The Hodge Laplacian

Author : Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor
Publisher : Walter de Gruyter GmbH & Co KG
Page : 528 pages
File Size : 41,7 Mb
Release : 2016-10-10
Category : Mathematics
ISBN : 9783110483390

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The Hodge-Laplacian by Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor Pdf

The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains. Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals. Contents: Preface Introduction and Statement of Main Results Geometric Concepts and Tools Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism Additional Results and Applications Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis Bibliography Index

Author : Detlef Muller,Marco M. Peloso,Fulvio Ricci
Publisher : American Mathematical Soc.
Page : 91 pages
File Size : 52,8 Mb
Release : 2014-12-20
Category : Mathematics
ISBN : 9781470409395

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Analysis of the Hodge Laplacian on the Heisenberg Group by Detlef Muller,Marco M. Peloso,Fulvio Ricci Pdf

The authors consider the Hodge Laplacian \Delta on the Heisenberg group H_n, endowed with a left-invariant and U(n)-invariant Riemannian metric. For 0\le k\le 2n+1, let \Delta_k denote the Hodge Laplacian restricted to k-forms. In this paper they address three main, related questions: (1) whether the L^2 and L^p-Hodge decompositions, 1

Author : Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor
Publisher : Walter de Gruyter GmbH & Co KG
Page : 528 pages
File Size : 46,9 Mb
Release : 2016-10-10
Category : Mathematics
ISBN : 9783110484380

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The Hodge-Laplacian by Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor Pdf

The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains. Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals. Contents: Preface Introduction and Statement of Main Results Geometric Concepts and Tools Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism Additional Results and Applications Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis Bibliography Index

Author : Dorina Mitrea,Marius Mitrea,Michael Taylor
Publisher : American Mathematical Soc.
Page : 137 pages
File Size : 44,8 Mb
Release : 2001
Category : Boundary value problems
ISBN : 9780821826591

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Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds by Dorina Mitrea,Marius Mitrea,Michael Taylor Pdf

The general aim of the present monograph is to study boundary-value problems for second-order elliptic operators in Lipschitz sub domains of Riemannian manifolds. In the first part (ss1-4), we develop a theory for Cauchy type operators on Lipschitz submanifolds of co dimension one (focused on boundedness properties and jump relations) and solve the $Lp$-Dirichlet problem, with $p$ close to $2$, for general second-order strongly elliptic systems. The solution is represented in the form of layer potentials and optimal non tangential maximal function estimates are established.This analysis is carried out under smoothness assumptions (for the coefficients of the operator, metric tensor and the underlying domain) which are in the nature of best possible. In the second part of the monograph, ss5-13, we further specialize this discussion to the case of Hodge Laplacian $\Delta: =-d\delta-\delta d$. This time, the goal is to identify all (pairs of) natural boundary conditions of Neumann type. Owing to the structural richness of the higher degree case we are considering, the theory developed here encompasses in a unitary fashion many basic PDE's of mathematical physics. Its scope extends to also cover Maxwell's equations, dealt with separately in s14. The main tools are those of PDE's and harmonic analysis, occasionally supplemented with some basic facts from algebraic topology and differential geometry.

Author : Steven Rosenberg
Publisher : Cambridge University Press
Page : 190 pages
File Size : 46,9 Mb
Release : 1997-01-09
Category : Mathematics
ISBN : 0521468310

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The Laplacian on a Riemannian Manifold by Steven Rosenberg Pdf

This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Author : Nelia Sofocli Charalambous
Publisher : Unknown
Page : 212 pages
File Size : 42,8 Mb
Release : 2004
Category : Electronic
ISBN : CORNELL:31924100426323

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On the LP Spectrum of the Hodge Laplacian and Logarithmic Sobolev Inequalities on Non-compact Manifolds by Nelia Sofocli Charalambous Pdf

Author : Eberhard Zeidler
Publisher : Springer Science & Business Media
Page : 1141 pages
File Size : 54,7 Mb
Release : 2011-08-17
Category : Mathematics
ISBN : 9783642224218

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Quantum Field Theory III: Gauge Theory by Eberhard Zeidler Pdf

