Volume Doubling Measures And Heat Kernel Estimates On Self Similar Sets

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Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets

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Author : Jun Kigami
Publisher : American Mathematical Soc.
ISBN 13 : 0821842927
Total Pages : 110 pages
Book Rating : 4.8/5 (218 download)

Book Synopsis Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets by : Jun Kigami

Download or read book Volume Doubling Measures and Heat Kernel Estimates on Self-Similar Sets written by Jun Kigami and published by American Mathematical Soc.. This book was released on 2009-04-10 with total page 110 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper studies the following three problems. 1. When does a measure on a self-similar set have the volume doubling property with respect to a given distance? 2. Is there any distance on a self-similar set under which the contraction mappings have the prescribed values of contractions ratios? 3. When does a heat kernel on a self-similar set associated with a self-similar Dirichlet form satisfy the Li-Yau type sub-Gaussian diagonal estimate? These three problems turn out to be closely related. The author introduces a new class of self-similar set, called rationally ramified self-similar sets containing both the Sierpinski gasket and the (higher dimensional) Sierpinski carpet and gives complete solutions of the above three problems for this class. In particular, the volume doubling property is shown to be equivalent to the upper Li-Yau type sub-Gaussian diagonal estimate of a heat kernel.

Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates

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Author : Jun Kigami
Publisher : American Mathematical Soc.
ISBN 13 : 082185299X
Total Pages : 145 pages
Book Rating : 4.8/5 (218 download)

Book Synopsis Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates by : Jun Kigami

Download or read book Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates written by Jun Kigami and published by American Mathematical Soc.. This book was released on 2012-02-22 with total page 145 pages. Available in PDF, EPUB and Kindle. Book excerpt: Assume that there is some analytic structure, a differential equation or a stochastic process for example, on a metric space. To describe asymptotic behaviors of analytic objects, the original metric of the space may not be the best one. Every now and then one can construct a better metric which is somehow ``intrinsic'' with respect to the analytic structure and under which asymptotic behaviors of the analytic objects have nice expressions. The problem is when and how one can find such a metric. In this paper, the author considers the above problem in the case of stochastic processes associated with Dirichlet forms derived from resistance forms. The author's main concerns are the following two problems: (I) When and how to find a metric which is suitable for describing asymptotic behaviors of the heat kernels associated with such processes. (II) What kind of requirement for jumps of a process is necessary to ensure good asymptotic behaviors of the heat kernels associated with such processes.

Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance

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Author : Jun Kigami
Publisher : American Mathematical Soc.
ISBN 13 : 1470436205
Total Pages : 118 pages
Book Rating : 4.4/5 (74 download)

Book Synopsis Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance by : Jun Kigami

Download or read book Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance written by Jun Kigami and published by American Mathematical Soc.. This book was released on 2019-06-10 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including n-dimensional cube [0,1]n are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on [0,1]n, density of the medium is homogeneous and represented by the Lebesgue measure. The author's study includes densities which are singular to the homogeneous one. He establishes a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on [0,1]2 and self-similar measures. The author shows the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, he obtains diagonal lower and upper estimates of the heat kernel as time tends to 0. In particular, to express the principal part of the lower diagonal heat kernel estimate, he introduces “protodistance” associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

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Author : David Carfi
Publisher : American Mathematical Soc.
ISBN 13 : 0821891480
Total Pages : 384 pages
Book Rating : 4.8/5 (218 download)

Book Synopsis Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II by : David Carfi

Download or read book Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II written by David Carfi and published by American Mathematical Soc.. This book was released on 2013-10-24 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings from three conferences: the PISRS 2011 International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics, held November 8-12, 2011 in Messina, Italy; the AMS Special Session on Fractal Geometry in Pure and Applied Mathematics, in memory of Benoît Mandelbrot, held January 4-7, 2012, in Boston, MA; and the AMS Special Session on Geometry and Analysis on Fractal Spaces, held March 3-4, 2012, in Honolulu, HI. Articles in this volume cover fractal geometry and various aspects of dynamical systems in applied mathematics and the applications to other sciences. Also included are articles discussing a variety of connections between these subjects and various areas of physics, engineering, computer science, technology, economics and finance, as well as of mathematics (including probability theory in relation with statistical physics and heat kernel estimates, geometric measure theory, partial differential equations in relation with condensed matter physics, global analysis on non-smooth spaces, the theory of billiards, harmonic analysis and spectral geometry). The companion volume (Contemporary Mathematics, Volume 600) focuses on the more mathematical aspects of fractal geometry and dynamical systems.

