Search Results for Borel


  1. Émile Borel biography
    • Felix Edouard Justin Emile Borel .
    • Emile Borel's father was Honore Borel who was a Protestant minister.
    • Honore Borel himself was the son of a craftsman from Montauben, the capital of Tarn-et-Garonne in the Midi-Pyrenees region.
    • At the time Emile was born, his parents were living in a fine eighteenth century house in which Honore Borel had a school for the children of Protestant families in the neighbourhood.
    • Honore Borel was an intelligent man and for the first years of his son's life he educated him at home.
    • However [',' E F Collingwood, Emile Borel, J.
    • Family friends urged Borel to enter the Ecole Polytechnique, which was considered the more prestigious establishment, but he had other ideas.
    • Borel was advised that a degree from the Ecole Polytechnique would give him the best opportunities for a job in industry or business.
    • The two became firm friends and through the Darboux family Borel came to know leading mathematicians of the day, in particular Emile Picard who he later remarked was a major influence on him at this time.
    • Collingwood writes [',' E F Collingwood, Emile Borel, J.
    • the theory of measure, Borel's theory of divergent series, his theory of non-analytic continuation and the theory of quasi-analytic functions all derive from ideas which make their first appearance in this paper.
    • And it contained the explicit statement and proof of the famous covering theorem which, quite inappropriately, acquired the name of the Heine-Borel theorem ..
    • Later Borel spoke about the mathematicians who had influenced him most in his early years mentioning, among others, Camille Jordan, Emile Picard, Paul Appell, Edouard Goursat, Paul Painleve and Marcel Brillouin.
    • At almost exactly the same time that he was receiving his doctorate, when still only 22 years of age, Borel was appointed Maitre de Conference at the University of Lille.
    • Borel achieved much over the next years, both in his career and in the outstanding mathematics which he produced.
    • They had no children but adopted one of Borel's nephews, Fernand Lebeau, the son of his eldest sister who he had lived with while a pupil at the Lycee at Montauben.
    • Marguerite Borel was an outstanding author, writing under the pen name Camille Marbo (Marbo being the first three letters of Marguerite and the first two of Borel), and received the distinction of being awarded the Prix Femina in 1913 for La Statue voilee Ⓣ.
    • In 1909 Borel was appointed to a chair of Theory of Functions created specially for him at the Sorbonne and he went on to hold this professorship until 1941.
    • During World War I he volunteered for military service and commanded an artillery battery [',' E F Collingwood, Emile Borel, J.
    • On the fall of Painleve in 1917, Borel returned for a time to the front ..
    • In 1928, with financial support from Rockefeller and Rothschild, he set up the Institut Henri Poincare (the Centre Emile Borel is now part of the Institute) and he ran the Institute for thirty years.
    • In [',' M Frechet, La vie et l’oeuvre d’Emile Borel, Enseignement mathematique 11 (1965), 1-95.','8] Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics.
    • Borel created the first effective theory of the measure of sets of points.
    • Borel, although not the first to define the sum of a divergent series, was the first to develop a systematic theory for a divergent series which he did in 1899.
    • His contributions to this area are described in detail in [',' E Knobloch, Emile Borel as a probabilist, in The probabilist revolution Vol 1 (Cambridge Mass., 1987), 215-233.','9] and we quote here part of the abstract of that paper:- .
    • In addition to many textbooks, Borel published more than fifty papers between 1905 and 1950 on the calculus of probability.
    • In addition, between 1921 and 1927, Borel published a series of papers on game theory and became the first to define games of strategy.
    • After 1924, Borel became active in the French government serving in the French Chamber of Deputies (1924-36) and as Minister of the Navy (1925-40).
    • In 1946, when he was 75 years old, Borel published the fascinating book Les paradoxes de l'infini Ⓣ.
    • A Poster of Emile Borel .
    • Honours awarded to Emile Borel .
    • 4.nLunar featuresnCrater Borel .
    • 5.nParis street namesnRue Borel and Square Borel (17th Arrondissement) .
    • .

  2. Armand Borel biography
    • Armand Borel .
    • Armand Borel attended secondary school in Geneva but was also educated at a number of private schools.
    • As well as Stiefel, Borel had attended lectures at the Ecole Polytechnique Federale by Hopf who played an important role in influencing Borel's mathematical tastes.
    • Following his graduation, Borel was appointed as a teaching assistant at the Ecole Polytechnique Federale in Zurich.
    • Jean Leray became Borel's thesis supervisor and he attended courses which he gave at the College de France.
    • Borel wrote [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
    • After his year in Paris, Borel went to Geneva where he substituted for the professor of algebra from 1950 to 1952.
    • In 1952 Borel married Gabrielle Aline Pittet; they had two daughters Dominique Odette Susan and Anne Christine.
    • In the autumn of 1952 Borel, and his new wife Gaby (as she was always known) set off for the United States.
    • Borel had been invited to spend a year at the Institute for Advanced Study at Princeton and this was extended to a second year.
    • Haefliger writes [',' A Haefliger, Armand Borel (1923-2003) (French), Gaz.
    • This was an opportunity for Borel to learn a great deal about algebraic geometry and number theory from Weil.
    • We should note at this point the major contribution that Borel made to Bourbaki.
    • He describes his experiences in [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
    • Borel writes in [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
    • Pierre Cartier writes [',' N Mok, Armand Borel in Hong Kong, Asian J.
    • I think this was basically the influence of one person, Armand Borel.
    • Haefliger sums up his contributions in [',' A Haefliger, Armand Borel (1923-2003) (French), Gaz.
    • Borel's work, apart from a dozen books, lecture notes ..
    • Among his books are Topics in the homology theory of fibre bundles (1967), which is based on lectures Borel gave at the University of Chicago in 1954 in which he described the state of the topic at that time adopting the same methods and points of view as in his thesis.
    • Also in 1969 Linear algebraic groups was published based on a graduate course given by Borel at Columbia University in the spring of 1968.
    • One book which does not seem to be based on a lecture course is Automorphic forms on SL(R) which Borel himself says would have been better titled "Introduction to some aspects of the analytic theory of automorphic forms on SL(R) and the upper half-plane X." .
    • Borel received many honours for his outstanding contributions to mathematics.
    • The citation states that Borel's results:- .
    • Borel also received the Balzan prize in 1992:- .
    • In fact we learn much of Borel's view of mathematics in the reply he made on receiving the Balzan prize:- .
    • Among his interests we mention music in particular [',' J Arthur, E Bombieri, K Chandrasekharan, F Hirzebruch, G Prasad, J-P Serre, T A Springer, J Tits, Armand Borel (1923-2003), Notices Amer.
    • Also, as Bombieri writes in [',' J Arthur, E Bombieri, K Chandrasekharan, F Hirzebruch, G Prasad, J-P Serre, T A Springer, J Tits, Armand Borel (1923-2003), Notices Amer.
    • Chandrasekharan writes [',' J Arthur, E Bombieri, K Chandrasekharan, F Hirzebruch, G Prasad, J-P Serre, T A Springer, J Tits, Armand Borel (1923-2003), Notices Amer.
    • Borel was an astute observer: he had an uncanny eye for artistic detail and would reflect on the influence of literature and culture on human outlook.
    • Borel loved to travel and made visits to many countries including India, Mexico and China.
    • Honours awarded to Armand Borel .
    • Zhejiang University (Pictures of Armand Borel) .
    • .

