highest quality since his solution of the problem to determine the number of conics tangent to five given conics which he found in 1864 was remarkable, particularly as it corrected a previous incorrect solution by the outstanding mathematician Steiner.

  45. Pell's equation references
    • 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 xŰ - nyŰ = 1 in integers (Russian), Istor.-Mat.

  46. Burnside problem references
    • I N Sanov, Solution of Burnside's problem for n = 4, Leningrad State University Annals (Uchenyi Zapiski) Math.
    • M Hall Jr., Solution of the Burnside Problem for Exponent Six, Illinois J.

  47. Pell's equation references
    • 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 xŰ - nyŰ = 1 in integers (Russian), Istor.-Mat.

  48. Harriot's manuscripts
    • He confessed that he had been listening to his master and a guest arguing about the solution to a mathematical problem.
    • Both, he politely pointed out, were wrong in their opinions and he proceeded to explain the correct solution.

  49. Tait's scrapbook
    • The Scrapbook contains information about Tait's solution of the 15 puzzle.
    • Tait had solved the puzzle and submitted a paper on the solution when he saw two articles in the American Journal of Mathematics in 1879, one by W W Johnson and one by W E Story.

  50. Burnside problem
    • If the Restricted Burnside Problem has a positive solution for some m, n then we may factor B(m, n) by the intersection of all subgroups of finite index to obtain B0(m,n), the universal finite m-generator group of exponent n having all other finite m-generator groups of exponent n as homomorphic images.
    • 1994Zelmanov was awarded a Fields medal for his positive solution of the Restricted Burnside Problem.

  51. Ring Theory
    • The equation xn+ yn= zn has no solution for positive integers x, y, z when n > 2.
    • In 1847 Lame announced that he had a solution of Fermat's Last Theorem and sketched out a proof.

  52. Nine chapters
    • Essentially linear equations are solved by making two guesses at the solution, then computing the correct answer from the two errors.
    • Then the correct solution is .

  53. Babylonian Pythagoras
    • ',17)">17], claims that the tablet is connected with the solution of quadratic equations and has nothing to do with Pythagorean triples:- .
    • 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.

  54. Burnside problem references
    • I N Sanov, Solution of Burnside's problem for n = 4, Leningrad State University Annals (Uchenyi Zapiski) Math.
    • M Hall Jr., Solution of the Burnside Problem for Exponent Six, Illinois J.

  55. Ten classics
    • In fact the solution given, although in a special case, gives exactly the modern method.

  56. 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.

  57. Indian Sulbasutras
    • This has the solution x = 1/(3 × 4 × 34) which is approximately 0.002450980392.

  58. Golden ratio
    • multiplied the number of propositions concerning the section which had their origin in Plato, employing the method of analysis for their solution.

  59. Gravitation
    • There was no simple solution to the problems that the different theories posed.

  60. Real numbers 1
    • He then went on to compute an approximate solution.

  61. Cartography
    • Al-Biruni wrote a textbook on the general solution of spherical triangles around 1000 then, some time after 1010, he applied these methods on spherical triangles to geographical problems.

  62. Set theory
    • 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.
      Go directly to this paragraph

  63. 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

  64. Knots and physics
    • Its solution will be the proposed new kinetic theory of gases.

  65. Egyptian mathematics
    • Some problems ask for the solution of an equation.

  66. 20th century time
    • My solution was really for the very concept of time, that is, that time is not absolutely defined but there is an inseparable connection between time and the velocity of light.

  67. Ledermann interview
    • He went over the problems from the previous week and very seldom, when somebody had given a really good solution, he was called on to go to the blackboard and show it to the other people.

  68. Measurement
    • Diderot and d'Alembert in their Encyclopedie greatly regretted the diversity, but saw no possible acceptable solution to the problem.

  69. Infinity
    • Fermat used his method to prove that there were no positive integer solutions to .

  70. Group theory
    • 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

  71. Real numbers 2
    • 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.

  72. Squaring the circle references
    • T Albertini, La quadrature du cercle d'ibn al-Haytham : solution philosophique ou mathematique?, J.

  73. Chrystal and the RSE
    • Scientifically speaking, uneducated themselves, they seem to think that they will catch the echo of the fact or the solution of an arithmetical problem by putting their ears to the sounding-shell of uneducated public opinion.

  74. 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.

  75. Brachistochrone problem references
    • H Erlichson, Johann Bernoulli's brachistochrone solution using Fermat's principle of least time, European J.

  76. Brachistochrone problem references
    • H Erlichson, Johann Bernoulli's brachistochrone solution using Fermat's principle of least time, European J.

  77. Cosmology
    • This solution implied that the Universe had been born at one moment, about ten thousand million years ago in the past and the galaxies were still travelling away from us after that initial burst.
      Go directly to this paragraph

  78. Indian Sulbasutras references
    • A E Raik and V N Ilin, A reconstruction of the solution of certain problems from the Apastamba Sulbasutra of Apastamba (Russian), in A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin, Studies in the history of mathematics 19 "Nauka" (Moscow, 1974), 220-222; 302.

  79. Squaring the circle references
    • T Albertini, La quadrature du cercle d'ibn al-Haytham : solution philosophique ou mathematique?, J.

  80. Real numbers 3
    • Solutions proposed by some mathematicians would only allow mathematics to treat objects which could be constructed.

  81. Brachistochrone problem references
    • H Erlichson, Johann Bernoulli's brachistochrone solution using Fermat's principle of least time, European J.

  82. Indian Sulbasutras references
    • A E Raik and V N Ilin, A reconstruction of the solution of certain problems from the Apastamba Sulbasutra of Apastamba (Russian), in A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin, Studies in the history of mathematics 19 "Nauka" (Moscow, 1974), 220-222; 302.

  83. 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.

  84. Babylonian numerals
    • How do we know this? Well if they had really found that the system presented them with real ambiguities they would have solved the problem - there is little doubt that they had the skills to come up with a solution had the system been unworkable.

  85. Newton's bucket
    • In 1918 Joseph Lense and Hans Thirring obtained approximate solutions of the equations of general relativity for rotating bodies.

  86. Poincaré - Inspector of mines
    • The Davy safety lamp, invented about 1815, provided the solution at the time Poincare worked.

  87. Mathematical classics
    • In fact the solution given, although in a special case, gives exactly the modern method.


Famous Curves

  1. Cycloid
    • Pascal published a challenge (not under his own name but under the name of Dettonville) offering two prizes for solutions to these problems.
    • Wallis and Lalouere entered but Lalouere's solution was wrong and Wallis was also not successful.
    • Pascal published his own solutions to his challenge problems together with an extension of Wren's result which he was able to find.
    • He already knew the brachistochrone property of the cycloid and published his solution in 1697.

  2. Involute
    • Finding a clock which would keep accurate time at sea was a major problem and many years were spent looking for a solution.

  3. Trisectrix
    • Like so many curves it was studied to provide a solution to one of the ancient Greek problems, this one is in relation to the problem of trisecting an angle.


Societies etc

  1. European Mathematical Society Prize
    • This problem deals with a variational problem with a singular boundary set, and proposes a finite representation of the optimum solution.
    • He succeeded in computing explicit solutions for "Asian options" where the pay-off is given by a time-average of geometric Brownian motion.
    • Thereby he gave a solution to a long-standing problem, open for more than 30 years, that has resisted the efforts of the greatest specialists of elliptic curves.
    • This last problem attracted the attention and efforts of many geometers for more than 20 years, and the method developed by Perelman yielded an astonishingly short solution.
    • Group invariant valuations were studied since Dehn's solution of Hilbert's third problem, with later contributions by Blaschke and others, and culminating in Hadwiger's celebrated characterization theory for the intrinsic volumes.
    • Further contributions include discovery of a functional form of isoperimetric inequalities and a recent solution (with Artstein, Ball and Naor) of a long-standing Shannon's problem on entropy production in random systems.
    • has introduced an entirely new perspective to the theory of discontinuous solutions of one-dimensional hyperbolic conservation laws, representing solutions as local superposition of travelling waves and introducing innovative Glimm functionals.
    • His ideas have led to the solution of the long standing problem of stability and convergence of vanishing viscosity approximations.
    • 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.
    • One of the earlier contributions is his surprising solution of the symplectic packing problem, completing work of Gromov, McDuff and Polterovich, showing that compact symplectic manifolds can be packed by symplectic images of equally sized Euclidean balls without wasting volume if the number of balls is not too small.
    • Answering affirmatively Melnikov's conjecture, Tolsa provides a solution of the Painleve problem in terms of the Menger curvature.

  2. Young Mathematician prize
    • for a study of short-wave asymptotics of the solution of the diffraction problem on a convex cylinder.
    • for work on the stability of solutions of operator Hamiltonian equations with periodic coefficients.
    • for works on the properties of a solution of non-homogeneous Cauchy-Riemann system.
    • for giving a solution of J-P Serre's problem.
    • for giving periodic solutions in control systems.
    • for finite-gap and isodromic solutions of equations of nonlinear Schrodinger type.
    • for giving fast algorithms for polynomial expansions and solutions of algebraic systems.
    • for work in representation theory for solutions of the Young-Baxter equation.
    • for work on regularity of solutions of some problems in mechanics.

  3. 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".
    • He is one of the founders of the modern theory of transformation groups and is particularly known for his contributions to the solution of Hilbert's fifth problem.
    • for his paper "Solution in the large for nonlinear hyperboic systems of conservation laws".
    • 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".
    • Methods for exact solution".

  4. Clay Award
    • 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.
    • for their work on local and global Galois representations, partly in collaboration with Clozel and Shepherd-Barron, culminating in the solution of the Sato-Tate conjecture for elliptic curves with non-integral j-invariants.
    • 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.
    • for their solutions of the Marden Tameness Conjecture, and, by implication through the work of Thurston and Canary, of the Ahlfors Measure Conjecture.

  5. AMS Bôcher Prize
    • for his memoirs "Green's function and the problem of Plateau", "The most general form of the problem of Plateau", and "Solution of the inverse problem of the calculus of variations".
    • for his solution of several outstanding problems in diffraction theory and scattering theory and for developing the analytical tools needed for their resolution.
    • 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 profound contributions toward understanding the structure of singular sets for solutions of variational problems.

  6. Paris Academy of Sciences
    • The 1857 prize was offered for a solution to Fermat's Last Theorem and, not surprisingly, no solutions were submitted even when the deadline was extended.

  7. Norwegian Mathematical Society
    • That problem found a temporary solution when Heegaard succeeded in obtaining funds for a series of pamphlets, Norsk matematisk forenings skrifter ..
    • For many years starting in 1922, Crown Prince Olav awarded a prize for the best solutions to a series of problems posed in the Journal.

  8. Sylvester Medal
    • for his solution of several outstanding problems of algebraic topology and of the methods he invented for this purpose which have proved of prime importance in the theory of the subject.
    • for his many contributions to number theory and in particular his solution of the famous problem concerning approximating algebraic numbers by rationals.

  9. AMS Fulkerson Prize
    • for 'Informational complexity and effective methods of solution for convex extremal problems', Ekonomika i Matematicheskie Metody 12 (1976), 357-369.
    • for 'The solution of van der Waerden's problem for permanents', Akademiia Nauk SSSR.

  10. NAS Award in Applied Mathematics
    • for his innovative and imaginative use of mathematics in the solution of a wide variety of challenging and significant scientific and engineering problems.
    • for his profound and penetrating solution of outstanding problems of statistical mechanics.

  11. Rolf Schock Prize
    • .for his outstanding work in mathematical physics, particularly for his contribution to the mathematical understanding of the quantum-mechanical many-body theory and for his work on exact solutions of models in statistical mechanics and quantum mechanics.

  12. Abel Prize
    • for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.

  13. AMS/SIAM Birkhoff Prize
    • for his contributions to the theory of Hamiltonian dynamical systems, especially his proof of the stability of periodic solutions of Hamiltonian systems having two degrees of freedom and his specific applications of the ideas in connection with this work.

  14. RSS Guy Medal in Gold
    • intended to encourage the cultivation of statistics in their scientific aspects and promote the application of numbers to the solution of important problems in all the relations of life in which the numerical method can be employed, with a view to determining the laws which regulate them.

  15. New York Mathematical Society
    • It is believed that the meetings of the Society may be rendered interesting by the discussion of mathematical subjects, the criticism of current mathematical literature, and solutions to problems proposed by its members and correspondents.

  16. 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.

  17. AMS Satter Prize
    • for her deep contributions to algebraic geometry, and in particular for her recent solutions to two long-standing open problems: the Kodaira problem ("On the homotopy types of compact Kahler and complex projective manifolds") and Green's Conjecture ("Green's canonical syzygy conjecture for generic curves of odd genus"; and "Green's generic syzygy conjecture for curves of even genus lying on a K3 surface".

  18. AMS Wiener Prize
    • 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.

  19. AMS Cole Prize in Algebra
    • for their solution of Abhyankar's conjecture.

  20. 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.

  21. Wilks Award of the ASS
    • His originality and ability to provide practical solutions to real-world statistical problems illuminate his extensive writings; a notable example is his classic text Survey Sampling, which is widely consulted and referenced by practitioners of statistics everywhere.

  22. History of the EMS
    • Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussions of new problems or new solutions, an comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.

  23. Serbian Academy of Sciences
    • The Minister of Education saw that the only solution was to merge the two, which he did in 1892.

  24. AMS Veblen Prize
    • for his work in differential geometry and, in particular, the solution of the four-dimensional Poincare conjecture.

  25. International Congress Speaker
    • Carlos Kenig, The Global Behavior of Solutions to Critical Non-linear Dispersive Equations .

  26. RSS Guy Medal in Bronze
    • intended to encourage the cultivation of statistics in their scientific aspects and promote the application of numbers to the solution of important problems in all the relations of life in which the numerical method can be employed, with a view to determining the laws which regulate them.

  27. Pioneer Prize
    • for his work on the applications of theoretical work in inverse problems to the solution of a wide range of industrial problems; for his promotion worldwide of industrial/applied mathematics problem solving; for his initiative to include very active applied mathematics components in the Austrian Mathematical Community; and for the founding of the Austrian Academy of Sciences sponsored RICAM, the Radon Institute for Computational and Applied Mathematics.

  28. RSS Guy Medal in Silver
    • intended to encourage the cultivation of statistics in their scientific aspects and promote the application of numbers to the solution of important problems in all the relations of life in which the numerical method can be employed, with a view to determining the laws which regulate them.

  29. Edinburgh Mathematical Society
    • Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussions of new problems or new solutions, a comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.

  30. BMC 1973

  31. BMC 1956
    • de Rham, G WElementary solutions of certain differential equations .

  32. BMC 2001
    • Bournaveas, N Low regularity solutions of the Klein-Gordon-Dirac equations .

  33. Scientific Committee 2007
    • Using hotels, or moving the BMC to September are possible solutions.

  34. BMC 1984
    • Read, C JA solution to the invariant subspace problem .

  35. Minutes for 2006
    • Edmund Robertson offered to come to York to find a reasonable financial solution of this question.

  36. BMC 2007
    • Qian, Z On strong solutions of the 3D-Navier-Stokes equations .

  37. BMC 1964
    • Lighthill, M JAsymptotic properties of Fourier integrals and of solutions of partial differential equations .

