Uniform Rectifiability And Quasiminimizing Sets Of Arbitrary Codimension
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Uniform Rectifiability And Quasiminimizing Sets Of Arbitrary Codimension
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Author : Guy David
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
Uniform Rectifiability And Quasiminimizing Sets Of Arbitrary Codimension written by Guy David and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.
This book is intended for graduate students and research mathematicians interested in calculus of variations and optimal control; optimization.
Uniform Rectifiability And Quasiminimizing Sets Of Arbitrary Codimension
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Author :
language : en
Publisher: American Mathematical Soc.
Release Date : 2000-03-03
Uniform Rectifiability And Quasiminimizing Sets Of Arbitrary Codimension written by and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-03-03 with Mathematics categories.
Roughly speaking, a $d$-dimensional subset of $\mathbf R^n$ is minimizing if arbitrary deformations of it (in a suitable class) cannot decrease its $d$-dimensional volume. For quasiminimizing sets, one allows the mass to decrease, but only in a controlled manner. To make this precise we follow Almgren's notion of ``restricted sets'' [2]. Graphs of Lipschitz mappings $f\:\mathbf R^d \to \mathbf R^{n-d}$ are always quasiminimizing, and Almgren showed that quasiminimizing sets are rectifiable. Here we establish uniform rectifiability properties of quasiminimizing sets, which provide a more quantitative sense in which these sets behave like Lipschitz graphs. (Almgren also established stronger smoothness properties under tighter quasiminimality conditions.) Quasiminimizing sets can arise as minima of functionals with highly irregular ``coefficients''. For such functionals, one cannot hope in general to have much more in the way of smoothness or structure than uniform rectifiability, for reasons of bilipschitz invariance. (See also [9].) One motivation for considering minimizers of functionals with irregular coefficients comes from the following type of question. Suppose that one is given a compact set $K$ with upper bounds on its $d$-dimensional Hausdorff measure, and lower bounds on its $d$-dimensional topology. What can one say about the structure of $K$? To what extent does it behave like a nice $d$-dimensional surface? A basic strategy for dealing with this issue is to first replace $K$ by a set which is minimizing for a measurement of volume that imposes a large penalty on points which lie outside of $K$. This leads to a kind of regularization of $K$, in which cusps and very scattered parts of $K$ are removed, but without adding more than a small amount from the complement of $K$. The results for quasiminimizing sets then lead to uniform rectifiability properties of this regularization of $K$. To actually produce minimizers of general functionals it is sometimes convenient to work with (finite) discrete models. A nice feature of uniform rectifiability is that it provides a way to have bounds that cooperate robustly with discrete approximations, and which survive in the limit as the discretization becomes finer and finer.
Rectifiability
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Author : Pertti Mattila
language : en
Publisher: Cambridge University Press
Release Date : 2023-01-12
Rectifiability written by Pertti Mattila and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-01-12 with Mathematics categories.
A broad survey of the theory of rectifiability and its deep connections to numerous different areas of mathematics.
Analysis And Geometry Of Metric Measure Spaces
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Author : Galia Devora Dafni
language : en
Publisher: American Mathematical Soc.
Release Date : 2013
Analysis And Geometry Of Metric Measure Spaces written by Galia Devora Dafni and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Mathematics categories.
Contains lecture notes from most of the courses presented at the 50th anniversary edition of the Seminaire de Mathematiques Superieure in Montreal. This 2011 summer school was devoted to the analysis and geometry of metric measure spaces, and featured much interplay between this subject and the emergent topic of optimal transportation.
Singular Sets Of Minimizers For The Mumford Shah Functional
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Author : Guy David
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-03-10
Singular Sets Of Minimizers For The Mumford Shah Functional written by Guy David and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-03-10 with Mathematics categories.
The Mumford-Shah functional was introduced in the 1980s as a tool for automatic image segmentation, but its study gave rise to many interesting questions of analysis and geometric measure theory. The main object under scrutiny is a free boundary K where the minimizer may have jumps. The book presents an extensive description of the known regularity properties of the singular sets K, and the techniques to get them. It is largely self-contained, and should be accessible to graduate students in analysis. The core of the book is composed of regularity results that were proved in the last ten years and which are presented in a more detailed and unified way.
Some Novel Types Of Fractal Geometry
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Author : Stephen Semmes
language : en
Publisher: Oxford University Press
Release Date : 2001
Some Novel Types Of Fractal Geometry written by Stephen Semmes and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.
The present book deals with fractal geometries which have features similar to ones of ordinary Euclidean spaces, while at the same time being quite different from Euclidean spaces in other ways. A basic type of feature being considered is the presence of Sobolev or Poincaré inequalities, concerning the relationship between the average behaviour of a function and the average behaviour of its small-scale oscillations. Remarkable results in the last few years of Bourdon-Pajot and Laakso have shown that there is much more in the way of geometries like this than has been realized. Examples related to nilpotent Lie groups and Carnot metrics were known previously. On the other hand, 'typical' fractals that might be seen in pictures do not have these same kinds of features. 'Some Novel Types of Fractal Geometry' will be of interest to graduate students and researchers in mathematics, working in various aspects of geometry and analysis.
