Search Results for Liouville


  1. Joseph Liouville (1809-1882)
    • Joseph Liouville .
    • Joseph Liouville's father was an army captain in Napoleon's army so Joseph had to spend the first few years of his life with his uncle.
    • Liouville entered the Ecole Polytechnique in 1825 and attended Ampere's Cours d'analyse et de mecanique Ⓣ in session 1825-26.
    • Although Liouville does not seem to have attended any of Cauchy's courses, it is clear that Cauchy must have had a strong influence on him.
    • Liouville graduated in 1827 with de Prony and Poisson among his examiners.
    • After graduating from the Ecole Polytechnique Liouville entered the Ecole des Ponts et Chaussees.
    • By now Liouville was set on an academic career and he found it impossible to study away from Paris.
    • In 1831 Liouville was appointed to his first academic post, as assistant to Claude Mathieu who had been appointed to Ampere's chair at the Ecole Polytechnique.
    • It is remarkable that during this period of his life Liouville taught between 35 and 40 hours a week at the different institutions.
    • In 1836 Liouville founded a mathematics journal Journal de Mathematiques Pures et Appliquees.
    • This journal, sometimes known as Journal de Liouville, did much for mathematics in France throughout the 19th century.
    • Liouville had already gained an international reputation with papers published in Crelle's Journal but at the same time the quality of Crelle's Journal made him aware of deficiencies in the avenues for mathematical publications which there were in France.
    • Liouville became favourite to fill the chair at the Ecole Polytechnique which fell vacant when Navier died in 1836.
    • In 1837 Liouville was appointed to lecture at the College de France as a substitute for Biot.
    • In 1838 Liouville was appointed Professor of Analysis and Mechanics at the Ecole Polytechnique.
    • In fact the quarrel between Liouville and Libri intensified after his election to the Academie.
    • In 1840, after a vacancy resulting from the death of Poisson, Liouville was elected to the Bureau des Longitudes.
    • In many ways 1840 was a turning point in Liouville's career.
    • As Lutzen writes in [',' B Belhoste and J Lutzen, Joseph Liouville et le College de France, Rev.
    • Before 1840, Liouville had pursued some clear paths to secure his own career; during the following twenty years, the promotion and development of other mathematicians' ideas became the central issue.
    • Life for Liouville developed into a year with two distinct parts.
    • Not everything went Liouville's way however.
    • When Lacroix died in 1843, Liouville applied for his chair at the College de France where he lectured only as a substitute for Biot.
    • Liouville immediately resigned from the College de France, writing in his resignation letter:- .
    • Another aspect of Liouville's life was his involvement in politics.
    • Other mathematical colleagues had also become involved with the political events of the time, for example Catalan, whose political views were similar to the republican views of Liouville, had damaged his mathematical career.
    • Liouville certainly never let his political views hold him back as he advanced his mathematical career, unlike Cauchy who had refused to swear the oaths of allegiance to the King that Liouville and even Arago had been prepared to do.
    • Encouraged by Arago, Liouville stood for election to the Constituting Assembly in 1848.
    • In his recommendation of Liouville as a candidate Arago wrote:- .
    • Mr Liouville is one of my best friends.
    • Elected on 23 April 1848, Liouville took his seat among the moderate republican majority.
    • Liouville continued his political career by being renominated for the Assembly elections in 1849 but the tide had turned against the moderate republicans and he was not elected.
    • The election defeat proved another turning point in Liouville's life.
    • As Lutzen writes in [',' B Belhoste and J Lutzen, Joseph Liouville et le College de France, Rev.
    • The political defeat changed Liouville's personality.
    • His chair at the College de France was declared vacant in 1850 and Cauchy and Liouville competed for the post.
    • In a close contest Liouville triumphed and began his lectures at the College de France in 1851.
    • Although Liouville's mathematical output had been greatly reduced while he was involved with politics, it picked up again in the 1850s despite health problems.
    • In fact 1856 and 1857 were two of Liouville's most productive years.
    • Not only did he have a high teaching load but Liouville was a perfectionist which meant that when he felt that he could not devote all the time necessary to give the best possible lectures he began to suffer.
    • Another blow to Liouville was the death of Dirichlet in 1859.
    • Liouville's mathematical work was extremely wide ranging, from mathematical physics to astronomy to pure mathematics.
    • Usually t is an integer but in this theory developed by Liouville in papers between 1832 and 1837, t could be a rational, an irrational or most generally of all a complex number.
    • Liouville investigated criteria for integrals of algebraic functions to be algebraic during the period 1832-33.
    • Having established this in four papers, Liouville went on to investigate the general problem of integration of algebraic functions in finite terms.
    • Another important area which Liouville is remembered for today is that of transcendental numbers.
    • Liouville's interest in this stemmed from reading a correspondence between Goldbach and Daniel Bernoulli.
    • Liouville certainly aimed to prove that e is transcendental but he did not succeed.
    • His work on boundary value problems on differential equations is remembered because of what is called today Sturm-Liouville theory which is used in solving integral equations.
    • Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.
    • Liouville contributed to differential geometry studying conformal transformations.
    • In 1842 Liouville began to read Galois's unpublished papers.
    • Liouville was therefore a major influence in bringing Galois's work to general notice when he published this work in 1846 in his Journal.
    • Liouville also lectured on Galois's work and Serret, possibly together with Bertrand and Hermite, attended the course.
    • In number theory Liouville wrote around 200 papers, working on quadratic reciprocity and many other topics.
    • A Poster of Joseph Liouville .
    • Honours awarded to Joseph Liouville .
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  2. Eugène Catalan biography
    • At the Ecole Polytechnique, Catalan attended mathematics courses given by Joseph Liouville and Gabriel Lame.
    • 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.
    • He wrote to Liouville in January 1837 who replied encouraging him to come to Paris in the Easter holidays to discuss his future with him.
    • Along with Charles-Francois Sturm and Joseph Liouville, he founded the Ecole Sainte-Barbe near the Sorbonne.
    • As soon as he had returned to Paris, in addition to these other tasks, Catalan had taken Liouville's advice regarding his baccalaureate, and began studying hard.
    • Catalan was, however, still supported by Liouville who proposed him for membership of the Societe Philomatique.
    • Chebyshev's paper appeared in Liouville's Journal de mathematiques pures et appliquees in May 1843.
    • Catalan was not the only one to leave the Ecole Polytechnique following the report of Le Verrier's commission; Michel Chasles, Joseph Liouville and Charles-Francois Sturm also immediately resigned in protest.
    • He had published many articles in Liouville's Journal de mathematiques pures et appliquees but in 1854 he stopped publishing there, preferring to publish in the Comptes rendus of the Academie des Sciences.
    • He was asked to submit a new version of his memoir in 1862 which was considered by a panel consisting of Bertrand, Chasles, Liouville and Serret.
    • Chasles and Liouville recommended that he receive the prize, but the others proposed that the prize should not be awarded.

