Félix Édouard Justin Émile Borel was a French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Borel was born in Saint-Affrique, the son of a Protestant pastor, he studied at the Collège Sainte-Barbe and Lycée Louis-le-Grand before applying to both the École normale supérieure and the École Polytechnique. Although he qualified in the first position for both, he chose to attend the former institution in 1889; that year he won an annual national mathematics competition. After graduating in 1892, he placed first in the agrégation, a competitive civil service examination leading to the position of professeur agrégé, his thesis, published in 1893, was titled Sur quelques points de la théorie des fonctions. That year, Borel started a four-year stint as a lecturer at the University of Lille, during which time he published 22 research papers, he returned to the École normale in 1897, was appointed to the chair of theory of function, which he held until 1941.
In 1901, Borel married the daughter of colleague Paul Émile Appel. Émile Borel died in Paris on 3 February 1956. Along with René-Louis Baire and Henri Lebesgue, Émile Borel was among the pioneers of measure theory and its application to probability theory; the concept of a Borel set is named in his honor. One of his books on probability introduced the amusing thought experiment that entered popular culture under the name infinite monkey theorem or the like, he published a series of papers that first defined games of strategy. With the development of statistical hypothesis testing in the early 1900s various tests for randomness were proposed. Sometimes these were claimed to have some kind of general significance, but they were just viewed as simple practical methods. In 1909, Borel formulated the notion that numbers picked randomly on the basis of their value are always normal, with explicit constructions in terms of digits, it is quite straightforward to get numbers that are normal. In 1913 and 1914 he bridged the gap between hyperbolic geometry and special relativity with expository work.
For instance, his book Introduction Geometrique à quelques Théories Physiques described hyperbolic rotations as transformations that leave a hyperbola stable just as a circle around a rotational center is stable. In 1928 he co-founded Institut Henri Poincaré in Paris. In the 1920s, 1930s, 1940s, he was active in politics. In 1922, he founded Paris Institute of the oldest French school for statistics. From 1924 to 1936, he was a member of the French National Assembly. In 1925, he was Minister of Marine in the cabinet of fellow mathematician Paul Painlevé. During the Second World War, he was a member of the French Resistance. Besides the Centre Émile Borel at the Institut Henri Poincaré in Paris and a crater on the Moon, the following mathematical notions are named after him: Borel described a poker model which he coins La Relance in his 1938 book Applications de la théorie des probabilités aux Jeux de Hasard. Borel was awarded the Resistance Medal in 1950. "La science est-elle responsable de la crise mondiale?", Scientia: rivista internazionale di sintesi scientifica, 51, 1932, pp. 99–106.
"La science dans une société socialiste", Scientia: rivista internazionale di sintesi scientifica, 31, 1922, pp. 223–228. "Le continu mathématique et le continu physique", Rivista di scienza, 6, 1909, pp. 21–35. 8. Michel Pinault, Emile Borel, une carrière intellectuelle sous la 3ème République, Paris, L'Harmattan, 2017. Voir: michel-pinault.over-blog.com Quotations related to Émile Borel at Wikiquote French Wikisource has original text related to this article: Auteur:Émile Borel Media related to Émile Borel at Wikimedia Commons Wikilivres has original media or text related to this article: Émile Borel Works by or about Émile Borel at Internet Archive O'Connor, John J.. Author profile in the database zbMATH
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system; the Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute metaphysical, sense.
Today, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, Euclidean space is a good approximation for it only over short distances. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects; this is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas. The Elements is a systematization of earlier knowledge of geometry, its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones, they are now nearly all lost. There are 13 books in the Elements: Books I–IV and VI discuss plane geometry.
Many results about plane figures are proved, for example "In any triangle two angles taken together in any manner are less than two right angles." and the Pythagorean theorem "In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced, it is proved. Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base; the platonic solids are constructed. Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of simple axioms; until the advent of non-Euclidean geometry, these axioms were considered to be true in the physical world, so that all the theorems would be true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.
