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- Mathematics In Europe - Andrei Nikolaevich Kolmogorov (April 4, 1903 -- October 20, 1987); by Heinz Klaus Strick, Germany

an example of an integrable function whose associated Fourier series diverges almost everywhere A Fourier series is a particular infinite sum whose terms are trigonometric functions Before he had taken his examinations in 1925 Kolmogorov had published eight papers on a variety of topics including in collaboration with Aleksandr Yakovlevich Khinchin 1894 1959 his first contribution to probability theory dealing with the weak law of large numbers Around 1700 Jacob Bernoulli 1654 1705 expounded a law that today is known as Bernoulli s law of large numbers It states that the probability that in a series of Bernoulli trials the difference between the relative frequency X n and the underlying success probability p is equal to at most a given positive number e converges to 1 as n goes to infinity lim n to infty P left left vert frac X n p right vert le varepsilon right 1 This law is what appears on the Swiss postage stamp pictured The mean value of the results of a number of trials tends to the expected value of the random variable In 1866 Pafnuty Chebyshev generalized Bernoulli s result to sums of independent random variables and gave also an ingeniously simple proof the Chebyshev inequality Kolmogorov s 1925 contribution gives three conditions under which one has lim n to infty P left left vert frac 1 n cdot left X 1 cdots X n right frac1 n cdot big rm E X 1 cdots rm E X n big right vert le varepsilon right 1 The above mentioned conditions relate to the sequence of sums of the random variables the sequence of associated expectations and the variances for this reason the theorem is sometimes called the three series theorem In the following years Kolmogorov published further papers on probability theory but also in other areas of mathematics With Pavel Sergeevich Alexandrov 1896 1982 he travelled through Europe visiting universities in Berlin Göttingen and Paris In 1930 he accepted a chair in mathematics at Moscow State University As a university professor Kolmogorov had an enormous effect on his students He took a personal interest in them on the regular hiking trips that he personally conducted mathematics was the primary topic of conversation Kolmogorov also authored a number of textbooks and encouraged mathematically talented students With his 1933 work Grundbegriffe der Wahrscheinlichkeitsrechnung Fundamentals of the Theory of Probability published in German Kolmogorov exercised a profound effect on the further development of the theory of probability In 1900 at the International Congress of Mathematicians in Paris David Hilbert 1862 1943 named what were in his opinion the 23 most important outstanding unsolved problems in mathematics The sixth of these problems asked how mechanics and probability theory which at the time was considered part of physics because of the applied problems in the field could be axiomatized In an axiomatized formulation one begins with a set of fundamental axioms from which further laws are then derived an analogy is the axiomatic system that Euclid gave

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Open archived version from archive - Mathematics In Europe - Sonya Kovalevskaja (January 15, 1850–February 10, 1891); by Heinz Kaus Strick, Germany

rights of women to obtain an education and saw it a point of honor to free Russia s daughters The unsuspecting parents accepted the budding paleontologist as their daughter s husband In the following year the newlyweds travelled to Heidelberg so that SONYA KOVALEVSKAYA could take up studies in mathematics and the natural sciences However at the time women could not officially enroll in German universities After many futile attempts she was finally allowed to petition individual lecturers to audit their lectures The professors including LEO KÖNIGSBERGER a student of WEIERSTRASS HERMANN HELMHOLTZ and GUSTAV ROBERT KIRCHHOFF quickly recognized the young woman s exceptional talent After three semesters she moved to Berlin on the recommendation of KÖNIGSBERGER in order to continue her studies under the supervision of KARL WEIERSTRASS himself whose lectures on analysis had become renowned for their intellectual rigor At first WEIERSTRASS ignored the letters of recommendation that she had produced but gave the supplicant a problem which to his great surprise she was quickly able to solve Since the university despite WEIERSTRASS s petition would not give permission for SONYA even to audit the lectures WEIERSTRASS saw only one way to help he taught her twice a week in private sessions In 1874 SOFIA KOVALEVSKAYA completed three papers WEIERSTRASS judged each one of them sufficient for the granting of a doctoral degree The first paper brought her research on the solvability of partial differential equations to a provisional conclusion Differential equations are equations that describe relationships between functions and their derivatives To solve a differential equation means to find all functions that satisfy the conditions of the equation In partial differential equations several variables are involved In the second paper KOVALEVSKAYA dealt with so called abelian integrals named in honor of NIELS HENRIK ABEL and provided methods for reducing these integrals to simpler integrals In the third paper she improved on a theory of PIERRE SIMON LAPLACE on the physics of the rings of Saturn WEIERSTRASS had a hard time finding a university in Germany that would recognize this work as the basis for granting a doctoral degree Finally the University of Göttingen expressed its willingness to do so and granted KOVALEVSKAYA the title of doctor in absentia with the addition of summa cum laude In support of his application WEIERSTRASS went so far as to cite the great GAUSS who in 1837 expressed his regret that the German universities had failed to grant the mathematician SOPHIE GERMAIN a doctorate in her lifetime It was many more years before women were granted the right and opportunity to pursue scientific work For example the physician and neurologist PAUL MÖBIUS insisted that there was no originality in the ideas and scientific work of SOFIA KOVALEVSKAYA this judgment appeared in the chapter On Women in Mathematics in a book that MÖBIUS published in 1900 with the title On the Natural Aptitude for Mathematics When an error was found in one of KOVALEVSKAYA s later articles on the refraction of light

