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- Mathematics In Europe - Constantin Carathéodory (September 13, 1873–February 2, 1950); by Heinz Klaus Strick, Germany

by announcing his intention to study mathematics and to do so at the place of his birth Berlin But he soon moved from Berlin to Göttingen which at the time enjoyed the reputation of being the capital of mathematical research He was especially fascinated with the calculus of variations a field established by JAKOB and JOHANN BERNOULLI through their solution of the brachistochrone problem and further developed by LEONHARD EULER CARL GUSTAV JACOBI and ROWAN HAMILTON He generalized the solution of a seemingly simple problem A lamp in the interior of a sphere projects points of the sphere onto a plane surface Find a curve on the sphere of prescribed length such that its shadow is as short or as long as possible In 1904 he received his doctorate with a thesis On discontinuous solutions in the calculus of variations His dissertation advisor was HERMANN MINKOWSKI one of the founders of the special theory of relativity His oral examination in applied mathematics was given by FELIX KLEIN and the examination in astronomy by KARL SCHWARZSCHILD As a way of keeping especially gifted mathematicians in Göttingen he was allowed to submit his habilitation without the usual waiting period which he did with a thesis On strong maxima and minima of simple integrals After a period as privatdozent at the University of Bonn he accepted professorship in Hanover In 1910 he was called to a professorship at the newly established Technical University of Breslau In 1913 CARATHÉODORY was named as FELIX KLEIN s successor in Göttingen He assumed the editorship of the journal Mathematische Annalen as well as of an Italian journal With the outbreak of World War I most of his Göttingen students and colleagues were called to military service and he felt uneasy in Göttingen CARATHÉODORY had married a close relative in Constantinople in 1909 with his wife EUPHROSYNE he had two children In 1918 he was appointed to a chair in Berlin and once again he moved MAX PLANCK held the laudatory oration when in 1919 he was inducted into the Prussian Academy of Sciences together with ALBERT EINSTEIN CARATHÉODORY had by that time maintained a correspondence with EINSTEIN for many years In 1915 he gave EINSTEIN significant pointers on the use of the calculus of variations in the development of the general theory of relativity In 1917 Greece had joined the Entente powers with the particular goal of being able to enlarge the territory of Greece following a successful conclusion of the war The treaty of Sèvres granted Thrace and region around Smyrna today Izmir to Greece CARATHÉODORY was then asked by the Greek government to establish a Greek university in Smyrna He then travelled throughout Europe to secure an adequate supply of books for the university library In the meantime however Greek forces attempted to alter the status quo by conquering the entire region up to Constantinople Istanbul This military enterprise ended in catastrophe for Greece The Turkish forces under the generalship of MUSTAFA KEMAL PASCHA who

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Open archived version from archive - Mathematics In Europe - Augustin Cauchy (August 8, 1789-May 5, 1857); by Heinz Klaus Strick, Germany

Cauchy is mad and there is nothing to be done about it yet he is the only one among us who knows how mathematics should be done Cauchy gave lectures on analysis defining in a stricter sense than theretofore the notion of limit and the conditions for the convergence of series which made possible a precise definition of the integral Already in 1814 he introduced functions of a complex variable being the first to do so and he gave conditions for their differentiability the Cauchy Riemann differential equations He extended the notion of an interval in mathbb R to encompass paths in the complex plane and derived theorems for computing integrals along such paths Cauchy integral theorem Elaborations of his pathbreaking lectures were quickly translated into other languages and they became standard works for decades His students were not enamoured of his efforts to make mathematical concepts precise They were unaccustomed to such rigour and considered his frequently excursive explanations too theoretical Above all his conservative political stance was alienating as were his religious views which he proclaimed with a missionary zeal The overthrow of the king in 1830 led Cauchy to emigrate he refused to take an oath of loyalty to the new citizen king Louis Philippe In Turin the king of Sardinia offered him a chair in theoretical physics He then accepted a position in Prague as teacher of mathematics and physics to the grandson of the former king Charles X For this not very successful service he was made a baron a rank on which subsequently he placed great importance The extent to which he came into contact there with Bernard Bolzano who also believed in a more rigorous approach to mathematical proof is unclear On his return to Paris he again became a member of the Academy However he was unable to attain any of the posts to which he applied The period from 1839 to 1848 was Cauchy s most productive time He published around 300 out of a total oeuvre of 789 papers in the journal of the Academy which led to a restriction on the number of articles that an individual could submit Following the February Revolution of 1848 the Bourbons contrary to Cauchy s hope did not assume power but rather Napoleon III Nevertheless Cauchy obtained a professorship due to his unassailable academic competence this time without the requirement of a loyalty oath His last years were marred by a priority dispute on which he dogmatically refused to admit that someone else could have had an idea before he did We have Cauchy to thank for an abundance of mathematical theorems and criteria for example the convergence criterion Cauchy s convergence criterion A sequence a n n in mathbb N is convergent if and only if for every varepsilon 0 there exists a number n 0 such that for all n m ge n 0 a n a m varepsilon He proved the convergence of geometric series for q 1 and from

