Multidimensional Analysis and Discrete Models

Aleksei A. Dezin

• Publisher: CRC Press
• Year: 2018
• Print ISBN: 9781315895765
• eBook ISBN: 9781351091763
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Multidimensional Analysis and Discrete Models, a thorough and detailed reference, covers the main structures of multidimensional analysis and the intrinsically defined discrete models in applied mathematics, mathematical physics, and related fields. The material is presented in a clear and straightforward manner, with background information provided to define finite models and to clarify the concepts of multidimensional analysis. The book covers special difference models of the mathematical physics equations, models of boundary value problems, and objects of quantum mechanics. Considerable attention is also given to differential operators on Riemannian manifolds and the interpretation of classical vector analysis. The primary focus of Multidimensional Analysis and Discrete Models is on the description of regular methods of constructing intrinsically defined discrete models for special classes of continual objects, but emphasis is also given to the interaction of ideas and methods that exist throughout the field of mathematics. For example, the connections between theories derived from classical and functional analysis, Riemannian geometry, and algebraic topology are illustrated, and are discussed in terms of their relevance to computing solutions.

Chapter 1
Formal Structures

0. Introductory Remarks

By no means can this chapter replace textbooks and monographs that contain a detailed study of the structures considered below. I can only hope that the exposition I present can serve as a summary of necessary data, rather formal in places, but, at the same time, containing a number of informal remarks that emphasize the specific features of the chosen point of view. Additional explanations are given in the "0" items prefacing every section.

In the definitions I repeatedly use the terms a "set" and a "correspondence" (or their synonyms such as a "collection", a "tuple" for a set, and a "function" and a "mapping" for a correspondence). Proceeding from the "naive" viewpoint, I consider these terms to be primary, intuitively clear, that are not subject to formalization or any further logical analysis at this level. The remarks concerning the set-theoretic symbols which are used are given in 1.0.

Two classical sets, the sets ℝ and ℂ of real and complex numbers, play an exceptionally significant part in constructing one or another type of abstract objects or in defining some additional structure on it (that makes the object more informative). I suppose that their properties are well known.

The references to papers that I consider to be the most suitable are given inside the sections.

1. Topology and Metrics

1.0. Preliminary notes

Since the main subject matter of this book is the classical (multidimensional) analysis using functions defined on subsets of the Euclidean space (or the Riemannian manifold which is its direct generalization), the major part of the discussion that follows deals with structures that are ...

Table of contents

• Cover
• Title
• Copyright
• Preface
• Contents
• To the Reader
• Chapter 0. One-Dimensional Models
• Chapter 1. Formal Structures
• Chapter 2. Analysis on Riemannian Manifolds
• Chapter 3. The Model of Euclidean Space and Difference Operators
• Chapter 4. Models in Quantum Physics
• Chapter 5. Structural Analysis of Discrete Equations
• References
• Index