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Open archived version from archive - Mathematics In Europe - Pierre Simon Laplace (March 3, 1749--March 5, 1827); by Heinz Klaus Strick, Germany

Alembert France 1959 In 1790 he became a member of a commission with the charge of standardizing the units of measurement and at the same time to introduce units based on the decimal system In 1793 he fled Paris with his wife who was 20 years his junior and their two children thereby avoiding the fate of his colleague Antoine Laurent de Lavoisier 1743 1794 who was beheaded After the end of the Jacobin Terror he returned and took over the leadership of the Bureau of Longitude and the Paris Observatory The Académie des Sciences was reopened as the Institut National des Sciences et des Arts Antoine Laurent de Lavoisier France 1943 Together with the chemist Claude Louis Berthollet 1748 1822 he began in 1806 to assemble in his hometown of Arcueil near Paris together with the Société d Arcueil a circle of researchers to which belonged Alexander von Humboldt During Napoleon s reign Laplace and Berthollet controlled the academic life in France thanks to their closeness to the emperor Claude Louis Berthollet France 1958 Laplace focused much of his research effort on investigating the application of mathematics to questions in astronomy Observations of the orbits of Jupiter and Saturn indicated a destabilizing influence in the solar system Isaac Newton had hypothesized that the solar system can maintain its equilibrium through regular divine intervention In 1776 Laplace discovered that the equations of motion established by Euler and Lagrange for the motions of the planets were not sufficiently precise and that the terms omitted in their calculations could have a significant effect over the course of a number of years He proved that from a long term viewpoint the solar system is indeed in equilibrium In 1796 there appeared a general work the Exposition du système du monde which contains the following hypothesis on the origin of the solar system Gas clouds broke away from the Sun and condensed to form the planets Similar ideas had been formulated by Immanuel Kant 1724 1804 in the year 1755 Alexander von Humboldt Federal Republic of Germany 1959 After an intensive involvement with a variety of problems in astronomy Laplace finally collected the results of his research in a comprehensive book Beginning in 1799 he published the five volumes of Mécanique céleste in which as the physicist Jean Baptiste Biot 1744 1862 noted Newton s Principia was translated into the language of calculus When Napoleon observed that God is nowhere to be found in the Mécanique céleste Laplace is said to have answered Sire je n avais pas besoin de cette hypothèse Your Majesty I had no need of such a hypothesis Laplace has also to his credit important advances in the theory of probability In his Théorie analytique des probabilités 1812 and in the more generalist Essai philosophique sur les probabilités 1814 he adopts the definition today considered classical of probability given by Abraham de Moivre 1667 1754 a Frenchman who had emigrated to England Doctrine of Chances 1738 probability of an event

