Fixed-point theory initially emerged in the article demonstrating existence of solutions of differential equations, which appeared in the second quarter of the 18th century (Joseph Liouville, 1837). Later on, this technique was improved as a method of successive approximations (Charles Emile Picard, 1890) which was extracted and abstracted as a fixed-point theorem in the framework of complete normed space (Stefan Banach, 1922). It ensures presence as well as uniqueness of a fixed point, gives an approximate technique to really locate the fixed point and the a priori and a posteriori estimates for the rate of convergence. It is an essential device in the theory of metric spaces. Subsequently, it is stated that fixed-point theory is initiated by Stefan Banach. Fixed-point theorems give adequate conditions under which there exists a fixed point for a given function and enable us to ensure the existence of a solution of the original problem. In an extensive variety of scientific issues, beginning from different branches of mathematics, the existence of a solution is comparable to the existence of a fixed point for a suitable mapping. The book "Fixed Point Theory & its Applications to Real World Problems" is an endeavour to present results in fixed point theory which are extensions, improvements and generalizations of classical and recent results in this area and touches on distinct research directions within the metric fixed-point theory. It provides new openings for further exploration and makes for an easily accessible source of knowledge. This book is apposite for young researchers who want to pursue their research in fixed-point theory and is the latest in the field, giving new techniques for the existence of a superior fixed point, a fixed point, a near fixed point, a fixed circle, a near fixed interval circle, a fixed disc, a near fixed interval disc, a coincidence point, a common fixed point, a coupled common fixed point, amiable fixed sets, strong coupled fixed points and so on, utilizing minimal conditions. It offers novel applications besides traditional applications which are applicable to real world problems. The book is self-contained and unified which will serve as a reference book to researchers who are in search of novel ideas. It will be a valued addition to the library.

This book provides an introduction to mathematical logic and the foundations of mathematics. It will help prepare students for advanced study in set theory and mathematical logic as well as other areas of mathematics, such as analysis, topology, and algebra. The presentation of finite state and Turing machines leads to the Halting Problem and Goedel's Incompleteness Theorem, which have broad academic interest, particularly in computer science and philosophy.

This innovative book introduces finite and transfinite interpolation methods at a general level in a unifying mathematical style before covering dynamical interpolation methods, which emphasize the underlying Eulerian/Lagrangian dynamics. Transfinite Interpolations and Eulerian/Lagrangian Dynamics: Considers the support of the data set as a geometrically structured set as opposed to an unstructured cloud of points. Is a self-contained graduate-level text, integrating theory, applications, numerical approximations, and computational techniques. Tackles transfinite interpolation methods applied to finite element meshes adaptation and ALE fluid-structure interaction and to the construction of velocity fields from the boundary expression of shape derivatives. Specialists in applied mathematics, physics, mechanics, computational sciences, imaging sciences, and engineering will find this book of interest.

"From the Calculus to Set Theory" traces the development of the calculus from the early seventeenth century through its expansion into mathematical analysis to the developments in set theory and the foundations of mathematics in the early twentieth century. It chronicles the work of mathematicians from Descartes and Newton to Russell and Hilbert and many, many others while emphasizing foundational questions and underlining the continuity of developments in higher mathematics. The other contributors to this volume are H. J. M. Bos, R. Bunn, J. W. Dauben, T. W. Hawkins, and K. Moller-Pedersen."

This volume presents some exciting new developments occurring on the interface between set theory and computability as well as their applications in algebra, analysis and topology. These include effective versions of Borel equivalence, Borel reducibility and Borel determinacy. It also covers algorithmic randomness and dimension, Ramsey sets and Ramsey spaces. Many of these topics are being discussed in the NSF-supported annual Southeastern Logic Symposium.

This volume contains the proceedings of Simon Fest, held in honor of Simon Thomas's 60th birthday, from September 15-17, 2017, at Rutgers University, Piscataway, New Jersey. The topics covered showcase recent advances from a variety of main areas of set theory, including descriptive set theory, forcing, and inner model theory, in addition to several applications of set theory, including ergodic theory, combinatorics, and model theory.

This book explores and highlights the fertile interaction between logic and operator algebras, which in recent years has led to the resolution of several long-standing open problems on C*-algebras. The interplay between logic and operator algebras (C*-algebras, in particular) is relatively young and the author is at the forefront of this interaction. The deep level of scholarship contained in these pages is evident and opens doors to operator algebraists interested in learning about the set-theoretic methods relevant to their field, as well as to set-theorists interested in expanding their view to the non-commutative realm of operator algebras. Enough background is included from both subjects to make the book a convenient, self-contained source for students. A fair number of the exercises form an integral part of the text. They are chosen to widen and deepen the material from the corresponding chapters. Some other exercises serve as a warmup for the latter chapters.

This comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff's classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff's initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics.The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2. Contents Fundamentals of the theory of classes, sets, and numbers Characterization of all natural models of Neumann - Bernays - Godel and Zermelo - Fraenkel set theories Local theory of sets as a foundation for category theory and its connection with the Zermelo - Fraenkel set theory Compactness theorem for generalized second-order language

This comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff's classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff's initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics. The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2. Contents Historical foreword on the centenary after Felix Hausdorff's classic Set Theory Fundamentals of the theory of functions Fundamentals of the measure theory Historical notes on the Riesz - Radon - Frechet problem of characterization of Radon integrals as linear functionals

The contents in this volume are based on the program Sets and Computations that was held at the Institute for Mathematical Sciences, National University of Singapore from 30 March until 30 April 2015. This special collection reports on important and recent interactions between the fields of Set Theory and Computation Theory. This includes the new research areas of computational complexity in set theory, randomness beyond the hyperarithmetic, powerful extensions of Goodstein's theorem and the capturing of large fragments of set theory via elementary-recursive structures.Further chapters are concerned with central topics within Set Theory, including cardinal characteristics, Fraisse limits, the set-generic multiverse and the study of ideals. Also Computation Theory, which includes computable group theory and measure-theoretic aspects of Hilbert's Tenth Problem. A volume of this broad scope will appeal to a wide spectrum of researchers in mathematical logic.

