Resistance Forms Quasisymmetric Maps And Heat Kernel Estimates

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Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates

Jun Kigami
Resistance Forms Quasisymmetric Maps and Heat Kernel Estimates Book PDF

Author : Jun Kigami
Publisher : American Mathematical Soc.
Page : 145 pages
File Size : 46,6 Mb
Release : 2012-02-22
Category : Mathematics
ISBN : 9780821852996

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Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates by Jun Kigami Pdf

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

Jun Kigami
Time Changes of the Brownian Motion Poincar Inequality Heat Kernel Estimate and Protodistance Book PDF

Author : Jun Kigami
Publisher : American Mathematical Soc.
Page : 118 pages
File Size : 55,5 Mb
Release : 2019-06-10
Category : Brownian motion processes
ISBN : 9781470436209

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

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.

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

Zhen-Qing Chen,Takashi Kumagai,Jian Wang
Stability of Heat Kernel Estimates for Symmetric Non Local Dirichlet Forms Book PDF

Author : Zhen-Qing Chen,Takashi Kumagai,Jian Wang
Publisher : American Mathematical Society
Page : 89 pages
File Size : 49,6 Mb
Release : 2021-09-24
Category : Mathematics
ISBN : 9781470448639

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Stability of Heat Kernel Estimates for Symmetric Non-Local Dirichlet Forms by Zhen-Qing Chen,Takashi Kumagai,Jian Wang Pdf

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

David Carfi,Michel Laurent Lapidus,Erin P. J. Pearse,Machiel Van Frankenhuysen
Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics Fractals in pure mathematics Book PDF

Author : David Carfi,Michel Laurent Lapidus,Erin P. J. Pearse,Machiel Van Frankenhuysen
Publisher : American Mathematical Soc.
Page : 410 pages
File Size : 48,9 Mb
Release : 2013-10-22
Category : Mathematics
ISBN : 9780821891476

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Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics: Fractals in pure mathematics by David Carfi,Michel Laurent Lapidus,Erin P. J. Pearse,Machiel Van Frankenhuysen Pdf

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.

Geometry and Analysis of Metric Spaces via Weighted Partitions

Jun Kigami
Geometry and Analysis of Metric Spaces via Weighted Partitions Book PDF

Author : Jun Kigami
Publisher : Springer Nature
Page : 164 pages
File Size : 45,8 Mb
Release : 2020-11-16
Category : Mathematics
ISBN : 9783030541545

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Geometry and Analysis of Metric Spaces via Weighted Partitions by Jun Kigami Pdf

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.

Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs

Alexander Grigor'yan,Yuhua Sun
Analysis and Partial Differential Equations on Manifolds Fractals and Graphs Book PDF

Author : Alexander Grigor'yan,Yuhua Sun
Publisher : Walter de Gruyter GmbH & Co KG
Page : 337 pages
File Size : 44,7 Mb
Release : 2021-01-18
Category : Mathematics
ISBN : 9783110700855

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Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs by Alexander Grigor'yan,Yuhua Sun Pdf

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.

Finite Order Automorphisms and Real Forms of Affine Kac-Moody Algebras in the Smooth and Algebraic Category

Ernst Heintze,Christian Gross
Finite Order Automorphisms and Real Forms of Affine Kac Moody Algebras in the Smooth and Algebraic Category Book PDF

Author : Ernst Heintze,Christian Gross
Publisher : American Mathematical Soc.
Page : 66 pages
File Size : 46,6 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821869185

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Finite Order Automorphisms and Real Forms of Affine Kac-Moody Algebras in the Smooth and Algebraic Category by Ernst Heintze,Christian Gross Pdf

Let $\mathfrak{g}$ be a real or complex (finite dimensional) simple Lie algebra and $\sigma\in\mathrm{Aut}\mathfrak{g}$. The authors study automorphisms of the twisted loop algebra $L(\mathfrak{g},\sigma)$ of smooth $\sigma$-periodic maps from $\mathbb{R}$ to $\mathfrak{g}$ as well as of the ``smooth'' affine Kac-Moody algebra $\hat L(\mathfrak{g},\sigma)$, which is a $2$-dimensional extension of $L(\mathfrak{g},\sigma)$. It turns out that these automorphisms which either preserve or reverse the orientation of loops, and are correspondingly called to be of first and second kind, can be described essentially by curves of automorphisms of $\mathfrak{g}$. If the order of the automorphisms is finite, then the corresponding curves in $\mathrm{Aut}\mathfrak{g}$ allow us to define certain invariants and these turn out to parametrize the conjugacy classes of the automorphisms. If their order is $2$ the authors carry this out in detail and deduce a complete classification of involutions and real forms (which correspond to conjugate linear involutions) of smooth affine Kac-Moody algebras.