  38. BMC 2005
    • Maz'ya, V Unsolved mysteries of solutions to PDEs near the boundary .

  39. Mathematical Association of America
    • Most of our existing journals deal almost exclusively with subjects beyond the reach of the average student or teacher of mathematics or at least with subjects with which they are familiar, and little, if any, space, is devoted to the solution of problems É No pains will be spared on the part of the Editors to make this the most interesting and most popular journal published in America.

  40. BMC 2001


References

  1. References for Newton
    • E J Aiton, The solution of the inverse-problem of central forces in Newton's 'Principia', Arch.
    • J B Brackenridge, Newton's mature dynamics and the 'Principia' : a simplified solution to the Kepler problem, Historia Math.
    • H Erlichson, Newton's first inverse solutions, Centaurus 34 (4) (1991), 345-366.
    • H Erlichson, Newton's solution to the equiangular spiral problem and a new solution using only the equiangular property, Historia Math.
    • H Erlichson, Newton's 1679/80 solution of the constant gravity problem, Amer.
    • B Pourciau, Newton's solution of the one-body problem, Arch.

  2. References for Adleman
    • L Adleman, 'Molecular Computation of Solutions to Combinatorial Problems', Science, 266, November 11, 1994, pp.
    • R J Lipton, 'DNA Solution of Hard Computational Problems', Science, 268, April 28 1994, pp.

  3. References for Cardan
    • E Kenney, Cardano : 'Arithmetic subtlety' and impossible solutions, Philosophia Mathematica II (1989), 195-216.
    • E Kenney, Cardano : 'arithmetic subtlety' and impossible solutions, Philos.
    • C Romo Santos, Cardano's 'Ars magna' and the solutions of cubic and quartic equations (Spanish), Rev.

  4. References for Euler
    • H H Frisinger, The solution of a famous two-centuries-old problem : The Leonhard Euler Latin square conjecture, Historia Math.
    • E Knobloch, Leibniz and Euler : problems and solutions concerning infinitesimal geometry and calculus, Conference on the History of Mathematics (Rende, 1991), 293-313.

  5. References for Huygens
    • P Dupont and C S Roero, The treatise 'De ratiociniis in ludo aleae' of Christiaan Huygens, with the 'Annotationes' of Jakob Bernoulli ('Ars conjectandi', Part I), presented in an Italian translation, with historical and critical commentary and modern solutions (Italian), Mem.
    • E Shoesmith, Huygens' solution to the gambler's ruin problem, Historia Math.

  6. References for De Moivre
    • P Dupont, Critical elaboration of de Moivre's solutions of the 'jeu de rencontre' (Italian), Atti Accad.
    • P Dupont, On the 'gamblers' ruin' problem : critical review of the solutions of De Moivre and Todhunter of a classical example (Italian), Atti Accad.
    • A Hald, On de Moivre's solutions of the problem of duration of play, 1708-1718, Arch.

  7. References for Chazy
    • K G Atrokhov and V I Gromak, Solution of the Chazy system (Russian), Differ.
    • K G Atrokhov and V I Gromak, Solution of the Chazy system, Differ.

  8. References for Harriot
    • M Kalmar, Thomas Hariot's 'De reflexione corporum rotundorum' : an early solution to the problem of impact, Arch.
    • J V Pepper, Some clarifications of Harriot's solution of Mercator's problem, Hist.

  9. References for Schauder
    • W Forster, J Schauder : Fragments of a portrait, Numerical Solution of Highly Nonlinear Problems (Amsterdam, 1980), 417-425.
    • J Leray, My friend Julius Schauder, Numerical Solution of Highly Nonlinear Problems (Amsterdam, 1980), 427-439.

  10. References for Al-Haytham
    • T Albertini, La quadrature du cercle d'ibn al-Haytham : solution philosophique ou mathematique?, J.
    • A S Dallal, ibn al-Haytham's universal solution for finding the direction of the qibla by calculation, Arabic Sci.

  11. References for Al-Haytham
    • T Albertini, La quadrature du cercle d'ibn al-Haytham : solution philosophique ou mathematique?, J.
    • A S Dallal, ibn al-Haytham's universal solution for finding the direction of the qibla by calculation, Arabic Sci.

  12. References for Delamain
    • J Dawplucker, Critical retrospect of works on the sliding-rule, and various mathematical problems and solutions in the 'Mechanics' magazine', Mechanics' Magazine (Saturday, 4 September 1830).
    • J Dawplucker, Critical retrospect of works on the sliding-rule, and various mathematical problems and solutions in the 'Mechanics' magazine', Iron: An illustrated weekly journal for iron and steel manufacturers, metallurgists, mine proprietors, engineers, shipbuilders, scientists, capitalists 14 (1931), 5-6.

  13. References for Wallis
    • 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 xŰ - nyŰ = 1 in integers (Russian), Istor.-Mat.

  14. References for Forsythe
    • A S Householder, Review: Computer Solution of Linear Algebraic Systems by George E Forsythe and Cleve B Moler, Mathematics of Computation 24 (110) (1970), 482.
    • R L Johnson, Review: Computer Solution of Linear Algebraic Systems by George E Forsythe and Cleve B Moler, SIAM Review 10 (3) (1968), 384-385.

  15. References for Gini
    • G Favero, A Totalitarian Solution: Corrado Gini and Italian Economic Statistics.

  16. References for Kublanovskaya
    • N K Jain and K Singhal, On Kublanovskaya's approach to the solution of the generalized latent value problem for functional-matrices, SIAM J.

  17. 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.

  18. References for Todhunter
    • P Dupont, On the 'gamblers' ruin' problem: critical review of the solutions of De Moivre and Todhunter of a classical example (Italian), Atti Accad.

  19. References for Apastamba
    • A E Raik and V N Ilin, A reconstruction of the solution of certain problems from the Apastamba Sulba Sutra Apastamba (Russian), in A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin, Studies in the history of mathematics 19 'Nauka' (Moscow, 1974), 220-222; 302.

  20. References for Galileo
    • S Quan, Galileo and the problem of concentric circles : A refutation, and the solution, Ann.

  21. References for Plancherel
    • Sur la convergence en moyenne des suites de solutions d'une equation aux derivees partielles du second ordre lineaire et de type elliptique.

  22. References for Heun
    • A Ishkhanyan and K-A Suominen, New solutions of Heun's general equation, J.

  23. References for De Beaune
    • A B Stykan and O A Petrova, Solution of a problem that was presented to Descartes by de Beaune (Russian), Istor.-Mat.

  24. References for Al-Khwarizmi
    • Y Dold-Samplonius, Developments in the solution to the equation cxŰ + bx = a from al-Khwarizmi to Fibonacci, in From deferent to equant (New York, 1987), 71-87.

  25. References for Al-Samawal
    • Y Dold-Samplonius, The solution of quadratic equations according to al-Samaw'al, in Mathemata, Boethius : Texte Abh.

  26. References for Finkel
    • R F Davis, Review: A Mathematical Solution Book, by B F Finkel, The Mathematical Gazette 2 (41) (1903), 342-343.

  27. References for Sripati
    • R S Lal and R Prasad, Contributions of Sripati (1039 A.D.) in the solution of first degree simultaneous indeterminate equations, Math.

  28. References for Horner
    • M H Bektasova, From the history of numerical methods for the solution of equations (Russian), in Collection of questions on mathematics and mechanics No.

  29. References for Torricelli
    • J Krarup and S Vajda, On Torricelli's geometrical solution to a problem of Fermat, Duality in practice, IMA J.

  30. References for Banneker
    • B Lumpkin, From Egypt to Benjamin Banneker : African origins of false position solutions, in R Calinger (ed.), Vita mathematica (Washington, DC, 1996), 279-289.

  31. References for Leonardo
    • E B Thro, Leonardo da Vinci's solution to the problem of the pinhole camera, Arch.

  32. References for Qin Jiushao
    • Pai Shang-shu, An inquiry into the solution techniques of nine surveying problems raised by Chin Chiu-shao, in Discourses on the history of mathematics of the Sung and Yuan period (Peking 1966).

  33. References for Steiner
    • H Dorrie, One Hundred Great Problems of Elementary Mathematics, Their History and Solution (Dover, 1965).

  34. References for Leibniz
    • E Knobloch, Leibniz and Euler : problems and solutions concerning infinitesimal geometry and calculus, in Conference on the History of Mathematics (Rende, 1991), 293-313.

  35. References for Brioschi
    • Zappa, History of the solution of fifth- and sixth-degree equations, with an emphasis on the contributions of Francesco Brioschi (Italian), Rend.

  36. References for Bombelli
    • G S Smirnova, Geometric solution of the cubic equations in Raffaele Bombelli's 'Algebra' (Russian), Istor.

  37. References for Ceva Giovanni
    • A Brigaglia and P Nastasi, The solutions of Girolamo Saccheri and Giovanni Ceva to Ruggero Ventimiglia's 'Geometram quaero' : Italian projective geometry in the late seventeenth century (Italian), Arch.

  38. References for Faedo
    • M Benzi and E Toscano, Mauro Picone, Sandro Faedo, and the Numerical Solution of Partial Differential Equations in Italy (1928-1953), Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia.

  39. References for Democritus
    • D E Hahm, Chrysippus' solution to the Democritean dilemma of the cone, Isis 63 (217) (1972), 205-220.

  40. References for Crank
    • A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc.

  41. References for Courant
    • C A Felippa, 50 year classic reprint : an appreciation of R Courant's 'Variational methods for the solution of problems of equilibrium and vibrations', Internat.

  42. References for Mollweide
    • C N Mills, Discussions: On Checking the Solution of a Triangle, Amer.

  43. References for Rejewski
    • M Rejewski, Mathematical solution of the Enigma cipher, Cryptologia 6 (1) (1982), 1-18.

  44. References for Chrysippus
    • D E Hahm, Chrysippus' solution to the Democritean dilemma of the cone, Isis 63 (217) (1972), 205-220.

  45. References for Khayyam
    • P D Yardley, Graphical solution of the cubic equation developed from the work of Omar Khayyam, Bull.

  46. References for D'Adhemar
    • J Hadamard, Recherches sur les solutions fondamentales et l'integration des equations lineaires aux derivees partielles (deuxieme memoire), Annales Scientifiques de l'ENS, troisieme serie 22, (1905), 101-141.

  47. References for Cholesky
    • L Fox, H D Huskey and J H Wilkinson, Notes on the solutions of algebraic linear simultaneous equations, Quart.

  48. References for Al-Kashi
    • E M Bruins, Numerical solution of equations before and after al-Kashi, in Mathemata, Boethius : Texte Abh.

  49. References for Gauss
    • W Benham, The Gauss anagram : an alternative solution, Ann.

  50. References for Mikusinski
    • R Hilfer, Y Luchko and Z Tomovski, Operational method for the solution of fractional differential equations with generalised Riemann-Liouville fractional derivatives, Fractional Calculus and Applied Analysis 12 (3) (2009), 299-318.

  51. References for Mersenne
    • J MacLachlan, Mersenne's solution for Galileo's problem of the rotating earth, Historia Math.

  52. References for Nicolson
    • A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc.

  53. References for Seitz
    • B F Finkel, Biography: E B Seitz, in B F Finkel, A mathematical solution book (1893), 440-441.

  54. References for Dudeney
    • M Kobayashi, Kiyasu-Zen'iti and G Nakamura, A solution of Dudeney's round table problem for an even number of people, J.

  55. References for Banachiewicz
    • T Angelitch, The solution of systems of algebraic linear equations by the method of Banachiewicz (Serbo-Croat), Srpska Akad.

  56. References for Mahavira
    • B Datta, On Mahavira's solution of rational triangles and quadrilaterals, Bull.

  57. References for Von Staudt
    • E John Hornsby, Geometrical and Graphical Solutions of Quadratic Equations, The College Mathematics Journal 21 (5) (1990), 362-369.

  58. References for Saccheri
    • A Brigaglia and P Nastasi, The solutions of Girolamo Saccheri and Giovanni Ceva to Ruggero Ventimiglia's 'Geometram quaero' : Italian projective geometry in the late seventeenth century (Italian), Arch.

  59. References for Sridhara
    • K Shankar Shukla, On Sridhara's rational solution of Nx^2+1=y^2, Ganita 1 (1950), 1-12.

  60. References for Delaunay
    • 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.

  61. References for Roomen
    • P P A Henry, La solution de Francois Viete au probleme d'Adriaan van Roomen, Ecole Polytechnique Federale de Lausanne.


Additional material

  1. Finkel's Solution Book
    • B F Finkel's Mathematical Solution Book .
    • A mathematical solution book containing systematic solutions of many of the most difficult problems.
    • Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions.
    • This work is the outgrowth of eight years' experience in teaching in the Public Schools, during which time I have observed that a work presenting a systematic treatment of solutions of problems would be serviceable to both teachers and pupils.
    • Anyone who can write out systematic solutions of problems can resort to "Short Cuts" at pleasure; but, on the other hand, let a student who has done all his work in mathematics by formulae, "Short Cuts," and "Lightning Methods" attempt to write out a systematic solution - one in which the work explains itself - and he will soon convince one of his inability to express his thoughts in a logical manner.
    • One solution, thoroughly analysed and criticised by a class, is worth more than a dozen solutions the difficulties of which are seen through a cloud of obscurities.
    • It has been the aim to give a solution of every problem presenting anything peculiar, and those which go the rounds of the country.
    • In this edition I have added a chapter on Longitude and Time, the biographies of a few more mathematicians, several hundred more problems for solution, an introduction to the study of Geometry, and an introduction to the study of Algebra.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Finkel_solution.html .

  2. David Hilbert: 'Mathematical Problems
    • If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future.
    • It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.
    • It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.
    • The mathematicians of past centuries were accustomed to devote themselves to the solution of difficult particular problems with passionate zeal.
    • The fruitful methods and the far-reaching principles which Poincare has brought into celestial mechanics and which are today recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that difficult problem and to approach nearer a solution.
    • The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential - to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics.
    • But, in the further development of a branch of mathematics, the human mind, encouraged by the success of its solutions, becomes conscious of its independence.
    • It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem.
    • I should say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated.
    • While insisting on rigour in the proof as a requirement for a perfect solution of a problem, I should like, on the other hand, to oppose the opinion that only the concepts of analysis, or even those of arithmetic alone, are susceptible of a fully rigorous treatment.
    • Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed.
    • The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated.
    • In later mathematics, the question as to the impossibility of certain solutions plays a pre-eminent part, and we perceive in this way that old and difficult problems, such as the proof of the axiom of parallels, the squaring of the circle, or the solution of equations of the fifth degree by radicals have finally found fully satisfactory and rigorous solutions, although in another sense than that originally intended.
    • It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts.
    • However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes.
    • Seek its solution.