  3. Eduard Heine (1821-1881)
    • He is best remembered for the Heine-Borel theorem:- .
    • The paper [',' P Dugac, Sur la correspondance de Borel et le theoreme de Dirichlet- Heine- Weierstrass- Borel- Schoenflies- Lebesgue, Arch.
    • The second part of this paper covers the history of the Heine-Borel theorem and is summarised in the following review:- .
    • The last half of the paper is devoted to a more systematic account of the gradual discovery and formulation of the so-called Heine-Borel theorem.
    • Borel formulated his theorem for countable coverings in 1895 and Schonflies and Lebesgue generalized it to any type of covering in 1900 and 1898 (published 1904), respectively.
    • The priority questions are nicely illustrated with quotes from the correspondence between Lebesgue and Borel and other letters.

  4. André Weil (1906-1998)
    • He took every opportunity to make the most of these years [',' A Borel, Andre Weil, 6 May 1906 - 6 August 1998, Proceedings of the American Philosophical Society 145 (1) (2001), 107-114.','11]:- .
    • Borel writes [',' A Borel, Andre Weil, Bull.
    • Shiing-Shen Chern writes [',' A Borel, P Cartier, K Chandrasekharan, S-S Chern and S Iyanaga, Andre Weil (1906-1998), Notices Amer.
    • Komaravolu Chandrasekharan writes in [',' A Borel, P Cartier, K Chandrasekharan, S-S Chern and S Iyanaga, Andre Weil (1906-1998), Notices Amer.

  5. Edward Van Vleck biography
    • In [',' A Novikoff and J Barone, The Borel law of normal numbers, the Borel zero-one law, and the work of Van Vleck, Historia Math.
    • It is argued that Van Vleck proved the first zero-one law, anticipating the zero-one law of Borel and, more strikingly, that of Kolmogorov.
    • By following Van Vleck's own steps in deriving consequences of his zero-one law, a result ("the extended Van Vleck theorem") is given which is directly comparable to Borel's law of normal numbers.
    • Finally, it is shown that the Van Vleck zero-one law, which in generality falls between that of Borel and that of Kolmogorov, is further distinguished in that it provides the key step in establishing what may be the earliest example in ergodic theory of a metrically transitive transformation.

  6. Lyudmila Vsevolodovna Keldysh biography
    • This was a construction of an arithmetic example (using continued fractions) of a set belonging to the fourth Borel class.
    • It was the first significant advance in the study of the Borel classification since the first non-trivial example of a third-class set was given by Baire in 1905.
    • Keldysh continued to undertake research on the structure of Borel sets and in 1941 she defended her doctoral thesis Structure of B-Sets (Russian).
    • First she published the paper Sur la structure des ensembles mesurables B Ⓣ (1944) which summarised her very considerable contributions to the structure of sets of higher Borel class.
    • A Russian monograph published in 1945 also gave a detailed description of the results she had obtained over several years concerning the classification of Borel sets in the space of irrational numbers lying in (0, 1).

  7. Georges Darmois biography
    • He was then given the position of 'Agrege preparateur' which was reserved for the most brilliant students, assisting Emile Borel who was director of the Ecole Normale Superieure.
    • This came about because Emile Borel, who had set up the course in 1923, had been elected to the Chamber of Deputies in 1925 and looked for someone he could trust to give the course for him.
    • At the International Congress of Mathematicians held at Bologna in September 1928, one of the plenary talks was given by Emile Borel who gave the address Le calcul des probabilites et les sciences exactes.
    • As well as the Institute of Statistics in Paris, there was also the Institut Henri Poincare set up in 1928, with Borel as its head, to facilitate interactions between researchers in probability and mathematical physics.
    • In the second part the general space of states, constituting a Borel field, is considered, with probability defined, as usual, as an additive function of states.

  8. Georges Valiron biography
    • Valiron was awarded a bursary to enable him to study for his doctorate at the Faculty of Science in Paris and he spent the years 1912-14 undertaking research advised by Emile Borel.
    • Further publications appeared: Sur les fonctions entieres d'ordre nul et d'ordre fini et en particulier les fonctions a correspondance reguliere Ⓣ (1913); Sur quelques theoremes de M Borel Ⓣ (1914) and Sur le calcul approche de certaines fonctions entieres Ⓣ (1914).
    • Among the papers that Valiron published in the years following World War I, we mention: Les theoremes generaux de M Borel dans la theorie des fonctions entieres Ⓣ (1920); Recherches sur le theoreme de M Picard Ⓣ (1921); Recherches sur le theoreme de M Picard dans la theorie des fonctions entieres Ⓣ (1922); Sur les fonctions entieres verifiant une classe d'equations differentielles Ⓣ (1923); Sur l'abscisse de convergence des series de Dirichlet Ⓣ (1924); Sur les surfaces qui admettent un plan tangent en chaque point Ⓣ (1926); and Sur la distribution des valeurs des fonctions meromorphes Ⓣ (1926).
    • At the Congress, he gave the lecture Le theoreme de Borel-Julia dans la theorie des fonctions meromorphes Ⓣ.

