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Weierstrass Approximation Theorem
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching mathematics, physics, botany and gymnastics. He later received an honorary doctorate and became professor of mathematics in Berlin. Among many other contributions, Weierstrass formalized the definition of the continuity of a function, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals. Biography Weierstrass was born into a Roman Catholic family in Ostenfelde, a village near Ennigerloh, in the Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst both of whom were catholic Rhinelanders. His inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ennigerloh
Ennigerloh () is a town in the Warendorf (district), district of Warendorf, in North Rhine-Westphalia, Germany. It is situated approximately 25 km northeast of Hamm and 30 km southeast of Münster. The town, located in an agricultural area and with a well-preserved medieval quarter, became more industrial in the 20th century as several cement factories were installed. Some of these closed towards the end of the century. Furniture manufacturing was also a significant industry. Geography Subdivisions * Enniger * Westkirchen * Ostenfelde (Ennigerloh), Ostenfelde Notable people * Alois Hanslian (born 1943), painter * Willy Hartner (1905–1981), professor, founded the Institute for the History of Natural Sciences in Frankfurt am Main * Karl Weierstrass (1815–1897), mathematician often described as "the father of analysis" References External links * {{Warendorf-geo-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathias Lerch
Mathias Lerch (''Matyáš Lerch'', ) (20 February 1860, Milínov – 3 August 1922, Sušice) was a Czech mathematician who published about 250 papers, largely on mathematical analysis and number theory. He studied in Prague and Berlin, and held teaching positions at the Czech Technical Institute in Prague, the University of Fribourg in Switzerland, the Czech Technical Institute in Brno, and Masaryk University in Brno; he was the first mathematics professor at Masaryk University when it was founded in 1920. In 1900, he was awarded the Grand Prize of the French Academy of Sciences for his number-theoretic work. The Lerch zeta function is named after him, as is the Appell–Lerch sum. His doctoral students include Michel Plancherel and Otakar Borůvka Otakar Borůvka (10 May 1899 in Uherský Ostroh – 22 July 1995 in Brno) was a Czech mathematician best known today for his work in graph theory.. Education and career Borůvka was born in Uherský Ostroh, a town in M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Theorem (other)
Several theorems are named after Karl Weierstrass. These include: *The Weierstrass approximation theorem, of which one well known generalization is the Stone–Weierstrass theorem *The Bolzano–Weierstrass theorem, which ensures compactness of closed and bounded sets in R''n'' *The Weierstrass extreme value theorem, which states that a continuous function on a closed and bounded set obtains its extreme values *The Weierstrass–Casorati theorem describes the behavior of holomorphic functions near essential singularities *The Weierstrass preparation theorem describes the behavior of analytic functions near a specified point *The Lindemann–Weierstrass theorem concerning the transcendental numbers *The Weierstrass factorization theorem In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an e .. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass–Erdmann Condition
The Weierstrass–Erdmann condition is a mathematical result from the calculus of variations, which specifies sufficient conditions for broken extremals (that is, an extremal which is constrained to be smooth except at a finite number of "corners"). Conditions The Weierstrass-Erdmann corner conditions stipulate that a broken extremal y(x) of a functional J=\int\limits_a^b f(x,y,y')\,dx satisfies the following two continuity relations at each corner c\in ,b/math>: Applications The condition allows one to prove that a corner exists along a given extremal. As a result, there are many applications to differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili .... In calculations of the Weierstrass E-Function, it is often helpful to find where corners exist along the cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
(ε, δ)-definition Of Limit
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function ''f'' assigns an output ''f''(''x'') to every input ''x''. We say that the function has a limit ''L'' at an input ''p,'' if ''f''(''x'') gets closer and closer to ''L'' as ''x'' moves closer and closer to ''p''. More specifically, when ''f'' is applied to any input ''sufficiently'' close to ''p'', the output value is forced ''arbitrarily'' close to ''L''. On the other hand, if some inputs very close to ''p'' are taken to outputs that stay a fixed distance apart, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Product Inequality
In mathematics, the Weierstrass product inequality states that for any real numbers 0 ≤ ''a1'', ''..., an'' ≤ 1 we have :(1-a_1)(1-a_2)(1-a_3)(1-a_4)....(1-a_n) \geq 1-S_n, :(1+a_1)(1+a_2)(1+a_3)(1+a_4)....(1+a_n) \geq 1+S_n, where S_n=a_1+a_2+a_3+a_4+....+a_n. The inequality is named after the German mathematician Karl Weierstrass. It can be proven easily via mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help .... References * {{cite book , last1=Honsberger , first1=Ross , title=More mathematical morsels , date=1991 , publisher=Mathematical Association of America , location= ashington, D.C., isbn=978-1-4704-5838-6 Inequalities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Function
