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Topic: Weierstrass

	Stone–Weierstrass theorem - Wikipedia, the free encyclopedia
		In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function.
		The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a,b], an arbitrary compact Hausdorff space K is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C(K) is investigated.
		As a consequence of the Weierstrass approximation theorem, one can show that the space C[a,b] is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients.
	en.wikipedia.org /wiki/Stone-Weierstrass_theorem (951 words)

	Weierstrass (Site not responding. Last check: 2007-11-06)
		Weierstrass had made a decision to become a mathematician but he was still supposed to be on a course studying public finance and administration.
		Weierstrass attended Gudermann's lectures on elliptic functions, some of the first lectures on this topic to be given, and Gudermann strongly encouraged Weierstrass in his mathematical studies.
		Weierstrass began his career as a qualified teacher of mathematics at the Pro-Gymnasium in Deutsch Krone in West Prussia (now Poland) in 1842 where he remained until he moved to the Collegium Hoseanum in Braunsberg in 1848.
	www-history.mcs.st-andrews.ac.uk /history/Mathematicians/Weierstrass.html (2473 words)

	[No title]
		Weierstrass gained enough confidence to apply for the post at the University in Breslau which was vacated by Kummer's appointment as professor in Berlin.
		Weierstrass even considered the possibility of leaving Berlin for Switzerland to avoid the continuing conflict; but, since he did not wish his successor at the university to be chosen by Kronecker, he decided to stay.
		Weierstrass himself was no longer able to supervise the whole process so his students undertook the task of gathering and polishing up his lectures based on their own notes or transcripts.
	math.nist.gov /opsf/personal/weierstrass.html (3054 words)

	PlanetMath: Weierstrass function
		The Weierstrass function is a continuous function that is nowhere differentiable, and hence is not an analytic function.
		The function is named after Karl Weierstrass who presented it in a lecture for the Berlin Academy in 1872 [1].
		This is version 15 of Weierstrass function, born on 2002-01-03, modified 2005-02-28.
	www.planetmath.org /encyclopedia/WeierstrassFunction.html (221 words)

	Weierstrass, Karl Theodor Wilhelm - Hutchinson encyclopedia article about Weierstrass, Karl Theodor Wilhelm
		Weierstrass was born in Ostenfelde, Westphalia, and trained as a teacher.
		Weierstrass' breakthrough came with a paper in 1854 that solved the inversion of hyperelliptic integrals.
		In the 1890s Weierstrass planned the publication of his life's work, again to be compiled from lecture notes.
	encyclopedia.farlex.com /Weierstrass%2c+Karl+Theodor+Wilhelm (210 words)

	10.11. Weierstrass, Karl (1815-1897)
		Karl Weierstrass was one of the leaders in rigor in analysis and was known as the "father of modern analysis." In addition, he is considered one of the greatest mathematics teachers of all-time.
		Karl Wilhelm Theodor Weierstrass was born October 31, 1815, in Ostenfelde, Westphalia, Germany.
		Weierstrass is famous in mathematics for numerous accomplishments.
	www.shu.edu /projects/reals/history/weierstr.html (1271 words)

	Weierstrass's elliptic functions - Wikipedia, the free encyclopedia
		The Weierstrass elliptic function can be given as the inverse of an elliptic integral.
		and not doubly-periodic, and a theta function called the Weierstrass sigma function, of which his zeta-function is the log-derivative.
		The Weierstrass sigma-function is an entire function; it played the role of 'typical' function in a theory of random entire functions of J.
	en.wikipedia.org /wiki/Weierstrass%27s_elliptic_functions (1249 words)

	Karl Theodor Wilhelm Weierstrass
		Weierstrass was torn between the subject he loved and the subject his father wanted for him, and he spent 4 years of intensive fencing and drinking.
		Eventually, Weierstrass made the decision to become a mathematicianHe convinced his father to let him study at the Theological and Philosophical Academy of Münster so that he could take the necessary examinations to become a secondary school teacher.
		Weierstrass published a full version of his theory of inversion of hyperelliptic integrals in his next paper in 1856.
	www.stetson.edu /~efriedma/periodictable/html/W.html (835 words)