  3. Vladimir Aleksandrovich Marchenko biography
    • Also in the 1950s he studied the asymptotic behaviour of the spectral measure and of the spectral function for the Sturm-Liouville equation.
    • He also obtained fundamental results in the theory of inverse problems in spectral analysis for the Sturm-Liouville and more general equations.
    • Later Marchenko published monographs on this work: Spectral theory of Strum-Liouville operators (1972); and Sturm-Liouville operators and their applications (1977).
    • The spectral theory of Sturm-Liouville operators is a classical domain of analysis, comprising a wide variety of problems at the present time.
    • Besides the basic results on the structure of the spectrum and the eigenfunction expansion of regular and singular Sturm-Liouville problems, it is in this domain that one-dimensional quantum scattering theory, inverse spectral problems and, of course, the surprising connections of the theory with nonlinear evolution equations first become related.

  4. Pafnuty Chebyshev biography
    • He submitted the paper to Liouville in late 1842 and the paper appeared in Liouville's journal in 1843.
    • Theory 96 (1) (1999), 111-138.','12] the authors suggest that Chebyshev may have visited Paris in 1842 accompanying the Russian geographer Chikhachev who certainly met Catalan (who assisted Liouville in producing his journal) in December of that year.
    • There is no conclusive evidence, but it must be highly likely that if Chebyshev did not personally visit Paris in 1842 then he sent his paper to Liouville via Chikhachev.
    • Liouville and Hermite suggested the idea of developing the ideas on which my thesis had been based.
    • His report covers his studies of applied mechanics as well as his discussions with French mathematicians including Liouville, Bienayme, Hermite, Serret, Poncelet, and English mathematicians including Cayley and Sylvester.

  5. Camille Jordan (1838-1922)
    • It was not until Liouville republished Galois's original work in 1846 that its significance was noticed at all.
    • Serret, Bertrand and Hermite had attended Liouville's lectures on Galois theory and had begun to contribute to the topic but it was Jordan who was the first to formulate the direction the subject would take.
    • It was usually known as the Journal de Liouville since Liouville had founded the journal in 1836.
    • Liouville died in 1882 and in 1885 Jordan became editor of the Journal, a role he kept for over 35 years until his death.

  6. Augustin-Louis Cauchy biography
    • Liouville and Libri were also candidates.
    • Libri was chosen, clearly by far the weakest of the three mathematically, and Liouville wrote the following day that he was:- .
    • Liouville and Cauchy were candidates for the chair again in 1850 as they had been in 1843.
    • After a close run election Liouville was appointed.
    • Subsequent attempts to reverse this decision led to very bad relations between Liouville and Cauchy.

  7. Carl Borchardt biography
    • The year 1846-47 he spent in Paris where he met Chasles, Hermite and Liouville.
    • He attended a course by Liouville on doubly periodic functions and although Liouville intended to publish the notes which Borchardt took of his lectures, in the end they were not published due to a priority dispute between Liouville and Hermite.