The resulting classification can be seen as an extension of Cartan's classification of symmetric spaces, i.e. of involutions on $\mathfrak{g}$. If $\mathfrak{g}$ is compact, then conjugate linear extensions of involutions from $\hat L(\mathfrak{g},\sigma)$ to conjugate linear involutions on $\hat L(\mathfrak{g}_{\mathbb{C}},\sigma_{\mathbb{C}})$ yield a bijection between their conjugacy classes and this gives existence and uniqueness of Cartan decompositions of real forms of complex smooth affine Kac-Moody algebras.

The authors show that their methods work equally well also in the algebraic case where the loops are assumed to have finite Fourier expansions.

N-harmonic Mappings Between Annuli

Tadeusz Iwaniec,Jani Onninen
N harmonic Mappings Between Annuli Book PDF

Author : Tadeusz Iwaniec,Jani Onninen
Publisher : American Mathematical Soc.
Page : 105 pages
File Size : 48,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821853573

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N-harmonic Mappings Between Annuli by Tadeusz Iwaniec,Jani Onninen Pdf

The central theme of this paper is the variational analysis of homeomorphisms $h: {\mathbb X} \overset{\textnormal{\tiny{onto}}}{\longrightarrow} {\mathbb Y}$ between two given domains ${\mathbb X}, {\mathbb Y} \subset {\mathbb R}^n$. The authors look for the extremal mappings in the Sobolev space ${\mathscr W}^{1,n}({\mathbb X},{\mathbb Y})$ which minimize the energy integral ${\mathscr E}_h=\int_{{\mathbb X}} \,|\!|\, Dh(x) \,|\!|\,^n\, \textrm{d}x$. Because of the natural connections with quasiconformal mappings this $n$-harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal $n$-harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians, becomes truly a work of art.

The Kohn-Sham Equation for Deformed Crystals

Weinan E,Jianfeng Lu
The Kohn Sham Equation for Deformed Crystals Book PDF

Author : Weinan E,Jianfeng Lu
Publisher : American Mathematical Soc.
Page : 97 pages
File Size : 53,5 Mb
Release : 2013-01-25
Category : Mathematics
ISBN : 9780821875605

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The Kohn-Sham Equation for Deformed Crystals by Weinan E,Jianfeng Lu Pdf

The solution to the Kohn-Sham equation in the density functional theory of the quantum many-body problem is studied in the context of the electronic structure of smoothly deformed macroscopic crystals. An analog of the classical Cauchy-Born rule for crystal lattices is established for the electronic structure of the deformed crystal under the following physical conditions: (1) the band structure of the undeformed crystal has a gap, i.e. the crystal is an insulator, (2) the charge density waves are stable, and (3) the macroscopic dielectric tensor is positive definite. The effective equation governing the piezoelectric effect of a material is rigorously derived. Along the way, the authors also establish a number of fundamental properties of the Kohn-Sham map.

Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II

David Carfi,Michel L. Lapidus,Erin P. J. Pearse,Machiel van Frankenhuijsen
Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II Book PDF

Author : David Carfi,Michel L. Lapidus,Erin P. J. Pearse,Machiel van Frankenhuijsen
Publisher : American Mathematical Soc.
Page : 384 pages
File Size : 41,9 Mb
Release : 2013-10-24
Category : Mathematics
ISBN : 9780821891483

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Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II by David Carfi,Michel L. Lapidus,Erin P. J. Pearse,Machiel van Frankenhuijsen Pdf

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.