  3. Euler Elogium.html.html
    • These discoveries led Euler to an important discovery by observing the unique characteristics of exponential and logarithmic quantities born within the circle and following methods by which the solutions make the problems disappear, the terms of the imaginaries which would then be present and which would have complicated the calculation, even though the are known to collapse, reduced the formulas to simpler and more convenient expressions.
    • They are quickly enriched by the solutions to a great number of problems that the Mathematicians dared not deal with because of the difficulty and the physical impossibility to conduct their calculations to a satisfactory conclusion.
    • His novel research into the series of indefinite products provided the necessary resources into solutions to a great many useful and curious questions.
    • 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.
    • Often he preferred not to reveal the process of his thinking rather than to be exposed to the suspicion of a slight of hand and that he arrived at the solution only after the fact.
    • Euler arrived at the conclusion that differential equations are susceptible to particular solutions which are not included in general solutions in which Mr.
    • Euler who has shown why these particular integrals are excluded from the general solution and he is the first to have devoted some time to this theory, which has since been perfected by other celebrated geometers and the memoire in which Mr.
    • de la Grange has left nothing unknown concerning the nature of these integrals and their use in the solution of problems.
    • The solutions to the problems of the least resistant solid, and the curve of quickest descent and the problem of largest isoperimetric areas were much celebrated in Europe.
    • The general method for solving the problem was hidden in these solutions, especially in those of Jakob Bernoulli who had found the answer to the isoperimetric question which provided him with an advantage over his brother since so many masterpieces subsequently fathered by Johann Bernoulli.
    • 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.
    • 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.
    • At other times it would be a problem that appeared insurmountable that he resolved in an instant by a very simple method or an elementary problem with a very difficult solution that could only be overcome with the greatest efforts.

  4. ELOGIUM OF EULER
    • These discoveries led Euler to an important discovery by observing the unique characteristics of exponential and logarithmic quantities born within the circle and following methods by which the solutions make the problems disappear, the terms of the imaginaries which would then be present and which would have complicated the calculation, even though the are known to collapse, reduced the formulas to simpler and more convenient expressions.
    • They are quickly enriched by the solutions to a great number of problems that the Mathematicians dared not deal with because of the difficulty and the physical impossibility to conduct their calculations to a satisfactory conclusion.
    • His novel research into the series of indefinite products provided the necessary resources into solutions to a great many useful and curious questions.
    • 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.
    • Often he preferred not to reveal the process of his thinking rather than to be exposed to the suspicion of a slight of hand and that he arrived at the solution only after the fact.
    • Euler arrived at the conclusion that differential equations are susceptible to particular solutions which are not included in general solutions in which Mr.
    • Euler who has shown why these particular integrals are excluded from the general solution and he is the first to have devoted some time to this theory, which has since been perfected by other celebrated geometers and the memoire in which Mr.
    • de la Grange has left nothing unknown concerning the nature of these integrals and their use in the solution of problems.
    • The solutions to the problems of the least resistant solid, and the curve of quickest descent and the problem of largest isoperimetric areas were much celebrated in Europe.
    • The general method for solving the problem was hidden in these solutions, especially in those of Jakob Bernoulli who had found the answer to the isoperimetric question which provided him with an advantage over his brother since so many masterpieces subsequently fathered by Johann Bernoulli.
    • 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.
    • 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.
    • At other times it would be a problem that appeared insurmountable that he resolved in an instant by a very simple method or an elementary problem with a very difficult solution that could only be overcome with the greatest efforts.

  5. James Gregory's manuscripts
    • James Gregory's manuscripts on algebraic solutions 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.
    • A comparison between these notes and the allusions contained in the letters of Gregory and Collins, from May to October, 1675, make it evident that Gregory believed he had hit upon a general algebraic solution for all orders.
    • "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.
    • 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.

  6. L R Ford - Differential Equations
    • The first three chapters lead up to the later chapters by their discussions of direction fields, of solutions in series, of the Wronskian and linear dependence.
    • The method of successive approximations leads naturally into the Chapters VI and VII on interpolation and numerical integration and solutions.
    • Attention should be called to the treatments of finite differences and of the symbolic operators, also to the emphasis placed on whole families of solutions.
    • 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.
    • What has been attempted here has been the presentation of a compact, connected, and (it is believed) teachable body of material which exhibits those elementary methods of solution which are of commonest use." Here again the author has succeeded.
    • 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.
    • From the beginning the student learns that successive approximation to a solution may be the best we can do and that singular solutions may exist.
    • 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.
    • Numerical solutions are preceded by a good chapter on finite differences, including approximate differentiation and integration and the algebra of operators i and E.
    • 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.

  7. De Montmort: 'Essai d'Analyse
    • Having set down the rules, he solves simple cases in a method somewhat reminiscent of Huygens, and then takes a plunge into a general solution which appears to be correct but is not always demonstrably so.
    • The preceding solution furnishes a singular use of the figurate numbers (of which I shall speak later), for I find in examining the formula, that Pierre's chance is expressible by an infinite series of terms which have alternate + and - signs, and such that the numerator is the series of numbers which are found in the Table (i.e.
    • He doesn't bother with the rules (they must have been entirely established by this time) and he calculates several simple chances but remarks that in the majority of situations the solution cannot be found.
    • I think I should add that this problem was posed by me to a Lady, who gave me almost immediately the correct solution using the Arithmetic Triangle.
    • He had a wide circle of correspondents among mathematicians of all countries, including Newton and Leibniz, exchanging with them news about mathematical problems and discussing solutions to the problems of the day (and fray).
    • who, upon occasion of a French tract, called l'Analyse des Jeux de Hasard, which had lately been published, was pleased to propose to me some Problems of much greater difficulty than any he had found in that Book; which having solved to his satisfaction, he engaged me to methodise these Problems, and to lay down the Rules which had led me to their Solution ..
    • Huygens, first, as I know, set down rules for the solution of the same kind of problem as those which the new French author illustrates freely with diverse examples.
    • Montmort has come to maturity and discusses, wherever he is able, the general solution to the problem he sets himself, rather than beginning with special cases and then plunging into a general statement which is often unsupported by mathematical argument.
    • The expansion involved in the solution of his Proposition XVI:- .
    • De Moivre probably took the problem from the first edition, generalised it, and gave the solution with no indication of method of proof.
    • Montmort reached the solution by himself also, since a letter to Johann Bernoulli in 1710 shows that he had already obtained it.
    • The matching distribution is presented with a proof of the general case; this proof was not given in the first edition, implying that Montmort either guessed the original solution or was dissatisfied with his first method of proof.
    • Both Johann and Nicolaus take as their focus of comment the first edition of the Essai d'Analyse, but it is Nicolaus who comes forward with helpful and sometimes new solutions.
    • Nicolaus is soothing to Montmort but fair to de Moivre, pointing out in several places in his commentary that de Moivre had shown to him his general solutions to various problems when he was in London, and trying to explain that de Moivre had not intended to slight Montmort by his introduction.

  8. Kantorovich books.html
    • The object of the present work, according to the author, is to show that the ideas and methods of functional analysis can be used for the development of effective, practical algorithms for the explicit solutions of practical problems with just as much success as that with which they have been used for the theoretical study of these problems.
    • this book by Kantorovich and Krylov, originally titled Methods for the Approximate Solution of Partial Differential Equations, ..
    • This book is concerned with approximate methods used in the solution of partial differential equations, conformal mapping and the approximate solution of integral equations.
    • The book itself is concerned mainly with the numerical solution of partial differential equations, as the title to the first edition (1936) indicated.
    • Next come methods of solution of Fredholm integral equations with applications to the Dirichlet problem.
    • The work is concerned with what Lanczos calls 'parexic analysis', a study of processes which lead to approximate solutions of the problems of mathematical physics by rigorous methods.
    • He lays particular emphasis on the use of efficient prices, derived from the solution of a linear program, to bring about marginal improvements in resource allocation, without the need to resort to a recomputation of the entire program.
    • Tables for the numerical solution of boundary value problems of the theory of harmonic functions (1963), by L V Kantorovich, V I Krylov and K Ye Chernin.
    • An approximate solution of the Dirichlet problem for a fixed region with arbitrarily prescribed boundary values [is] be obtained from tabulated values of [certain] coefficient functions ..
    • 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).
    • The first of these contained fundamental advances and determined the content and further development of this discipline: it examined the mathematically new type of "extremal" problems; it evolved a universal method for their solution (method of solution multipliers) as well as various efficient numerical algorithms derived from it; it indicated the more important fields of technical-economic problems where these methods could be most usefully applied; and it brought out the economic significance of indicators resulting from an analysis of problems by this method which is particularly essential in problems of a socialist economy.
    • The discussion in academic circles in the Soviet Union around the proposals of Kantorovich has shown that his solution to some is quite unthinkable and unmarxist, while others with a knack for higher mathematics and less Marx tend to support some of his main propositions.

  9. Valiant Turing Award
    • This difficulty was characterized by complexity classes, such as P (tractable problems) and NP (problems for which a solution can easily be checked once it has been produced, but perhaps not easily found in the first place).
    • For example, he and Vijay Vazirani wrote the influential paper "NP is as easy as detecting single solutions" (Theoretical Computer Science, 1986).
    • Before this paper, many researchers believed that the hardest search problems were those with many solutions (sparsely embedded among far more non-solutions), because the multiplicity of solutions could confuse search algorithms and keep them from being able to narrow in on a single solution.
    • Valiant and Vazirani gave a dramatic and beautiful demonstration that this idea was completely wrong, by showing how a good algorithm for finding unique solutions could be used to solve any search problem.
    • Valiant discovered a brilliant and simple randomized solution to the problem.
    • Following this articulation of the challenge, Valiant proposed the BSP model as a candidate solution.

  10. Charles Bossut on Leibniz and Newton Part 2
    • It produced challenges of very difficult problems, the solution of which gave rise to new theories, and considerably extended the domain of geometry.
    • The English already triumphed: but Johann Bernoulli, taking up the cause of Leibniz who had just died, laughed at this scheme of a solution.
    • However, they all agreed in considering Newton's solution as insufficient and of no use.
    • Without stopping to develop Newton's solution, he gave one of his own in the Philosophical Transactions for 1717 which answered the question as proposed by Leibniz in it's full extent.
    • Had he contented himself with this he would have merited only praise: but, urged on by his resentment against Johann Bernoulli who had spoken a little slightingly of him on another occasion, he prefixed to his solution some insulting reflections on the partisans of Leibniz, having principally in view Johann Bernoulli their leader.
    • Among other things, he said that if they did not perceive how Newton's solution led to the equations of the problem, it must be attributed to their ignorance: illorum imperitiae tribuendum.
    • In a dissertation on orthogonal trajectories in the Leipzig Transactions for 1718, composed jointly by Johann Bernoulli and his son Nicholas, it was agreed that Dr Taylor's solution was accurate, and even evinced some sagacity; but then it was shown that it was far from being sufficiently general and that there existed a great number of resolvable cases to which it could not be applied.
    • Of this he complained with acrimony; and at the same time retorted the accusation by showing that Johann Bernoulli, in his last solution of the isoperimetrical problem, had travestied the solution of his brother, and that all the simplifications he had made in it had not changed it's nature.
    • We are again obliged to say that Johann Bernoulli retained his superiority here by the simplicity and elegance of his solutions.
    • This case Keill proposed to Johann Bernoulli, who not only resolved it in a very short time, but extended the solution to the general hypothesis in which the resistance of the medium should be as any power of the velocity of the projectile.
    • When he had discovered this theory, he offered repeatedly to send it to a confidential person in London on condition that Keill would give up his solution likewise; but Keill, though strongly urged, maintained a profound silence.
    • In this conjecture he was cruelly mistaken: and his challenge, which was something more than indiscreet, drew on him a reprimand from the Swiss geometrician that was so much the more poignant as the only mode of answering it satisfactorily was by a solution of the problem which he could neither effect by his own skill nor by the assistance of his friends.

  11. Eulogy to Euler by Fuss
    • His superiority in analysis provided the necessary recognition, however what truly made his glory was the solution to the isoperimetric problem, so famous by its controversy between the two Bernoulli brothers, Johann and Jakob each of whom pretended to have found the solution but neither knew of it in its entirety.
    • Bernoulli possessed the greater ability in physical principals combined with a patience to help in the solution of the problems which calculations brought about by experiments conducted with the utmost focus and manipulation.
    • While reading through the fifty-four letters that the King wrote to him between 1741 and 1777, amongst which there are letters in the King's own handwriting and I have noted that more often that not his particularly brilliant solutions were used.
    • He had provided real and immediate solutions to problems concerning the salt works at Schonebeck, the fountain pumps at Sans-Souci and various financial projects.
    • Bernoulli's solution taken from the Taylorian troichoides is not general and certainly not sufficient to explain it.
    • The solution to the important problem regarding the precession of the equinox and the nutation of the earth's axis that Mr.
    • In his last memoire he had found a solution as to how to understand the number of eccentricities in lunar motion, which he had not been able to determine in his first theory due to the complicated calculations and the incomplete method that was available to him at that time.
    • One finds the most felicitous integrations and a multitude of contrivances and refinements of the most sublime analysis, truly deep research on the nature and the properties of numbers, the ingenious proofs of a numbers of Fermat's theorems, the solution to a number of very difficult problems concerning equilibrium and the motion of solid bodies both flexible and elastic and the unraveling of a number of apparent paradoxes.

  12. Eulogy to Euler by Fuss
    • His superiority in analysis provided the necessary recognition, however what truly made his glory was the solution to the isoperimetric problem, so famous by its controversy between the two Bernoulli brothers, Johann and Jakob each of whom pretended to have found the solution but neither knew of it in its entirety.
    • Bernoulli possessed the greater ability in physical principals combined with a patience to help in the solution of the problems which calculations brought about by experiments conducted with the utmost focus and manipulation.
    • While reading through the fifty-four letters that the King wrote to him between 1741 and 1777, amongst which there are letters in the King's own handwriting and I have noted that more often that not his particularly brilliant solutions were used.
    • He had provided real and immediate solutions to problems concerning the salt works at Schonebeck, the fountain pumps at Sans-Souci and various financial projects.
    • Bernoulli's solution taken from the Taylorian troichoides is not general and certainly not sufficient to explain it.
    • The solution to the important problem regarding the precession of the equinox and the nutation of the earth's axis that Mr.
    • In his last memoire he had found a solution as to how to understand the number of eccentricities in lunar motion, which he had not been able to determine in his first theory due to the complicated calculations and the incomplete method that was available to him at that time.
    • One finds the most felicitous integrations and a multitude of contrivances and refinements of the most sublime analysis, truly deep research on the nature and the properties of numbers, the ingenious proofs of a numbers of Fermat's theorems, the solution to a number of very difficult problems concerning equilibrium and the motion of solid bodies both flexible and elastic and the unraveling of a number of apparent paradoxes.

  13. Finkel and The American Mathematical Monthly
    • Most of our existing Journals deal almost exclusively with subjects beyond the reach of the average student or teacher of Mathematics or at least with subjects with which they are not familiar, and little, if any space, is devoted to the solution of problems.
    • While realizing that the solution of problems is one of the lowest forms of Mathematical research, and that, in general, it has no scientific value, yet its educational value can not be over estimated.
    • 'The American Mathematical Monthly' will, therefore, devote a due portion of its space to the solution of problems, whether they be the easy problems in Arithmetic, or the difficult problems in the Calculus, Mechanics, Probability, or Modern Higher Mathematics.
    • Teachers, students and all lovers of mathematics are, therefore, cordially invited to contribute problems, solutions and papers on interesting and important subjects in mathematics.
    • All problems, solutions, and articles intended for publication in the February Number, should be received on or before February 1st, 1894.
    • Solutions to problems in this Number will appear in March Number, but should be mailed to Editors before February 15th.
    • Professor Finkel remained one of the editors of the department of Problems and Solutions through 1933, and was still a member of the board of editors at the time of his death.