Near the beginning of the first book of the Elements, Euclid gives five postulates for plane geometry, stated in terms of constructions: Let the following be postulated:To draw a straight line from any point to any point. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance; that all right angles are equal to one another.: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique; the Elements include the following five "common notions": Things that are equal to the same thing are equal to one another. If equals are added to equals the wholes are equal. If equals are subtracted from equals the differences are equal.
Things that coincide with one another are equal to one another. The whole is greater than the part. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than the others, they aspired to create a system of certain propositions, to them it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible, since one can construct consistent systems of geometry in which the parallel postulate is true, others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated. For example, Playfair's axiom states: In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the giv
Andrey Nikolaevich Kolmogorov was a Soviet mathematician who made significant contributions to the mathematics of probability theory, intuitionistic logic, classical mechanics, algorithmic information theory and computational complexity. Andrey Kolmogorov was born in Tambov, about 500 kilometers south-southeast of Moscow, in 1903, his unmarried mother, Maria Y. Kolmogorova, died giving birth to him. Andrey was raised by two of his aunts in Tunoshna at the estate of his grandfather, a well-to-do nobleman. Little is known about Andrey's father, he was named Nikolai Matveevich Kataev and had been an agronomist. Nikolai had been exiled from St. Petersburg to the Yaroslavl province after his participation in the revolutionary movement against the czars, he was presumed to have been killed in the Russian Civil War. Andrey Kolmogorov was educated in his aunt Vera's village school, his earliest literary efforts and mathematical papers were printed in the school journal "The Swallow of Spring". Andrey was the "editor" of the mathematical section of this journal.
Kolmogorov's first mathematical discovery was published in this journal: at the age of five he noticed the regularity in the sum of the series of odd numbers: 1 = 1 2. In 1910, his aunt adopted him, they moved to Moscow, where he graduated from high school in 1920; that same year, Kolmogorov began to study at the Moscow State University and at the same time Mendeleev Moscow Institute of Chemistry and Technology. Kolmogorov writes about this time: "I arrived at Moscow University with a fair knowledge of mathematics. I knew in particular the beginning of set theory. I studied many questions in articles in the Encyclopedia of Brockhaus and Efron, filling out for myself what was presented too concisely in these articles."Kolmogorov gained a reputation for his wide-ranging erudition. While an undergraduate student in college, he attended the seminars of the Russian historian S. V. Bachrushin, he published his first research paper on the fifteenth and sixteenth centuries' landholding practices in the Novgorod Republic.
During the same period, Kolmogorov worked out and proved several results in set theory and in the theory of Fourier series. In 1922, Kolmogorov gained international recognition for constructing a Fourier series that diverges everywhere. Around this time, he decided to devote his life to mathematics. In 1925, Kolmogorov graduated from the Moscow State University and began to study under the supervision of Nikolai Luzin, he formed a lifelong close friendship with a fellow student of Luzin. Kolmogorov became interested in probability theory. In 1925, he published his work in intuitionistic logic — On the principle of the excluded middle, in which he proved that under a certain interpretation, all statements of classical formal logic can be formulated as those of intuitionistic logic. In 1929, Kolmogorov earned his Doctor of Philosophy degree, from Moscow State University. In 1930, Kolmogorov went on his first long trip abroad, traveling to Göttingen and Munich, to Paris, he had various scientific contacts in Göttingen.
First of all with Richard Courant and his students working on limit theorems, where diffusion processes turned out to be the limits of discrete random processes with Hermann Weyl in intuitionistic logic, lastly with Edmund Landau in function theory. His pioneering work, About the Analytical Methods of Probability Theory, was published in 1931. In 1931, he became a professor at the Moscow State University. In 1933, Kolmogorov published his book, Foundations of the Theory of Probability, laying the modern axiomatic foundations of probability theory and establishing his reputation as the world's leading expert in this field. In 1935, Kolmogorov became the first chairman of the department of probability theory at the Moscow State University. Around the same years Kolmogorov contributed to the field of ecology and generalized the Lotka–Volterra model of predator-prey systems. In 1936, Kolmogorov and Alexandrov were involved in the political persecution of their common teacher Nikolai Luzin, in the so-called Luzin affair.