  8. Émile Mathieu (1835-1890)
    • He defended his thesis before the examining committee consisting of Gabriel Lame (chairman), Joseph Liouville and Joseph Alfred Serret on 28 March.
    • Mathieu published the results from his thesis in two papers Memoire sur le nombre de valeurs que peut acquerir une fonction quand on y permute ses variables de toutes les manieres possibles Ⓣ (1860) and Memoire sur l'etude des fonctions de plusieurs quantites, sur la maniere de les former et sur les substitutions qui les laissent invariables Ⓣ (1861) both of which were published in Liouville's Journal de mathematiques pures et appliquees.
    • Nor was Lame alone with his opinion in the Academy; Liouville fully approved it.
    • Mathieu presented to the Minister a list of recommendations signed by Joseph Alfred Serret, Jean Victor Poncelet, Jean-Marie Duhamel, Joseph Liouville, Michel Chasles, Charles Delaunay, and Victor Puiseux.

  9. André-Marie Ampère biography
    • Ampere therefore taught electrodynamics at the College de France and this course was taken by Liouville in 1826-27.
    • This was the second time Ampere had taught Liouville since Liouville had taken Ampere's courses at the Ecole Polytechnique in the previous session.
    • Liouville made an important contribution to Ampere's electrodynamics course by editing a set of notes taken from Ampere's lectures.

  10. Joseph Ritt biography
    • Liouville's Theory of Elementary Methods which was published in 1948.
    • This little book is the first treatise to deal with the theory of elementary integrals according to Abel and Liouville, i.e.
    • Among his heroes were Niels Henrik Abel, Augustin Louis Cauchy, David Hilbert, Carl G J Jacobi, Joseph-Louis Lagrange, the marquis Pierre Simon de Laplace, Joseph Liouville and Jules Henri Poincare.

  11. Pierre Laurent (1813-1854)
    • Being late, the memoir was never seriously considered for the Grand Prix, which was won by Pierre Frederic Sarrus (1798-1861), a mathematician working at Strasbourg, with Charles Delaunay's entry receiving an honourable mention, but Cauchy and Liouville were asked to review Laurent's paper and consider it for publication.
    • The Academy of Sciences published the entries of Sarrus and Delaunay but they ignored Cauchy and Liouville's recommendation concerning Laurent and his memoir was not published.
    • Only on 30 October did he present his (and Liouville's) report on Laurent's memoir to the Academy.

  12. Charles Hermite (1822-1901)
    • Hermite may have still been an undergraduate but it is likely that his ideas from around 1843 helped Liouville to his important 1844 results which include the result now known as Liouville's theorem.
    • Sturm and Cauchy gave a good report on this memoir in 1851 but a priority dispute with Liouville seems to have prevented its publication.

  13. William Thomson (1824-1907)
    • There he worked in the physical laboratory of Henri-Victor Regnault and he was soon taking part in deep discussions with Biot, Cauchy, Liouville, Dumas, and Sturm.
    • Perhaps the most profitable discussions that Thomson had in Paris were with Liouville.
    • It was at Liouville's request that Thomson began to try to bring together the ideas of Faraday, Coulomb and Poisson on electrical theory.

  14. Charles-François Sturm (1803-1855)
    • Liouville was also working on differential equations derived from the theory of heat.
    • Papers of 1836-1837 by Sturm and Liouville on differential equations involved expansions of functions in series and is today well-known as the Sturm-Liouville problem, an eigenvalue problem in second order differential equations.

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

  16. Guglielmo Libri biography
    • One such powerful figure was Liouville who worked for many years against Libri and the two would attack each other at every opportunity in meetings of the Academy.
    • Liouville was in many ways someone who Libri should not have competed with, for he was an outstanding mathematician who could usually come up with a more elegant proof of Libri's results than he could himself.
    • A little-known consequence of these disputes is that Liouville made his famous announcement of Evariste Galois's important work on the theory of equations in response to an attack by Libri in 1843.

  17. Jacques Binet biography
    • This journal, known as Liouville's Journal, had been set up by Joseph Liouville in 1836 and he actively sought papers for the journal by encouraging mathematicians to write them.
    • Liouville thought the problem deserved further work and approached Binet encouraging him to investigate.

  18. Vladimir Maz'ya biography
    • In 1997 (with Vladimir Kozlov) Maz'ya published Theory of a higher-order Sturm-Liouville equation which Eastham summarises by writing that:- .
    • the authors have identified a special type of higher-order analogue of the hyperbolic Sturm-Liouville equation and they have developed a coherent theory based on the Green's function.

  19. Naum Il'ich Feldman biography
    • In addition to his work on the measure of transcendence of numbers, Feldman also produced many results strengthening Liouville's theorem on the rational approximation of algebraic numbers.
    • In An effective power sharpening of a theorem of Liouville in 1971 he proved the following theorem:- .

  20. Évariste Galois (1811-1832)
    • However the papers reached Liouville who, in September 1843, announced to the Academy that he had found in Galois' papers a concise solution .
    • Liouville published these papers of Galois in his Journal in 1846.

  21. Victor Puiseux (1820-1883)
    • During this period Puiseux published a series of more than ten papers in Liouville's Journal.
    • His task consisted of supplementing the lectures of the professors, such as Duhamel, Liouville, and Cauchy, set exercises for students, correct them, explained them, and show where the ideas lead.

  22. Marcel Riesz (1886-1969)
    • In 1949, Riesz published a 223 page paper L'integrale de Riemann-Liouville et le probleme de Cauchy Ⓣ in which he introduced a multiple integral of Riemann-Liouville type and showed how important this idea is in the theory of the wave equation.

