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Lipschitz biography Rudolf Lipschitz's father was a landowner and Rudolf was born his father's estate at Bönkein which was near Königsberg. This was not a particularly easy time for Lipschitz whose health was rather poor and caused him to take a year away from his studies to recover. Lipschitz is remembered for the 'Lipschitz condition', an inequality that guarantees a unique solution to the www-history.mcs.st-andrews.ac.uk /Biographies/Lipschitz.html   (595 words)

The Lipschitz Constant   (Site not responding. Last check: 2007-10-13) The lipschitz constant (biography) is not a universal constant like E or π. Instead, the lipschitz constant is a property of a function f from one metric space into another. The nonnegative real number k acts as a lipschitz constant for the function f if, for every x and y in the domain, the distance from f(x) to f(y) is no larger than k times the distance from x to y. www.mathreference.com /top-ms,lip.html   (481 words)

PlanetMath: Lipschitz condition and differentiability "Lipschitz condition and differentiability" is owned by Mathprof. Cross-references: differentiable, function, QED, Lipschitz constant, compact set, continuously differentiable, mapping, secant, restricted, upper bound, finite, compact, continuous, derivative, linear mapping, norm, bounded linear maps, satisfies, restriction, compact subset, limit, converge, bounded, ratio, Lipschitz condition, relation, Banach spaces This is version 30 of Lipschitz condition and differentiability, born on 2001-11-12, modified 2006-09-13. planetmath.org /encyclopedia/CalculatingLipschitzRatios.html   (140 words)

CVGMT: The coarea formula for real-valued Lipschitz maps on stratified groups   (Site not responding. Last check: 2007-10-13) Abstract: We establish a coarea formula for real-valued Lipschitz maps on stratified groups when the domain is endowed with a homogeneous distance and level sets are measured by the Q-1 dimensional spherical Hausdorff measure. We construct a Lipschitz map on the Heisenberg group which is not approximately differentiable on a set of positive measure, provided that the Euclidean notion of differentiability is adopted. This phenomenon shows that the coarea formula holds for the natural class of Lipschitz maps which arises from the geometry of the group and that this class may be strictly larger than the usual one. cvgmt.sns.it /papers/mag03   (161 words)

Brian White's Research Papers This paper shows that the infimum of X(g) among all lipschitz maps g: M --> N homotopic to a given map f does not depend on the homotopy class of f; it only depends on the [p]-homotopy class of f, i. It follows that the identity map on M is homotopic to maps f with X(f) arbitrarily small if and only if the kth homotopy group of M vanishes for k=1,2,...,[p]. Let f be a harmonic map from the three dimensional unit ball to a compact smooth two dimensional manifold N, such that f has an isolated singularity at 0. math.stanford.edu /~white/biblio.htm   (3414 words)

Lipschitz continuity - Wikipedia, the free encyclopedia This is an example of a Lipschitz continuous function that is not differentiable. A function with a bounded first derivative is Lipschitz continuous, which follows from the mean value theorem. This structure is intermediate between that of a piecewise-linear manifold and a smooth manifold. en.wikipedia.org /wiki/Lipschitz_continuity   (634 words)

Most Recent Preprints Sublinear Higson corona and Lipschitz extensions by M.Cencelj, J.Dydak, J.Smrekar, and A.Vavpetic. On Gateaux differentiability of pointwise Lipschitz mappings by Jakub Duda. Integral mappings and the principle of local reflexivity for noncommutative L^1-spaces by Edward G. Effros, Marius Junge and Zhong-Jin Ruan. www.math.okstate.edu /~alspach/banach/recent.html   (5401 words)

Abstracts for publications and preprints of J. T. Tyson Using Cheeger's differentiability theorem for Lipschitz functions on metric measure spaces, we construct a conformal analogue of the Martin boundary for relatively compact domains in locally compact metric measure spaces which are locally $Q$-regular for some $Q>1$ and support a $(1,p)$-Poincaré inequality for some $p<Q$. For a general Carnot group $G$ with homogeneous dimension $Q$ we prove the existence of a fundamental solution of the $Q$-Laplacian whose exponential is a homogeneous norm on G. This implies a representation formula for smooth functions on $G$ which is used to prove the sharp Carnot group version of the celebrated Moser-Trudinger inequality. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincaré inequality. www.math.uiuc.edu /~tyson/abstracts.html   (2392 words)