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Open archived version from archive - Mathematics In Europe - Joseph-Louis Lagrange (January 25, 1736–April 10, 1813); Heinz Klaus Strick, Germany

was London but he became ill en route and had to break his journey in Paris There he met Jean Baptiste le Rond d Alembert who offered him a more suitable position than the one he held in Turin Lagrange won a number of prizes of the Paris Académie des Sciences for work on the motion of the moon as well as the orbits of Jupiter s moons In 1766 Lagrange was encouraged by d Alembert to move to Berlin however he accepted the offer of the Prussian king Frederick II only after it became clear that Euler would be returning to St Petersburg Thus Lagrange became Euler s successor as director of the Mathematics Division of the Prussian Academy of Sciences During his twenty year stay in Berlin Lagrange further developed methods for dealing with functions of several variables and for the solution of differential equations After the death of Frederick II in 1786 Lagrange received numerous offers and in the end he decided to accept a position in Paris where he was able to complete work on his Mécanique analytique analytic mechanics with the support of Adrien Marie Legendre in this work numerous physical problems were solved using exclusively mathematical methods During the French Revolution he was appointed to the Commission for the Reform of Weights and Measures which had the task of introducing a decimal system of weights and measures as well as to the Bureau des Longitudes When a law was promulgated requiring all foreign residents to leave the country Antoine Laurent de Lavoisier intervened and obtained an explicit exemption for Lagrange A few months later Lavoisier would become a victim of the guillotine during the Reign of Terror Under Napoleon Lagrange was again rehabilitated and loaded with titles and honours His last works provided a decisive impetus for the further development of differential geometry the study of curves and surfaces in space and complex analysis the study of complex valued functions of a complex variable Today many theorems and concepts recall in their names the achievements of Lagrange The four squares theorem of Lagrange states that every natural number can be written as the sum of at most four perfect squares Lagrange proved that a natural number n is a prime number if and only if n 1 1 is divisible by n Moreover he showed that the continued fraction developments of square roots are always infinite and periodic Euler proved the converse sqrt 2 1 frac 1 2 frac 1 2 frac 1 2 frac 1 2 dots qquad sqrt 3 1 frac 1 1 frac 1 2 frac 1 1 frac 1 2 dots qquad sqrt 11 3 frac 1 3 frac 1 6 frac 1 3 frac 1 6 frac 1 3 dots qquad sqrt 5 2 frac 1 4 frac 1 4 frac 1 4 frac 1 4 dots In addition Lagrange proved that the equation Dx 2 1 y 2 can be solved in the set of integers if D

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Open archived version from archive - Mathematics In Europe - Henri Léon Lebesgue (June 28, 1875–July 26, 1941); by Heinz Klaus Strick, Germany

continuous function f defined on a closed interval I is differentiable and that its derivative is equal to the function f being integrated The precise conceptualization and proofs of this were given first by AUGUSTIN CAUCHY BERNHARD RIEMANN later developed the concept of integrability of a function by dividing the interval I into subintervals and determining the areas of rectangles with base the length of the subinterval and height the value of the function at a point in the subinterval that is sums of the form sum i f t i x i 1 x i where i is the index of the subinterval from x i to x i 1 and t i is a point in the subinterval If the sums converge to some fixed number for every choice of decomposition into subintervals and choice of points t i then that number is called the RIEMANN integral and is denoted by int a b f x dx Inspired by publications of ÉMILE BOREL professor at the École Normale Supérieure LEBESGUE took on in his doctoral research the problem of measurability of sets and thereby arrived at a new definition of integrability While with RIEMANN the domain of definition of a function stands in the foreground LEBESGUE began with the set of the function s values which he divided into subintervals An approximation of the LEBESGUE integral is achieved by determining the areas of subsets of the plane whose height is taken from the length of a subinterval of the set of values and whose width is determined by the total length of the associated set of the domain of definition More precisly an approximation to the integral is the sum over the areas of the sets f 1 x i x i 1 times x i x i 1 Picture from Wikipedia LEBESGUE explains his approach in a vivid way thus Suppose that I have to pay a certain sum I look through my pockets and there I find coins and currency notes of various values I give then to my creditor in the order in which I find them until I have reached the total amount of my debt That is the RIEMANN integral But I proceed otherwise After taking all the money out of my pockets I place all of the notes of the same value together and I do the same with the coins and I make my payment by handing over in sequence all the money of a given value That is my integral The method introduced by LEBESGUE is a generalization of that of RIEMANN every bounded Riemann integrable function on a bounded interval is Lebesgue integrable but there are Lebesgue integrable functions that are not Riemann integrable An example is given by the DIRICHLET function D on the interval 0 1 which is defined for all rational numbers by 1 and by 0 for the irrational numbers This function is discontinuous at every point of its domain It is not RIEMANN integrable