Original URL path: http://mathematics-in-europe.eu/tr/76-enjoy-maths/strick/602-augustin-cauchy-august-8-1789-may-5-1857-by-heinz-klaus-strick-germany (2013-11-18)

Open archived version from archive - Mathematics In Europe - Pafnuty L. Chebyshev (May 16, 1821–December 8, 1894); by Heinz Klaus Strick, Germany

of the arithmetic mean of random variables X 1 X 2 ldots to the common expectation E X of these random variables P frac X 1 cdots X n n E X epsilon rightarrow 0 for all epsilon 0 What is particularly noteworthy about CHEBYSHEV s inequality is that it is described by a simple relationship between a random variable X its expectation E X and the associated variance V X or standard deviation sigma X which moreover can be proved using comparatively elementary arguments The inequality can be formulated in a variety of ways For an arbitrary epsilon 0 one has P X E X ge epsilon le sigma epsilon 2 or if epsilon is replaced by k sigma P X E X ge k sigma le 1 k 2 or P X E X k sigma ge 1 1 k 2 For arbitrary distributions for example one has that with probability at least 75 the values of a random variable X will lie within an interval of width 2 sigma about the expectation E X while with probability at least about 89 it will lie in a 3 sigma interval This holds for an arbitrary probability distribution with finite values of E X and sigma while for binomial and normal distributions one can replace these values as is well known with even higher probabilities 95 5 and 99 7 respectively Euler and Gauß In the course of writing his doctoral dissertation CHEBYSHEV considered the question of the distribution of prime numbers The motivation for this came from the number theoretic work of LEONHARD EULER that had been published jointly with VIKTOR YAKOVLEVICH BUNYAKOVSKY Already at the end of the eighteenth century CARL FRIEDRICH GAUSS and ADRIEN MARIE LEGENDRE had conjectured a relationship between the prime number function pi x which represents the number of prime numbers less than x and the values of the function and had made the first estimates of the function pi x In his paper Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée On the function that gives the number of primes less than a given limit he was able to show that for the quotient of the two functions one has the relationship 0 92929 le pi x x log x le 1 1056 This means that the difference between the two functions is limited to an amount of about 10 above and below In investigating the function f x x log x as well as the function Li x int 2 x 1 log t dt he showed that if the limit lim x rightarrow infty pi x x log x exists then it must equal 1 A proof of the prime number theorem namely that lim pi x x log x 1 was obtained only in 1896 two years after CHEBYSHEV s death Proofs were obtained independently by the French mathematician JACQUES SALOMON HADAMARD and the Belgian mathematician CHARLES JEAN DE LA VALLÉE POUSSIN