  14. Percy MacMahon addresses the British Association in 1901, Part 2
    • Their solutions did not further the general progress, but were merely valuable in connection with the special problems.
    • Starting with this notion, Euler developed a theory of generating functions on the expansion of which depended the formal solutions of many problems.
    • I propose to give some account of these problems, and to add a short history of the way in which a method of solution has been reached.
    • One of the most important questions awaiting solution in connection with the theory of finite discontinuous groups is the enumeration of the types of groups of given order, or of Latin Squares which satisfy additional conditions.
    • For a general investigation, however, it is more scientific to start by designing functions and operations, and then to ascertain the problems of which the solution is furnished.
    • a further condition being that one solution only is given by a group of numbers satisfying the equation; that in fact permutations amongst the quantities a, b, g ..
    • A generating function can be formed which involves in its construction the Diophantine equation and inequalities, and leads after treatment to a representative, as well as enumerative, solution of the problem.

  15. John Williamson papers
    • P S Dwyer,E P Starke, J Williamson and J H M Wedderburn, Problems and Solutions: Advanced Problems: Solutions: 3645, Amer.
    • R Robinson and J Williamson, Problems and Solutions: Advanced Problems: Solutions: 3667, Amer.
    • J W Cell and J Williamson, Problems and Solutions: Elementary Problems: Solutions: E280, Amer.
    • J H M Wedderburn and J Williamson, Problems and Solutions: Advanced Problems: Solutions: 3856, Amer.
    • J A Greenwood and J Williamson, Problems and Solutions: Advanced Problems: Solutions: 4023, Amer.
    • C E Springer and J Williamson, Problems and Solutions: Advanced Problems: Solutions: 3994, Amer.
    • F J Duarte and J Williamson, Problems and Solutions: Advanced Problems: Solutions: 4159, Amer.
    • D H Lehmer, D M Smiley, M F Smiley and J Williamson, Problems and Solutions: Elementary Problems: Solutions: E710, Amer.

  16. Poincaré on the future of mathematics
    • Many times already men have thought that they had solved all the problems, or at least that they had made an inventory of all that admit of solution.
    • And then the meaning of the word solution has been extended; the insoluble problems have become the most interesting of all, and other problems hitherto undreamed of have presented themselves.
    • For the Greeks a good solution was one that employed only rule and compass; later it became one obtained by the extraction of radicals, then one in which algebraical functions and radicals alone figured.
    • An algebraical formula which gives us the solution of a type of numerical problems, if we finally replace the letters by numbers, is the simple example which occurs to one's mind at once.
    • What is it that gives us the feeling of elegance in a solution or a demonstration? It is the harmony of the different parts, their symmetry, and their happy adjustment; it is, in a word, all that introduces order, all that gives them unity, that enables us to obtain a clear comprehension of the whole as well as of the parts.
    • Briefly stated, the sentiment of mathematical elegance is nothing but the satisfaction due to some conformity between the solution we wish to discover and the necessities of our mind, and it is on account of this very conformity that the solution can be an instrument for us.
    • And since it enables us to foresee whether the solution of these problems will be simple, it shows us at least whether the calculation is worth undertaking.
    • Formerly an equation was not considered to have been solved until the solution had been expressed by means of a finite number of known functions.
    • It then remains to find the exact solution of the problem.
    • Can this be regarded as a true solution? The story goes that Newton once communicated to Leibnitz an anagram somewhat like the following: aaaaabbbeeeeii, etc.
    • To-day a similar solution would no longer satisfy us, for two reasons - because the convergence is too slow, and because the terms succeed one another without obeying any law.
    • Nevertheless an imperfect solution may happen to lead us towards a better one.
    • When the problems relating to congruents with several variables have been solved, we shall have made the first step towards the solution of many questions of indeterminate analysis.
    • Or rather I am wrong, for they would certainly have presented themselves, since their solution is necessary for a host of questions of analysis, but they would have presented themselves isolated, one after the other, and without our being able to perceive their common link.

  17. Taylor versus Continental mathematicians
    • Jacob Bernoulli had published a correct solution in 1701 but Johann Bernoulli's solution, obtained at the same time, was not satisfactory.
    • The author has confined himself solely to his subject; he has avoided express mention of what has been done by others because that would necessarily have forced him to note several mistakes and imperfections found in their solutions: for that reason he has not judged it apropos to speak of the solut)rowed from these solutions the analysis he uses to resolve these problems, these solutions having the defect of being restricted to particular cases, although the problems are proposed in general terms.
    • would be regarded as a favour done to the author of this solution, because, if he had mentioned it, he would not have been able to dispense with censuring three or four very considerable mistakes which one encounters there.

  18. Pólya: 'How to solve it' Preface
    • A great discovery solves a great problem but there is a grain of discovery in the solution of any problem.
    • He listened to lectures, read books, tried to take in the solutions and facts presented, but there was a question that disturbed him again and again: "Yes, the solution seems to work, it appears to be correct; but how is it possible to invent such a solution? Yes, this experiment seems to work, this appears to be a fact; but how can people discover such facts? And how could I invent or discover such things by myself?" Today the author is teaching mathematics in a university; he thinks or hopes that some of his more eager students ask similar questions and he tries to satisfy their curiosity.
    • Trying to understand not only the solution of this or that problem but also the motives and procedures of the solution, and trying to explain these motives and procedures to others, he was finally led to write the present book.
    • Behind the desire to solve this or that problem that confers no material advantage, there may be a deeper curiosity, a desire to understand the ways and means, the motives and procedures, of solution.
    • The following pages are written somewhat concisely, but as simply as possible, and are based on a long and serious study of methods of solution.

  19. Vajda books
    • In fact, this book would better be described as a digest of articles which deal with the solution of problems by linear programming, with emphasis on the use of the transportation method.
    • The book is rich in exercises and illustrative examples, and it covers a wide variety of theorems ranging from existence of solutions to their construction.
    • The only significant change is the addition of seven new chapters devoted to brief introductions to and examples of Discrete Linear Programming, Dynamic Programming, Stochastic Programming and Quadratic Programming, and also of a section on the solutions to the problems posed at the ends of some chapters.
    • There is no increase in the size of the book; this has been achieved partly by the regrettable removal of problems for the reader and their solutions.
    • However, unless the reader already has a good working knowledge of the subjects, he will find the descriptions of the techniques and the solutions to the problems rather brief.
    • This is a collection of 236 problems, with answers or full solutions.
    • Some hypothetical situations are presented as exercises for each of the four chapters, and for the serious student their solutions are presented.

  20. Reviews Landau Lifshitz.html
    • For example, while a basic treatment of the Dirac equation will discuss its solution for spherically symmetric potentials, ending with the bound-state solutions to the Coulomb problem, this book goes on to give the scattering states in a Coulomb potential, a treatment of ultra-relativistic scattering, the solution to the Dirac equation in an external electromagnetic plane wave, and more.
    • Quite a number of examples, in the form of problems with solutions, appear throughout the text.
    • No problems for solution by the student are given.

  21. Halmos: creative art
    • The modern angle trisector either doesn't know those rules, or he knows them but thinks that the idea is to get a close approximation, or he knows the rules and knows that an exact solution is required but lets wish be father to the deed and simply makes a mistake.
    • Let me, therefore, conclude this particular tack by mentioning a tiny and trivial mathematical problem and describing its solution-possibly you'll then get (if you don't already have) a little feeling for what attracts and amuses mathematicians and what is the nature of the inspiration I have been talking about.
    • All cognoscenti know, therefore, that the presence in the statement of a problem of a number like 1 + 210 is bound to be a strong hint to its solution; the chances are, and this can be guessed even before the statement of the problem is complete, that the solution will depend on doubling - or halving - something ten times.
    • That's a simple job that pencil and paper can accomplish in a few seconds; the answer (and hence the solution of the problem) is 1024.
    • The trouble with this solution is that it's much too special.
    • His solution works, but it is as free of inspiration as the layman's.
    • The problem has also an inspired solution, that requires no computation, no formulas, no numbers - just pure thought.
    • In mathophysics the question always comes from outside, from the 'real world', and the satisfaction the scientist gets from the solution comes, to a large extent, from the light it throws on facts.

  22. Kepler's Planetary Laws
    • Kepler's principal aim was to find a solution that would satisfy observations - and in that respect he possessed the outlook of a modern scientist.
    • And because it was expressed geometrically, the solution would potentially be exact - the closed orbit of a single planet in a plane round the fixed Sun.
    • This is the process that was described (in Section 4) as idealization because it ensured an exact solution (of the one-body problem) which was uniquely simple.
    • The solutions reached in each case are in some senses provisional, but they are certainly vital steps on the way to the presentday solution.

  23. Kepler's Planetary Laws
    • Kepler's principal aim was to find a solution that would satisfy observations - and in that respect he possessed the outlook of a modern scientist.
    • And because it was expressed geometrically, the solution would potentially be exact - the closed orbit of a single planet in a plane round the fixed Sun.
    • This is the process that was described previously as idealization because it ensured an exact solution (of the one-body problem) which was uniquely simple.
    • The solutions reached in each case are in some senses provisional, but they are certainly vital steps on the way to a modern solution.

  24. Al-Khwarizmi and quadratic equations
    • In this book, which has given us the word 'algebra', al-Khwarizmi gives a complete solution to all possible types of quadratic equation.
    • Here is al-Khwarizmi's solution of the equation .
    • What is most remarkable is that in this case he knows that the quadratic has two solutions:- .
    • Now it is clear that al-Khwarizmi is intending to teach his readers general methods of solution and not just how to solve specific examples.
    • When you meet an instance which refers you to this case, try its solution by addition, and if that does not work subtraction will.

  25. Carathéodory: 'Conformal representation
    • This work of Gauss appeared to give the whole inquiry its final solution; actually it left unanswered the much more difficult question whether and in what way a given finite portion of the surface can be represented on a portion of the plane.
    • In the proof of this theorem, which forms the foundation of the whole theory, he assumes as obvious that a certain problem in the calculus of variations possesses a solution, and this assumption, as Weierstrass (1815-1897) first pointed out, invalidates his proof Quite simple, analytic, and in every way regular problems in the calculus of variations axe now known which do not always possess solutions.
    • Nevertheless, about fifty years after Riemann, Hilbert was able to prove rigorously that the particular problem which arose in Riemann's work does possess a solution; this theorem is known as Dirichlet's Principle.

  26. Henry Baker addresses the British Association in 1913, Part 2
    • 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.
    • 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.

  27. N S Krylov's monograph - Introduction
    • Closely connected with the first type of difficulties are the problem of the mechanical interpretation of irreversibility and, among other things, all the well-known objections to Boltzmann's treatment of the H-theorem, and all the attempts still being made at achieving a quantum-mechanical solution of this problem.
    • The purpose of this work is, in considering together the two above-mentioned groups of difficulties, to give a solution to the problem that would explicitly introduce the basic concept of statistical physics - the concept of relaxation - and would make possible the quantitative evaluation of relaxation time.
    • Finally, it should form a logically well built structure devoid, at any rate, of those contradictions that are characteristic, as will be shown further, of all the solutions that have ever been proposed to the problem taken in all its generality (that is, comprising the problem of introducing irreversibility, ergodicity, and finite relaxation time into the theory).
    • Although the solution of the said basic problem must be brought about by the work as a whole, each of the above-mentioned parts may be regarded as constituting a more or less self-contained unit.

  28. The Dundee Numerical Analysis Conferences
    • This included an Honours course in numerical analysis, taught mainly by Jim Fulton, although Mike Osborne contributed some lectures on the numerical solution of differential equations.
    • If there were any records of the meeting, they have not survived, but I still possess a folder (unfortunately the original contents are long gone) with "Symposium on the solution of differential equations, St Andrews, June 1965" written on it.
    • Mike talked about the recently described Nordsieck's method being equivalent to a multi-step method, Ron gave a survey talk on ADI methods, and Jack Lambert spoke on some aspect of the numerical solution of ordinary differential equations.
    • This was held from 26-30 June, and called "Colloquium on the numerical solution of differential equations".
    • It was called "Conference on the numerical solution of differential equations", attracted 148 participants, and there were eight invited speakers, all from outside the UK: J Albrecht, E G D'Jakonov, B Noble, K Nickel, G Strang, M Urabe, E Vitasek, O Widlund.
    • There was a Seminar on Ritz-Galerkin and the Finite Element Method from 8 - 9 July, 1971, and finally a Conference on the Numerical Solution of Ordinary Differential Equations from 5 - 6 August, 1971.
    • In 1973 a Conference on the Numerical Solution of Differential Equations was held from July 3 - 6.

  29. Dickson: 'Theory of Equations
    • The exercises are so placed that a reasonably elegant and brief solution may be expected, without resort to tedious multiplications and similar manual labour.
    • 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.
    • In the first chapter we are given also the graphical solution of a quadratic.
    • These puzzle students and often teachers, partly because the problem is not clearly understood, and partly because there is so obviously a solution; and yet their impossibility may readily be made plausible to a student familiar with coordinate geometry and is here rigorously proved in an elementary way.
    • 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.

  30. Truesdell's books
    • The reader must constantly go back and forth between the actors on the stage, who, of course, do not yet know the solution of the knots, and Truesdell, who does.
    • It contains many recently determined solutions to the equations of motion for nonlinear fluids and uses these solutions to evaluate the fluid models.
    • The book is written very clearly and contains a large number of exercises and their solutions.

  31. Whittaker EMS Obituary.html
    • It became particularly useful when wave mechanics was being developed in the years 1925 and 1926 and solutions of Schrodinger's equation were being ardently sought after.
    • Then in 1902 and 1903 he published two papers on the partial differential equations of mathematical physics in which he obtained the solution of Laplace's equation with which his name is associated.
    • The advertisement intimating the opening of the laboratory in October 1913 specified the subjects to be taught (some of which were interpolation, method of least squares, solution of systems of linear equations, evaluation of determinants, determination of roots of transcendental equations, practical Fourier analysis, evaluation of definite integrals, numerical solution of differential equations, construction of tables of functions not previously tabulated such as parabolic cylinder functions) and also indicated that facilities were available for original research and that the University would grant recognition, under certain conditions, to research students who would be permitted to offer themselves for the degree of D.Sc.

  32. Todd: 'Basic Numerical Mathematics
    • Throughout both volumes we emphasize the idea of "controlled computational experiments": we try to check our programs and get some idea of errors by using them on problems of which we already know the solution such experiments can in some way replace the error analyses which are not appropriate in beginning courses.
    • Then the direct solution of the inversion problem is taken up, first in the context of theoretical arithmetic (i.e., when round-off is disregarded) and then in the context of practical computation.
    • Next, several iterative methods for the solution of systems of linear equations are examined.
    • It is then feasible to discuss two applications: the first, the solution of a two-point boundary value problem, and the second, that of least squares curve fitting.
    • We have not considered it necessary to give the machine programs required in the solution of the problems: the programs are almost always trivial and when they are not, the use of library subroutines is intended.
    • A typical problem later in Volume 2 will require, e.g., the generation of a special matrix, a call to the library for a subroutine to operate on the matrix and then a program to evaluate the error in the alleged solution provided by the machine.

  33. Carl Runge: 'Graphical Methods
    • We may distinguish different stages in the solution of a problem.
    • Or to give another instance take Fermat's problem, for the solution of which the late Mr Wlolfskehl, of Darmstadt, has left $25,000 in his will.
    • So the solution of the problem may or may not end in its first stage.
    • In many other cases the first stage of the solution may be so easy, that we immediately pass on to the second stage of finding methods to calculate the unknown quantities sought for.
    • Or even if the first stage of the solution is not so easy, it may be expedient to pass on to the second stage.
    • So there arises a third stage of the solution of a mathematical problem in which the object is to develop methods for finding the result with as little trouble as possible.