  9. René Eugène Gateaux biography
    • Volterra himself, invited by Borel and Hadamard, came to Paris to give a series of lectures on functional analysis, published in 1913 ([',' Vito Volterra, Lecons sur les fonctions de lignes (Gauthier-Villars, Paris, 1913).','23]) and whose redaction was precisely made by Peres.
    • On 18 April 1913, Emile Borel wrote to Volterra that:- .
    • Moreover, on the postcard sent by Borel to Volterra on 1 January 1914 with his best wishes for the new 1914 year (a sentence which sounds strange to the ears of one knowing what was going to happen soon ..
    • .), Borel mentioned how he was glad to learn that Volterra was absolutely satisfied with Gateaux in Rome.

  10. Carlo Bonferroni biography
    • While an undergraduate, he read Emile Borel's Elements de la theorie des probabilites Ⓣ and noticed an error that Borel had made in a coin tossing problem.
    • He wrote to Borel but never received a reply, although he noticed that Borel corrected the error in the second edition of his book.

  11. Rolf Nevanlinna (1895-1980)
    • He starts with the construction by Weierstrass of an entire function with a preassigned sequence of zeros, the introduction by Laguerre of the genus and the results by Poincare and Hadamard on the link between the exponent of convergence of the zeros and the order of the function which had first been defined by Borel.
    • Borel also proved the extension of Picard's theorem which states that the exponent of convergence of the a-points of any entire function is equal to its order except for at most one value of a.
    • through R Nevanlinna's successive papers culminating in the monograph 'Le Theoreme de Picard-Borel et la theorie des fonctions meromorphes' Ⓣ(1929).

  12. Louis Bachelier biography
    • It is known, however, that he received occasional scholarships to continue his studies (on the recommendation of Emile Borel (1871-1956)) and he gave lectures as a 'free professor' at the Sorbonne between 1909 and 1914.
    • Borel, however, must have known Bachelier (he had approved the scholarships to Bachelier).
    • It seems that Bachelier, was regarded as being of lesser importance in the eyes of the French mathematical elite (Hadamard, Borel, Lebesgue, Levy, Baire).

  13. Harish-Chandra (1923-1983)
    • Armand Borel describes some of Harish-Chandra's contributions in [',' A Borel, Some recollections of Harish-Chandra, in The mathematical legacy of Harish-Chandra, Baltimore, MD, January 9-10, 1998 (Amer.
    • In October 1983 a conference in honour of Armand Borel was held in Princeton [',' R P Langlands, Harish-Chandra, Biographical Memoirs of Fellows of the Royal Society of London 31 (1985), 199-225.','14]:- .

  14. Simion Stoilow biography
    • He was able to attend lectures by Picard, Poincare, Goursat, Hadamard, Borel and Lebesgue.
    • Later in his career he wrote articles on some of these outstanding French mathematicians (for example Mathematical work of Henri Lebesgue (Romanian) (1942) and Emile Borel and modern mathematical analysis (Romanian) (1956)).
    • His book Lecons sur les principes topologiques de la theorie des fonctions analytiques Ⓣ, published in the prestigious Collection Borel (Paris, 1937), became a classical reference in the 1940s.

  15. Lee Lorch biography
    • in 1941 for his thesis Some Problems on the Borel Summability of Fourier Series.
    • The estimation of such constants for various summability methods was calculated by Lorch and his first published paper, following on from his thesis work, was The Lebesgue constants for Borel summability (1944) which, as the title indicates made a study in the case of Borel summability.

  16. Joseph Pérès (1890-1962)
    • At the Ecole Normale Superieure, the professors who had the most influence on Peres were Ernest Vessiot, Emile Borel and Jacques Hadamard.
    • Emile Borel introduced him to Vito Volterra when Volterra made one of his many journeys to France to promote scientific collaboration.
    • Peres was examined in May 1915 by three examiners, Emile Borel (President), Ernest Vessiot and Jacques Hadamard.

  17. Mikhail Yakovlevich Suslin biography
    • Luzin suggested that his students work on Borel sets and asked Suslin to read Henri Lebesgue's 1905 paper Sur les functions representables analytiquement Ⓣ.
    • Aleksandrov had proved that every Borel set can be obtained by applying an operation (named an A-operation by Suslin) to a closed set.
    • He emphasized that he was proposing this terminology in my honour, by analogy with Borel sets, which by then were usually called B-sets.

  18. Francesco Cantelli biography
    • Although his name is frequently connected with the name of E Borel, Cantelli's approach to probability is very different from that of Borel.
    • Around the time that Cantelli worked on the law of large numbers, Borel was also interested in the topic.

  19. Naum Il'ich Feldman biography
    • This result was strengthened by Borel in 1899 when he proved a lower bound for P(e), where P is a polynomial with integer coefficients, depending on the maximum modulus of the integer coefficients of P.
    • Gelfond, Feldman's supervisor, had extended Borel's result to numbers of the form αβ, where α, β are algebraic numbers.
    • Feldman proved in his thesis Borel type results (called the measure of transcendence) for logarithms of algebraic numbers, obtaining estimates for the lower bound depending (as did Gelfond) on both the degree of P and the maximum modulus of its coefficients.

  20. Anders Wiman biography
    • He was the first to do so and his work influenced Emile Borel to work on such questions.
    • For more details, see [','levnadsteckningar 10 (1968-74), 181-194.','6] and [',' L Mazliak, How Paul Levy saw Jean Ville and Martingales, Electronic Journal for History of Probability and Statistics 5 (1) (June 2009).','7] where theories are put forward as to how Borel learnt of Wiman's paper.
    • Trygve Nagell (1895-1988), who succeeded to Wiman's chair at Uppsala when he retired, writes [',' L Mazliak and M Sage, Au dela des reels Borel et l’approche probabiliste de la realite.','8]:- .

  21. Deane Montgomery biography
    • I was especially interested in Borel sets, analytic sets, and projective sets ..
    • Armand Borel expresses similar views in [',' A Borel, Deane Montgomery (1909-1992), Notices Amer.

  22. Abraham Plessner biography
    • In Eine Kennzeichunung der totalstetigen Funktionen Ⓣ (1929), Plessner characterised the absolutely continuous measures among the class of Borel measures.
    • In point of difficulty, it stands between these and de la Vallee Poussin's monograph in the Borel series.