In this third volume of his modern introduction to quantum field theory, Eberhard Zeidler examines the mathematical and physical aspects of gauge theory as a principle tool for describing the four fundamental forces which act in the universe: gravitative, electromagnetic, weak interaction and strong interaction. Volume III concentrates on the classical aspects of gauge theory, describing the four fundamental forces by the curvature of appropriate fiber bundles. This must be supplemented by the crucial, but elusive quantization procedure. The book is arranged in four sections, devoted to realizing the universal principle force equals curvature: Part I: The Euclidean Manifold as a Paradigm Part II: Ariadne's Thread in Gauge Theory Part III: Einstein's Theory of Special Relativity Part IV: Ariadne's Thread in Cohomology For students of mathematics the book is designed to demonstrate that detailed knowledge of the physical background helps to reveal interesting interrelationships among diverse mathematical topics. Physics students will be exposed to a fairly advanced mathematics, beyond the level covered in the typical physics curriculum. Quantum Field Theory builds a bridge between mathematicians and physicists, based on challenging questions about the fundamental forces in the universe (macrocosmos), and in the world of elementary particles (microcosmos).

Author : Edward Lee Bueler
Publisher : Unknown
Page : 152 pages
File Size : 45,9 Mb
Release : 1997
Category : Electronic
ISBN : CORNELL:31924079838987

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The Heat Kernel Weighted Hodge Laplacian on Noncompact Manifolds by Edward Lee Bueler Pdf

Author : Douglas N. Arnold
Publisher : SIAM
Page : 120 pages
File Size : 51,9 Mb
Release : 2018-12-12
Category : Mathematics
ISBN : 9781611975543

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Finite Element Exterior Calculus by Douglas N. Arnold Pdf

Computational methods to approximate the solution of differential equations play a crucial role in science, engineering, mathematics, and technology. The key processes that govern the physical world—wave propagation, thermodynamics, fluid flow, solid deformation, electricity and magnetism, quantum mechanics, general relativity, and many more—are described by differential equations. We depend on numerical methods for the ability to simulate, explore, predict, and control systems involving these processes. The finite element exterior calculus, or FEEC, is a powerful new theoretical approach to the design and understanding of numerical methods to solve partial differential equations (PDEs). The methods derived with FEEC preserve crucial geometric and topological structures underlying the equations and are among the most successful examples of structure-preserving methods in numerical PDEs. This volume aims to help numerical analysts master the fundamentals of FEEC, including the geometrical and functional analysis preliminaries, quickly and in one place. It is also accessible to mathematicians and students of mathematics from areas other than numerical analysis who are interested in understanding how techniques from geometry and topology play a role in numerical PDEs.

Author : De-Nian Yang
Publisher : Springer Nature
Page : 490 pages
File Size : 44,7 Mb
Release : 2024-05-18
Category : Electronic
ISBN : 9789819722532

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Advances in Knowledge Discovery and Data Mining by De-Nian Yang Pdf

Author : Frank W. Warner
Publisher : Springer Science & Business Media
Page : 283 pages
File Size : 50,6 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9781475717990

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Foundations of Differentiable Manifolds and Lie Groups by Frank W. Warner Pdf

Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.

Author : Jean Gallier,Jocelyn Quaintance
Publisher : Springer Nature
Page : 627 pages
File Size : 43,7 Mb
Release : 2020-08-18
Category : Mathematics
ISBN : 9783030460471

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Differential Geometry and Lie Groups by Jean Gallier,Jocelyn Quaintance Pdf

This textbook explores advanced topics in differential geometry, chosen for their particular relevance to modern geometry processing. Analytic and algebraic perspectives augment core topics, with the authors taking care to motivate each new concept. Whether working toward theoretical or applied questions, readers will appreciate this accessible exploration of the mathematical concepts behind many modern applications. Beginning with an in-depth study of tensors and differential forms, the authors go on to explore a selection of topics that showcase these tools. An analytic theme unites the early chapters, which cover distributions, integration on manifolds and Lie groups, spherical harmonics, and operators on Riemannian manifolds. An exploration of bundles follows, from definitions to connections and curvature in vector bundles, culminating in a glimpse of Pontrjagin and Chern classes. The final chapter on Clifford algebras and Clifford groups draws the book to an algebraic conclusion, which can be seen as a generalized viewpoint of the quaternions. Differential Geometry and Lie Groups: A Second Course captures the mathematical theory needed for advanced study in differential geometry with a view to furthering geometry processing capabilities. Suited to classroom use or independent study, the text will appeal to students and professionals alike. A first course in differential geometry is assumed; the authors’ companion volume Differential Geometry and Lie Groups: A Computational Perspective provides the ideal preparation.