Convergence of discrete approximations for constrained minimization   (Site not responding. Last check: 2007-10-13) If a constrained minimization problem, under Lipschitz or uniformly continuous hypotheses on the functions, has a strict local minimum, then a small perturbation of the functions leads to a minimum of the perturbed problem, close to the unperturbed minimum. Conditions are given for the perturbed minimum point to be a Lipschitz function of a perturbation parameter. This is used to study convergence rate for a problem of continuous programming, when the variable is approximated by step-functions. anziamj.austms.org.au /V36/part1/Craven.html   (109 words)

PlanetMath: proof of Hartman-Grobman theorem where we use the fact that by the Lipschitz inverse mapping theorem, if This is a direct consequence of the mean value inequality and the fact that Cross-references: extension, conjugate, derivative, differentiable map, constant, mean value inequality, ball, neighborhood, open, equation, solution, implies, homeomorphism, proposition, Lipschitz constant, contraction, inverse, Lipschitz, invertible, Lipschitz inverse mapping theorem, conjugation, restriction, projection, contained, range, linear operator, operator, induced, supremum, continuous maps, bounded, Banach space, skewness, equivalent, norm, topologically conjugate, maps, hyperbolic isomorphism planetmath.org /encyclopedia/ProofOfHartmanGrobmanTheorem.html   (210 words)

Lipschitz domain - Wikipedia, the free encyclopedia In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. The term is named after the German mathematician Rudolf Lipschitz. Many of the Sobolev embedding theorems require that the domain of study be a Lipschitz domain. en.wikipedia.org /wiki/Lipschitz_domain   (175 words)

Citebase - Minimal stretch maps between hyperbolic surfaces   (Site not responding. Last check: 2007-10-13) Authors: Thurston, William P. This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured laminations, which is attained with probability one on a simple closed curve. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9801039   (169 words)

DIMACS Workshop on Discrete Metric Spaces and their Algorithmic Applications A map f between metric spaces is said to be co-Lipschitz provided the image of every ball B(x,r) in the domain contains a ball B(f(x),r/C) of proportional radius in the range, where the constant C is independent of the ball. This is the natural notion of quotient map in the category of metric spaces when the Lipschitz maps are the morphisms. When X and Y are normed spaces and Y is a Lipschitz quotient of X, the most basic question is: What additional condition (if any) will guarantee that Y is also a linear quotient of X? I'll survey what is known on this question. dimacs.rutgers.edu /Workshops/MetricSpaces/abstracts.html   (2823 words)

UNT Department of Mathematics: Graduate Seminar Georganopoulos has sown that a continuous function f: X \rightarrow B, where X is a compact metric space and B a convex subset of a real normed space Y, is a uniform limit of Lipschitz maps from X to B. This result is obtained using a Lipschitz partition of unity. The class of Lipschitz functions is of particular interest since the problem of extension of such functions comes up in geometry also. Namely a Lipschitz function from a subset of a metric space to R, or from a subset of R^n to R^m can be extended to the whole space conserving the Lipschitz constant. www.math.unt.edu /seminars/grad.shtml   (3438 words)

Citebase - On Fréchet differentiability of Lipschitz maps between Banach spaces   (Site not responding. Last check: 2007-10-13) A well-known open question is whether every countable collection of Lipschitz functions on a Banach space X with separable dual has a common point of Frechet differentiability. Our main result states that a Lipschitz map between separable Banach spaces is Fréchet differentiable Γ-almost everywhere provided that it is regularly Gateaux differentiable Γ-almost everywhere and the Gateaux derivatives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of X to spaces with the Radon-Nikodym property are Gateaux differentiable Γ-almost everywhere. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0402160   (326 words)

On Fréchet differentiability of Lipschitz maps between Banach spaces, Joram Lindenstrauss, David Preiss A well-known open question is whether every countable collection of Lipschitz functions on a Banach space $X$ with separable dual has a common point of Fréchet differentiability. Our main result states that a Lipschitz map between separable Banach spaces is Fréchet differentiable $\Gamma$-almost everywhere provided that it is regularly Gâteaux differentiable $\Gamma$-almost everywhere and the Gâteaux derivatives stay within a norm separable space of operators. It is easy to see that Lipschitz maps of $X$ to spaces with the Radon-Nikodým property are Gâteaux differentiable $\Gamma$-almost everywhere. projecteuclid.org /getRecord?id=euclid.annm/1044909498   (275 words)

Quasiconformal Lipschitz Maps, Sullivan's Convex Hull Theorem And Brennan's Conjecture (ResearchIndex)   (Site not responding. Last check: 2007-10-13) Quasiconformal Lipschitz Maps, Sullivan's Convex Hull Theorem And Brennan's Conjecture (ResearchIndex) Quasiconformal Lipschitz Maps, Sullivan's Convex Hull Theorem And Brennan's Conjecture 3 The integrability of the derivative in conformal mapping (context) - Brennan - 1978 citeseer.ist.psu.edu /449410.html   (665 words)