  23. Henri Lebesgue (1875-1941)
    • Building on the work of others, including that of Emile Borel and Camille Jordan, Lebesgue formulated the theory of measure in 1901 and in his famous paper Sur une generalisation de l'integrale definie Ⓣ, which appeared in the Comptes Rendus on 29 April 1901, he gave the definition of the Lebesgue integral that generalises the notion of the Riemann integral by extending the concept of the area below a curve to include many discontinuous functions.
    • During the first world war he worked for the defence of France, and at this time he fell out with Borel who was doing a similar task.

  24. Jean Dieudonné biography
    • Armand Borel writes in [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.

  25. Jerzy Neyman (1894-1981)
    • He arrived in Paris in the summer of 1926 to visit Borel.
    • In Paris for session 1926-27 Neyman attended lectures by Borel, Lebesgue (whose lectures he particularly enjoyed) and Hadamard and his interests began to move back towards sets, measure and integration.

  26. Tomás Rodríguez Bachiller biography
    • In particular he attended Emile Borel's course on elasticity, Jules Drach's course on contact transformations, Emile Picard's course on algebraic curves and surfaces, Elie Cartan's course on fluid mechanics, Jacques Hadamard's course on differential equations, the course that Claude Guichard (1861-1924) delivered on differential geometry and Laplace transformations, Henri Lebesgue's course on topology and Ernest Vessiot's courses on partial differential equations and on group theory.
    • He reviewed work of mathematicians across the whole of Europe, in particular writings articles on the topology of Maurice Frechet, the doctoral theses of Luigi Fantappie on algebraic geometry and of Louis Antoine on topology, and the monograph L'Analysis situs et la Geometrie algebrique Ⓣ written by Solomon Lefschetz for the Borel Collection.

  27. Felix Bernstein biography
    • When Cantor saw Bernstein's proof he was so impressed that he communicated it to Emile Borel and it was published in Borel's Lecons sur la theorie des fonctions in 1898.

  28. Friedrich Hirzebruch biography
    • After serving as a Scientific Assistant at the University of Erlangen during 1951-52, he spent the two years 1952-54 at the Institute for Advanced Study in Princeton in the United States working with Armand Borel, Kunihiko Kodaira, and D C Spencer on topics such as sheaf theory, vector bundles, characteristic classes and Thom cobordism.
    • the proportionality theorem for complex homogeneous manifolds and (with Armand Borel) the general theory of characteristic classes of homogeneous spaces of compact Lie groups, .

  29. Iossif Vladimirovich Ostrovskii biography
    • In the late 1970s he studied complex-valued Borel measures on the real axis.
    • He continued to work on this topic with his student Alexandr M Ulanovskii and they surveyed their results in this area in the joint paper Classes of complex-valued Borel measures than can be uniquely determined by restrictions (1989).

  30. René Baire biography
    • The letters written to Baire by de la Vallee Poussin, and reproduced in [',' H Gispert, La theorie des ensembles en France avant la crise de 1905 : Baire, Borel, Lebesgue ..
    • While on the topic of letters, we should remark that [',' A Buhl and G Bouligand, En memoire de Rene Baire, L’enseignement mathematique 31 (1932), 5-13.','4] contains fifty letters written by Baire to Emile Borel.

  31. Henri Padé biography
    • In this post he succeeded Emile Borel who had just left Lille to take up an appointment at the Ecole Normale Superieure in Paris.
    • Although the theory of Pade approximants which he had developed in his thesis, and in many later papers, was not quick to be taken up by many other mathematicians, it did become well known after Borel presented Pade approximants in his 1901 book on divergent series.

  32. Maurice Fréchet (1878-1973)
    • Another task undertaken by Frechet around this time was writing up Borel's lectures for publication.
    • It was Borel who encouraged Frechet to seek positions in Paris and he supported his candidacy.

  33. Arthur Schönflies (1853-1928)
    • In this work he was the first to give the name 'Heine-Borel theorem' to the theorem which today is always known by this name.
    • Schonflies tells the reader that the proof of the Heine-Borel theorem is one of the most significant applications of transfinite numbers.

  34. Jean Leray biography
    • For Leray [',' A Borel, G M Henkin and P D Lax, Jean Leray (1906-1998), Notices Amer.
    • In 1959 he [',' A Borel, G M Henkin and P D Lax, Jean Leray (1906-1998), Notices Amer.

  35. Dmitrii Menshov biography
    • Menshov attended Luzin's lecture course, and when Luzin posed the open problem of whether the Denjoy integral and the Borel integral were equivalent, he was able to solve the problem.
    • It appeared as the paper The relationship between the definitions of the Denjoy and Borel integrals in 1916.

  36. Arnaud Denjoy biography
    • In 1902 Denjoy entered the Ecole Normale Superieure where he studied under Borel, Painleve and Emile Picard.
    • Denjoy worked on functions of a real variable in the same areas as Borel, Baire and Lebesgue.

  37. Felix Hausdorff (1868-1942)
    • Hausdorff proved further results on the cardinality of Borel sets in 1916.

  38. Kenneth May biography
    • However, his interests changed in the 1950 towards mathematical education and, in the 1960s, to the history of mathematics with papers such as The origin of the four-color conjecture (1965) and biographies of Paul Appell, Eric Temple Bell and Emile Borel in the Dictionary of Scientific Biography.

  39. Charles De la Vallée Poussin (1866-1962)
    • Further important texts published by him were his Borel tract on the Lebesgue integral (1916), approximation theory (1919), mechanics (1924), and potential theory (1937).

  40. Duro Kurepa biography
    • Chapter 4 is on topological and metric spaces, with the fifth and final chapter on limiting processes in analysis, measure theory, Borel and Souslin sets.

  41. Samuel Verblunsky biography
    • The 1930 Mathematical Proceedings of the Cambridge Philosophical Society contains eight papers by Verblunsky, namely: A property of continuous arcs (submitted May 1929), The relation between Riemann's method of summation and Cesaro's (submitted May 1929), Note on the sum of an oscillating series (submitted October 1929), Note on the Gibbs Phenomenon (submitted November 1929), The convergence of singular integrals (submitted February 1930), Note on the modified Heine-Borel theorem (submitted April 1930), A property of continuous arcs II (submitted July 1930), and Note on the sum of an oscillating series II (submitted July 1930).

  42. Ernest Hobson (1856-1933)
    • His book Theory of Functions of a Real Variable published in 1907 was the first English book on the measure and integration developed by Baire, Borel and Lebesgue.