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Open archived version from archive - Mathematics In Europe - Richard Dedekind (October 6, 1831–February 12, 1916); by Heinz Kaus Strick, Germany

to deal with the childish behaviour of some of them When in 1861 the Collegium Carolinum in Braunschweig was expanded into an institute of technology and advertised for a professor of mathematics he applied for the position in order to be able to return to his hometown with the express request that he never again have to teach lower mathematics His conditions were accepted and from then on until his retirement in 1894 DEDEKIND worked in Braunschweig He declined all offers even from Göttingen In the early 1870s the Collegium Carolinum was transformed into the Herzogliche Technische Hochschule Carolo Wilhelmina DEDEKIND was named its first director When the duke of Braunschweig gave his consent for the new university DEDEKIND took over direction of the building commission In 1872 DEDEKIND published a paper on Continuity and Irrational Numbers Stetigkeit und Irrationale Zahlen in which he presented his idea developed earlier in Zurich of cuts What DEDEKIND meant here by continuity is today known as completeness The core of this idea is the following consideration The number line is apparently complete that is without any holes If one chooses some point say P then this point will correspond to either a rational or irrational number Every point divides the number line into two parts All points of the line are decomposed into two classes in such a way that every point of the first class lies to the left of every point of the second class and so there exists one and only one point that brings about this division of all points into two classes this cutting of the line into two pieces By this method a rational number a divides the set Q of rational numbers into two subsets A1 and A2 all elements of the lower class A1 are smaller than all elements of the upper class A2 Regardless of whether one considers the rational number a as belonging to the lower class as the largest number in A1 or to the upper class as the smallest number in A2 the result is that the set Q of rational numbers is divided in two subsets by means of the rational number a Moreover every irrational number b for example sqrt 2 divides the set Q of rational numbers into two subsets A1 and A2 the elements of the lower class A1 are all smaller than the irrational number b under consideration and it in turn is smaller than all the elements of the upper class A2 A1 e g is the collection of all rational x such that x is negative or x² 2 The lower class associated with an irrational number b however has no largest element while the upper class has no smallest element The irrational numbers thus fill the holes between all the pairs of subsets of rational numbers It is in this property that not all cuts are brought about by rational numbers that the incompleteness or discontinuity of the set R of all rational

Original URL path: http://mathematics-in-europe.eu/tr/76-enjoy-maths/strick/771-richard-dedekind-october-6-1831-february-12-1916-by-heinz-kaus-strick-germany (2013-11-18)

Open archived version from archive - Mathematics In Europe - René Descartes (1596 - 1650); by Heinz Klaus Strick, Germany

in the by now republican Netherlands hoping to have there greater freedom of expression and began work on a book that was to have the title Traité du monde The World however he abandoned the project after learning about the problems faced by Galileo with the Inquisition Eventually his friends convinced Descartes to publish his philosophical ideas Finally there appeared in 1637 at first anonymously his Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences Discourse on the method of rightly conducting one s reason and of seeking truth in the sciences with three appendices La Dioptrique On refraction Les Météores On meteors and La Géométrie In 1641 his Meditationes appeared with the famous sentence Cogito ergo sum I think therefore I am first in Latin and later in French as well The full title in English reads Meditations on first philosophy in which the existence of God and the immortality of the soul are demonstrated Descartes s method of philosophical reasoning included the following precepts Accept nothing as true that admits of doubt Break difficult problems into smaller subproblems begin with the simple and progress to the difficult determine whether one s investigation is complete He was convinced that all natural phenomena admit of a rational formulation and explanation For example he gives a correct explanation for the appearance of rainbows Descartes was the first to formulate the law of conservation of momentum he explained the origin of the solar system as resulting from a vortex of matter being set in motion by God from which arose the Sun the planets and the comets His theories however allow for no interactions without immediate material contact which rules out the possibility of magnetism and gravitation Descartes s approach stands in contrast to the previously uncontested worldview of Aristotle His physical theories however were largely speculative and were only gradually superseded by Newtonian physics which was based on the scientific method of observation and scientific deduction In 1649 Descartes accepted an invitation from Queen Christina of Sweden to move to Stockholm where he was forced to abandon his custom of remaining late in bed since the queen expected him to appear at her five o clock breakfast to discuss mathematical and philosophical problems Descartes did not survive his first Nordic winter from his early morning walks with the queen he came down with a lung infection and he died soon thereafter With the work La Géométrie a new branch of mathematics was born analytic geometry Descartes showed that algebraic equations can be solved with geometric constructions and that geometric objects can be described in terms of algebraic equations Even though he did not employ the coordinate system that is called Cartesian in his honor his methods revolutionized mathematics Geometry and algebra support each other and since Descartes the one has been inseparable from the other Descartes was the first mathematician to use consistently the signs and as well as exponential notation and the square