  23. Joseph Saurin (1659-1737)
    • He defended Huygens' theory of the pendulum after it was attacked by le Chevalier de Liouville in eclaircissement sur une difficulte propose aux mathematiciens par M le Chevalier de Liouville (1722).

  24. Boris Yakovlevich Levin biography
    • He was already publishing papers and, in addition to the one we mentioned above, he had published The arithmetic properties of holomorphic functions (1933), The intersection of algebraic curves (1934), Entire functions of irregular growth (1936), and The growth of the Sturm-Liouville integral equation (1936).
    • He became interested in almost periodic functions, quasi-analytic classes and related problems of completeness and approximation, algebraic problems of the theory of entire functions, and Sturm-Liouville operators.

  25. Nicolai Vasilievich Bugaev biography
    • Over a period of two and a half years he studied under Kummer and Weierstrass in Berlin and Liouville in Paris.
    • Bugaev gave proofs of theorems stated without proof by Liouville.

  26. Ernst Mohr biography
    • He continued to examine Sturm-Liouville problems, in particular the limit circle case for the Sturm-Liouville problem.

  27. Louis Lefébure de Fourcy biography
    • There he taught the special class and among his pupils we mention in particular Joseph Liouville.

  28. Jules Bienaymé biography
    • The French paper was published in Liouville's Journal de Mathematiques Pures et Appliquees and the editor clearly realised that the inequality had been given by Bienayme fourteen years earlier since he reprinted Bienayme's 1853 paper immediately before Chebyshev's paper [',' E Seneta, I J Bienayme [1796-1878]: Criticality, Inequality, and Internationalization, International Statistical Review 66 (3) (1998), 291-301.','14]:- .

  29. Paulette Libermann biography
    • For example she gave a survey of various geometric concepts and results used in analytical mechanics in her lecture Liouville forms, parallelisms and Cartan connections to the Jean Leray '99 Conference, and reviewed and summarized the theory of Cartan connections in her lecture Cartan connections and momentum maps given at the Classical and Quantum Integrability conference held in Warsaw in 2001.

  30. Henri Brocard biography
    • Henri Brocard's parents were Elizabeth Auguste Liouville and Jean Sebastien Brocard.

  31. Alfréd Haar (1885-1933)
    • He examined the standard systems of orthonormal trigonometric functions and also orthonormal systems related to Sturm-Liouville differential equations.

  32. István Feny biography
    • This covers topics such as the Green function technique for a one-dimensional Sturm-Liouville problem, and Dirichlet and Carl Neumann problems for the two- and three-dimensional Laplacian operator.

  33. Pierre Wantzel biography
    • In 1837 Wantzel published proofs of what are some of the most famous mathematical problems of all time in a paper in Liouville's Journal on .

  34. Aleksandr Nikolaevich Korkin biography
    • Korkin attended lectures by Liouville, Lame and Bertrand in Paris, returned briefly to Russia in May 1863, then went to Germany where he attended lectures by Kummer, Weierstrass and others in Berlin.

  35. Georg Cantor biography
    • Liouville established in 1851 that transcendental numbers exist.

  36. Edvard Phragmén biography
    • Phragmen's first progress towards this was his paper Sur une extension d'un theoreme classique de la theorie des fonctions (1904) in which he extended Liouville's result that every entire bounded function must be constant to show that the same result could be obtained for entire functions with controlled growth in a given sector that were bounded outside the sector.

  37. Adriaan Cornelis Zaanen biography
    • In this thesis he studied the theory of the Sturm-Liouville two boundary value problem [',' C D Aliprantis, In memoriam: Yuri Alexander Abramovich and Adriaan Cornelis Zaanen, Positivity 9 (3) (2005), 269-272.','1]:- .

  38. Alfredo Capelli biography
    • In Sulla limitata possibilita di trasformazioni conformi nello spazio Ⓣ (1886), Capelli gives a geometrical proof of Liouville's theorem on when conformal mappings are Mobius transformations.

  39. Thomas Hirst (1830-1892)
    • From Berlin Hirst made the journey to Paris where he spent two months attending lectures by Liouville and Lame.

  40. James Ivory (1765-1842)
    • His work on the ellipsoidal equilibrium configuration of self-gravitating fluids was an extension of that of Laplace, and it influenced the achievements of Jacobi and Liouville which followed.

  41. Viktor Vladimirovich Vagner biography
    • Among Vagner's early papers we mention Differential geometry of non-linear non-holonomic manifolds in the three-dimensional Euclidean space (1940), The geometry of an (n-1)-dimensional non-holonomic manifold in an n-dimensional space (Russian) (1941), Geometric interpretation of the motion of non-holonomic dynamical systems (Russian) (1941), On the problem of determining the invariant characteristics of Liouville surfaces (Russian) (1941), and On the Cartan group of holonomicity for surfaces (1942).

  42. Gaston Darboux biography
    • From 1873 to 1878 he was suppleant to Liouville in the chair of rational mechanics at the Sorbonne.