  43. Gustave Choquet biography
    • He studied Emile Borel's Lecons sur la theorie des fonctions Ⓣ which fascinated him.

  44. Pavel Aleksandrov biography
    • Aleksandrov proved his first important result in 1915, namely that every non-denumerable Borel set contains a perfect subset.

  45. Paul Appell biography
    • One of his three daughters was to marry Borel.

  46. Claude Hardy (1598-1678)
    • Finally, we know Hardy undertook chemistry experiments with Annibal Barlet, a physician who taught alchemy in Paris, and with Pierre Borel, before he became physician to Louis XIV in 1654.

  47. Georges de Rham biography
    • Dumas encouraged de Rham to read the works of Henri Poincare while Mirimanoff advised him to read books by Emile Borel, Rene Baire, Henri Lebesgue and Joseph Serret's Cours d'algebre superieure Ⓣ [',' Georges de Rham, The First Century of the International Commission on Mathematical Instruction (1908-2008).','16]:- .

  48. Vito Volterra (1860-1940)
    • In 1897 he attended the first International Congress of Mathematicians in Zurich and there met Paul Painleve and Emile Borel who invited him to Paris in the following year.

  49. Jules Drach (1871-1949)
    • Drach was a friend of Borel, and together they published lectures by Poincare Lecons sur la theorie de l'elasticite Ⓣ (1892) and by Jules Tannery Introduction a l'etude de la theorie des nombres et de l'algebre superieure Ⓣ (1895) while Drach was a student at the Ecole Normale Superieure.

  50. Cesare Arzelà biography
    • Those produced in his last few years include: Sul limite di un integrale doppio Ⓣ (1907); Sul teorema di Borel Ⓣ (1909); and Su alcune questioni di calcolo funzionale Ⓣ (1910).

  51. Gheorghe Calugrenu biography
    • His investigations on Picard's fundamental theorem, on theorems of Borel and Nevanlinna in connection with the study of exceptional values of meromorphic functions of finite genus, made him already in the third and fourth decades of this century one of the first important Romanian mathematicians as well as a mathematician of European stature and distinguished member of the Romanian school of complex analysis founded by David Emmanuel and Dimitrie Pompeiu.

  52. Evan (Ianto) Davies biography
    • Now the person who influenced Davies most during these years was Paul Dienes, a Hungarian who had worked under Emile Borel and Jacques Hadamard in Paris.

  53. Paul-André Meyer biography
    • France, which had been the cradle of probability with Pascal, was running the risk of seeing the line of her famous theoreticians (Borel, Frechet, Levy ..

  54. J C Burkill (1900-1993)
    • This was a particularly active area of research in the early decades of this century after the pioneering work of Lebesgue, Borel and their contemporaries in establishing the concepts of measure and the Lebesgue integral associated with it.

  55. François Bruhat biography
    • Finally we should mention that Bruhat was a member of Bourbaki being a third generation member along with Armand Borel, Alexandre Grothendieck, Pierre Cartier, Serge Lang, and John Tate.

  56. Wacaw Sierpiski (1882-1969)
    • Borel had proved such numbers exist but Sierpiński was the first to give an example.

  57. Nikolai Luzin (1883-1950)
    • Luzin proceeded from the point of view of the French school (Borel, Lebesgue), which greatly influenced him.

  58. John von Neumann (1903-1957)
    • Early in his work, a paper by Borel on the minimax property led him to develop ..

  59. Eliakim Moore (1862-1932)
    • He brought to culmination the study of improper integrals before the appearance of the more effective integration theories of Borel and Lebesgue.

  60. Otto Blumenthal biography
    • After this he went to Paris where he spent the winter of 1899-1900 studying under Borel and Jordan.

  61. Ferran Sunyer biography
    • The bulk of his work, published between 1939 and 1970, is on topics in the theory of entire and meromorphic functions, where his main tendency is that of generalizing important results, and is grouped in the volume under review into eight sections: (i) "miscellaneous papers", one of which contains generalizations of the theorems of Landau, Schottky, Picard, and of Montel's normality criterion, another on a new method for the summation of power series, but also a paper that falls entirely outside of the realm of analysis (to which all other papers are devoted), answering a question raised by Sierpinski in 1951 on order types; (ii) and (iii), spread over 220 pages, the study of exceptional values of an entire function as influenced by the lacunarity of the Taylor (or Dirichlet) series defining it, (ii) containing F Sunyer i Balaguer's 'Acta Mathematica' paper of 1952; (iv) work on the behaviour of an entire function and its derivatives and primitives along Borel-Valiron directions of maximal type; (v) generalizations of S Mandelbrojt's fundamental inequality (vi) topics in the theory of entire functions and overconvergence; (vii) generalizations of quasi-periodic and elliptic functions; (viii) differentiable functions of a real variable, containing a lovely result (the only one with a co-author, E Corominas) published in 1954) ..

  62. Julio Rey Pastor (1888-1962)
    • He also brought important foreign mathematicians to the university to give short courses, including: Frederigo Enriques (1925), Francesco Severi (1930), Tullio Levi-Civita (1937), Emile Borel (1928) and Jacques Hadamard (1930).

  63. Zoárd Geöcze biography
    • He spent 1908 in Paris where he learnt of the effective theory of the measure of sets of points being developed by Borel, Baire and Lebesgue.

  64. Jean Chazy (1882-1955)
    • This laboratory was under the direction of Jacques Duclaux who was married to the radiologist Germaine Berthe Appell, making Duclaux the son-in-law of Paul Appell and the brother-in-law of Emile Borel.

  65. Raymond Paley biography
    • Zygmund discovered Paley's extraordinary talent and the two worked jointly on existence proofs, brilliantly applying ideas from Borel's Calcul des probabilites denombrables.

  66. Pia Nalli biography
    • In this work she studied the theory of the integral, in particular bringing together recent work on the subject by Emile Borel, Henri Lebesgue, Charles de la Vallee Poussin, Giuseppe Vitali and Arnaud Denjoy.

  67. Robert Geary biography
    • At the Sorbonne, Geary attended lectures by Emile Borel, who held the chair of Theory of Functions, Elie Cartan, who held the Chair of Differential and Integral Calculus, Edouard Goursat, an expert on differential equations, and Henri Lebesgue who was Professor of the Application of Geometry to Analysis.

  68. L E J Brouwer biography
    • A couple of months later he made an important visit to Paris, around Christmas 1909, and there met Poincare, Hadamard and Borel.

