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Functional Analysis
This concise text provides a gentle introduction to functional analysis. Chapters cover essential topics such as special spaces, normed spaces, linear functionals, and Hilbert spaces. Numerous examples and counterexamples aid in the understanding of key concepts, while exercises at the end of each chapter provide ample opportunities for practice with the material. Proofs of theorems such as the Uniform Boundedness Theorem, the Open Mapping Theorem, and the Closed Graph Theorem are worked through step-by-step, providing an accessible avenue to understanding these important results. The prerequisites for this book are linear algebra and elementary real analysis, with two introductory chapters providing an overview of material necessary for the subsequent text. Functional Analysis offers an elementary approach ideal for the upper-undergraduate or beginning graduate student. Primarily intended for a one-semester introductory course, this text is also a perfect resource for independent study or as the basis for a reading course.
The Mathematics of Preference, Choice and Order
Peter Fishburn has had a splendidly productive career that led to path-breaking c- tributions in a remarkable variety of areas of research. His contributions have been published in a vast literature, ranging through journals of social choice and welfare, decision theory, operations research, economic theory, political science, mathema- cal psychology, and discrete mathematics. This work was done both on an individual basis and with a very long list of coauthors. The contributions that Fishburn made can roughly be divided into three major topical areas, and contributions to each of these areas are identi?ed by sections of this monograph. Section 1 deals with topics that are included in the general areas of utility, preference, individual choice, subjective probability, and measurement t- ory. Section 2 covers social choice theory, voting models, and social welfare. S- tion 3 deals with more purely mathematical topics that are related to combinatorics, graph theory, and ordered sets. The common theme of Fishburn’s contributions to all of these areas is his ability to bring rigorous mathematical analysis to bear on a wide range of dif?cult problems.
Real Analysis: Foundations
This textbook explores the foundations of real analysis using the framework of general ordered fields, demonstrating the multifaceted nature of the area. Focusing on the logical structure of real analysis, the definitions and interrelations between core concepts are illustrated with the use of numerous examples and counterexamples. Readers will learn of the equivalence between various theorems and the completeness property of the underlying ordered field. These equivalences emphasize the fundamental role of real numbers in analysis. Comprising six chapters, the book opens with a rigorous presentation of the theories of rational and real numbers in the framework of ordered fields. This is fol...
Aggregation and Fusion of Imperfect Information
This book presents the main tools for aggregation of information given by several members of a group or expressed in multiple criteria, and for fusion of data provided by several sources. It focuses on the case where the availability knowledge is imperfect, which means that uncertainty and/or imprecision must be taken into account. The book contains both theoretical and applied studies of aggregation and fusion methods in the main frameworks: probability theory, evidence theory, fuzzy set and possibility theory. The latter is more developed because it allows to manage both imprecise and uncertain knowledge. Applications to decision-making, image processing, control and classification are described.
Fuzzy Preference Modelling and Multicriteria Decision Support
This book provides in-depth coverage of the most important results about fuzzy logic including negations, conjunctions, disjunctions, implications and gives the interrelations between those different connectives. The work brings together multiple results about valued binary relations satisfying diverse transitivity-type conditions. The authors propose the first sound introduction to valued preference modelling through the systematic use of fuzzy set theory and functional equations and derive the possible foundations for multicriteria decision aid using aggregation, ranking and choice procedures on the basis of axiomatic results. The text presents a unified view of various multicriteria decision making tools that have been independently derived in the past, dealing with pairwise comparisons. The monograph is mathematically oriented but the results will be of the greatest interest for engineers and economists who design and implement decision support systems in practice. It is also supplied with a sufficient number of examples to make it attractive to nonspecialists.
Introduction to Analysis in One Variable
This is a text for students who have had a three-course calculus sequence and who are ready to explore the logical structure of analysis as the backbone of calculus. It begins with a development of the real numbers, building this system from more basic objects (natural numbers, integers, rational numbers, Cauchy sequences), and it produces basic algebraic and metric properties of the real number line as propositions, rather than axioms. The text also makes use of the complex numbers and incorporates this into the development of differential and integral calculus. For example, it develops the theory of the exponential function for both real and complex arguments, and it makes a geometrical study of the curve (expit) (expit), for real t t, leading to a self-contained development of the trigonometric functions and to a derivation of the Euler identity that is very different from what one typically sees. Further topics include metric spaces, the Stone–Weierstrass theorem, and Fourier series.
Pop Culture Russia!
A revealing look at contemporary Russian popular culture, exploring the historical and social influences that make it unique. Pop music is only one aspect of contemporary Russian culture that has taken some unexpected turns in the chaotic aftermath of the Soviet Union's collapse. Television and advertising, theater and cinema, athletics and religion, even fashion and food now reflect more exposure to the West, yet remain in essence distinctively Russian. Pop Culture Russia! introduces readers to the fascinating, often surprising, post-Soviet cultural landscape. With chapters on media, the arts, recreation, religion, and consumerism, the book offers an insightful survey of Russian mass culture from the death of Stalin in 1953 to the present, exploring the historical significance of important events and trends, as well as the social and political contexts from which they emerged.
Fuzzy Reasoning in Decision Making and Optimization
Many decision-making tasks are too complex to be understood quantitatively, however, humans succeed by using knowledge that is imprecise rather than precise. Fuzzy logic resembles human reasoning in its use of imprecise informa tion to generate decisions. Unlike classical logic which requires a deep under standing of a system, exact equations, and precise numeric values, fuzzy logic incorporates an alternative way of thinking, which allows modeling complex systems using a higher level of abstraction originating from our knowledge and experience. Fuzzy logic allows expressing this knowledge with subjective concepts such as very big and a long time which are mapped into exact numeric ranges. S...
Introduction to Proof Through Number Theory
Lighten up about mathematics! Have fun. If you read this book, you will have to endure bad math puns and jokes and out-of-date pop culture references. You'll learn some really cool mathematics to boot. In the process, you will immerse yourself in living, thinking, and breathing logical reasoning. We like to call this proofs, which to some is a bogey word, but to us it is a boogie word. You will learn how to solve problems, real and imagined. After all, math is a game where, although the rules are pretty much set, we are left to our imaginations to create. Think of this book as blueprints, but you are the architect of what structures you want to build. Make sure you lay a good foundation, for otherwise your buildings might fall down. To help you through this, we guide you to think and plan carefully. Our playground consists of basic math, with a loving emphasis on number theory. We will encounter the known and the unknown. Ancient and modern inquirers left us with elementary-sounding mathematical puzzles that are unsolved to this day. You will learn induction, logic, set theory, arithmetic, and algebra, and you may one day solve one of these puzzles.
Handbook of Fuzzy Computation
Initially conceived as a methodology for the representation and manipulation of imprecise and vague information, fuzzy computation has found wide use in problems that fall well beyond its originally intended scope of application. Many scientists and engineers now use the paradigms of fuzzy computation to tackle problems that are either intractable