Author : Hayit Greenspan,Anant Madabhushi,Parvin Mousavi,Septimiu Salcudean,James Duncan,Tanveer Syeda-Mahmood,Russell Taylor
Publisher : Springer Nature
Page : 844 pages
File Size : 46,8 Mb
Release : 2023-09-30
Category : Computers
ISBN : 9783031439049

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Medical Image Computing and Computer Assisted Intervention – MICCAI 2023 by Hayit Greenspan,Anant Madabhushi,Parvin Mousavi,Septimiu Salcudean,James Duncan,Tanveer Syeda-Mahmood,Russell Taylor Pdf

The ten-volume set LNCS 14220, 14221, 14222, 14223, 14224, 14225, 14226, 14227, 14228, and 14229 constitutes the refereed proceedings of the 26th International Conference on Medical Image Computing and Computer-Assisted Intervention, MICCAI 2023, which was held in Vancouver, Canada, in October 2023. The 730 revised full papers presented were carefully reviewed and selected from a total of 2250 submissions. The papers are organized in the following topical sections: Part I: Machine learning with limited supervision and machine learning – transfer learning; Part II: Machine learning – learning strategies; machine learning – explainability, bias, and uncertainty; Part III: Machine learning – explainability, bias and uncertainty; image segmentation; Part IV: Image segmentation; Part V: Computer-aided diagnosis; Part VI: Computer-aided diagnosis; computational pathology; Part VII: Clinical applications – abdomen; clinical applications – breast; clinical applications – cardiac; clinical applications – dermatology; clinical applications – fetal imaging; clinical applications – lung; clinical applications – musculoskeletal; clinical applications – oncology; clinical applications – ophthalmology; clinical applications – vascular; Part VIII: Clinical applications – neuroimaging; microscopy; Part IX: Image-guided intervention, surgical planning, and data science; Part X: Image reconstruction and image registration.

Author : Michael E. Taylor
Publisher : Springer Science & Business Media
Page : 590 pages
File Size : 52,8 Mb
Release : 1996-06-25
Category : Mathematics
ISBN : 0387946543

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Partial Differential Equations by Michael E. Taylor Pdf

This text provides an introduction to the theory of partial differential equations. It introduces basic examples of partial differential equations, arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, including particularly Fourier analysis, distribution theory, and Sobolev spaces. These tools are applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. Companion texts, which take the theory of partial differential equations further, are AMS volume 116, treating more advanced topics in linear PDE, and AMS volume 117, treating problems in nonlinear PDE. This book is addressed to graduate students in mathematics and to professional mathematicians, with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.

Author : Federico Battiston,Giovanni Petri
Publisher : Springer Nature
Page : 436 pages
File Size : 47,7 Mb
Release : 2022-04-26
Category : Science
ISBN : 9783030913748

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Higher-Order Systems by Federico Battiston,Giovanni Petri Pdf

The book discusses the potential of higher-order interactions to model real-world relational systems. Over the last decade, networks have emerged as the paradigmatic framework to model complex systems. Yet, as simple collections of nodes and links, they are intrinsically limited to pairwise interactions, limiting our ability to describe, understand, and predict complex phenomena which arise from higher-order interactions. Here we introduce the new modeling framework of higher-order systems, where hypergraphs and simplicial complexes are used to describe complex patterns of interactions among any number of agents. This book is intended both as a first introduction and an overview of the state of the art of this rapidly emerging field, serving as a reference for network scientists interested in better modeling the interconnected world we live in.