  69. Tadeusz Waewski biography
    • His doctoral dissertation, on topological results relating to dendrites, was examined in 1923 by the powerful examining committee consisting of Borel, Denjoy and Montel.

  70. Ambrose Rogers biography
    • His later work covered a wide range of different topics in geometry and analysis including Borel functions, Hausdorff measure and local measure, topological properties of Banach spaces and upper semicontinuous functions.

  71. Mikhail Alekseevich Lavrent'ev biography
    • This was an important time for the young mathematician who was advised by Paul Montel, attended lectures by Emile Borel, Gaston Julia and Henri Lebesgue, and participated in Jacques Hadamard's seminar.

  72. Gilbert Hunt biography
    • In the last section potential theory is reached and some important results are proved: the completed maximum principle for potentials, balayage, and the almost-Borel measurability of excessive functions.

  73. Gheorghe ieica biography
    • Among his lecturers were a whole host of leading mathematicians including Darboux, Picard, Poincare, Appell, Goursat, Hadamard, and Borel.

  74. Jules Tannery (1848-1910)
    • Through his lectures and supervisory duties at the Ecole Normale this gifted teacher gave valuable guidance to many students and inspired a number of them to seek careers in science (for example, Paul Painleve, Jules Drach, and Emile Borel).

  75. Hans Hahn biography
    • In this area he studied a construction of the Lebesgue integral as a limit of Riemann sums, an integral proposed by Borel around 1910, and worked on the theory of abstract measures, in particular product measures.

  76. Aleksandr Yakovlevich Khinchin (1894-1959)
    • With these ideas he also strengthened the law of large numbers due to Borel.

History Topics

  1. Real numbers 3
    • Emile Borel introduced the concept of a normal real number in 1909.
    • Borel called a number normal (in base 10) if every k-digit number occurred among all the k digit blocks about 1/10k of the time.
    • Now Borel was able to prove that, in one sense, almost every real number was normal.
    • However despite proving these facts, Borel couldn't show that any specific number was absolutely normal.
    • In 1927 Borel came up with his "know-it-all" number.
    • Borel describes k as an unnatural real number, or an "unreal" real.
    • Borel devotes a whole book [',' E Borel, Les nombre inaccessible (Gauthier-Villiars, Paris, 1952).','1], which he published in 1952, to discuss another idea, namely that of an "inaccessible number".
    • An accessible number, to Borel, is a number which can be described as a mathematical object.
    • However, as Borel pointed out, there are a countable number of such descriptions.

  2. Bourbaki 2
    • Armand Borel first became acquainted with the Bourbaki team in 1949.
    • He described [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
    • Armand Borel, Francois Bruhat, Pierre Cartier, Alexander Grothendieck, Serge Lang, and John Tate are considered the younger members of this third generation.
    • Armand Borel explains in [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
    • Grothendieck presented to the next Congress a detailed draft of two chapters, the first containing preliminaries to the book on manifolds and categories of manifolds, while the second contained differentiable manifolds, and the differential formalism [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
    • For this Armand Borel must take the bulk of the credit for he was the main driving force behind the style and content of this part.

  3. References for Bourbaki 2
    • A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
    • A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Mitt.
    • K Krickeberg, Comment: "Twenty-five years with Nicolas Bourbaki 1949-1973" by A Borel, Mitt.

  4. References for Bourbaki 1
    • A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
    • A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Mitt.
    • K Krickeberg, Comment: "Twenty-five years with Nicolas Bourbaki 1949-1973" by A Borel, Mitt.

  5. Bourbaki 1
    • Armand Borel explains the subtle title that was chosen for the whole work [',' A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.

  6. References for Mathematical games
    • R W Dimand and M A Dimand, The early history of the theory of strategic games from Waldegrave to Borel, in Toward a history of game theory (Durham, NC, 1992), 15-27.

  7. References for Real numbers 3
    • E Borel, Les nombre inaccessible (Gauthier-Villiars, Paris, 1952).

  8. Set theory

  9. Pi history
    • Is π normal ? That is does every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense? The concept was introduced by Borel in 1909.

  10. References for Set theory
    • H Gispert, La theorie des ensembles en France avant la crise de 1905 : Baire, Borel, Lebesgue ..

Societies etc

  1. French Mathematical Society
    • 1905 E Borel .


  1. Rue Emile Borel
    • Rue Emile Borel .

  2. International Congress Speaker
    • Emile Borel, Definition et domaine d'existence des fonctions monogenes uniformes.
    • Emile Borel, Le calcul des probabilites et les sciences exactes.
    • Georges Valiron, Le theoreme de Borel-Julia dans la theorie des fonctions meromorphes.
    • Armand Borel, Arithmetic Properties of Linear Algebraic Groups.

  3. Karp Prize
    • for their work on Borel equivalence relations, in particular for their results on turbulence and countable Borel equivalence relations.

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

  5. MAA Chauvenet Prize
    • The Borel Theorem and Its Generalizations, Bull.

  6. LMS Honorary Member
    • 1939 E Borel .

  7. Times Obituaries
    • Borel's biographyThe obituary (1956) .

  8. AMS Steele Prize
    • 1991 Armand Borel .

  9. Brouwer Medal
    • 1978 A Borel .

  10. AMS Colloquium Lecturer
    • 1971 Armand Borel .

  11. Lunar features
    • Borel .

  12. Groups St Andrews.html
    • Superrigidity and Borel equivalence relations .

  13. Paris street names
    • Rue Emile Borel (17th Arrondissement) WnMn .


  1. References for Émile Borel
    • References for Emile Borel .
    • .
    • S Callens, Ensemble, mesure et probabilite selon Emile Borel, Math.
    • E F Collingwood, Emile Borel, J.
    • E F Collingwood, Addendum: Emile Borel, J.
    • L de Broglie, Notice sur la vie et l'oeuvre de Emile Borel, Academie des Sciences (9 December 1957).
    • M Frechet, La vie et l'oeuvre d'Emile Borel, Enseignement mathematique 11 (1965), 1-95.
    • E Knobloch, Emile Borel as a probabilist, in The probabilist revolution Vol 1 (Cambridge Mass., 1987), 215-233.
    • Lettres de Rene Baire a Emile Borel, Cahiers du Seminaire d'Histoire des Mathematiques 11 (Univ.
    • B Maurey and J-P Tacchi, La genese du theoreme de recouvrement de Borel, Rev.
    • F A Medvedev, The Du Bois-Reymond theorem and ordinal transfinite numbers in the investigations of E Borel (Russian), Istor.-Mat.
    • P Montel, Notice necrologique sur Emile Borel, C.
    • P Montel, Necrologie: Emile Borel, Rev.
    • O Onicescu, Emile Borel (1871-1956), the creator of the theory of measure (Romanian), Gaz.
    • B Penkov, Emile Borel (1871-1956) (on the occasion of the 100th anniversary of his birth) (Bulgarian), Fiz.-Mat.
    • S Stoilow, Emile Borel and modern mathematical analysis (Romanian), Gaz.

