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List Of Things Named After Karl Weierstrass
This is a list of things named after the German mathematician Karl Weierstrass. Mathematical concepts, theorems, and the like Named after Weierstrass and other persons Named after Weierstrass alone {{columns-list, colwidth=20em, * Weierstrass approximation theorem * Weierstrass coordinates * Weierstrass's elliptic functions * Weierstrass equation * Weierstrass factorization theorem * Weierstrass function * Weierstrass functions * Weierstrass M-test * Weierstrass point * Weierstrass preparation theorem * Weierstrass product inequality * Weierstrass ring * Weierstrass substitution * Weierstrass theorem (other) – any of several theorems * Weierstrass transform Typography * Weierstrass p, a form of the letter ''p'' used to denote the Weierstrass elliptic function Celestial bodies or features of them * Weierstrass (crater) * 14100 Weierstrass Research institutes * Weierstrass Institute for Applied Analysis and Stochastics The Weierstrass Institute f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Weierstrass
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 int ... [...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] |
14100 Weierstrass , a year of the pre-Julian Roman calendar
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141 may refer to: * 141 (number), an integer * AD 141, a year of the Julian calendar * 141 BC __NOTOC__ Year 141 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Caepio and Pompeius (or, less frequently, year 613 '' Ab urbe condita''). The denomination 141 BC for this year has been ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass (crater)
Weierstrass is a small lunar impact crater that is attached to the northern rim of the walled plain Gilbert, in the eastern part of the Moon. It also lies very near the crater Van Vleck, a similar formation just to the southeast that is almost attached to the outer rim. Due to its location, the crater appears foreshortened as seen from the Earth. The crater has an oval-shaped outer rim that is longer along an east–west axis. There are some slumped shelves along the inner walls to the north and south. The interior floor is nearly featureless, with only a few tiny impacts. Neither the rim nor the interior are marked by impact craters of significance. This crater was designated Gilbert N prior to being named by the IAU in 1976. Gazetteer of Planetary Nomenclature, International Astronomical Union (IAU) Working Group for Planetary System Nome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass P
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script ''p''. They play an important role in the theory of elliptic functions. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice. Symbol for Weierstrass \wp-function Definition Let \omega_1,\omega_2\in\mathbb be two complex numbers that are linearly independent over \mathbb and let \Lambda:=\mathbb\omega_1+\mathbb\omega_2:=\ be the lattice generated by those numbers. Then the \wp-function is defined as follows: \weierp(z,\omega_1,\omega_2):=\weierp(z,\Lambda) := \frac + \sum_\left(\frac 1 - \frac 1 \right). This series converges locally uniformly absolutely in \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Transform
In mathematics, the Weierstrass transform of a function , named after Karl Weierstrass, is a "smoothed" version of obtained by averaging the values of , weighted with a Gaussian centered at ''x''. Specifically, it is the function defined by :F(x)=\frac\int_^\infty f(y) \; e^ \; dy = \frac\int_^\infty f(x-y) \; e^ \; dy~, the convolution of with the Gaussian function :\frac e^~. The factor 1/√(4 π) is chosen so that the Gaussian will have a total integral of 1, with the consequence that constant functions are not changed by the Weierstrass transform. Instead of one also writes . Note that need not exist for every real number , when the defining integral fails to converge. The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity equal to 1, the ... [...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 Substitution
In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of x into an ordinary rational function of t by setting t = \tan \tfrac x2. This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line. The general transformation formula is: \int f(\sin x, \cos x)\, dx =\int f \frac. The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent. Leonhard Euler used it to evaluate the integral \int dx / (a + b\cos x) in his 1768 integral calculus textbook, and Adrien-Marie Legendre described the general method in 1817. The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. It is known in Russia as the universal trigonometric substitution, and also known ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Ring
In mathematics, a Weierstrass ring, named by Nagata after Karl Weierstrass, is a commutative local ring that is Henselian, pseudo-geometric, and such that any quotient ring by a prime ideal is a finite extension of a regular local ring. Examples *The Weierstrass preparation theorem can be used to show that the ring of convergent power series over the complex numbers in a finite number of variables is a Wierestrass ring. The same is true if the complex numbers are replaced by a perfect field with a valuation. *Every ring that is a finitely-generated module In mathematics, a finitely generated module is a module (mathematics), module that has a Finite set, finite generating set. A finitely generated module over a Ring (mathematics), ring ''R'' may also be called a finite ''R''-module, finite over ''R' ... over a Weierstrass ring is also a Weierstrass ring. References Bibliography * * Commutative algebra {{commutative-algebra-stub ... [...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 Preparation Theorem
In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point ''P''. It states that such a function is, up to multiplication by a function not zero at ''P'', a polynomial in one fixed variable ''z'', which is monic, and whose coefficients of lower degree terms are analytic functions in the remaining variables and zero at ''P''. There are also a number of variants of the theorem, that extend the idea of factorization in some ring ''R'' as ''u''·''w'', where ''u'' is a unit and ''w'' is some sort of distinguished Weierstrass polynomial. Carl Siegel has disputed the attribution of the theorem to Weierstrass, saying that it occurred under the current name in some of late nineteenth century ''Traités d'analyse'' without justification. Complex analytic functions For one variable, the local form of an analytic function ''f''(''z'') near 0 is ''z''''k''''h''(''z'') where ''h''(0) is not 0, and ''k' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Point
In mathematics, a Weierstrass point P on a nonsingular algebraic curve C defined over the complex numbers is a point such that there are more functions on C, with their poles restricted to P only, than would be predicted by the Riemann–Roch theorem. The concept is named after Karl Weierstrass. Consider the vector spaces :L(0), L(P), L(2P), L(3P), \dots where L(kP) is the space of meromorphic functions on C whose order at P is at least -k and with no other poles. We know three things: the dimension is at least 1, because of the constant functions on C; it is non-decreasing; and from the Riemann–Roch theorem the dimension eventually increments by exactly 1 as we move to the right. In fact if g is the genus of C, the dimension from the k-th term is known to be :l(kP) = k - g + 1, for k \geq 2g - 1. Our knowledge of the sequence is therefore :1, ?, ?, \dots, ?, g, g + 1, g + 2, \dots. What we know about the ? entries is that they can increment by at most 1 each time (this is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
