Geometry and Analysis on Finsler Spaces (1版)

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【簡介】 Finsler geometry is just Riemannian geometry without a quadratic restriction. It has applications in many fields of natural sciences, including physics, psychology, and ecology. The book is intended to provide basic materials on Finsler geometry for readers and to bring them to the frontiers of active research on related topics.This book is comprised of three parts. In Part I (Chapters 1-4), the author introduces the basics, such as Finsler metrics, the Chern connection, geometric invariant quantities, etc., and gives some rigidity results on Finsler manifolds with certain curvature properties. Part II (Chapters 5-6) covers the theory of geodesics, using which the author establishes some comparison theorems, which are fundamental tools to study global Finsler geometry. In Part III (Chapters 7-9), the author presents recent developments in nonlinear geometric analysis on Finsler spaces, partly based on the author’s recent works on Finsler harmonic functions, the eigenvalue problem, and heat flow. The author has made efforts to ensure that the contents are accessible to advanced undergraduates, graduate students, and researchers who are interested in Finsler geometry.

Analysis in Banach Spaces: Volume III: Harmonic Analysis and Spectral Theory

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This third volume of Analysis in Banach Spaces offers a systematic treatment of Banach space-valued singular integrals, Fourier transforms, and function spaces. It further develops and ramifies the theory of functional calculus from Volume II and describes applications of these new notions and tools to the problem of maximal regularity of evolution equations. The exposition provides a unified treatment of a large body of results, much of which has previously only been available in the form of research papers. Some of the more classical topics are presented in a novel way using modern techniques amenable to a vector-valued treatment. Thanks to its accessible style with complete and detailed proofs, this book will be an invaluable reference for researchers interested in functional analysis, harmonic analysis, and the operator-theoretic approach to deterministic and stochastic evolution equations.

Metric Spaces and Related Analysis

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This book offers the comprehensive study of one of the foundational topics in Mathematics, known as Metric Spaces. The book delivers the concepts in an appropriate and concise manner, at the same time rich in illustrations and exercise problems. Special focus has been laid on important theorems like Baire\'s Category theorem, Heine–Borel theorem, Ascoli–Arzela Theorem, etc, which play a crucial role in the study of metric spaces. The additional chapter on Cofinal completeness, UC spaces and finite chainability makes the text unique of its kind. This helps the students in: taking the secondary step towards analysis on metric spaces, realizing the connection between the two most important classes of functions, continuous functions and uniformly continuous functions, understanding the gap between compact metric spaces and complete metric spaces. Readers will also find brief discussions on various subtleties of continuity like subcontinuity, upper semi-continuity, lower semi-continuity, etc. The interested readers will be motivated to explore the special classes of functions between metric spaces to further extent. Consequently, the book becomes a complete package: it makes the foundational pillars strong and develops the interest of students to pursue research in metric spaces. The book is useful for third and fourth year undergraduate students and it is also helpful for graduate students and researchers. Sample Chapter(s) Foreword Chapter 1: Fundamentals of Analysis Contents: Fundamentals of Analysis Continuity and Some Stronger Notions Complete Metric Spaces Compactness Weaker Notions of Compactness Real-Valued Functions on Metric Spaces Connectedness Readership: Undergraduate and graduate students, researchers in the areas of real analysis, analysis on metric spaces, real functions, topology, functional analysis.