In mathematics, the Weierstrass function is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstrass function has historically served the role of a pathological (mathematics), pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness. These types of functions were denounced by contemporaries: Henri Poincaré famously described them as "monsters" and called Weierstrass' work "an outrage against common sense", while Charles Herm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ludwig Stickelberger
Ludwig Stickelberger (18 May 1850 – 11 April 1936) was a Swiss mathematician who made important contributions to linear algebra (theory of elementary divisors) and algebraic number theory (Stickelberger relation in the theory of cyclotomic fields). Short biography Stickelberger was born in Buch in the canton of Schaffhausen into a family of a pastor. He graduated from a gymnasium in 1867 and studied next in the University of Heidelberg. In 1874 he received a doctorate in Berlin under the direction of Karl Weierstrass for his work on the transformation of quadratic forms to a diagonal form. In the same year, he obtained his Habilitation from Polytechnicum in Zurich (now ETH Zurich). In 1879 he became an extraordinary professor in the Albert Ludwigs University of Freiburg. From 1896 to 1919 he worked there as a full professor, and from 1919 until his return to Basel in 1924 he held the title of a distinguished professor ("ordentlicher Honorarprofessor"). He was married in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Schwarz
Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis. Life Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kummer, who was the daughter to the mathematician Ernst Eduard Kummer and Ottilie née Mendelssohn (a daughter of Nathan Mendelssohn's and granddaughter of Moses Mendelssohn). Schwarz and Kummer had six children, including his daughter Emily Schwarz. Schwarz originally studied chemistry in Berlin but Ernst Eduard Kummer and Karl Theodor Wilhelm Weierstrass persuaded him to change to mathematics. He received his Ph.D. from the Universität Berlin in 1864 and was advised by Kummer and Weierstrass. Between 1867 and 1869 he worked at the University of Halle, then at the Swiss Federal Polytechnic. From 1875 he worked at Göttingen University, dealing with the subjects of complex analysis, differential geometry and the calculus of variations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedrich Schottky
Friedrich Hermann Schottky (24 July 1851 – 12 August 1935) was a German mathematician who worked on elliptic, abelian, and theta functions and introduced Schottky groups and Schottky's theorem. He was born in Breslau, Germany (now Wrocław, Poland) and died in Berlin. Schottky was a professor at the University of Zurich from 1882-1892. He is also the father of Walter H. Schottky, the German physicist and inventor of a variety of semiconductor concepts. See also *Prime form *Prym variety *Walter H. Schottky Walter Hans Schottky (23 July 1886 – 4 March 1976) was a German physicist who played a major early role in developing the theory of electron and ion emission phenomena, invented the screen-grid vacuum tube in 1915 while working at Siemens ... External links * * * * 1851 births 1935 deaths 19th-century German mathematicians 20th-century German mathematicians Scientists from Wrocław People from the Province of Silesia ETH Zurich faculty German expa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur Schoenflies
Arthur Moritz Schoenflies (; 17 April 1853 – 27 May 1928), sometimes written as Schönflies, was a German mathematician, known for his contributions to the application of group theory to crystallography, and for work in topology. Schoenflies was born in Landsberg an der Warthe (modern Gorzów, Poland). Arthur Schoenflies married Emma Levin (1868–1939) in 1896. He studied under Ernst Kummer and Karl Weierstrass, and was influenced by Felix Klein. The Schoenflies problem is to prove that an (n - 1)-sphere in Euclidean ''n''-space bounds a topological ball, however embedded. This question is much more subtle than it initially appears. He studied at the University of Berlin from 1870 to 1875. He obtained a doctorate in 1877, and in 1878 he was a teacher at a school in Berlin. In 1880, he went to Colmar to teach. Schoenflies was a frequent contributor to Klein's encyclopedia: In 1898 he wrote on set theory, in 1902 on kinematics, and on projective geometry in 1910. He was a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Runge
Carl David Tolmé Runge (; 30 August 1856 – 3 January 1927) was a German mathematician, physicist, and spectroscopist. He was co-developer and co-eponym of the Runge–Kutta method (German pronunciation: ), in the field of what is today known as numerical analysis. Life and work Runge spent the first few years of his life in Havana, where his father Julius Runge was the Danish consul. His mother was Fanny Schwartz Tolmé. The family later moved to Bremen, where his father died early (in 1864). In 1880, he received his Ph.D. in mathematics at Berlin, where he studied under Karl Weierstrass. In 1886, he became a professor at the Technische Hochschule Hannover in Hanover, Germany. His interests included mathematics, spectroscopy, geodesy, and astrophysics. In addition to pure mathematics, he did experimental work studying spectral lines of various elements (together with Heinrich Kayser), and was very interested in the application of this work to astronomical spectroscopy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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