In a 1938 paper, Kolmogorov "established the basic theorems for smoothing and predicting stationary stochastic processes"—a paper that had major military applications during the Cold War. In 1939, he was elected a full member of the USSR Academy of Sciences. During World War II Kolmogorov contributed to the Russian war effort by applying statistical theory to artillery fire, developing a scheme of stochastic distribution of barrage balloons intended to help protect Moscow from German bombers. In his study of stochastic processes Markov processes and the British mathematician Sydney Chapman independently developed the pivotal set of equations in the field, which have been given the name of the Chapman–Kolmogorov equations. Kolmogorov focused his research on turbulence, where his publications influenced the field. In classical mechanics, he is best known for the Kolmogorov–Arnold–Moser theorem, first presented in 1954 at the International Congress of Mathematicians. In 1957, working jointly with his student Vladimir Arnold, he solved a particular interpretation of Hilbert's thirteenth problem.
In mathematics, a monotonic function is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, was generalized to the more abstract setting of order theory. In calculus, a function f defined on a subset of the real numbers with real values is called monotonic if and only if it is either non-increasing, or non-decreasing; that is, as per Fig. 1, a function that increases monotonically does not have to increase, it must not decrease. A function is called monotonically increasing, if for all x and y such that x ≤ y one has f ≤ f, so f preserves the order. A function is called monotonically decreasing if, whenever x ≤ y f ≥ f, so it reverses the order. If the order ≤ in the definition of monotonicity is replaced by the strict order < one obtains a stronger requirement. A function with this property is called increasing. Again, by inverting the order symbol, one finds a corresponding concept called decreasing. Functions that are increasing or decreasing are one-to-one If it is not clear that "increasing" and "decreasing" are taken to include the possibility of repeating the same value at successive arguments, one may use the terms weakly increasing and weakly decreasing to stress this possibility.
The terms "non-decreasing" and "non-increasing" should not be confused with the negative qualifications "not decreasing" and "not increasing". For example, the function of figure 3 first falls rises falls again, it is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing. A function f is said to be monotonic over an interval if the derivatives of all orders of f are nonnegative or all nonpositive at all points on the interval; the term monotonic transformation can possibly cause some confusion because it refers to a transformation by a increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform. In this context, what we are calling a "monotonic transformation" is, more called a "positive monotonic transformation", in order to distinguish it from a “negative monotonic transformation,” which reverses the order of the numbers; the following properties are true for a monotonic function f: R → R: f has limits from the right and from the left at every point of its domain.
F can only have jump discontinuities. The discontinuities, however, do not consist of isolated points and may be dense in an interval; these properties are the reason. Two facts about these functions are: if f is a monotonic function defined on an interval I f is differentiable everywhere on I, i.e. the set of numbers x in I such that f is not differentiable in x has Lebesgue measure zero. In addition, this result cannot be improved to countable: see Cantor function. If f is a m
Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat, it is the two-dimensional analog of the volume of a solid. The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units, the standard unit of area is the square metre, the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, the area of any other shape or surface is a dimensionless real number. There are several well-known formulas for the areas of simple shapes such as triangles and circles.
Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus. For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape requires multivariable calculus. Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.
Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved. An approach to defining what is meant by "area" is through axioms. "Area" can be defined as a function from a collection M of special kind of plane figures to the set of real numbers which satisfies the following properties: For all S in M, a ≥ 0. If S and T are in M so are S ∪ T and S ∩ T, a = a + a − a. If S and T are in M with S ⊆ T T − S is in M and a = a − a. If a set S is in M and S is congruent to T T is in M and a = a; every rectangle R is in M. If the rectangle has length h and breadth k a = hk. Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i.e. S ⊆ Q ⊆ T. If there is a unique number c such that a ≤ c ≤ a for all such step regions S and T a = c, it can be proved that such an area function exists. Every unit of length has a corresponding unit of area, namely the area of a square with the given side length.