	Biographie : Karl Theodor Weierstrass (31 octobre 1815 [Ostenfelde] - 19 février 1897 [Berlin]) (Site not responding. Last check: 2007-11-06)
		A Munster, Weierstrass rencontre Guddermann, qui l'éveillera complètement aux mathématiques.
		Weierstrass se signale aussi par sa volonté d'algébrisation de l'analyse.
		Weierstrass est très affecté, et brûle même toutes ses lettres.
	www.bibmath.net /bios/index.php3?action=affiche&quoi=weierstrass (713 words)

	Stone-Weierstrass theorem (Site not responding. Last check: 2007-11-06)
		In mathematical analysis the Weierstrass approximation theorem states that every continuous function defined on an interval [ a b ] can be uniformly approximated as closely as desired by a polynomial function.
		The Stone-Weierstrass theorem generalizes the Weierstrass theorem in two directions: instead of the interval [ a b ] an arbitrary compact Hausdorff space K is considered and instead of the algebra of polynomial functions approximation with elements more general subalgebras of C(K) is investigated.
		The set of all polynomial functions a subalgebra of C[ a b ] and the content of the Weierstrass theorem is that this subalgebra is dense in C[ a b ].
	www.freeglossary.com /Weierstrass_approximation_theorem (1026 words)

	Weierstrass preparation theorem - Wikipedia, the free encyclopedia
		w, where u is a unit and w is some sort of distinguished Weierstrass polynomial.
		C.L. 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.
		This has the immediate consequence that the set of zeroes of f, near (0,..., 0), can be found by fixing any small value of z and then solving W(z).
	en.wikipedia.org /wiki/Weierstrass_preparation_theorem (243 words)

	Geometry.Net - Scientists: Weierstrass Karl
		Weierstrass was the oldest son of a minor customs official, and his childhood is not particularly unusual.
		Weierstrass uppfann också en [kontinuerlig funktion] som saknade derivata i varje punkt.
		Weierstrass, Karl (1815-1897) IRA Karl Weierstrass was one of the leaders in rigor in analysis and was known as the "father of modern analysis." In addition, he is considered one of the greatest mathematics teachers of all-time.
	www.geometry.net /scientists/weierstrass_karl.php (2078 words)

	WEIERSTRASS, K.T.W.(1815-1897) and RIEMANN, G.F.B.(1826-1866)
		Weierstrass wrote a number of early papers on hyperelliptic integrals, Abelian functions, and algebraic differntial equations, but his widest known contribution to mathematics is his construcition of complex functions by means of power series.
		In algebra, Weierstrass was perhaps the first to give a so-called posrulational definition of a determinsant.
		He defuned the determinant of a square matrix Aas a polynomial in the elements of A, which is homogeneous and linear in the elements of each row of A, which merely changes sign when two rows of Aare permuted, and which reduces to I when A is the correspending identity matrix.
	library.thinkquest.org /22584/temh3021.htm (518 words)

	CCSD thèses-EN-ligne: Points de Weierstrass et jacobienne de courbes algebriques de genre 3
		More precisely, the object of this thesis is the study of the group generated by the Weierstrass points in the Jacobian for some smooth plane curves of genus three.
		Moreover, we compute this group for the only plane quartic, apart for Fermat's quartic, possessing the minimal number of Weierstrass points, that is twelve; also in this case, the geometry of the Jacobian is useful to compute this group.
		These computations enable us to give an estimation on the rank of this group and on the torsion part for a generic quartic, depending on the number of hyperflexes (that is points where the tangent meets the curve with multiplicity four).
	tel.ccsd.cnrs.fr /documents/archives0/00/00/11/37/index_fr.html (573 words)

	Simple genus-2 Jacobians with rational points of high order
		This has a rational non-Weierstrass point (x,y)=(0,10) that differs from the Weierstrass point (10,0) by a 17-torsion divisor; since the difference between (10,0) and the Weierstrass point (-2/3,0) must be 2-torsion, we conclude that (0,10) - (-2/3,0) is a torsion divisor of order 34.
		He confirmed that for the N=39 curve these four pairs of points, together with the six Weierstrass points, are the only points of C, even over the complex numbers, that differ from their hyperelliptic conjugates by torsion divisors.
		The difference between this point and either of the Weierstrass points is a torsion divisor of order 40, the highest known (as of 6/2001) for a simple genus-2 Jacobian.
	www.math.harvard.edu /~elkies/g2_tors.html (1731 words)