Dirichlet Forms and Related Topics

Zhen-Qing Chen,Masayoshi Takeda,Toshihiro Uemura
Dirichlet Forms and Related Topics Book PDF

Author : Zhen-Qing Chen,Masayoshi Takeda,Toshihiro Uemura
Publisher : Springer Nature
Page : 572 pages
File Size : 45,6 Mb
Release : 2022-09-04
Category : Mathematics
ISBN : 9789811946721

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Dirichlet Forms and Related Topics by Zhen-Qing Chen,Masayoshi Takeda,Toshihiro Uemura Pdf

This conference proceeding contains 27 peer-reviewed invited papers from leading experts as well as young researchers all over the world in the related fields that Professor Fukushima has made important contributions to. These 27 papers cover a wide range of topics in probability theory, ranging from Dirichlet form theory, Markov processes, heat kernel estimates, entropy on Wiener spaces, analysis on fractal spaces, random spanning tree and Poissonian loop ensemble, random Riemannian geometry, SLE, space-time partial differential equations of higher order, infinite particle systems, Dyson model, functional inequalities, branching process, to machine learning and Hermitizable problems for complex matrices. Researchers and graduate students interested in these areas will find this book appealing.

Hopf Algebras and Congruence Subgroups

Yorck Sommerhäuser,Yongchang Zhu
Hopf Algebras and Congruence Subgroups Book PDF

Author : Yorck Sommerhäuser,Yongchang Zhu
Publisher : American Mathematical Soc.
Page : 134 pages
File Size : 49,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821869130

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Hopf Algebras and Congruence Subgroups by Yorck Sommerhäuser,Yongchang Zhu Pdf

The authors prove that the kernel of the action of the modular group on the center of a semisimple factorizable Hopf algebra is a congruence subgroup whenever this action is linear. If the action is only projective, they show that the projective kernel is a congruence subgroup. To do this, they introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.

Elliptic Integrable Systems

Idrisse Khemar
Elliptic Integrable Systems Book PDF

Author : Idrisse Khemar
Publisher : American Mathematical Soc.
Page : 217 pages
File Size : 40,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821869253

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Elliptic Integrable Systems by Idrisse Khemar Pdf

In this paper, the author studies all the elliptic integrable systems, in the sense of C, that is to say, the family of all the $m$-th elliptic integrable systems associated to a $k^\prime$-symmetric space $N=G/G_0$. The author describes the geometry behind this family of integrable systems for which we know how to construct (at least locally) all the solutions. The introduction gives an overview of all the main results, as well as some related subjects and works, and some additional motivations.

Infinite-dimensional Representations of 2-groups

John C. Baez
Infinite dimensional Representations of 2 groups Book PDF

Author : John C. Baez
Publisher : American Mathematical Soc.
Page : 120 pages
File Size : 53,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821872840

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Infinite-dimensional Representations of 2-groups by John C. Baez Pdf

A “$2$-group'' is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, $2$-groups have representations on “$2$-vector spaces'', which are categories analogous to vector spaces. Unfortunately, Lie $2$-groups typically have few representations on the finite-dimensional $2$-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional $2$-vector spaces called ``measurable categories'' (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie $2$-groups. Here they continue this work.

They begin with a detailed study of measurable categories. Then they give a geometrical description of the measurable representations, intertwiners and $2$-intertwiners for any skeletal measurable $2$-group. They study tensor products and direct sums for representations, and various concepts of subrepresentation. They describe direct sums of intertwiners, and sub-intertwiners--features not seen in ordinary group representation theory and study irreducible and indecomposable representations and intertwiners. They also study “irretractable'' representations--another feature not seen in ordinary group representation theory. Finally, they argue that measurable categories equipped with some extra structure deserve to be considered “separable $2$-Hilbert spaces'', and compare this idea to a tentative definition of $2$-Hilbert spaces as representation categories of commutative von Neumann algebras.

Extended Graphical Calculus for Categorified Quantum Sl(2)

Mikhail Khovanov
Extended Graphical Calculus for Categorified Quantum Sl 2 Book PDF

Author : Mikhail Khovanov
Publisher : American Mathematical Soc.
Page : 87 pages
File Size : 47,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821889770

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Extended Graphical Calculus for Categorified Quantum Sl(2) by Mikhail Khovanov Pdf

A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in a paper (arXiv:0803.3652) by Aaron D. Lauda. Here the authors enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. They obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements.

These formulas have integral coefficients and imply that one of the main results of Lauda's paper--identification of the Grothendieck ring of his 2-category with the idempotented quantum sl(2)--also holds when the 2-category is defined over the ring of integers rather than over a field. A new diagrammatic description of Schur functions is also given and it is shown that the the Jacobi-Trudy formulas for the decomposition of Schur functions into elementary or complete symmetric functions follows from the diagrammatic relations for categorified quantum sl(2).