Analysis, Probability And Mathematical Physics On Fractals

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Author : Patricia Alonso Ruiz
Publisher : World Scientific
ISBN 13 : 9811215545
Total Pages : 594 pages
Book Rating : 4.8/5 (112 download)

Book Synopsis Analysis, Probability And Mathematical Physics On Fractals by : Patricia Alonso Ruiz

Download or read book Analysis, Probability And Mathematical Physics On Fractals written by Patricia Alonso Ruiz and published by World Scientific. This book was released on 2020-02-26 with total page 594 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Using classical and abstract analytic tools developed by Cantor, Hausdorff, and Sierpinski, they have sought to address fundamental questions: How can we measure the size of a fractal set? How do waves and heat travel on irregular structures? How are analysis, geometry and stochastic processes related in the absence of Euclidean smooth structure? What new physical phenomena arise in the fractal-like settings that are ubiquitous in nature?This book introduces background and recent progress on these problems, from both established leaders in the field and early career researchers. The book gives a broad introduction to several foundational techniques in fractal mathematics, while also introducing some specific new and significant results of interest to experts, such as that waves have infinite propagation speed on fractals. It contains sufficient introductory material that it can be read by new researchers or researchers from other areas who want to learn about fractal methods and results.

Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs

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Author : Alexander Grigor'yan
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110700859
Total Pages : 337 pages
Book Rating : 4.1/5 (17 download)

Book Synopsis Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs by : Alexander Grigor'yan

Download or read book Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs written by Alexander Grigor'yan and published by Walter de Gruyter GmbH & Co KG. This book was released on 2021-01-18 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book covers the latest research in the areas of mathematics that deal the properties of partial differential equations and stochastic processes on spaces in connection with the geometry of the underlying space. Written by experts in the field, this book is a valuable tool for the advanced mathematician.

Geometry and Analysis of Metric Spaces via Weighted Partitions

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Author : Jun Kigami
Publisher : Springer Nature
ISBN 13 : 3030541541
Total Pages : 164 pages
Book Rating : 4.0/5 (35 download)

Book Synopsis Geometry and Analysis of Metric Spaces via Weighted Partitions by : Jun Kigami

Download or read book Geometry and Analysis of Metric Spaces via Weighted Partitions written by Jun Kigami and published by Springer Nature. This book was released on 2020-11-16 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space. Metrics and measures on the space are then studied from an integrated point of view as weights of the partition. In the course of the text: It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic. Various relations between metrics and measures such as bilipschitz equivalence, quasisymmetry, Ahlfors regularity, and the volume doubling property are translated to relations between weights. In particular, it is shown that the volume doubling property between a metric and a measure corresponds to a quasisymmetry between two metrics in the language of weights. The Ahlfors regular conformal dimension of a compact metric space is characterized as the critical index of p-energies associated with the partition and the weight function corresponding to the metric. These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.

Fractal Zeta Functions and Fractal Drums

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Author : Michel L. Lapidus
Publisher : Springer
ISBN 13 : 3319447068
Total Pages : 685 pages
Book Rating : 4.3/5 (194 download)

Book Synopsis Fractal Zeta Functions and Fractal Drums by : Michel L. Lapidus

Download or read book Fractal Zeta Functions and Fractal Drums written by Michel L. Lapidus and published by Springer. This book was released on 2017-06-07 with total page 685 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph gives a state-of-the-art and accessible treatment of a new general higher-dimensional theory of complex dimensions, valid for arbitrary bounded subsets of Euclidean spaces, as well as for their natural generalization, relative fractal drums. It provides a significant extension of the existing theory of zeta functions for fractal strings to fractal sets and arbitrary bounded sets in Euclidean spaces of any dimension. Two new classes of fractal zeta functions are introduced, namely, the distance and tube zeta functions of bounded sets, and their key properties are investigated. The theory is developed step-by-step at a slow pace, and every step is well motivated by numerous examples, historical remarks and comments, relating the objects under investigation to other concepts. Special emphasis is placed on the study of complex dimensions of bounded sets and their connections with the notions of Minkowski content and Minkowski measurability, as well as on fractal tube formulas. It is shown for the first time that essential singularities of fractal zeta functions can naturally emerge for various classes of fractal sets and have a significant geometric effect. The theory developed in this book leads naturally to a new definition of fractality, expressed in terms of the existence of underlying geometric oscillations or, equivalently, in terms of the existence of nonreal complex dimensions. The connections to previous extensive work of the first author and his collaborators on geometric zeta functions of fractal strings are clearly explained. Many concepts are discussed for the first time, making the book a rich source of new thoughts and ideas to be developed further. The book contains a large number of open problems and describes many possible directions for further research. The beginning chapters may be used as a part of a course on fractal geometry. The primary readership is aimed at graduate students and researchers working in Fractal Geometry and other related fields, such as Complex Analysis, Dynamical Systems, Geometric Measure Theory, Harmonic Analysis, Mathematical Physics, Analytic Number Theory and the Spectral Theory of Elliptic Differential Operators. The book should be accessible to nonexperts and newcomers to the field.