	Weierstrass's elliptic functions (Site not responding. Last check: 2007-11-06)
		In mathematics, Weierstrass introduced some particular ellipticfunctions that have become the basis for the most standard notations used.
		and not doubly-periodic, and a theta function called the Weierstrass sigma-function, of whichhis zeta-function is the log-derivative.
		The Weierstrass sigma-function is an entire function ; it playedthe role of 'typical' function in a theory of random entire functions of J.
	www.therfcc.org /weierstrass%27s-elliptic-functions-153227.html (412 words)

	The Weierstrass representation of closed surfaces in R 3 - Taimanov (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: In the present paper which a sequel to [19] and [20] a global Weierstrass representation of an arbitrary closed oriented surface of genus 1 in the the three-space is constructed.
		The Weierstrass spectrum of a torus immersed into R 3 is introduced and finite-zone planes as well as finite-zone solutions to the modified Novikov-Veselov equations are constructed.
		The Weierstrass representation of spheres in R³, the..
	citeseer.csail.mit.edu /228724.html (596 words)

	PlanetMath: Weierstrass preparation theorem
		The following theorem is known as the Weierstrass preparation theorem, though sometimes that name is reserved for the corollary and this theorem is then known as the Weierstrass division theorem.
		Cross-references: Weierstrass polynomial, linear combinations, finite, expansions, power series, representation, independent, constant, degree, Variable, polynomial, bounded, holomorphic, polydisc, order, integer, positive, origin, neighbourhood, analytic, function, coordinates, corollary, theorem
		This is version 2 of Weierstrass preparation theorem, born on 2005-02-22, modified 2005-03-07.
	www.planetmath.org /encyclopedia/WeierstrassDivisionTheorem.html (202 words)

	Teaching
		Preliminaries, power series, elementary trascendental functions, integration along path, Cauchy theorem for star regions, Cauchy integral formula, Taylor series, properties of holomorphic functions, isolated singularities, meromorphic functions, Casorati-Weierstrass theorem, Laurent series in the annulus, the general Cauchy theorem, residus and residus theorem, calculus of definite integrals.
		, the addition theorem for the Weierstrass pi function, analitic prolongation and the concept of analitic function, Algebraic functions.
		Weierstrass points: weight and number, hyperelliptic RS, Hurwitz's theorem on Aut(X), Abel's theorem and Jacobi's inversion, the Jacobian, embeddings into projective spaces and algebraic curves, the canonical model, the geometry of curves of low genus.
	web.unife.it /utenti/andrea.delcentina/teaching.html (152 words)

	weierstrass (Site not responding. Last check: 2007-11-06)
		Karl Weierstrass más conocido por su construcción de la teoría de las funciones complejas por medio de series.
		Después que Weierstrass había ocupado varias posiciones de enseñanza menor, llegó a ser reconocido después que publicó una gran cantidad de escritos de las funciones abellacas en el periódico CRELLE.
		En sus conferencias el año 1863 Weierstrass probó que los números complejos son sólo conmutativos en su extensión algebraica de los números reales.
	www.sectormatematica.cl /biografias/weierstrass.htm (307 words)

	WEIERSTRASS (Site not responding. Last check: 2007-11-06)
		En cuanto a su faceta como estudiante siempre despuntó adjuducándose premios de final de curso e incluso saltándose un curso en el Instituto y al no ingresar en la facultad de matemáticas (por expreso deseo de su padre) comenzó su labor autodidacta en el mundo de las matemáticas.
		Cabe destacar que cuando Weierstrass reorientó su vida estudiantil hacia las matemáticas elaboró un estudio sobre las funciones elípticas.
		Se deben a Weierstrass una nueva teoría de las funciones elípticas, el teorema de la aproximación uniforme de una función cualquiera por polinomios, y la importante teoría de las "Funciones analíticas" de variable compleja, con los conceptos de prolongación analítica, trascendentes enteras, factores primarios,etc.
	estudiantes.uam.es /david.manras/weierstrass.html (343 words)