  43. Nikolay Sonin (1849-1915)
    • In the middle of this period Nicolai Vasilievich Bugaev, after studying for a period of two and a half years with Kummer and Weierstrass in Berlin and Liouville in Paris, was appointed as a professor at Moscow University.

  44. Gösta Mittag-Leffler (1846-1927)
    • Although Mittag-Leffler met many mathematicians in Paris, such as Bouquet, Briot, Chasles, Darboux, and Liouville, the main aim of the visit was to learn from Hermite.

  45. Hermann Weyl (1885-1955)
    • His habilitation thesis Uber gewohnliche Differentialgleicklungen mit Singularitaten und die zugehorigen Entwicklungen willkurlicher Funktionen Ⓣ investigated the spectral theory of singular Sturm-Liouville problems.

  46. Israil Gelfand biography
    • Another important area of his work is that on differential equations where he worked on the inverse Sturm-Liouville problem.

  47. Klaus Roth biography
    • Liouville showed in 1844 that if r is an algebraic number of degree n then μ(r) ≤ n.

  48. Mikhail Vasilevich Ostrogradski (1801-1862)
    • Liouville had produced similar results.

  49. Charles Delaunay biography
    • He suffered a great tragedy in 1849 when his young wife died and it was largely due to Liouville's efforts to support him at this time that he was able to continue with his academic work.

  50. Georges Reeb biography
    • The paper of this talk contains an intrinsic definition of the Liouville form.

  51. Edgar Raymond Lorch biography
    • In a series of remarkable memoirs, Liouville demonstrated the impossibility of evaluating certain indefinite integrals, and of solving certain differential equations, in terms of elementary functions.

  52. Karl Heinrich Weise biography
    • In years soon after the war Weise published a small but concise book Gewohnliche Differentialgleichungen Ⓣ (1948) in which he discusses Legendre, Bessel, and Sturm-Liouville equations.

  53. Ulisse Dini biography
    • He studied surfaces and developed ideas related to those of Liouville and Beltrami.

  54. Carl Siegel (1896-1981)
    • It extended an idea first noted by Liouville, then pushed forward by Thue who proved that, given a rational number q and any e > 0 there are only finitely many rational numbers p/q (in their lowest terms) such that .

  55. Heinz Prüfer biography
    • He also published a paper on Sturm-Liouville theory; Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen Ⓣ (1926).

  56. Antonio Bordoni biography
    • When he became acquainted with the work of Liouville and the ideas of Gauss, he encouraged his colleagues and students at the University of Pavia to develop them.

  57. Paul Mansion (1844-1919)
    • During these years Mansion published 15 papers in the journal, for example: Demonstration d'un theoreme de Liouville Ⓣ (1874); Theorie analytique des transformations lineaires Ⓣ (1874, 1876, 1887, 1888); Sur les carres magiques Ⓣ (1876); Remarques sur les theoremes arithmetiques de Fermat Ⓣ (1879); and Derivees des fonctions elementaires d'une variable imaginaire Ⓣ (1880).

  58. Jan Mikusinski biography
    • Rudolf Hilfer, Yury Luchko and Zivorad Tomovski write in [',' 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.','7]:- .

  59. Jean-Marie Duhamel (1797-1872)
    • However, from 1851, he again filled the analysis chair at the Ecole Polytechnique after Liouville was appointed to the vacant chair at the College de France.

  60. Felice Casorati biography
    • In fact the first to make an investigation relevant to this problem was Liouville who began the attempt to characterize analytic functions in terms of their singularities and to investigate spherical representation.

  61. Georges de Rham biography
    • In April 1930 his thesis was complete and he sent a copy of it to Lebesgue who helped him to publish it in Liouville's journal, the Journal de mathematiques pures et appliquees.

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

  63. Enrico Magenes biography
    • This is a nonlinear Sturm-Liouville problem.

  64. Jean Claude Saint-Venant (1797-1886)
    • Saint-Venant attended lectures at the College de France and the lecture notes he took in Liouville's 1839-40 class still survive.

  65. Eugène Rouché biography
    • An oral examination was held on 8 November 1858 with Louis Lefebure de Fourcy as President, and Gabriel Lame and Joseph Liouville as examiners.

  66. Alan Baker biography
    • The theory of transcendental numbers, initiated by Liouville in 1844, has been enriched greatly in recent years.

  67. Adolf Kneser (1862-1930)
    • One of these areas is that of linear differential equations; in particular he worked on the Sturm-Liouville problem and integral equations in general.

  68. Charles Pisot biography
    • French mathematics is filled with the names of leaders in the Theory of Numbers: Fermat, Galois, Lagrange, Liouville and others.

  69. George Green (1793-1841)
    • Sixty years later Thomson recalled his excitement and that of Liouville and Sturm, to whom he showed the work in Paris in the summer of 1845.

  70. Ludwig Sylow (1832-1918)
    • In Paris he attended lectures by Michel Chasles on the theory of conics, by Joseph Liouville on rational mechanics and by Jean-Marie Duhamel on the theory of limits.

  71. Carl Jacobi (1804-1851)
    • This result prompted much further work in this area, in particular by Liouville and Cauchy.

