This book is devoted to some results from the classical Point Set Theory and their applications to certain problems in mathematical analysis of the real line. Notice that various topics from this theory are presented in several books and surveys. From among the most important works devoted to Point Set Theory, let us first of all mention the excellent book by Oxtoby [83] in which a deep analogy between measure and category is discussed in detail. Further, an interesting general approach to problems concerning measure and category is developed in the well-known monograph by Morgan [79] where a fundamental concept of a category base is introduced and investigated. We also wish to mention that the monograph by Cichon, W«;glorz and the author [19] has recently been published. In that book, certain classes of subsets of the real line are studied and various cardinal valued functions (characteristics) closely connected with those classes are investigated. Obviously, the IT-ideal of all Lebesgue measure zero subsets of the real line and the IT-ideal of all first category subsets of the same line are extensively studied in [19], and several relatively new results concerning this topic are presented. Finally, it is reasonable to notice here that some special sets of points, the so-called singular spaces, are considered in the classi

Set Theoretical Aspects of Real Analysis is built around a number of questions in real analysis and classical measure theory, which are of a set theoretic flavor. Accessible to graduate students, and researchers the beginning of the book presents introductory topics on real analysis and Lebesgue measure theory. These topics highlight the boundary between fundamental concepts of measurability and nonmeasurability for point sets and functions. The remainder of the book deals with more specialized material on set theoretical real analysis. The book focuses on certain logical and set theoretical aspects of real analysis. It is expected that the first eleven chapters can be used in a course on Lebesque measure theory that highlights the fundamental concepts of measurability and non-measurability for point sets and functions. Provided in the book are problems of varying difficulty that range from simple observations to advanced results. Relatively difficult exercises are marked by asterisks and hints are included with additional explanation. Five appendices are included to supply additional background information that can be read alongside, before, or after the chapters. Dealing with classical concepts, the book highlights material not often found in analysis courses. It lays out, in a logical, systematic manner, the foundations of set theory providing a readable treatment accessible to graduate students and researchers.

This unusually clear and interesting classic offers a thorough and reliable treatment of an important branch of higher analysis. The work covers real numbers and sequences, foundations of the theory of infinite series, and development of the theory (series of valuable terms, Euler's summation formula, asymptotic expansions, and other topics). Exercises throughout. Ideal for self-study.

Strange Functions in Real Analysis, Third Edition differs from the previous editions in that it includes five new chapters as well as two appendices. More importantly, the entire text has been revised and contains more detailed explanations of the presented material. In doing so, the book explores a number of important examples and constructions of pathological functions. After introducing basic concepts, the author begins with Cantor and Peano-type functions, then moves effortlessly to functions whose constructions require what is essentially non-effective methods. These include functions without the Baire property, functions associated with a Hamel basis of the real line and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum. Finally, the author considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms. On the whole, the book is devoted to strange functions (and point sets) in real analysis and their applications.

There are many mathematics textbooks on real analysis, but they focus on topics not readily helpful for studying economic theory or they are inaccessible to most graduate students of economics. Real Analysis with Economic Applications aims to fill this gap by providing an ideal textbook and reference on real analysis tailored specifically to the concerns of such students. The emphasis throughout is on topics directly relevant to economic theory. In addition to addressing the usual topics of real analysis, this book discusses the elements of order theory, convex analysis, optimization, correspondences, linear and nonlinear functional analysis, fixed-point theory, dynamic programming, and calculus of variations. Efe Ok complements the mathematical development with applications that provide concise introductions to various topics from economic theory, including individual decision theory and games, welfare economics, information theory, general equilibrium and finance, and intertemporal economics. Moreover, apart from direct applications to economic theory, his book includes numerous fixed point theorems and applications to functional equations and optimization theory. The book is rigorous, but accessible to those who are relatively new to the ways of real analysis. The formal exposition is accompanied by discussions that describe the basic ideas in relatively heuristic terms, and by more than 1,000 exercises of varying difficulty. This book will be an indispensable resource in courses on mathematics for economists and as a reference for graduate students working on economic theory.

Weierstrass and Blancmange nowhere differentiable functions, Lebesgue integrable functions with everywhere divergent Fourier series, and various nonintegrable Lebesgue measurable functions. While dubbed strange or "pathological," these functions are ubiquitous throughout mathematics and play an important role in analysis, not only as counterexamples of seemingly true and natural statements, but also to stimulate and inspire the further development of real analysis. Strange Functions in Real Analysis explores a number of important examples and constructions of pathological functions. After introducing the basic concepts, the author begins with Cantor and Peano-type functions, then moves to functions whose constructions require essentially noneffective methods. These include functions without the Baire property, functions associated with a Hamel basis of the real line, and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum. Finally, he considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms and demonstrates that their existence follows from certain set-theoretical hypotheses, such as the Continuum Hypothesis.

One of the best-written, most skillful expositions of group theory and its physical applications, directed primarily to advanced undergraduate and graduate students in physics, especially quantum physics. With problems.

An introduction to the methods used to study monomial algebras and their presentation ideals, including Stanley-Reisner rings, subrings, and toric varieties. It emphasizes square-free quadratics and effective computational methods, and applies a combinatorial description of the integral closure of the corresponding monomial subring to graph theory.

"A treatise on finite difference ineuqalities that have important applications to theories of various classes of finite difference and sum-difference equations, including several linear and nonlinear finite difference inequalities appearing for the first time in book form."

In one exceptional volume, Abstract Algebra covers subject matter typically taught over the course of two or three years and offers a self-contained presentation, detailed definitions, and excellent chapter-matched exercises to smooth the trajectory of learning algebra from zero to one. Field-tested through advance use in the ERASMUS educational project in Europe, this ambitious, comprehensive book includes an original treatment of representation of finite groups that avoids the use of semisimple ring theory and explains sets, maps, posets, lattices, and other essentials of the algebraic language; Peano's axioms and cardinality; groupoids, semigroups, monoids, groups; and normal subgroups.