	Karl Weierstrass (Site not responding. Last check: 2007-11-06)
		At 26 he took his examinations for a teaching certificate and did so well on a piece of original mathematics that Gudermann recommended that he teach at a university instead of a secondary school.
		When Sonja Kovalevskaia came to Germany from Russia to study mathematics, she was not allowed to attend classes at the university because she was a woman, so Weierstrass privately tutored her for several years.
		Weierstrass' work had a great influence on 19th and 20th century mathematics, and several of his results are now a standard part of a typical calculus course, for example, the Weierstrass M-test for series and his example of a continuous but nowhere differentiable function.
	www.scidiv.bcc.ctc.edu /Math/Weierstrass.html (515 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-06)
		Date: 09/13/2002 at 01:11:24 From: Colin Subject: Weierstrass curve I read in an old article written by Alfred Adler that about 100 years ago the mathematician Weierstrass gave an example of a curve consisting of angles, or corners, and nothing else.
		Date: 09/13/2002 at 10:26:59 From: Doctor Fenton Subject: Re: Weierstrass curve Hi Colin, Weierstrass's original example was a trigonometric series, and checking the literature, I found several different versions of it, so I'm not sure exactly which version was Weierstrass's.
		I did a quick Web search using keywords "Weierstrass" and "nondifferentiable" and found many references, but no explicit formulas, but then I didn't check many sites.
	www.mathforum.org /library/drmath/view/61209.html (318 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Weierstrass' paper with his proof of the Weierstrass Theorem on density on algebraic polynomials in the space of continuous real-valued functions on any finite closed interval.
		Weierstrass, K., Sur la possibilité d'une représentation analytique des fonctions dites arbitraires d'une variable réelle, J.
		This is the translation of the Weierstrass 1885 paper and, as the original, it appeared in two parts and in subsequent issues, but under the same title.
	www.math.technion.ac.il /hat/wei.html (189 words)

	Karl Weierstrass (Site not responding. Last check: 2007-11-06)
		Karl Theodor Wilhelm Weierstrass (de octubre el 31 de 1815 - de febrero el 19 de 1897) era un matemático alemán que se cita a menudo como el "padre del análisis moderno".
		(la letra ss se puede transcribir como ss;uno escribe a menudo a Weierstrass.) Él nació en Ostenfelde, Westfalia (hoy Alemania) y murió en Berlín, Alemania.
		Las versiones numeradas de las publicaciones originales de Weierstrass son libremente accesibles en línea de la biblioteca del der Wissenschaften de Berlín Brandenburgische Akademie.
	www.yotor.net /wiki/es/ka/Karl%20Weierstrass.htm (100 words)

	AMCA: The density of elliptic curves having a global minimal Weierstrass equation by Ebru Bekyel (Site not responding. Last check: 2007-11-06)
		We show that a positive density of elliptic curves over a number field counted using their short Weierstrass equations belong to a given Weierstrass class and in particular, a positive density of elliptic curves have a global minimal Weierstrass equation.
		The density is given by a ratio of partial zeta functions of the number field K evaluated at 10 with some extra factors for the bad primes.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /cgi-bin/amca/calp-16 (139 words)

	Reports of the Mathematical Institute Leiden (Site not responding. Last check: 2007-11-06)
		The group generated by the Weierstrass points of a smooth curve in its Jacobian is an interesting intrinsic invariant of the curve.
		Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space M
		In particular, we can conclude that for a generic quartic, it is a free abelian group of rank at least 11.
	www.math.leidenuniv.nl /reports/2002-15.shtml (136 words)

	Citations: Neuer Beweis des Fundamentalsatzes der Algebra - WEIERSTRASS (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Weierstrass, Neuer Beweis des Fundamentalsatzes der Algebra, Mathematische Werker, Tome III, Mayer und Mueller, Berlin, 1903, pp.
		P97] BP,a] This method is a multivariate version of Newton s iteration, which converges simultaneously to all the n zeros of p(x) by using order of n 2 ops in each iteration.
		In particular, Durand Kerner s algorithm of [D60] and [Ke66] also justly called the Weierstrass algorithm, cf.
	citeseer.ist.psu.edu /context/829939/0 (312 words)

	A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory, by Otmar Venjakob (Site not responding. Last check: 2007-11-06)
		In this paper and a forthcoming joint one with Y. Hachimori we study Iwasawa modules over an infinite Galois extension K of a number field k whose Galois group G=G(K/k) is isomorphic to the semidirect product of two copies of the p-adic numbers.
		As a main tool we prove a Weierstrass preparation theorem for certain skew power series rings.
		Finally we show that the completed group algebra with coefficients in the finite field of p elements is a unique factorization domain in the sense of Chatters.
	www.math.uiuc.edu /Algebraic-Number-Theory/0343 (165 words)

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