  72. George Birkhoff biography
    • His research concentrated on asymptotic expansions, boundary value problems, and Sturm-Liouville type problems but his thesis advisor Eliakim Moore appears to have been a less influential guide to Birkhoff than was Poincare.

  73. Pierre Bonnet biography
    • In 1878 Bonnet succeeded Le Verrier to the chair at the Sorbonne, then in 1883 he succeeded Liouville as a member of the Bureau des Longitudes.

  74. Yuri Ivanovich Manin biography
    • For example, a mathematician represents the motion of planets of the solar system by a flow line of an incompressible fluid in a 54-dimensional phase space, whose volume is given by the Liouville measure ..

  75. Józeph Petzval biography
    • He was influenced by the work of Liouville and wrote both a long paper and a two volume treatise on the Laplace transform and its application to ordinary linear differential equations.

  76. William Birnbaum biography
    • He published results he obtained on Sturm-Liouville theorems in Abschatzung der Eigenwerte eines Sturm-Liouvilleschen Eigenwert-Problems mit Koeffizienten von beschraenkter Schwankung Ⓣ (1930) and work on approximation theorems in function space in two papers Uber Approximation im Mittel Ⓣ (1930) (the first written with Władysław Orlicz, the second being single-authored).

  77. Urbain Le Verrier (1811-1877)
    • When the recommendations were approved, some reactions at the school were stark: Liouville and Chasles resigned immediately in protest at this rejection of the emphasis on teaching purish mathematics that they had inherited and continued from predecessors such as Cauchy.

History Topics

  1. Hirst's diary
    • Liouville .
    • Joseph Liouville: .
    • (18 May 1879) A little shrivelled gouty old man [Liouville] has become and very garrulous.

  2. Orbits
    • In 1836 Liouville studied planetary theory, the three body problem and the motion of the minor planets Ceres and Vesta.
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    • Liouville made a number of very important mathematical discoveries while working on the theory of perturbations including the discovery of Liouville's theorem "when a bounded domain in phase space evolves according to Hamilton's equations its volume is conserved".
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  3. Fermat's last theorem
    • Lame acknowledged that the idea was suggested to him by Liouville.
      Go directly to this paragraph
    • However Liouville addressed the meeting after Lame and suggested that the problem of this approach was that uniqueness of factorisation into primes was needed for these complex numbers and he doubted if it were true.
      Go directly to this paragraph
    • On 24 May Liouville read a letter to the Academie which settled the arguments.
      Go directly to this paragraph

  4. Group theory

  5. Abstract groups
    • Galois defined a group in 1832 although it did not appear in print until Liouville published Galois' papers in 1846.
    • Now in 1845, one year before Liouville published the above definition by Galois, Cauchy gave a definition.

  6. Real numbers 2
    • Liouville's interest in transcendental numbers stemmed from reading a correspondence between Goldbach and Daniel Bernoulli.
    • Liouville certainly aimed to prove that e is transcendental but he did not succeed.

  7. Trisecting an angle
    • In 1837 Wantzel published proofs in Liouville's Journal of:- .

  8. Doubling the cube
    • In 1837 Wantzel published proofs in Liouville's Journal of:- .

  9. Set theory
    • Cantor now remarks that this proves a theorem due to Liouville, namely that there are infinitely many transcendental (i.e.
      Go directly to this paragraph

  10. Ring Theory
    • Liouville suggested that the proof depended on a unique decomposition into primes which was unlikely to be true.

  11. Real numbers 3
    • We can specify Liouville's transcendental number easily enough as having a 1 in place n! and 0 elsewhere.

Societies etc

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  1. Rue Joseph Liouville
    • Rue Joseph Liouville .

  2. Liouville
    • Joseph Liouville .

  3. Fellow of the Royal Society
    • Joseph Liouville 1850 .

  4. Lunar features
    • (W) (L) Liouville .

  5. Fellows of the RSE
    • Joseph Liouville1875More infoMacTutor biography .

  6. Fellows of the RSE
    • Joseph Liouville1875More infoMacTutor biography .

  7. Lunar features
    • Liouville .

  8. Paris street names
    • Rue Joseph Liouville (15th Arrondissement) WnMn .


  1. References for Joseph Liouville
    • References for Joseph Liouville .
    • .
    • J Lutzen, Joseph Liouville 1809-1882: Master of Pure and Applied Mathematics (New York- Berlin, 1990).
    • B Belhoste and J Lutzen, Joseph Liouville et le College de France, Rev.
    • C Berg and J Lutzen, J Liouville's unpublished work on an integral operator in potential theory : A historical and mathematical analysis, Exposition.
    • G Chrystal, Joseph Liouville, Proc.
    • J Lutzen, Joseph Liouville's Contribution to the Theory of Integral Equations, Historia Mathematica 9 (1982), 371-391.
    • J Lutzen, Joseph Liouville's Work on the Figures of Equilibrium of a Rotating Mass of fluid, Rev.
    • J Lutzen, The birth of spectral theory - Joseph Liouville's contributions, in Proceedings of the International Congress of Mathematicians (Tokyo, 1991), 1651-1663.
    • J Lutzen, Joseph Liouville und die nach ihm benannten Satze, NTM Schr.
    • J Lutzen, The geometrization of analytical mechanics : a pioneering contribution by Joseph Liouville (ca.
    • J Lutzen, Sturm and Liouville's work on ordinary linear differential equations : The emergence of Sturm - Liouville theory, Arch.
    • E Neuenschwander, The unpublished papers of Joseph Liouville in Bordeaux, Historia Math.
    • E Neuenschwander, Joseph Liouville (1809-1882) : correspondance inedite et documents biographiques provenant de differentes archives parisiennes, Boll.
    • J Peiffer, Joseph Liouville (1809-1882) : ses contributions a la theorie des fonctions d'une variable complexe, Rev.
    • D Stander, Makers of modern mathematics : Joseph Liouville, Bull.