UCL > The Department of Mathematics > Research Lipschitz maps, problems concerning the complexity of sets and functions.e.g. Is every Borel mapping of a metric space X to a metric space Y of bounded class? Are the ball and the sphere of the Hilbert space Lipschitz isomorphic? www.ucl.ac.uk /maths/staff/MC.html   (242 words)

Papers Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps Quasiconformal homeomorphisms and dynamics III: The Teichmüller space of a rational map Families of rational maps and iterative root-finding algorithms www.math.harvard.edu /~ctm/papers/index.html   (257 words)

[No title] The extended map is differentiable almost everywhere and at almost all points of $\phi(C_1\cap C_2)$ the differential is independent of the extension. Note however, that if we chose Lipschitz extentions, $\tilde{f}_{j,\alpha}$, and denote by $\tilde{\omega}$, the corresponding extension of $\omega$, then by a standard regularization argument, it follows that at almost all points of $\phi (C)$, the distributional exterior derivative of $\tilde{\omega}$ is independent of the particular extension, and is given by the expression in \eqr{e:i'.7}. Let $f_i$ denote the Lipschitz function determined by the following conditions: (a) The restriction of $f_i$ to each closed interval of the set, $J_i$, is the linear function which vanishes at the left hand end point and has derivative $\equiv 1$. www.intlpress.com /JDG/archive/vol.54/issue1/3P/aa3chco1.tex   (6527 words)

Martin Väth - List of Publications Dirr, G. and Väth, M., Continuity of near-duality maps and characterizations of ideal spaces of measurable functions, Recent Trends in Nonlinear Analysis (Appell, J., ed.), Festschrift Dedicated to Alfonso Vignoli on the Occassion of His Sixtieth Birthday, Birkhäuser, 2000, 139-148. Giorgieri, E. and Väth, M., A characterization of 0-epi maps with a degree, J. Funct. Väth, M., Coepi maps and generalizations of the Hopf extension theorem, Topology Appl. www.mathematik.uni-wuerzburg.de /~vaeth/bibliogr.html   (1069 words)

HJM, Vol. 27, No. 2, 2001 A suitable notion of jacobian is given for differential maps, called G-linear maps, finding relations with the classical definition of jacobian. We derive a criterion for the minimality and for the harmonicity of such vector fields by means of the infinitesimal models which correspond to (locally) homogeneous spaces and which are determined by using homogeneous structures. This leads to the construction of a lot of new examples of unit vector fields which are minimal or harmonic or which determine a harmonic map from (M,g) into its unit tangent sphere bundle equipped with the Sasaki metric. www.math.uh.edu /~hjm/Vol27-2.html   (1518 words)

Fractals The overall structure of the course is this: we begin by studying the construction of fractals as self-similar sets; then we study dimension theory; finally, the two strands are brought together in Hutchinson's Theorem which allows one to compute the dimension from the self-similarity data for a wide class of fractals. Indeed, whilst the primary aim of this course is the study of dimension and self-similarity, an important secondary aim is to consolidate your knowledge of the basic ideas of abstract analysis. Properties of Hausdorff dimension with regard to Lipschitz mappings, subsets and countable unions. www.peter-dixon.staff.shef.ac.uk /teaching/PMA443/PMA443.HTM   (1169 words)

Seminars: Rutgers Center for Systems and Control - SYCON After observing a random sample of inputs and the corresponding outputs, the aim is to learn a mapping from inputs to outputs given by an unknown control system. The first lecture will deal with the issue of how to develop a theory of "generalized differentials" at a point for maps that are not differentiable in the ordinary sense. The first two lectures deal with the issue of how to develop a theory of ``generalized differentials'' at a point for maps that are not differentiable in the ordinary sense. www.math.rutgers.edu /~sontag/sycon/98fall-seminars.html   (1368 words)

Lipschitz maps and nets in Euclidean space - McMullen (ResearchIndex) Lipschitz maps and nets in Euclidean space (1999) 2, and the (homogeneous) Holder analogs of the mapping problems in Theorems 1.1 and... 0.3: Quasiconformal Lipschitz Maps, Sullivan's Convex Hull Theorem And.. citeseer.ist.psu.edu /342633.html   (419 words)

Real analysis - Wikipedia, the free encyclopedia This is done in point set topology and using metric spaces. Concepts such as compactness, completeness, connectedness, uniform continuity, separability, Lipschitz maps, contractive maps are defined and investigated. One can take limits of functions and attempt to change the orders of integrals, derivatives and limits. en.wikipedia.org /wiki/Real_analysis   (459 words)

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