Thus areas can be measured in square metres, square centimetres, square millimetres, square kilometres, square feet, square yards, square miles, so forth. Algebraically, these units can be thought of as the squares of the corresponding length units; the SI unit of area is the square metre, considered an SI derived unit. Calculation of the area of a square whose length and width are 1 metre would be: 1 metre x 1 metre = 1 m2and so, a rectangle with different sides would have an area in square units that can be calculated as: 3 metres x 2 metres = 6 m2; this is equivalent to 6 million square millimetres. Other useful conversions are: 1 square kilometre = 1,000,000 square metres 1 square metre = 10,000 square centimetres = 1,000,000 square millimetres 1 square centimetre = 100 square millimetres. In non-metric units, the conversion between two square units is the square of the conversion between the corresponding length units. 1 foot = 12 inches,the relationship between square feet and square inches is 1 square foot = 144 square inches,where 144 = 122 = 12 × 12.
Similarly: 1 square yard = 9 square feet 1 square mile = 3,097,600 square yards = 27,878,400 square feetIn addition, conversion factors include: 1 square inch = 6.4516 square centimetres 1 square foot = 0.09290304 square metres 1 square yard = 0.83612736 square metres 1 square mile = 2.589988110336 square kilometres There are several other common units for area. The are was the original unit of area in the metric system, with: 1 are = 100 square metresThough the are has fallen out of use, the hectare is still used to measure land: 1 hectare = 100 ares = 10,000 square metres = 0.01 square kilometresOther uncommon metric units of area include the tetrad, the hectad, the myriad. The acre is commonly used to measure land areas, where 1 acre = 4,840 square yards = 43,560 square feet. An acre is 40% of a hectare. On the atomic scale, area is measured in units of barns, such that: 1 barn = 10−28 square meters; the barn is used in describing the cross-sectional area of interaction in nuclear physics.
In India, 20 dhurki = 1 dhur 20 dhur = 1 khatha 20 khata = 1 bigha 32 khata = 1 acre In the 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of
Johann Karl August Radon was an Austrian mathematician. His doctoral dissertation was on the calculus of variations. Radon was born in Tetschen, Austria-Hungary, now Děčín, Czech Republic, he received his doctoral degree at the University of Vienna in 1910. He spent the winter semester 1910/11 at the University of Göttingen he was an assistant at the German Technical University in Brno, from 1912 to 1919 at the Technical University of Vienna. In 1913/14, he passed his habilitation at the University of Vienna. Due to his near-sightedness, he was exempt from the draft during wartime. In 1919, he was called to become Professor extraordinarius at the newly founded University of Hamburg, he was Ordinarius at the University of Breslau from 1928 to 1945. After a short stay at the University of Innsbruck he became Ordinarius at the Institute of Mathematics of the University of Vienna on 1 October 1946. In 1954/55, he was rector of the University of Vienna. In 1939, Radon became corresponding member of the Austrian Academy of Sciences, in 1947, he became a member.
From 1952 to 1956, he was Secretary of the Class of Science of this Academy. From 1948 to 1950, he was president of the Austrian Mathematical Society. Johann Radon married Maria Rigele, a secondary school teacher, in 1916, they had three sons who died young or young. Their daughter Brigitte, born in 1924, obtained a Ph. D. in mathematics at the University of Innsbruck and married the Austrian mathematician Erich Bukovics in 1950. Brigitte lives in Vienna. Radon, as Curt C. Christian described him in 1987 at the occasion of the unveiling of his brass bust at the University of Vienna, was a friendly, good-natured man esteemed by students and colleagues alike, a noble personality, he did make the impression of a quiet scholar, but he was sociable and willing to celebrate. He loved music, he played music with friends at home, being an excellent violinist himself, a good singer, his love for classical literature lasted through all his life. In 2003, the Austrian Academy of Sciences founded an Institute for Computational and Applied Mathematics and named it after Johann Radon.