  2. References for Charles-François Sturm
    • J Lutzen, Sturm and Liouville's work on ordinary linear differential equations : The emergence of Sturm - Liouville theory, Arch.

  3. References for Jan 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.

Additional material

  1. V Lebesgue publications
    • Journal de Liouville (1) I (1836), 266-268.
    • Journal de Liouville (1) II (1837), 253-292.
    • Journal de Liouville (1) III (1838), 113-144.
    • Journal de Liouville (1) IV (1839), 9-59.
    • Journal de Liouville (1) II (1837), 337-365.
    • Journal de Liouville (1) IV (1839), 60-62.
    • Journal de Liouville (1) V (1840), 42-71.
    • Journal de Liouville (1) V (1840), 184-185.
    • Journal de Liouville (1) V (1840), 186-188.
    • Journal de Liouville (1) V (1840), 276-279.
    • Journal de Liouville (1) V (1840), 281-310.
    • Journal de Liouville (1) V (1840), 348-349.
    • Journal de Liouville (1) VI (1841), 17-35.
    • Journal de Liouville (1) VII (1842), 137-159.
    • Journal de Liouville (1) VIII (1843), 49-70.
    • Journal de Liouville (1) X (1845), 316-319.
    • Journal de Liouville (1) XI (1846), 76-80.
    • Journal de Liouville (1) XI (1846), 331-335.
    • Victor Amedee Lebesgue, Extrait d'une Lettre adressee a M Liouville, Journal de Mathematique pures et appliquees.
    • Journal de Liouville (1) XI (1846), 336-337.
    • Journal de Liouville (1) XI (1846), 338-340.
    • Journal de Liouville (1) XII (1847), 457-473.
    • Journal de Liouville (1) XII (1847), 497-517.
    • Journal de Liouville (1) XV (1850), 215-237.
    • Journal de Liouville (1) XVIII (1853), 73-86.
    • Journal de Liouville (1) XIX (1854), 289-300.
    • Journal de Liouville (1) XIX (1854), 334-336.
    • Journal de Liouville (2) I (1856), 377-378.
    • Journal de Liouville (2) I (1856), 401-410.
    • Journal de Liouville (2) III (1857), 149-152.
    • Journal de Liouville (2) III (1858), 391-394.
    • Journal de Liouville (2) IV (1859), 105-110.
    • Journal de Liouville (2) IV (1859), 366.
    • Journal de Liouville (2) IV (1859), 389-398.
    • Journal de Liouville (2) IV (1859).
    • Victor Amedee Lebesgue, Extrait d'une Lettre de M Le Besgue a M Liouville, Journal de Mathematique pures et appliquees.
    • Journal de Liouville (2) VII (1862), 417-420.
    • Victor Amedee Lebesgue, Notice sur les Notes et Memoires inseres dans le Journal de Mathematiques de M Liouville, jusqu'a present (1860), par V-A Le Besgue (Ouvrage posthume), Bullettino di Bibliografia e di Storia delle Scienze matematiche e fisiche, pubblicato da B Boncompagni IX (1876), 574-582.

  2. Hardy Inaugural Lecture
    • The next advance towards the solution of the problem was made by Liouville, who established the existence of g(4).
    • Liouville's proof, which was first published in 1859, is quite simple and, as the simplest example of an important type of argument, is worth reproducing here.
    • It has not the striking simplicity of Liouville's proof; but it has enabled successive investigators to reduce the number of cubes, until finally Wieferich, in 1909, proved that g(3) ≤ 9.
    • Starting from this formula he was able, by an exceedingly ingenious process based upon the definition of a definite integral as the limit of a finite sum, to prove the existence in the general case of algebraical identities analogous to that used by Liouville and his followers when k is 4.

  3. Who was who 1852
    • At the College de France in Paris we would have found Joseph Liouville (1809-1882), the man who started the mathematical theory of boundary value problems for linear second order differential equations, who produced the first integral equation and the first resolvent, but also the founder of the theory of transcendental numbers.
    • He was the editor of the Journal de Mathematiques Pures et Appliquees, started in 1835 and still known as the Journal de Liouville.
    • In connection with Liouville we often think of Jacques-Charles-Francois Sturm (1805-1855); he was Poisson's successor as professor of mechanics at the Sorbonne in 1850.
    • Liouville started his journal in 1835.

  4. Hille publications
    • On the zeros of Sturm-Liouville functions, Ark.
    • Convex distribution of the zeros of Sturm-Liouville functions, Bull.
    • A correction: Convex distribution of the zeros of Sturm-Liouville functions, Bull.