  2. References for Armand Borel
    • References for Armand Borel .
    • A Borel, Oeuvres : collected papers, Vol.
    • A Borel, Oeuvres : collected papers, Vol.
    • A Borel, Oeuvres : collected papers, Vol.
    • A Borel, Oeuvres : collected papers, Vol.
    • J Arthur, E Bombieri, K Chandrasekharan, F Hirzebruch, G Prasad, J-P Serre, T A Springer, J Tits, Armand Borel (1923-2003), Notices Amer.
    • A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
    • B Casselman, Some of my memories of Armand Borel, Asian J.
    • A Haefliger, Armand Borel (1923-2003) (French), Gaz.
    • A Haefliger, Armand Borel (1923-2003), European Math.
    • L Ji, Armand Borel as a mentor, Asian J.
    • A W Knapp, J-P Serre, K Chandrasekharan, E Bombieri, F Hirzebruch, T A Springer, J Tits, J Arthur, G Prasad and M Goresky, Armand Borel, Asian J.
    • N Mok, Armand Borel in Hong Kong, Asian J.
    • J-P Serre, Discours prononce en seance publique le 30 septembre 2003 en hommage a Armand Borel (1923-2003), Gaz.
    • N R Wallach, Armand Borel: A reminiscence, Asian J.

  3. References for André Weil
    • A Borel, Andre Weil (6 May 1906 - 6 August 1998).
    • A Borel, Andre Weil, 6 May 1906 - 6 August 1998, Proceedings of the American Philosophical Society 145 (1) (2001), 107-114.
    • A Borel, Andre Weil, Bull.
    • A Borel, Andre Weil and Algebraic Topology, Notices Amer.
    • .
    • A Borel, P Cartier, K Chandrasekharan, S-S Chern and S Iyanaga, Andre Weil (1906-1998), Notices Amer.
    • A Borel, Andre Weil: quelques souvenirs, Gaz.
    • A Borel, Andre Weil and algebraic topology, Gaz.

  4. References for René Eugène Gateaux
    • Emile Borel, Introduction geometrique a quelques theories physiques (Gauthier-Villars, 1914).
    • Emile Borel, Sur les principes de la theorie cinetique des gaz, Ann.
    • Emile Borel, Mecanique statistique, d'apres l'article allemand the P Ehrenfest et T Ehrenfest, Encyclopedie des Sciences Mathematiques, Tome IV, Vol.

  5. References for Nicolas Bourbaki
    • A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.
    • A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Mitt.
    • K Krickeberg, Comment: 'Twenty-five years with Nicolas Bourbaki 1949-1973' by A Borel, Mitt.

  6. References for Eduard Heine
    • P Dugac, Sur la correspondance de Borel et le theoreme de Dirichlet- Heine- Weierstrass- Borel- Schoenflies- Lebesgue, Arch.

  7. References for Edward Van Vleck
    • A Novikoff and J Barone, The Borel law of normal numbers, the Borel zero-one law, and the work of Van Vleck, Historia Math.

  8. References for René Baire
    • R Baire, Lettres de Rene Baire a Emile Borel, Cahiers du Seminaire d'Histoire des Mathematiques 11 (Paris, 1990), 33-120.
    • H Gispert, La theorie des ensembles en France avant la crise de 1905 : Baire, Borel, Lebesgue ..

  9. References for Harish-Chandra
    • A Borel, Some recollections of Harish-Chandra, in The mathematical legacy of Harish-Chandra, Baltimore, MD, January 9-10, 1998 (Amer.
    • A Borel, Some recollections of Harish-Chandra, Current Sci.

  10. References for Henri Poincaré
    • A Borel, Henri Poincare and special relativity, Enseign.
    • A Chatelet, G Valiron, E LeRoy and E Borel, Hommage a Henri Poincare, Congres International de Philosophie des Sciences, Paris, 1949 Vol I (Paris, 1951), 37-64.

  11. References for Édouard Benjamin Baillaud
    • E Borel, Edouard Benjamin Baillaud,nComptes Rendue hebd.
    • E Borel, R Deltheil and E Esclangon, Benjamin Baillaud, 1848-1934, Extraits des rapports annuels presentes au Conseil de l'Observatoire de Paris (Toulouse, 1937).

  12. References for Jean Leray
    • A Borel, G M Henkin and P D Lax, Jean Leray (1906-1998), Notices Amer.

  13. References for Henri Lebesgue
    • B Bru and P Dugac (eds.), Lettres d'Henri Lebesgue a Emile Borel, in Cahiers du Seminaire d'Histoire des Mathematiques 12 (Paris, 1991), 1-511.

  14. References for Jacques Dixmier
    • A Borel, Twenty-five years with Nikolas Bourbaki (1949-1973), Notices Amer.

  15. References for Claude Chevalley
    • A Borel, The work of Chevalley in Lie groups and algebraic groups, in Proceedings of the Hyderabad Conference on Algebraic Groups, Hyderabad, 1989 (Manoj Prakashan, Madras, 1991), 1-22.

  16. References for Paul Montel
    • S Domoradzki, Among the teachers of Tadeusz Wazewski were E Borel, A Denjoy and P Montel (Polish), in School of the history of mathematics (Polish) (Miedzyzdroje, 1996), Zesz.

  17. References for Jean Dieudonné
    • A Borel, Twenty-Five Years with Nicolas Bourbaki, (1949-1973), Notices Amer.

  18. References for Deane Montgomery
    • A Borel, Deane Montgomery (1909-1992), Notices Amer.

  19. References for Hermann Weyl
    • A Borel, Hermann Weyl and Lie groups, Hermann Weyl, 1885-1985 (Eidgenossische Tech.