Original URL path: http://mathematics-in-europe.eu/tr/76-enjoy-maths/strick/596-rene-descartes-1596-1650-by-heinz-klaus-strick-germany (2013-11-18)

Open archived version from archive - Mathematics In Europe - Gustav Lejeune Dirichlet (February 13, 1805–May 5, 1859); by Heinz Klaus Strick, Germany

university faculty finally arrived at the solution of awarding DIRICHLET an honorary doctorate In the spring of 1827 DIRICHLET was able to take up his duties as lecturer in Breslau On his way there he visited GAUSS in Göttingen GAUSS received him graciously and listened to news about the current state of mathematical research in Paris DIRICHLET was able to satisfy only two of the three requirements for his habilitation He presented a demonstration lecture on the irrationality of pi and wrote an article on the relationships between b in Bbb Q sqrt b notin Bbb Q and the coefficients u v in the development of a sqrt b n u v sqrt b for arbitrary a b n VON ALTENSTEIN was willing to forgo the defence being held in Latin DIRICHLET was not very happy in the Silesian province He missed the exchange of ideas with other scientists to which he had become accustomed in Paris Nonetheless he used his time there somewhat over a year to produce some noteworthy publications In 1825 and 1831 GAUSS published articles on what is known as the law of biquadratic reciprocity a characterization of equations of the fourth degree in terms of arithmetic modulo p DIRICHLET found shorter proofs to some of the theorems found by GAUSS and thereby supplemented GAUSS s work FRIEDRICH WILHELM BESSEL an astronomer and mathematician in Königsberg was so impressed with this work that he wrote a letter to ALEXANDER VON HUMBOLDT who took the letter again to the minister VON ALTENSTEIN to encourage him finally to secure a position for DIRICHLET in Berlin At first this was nothing more than a position as teacher at a military academy army college in Berlin at which Prussian officers were educated Within three years he had taught courses on the theory of equations solution of equations up to degree 4 series and sequences descriptive geometry trigonometry conic sections the geometry of space mechanics and geodesy Over the years DIRICHLET saw to it that topics from calculus and its application to mechanics were added to the instructional program DIRICHLET was extremely effective as a teacher with the result that over a period of twenty eight years the military administration was unwilling to relieve him of his teaching duties of up to eighteen hours per week From the very beginning DIRICHLET gave supplementary mathematical lectures at the University of Berlin which in 1831 invited him to join the philosophical faculty though only as a professor designatus since the university insisted on compliance with the rules of habilitation In 1832 he was inducted into the Prussian Academy of Sciences as its youngest member DIRICHLET s reputation grew from year to year even without a formal habilitation His students were deeply impressed with the clarity of his lectures which he presented with great enthusiasm and without notes only rarely making use of the chalkboard When the prescribed time for his lecture had elapsed he made a note and then resumed the next lecture

Original URL path: http://mathematics-in-europe.eu/tr/76-enjoy-maths/strick/982-gustav-lejeune-dirichlet-february-13-1805-may-5-1859-by-heinz-klaus-strick-germany (2013-11-18)

Open archived version from archive - Mathematics In Europe - Paul Erdös (March 26, 1913–September 20, 1996); by Heinz Klaus Strick, Germany