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Open archived version from archive - Mathematics In Europe - Adrien-Marie Legendre (September 18, 1752–January 10, 1833); by Heinz Klaus Strick, Germany

of this conjecture was finally given in 1837 by DIRICHLET and today it is referred to as DIRICHLET s theorem on primes in arithmetic progressions without any reference to the contribution made by LEGENDRE Elsewhere in the paper LEGENDRE dealt with the conjecture of EULER known as the law of quadratic reciprocity if p and q are distinct prime numbers then the congruences x 2 p mod q and x 2 q bmod p are either both soluble or both insoluble unless p and q both yield a remainder of 3 on division by 4 that is in mathematical terminology both are congruent to 3 modulo 4 in which case exactly one of the two congruences in soluble In formulation the theorem one uses today the notation introduced by LEGENDRE namely the LEGENDRE symbol bigl frac ap bigr for an integer a and prime number p One writes bigl frac ap bigr 1 if a is a quadratic residue modulo p that is if the congruence x 2 a bmod p is soluble The congruence x 2 2 bmod 7 for example has the solutions 3 and 4 Therefore one writes bigl frac 27 bigr 1 On the other hand if a is not a quadratic residue modulo p that is if the congruence has no solution then one expresses this fact by writing bigl frac ap bigl 1 For instance the congruence x 2 5 bmod 7 has no solutions Therefore one writes bigl frac 57 bigr 1 If a is a multiple of p then one has by definition bigl frac ap bigr 0 With this notation in hand we may express the law of quadratic reciprocity for odd primes p and q in terms of the LEGENDRE symbol as follows bigl frac pq bigr bigl frac pq bigr 1 p 1 q 1 4 In 1787 LEGENDRE was given the assignment as a member of the Académie to bring the survey measurements of the observatories of Paris and Greenwich into alignment In recognition of his services he was inducted into the Royal Society In the same year he published a paper dealing with the errors and necessary adjustments of trigonometric measurements When in 1793 the French national convention introduced the metre as the compulsory unit of linear measurement defined as one ten millionth of the distance of the Paris Meridian the quarter of the Earth s circumference from the North Pole to the equator running along the meridian passing through the Paris Observatory this value was based on data with whose coordination he had been tasked In the following years LEGENDRE worked in a position of authority on the project of producing logarithmic tables in base 10 and associated tables of trigonometric functions During the turmoil of the French Revolution LEGENDRE who was newly married lost his entire inherited fortune and when the Académie des Sciences was closed following a decision of the national convention he found himself temporarily destitute After the reestablishment of the Académie des

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Open archived version from archive - Mathematics In Europe - Gottfried W. Leibniz (July 1, 1646-November 14, 1716); by Heinz Klaus Strick, Germany