  20. References for Luther Eisenhart
    • A Borel, The School of Mathematics at the Institute for Advanced Study, in A century of mathematics in America II (Amer.

  21. References for Vito Volterra
    • A Durand and L Mazliak, Revisiting the sources of Borel's interest in probability: continued fractions, social involvement, Volterra's prolusione, Centaurus 53 (4) (2011), 306-332.

  22. References for Georges Humbert
    • E Borel, Notice sur la vie et les travaux de Georges Humbert (Paris, 1922).

  23. References for Georges Darmois
    • L Mazliak, Borel, Frechet, Darmois: la decouverte des statistiques par les probabilistes francais, J.

  24. References for Robert d'Adhémar
    • Lettres de Henri Lebesgue a Emile Borel, Cahier du seminaire d'histoire des mathematiques 12 (1991), 1-506.

  25. References for Anders Wiman
    • L Mazliak and M Sage, Au dela des reels Borel et l'approche probabiliste de la realite.

  26. References for Jacques Hadamard
    • J D Gray, Comments on collected works, in particular those of Emile Borel and Jacques Hadamard, Historia Mathematica 3 (2) (1976), 203-206.

Additional material

  1. Stäckel's contribution to Mathematics Teaching
    • A similar reform movement was taking place in France at around the same time, largely due to Emile Borel, who had written several books in French on the subject.
    • Having, as he did, a good command of French, Stackel read, and was impressed by, Borel's books and wanted to make [','T Bohnet, Leben und Werk des Mathematikers Paul Stackel (Staatsexamensarbeit, Karlsruhe University, 1993).','2]:- .
    • He was, however, not content simply to translate Borel's work into German and instead set about reworking the French version into a textbook suitable for a German-speaking audience and for the German education system.

  2. Statistical Society of Paris
    • Note also the presence of Emile Borel who will become President of the Association in 1922 and to whose actions we will return later.
    • For two years Emile Borel will teach at the ISUP the calculation of probabilities and its applications to statistics.

  3. Peres books
    • The first half of the work leads us to Descartes and Fermat; The second to the work of Borel, Volterra, etc.
    • This volume and the other two in prospect, as well as the recent treatise, "Operations inflnitesimales lineaires," by Volterra and Hostinsky, form an amplification and modernization of the two volumes on functions of curves and integral equations published in the same Borel series on the theory of functions, some twenty-five years ago.

  4. Feller Reviews 4
    • The presentation is rigorous but can be studied with profit by someone who neither knows or cares what a Borel set is.

  5. Peres publications
    • Vito Volterra and Joseph Peres, Lecons sur la composition et los fonctions permutables (Collection E Borel) (Gauthier-Villars, Paris, 1924).

  6. W H Young addresses ICM 1928
    • The main invited lectures were by D Hilbert, J Hadamard, U Puppini, E Borel, O Veblen, G Castelnuovo, W H Young, V Volterra, H Weyl, T von Karman, L Tonelli, L Amoroso, M Frechet, R Marcolongo, N Luzin, F Enriques, G D Birkhoff.

  7. Oswald Veblen Publications
    • 1904 (a) "The Heine-Borel Theorem", Bull.

  8. Haupt calculus textbooks
    • 38) there appears the Heine-Borel-Lebesgue covering theorem.

  9. Rios publications
    • Sixto Rios Garcia, Probabilidad de que una serie de Taylor sea prolongable, solucion del problema de Borel, Revista Matematica Hispano-Americana 16 (1941), 1-3.

  10. Samuel Wilks' books
    • The books of the other category, some of them excellent [Borel Series, Cramer, Kolmogorov, Uspensky], deal with mathematical theory of probability with only occasional glimpses on some particular questions pertaining to statistical theory.

  11. Bell papers
    • Admitting that logically the position of the finitists is very strong, Borel justifies the use of the doubtful element as follows, and probably most students of mathematics will agree with him.

  12. Mathematics in France during World War II
    • This man could see in the list of names Borel, Montel, de Broglie, Valery, Brunschvicg, only a group of prisoners who wanted to give him the slip.

  13. Rios's books
    • In these lectures (written in collaboration with L Vigil) the author covers a wide variety of topics in the representation of analytic functions of a complex variable: Runge's theorems; analytic continuation by overconvergence and by rearrangement (he constructs, among other examples, a "universal" series of polynomials which can be rearranged to converge uniformly to any prescribed analytic function in any desired region); Mittag-Leffler, Borel and Painleve expansions; analytic continuation by summation of series; representation of functions by Laplace integrals and by Dirichlet, factorial, interpolation and Lambert series.

  14. Walk Around Paris
    • It was created in 1928 by two mathematicians, a French one, Emile Borel, and an American: one, George Birkhoff, who both thought that French science needed an international platform for learning and exchanging knowledge in mathematics and physics.

  15. Vivanti publications
    • Giulio Vivanti, Recensione di Borel.

  16. Moran reviews
    • Thus the reader will find, in addition to the usual material on distributions and their manipulation, such useful tools of statistics and applied probability as Cochran's theorem, Campbell's theorem ("shot effect"), Spitzer's identity, and the Borel-Tanner distribution.


  1. Quotations by Borel
    • Quotations by Emile Borel .

Famous Curves

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  1. Mathematical Chronology
    • Heine publishes a paper which contains the theorem now known as the "Heine-Borel theorem".
    • Borel introduces "Borel measure".
    • Borel publishes the first in a series of papers on game theory and becomes the first to define games of strategy.

  2. Chronology for 1890 to 1900
    • Borel introduces "Borel measure".

  3. Chronology for 1920 to 1930
    • Borel publishes the first in a series of papers on game theory and becomes the first to define games of strategy.

  4. Chronology for 1870 to 1880
    • Heine publishes a paper which contains the theorem now known as the "Heine-Borel theorem".

EMS Archive

  1. 1916-17 May meeting
    • Ford, Lester R: "A geometrical proof of a theorem by Hurwitz and Borel", [Title] .

  2. Edinburgh Mathematical Society Lecturers 1883-2016
    • (Edinburgh) A geometrical proof of a theorem by Hurwitz and Borel .

BMC Archive

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Gazetteer of the British Isles

  1. References
    • [Obituary:] Emile Borel.

Astronomy section

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