is open From 1941 until the liberation of Hungary he received no news from his homeland His father died in 1942 and a number of his relatives perished in concentration camps His mother survived the period of terror as if by a miracle It was not until 1948 that ERDOS was again able to visit the country of his birth In 1896 JACQUES SALOMON HADAMARD und CHARLES JEAN DE LA VALLÉE POUSSIN gave the first proofs of the prime number theorem which states that the function pi x which gives the number of primes less than the real number x satisfies the relation lim x rightarrow infty frac pi x x log x 1 In 1949 ERDOS and ATLE SELBERG simultaneously discovered elementary proofs of the prime number theorem here elementary means without the use of complex analysis The two mathematicians could not agree on how their results should be published since each had used work of the other in obtaining his result ERDOS wanted publication of a joint paper and SELBERG would not agree to that In the end separate papers appeared and the following year SELBERG was awarded the FIELDS Medal for his outstanding accomplishments the highest honour in mathematics comparable with the NOBEL Prize ERDOS accepted this outcome with equanimity In 1952 he accepted a generous offer from the University of Notre Dame in South Bend Indiana that gave him complete freedom regarding his teaching responsibilities On his return from a visit to Amsterdam he was subjected to an interrogation in which he was asked among other things his opinion of KARL MARX His reply that MARX was surely a man of importance might have tipped the balance in denying him entry into the country However in an FBI file on him there is also a notation from the year 1941 when he had unintentionally because he was deep in conversation on a mathematical problem trespassed on a military reservation As someone who was in regular contact with individuals living in Communist countries for example his mother and also a number theorist from Communist China he was suspected of being a Communist spy During the following ten years he spent most of his time in Israel which he called Is Real The Technion Israel Institute of Technology in Haifa named ERDOS a permanent visiting professor Despite numerous invitations from American universities it was not until 1963 that ERDOS was allowed to enter the United States His fear that following his return to Hungary he would not be allowed to leave never materialized However in the 1970s he refused for a long time to visit his homeland as a protest against its anti Israeli policies He received numerous honours for his accomplishments including at least fifteen honorary doctorates All the prize money that he received he used to fund his own awards offering cash prizes for the solutions of particular problems that he posed whereby he himself judged the monetary value of a problem between 25 and 5000

Original URL path: http://mathematics-in-europe.eu/tr/76-enjoy-maths/strick/810-paul-erdoes-march-26-1913-september-20-1996-by-heinz-klaus-strick-germany (2013-11-18)

Open archived version from archive - Mathematics In Europe - Leonhard Euler (1707 - 1783); by Heinz Klaus Strick, Germany

the scientific world with a sensational discovery That the harmonic series diverges had already been proved by a variety of methods It was conjectured that the analogous series of reciprocals of squares converged Indeed Johann Bernoulli had found a majorant for the series that converged to the value 2 However he was unable to determine the limiting value just as before him Leibniz Stirling de Moivre and others had also failed Euler exhibited a rare virtuosity in dealing with the power series developments of functions The sine function can be written as follows in a power series From this Euler wrote down the representation This function takes on the value 1 at x 0 and the zeros are located at π 2π 3π The function can also be written as an infinite product of linear factors If we now multiply out the product of linear factors and compare coefficients of x 2 we obtain that is If one considers only the even powers of this infinite series one sees that For the odd powers the result is Moreover one obtains a result that John Wallis 1616 1703 had obtained in 1650 by a completely different method In the seventeenth century we should mention it was still common practice to work with infinite sequences and their limits intuitively without worrying about proofs of convergence Using the same method of comparing coefficients Euler derived the result that the sum of the reciprocals of fourth powers of the natural numbers converges to π 4 90 and that the analogous sum for sixth powers converges to π 6 945 He continued these derivations up to the 26th power In contrast even today no method has yet been found to determine a closed form expression for the limits of the sums of reciprocal odd powers Spektrum der Wissenschaft special issue 2 2005 Unendlich plus eins p 19 Euler generalized the problem of sums of reciprocal powers by introducing the zeta function where only prime powers appear Spektrum der Wissenschaft 9 2008 p 86 Euler discovered that while the sequence of sums of fractions with 1 in the numerator the harmonic series diverges it does so very slowly and that one can compare the values of this sequence with the natural logarithmic function The sequence converges to γ 0 577 215 664 In 1781 Euler calculated this limit value to 16 decimal places Lorenzo Mascheroni 1750 1800 did the calculation to 32 places Today the number γ is called the Euler Mascheroni constant Simultaneously with Jean d Alembert 1717 1783 Euler developed a formal system of calculation with complex numbers and investigated functions of complex variables however complex numbers had no real significance for Euler In 1743 in studying power series with complex arguments he discovered the following simple relationship between the trigonometric functions and the exponential function Euler s formula For z π one obtains the remarkable relationship e iπ 1 0 while for z π 2 the result is i e i π 2

Original URL path: http://mathematics-in-europe.eu/tr/76-enjoy-maths/strick/597-leonhard-euler-1707-1783-by-heinz-klaus-strick-germany (2013-11-18)

Open archived version from archive