Radon is known for a number of lasting contributions, including: his part in the Radon–Nikodym theorem. He is the first to make use of the so-called Radon–Riesz property. O'Connor, John J.. Johann Radon at the Mathematics Genealogy Project Johann Radon Institute for Computational and Applied Mathematics
Maurice René Fréchet
Maurice Fréchet was a French mathematician. He made major contributions to the topology of point sets and introduced the entire concept of metric spaces, he made several important contributions to the field of statistics and probability, as well as calculus. His dissertation opened the entire field of functionals on metric spaces and introduced the notion of compactness. Independently of Riesz, he discovered the representation theorem in the space of Lebesgue square integrable functions, he was born to a Protestant family in Maligny, Yonne to Zoé Fréchet. At the time of his birth, his father was a director of a Protestant orphanage in Maligny and was in his youth appointed a head of a Protestant school. However, the newly established Third Republic was not sympathetic to religious education and so the laws were enacted requiring all education to be secular; as a result, his father lost his job. To generate some income his mother set up a boarding house for foreigners in Paris, his father was able to obtain another teaching position within the secular system – it was not a job of a headship and the family could not expect as high standards as they might have otherwise.
Maurice attended the secondary school Lycée Buffon in Paris where he was taught mathematics by Jacques Hadamard. Hadamard decided to tutor him on an individual basis. After Hadamard moved to the University of Bordeaux in 1894, Hadamard continuously wrote to Fréchet, setting him mathematical problems and harshly criticising his errors. Much Fréchet admitted that the problems caused him to live in a continual fear of not being able to solve some of them though he was grateful for the special relationship with Hadamard he was privileged to enjoy. After completing high-school Fréchet was required to enroll in the military service; this is the time when he was deciding whether to study mathematics or physics – he chose mathematics out of dislike of chemistry classes he would have had to take otherwise. Thus in 1900 he enrolled to École Normale Supérieure to study mathematics, he started publishing quite early, having published four papers in 1903. He published some of his early papers in the American Mathematical Society due to his contact with American mathematicians in Paris—particularly Edwin Wilson.
Fréchet served at many different institutions during his academic career. From 1907–1908 he served as a professor of mathematics at the Lycée in Besançon moved in 1908 to the Lycée in Nantes to stay there for a year. After that he served at the University of Poitiers between 1910–1919, he married in 1908 to Suzanne Carrive and had four children: Hélène, Henri and Alain. Fréchet was planning to spend a year in the United States at the University of Illinois but his plan was disrupted when the First World War broke out in 1914, he was mobilised on 4 August the same year. Because of his diverse language skills, gained when his mother ran the establishment for foreigners, he served as an interpreter for the British Army. However, this was not a safe job. French egalitarian ideals caused many academics to be mobilised, they served in the trenches and many of them were lost during the war. It is remarkable that during his service in the war, he still managed to produce cutting edge mathematical papers despite having little time to devote to mathematics.
After the end of the war, Fréchet was chosen to go to Strasbourg to help with the reestablishment of the university. He served as a professor of Director of the Mathematics Institute. Despite being burdened with administrative work, he was again able to produce a large amount of high quality research. In 1928 Fréchet decided to move back to Paris, thanks to encouragement from Borel, Chair in the Calculus of Probabilities and Mathematical Physics at the Sorbonne. Fréchet held a position of lecturer at the Sorbonne's Rockefeller Foundation and from 1928 was a Professor. Fréchet was promoted to tenured Chair of General Mathematics in 1933 and to Chair of Differential and Integral Calculus in 1935. In 1941 Fréchet succeeded Borel as Chair in the Calculus of Probabilities and Mathematical Physics, a position Fréchet held until he retired in 1949. From 1928 to 1935 Fréchet was put in charge of lectures at the École Normale Supérieure. Despite his major achievements, Fréchet was not overly appreciated in France.
As an illustration, while being nominated numerous times, he was not elected a member of the Academy of Sciences until the age of 78. In 1929 he became foreign member of the Polish Academy of Science and Arts and in 1950 foreign member of the Royal Netherlands Academy of Arts and Sciences. Fréchet was an Esperantist, publishing some articles in that constructed language, he served as president of the Internacia Scienca Asocio Esperantista from 1950–53. His first major work was his outstanding 1906 PhD thesis Sur quelques points du calcul fonctionnel, on the calculus of functionals. Here Fréchet introduced the concept of a metric space. Fréchet's level of abstraction is similar to that in group theory, proving theorems within a chosen axiomatic system which can be applied to a large array of particular cases. Here is a list of his most important works, in chronological order: Sur les opérations linéaires I-III, 1904–1907 Les Espaces abstraits, 1928 (Ab