Non Uniform Lattices On Uniform Trees
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Author : Lisa Carbone
language : en
Publisher: American Mathematical Soc.
Release Date : 2001
Non Uniform Lattices On Uniform Trees written by Lisa Carbone and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.
This title provides a comprehensive examination of non-uniform lattices on uniform trees. Topics include graphs of groups, tree actions and edge-indexed graphs; $Aut(x)$ and its discrete subgroups; existence of tree lattices; non-uniform coverings of indexed graphs with an arithmetic bridge; non-uniform coverings of indexed graphs with a separating edge; non-uniform coverings of indexed graphs with a ramified loop; eliminating multiple edges; existence of arithmetic bridges. This book is intended for graduate students and research mathematicians interested in group theory and generalizations.
Advances In Analysis
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Author : Charles Fefferman
language : en
Publisher: Princeton University Press
Release Date : 2014-01-05
Advances In Analysis written by Charles Fefferman and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-05 with Mathematics categories.
Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His fundamental contributions include the Kunze-Stein phenomenon, the construction of new representations, the Stein interpolation theorem, the idea of a restriction theorem for the Fourier transform, and the theory of Hp Spaces in several variables. Through his great discoveries, through books that have set the highest standard for mathematical exposition, and through his influence on his many collaborators and students, Stein has changed mathematics. Drawing inspiration from Stein’s contributions to harmonic analysis and related topics, this volume gathers papers from internationally renowned mathematicians, many of whom have been Stein’s students. The book also includes expository papers on Stein’s work and its influence. The contributors are Jean Bourgain, Luis Caffarelli, Michael Christ, Guy David, Charles Fefferman, Alexandru D. Ionescu, David Jerison, Carlos Kenig, Sergiu Klainerman, Loredana Lanzani, Sanghyuk Lee, Lionel Levine, Akos Magyar, Detlef Müller, Camil Muscalu, Alexander Nagel, D. H. Phong, Malabika Pramanik, Andrew S. Raich, Fulvio Ricci, Keith M. Rogers, Andreas Seeger, Scott Sheffield, Luis Silvestre, Christopher D. Sogge, Jacob Sturm, Terence Tao, Christoph Thiele, Stephen Wainger, and Steven Zelditch.
Special Groups
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Author : M. A. Dickmann
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
Special Groups written by M. A. Dickmann and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.
This monograph presents a systematic study of Special Groups, a first-order universal-existential axiomatization of the theory of quadratic forms, which comprises the usual theory over fields of characteristic different from 2, and is dual to the theory of abstract order spaces. The heart of our theory begins in Chapter 4 with the result that Boolean algebras have a natural structure of reduced special group. More deeply, every such group is canonically and functorially embedded in a certain Boolean algebra, its Boolean hull. This hull contains a wealth of information about the structure of the given special group, and much of the later work consists in unveiling it. Thus, in Chapter 7 we introduce two series of invariants "living" in the Boolean hull, which characterize the isometry of forms in any reduced special group. While the multiplicative series--expressed in terms of meet and symmetric difference--constitutes a Boolean version of the Stiefel-Whitney invariants, the additive series--expressed in terms of meet and join--, which we call Horn-Tarski invariants, does not have a known analog in the field case; however, the latter have a considerably more regular behaviour. We give explicit formulas connecting both series, and compute explicitly the invariants for Pfister forms and their linear combinations. In Chapter 9 we combine Boolean-theoretic methods with techniques from Galois cohomology and a result of Voevodsky to obtain an affirmative solution to a long standing conjecture of Marshall concerning quadratic forms over formally real Pythagorean fields. Boolean methods are put to work in Chapter 10 to obtain information about categories of special groups, reduced or not. And again in Chapter 11 to initiate the model-theoretic study of the first-order theory of reduced special groups, where, amongst other things we determine its model-companion. The first-order approach is also present in the study of some outstanding classes of morphisms carried out in Chapter 5, e.g., the pure embeddings of special groups. Chapter 6 is devoted to the study of special groups of continuous functions.
Invariant Measures For Unitary Groups Associated To Kac Moody Lie Algebras
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Author : Doug Pickrell
language : en
Publisher: American Mathematical Soc.
Release Date : 2000
Invariant Measures For Unitary Groups Associated To Kac Moody Lie Algebras written by Doug Pickrell and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Mathematics categories.
The main purpose of this paper is to prove the existence, and in some cases the uniqueness, of unitarily invariant measures on formal completions of groups associated to affine Kac-Moody algebras, and associated homogeneous spaces. The basic invariant measure is a natural generalization of Haar measure for a simply connected compact Lie group, and its projection to flag spaces is a generalization of the normalized invariant volume element. The other "invariant measures" are actually measures having values in line bundles over these spaces; these bundle-valued measures heuristically arise from coupling the basic invariant measure to Hermitian structures on associated line bundles, but in this infinite dimensional setting they are generally singular with respect to the basic invariant measure.