  34. Julia Robinson: Hilbert's 10th Problem
    • Julia Robinson and Martin Davis spent a large part of their lives trying to solve Hilbert's Tenth Problem: Does there exist an algorithm to determine whether a given Diophantine equation had a solution in rational numbers? .
    • I even worked in the opposite direction, trying to show that there was a positive solution to Hilbert's problem, but I never published any of that work.
    • It followed that the solution to Hilbert's tenth problem is negative - a general method for determining whether a given diophantine equation has a solution in integers does not exist.
    • I have been told that some people think that I was blind not to see the solution myself when I was so close to it.
    • At that time, in connection with the solution of Hilbert's problem and the role played in it by the Robinson hypothesis, Linnik told me that I was the second most famous Robinson in the Soviet Union, the first being Robinson Crusoe.

  35. Julia Robinson: Hilbert's 10th Problem
    • Julia Robinson and Martin Davis spent a large part of their lives trying to solve Hilbert's Tenth Problem: Does there exist an algorithm to determine whether a given Diophantine equation had a solution in rational numbers? .
    • I even worked in the opposite direction, trying to show that there was a positive solution to Hilbert's problem, but I never published any of that work.
    • It followed that the solution to Hilbert's tenth problem is negative - a general method for determining whether a given diophantine equation has a solution in integers does not exist.
    • I have been told that some people think that I was blind not to see the solution myself when I was so close to it.
    • At that time, in connection with the solution of Hilbert's problem and the role played in it by the Robinson hypothesis, Linnik told me that I was the second most famous Robinson in the Soviet Union, the first being Robinson Crusoe.

  36. Julia Robinson: Hilbert's 10th Problem
    • Julia Robinson and Martin Davis spent a large part of their lives trying to solve Hilbert's Tenth Problem: Does there exist an algorithm to determine whether a given Diophantine equation had a solution in rational numbers? .
    • I even worked in the opposite direction, trying to show that there was a positive solution to Hilbert's problem, but I never published any of that work.
    • It followed that the solution to Hilbert's tenth problem is negative - a general method for determining whether a given diophantine equation has a solution in integers does not exist.
    • I have been told that some people think that I was blind not to see the solution myself when I was so close to it.
    • At that time, in connection with the solution of Hilbert's problem and the role played in it by the Robinson hypothesis, Linnik told me that I was the second most famous Robinson in the Soviet Union, the first being Robinson Crusoe.

  37. Mathematicians and Music 3
    • Of this equation he found the solution .
    • Euler immediately raised the question of the generality of the solution and set forth his interpretation.
    • He started with Taylor's particular solution and found, in effect, that the function for determining the position of the string after starting from rest could naturally be expressed in a form later called a Fourier series.
    • Bernoulli remarked that since his solution was perfectly general it should include those of Euler and D'Alembert.
    • No mathematician would admit even the possibility of its solution till this was thoroughly demonstrated, in connection with certain problems in the flow of heat, by Fourier who gives due credit to the suggestiveness of the work of those in the previous century to whom I have referred.

  38. Planetary motion tackled kinematically
    • Nowadays astronomers accept that planetary motion has to be treated dynamically, as a many-body problem, for which there is bound to be no exact solution.
    • The astronomical solution to the one-body problem consists of the two laws: .
    • This composite solution represents what is in fact the earliest instance of a planetary orbit: it will be succinctly referred to in what follows as 'the Sun-focused ellipse'.
    • We shall now prove that, subject to its obvious external limitations, this unique solution is of universal applicability as a self-contained piece of mathematics.
    • As a consequence, our kinematical solution is qualitatively different from any later, dynamical one in that it is exact on the basis of geometry alone, as we shall demonstrate.

  39. L'Hôpital: 'Analyse des infiniment petits' Preface
    • But since he was mainly concerned with the solution of equations he was interested in curves only as a way to finding roots.
    • M Descartes' Geometry made it fashionable to solve geometrical problems by means of equations, and opened up many possibilities of obtaining such solutions.
    • The ninth consists of solutions to various problems arising out of the earlier work.

  40. John Couch Adams' account of the discovery of Neptune
    • I find among my papers the following memorandum, dated July 3, 1841: "Formed a design, in the beginning of this week, of investigating, as soon as possible after taking my degree, the irregularities in the motion of Uranus, which are yet unaccounted for, in order to find whether they may be attributed to the action of an undiscovered planet beyond it, and, if possible, thence to determine approximately the elements of its orbit, etc., which would probably lead to its discovery." Accordingly, in 1843, I attempted a first solution of the problem, assuming the orbit to be a circle, with a radius equal to twice the mean distance of Uranus from the sun.
    • Meanwhile the Royal Academy of Sciences of Gottingen had proposed the theory of Uranus as the subject of their mathematical prize, and although the little time which I could spare from important duties in my college prevented me from attempting the complete examination of the theory which a competition for the prize would have required, yet this fact, together with the possession of such a valuable series of observations, induced me to undertake a new solution of the problem.
    • After obtaining several solutions differing little from each other, by gradually taking into account more and more terms of the series expressing the perturbations, I communicated to Professor Challis, in September 1845, the final values which I had obtained for the mass, heliocentric longitude, and elements of the orbit of the assumed planet.

  41. H S Ruse papers
    • W Feller writes: The authors study Riemannian Vn in connection with the fundamental solution of the corresponding equation Δ2u = 0, where Δ2 stands for the second differential parameter.
    • General solutions of Laplace's equation in a simply harmonic manifold (1963).
    • T J Willmore writes: Explicit formulae are obtained for general solutions of Laplace's equation in a real n-cell equipped with a simply harmonic riemannian metric.

  42. R A Fisher: 'History of Statistics
    • We do know that the reason for his hesitation to publish was his dissatisfaction with the postulate required for the celebrated "Bayes' Theorem." While we must reject this postulate, we should also recognise Bayes' greatness in perceiving the problem to be solved, in making an ingenious attempt at its solution, and finally in realising more clearly than many subsequent writers the underlying weakness of his attempt.
    • Once the true nature of the problem was indicated, a large number of sampling problems were within reach of mathematical solution.
    • "Student" himself gave in this and a subsequent paper the correct solutions for three such problems - the distribution of the estimate of the variance, that of the mean divided by its estimated standard deviation, and that of the estimated correlation coefficient between independent variates.

  43. G A Miller - A letter to the editor
    • For instance, recent discoveries relating to the finding of at least one root by the ancient Babylonians of certain numerical quadratic and cubic equations throws new light on the history of algebra and on the contributions made by the Greeks and the Arabians towards the solution of algebraic equations.
    • It is to be emphasized that the complete solution of general quadratic and of general cubic equations could not be attained until our ordinary complex numbers began to be understood at about the beginning of the nineteenth century, although complete formal solutions were used earlier in Europe.

  44. John Couch Adams' account of the discovery of Neptune
    • I find among my papers the following memorandum, dated July 3, 1841: "Formed a design, in the beginning of this week, of investigating, as soon as possible after taking my degree, the irregularities in the motion of Uranus, which are yet unaccounted for, in order to find whether they may be attributed to the action of an undiscovered planet beyond it, and, if possible, thence to determine approximately the elements of its orbit, etc., which would probably lead to its discovery." Accordingly, in 1843, I attempted a first solution of the problem, assuming the orbit to be a circle, with a radius equal to twice the mean distance of Uranus from the sun.
    • Meanwhile the Royal Academy of Sciences of Gottingen had proposed the theory of Uranus as the subject of their mathematical prize, and although the little time which I could spare from important duties in my college prevented me from attempting the complete examination of the theory which a competition for the prize would have required, yet this fact, together with the possession of such a valuable series of observations, induced me to undertake a new solution of the problem.
    • After obtaining several solutions differing little from each other, by gradually taking into account more and more terms of the series expressing the perturbations, I communicated to Professor Challis, in September 1845, the final values which I had obtained for the mass, heliocentric longitude, and elements of the orbit of the assumed planet.

  45. G H Hardy addresses the British Association in 1922
    • It is impossible, for me to give you, in the time at my command, any general account of the problems of the theory of numbers, or of the progress that has been made towards their solution even during the last twenty years.
    • But there is no similar solution for our actual problem, nor, I need hardly say, for the analogous problems for fourth, fifth, or higher powers.
    • There is no case, except the simple case of squares, in which the solution is in any sense complete.
    • The first step towards a solution was made by Dirichlet, who proved for the first time, in 1837, that any such arithmetical progression contains an infinity of primes.

  46. James Clerk Maxwell on the nature of Saturn's rings
    • There are some questions in Astronomy, to which we are attracted rather on account of their peculiarity, as the possible illustration of some unknown principle, than from any direct advantage which their solution would afford to mankind.
    • There is a very general and very important problem in Dynamics, the solution of which would contain all the results of this Essay and a great deal more.
    • It is this: "Having found a particular solution of the equations of motion of any material system, to determine whether a slight disturbance of the motion indicated by the solution would cause a small periodic variation, or a total derangement of the motion." .

  47. Pólya on Fejér
    • Yet he could perceive the significance, the beauty, and the promise of a rather concrete not too large problem, foresee the possibility of a solution and work at it with intensity.
    • And, when he had found the solution, he kept on working at it with loving care, till each detail became fully transparent.
    • It is due to such care spent on the elaboration of the solution that Fejer's papers are very clearly written, and easy to read and most of his proofs appear very clear and simple.
    • Yet only the very naive may think that it is easy to write a paper that is easy to read, or that it is a simple thing to point out a significant problem that is capable of a simple solution.

  48. Horace Lamb addresses the British Association in 1904
    • The practical character of the mathematical work of Stokes and his followers is shown especially in the constant effort to reduce the solution of a physical problem to a quantitative form.
    • It is now generally accepted that an analytical solution of a physical question, however elegant it may be made to appear by means of a judicious notation, is not complete so long as the results are given merely in terms of functions defined by infinite series or definite integrals, and cannot be exhibited in a numerical or graphical form.
    • Moreover, the interest of the subject, whether mathematical or physical, is not yet exhausted; many important problems in Optics and Acoustics, for example, still await solution.
    • It would appear that there is an opening here for the mathematician; at all events, the numerical or graphical solution of any one of the numerous problems that could be suggested would be of the highest interest.

  49. James Clerk Maxwell on the nature of Saturn's rings
    • There are some questions in Astronomy, to which we are attracted rather on account of their peculiarity, as the possible illustration of some unknown principle, than from any direct advantage which their solution would afford to mankind.
    • There is a very general and very important problem in Dynamics, the solution of which would contain all the results of this Essay and a great deal more.
    • It is this: "Having found a particular solution of the equations of motion of any material system, to determine whether a slight disturbance of the motion indicated by the solution would cause a small periodic variation, or a total derangement of the motion." .

  50. Max Planck: 'The Nature of Light
    • I shall doubtless mention much that is familiar to each of you, but I shall also deal with newer problems still awaiting solution.
    • When and how the last step will be made, the linking up of mechanics and electro-dynamics, cannot be said, and though many clever physicists are at present occupied with this question, the time does not yet seem ripe for the solution.
    • It would have been possible to seek a solution by supposing that the theory would have been better had it abstained, in general, from making special hypotheses, based on immediate observations, and to limit oneself to the pure facts, i.e.
    • It is not possible today to predict with certainty when any definite solution to this problem will be obtained.

  51. Warga abstract
    • The first one is in well known problems of classical optimal control in which one approaches optimal solutions with rapidly oscillating control functions.
    • A limit solution is then obtained by imbedding the controls in the class of relaxed controls which are functions with values that are probability measures.

  52. Von Neumann Silliman lectures
    • Then, during the late thirties, he became interested in questions of theoretical hydrodynamics, particularly in the great difficulties encountered in obtaining solutions to partial differential equations by known analytical methods.
    • This was the period during which he became completely convinced, and tried to convince others in many varied fields, that numerical calculations done on fast electronic computing devices would substantially facilitate the solution of many difficult, unsolved, scientific problems.

  53. Edinburgh's tribute to A C Aitken
    • He found the solution of a sixth order difference equation which arose in Whittaker's Theory of Graduation, a problem which suited his special gifts in manipulative algebra and numerical skill.
    • They contribute to a practical approach to mathematics which has enabled him the more readily to see the solutions of mathematical problems as well as to suggest many new practical methods to his colleagues.

  54. Konrad Knopp: Texts
    • Solution hints are provided, plus complete solutions to all problems.

  55. Edward Sang on his tables
    • The quinquesection of these parts was effected by help of the method of the solution of equations of all orders, published by me in 1829; and the computation of the multiples of those parts was effected by the use of the usual formula for second differences.
    • Kepler's celebrated problem has ever since his time exercised mathematicians, and, sharing the ambition of many others, I also sought often, and in vain, for an easy solution of it.
    • Accident brought it again before me, and this time, considering not the relations of the lines connected with it, but the relations of the areas concerned, an exceedingly simple solution was found.

  56. Pappus on the trisection of an angle
    • Now it is considered a serious type of error for geometers to seek a solution to a plane problem by conics or linear curves and, in general, to seek a solution by a curve of the wrong type.
    • But later with the help of the conics they trisected the angle using the following 'vergings' for the solution.

  57. Mathematical and Physical Journal for Secondary Schools
    • Each issue of the Kozepiskolai Matematikai Lapok contained a number of selected exercises from mathematics and shortly thereafter from physics, as well as solutions to past months' problems and a list of those pupils who sent in correct solutions.
    • It regularly prints the best solutions of the students and the names of the best problems solvers.

  58. What do mathematicians do?
    • In the 1920's, for example, the discovery of quantum mechanics went a very long way toward reducing chemistry to the solution of well-defined mathematical problems.
    • Much more difficult to establish is the beautiful result that solutions exist for the prime 2 and for precisely those odd primes which leave a remainder of 1 when divided by 4.

  59. Ince obituary.html
    • The war years - a physical disability precluded active service - he spent in Edinburgh and Trinity College, Cambridge, as a research student; in various forms of national service (including a period at University College, London, where he had contacts with that enthusiast for singular solutions of differential equations, M J M Hill); and as a temporary lecturer at the University of Leeds.
    • The work on Mathieu's equation included a proof of the impossibility of the equation having more than one periodic solution, and culminated in a series of papers published in the Proceedings of the Royal Society of Edinburgh in which he developed methods for the computation of the functions and carried these methods to fruition by the actual formation of the tables themselves.

  60. André Weil: 'Algebraic Geometry
    • To take only one instance, a personal one, this book has arisen from the necessity of giving a firm basis to Severi's theory of correspondences on algebraic curves, especially in the case of characteristic p ≠ 0 (in which there is no transcendental method to guarantee the correctness of the results obtained by algebraic means), this being required for the solution of a long outstanding problem, the proof of the Riemann hypothesis in function-fields.
    • Our results include all that is required for a rigorous treatment of so-called "enumerative geometry", thus providing a complete solution of Hilbert's fifteenth problem.
    • IX; it contains such general comments as could not appropriately be made before, formulates problems, some of them of considerable importance, and, in some cases, makes tentative suggestions about what seems to be at present the best approach to their solution; it is hoped that these may be helpful to the reader, to whom the author, having acted as his pilot until this point, heartily wishes Godspeed on his sailing away from the axiomatic shore, further and further into open sea.

