Описание: The main goals of this paper are:(i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without calling into play any sort of analysis in charts, our assumptions being: the metric space is complete and separable and the measure is Radon and non-negative.(ii) To employ these notions of calculus to provide, via integration by parts, a general definition of distributional Laplacian, thus giving a meaning to an expression like $\Delta g=\mu$, where $g$ is a function and $\mu$ is a measure.(iii) To show that on spaces with Ricci curvature bounded from below and dimension bounded from above, the Laplacian of the distance function is always a measure and that this measure has the standard sharp comparison properties. This result requires an additional assumption on the space, which reduces to strict convexity of the norm in the case of smooth Finsler structures and is always satisfied on spaces with linear Laplacian, a situation which is analyzed in detail.
Описание: This fully updated new edition of Wilson Sutherland's classic text, Introduction to Metric and Topological Spaces, establishes the language of metric and topological spaces with continuity as the motivating concept, before developing its discussion to cover compactness, connectedness, and completeness.
Описание: The exposition, though not fully self contained, covers abstract measure theory, Hausdorff measures, Lipschitz functions, covering theorems, Sobolev spaces; The last chapter contains a brief, but very general, description of the theory of integration with respect to nondecreasing set functions.
Автор: Ambrosio Luigi, Serra Cassano Francesco Название: Lectures on analysis in metric spaces ISBN: 8876422552 ISBN-13(EAN): 9788876422553 Издательство: Springer Рейтинг: Цена: 1733 р. Наличие на складе: Поставка под заказ.
Автор: ? Searc?id Название: Metric Spaces ISBN: 1846283698 ISBN-13(EAN): 9781846283697 Издательство: Springer Рейтинг: Цена: 4037 р. Наличие на складе: Поставка под заказ.
Описание: The abstract concepts of metric spaces are often perceived as difficult. This book offers a unique approach to the subject which gives readers the advantage of a new perspective on ideas familiar from the analysis of a real line. Rather than passing quickly from the definition of a metric to the more abstract concepts of convergence and continuity, the author takes the concrete notion of distance as far as possible, illustrating the text with examples and naturally arising questions. Attention to detail at this stage is designed to prepare the reader to understand the more abstract ideas with relative ease.The book is designed for third- and fourth-year undergraduates and beginning graduates. Readers should have some practical knowledge of differential and integral calculus and have completed a first course in real analysis. With its many examples, careful illustrations, and full solutions to selected exercises, this book provides a gentle introduction that is ideal for self-study and an excellent preparation for applications.
Описание: Presents up-to-date Banach space results. Features an extensive bibliography for outside reading. Provides detailed exercises that elucidate more introductory material.
Автор: Agarwal, Ravi P. Karapinar, Erdal O`regan, Donal Название: Fixed point theory in metric type spaces ISBN: 3319240803 ISBN-13(EAN): 9783319240800 Издательство: Springer Рейтинг: Цена: 10971 р. Наличие на складе: Поставка под заказ.
Описание: Introduction with a Brief Historical Survey.- Preliminaries.- G-Metric Spaces.- Basic Fixed Point Results in the Setting of G-Metric Spaces.- Fixed Point Theorems in Partially Ordered G-Metric Spaces.- Further Fixed Point Results on G-Metric Spaces.- Fixed Point Theorems via Admissible Mappings.- New Approaches to Fixed Point Results on G-Metric Spaces.- Expansive Mappings.- Reconstruction of G-Metrics: G*-Metrics.- Multidimensional Fixed Point Theorems on G-Metric Spaces.- Recent Motivating Fixed Point Theory.
Описание: This volume is dedicated to Heinz Langer, a leading expert in spectral analysis and its applications, in particular to operators in spaces with an indefinite metric, on the occasion of his 60th birthday.The book begins with his biography and list of publications.
Автор: Heinonen Juha Название: Lectures on Analysis on Metric Spaces ISBN: 0387951040 ISBN-13(EAN): 9780387951041 Издательство: Springer Рейтинг: Цена: 8084 р. Наличие на складе: Поставка под заказ.
Описание: Analysis in spaces with no a priori smooth structure has progressed to include concepts from the first order calculus. In particular, there have been important advances in understanding the infinitesimal versus global behavior of Lipschitz functions and quasiconformal mappings in rather general settings; abstract Sobolev space theories have been instrumental in this development. The purpose of this book is to communicate some of the recent work in the area while preparing the reader to study more substantial, related articles. The material can be roughly divided into three different types: classical, standard but sometimes with a new twist, and recent. The author first studies basic covering theorems and their applications to analysis in metric measure spaces. This is followed by a discussion on Sobolev spaces emphasizing principles that are valid in larger contexts. The last few sections of the book present a basic theory of quasisymmetric maps between metric spaces. Much of the material is relatively recent and appears for the first time in book format. There are plenty of exercises. The book is well suited for self-study, or as a text in a graduate course or seminar. The material is relevant to anyone who is interested in analysis and geometry in nonsmooth settings.
Автор: Shirali Название: Metric Spaces ISBN: 1852339225 ISBN-13(EAN): 9781852339227 Издательство: Springer Рейтинг: Цена: 4614 р. Наличие на складе: Поставка под заказ.
Описание: Provides an introduction to metric space theory. This book covers the topology of metric spaces, continuity, connectedness, compactness and product spaces, and the Tietze-Urysohn extension theorem, Picard`s theorem on ordinary differential equations, and the set of discontinuities of the pointwise limit of a sequence of continuous functions.
Автор: Alexander Zaslavski Название: Optimization on Metric and Normed Spaces ISBN: 0387886206 ISBN-13(EAN): 9780387886206 Издательство: Springer Рейтинг: Цена: 19056 р. Наличие на складе: Поставка под заказ.
Описание: Covering recent work on Banach and complete metric spaces, this book uses the Baire approach and considers approximate solutions. It presents new results including penalty methods in constrained optimization and extant solutions in parametric optimization.
Автор: Heinonen Название: Sobolev Spaces on Metric Measure Spaces ISBN: 1107092345 ISBN-13(EAN): 9781107092341 Издательство: Cambridge Academ Рейтинг: Цена: 13283 р. Наличие на складе: Поставка под заказ.
Описание: Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincare inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincare inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger's stability theorem for Poincare inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincare inequalities.
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