  5. Thue speeches
    • Thue once told me that in Leipzig he had discovered a proof of the existence of transcendental numbers, but it turned out that Liouville had already come up with the same proof.
    • "When I went into the library and saw the book which contained Liouville's proof, which was almost identical with mine, I was so struck that I can still see before me the place on the shelf where the book stood," he said.
    • Uber Annaherungswerte algebraischer Zahlen: "Vergleiche den Beweis von Liouville fur die Existenz transzendenter Grossen; Journ.

  6. G H Hardy: 'Integration of functions
    • I have borrowed largely from the Cours d'Analyse of Hermite and Goursat, but my greatest debt is to Liouville, who published in the years 1830-40 a series of remarkable memoirs on the general problem of integration which appear to have fallen into an oblivion which they certainly do not deserve.
    • It was Liouville who first gave rigid proofs of whole series of theorems of the most fundamental importance in analysis - that the exponential function is not algebraical, that the logarithmic function cannot be expressed by means of algebraical and exponential functions, and that the standard elliptic, integrals cannot be expressed by algebraical, exponential and logarithmic functions.

  7. M Bôcher: 'Integral equations
    • Abel and Liouville, however, and after them others began the treatment of special integral equations in a perfectly conscious way, and many of them perceived clearly what an important place the theory was destined to fill.
    • This was true not merely in the early days of Abel and Liouville, but also more recently in the cases of Volterra and Fredholm.

  8. Prufer's publications
    • Neue Herleitung der Sturm-Liouvilleschen Reihenentwicklung stetiger Funktionen (New derivation of the storm-Liouville series expansion of continuous functions) (1926).

  9. Cafaro's papers
    • The established theory of linear diffusion phenomena is reformulated in the framework of non-Hamiltonian dynamical systems theory (Liouville's approach), through the introduction of a suitable characteristic velocity for the processes under consideration.

  10. Perron books
    • The final chapter discusses their construction with special reference to Liouville's numbers and the criteria for them.

  11. Publications of Eduard Heine
    • E Heine, Lettre a M Re'sal, Liouville J., (3) II (1876), 155-158.

  12. Hermite's works
    • This dates back to Hermite's younger days, and was only recently discovered among the papers of Joseph Liouville.

  13. Catalan retirement
    • Apres ma sortie de l'Ecole polytechnique, je suis devenu le disciple et l'ami de Liouville, de Sturm, de Lame, d'Arago, de Chasles.

  14. Catalan retirement
    • After I left the Ecole Polytechnique, I became the disciple and friend of Liouville, Sturm, Lame, Arago and Chasles.

  15. Galois book review
    • Some additional material, such as copies of some manuscripts by Auguste Chevalier and corrected proofs by Liouville when he worked on this material over 10 years after Galois' death, are also in the dossiers.

  16. Moiseiwitsch Variational Principles
    • After summarizing the theory of the small oscillations of a dynamical system at the beginning of the fourth chapter, Rayleigh's principle is proved and the Ritz variational method developed for the Sturm-Liouville equation.


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Famous Curves

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  1. Mathematical Chronology
    • Liouville founds a mathematics journal Journal de Mathematiques Pures et Appliquees.
    • This journal, sometimes known as Journal de Liouville, did much to advance mathematics in France throughout the 19th century.
    • Liouville discusses integral equations and gives the "Sturm-Liouville theory" which is used in solving such equations.
    • Liouville announces to the Academie des Sciences in Paris that he had found deep results in Galois's unpublished work and promises to publish Galois's papers together with his own commentary.
    • Liouville finds the first transcendental numbers - numbers that cannot be expressed as the roots of an algebraic equation with rational coefficients.
    • Liouville publishes Galois' papers on the solution of algebraic equations in Liouville's Journal.
    • Liouville publishes a second work on the existence of specific transcendental numbers which are now known as "Liouville numbers".

  2. Chronology for 1840 to 1850
    • Liouville announces to the Academie des Sciences in Paris that he had found deep results in Galois's unpublished work and promises to publish Galois's papers together with his own commentary.
    • Liouville finds the first transcendental numbers - numbers that cannot be expressed as the roots of an algebraic equation with rational coefficients.
    • Liouville publishes Galois' papers on the solution of algebraic equations in Liouville's Journal.

  3. Chronology for 1830 to 1840
    • Liouville founds a mathematics journal Journal de Mathematiques Pures et Appliquees.
    • This journal, sometimes known as Journal de Liouville, did much to advance mathematics in France throughout the 19th century.
    • Liouville discusses integral equations and gives the "Sturm-Liouville theory" which is used in solving such equations.

  4. Chronology for 1850 to 1860
    • Liouville publishes a second work on the existence of specific transcendental numbers which are now known as "Liouville numbers".

EMS Archive

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BMC Archive

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Gazetteer of the British Isles

  1. Cambridge Individuals
    • He was thrilled by it, as were Liouville and Sturm, and it was republished in J.

  2. References
    • Makers of modern mathematics: Joseph Liouville.

Astronomy section

  1. List of astronomers

  2. List of astronomers
    • Liouville, Joseph .

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