  61. A A Albert: 'Structure of Algebras
    • The theory of linear associative algebras probably reached its zenith when the solution was found for the problem of determining all rational division algebras.
    • Since that time it has been my hope that I might develop a reasonably self-contained exposition of that solution as well as of the theory of algebras upon which it depends and which contains the major portion of my own discoveries.
    • The results are also applied in the determination of the structure of the multiplication algebras of all generalized Riemann matrices, a result which is seen in Chapter XI to imply a complete solution of the principal problem on Riemann matrices.

  62. Heath: Everyman's Library 'Euclid' Introduction
    • 28 and 29: "These two problems, to the first of which the 27th proposition is necessary, are the most general and useful of all in the Elements, and are most frequently made use of by the ancient geometers in the solution of other problems; and, therefore, are very ignorantly left out by Tacquet and Dechales in their editions of the Elements, who pretend that they are scarce of any use." The important words here are those referring to the ancient geometers.
    • The propositions embody, in fact, the general method known as the "application of areas," which was of vital consequence to the Greek geometers, being the geometrical equivalent of the solution of the general quadratic equations ax ∓ bx2/c = S so far as they have real roots.
    • The simplest case of "application of areas," which is equivalent to the solution for x of the simple equation ax = S, can be read in this volume (Eucl.

  63. R A Fisher: 'Statistical Methods' Introduction
    • The solutions of problems of distribution (which may be regarded as purely deductive problems in the theory of probability) not only enable us to make critical tests of the significance of statistical results, and of the adequacy of the hypothetical distributions upon which our methods of numerical inference are based, but afford real guidance in the choice of appropriate statistics for purposes of estimation.
    • In view of the mathematical difficulty of some of the problems which arise it is also useful to know that approximations to the maximum likelihood solution are also in most cases efficient statistics.

  64. Berge books
    • Of the several appendices, one lists fourteen questions still awaiting solution.
    • 2-person 0-sum games are introduced and solved by linear programming and by successive approximations (without mentioning George W Brown, Iterative solution of games by fictitious play, Activity Analysis of Production and Allocation, T C Koopmans [Tjalling Charles Koopmans (1910-1985)], ed., John Wiley, New York, 1951, 374-376; Julia Robinson, An iterative method for solving a game, Ann.
    • Their contents are respectively: separation theorems for convex sets, the Farkas-Minkowski [named after Gyula Farkas and Hermann Minkowski] and von Neumann minimax theorems, with divers extensions and corollaries; the various forms of the minimization problem (with equivalence theorems); algorithms of "simplex" type for the solution of convex and quadratic programming problems.

  65. EMS 1914 Colloquium
    • (Professor of Mathematics in the University of Edinburgh), on THE SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS IN THE MATHEMATICAL LABORATORY.
    • This method bids fair to displace in practical application the older method of graphical solution equations, as it can very readily be applied to equations in any number of variables.
    • Professor Whittaker delivered the first of his two lectures on the solution of equations.

  66. G H Hardy addresses the British Association in 1922, Part 1
    • It is impossible, for me to give you, in the time at my command, any general account of the problems of the theory of numbers, or of the progress that has been made towards their solution even during the last twenty years.
    • But there is no similar solution for our actual problem, nor, I need hardly say, for the analogous problems for fourth, fifth, or higher powers.
    • There is no case, except the simple case of squares, in which the solution is in any sense complete.

  67. Gibson History 7 - Robert Simson
    • In the higher geometry he prelected from his own Conics, and he gave a small specimen of the linear problems of the ancients by explaining the properties sometimes of the conchoid, sometimes of the cissoid, with their application to the solution of such problems.
    • Till he produced his article in the Philosophical Transactions in 1723 there had been no elucidation of the mystery that had baffled every inquirer, and even then there was only an approach to a solution, not the solution itself.

  68. Phillip Griffiths Looks at 'Two Cultures' Today
    • Here at home, Congressman George Brown of California, a physicist and a former chair of the House Committee on Space, Science, and Technology, has been strongly influenced by Havel and now sees little evidence that "objective scientific knowledge leads to subjective benefits for humanity." Congressman Brown has written that he wants to "make strong attempts to involve ordinary citizens in processes of discussion and decision-making, including citizens who have not previously demonstrated expertise about such matters at all." To some, this recalls the tone of Germany's "volkische" solution, and even of Mao Tse Tung's cultural revolution.
    • Such scientists, he said, seldom see that many problems of application and engineering are as intellectually exacting as so-called "pure" problems, and that many of the solutions are as satisfying and beautiful.

  69. Miller graduation address
    • There can certainly be no interest more fundamental or of greater concern to the human family than the solution of the problem - how men may dwell together in peace and prosperity, under a stable social, civil, and political policy.
    • The Hebrews and Greeks have solved man's relation to the eternal mystery - the one in its religious, the other in its philosophical aspect; each has come as near the perfect solution as, perhaps, it is possible for the human mind to reach.
    • Where this adjustment is complicated by diverse physical peculiarities and by different inherited or acquired characteristics, the problem becomes one of the greatest intricacy that has ever taxed human wisdom and patience for solution.

  70. Durell and Robson: 'Advanced Trigonometry
    • A Key is published, for the convenience of teachers, in which solutions are given in considerable detail, and in some cases alternative methods of solution are supplied, so that to some extent the Key forms a supplementary teaching manual.

  71. Mathematicians and Music 2.2
    • Among those of the sixteenth century achieving a reputation in mathematics and medicine none was better known than he, whose greatest mathematical work, Ars Magna (1545), contains the first solution of the general cubic equation in print.
    • In them he suggests another solution of the problem of how suitably to arrive at a tempered scale.

  72. R L Wilder: 'Cultural Basis of Mathematics I
    • I don't mean that it can solve these problems, but that it can point the way to solutions as well as show the kinds of solutions that may be expected.

  73. Kline's books
    • This monograph was inspired by unpublished lecture notes of the late Rudolf Luneberg on the foundations of geometrical optics, based on solutions of the electromagnetic wave equation.
    • In his "proper direction for reform" he does not offer novel solutions.

  74. Malcev: 'Foundations of Linear Algebra' Introduction
    • For example, the fundamental idea behind the solution of a system of linear equations in several unknowns is that of replacing such a system by a chain of these simple equations.
    • The search in the 18th century for the general solution of n linear equations in n unknowns led Leibniz and Cramer to the notion of the determinant.

  75. John Walsh's delusions
    • Solution of Equations of the higher orders, 1845.
    • In his diary there is an entry: "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 7 July 1844; exactly 22 years after the invention of the Geometry of Partial Equations, and the expulsion of the differential calculus from Mathematical Science.] .

  76. Edmund Whittaker: 'Physics and Philosophy
    • If the transitions from mathematics to moral values are not firmly established, Whittaker's attempt does not succeed in remedying the defects of Descartes' solution.
    • At the end of the lecture Whittaker just mentions the problem of how God's foreknowledge can be reconciled with man's freewill, and perhaps it may not be out of place to give briefly the scholastics' solution of the problem.

  77. James Jeans addresses the British Association in 1934, Part 2
    • It may seem strange, and almost too good to be true, that nature should in the last resort consist of something we can really understand; but there is always the simple solution available that the external world is essentially of the same nature as mental ideas.
    • It is only a step from this to a solution of the problem which would have commended itself to many philosophers, from Plato to Berkeley, and is, I think, directly in line with the new world-picture of modern physics.

  78. Science at St Andrews
    • Despite its pugnacity the booklet is a tour de force, with a bewildering variety of novel ideas concealed in a verbiage of medieval geometry, yet containing a forthright solution of a problem on the rhumb line that had mystified men for a century.
    • This period of discovery culminated for Gregory in the central expansion theorems of interpolation and the differential calculus, the former of which he announced in a letter to Collins, November 1670, and the latter of which he exemplified in the following February by half a dozen examples and again a year later by the solution of Kepler's problem - on determining the theoretical position in its orbit of a planet at a given time - which Gregory solved by invoking the properties of the cycloid and repeated differentiation.

  79. EMS 1913 Colloquium
    • It was then shown that the electrodynamical equations were transformed in equations of the same type, and examples were given to show how from the solution of a problem in electrostatics the solution of a problem dealing with moving distributions of electricity could be obtained.

  80. Tietze: 'Famous Problems of Mathematics
    • In the discussion of the disputes over the priority of discovery of the solution of third-degree equations, a reference to the difference between objective and brachial methods of conflict had a certain timeliness in view of the battles-royal that were then frequently "organized" at political meetings.
    • The lectures regularly contain a few biographical data on individual mathematicians who have a special relationship with the problem under discussion, or who discovered its solution.

  81. The Works of Sir John Leslie
    • This is a sort of inverted form of solution.
    • He remarks quaintly: "The superior elegance and perspicuity with which the geometrical process unfolds the properties of these higher curves, may show that the fluxionary calculus should be more sparingly employed, if not reserved for the solution of problems of a more arduous nature." After that it comes quite as a shock to meet mere differential equations masquerading in such elegant geometrical company, but these are seen to be rank outsiders, members of the nouveaux riches.

  82. Gregory-Collins correspondence
    • In December 1670 Collins began discussing the solution of algebraic equations with Gregory.
    • 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.

  83. Muir on research in Scotland
    • Is it too Quixotic to suppose that had all mutual recriminations been laid aside, and a National Conference on the Higher Education been held several years ago, - a Conference including professors, secondary schoolmasters, and the many influential laymen who, from their connection with school boards and other public bodies, take an interest in the subject, -a basis of action could have been harmoniously arrived at, which would ere this have made a solution an immediate possibility? The evil is more clamant than ever now, when the far more difficult problem of providing a national system of elementary education has received so satisfactory a solution; and may we not reasonably expect that the intellects and wills which solved the one are capable of solving the other? .

  84. Charles Bossut on Leibniz and Newton
    • He contents himself with saying that he has deduced them from the solution of a general problem which he expresses enigmatically by transposing the letters and the sense of which, as explained after the business was known, is 'an equation containing flowing quantities being given, to find fluxions and inversely.' What light could Leibniz derive from such an anagram? All we can conclude from such a letter is that at the time it was written Newton was in possession of the method of fluxions; by which, however, is to be understood simply the method of tangents and quadratures; for the method of resolving differential equations was then out of the question, this not being invented till long after as has been said above.
    • Here then we have the clear and positive solution of the problem, the possession of which Newton so carefully endeavoured to reserve to himself.

  85. EMS 1913 Colloquium 3.html.html
    • It was then shown that the electrodynamical equations were transformed in equations of the same type, and examples were given to show how from the solution of a problem in electrostatics the solution of a problem dealing with moving distributions of electricity could be obtained.

  86. George Chrystal's First Promoter's Address
    • It would be merest affection to say that I have not learned much from all this experience; but it is quite within the truth to say that there are many practical questions of high importance, on the solution of which my experience throws no light whatever.
    • How one of them is to live by lecturing, say on definite integrals, is a mystery to the solution of which the advocates of pure and unadulterated extra-muralism have not addressed themselves.

  87. Pappus on analysis and synthesis in geometry
    • Analysis, then, takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we admit that which is sought as if it were already done and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards.
    • There are in all thirty-three Books, the contents of which up to the Conics of Apollonius I have set out for your consideration, including not only the number of the propositions, the diorismi [a statement in advance as to when, how, and in how many ways the problem will be capable of solution] and the cases dealt with in each Book, but also the lemmas which are required; indeed I have not, to the best of my belief, omitted any question arising in the study of the Books in question.

  88. Goursat: 'Cours d'analyse mathématique
    • - Solutions periodiques et asymptotiques.
    • Solutions discontinues.

  89. Max Planck: 'Quantum Theory
    • For me, such an object has, for a long time, been the solution of the problem of the distribution of energy in the normal spectrum of radiant heat.
    • Until this goal is attained the problem of the quantum of action will not cease to stimulate research and to yield results, and the greater the difficulties opposed to its solution, the greater will be its significance for the extension and deepening of all our knowledge of physics.

  90. NAS Award in Applied Mathematics and Numerical Analysis
    • for his innovative and imaginative use of mathematics in the solution of a wide variety of challenging and significant scientific and engineering problems.
    • for his profound and penetrating solution of outstanding problems of statistical mechanics.

  91. Muir on research in Scotland
    • Is it too Quixotic to suppose that had all mutual recriminations been laid aside, and a National Conference on the Higher Education been held several years ago, - a Conference including professors, secondary schoolmasters, and the many influential laymen who, from their connection with school boards and other public bodies, take an interest in the subject, -a basis of action could have been harmoniously arrived at, which would ere this have made a solution an immediate possibility? The evil is more clamant than ever now, when the far more difficult problem of providing a national system of elementary education has received so satisfactory a solution; and may we not reasonably expect that the intellects and wills which solved the one are capable of solving the other? .

  92. Kuratowski: 'Introduction to Topology
    • As a consequence, topology is often suitable for the solution of problems to which analysis cannot give the answer.
    • theorems on the existence of a solution of certain types of differential equations can be expressed as theorems on the existence of invariant points of a function space (the space of continuous functions) under continuous transformations; these theorems can be proved by topological methods in a more general form and in a simpler way than was formerly done without the aid of topology.

  93. W H Young addresses ICM 1928
    • The solution of many even of the problems of today lies far beyond the Mathematics of our time, or of any definite epoch.
    • Verily in the clear statement of the problem lies more than half its solution.

  94. Gowers laureation
    • Nevertheless, in 2009 he ventured to the other extreme, by posting a question on his blog 'Is massively collaborative mathematics possible?' He proposed a 'Polymath' project, where anyone who wished could contribute their ideas on a blog page aiming to reach the solution of a difficult research problem collectively.
    • The solution was written up and published in a top journal under the name of Polymath.

  95. Gauss: 'Disquisitiones Arithmeticae
    • The Analysis which is called indeterminate or Diophantine and which discusses the manner of selecting from an infinite set of solutions for an indeterminate problem those that are integral or at least rational (and especially with the added condition that they be positive) is not the discipline to which I refer but rather a special part of it, just as the art of reducing and solving equations (Algebra) is a special part of universal Analysis.
    • My greatest hope is that it pleases those who have at heart the development of science and that it proposes solutions that they have been looking for or at least opens the way for new investigations.

  96. H L F Helmholtz: 'Theory of Music' Introduction
    • The horizons of physics, philosophy, and art have of late been too widely separated, and, as a consequence, the language, the methods, and the aims of any one of these studies present a certain amount of difficulty for the student of any other of them; and possibly this is the principal cause why the problem here undertaken has not been long ago more thoroughly considered and advanced towards its solution.
    • Meanwhile musical aesthetics has made unmistakable advances in those points which depend for their solution rather on psychological feeling than on the action of the senses, by introducing the conception of movement in the examination of musical works of art.

  97. Born Inaugural
    • The justification for considering this special branch of science as a philosophical doctrine is not so much its immense object, the universe from the atom to the cosmic spheres, as the fact that the study of this object in its totality is confronted at every step by logical and epistemological difficulties; and although the material of the physical sciences is only a restricted section of knowledge, neglecting the phenomena of life and consciousness, the solution of these logical and epistemological problems is an urgent need of reason.
    • In spite of this difficulty, I shall try to outline the problem and its solution, called quantum mechanics.