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics: Fractals in pure mathematics

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Author : David Carfi
Publisher : American Mathematical Soc.
ISBN 13 : 0821891472
Total Pages : 410 pages
Book Rating : 4.8/5 (218 download)

Book Synopsis Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics: Fractals in pure mathematics by : David Carfi

Download or read book Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics: Fractals in pure mathematics written by David Carfi and published by American Mathematical Soc.. This book was released on 2013-10-22 with total page 410 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings from three conferences: the PISRS 2011 International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics, held November 8-12, 2011 in Messina, Italy; the AMS Special Session on Fractal Geometry in Pure and Applied Mathematics, in memory of Benoit Mandelbrot, held January 4-7, 2012, in Boston, MA; and the AMS Special Session on Geometry and Analysis on Fractal Spaces, held March 3-4, 2012, in Honolulu, HI. Articles in this volume cover fractal geometry (and some aspects of dynamical systems) in pure mathematics. Also included are articles discussing a variety of connections of fractal geometry with other fields of mathematics, including probability theory, number theory, geometric measure theory, partial differential equations, global analysis on non-smooth spaces, harmonic analysis and spectral geometry. The companion volume (Contemporary Mathematics, Volume 601) focuses on applications of fractal geometry and dynamical systems to other sciences, including physics, engineering, computer science, economics, and finance.

Stability of Heat Kernel Estimates for Symmetric Non-Local Dirichlet Forms

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Author : Zhen-Qing Chen
Publisher : American Mathematical Society
ISBN 13 : 1470448637
Total Pages : 89 pages
Book Rating : 4.4/5 (74 download)

Book Synopsis Stability of Heat Kernel Estimates for Symmetric Non-Local Dirichlet Forms by : Zhen-Qing Chen

Download or read book Stability of Heat Kernel Estimates for Symmetric Non-Local Dirichlet Forms written by Zhen-Qing Chen and published by American Mathematical Society. This book was released on 2021-09-24 with total page 89 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Operator Theory And Analysis Of Infinite Networks

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Author : Palle Jorgensen
Publisher : World Scientific
ISBN 13 : 9811265534
Total Pages : 449 pages
Book Rating : 4.8/5 (112 download)

Book Synopsis Operator Theory And Analysis Of Infinite Networks by : Palle Jorgensen

Download or read book Operator Theory And Analysis Of Infinite Networks written by Palle Jorgensen and published by World Scientific. This book was released on 2023-03-21 with total page 449 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume considers resistance networks: large graphs which are connected, undirected, and weighted. Such networks provide a discrete model for physical processes in inhomogeneous media, including heat flow through perforated or porous media. These graphs also arise in data science, e.g., considering geometrizations of datasets, statistical inference, or the propagation of memes through social networks. Indeed, network analysis plays a crucial role in many other areas of data science and engineering. In these models, the weights on the edges may be understood as conductances, or as a measure of similarity. Resistance networks also arise in probability, as they correspond to a broad class of Markov chains.The present volume takes the nonstandard approach of analyzing resistance networks from the point of view of Hilbert space theory, where the inner product is defined in terms of Dirichlet energy. The resulting viewpoint emphasizes orthogonality over convexity and provides new insights into the connections between harmonic functions, operators, and boundary theory. Novel applications to mathematical physics are given, especially in regard to the question of self-adjointness of unbounded operators.New topics are covered in a host of areas accessible to multiple audiences, at both beginning and more advanced levels. This is accomplished by directly linking diverse applied questions to such key areas of mathematics as functional analysis, operator theory, harmonic analysis, optimization, approximation theory, and probability theory.

Non-Divergence Equations Structured on Hormander Vector Fields: Heat Kernels and Harnack Inequalities

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Author : Marco Bramanti
Publisher : American Mathematical Soc.
ISBN 13 : 0821849034
Total Pages : 136 pages
Book Rating : 4.8/5 (218 download)

Book Synopsis Non-Divergence Equations Structured on Hormander Vector Fields: Heat Kernels and Harnack Inequalities by : Marco Bramanti

Download or read book Non-Divergence Equations Structured on Hormander Vector Fields: Heat Kernels and Harnack Inequalities written by Marco Bramanti and published by American Mathematical Soc.. This book was released on 2010 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: "March 2010, Volume 204, number 961 (end of volume)."