day s length is insufficient even to write them down Thus for example he busied himself trying to solve the problem of the flooding of silver mines in the Harz region and developed ideas for the construction of a submarine and for improvements in the security of door locks He proposed a system of pensions for widows and orphans and also dispensed medical advice Federal Republic of Germany 1966 Leibniz became involved in an effort to achieve a reconciliation between the Catholic and Protestant camps he worked out plans for coinage reform in order to simplify commerce Not least he encouraged the creation of learned academies on the French and English model in a number of European countries Prussia Saxony Russia and the Habsburg dominions Above all however he published numerous papers on topics in mathematics physics and philosophy Despite the great respect in which he was held in scientific circles he suffered from a lack of self esteem Perhaps this was due to his physical appearance perhaps he was ashamed of his strong Saxon accent Albania 1996 Moreover the dukes of Hanover valued the accomplishments of the polymath genius Leibniz not at all At the end of his life his health suffered from the stress of the priority dispute over the infinitesimal calculus Isaac Newton 1643 1727 had begun in 1666 to develop his fluxion calculus differential calculus but his first publications on the subject appeared only in 1687 On his part Leibniz published his Calculus independently of Newton in the year 1684 in the article Nova methodus pro maximis et minimis itemque tangentibus qua nec fractas nec irrationales quantitates moratur et singulare pro illi calculi genus This work contains all the derivative rules including the chain rule as well as conditions for the existence of extrema and points of inflection Two years later there followed De geometria recondita in which the integral sign int was used On the Continent the notation and terminology such as constant variable function developed by Leibniz spread quickly His adherents above all Johann and Jakob Bernoulli proved the value of Leibniz s notation in their works on calculus and its applications to physics Rumania 1996 In 1677 Newton attacked Leibniz accusing him of having stolen his methods The conflict escalated over the years and created a schism in the scientific world In 1713 a partisan commission appointed by the Royal Society upheld the charges of plagiarism Today however it is generally acknowledged that the two theories were developed independently of each other DDR 1950 Leibniz s ability to choose a suitable symbolism for dealing with scientific questions is seen as well in his extensive studies in formal logic He discovered the significance of the binary number system on which the modern computer is based But he linked it as well to his theological philosophical worldview In the credo Without God there is nothing he gives the value 1 to God and 0 to nothing His famous and widely ridiculed thesis that the world

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Open archived version from archive - Mathematics In Europe - John von Neumann (December 28, 1903–February 8, 1957); by Heinz Klaus Strick,Germany

set theory In 1901 BERTRAND RUSSELL 1872 1970 discovered a paradox that arises if one allows arbitrary sets into the theory what is the set of all sets that do not contain themselves as elements In his doctoral thesis VON NEUMANN produced in 1925 a consistent axiomatic development of set theory BERTRAND RUSSELL India 1972 With a fellowship from the Rockefeller Foundation he worked with DAVID HILBERT 1862 1943 in Göttingen then worked as at the Universities of Berlin and Hamburg as their youngest lecturer From 1930 he added to his European positions that of a guest lecturer at Princeton University His fame as a mathematical genius spread throughout the world In 1933 he became along with ALBERT EINSTEIN 1879 1955 one of the first five professors at the newly established Institute for Advanced Study in Princeton In 1937 he became an American citizen ALBERT EINSTEIN USA 1979 In 1926 he worked on the mutually contradictory seeming theories of the atomic physicists WERNER HEISENBERG 1901 1976 and ERWIN SCHRÖDINGER 1887 1961 and attempted to create a mathematical theory that encompassed both approaches His Mathematische Grundlagen der Quantenmechanik mathematical foundations of quantum mechanics appeared in 1932 Heisenberg Germany 2001 This work gave a new impetus to the field of functional analysis a branch of mathematics that deals generally with the properties of spaces of functions VON NEUMANN algebra In 1936 he wrote an article on a new logic of quantum mechanics Photons cannot pass through mutually perpendicular polarization filters and according to classical logic there should be no effect were one to add a third filter If one places the third filter diagonally to the other two either in front of or behind them in the path of the photons then indeed nothing changes However such is not the case if it is placed between the two filters JOHN VON NEUMANN is considered the founder of game theory In 1928 he published an article on the minimax theorem This mathematical theorem deals with a strategy by which the maximal losses of the players in a zero sum game that is the total amount won equals the total amount lost are kept to a minimum In 1937 he generalized this work to questions about the equilibrium of supply and demand and in 1944 he published together with OSKAR MORGENSTERN 1902 1977 the standard text on economy Theory of Games and Economic Behaviour in which among other things it is shown that even collective bargaining strategic business decisions and international conflicts can be described with the help of mathematical models In 1936 ALAN TURING 1912 1954 went to the Institute for Advanced Study in Princeton to complete his doctorate In his famous work On Computable Numbers with an Application to the Entscheidungsproblem he studied the computability of a mathematical problem TURING machine ALAN TURING St Vincent 2000 ALAN TURING and JOHN VON NEUMANN Portugal 2000 His presence in Princeton stimulated VON NEUMANN to work more energetically on automated computing machines He analysed the

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Open archived version from archive