  98. H Weyl: 'Theory of groups and quantum mechanics'Preface to Second Edition
    • At present no solution of the problem seems in sight; I fear that the clouds hanging over this part of the subject will roll together to form a new crisis in quantum physics.

  99. Gheorghe Mihoc's books
    • The authors give an exposition, at a not too advanced mathematical level, of the main types of mathematical programming problems and of the more effective methods for their solution.

  100. Ferrar: 'Textbook of Convergence
    • The majority are reasonably straightforward; hints for their solution are occasionally given.

  101. Ernest Hobson addresses the British Association in 1910
    • Ample opportunities for the full discussion of all the detailed problems, the solution of which forms a great and necessary part of the work of those who are advancing science in its various branches, are afforded by the special Societies which make those branches their exclusive concern.

  102. EMS 1914 Colloquium 3.html
    • Professor Whittaker delivered the first of his two lectures on the solution of equations.

  103. Rédei: Algebra
    • Galois theory, quadratic reciprocity, cyclic fields, solvability, the general equation, solution of cubic and quartic equations, geometric constructions, the normal basis theorem.

  104. Cajori: 'A history of mathematics' Introduction
    • The interest which pupils take in their studies may be greatly increased if the solution of problems and the cold logic of geometrical demonstrations are interspersed with historical remarks and anecdotes.

  105. EMS obituary
    • This he put right by bringing such forms into line with the signature test that is essential for real quadratic forms: but he generously gave the problem over to his pupil, the late G Richard Trott, who published the solution in his dissertation for a doctorate.

  106. EMS 1914 Colloquium 1.html.html
    • This method bids fair to displace in practical application the older method of graphical solution equations, as it can very readily be applied to equations in any number of variables.

  107. EMS obituary
    • Other titles were "On the uniqueness of the solution of the linear differential equation of the second order" (1903) and "The condition for the reality of the roots of an n-ic" (1906).

  108. Harold Jeffreys on Logic and Scientific Inference
    • We do not say that it is the solution of the present difficulty, but a priori knowledge exists, and we shall have occasion later to consider instances of it at length.

  109. Bartlett's reviews
    • Now there are a number of textbooks at all levels about the analytical solution of stochastic models, i.e.

  110. A I Khinchin: 'Statistical Mechanics' Introduction
    • This problem, originated by Boltzmann, apparently is far from its complete solution even at the present time.

  111. EMS obituary
    • Tweedie's brilliant solutions of the examples first brought him into notice.

  112. James Jeans addresses the British Association in 1934
    • It seemed likely that Heisenberg had unravelled the secret of the structure of matter, and yet his solution was so far removed from the concepts of ordinary life that another parable had to be invented to make it comprehensible.

  113. Max Planck and the quanta of energy
    • For many years, such an aim for me was to find the solution to the problem of the distribution of energy in the normal spectrum of radiating heat.

  114. Franklin's textbooks
    • It is designed to show them how to operate with complex quantities and how to solve problems for their solution on the use of Fourier series and integrals, and Laplace transforms.

  115. Mitchell Feigenbaum: the interviewer
    • Briefly, he discovered a universal quantitative solution characterized by specific measurable constants that describes the crossover from simple to chaotic behaviors in many complex systems.

  116. Gábor Szegó's student years
    • The names of the others who were in the same business were quickly known to me, and frequently I read with considerable envy how they had succeeded to solve some problems which I could not handle with complete success, or how they had found a better solution (simpler, more elegant, or wittier) than the one I had sent in.

  117. Oskar Bolza: 'Calculus of Variations
    • Weierstrass's discovery of the fourth necessary condition and his sufficiency proof for a so-called "strong" extremum, which gave for the first time a complete solution, at least for the simplest type of problems, by means of an entirely new method based upon what is now known as Weierstrass's construction." .

  118. Gibson History 10 - Matthew Stewart, John Stewart, William Trail
    • He undoubtedly obtained many important successes in this way; his solution of Kepler's problem being one of the most remarkable.

  119. Archimedes: 'Quadrature of the parabola
    • But I am not aware that any one of my predecessors has attempted to square the segment bounded by a straight line and a section of a right-angled cone [a parabola], of which problem I have now discovered the solution.

  120. Mathematics in Aberdeen
    • Geometry, Algebra, Trigonometry (Plane and Spherical), Conic Sections, Theory of Equations, Analytical Geometry of Two and Three Dimensions, and Differential and Integral Calculus, including the Solution of Differential Equations.

  121. Tullio Levi-Civita

  122. EMS obituary
    • Among longer papers may be mentioned "On the number and nature of the solutions of the Apollonian contact problem" in Vol.

  123. Van der Waerden (print-only)
    • These are solutions of the Burnside problem.

  124. Woodward (print-only)
    • This problem was one requiring for its solution mathematical work of the highest order and, in addition, the experience of the engineer, so to shape his formulas that they could be applied directly by the computer.

  125. Van der Waerden biography
    • These are solutions of the Burnside problem.

  126. Woodward biography
    • This problem was one requiring for its solution mathematical work of the highest order and, in addition, the experience of the engineer, so to shape his formulas that they could be applied directly by the computer.

  127. Finlay Freundlich's Inaugural Address, Part 2
    • Here again, astronomy holds the key position to a final solution.

  128. Turnbull lectures on Colin Maclaurin, Part 2
    • in such a manner as may suggest a synthetic demonstration that may serve to verify the solution.' .

  129. Menger on the Calculus of Variations
    • So Newton's solution had no great effect on the development of mathematics.

  130. Perelman's Fields Medal
    • In three preprints posted on the arXiv in 2002-2003 [The entropy formula for the Ricci flow and its geometric applications; Ricci flow with surgery on three-manifolds; Finite extinction time for the solutions to the Ricci flow on certain three-manifolds], Perelman presented proofs of the Poincare conjecture and the geometrization conjecture.

  131. Chrystal: 'Algebra' Preface
    • I suppose that the student has gone in this way the length of, say, the solution of problems by means of simple or perhaps even quadratic equations, and that he is more or less familiar with the construction of literal formulae, such, for example, as that for the amount of a sum of money during a given term at simple interest.

  132. Ernest Hobson addresses the British Association in 1910, Part 3
    • Only a firm grasp of the principles will give the necessary freedom in handling the methods of Mathematics required for the various practical problems in the solution of which they are essential.

  133. Bronowski and retrodigitisation
    • The topic was capped the following year by the then Editor of the Gazette, R L Goodstein (1912--1985), who noted [11] that Littlewood's approach gave a more succinct solution than Primrose's generalization.

  134. W Burnside: 'Theory of Groups of Finite Order
    • The student of the theory of groups will find here a rich storehouse of material, and the investigator will find numerous suggestions in regard to problems which await solution and methods of attacking them.

  135. Peter Lax's student years
    • The problems can be very difficult, they can look impossible to solve, but if a solution is found, it is unbelievably worthwhile.

  136. Paul Halmos: the Moore method
    • Often a student who hadn't yet found the proof of Theorem 11 would leave the room while someone else was presenting the proof of it - each student wanted to be able to give Moore his private solution, found without any help.

  137. G H Hardy addresses the British Association in 1922, Part 2
    • The first step towards a solution was made by Dirichlet, who proved for the first time, in 1837, that any such arithmetical progression contains an infinity of primes.

  138. K Ollerenshaw: 'The Girls' School
    • The book as a whole, though it leaves no doubt as to the author's own convictions, is a fair-minded presentation of the problem confronting the girls' schools at the present time and of the lines which their solution might be sought.

  139. P G Tait's obituary of Listing
    • The solution of all such questions depends at once on the enumeration of the points of the complex figure at which an odd number of lines meet.

  140. Anna Carlotta Leffler on Sonya Kovalevskaya
    • He did not believe her, but asked her to sit down beside him, after which he began to examine her solutions one by one.

  141. Sheppard Papers
    • (Solutions in terms of differences and sums.)" Ibid., pp.

  142. Somerville's Booklist
    • Mary submitted solutions to some of these and started a mathematical correspondence with him.

  143. Vajda citation
    • That book was a key reference work since it not only introduced the families of optimization problems and the algorithms for their solution, but also set out the scope and limitations of mathematical programming as normative models for managerial decisions.

  144. Eddington: 'Mathematical Theory of Relativity' Introduction
    • We adopt what seems to be the commonsense solution of the difficulty.

  145. Gaschutz' path
    • This is an example that shows how minor variations of the initial conditions can influence the solutions of an equation considerably.

  146. Collins by Wood
    • On the 13th of October 1667 he was elected fellow of the Royal Society upon the publication in the 'Philosophical Transactions' of his 'Solution of a Problem concerning Time, that is, about the Julian Period, with several different Perpetual Almanacks in single Verses; a Chronological Problem', and other things afterwards in the said 'Transactions' concerning 'Merchants Accounts, Compound Interest', and 'Annuities', etc.

  147. Archimedes Quadrature of the parabola

  148. EMS obituary
    • The generation of British mathematicians to which Steggall belonged delighted in proposing and working out problems whose solution might require the aid of any branch of pure or applied mathematics.

  149. Archimedes' 'Quadrature of the parabola
    • But I am not aware that any one of my predecessors has attempted to square the segment bounded by a straight line and a section of a right-angled cone [a parabola], of which problem I have now discovered the solution.

  150. Marie-Louise Dubreil-Jacotin
    • Thanks to new methods, she was able to find the exact solution to certain problems in rotational movement or the study of waves in homogenous and heterogeneous liquids.

  151. Pál Erdös's student years
    • They come from number theory, graph theory, geometry, set theory, and they range in difficulty from ingenious high school competition problems to the most difficult research problems- that defy, and will continue to defy for many years to come, all attempts at solution; their common feature is that they are all fascinatingly interesting.

  152. A D Aleksandrov's view of Mathematics
    • The same theory is useful, for example, in the solution of problems concerning the oozing of water under a dam, problems whose importance is obvious during the present period of construction of huge hydroelectric stations.

  153. Mathematicians and Music 2.1
    • Another of Paul Tannery's suggestions involves finding solutions of a Diophantine equation in three variables.

  154. De Thou on François Vičte
    • Adrian Romanus proposed a problem to all the mathematicians of Europe and Viete, who was the first to solve it, sent his solution to Romanus with corrections and a proof, together with Apollonius Gallus.

  155. University of Edinburgh Examinations
    • The solutions, when not integral, to be carried to two places of decimals.

  156. Airy's work in engineering
    • The latter set includes a subset, those who admire Airy's sensible arrival at a solution to a problem which puzzled him for, perhaps, as long as a milli-second or two.

  157. de Montessus publications
    • Solution du probleme fondamental de la statistique, Annales Societe sc.

  158. Heinrich Tietze on Numbers
    • The solution to this problem is clearly reflected in the nomenclature of numbers.

  159. Percy MacMahon addresses the British Association in 1901
    • The custom of offering prizes for the solutions of definite problems which are necessary to the general advance obtains more in Germany and in France than here, where, I believe, the Adams Prize stands alone.

  160. W H Young addresses ICM 1928 Part 2
    • The assumption of the wave- length as the characteristic of a monochromatic light, does not seem to have availed much towards the solution of the problem.

  161. Isaac Todhunter: 'Euclid' Introduction
    • The construction then usually follows, which states the necessary straight lines and circles which must be drawn in order to constitute the solution of the problem, or to furnish assistance in the demonstration of the theorem.

  162. R A Fisher: the life of a scientist' Preface
    • It had the vitality of his immense pleasure in the process of thinking, the play of ideas, the solution of puzzles.

  163. Sommerville obituary.html
    • The written solutions and comments went far beyond what was necessary for mere elucidation.

  164. Planck's quanta.html
    • For many years, such an aim for me was to find the solution to the problem of the distribution of energy in the normal spectrum of radiating heat.

  165. EMS session 5
    • Thereafter Mr J S Mackay read a paper on the solutions of Euclid's problems with one fixed aperture of the compasses by the Italian geometers of the 16th century; and communicated a note from Mr R Tucker giving some novel properties connected with the triangle.

  166. Mathematics at Aberdeen 4
    • When it was first awarded in 1795, for solutions of questions in geometry, the dies were made larger than intended, but, according to William Knight (a later Professor of Natural Philosophy), this was 'better, as there is less temptation in future time to give away a large than a small medal'.

  167. O Veblen's Opening Address to ICM 1950
    • The solution will not be to give up international mathematical meetings and organizations altogether, for there is a deep human instinct that brings them about.

  168. Napier Tercentenary 4.html.html
    • He hoped by such comparisons to lead to a satisfactory solution of tile problem how best to arrange tabular matter, on what colour of paper, and with what kind of type.

  169. James Jeans addresses the British Association in 1934, Part 3
    • It is only a step from this to a solution of the problem which would have commended itself to many philosophers, from Plato to Berkeley, and is, I think, directly in line with the new world-picture of modern physics.

  170. Schrödinger: 'Statistical Thermodynamics
    • This is the mathematical problem - always the same; we shall soon present its general solution, from which in the case of every particular kind of system every particular classification that may be desirable can be found as a special case.

  171. Coolidge: 'Origin of Polar Coordinates
    • Pascal used the same transformation to calculate the length of a parabolic are, a problem previously solved by Roberval, but his solution was not universally accepted as valid.

  172. G H Hardy: 'Integration of functions
    • My object has been to do what I can to show that this impression is mistaken, by showing that the solution of any elementary problem of integration may be sought in a perfectly definite and systematic way.

  173. Archimedes on mechanical and geometric methods
    • [All these propositions have already been] proved [the solution to the problem of the centre of gravity of a cone must be in a work by Archimedes which has not survived].

  174. EMS 1914 Colloquium 0.html.html
    • (Professor of Mathematics in the University of Edinburgh), on THE SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS IN THE MATHEMATICAL LABORATORY.

  175. EMS obituary
    • In 1 Baker showed that the imaginary part of the same function was also, expressible as a multiple integral, as therefore was the complete function, and, instigated by another paper of Poincare's published in 1902, gave in 2, using his result of 1, a simpler method of obtaining Poincare's solution of Weierstrass' problem.

  176. G C McVittie papers
    • Cahill and McVittie (the authors) write: A solution of Einstein's field equations for the motion of a spherically symmetric distribution of perfect fluid is investigated in an isotropic comoving coordinate system.

  177. Veblen's Opening Address to ICM 1950
    • The solution will not be to give up international mathematical meetings and organizations altogether, for there is a deep human instinct that brings them about.

  178. A N Whitehead addresses the British Association in 1916
    • The solution I am asking for is not a phrase however brilliant, but a solid branch of science, constructed with slow patience, showing in detail how the correspondence is effected.

  179. Collins and Gregory discuss Tschirnhaus
    • I received lately two of your letters, whereby I perceive you have fallen in acquaintance with a very learned gentleman [Walter von Tschirnhaus] and a great admirer of Descartes, whom I also admire so much that he or any other shall help him as to his solution of biquadratic and cubic equations.

  180. EMS obituary
    • Among the best known of these is the theorem of analytical dynamics called after him, namely that, to any set of m invariant relations of a Hamiltonian system, which are in involution, there corresponds a family of 8m particular solutions of the Hamiltonian system, whose determination depends upon the integration of a system of order (m - 1).

  181. Prefaces Landau Lifshitz.html
    • We have not included in this book the various theories of ordinary liquids and of strong solutions, which to us appear neither convincing nor useful.

  182. Laplace: 'Méchanique Céleste
    • The solution of this problem depends, at the same time, upon the accuracy of the observations, and upon the perfection of the analysis.