Geometry and Analysis of Fractals

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Author : De-Jun Feng
Publisher : Springer
ISBN 13 : 3662439204
Total Pages : 360 pages
Book Rating : 4.6/5 (624 download)

Book Synopsis Geometry and Analysis of Fractals by : De-Jun Feng

Download or read book Geometry and Analysis of Fractals written by De-Jun Feng and published by Springer. This book was released on 2014-08-01 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume collects thirteen expository or survey articles on topics including Fractal Geometry, Analysis of Fractals, Multifractal Analysis, Ergodic Theory and Dynamical Systems, Probability and Stochastic Analysis, written by the leading experts in their respective fields. The articles are based on papers presented at the International Conference on Advances on Fractals and Related Topics, held on December 10-14, 2012 at the Chinese University of Hong Kong. The volume offers insights into a number of exciting, cutting-edge developments in the area of fractals, which has close ties to and applications in other areas such as analysis, geometry, number theory, probability and mathematical physics.

Quantum Graphs and Their Applications

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Author : Gregory Berkolaiko
Publisher : American Mathematical Soc.
ISBN 13 : 0821837656
Total Pages : 322 pages
Book Rating : 4.8/5 (218 download)

Book Synopsis Quantum Graphs and Their Applications by : Gregory Berkolaiko

Download or read book Quantum Graphs and Their Applications written by Gregory Berkolaiko and published by American Mathematical Soc.. This book was released on 2006 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is a collection of articles dedicated to quantum graphs, a newly emerging interdisciplinary field related to various areas of mathematics and physics. The reader can find a broad overview of the theory of quantum graphs. The articles present methods coming from different areas of mathematics: number theory, combinatorics, mathematical physics, differential equations, spectral theory, global analysis, and theory of fractals. They also address various important applications, such as Anderson localization, electrical networks, quantum chaos, mesoscopic physics, superconductivity, optics, and biological modeling.

Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces

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Author : Volkmar Liebscher
Publisher : American Mathematical Soc.
ISBN 13 : 0821843184
Total Pages : 124 pages
Book Rating : 4.8/5 (218 download)

Book Synopsis Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces by : Volkmar Liebscher

Download or read book Random Sets and Invariants for (Type II) Continuous Tensor Product Systems of Hilbert Spaces written by Volkmar Liebscher and published by American Mathematical Soc.. This book was released on 2009-04-10 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: In a series of papers Tsirelson constructed from measure types of random sets or (generalised) random processes a new range of examples for continuous tensor product systems of Hilbert spaces introduced by Arveson for classifying $E_0$-semigroups upto cocycle conjugacy. This paper starts from establishing the converse. So the author connects each continuous tensor product system of Hilbert spaces with measure types of distributions of random (closed) sets in $[0,1]$ or $\mathbb R_+$. These measure types are stationary and factorise over disjoint intervals. In a special case of this construction, the corresponding measure type is an invariant of the product system. This shows, completing in a more systematic way the Tsirelson examples, that the classification scheme for product systems into types $\mathrm{I}_n$, $\mathrm{II}_n$ and $\mathrm{III}$ is not complete. Moreover, based on a detailed study of this kind of measure types, the author constructs for each stationary factorising measure type a continuous tensor product system of Hilbert spaces such that this measure type arises as the before mentioned invariant.

Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three

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Author : Robert C. Dalang
Publisher : American Mathematical Soc.
ISBN 13 : 0821842889
Total Pages : 83 pages
Book Rating : 4.8/5 (218 download)

Book Synopsis Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three by : Robert C. Dalang

Download or read book Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three written by Robert C. Dalang and published by American Mathematical Soc.. This book was released on 2009-04-10 with total page 83 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors study the sample path regularity of the solution of a stochastic wave equation in spatial dimension $d=3$. The driving noise is white in time and with a spatially homogeneous covariance defined as a product of a Riesz kernel and a smooth function. The authors prove that at any fixed time, a.s., the sample paths in the spatial variable belong to certain fractional Sobolev spaces. In addition, for any fixed $x\in\mathbb{R}^3$, the sample paths in time are Holder continuous functions. Further, the authors obtain joint Holder continuity in the time and space variables. Their results rely on a detailed analysis of properties of the stochastic integral used in the rigourous formulation of the s.p.d.e., as introduced by Dalang and Mueller (2003). Sharp results on one- and two-dimensional space and time increments of generalized Riesz potentials are a crucial ingredient in the analysis of the problem. For spatial covariances given by Riesz kernels, the authors show that the Holder exponents that they obtain are optimal.

Heat Eisenstein Series on $\mathrm {SL}_n(C)$

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Author : Jay Jorgenson
Publisher : American Mathematical Soc.
ISBN 13 : 0821840444
Total Pages : 146 pages
Book Rating : 4.8/5 (218 download)

Book Synopsis Heat Eisenstein Series on $\mathrm {SL}_n(C)$ by : Jay Jorgenson

Download or read book Heat Eisenstein Series on $\mathrm {SL}_n(C)$ written by Jay Jorgenson and published by American Mathematical Soc.. This book was released on 2009 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this Memoir is to define and study multi-variable Eisenstein series attached to heat kernels. Fundamental properties of heat Eisenstein series are proved, and conjectural behavior, including their role in spectral expansions, are stated.