  183. Dubreil-Jacotin on Maria Gaetana Agnesi
    • She left behind, under the name of Instituzioni analitiche, a "remarkable account of ordinary algebra, with the solution of several solved and unsolved geometric problems"; a second volume, entirely devoted to infinitesimal analysis, a science then quite new, was declared "the most complete and the best done in this field" by the commissioners of the Academy of Sciences of Paris, who were assigned to examine this work at their meeting of 6 December 1749.

  184. Proclus and the history of geometry to Euclid
    • Leon also first discussed the diorismi (distinctions), that is the determination of the conditions under which the problem posed is capable of solution, and the conditions under which it is not.

  185. Collected Papers of Paul Ehrenfest' Preface
    • Lorentz did not like to speak about a problem before he had arrived at a solution, and he reproved - always in an extremely polite and mild way - those who made remarks without due cogitation.

  186. Andrew Forsyth addresses the British Association in 1905, Part 2
    • This accumulation of facts is only one process in the solution of the universe: when the compelling genius is not at hand to transform knowledge into wisdom, useful work can still be done upon them by the construction of organised accounts which shall give a systematic exposition of the results, and shall place them as far as may be in relative significance.

  187. Levi-Civita.html
    • Among the best known of these is the theorem of analytical dynamics called after him, namely that, to any set of m invariant relations of a Hamiltonian system, which are in involution, there corresponds a family of 8m particular solutions of the Hamiltonian system, whose determination depends upon the integration of a system of order (m - 1).

  188. The Edinburgh Mathematical Society: the first hundred years
    • Amongst the methods by which this object might be attained may be mentioned: Reviews of works both British and Foreign, historical notes, discussion of new problems or new solutions, and comparison of the various systems of teaching in different countries, or any other means tending to the promotion of mathematical Education.

  189. Wolfgang Pauli and the Exclusion Principle
    • It is therefore not surprising that I could not find a satisfactory solution of the problem at that time.

  190. Henry Baker addresses the British Association in 1913
    • And, alas! to deal only with one of the earliest problems of the subject, though the finally sufficient conditions for a minimum of a simple integral seemed settled long ago, and could be applied, for example, to Newton's celebrated problem of the solid of least resistance, it has since been shown to be a general fact that such a problem cannot have any definite solution at all.

  191. Steggal obituary.html
    • The generation of British mathematicians to which Steggall belonged delighted in proposing and working out problems whose solution might require the aid of any branch of pure or applied mathematics.

  192. Charles Babbage and deciphering codes
    • I offered to give a few hours to the subject; and if I could see my way to a solution, to continue my researches; but if not on the road to success, to tell him I had given up the task.

  193. A N Whitehead: 'Autobiographical Notes
    • The only point on which I feel certain is that there is no widespread, simple solution.

  194. William Lowell Putnam Mathematical Competition
    • A full list of the questions from the Putnam competition (with solutions!) is available HERE .

  195. Proclus and the history of geometry as far as Euclid
    • Leon also first discussed the diorismi (distinctions), that is the determination of the conditions under which the problem posed is capable of solution, and the conditions under which it is not.

  196. Gibson History 4 - John Napier
    • The Descriptio defines a logarithm, lays down the rules for working with logarithms, illustrating their use particularly by applying them to the solution of triangles, and contains a Table of the logarithms.

  197. Library of Mathematics
    • Solutions of Laplace's equationD R Bland .

  198. Thomas Bromwich: 'Infinite Series
    • The arrangement of the first seven chapters, as well as of Chapter IX, has undergone very little alteration: to the eighth chapter a discussion of the solution of linear differential equations of the second order has been added.

  199. J L Synge and Hamilton
    • For that reason it is not necessary to defend the application of the method to problems which would admit shorter special solutions.

  200. Heinrich Tietze on Numbers, Part 2
    • The extension of the number system was required not only for geometric measurements, but also for the solution of algebraic equations.

  201. J A Schouten's Opening Address to ICM 1954
    • Also the so-called "applied mathematics" came to new life and asked for more men well trained in mathematics and physics, because modern computing machines had made it possible to make use of solutions that formerly only had theoretical value on account of the impossibility of doing the computing work in a reasonable time.

  202. Cochran: 'Sampling Techniques' Introduction
    • On the other hand, if information is wanted for many subdivisions or segments of the population, it may be found that a complete enumeration offers the best solution.

  203. Conforto Gottingen.html
    • Certainly the topic is not easy, but it is very likely that combining the ideas we have, for instance the theory of functionals, with the notions that are in common use here, we could find an easy solution.

  204. Napier Tercentenary
    • He hoped by such comparisons to lead to a satisfactory solution of tile problem how best to arrange tabular matter, on what colour of paper, and with what kind of type.

  205. Mathematics in Aberdeen.html

  206. Letters from Galileo' Preface
    • Only the third question, 'Why did it happen', still divides scholars and others to such an extent that no one can pretend that even an approximate solution has been reached.

  207. Proclus and the history of geometry to Euclid

  208. Whittaker RSE Prize
    • An early and brilliant example was his general solution of Laplace's equation, which might be considered the fundamental partial differential equation of the older physics.

  209. The Edinburgh Mathematical Society: the first hundred years (1883-1983) Part 2
    • His experience in teaching mathematics in the University convinced him that 'algebra, as we teach it, is neither an art nor a science, but an ill-digested farrago of rules whose object is the solution of examination problems.' This led him to write his monumental treatise of nearly 1200 pages on Algebra, which appeared in two parts in 1886 and 1889, and had a powerful effect on the teaching of algebra in Great Britain and abroad.

  210. M Bôcher: 'Integral equations
    • 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.

  211. Fermat's Journal des Sçavans obituary
    • He gave an introduction to loci, plane and three-dimensional, which is an analytical treatise on the solution of problems in two and three dimensions.

  212. EMS summer 1937.html
    • "What is sauce for the goose," he said, "is another man's poison." He proceeded to show how a calculating machine could be used for the rapid numerical solution of complicated problems of interpolation and of algebraic and differential equations.

  213. Coulson: 'Electricity
    • We therefore break off, in Chapters IX and X, to discuss in detail a selected number of such problems, and to illustrate the technique required in their solution.

  214. de Montessus publications
    • Solution du probleme fondamental de la statistique, Annales Societe sc.

  215. Studies presented to Richard von Mises' Introduction
    • "If this goes on", writes von Mises, "the predictions of those who believe that the next step toward the solution of the basic sociological problems must come from physical annihilation of one of the two groups of people will be borne out".

  216. Huygens: 'Traité de la lumičre
    • Since, however, the opinions offered, although ingenious, are not such that more intelligent people would need no further explanations of a more satisfying nature, I wish here to present my thoughts on the subject so that, to the best of my ability, I might contribute to a solution of that part of science which, not without reason, is considered to be one of the most difficult.

  217. Edinburgh Mathematics Examinations
    • The solutions, when not integral, to be carried to two places of decimals.

  218. George Gibson: 'Calculus
    • them; and with the object of encouraging the student to put himself through the drill that is absolutely necessary for the acquisition of facility and confidence in applying the Calculus, I have freely given hints towards the solution of the more important examples.

  219. Three Sadleirian Professors
    • Among Professor Forsyth's own researches Cajori's History of Mathematics specially mentions Differential Invariants and Reciprocants, and Singular Solutions.

  220. Footnote 12
    • He would become seriously upset with the attitude of indifference that my modest temperament made to assume when I told him the solution to a problem or a proof of a theorem that I was able to find.


Quotations

  1. Quotations by Polya
    • "In order to solve this differential equation you look at it till a solution occurs to you." .
    • Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else.
    • We have a natural opportunity to investigate the connections of a problem when looking back at its solution.
    • A GREAT discovery solves a great problem, but there is a grain of discovery in the solution of any problem.

  2. Quotations by Gauss
    • There are problems to whose solution I would attach an infinitely greater importance than to those of mathematics, for example touching ethics, or our relation to God, or concerning our destiny and our future; but their solution lies wholly beyond us and completely outside the province of science.

  3. Quotations by Bernoulli Johann
    • [After reading an anonymous solution to a problem that he realized was Newton's solution.] .

  4. Quotations by Abel
    • The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it.
    • Opening of Memoir on algebraic equations, proving the impossibility of a solution of the general equation of the fifth degree (1824) .

  5. Quotations by Poincare
    • What is it indeed that gives us the feeling of elegance in a solution, in a demonstration? It is the harmony of the diverse parts, their symmetry, their happy balance; in a word it is all that introduces order, all that gives unity, that permits us to see clearly and to comprehend at once both the ensemble and the details.
    • When one tries to depict the figure formed by these two curves and their infinity of intersections, each of which corresponds to a doubly asymptotic solution, these intersections form a kind of net, web or infinitely tight mesh .

  6. A quotation by Peano
    • Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution.

  7. Quotations by Klein
    • The answer to such arguments, however, is that the mathematician, even when he is himself operating with numbers and formulas, is by no means an inferior counterpart of the errorless machine, "thoughtless thinker" of Thomas; but rather, he sets for himself his problems with definite, interesting, and valuable ends in view, and he carries them to solution in appropriate and original manner.

  8. Quotations by Heath
    • One feature which will probably most impress the mathematician accustomed to the rapidity and directness secured by the generality of modern methods is the deliberation with which Archimedes approaches the solution of any one of his main problems.

  9. A quotation by Pacioli
    • And therefore when in your equations you find terms with different intervals without proportion, you shall say that the art, until now, has not given the solutions to this case ..

  10. A quotation by Dudeney
    • A good puzzle should demand the exercise of our best wit and ingenuity, and although a knowledge of mathematics and of logic are often of great service in the solution of these things, yet it sometimes happens that a kind of natural cunning and sagacity is of considerable value.

  11. Quotations by De Morgan
    • The imaginary expression √(-a) and the negative expression -b, have this resemblance, that either of them occurring as the solution of a problem indicates some inconsistency or absurdity.

  12. Quotations by Dirac
    • I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.

  13. Quotations by Eddington
    • The solution goes on famously; but just as we have got rid of all the other unknowns, behold! V disappears as well, and we are left with the indisputable but irritating conclusion: .

  14. Quotations by Fuller
    • But when I have finished, if the solution is not beautiful, I know that it is wrong.

  15. Quotations by Weyl
    • The question of the ultimate foundations and the ultimate meaning of mathematics remains open; we do not know in what direction it will find its final solution or even whether a final objective answer can be expected at all.

  16. Quotations by Descartes
    • The second, to divide each problem I examined into as many parts as was feasible, and as was requisite for its better solution.

  17. Quotations by Alfven
    • The technologists claim that if everything works [in a nuclear fission reactor] according to their blueprints, fission energy will be a safe and very attractive solution to the energy needs of the world.


Chronology

  1. Mathematical Chronology
    • Diophantus of Alexandria writes Arithmetica, a study of number theory problems in which only rational numbers are allowed as solutions.
    • Al-Khayyami (usually known as Omar Khayyam) writes Treatise on Demonstration of Problems of Algebra which contains a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.
    • It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones and is one of the best textbooks in the whole of medieval literature.
    • Later known as Fermat's last theorem, it states that the equation xn + yn = zn has no non-zero solutions for x, y and z when n > 2.
    • which gives solutions to some of Fermat's number theory challenges.
    • He considers integer solutions of ax - by = 1 where a, b are integers.
    • The book discusses 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.
    • He gives solutions for certain special cases to the equation which was first studied by Jacob Bernoulli.
    • The formula connecting surface and volume integrals, now known as "Green's theorem", appears for the first time in the work, as does the "Green's function" which would be extensively used in the solution of partial differential equations.
    • Galois submits his first work on the algebraic solution of equations to the Academie des Sciences in Paris.
    • Liouville publishes Galois' papers on the solution of algebraic equations in Liouville's Journal.
    • Peano proves that if f(x, y) is continuous then the first order differential equation dy/dx = f(x, y) has a solution.
    • Leray shows the existence of weak solutions to the Navier-Stokes equations.
    • This is a major contribution to the solution of the Goldbach conjecture.
    • Douglas gives a complete solution to the Plateau problem, proving the existence of a surface of minimal area bounded by a contour.
    • Matiyasevich shows that "Hilbert's tenth problem" is unsolvable, namely that there is no general method for determining when polynomial equations have a solution in whole numbers.
    • Quidong Wang finds infinite series solutions to the n-body problem (with minor exceptions).
    • A large prize is offered by banker Andrew Beal for a solution to the Beal Conjecture: the equation xp + yq = zr has no solutions for p, q, r > 2 and coprime integers x, y, z.

  2. Chronology for 1990 to 2000
    • Quidong Wang finds infinite series solutions to the n-body problem (with minor exceptions).
    • A large prize is offered by banker Andrew Beal for a solution to the Beal Conjecture: the equation xp + yq = zr has no solutions for p, q, r > 2 and coprime integers x, y, z.
    • A prize of seven million dollars is put up for the solution of seven famous mathematical problems.

  3. Chronology for 1930 to 1940
    • Leray shows the existence of weak solutions to the Navier-Stokes equations.
    • This is a major contribution to the solution of the Goldbach conjecture.
    • Douglas gives a complete solution to the Plateau problem, proving the existence of a surface of minimal area bounded by a contour.

  4. Chronology for 1820 to 1830
    • The formula connecting surface and volume integrals, now known as "Green's theorem", appears for the first time in the work, as does the "Green's function" which would be extensively used in the solution of partial differential equations.
    • Galois submits his first work on the algebraic solution of equations to the Academie des Sciences in Paris.

  5. Chronology for 1970 to 1980
    • Matiyasevich shows that "Hilbert's tenth problem" is unsolvable, namely that there is no general method for determining when polynomial equations have a solution in whole numbers.

  6. Chronology for 1720 to 1740
    • He gives solutions for certain special cases to the equation which was first studied by Jacob Bernoulli.

  7. Chronology for 1880 to 1890
    • Peano proves that if f(x, y) is continuous then the first order differential equation dy/dx = f(x, y) has a solution.

  8. Chronology for 1840 to 1850
    • Liouville publishes Galois' papers on the solution of algebraic equations in Liouville's Journal.

  9. Chronology for 1650 to 1675
    • which gives solutions to some of Fermat's number theory challenges.

  10. Chronology for 900 to 1100
    • Al-Khayyami (usually known as Omar Khayyam) writes Treatise on Demonstration of Problems of Algebra which contains a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.

  11. Chronology for 1700 to 1720
    • The book discusses 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.

  12. Chronology for 1675 to 1700
    • He considers integer solutions of ax - by = 1 where a, b are integers.

  13. Chronology for 1960 to 1970
    • Matiyasevich shows that "Hilbert's tenth problem" is unsolvable, namely that there is no general method for determining when polynomial equations have a solution in whole numbers.

  14. Chronology for 1AD to 500
    • Diophantus of Alexandria writes Arithmetica, a study of number theory problems in which only rational numbers are allowed as solutions.

  15. Chronology for 1300 to 1500
    • It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones and is one of the best textbooks in the whole of medieval literature.

  16. Chronology for 1625 to 1650
    • Later known as Fermat's last theorem, it states that the equation xn + yn = zn has no non-zero solutions for x, y and z when n > 2.


This search was performed by Kevin Hughes' SWISH and Ben Soares' HistorySearch Perl script

Another Search Search suggestions
Main Index Biographies Index

JOC/BS August 2001