Type-2 fuzzy sets extend both ordinary and interval-valued fuzzy sets to allow distributions, rather than single values, as degrees of membership. Computations with these truth values are governed by the truth value algebra of type-2 fuzzy sets. The Truth Value Algebra of Type-2 Fuzzy Sets: Order Convolutions of Functions on the Unit Interval explores the fundamental properties of this algebra and the role of these properties in applications. Accessible to anyone with a standard undergraduate mathematics background, this self-contained book offers several options for a one- or two-semester course. It covers topics increasingly used in fuzzy set theory, such as lattice theory, analysis, category theory, and universal algebra. The book discusses the basics of the type-2 truth value algebra, its subalgebra of convex normal functions, and their applications. It also examines the truth value algebra from a more algebraic and axiomatic view.

The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

In Parts of Classes, David Lewis outlined a reduction of ZFC to a second order mereology. His conclusion takes on the following form in this reconstruction: ZFC is susceptible to parameterized interpretation in M (classical second order mereology) plus, there is a strongly unreachable partition. The proof makes use of the fact that ordered pairs in M plus an infinite partition are susceptible to parameterized interpretation."

This book is designed for readers who know elementary mathematical logic and axiomatic set theory, and who want to learn more about set theory. The primary focus of the book is on the independence proofs. Most famous among these is the independence of the Continuum Hypothesis (CH); that is, there are models of the axioms of set theory (ZFC) in which CH is true, and other models in which CH is false. More generally, cardinal exponentiation on the regular cardinals can consistently be anything not contradicting the classical theorems of Cantor and Konig. The basic methods for the independence proofs are the notion of constructibility, introduced by Godel, and the method of forcing, introduced by Cohen. This book describes these methods in detail, verifi es the basic independence results for cardinal exponentiation, and also applies these methods to prove the independence of various mathematical questions in measure theory and general topology. Before the chapters on forcing, there is a fairly long chapter on "infi nitary combinatorics." This consists of just mathematical theorems (not independence results), but it stresses the areas of mathematics where set-theoretic topics (such as cardinal arithmetic) are relevant. There is, in fact, an interplay between infi nitary combinatorics and independence proofs. Infi nitary combinatorics suggests many set-theoretic questions that turn out to be independent of ZFC, but it also provides the basic tools used in forcing arguments. In particular, Martin's Axiom, which is one of the topics under infi nitary combinatorics, introduces many of the basic ingredients of forcing.

This book is about "diamond," a logic of paradox. In diamond, a statement can be true yet false; an "imaginary" state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued boolean logic. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book has three sections: Paradox Logic, which covers the classic paradoxes of mathematical logic, shows how they can be resolved in this new system; The Second Paradox, which relates diamond to Boolean logic and the Spencer-Brown "modulator"; and Metamathematical Dilemma, which relates diamond to Gdelian meta-mathematics and dilemma games.

Nonlinear systems with stationary sets are important because they cover a lot of practical systems in engineering. Previous analysis has been based on the frequency-domain for this class of systems. However, few results on robustness analysis and controller design for these systems are easily available.This book presents the analysis as well as methods based on the global properties of systems with stationary sets in a unified time-domain and frequency-domain framework. The focus is on multi-input and multi-output systems, compared to previous publications which considered only single-input and single-output systems. The control methods presented in this book will be valuable for research on nonlinear systems with stationary sets.

This text unites the logical and philosophical aspects of set theory in a manner intelligible both to mathematicians without training in formal logic and to logicians without a mathematical background. It combines an elementary level of treatment with the highest possible degree of logical rigor and precision.
Starting with an explanation of all the basic logical terms and related operations, the text progresses through a stage-by-stage elaboration that proves the fundamental theorems of finite sets. It focuses on the Bernays theory of finite classes and finite sets, exploring the system's basis and development, including Stage I and Stage II theorems, the theory of finite ordinals, and the theory of finite classes and finite sets. This volume represents an excellent text for undergraduates studying intermediate or advanced logic as well as a fine reference for professional mathematicians.

Datenqualitat, Genauigkeit bzw. Ungenauigkeit von Daten und anderen Informationen sind grundlegende Aspekte von Messungen und Beobachtungen, die quantitativ beschrieben werden mussen, um unrealistische Resultate von Analysen zu vermeiden. In vielen praktischen Anwendungen erscheint die Angabe reeller Zahlen als vorliegende Datenelemente fragwurdig. Die Verwendung von unscharfen Zahlen ermoglicht es, die Unscharfe in die Modellbildung miteinzubeziehen und erlaubt somit eine realistischere Beschreibung von Daten.

Das Buch ist fur Leser geschrieben, die mit elementaren stochastischen Modellen und statistischen Verfahren vertraut sind.

Ziel ist es, Methoden der quantitativen Beschreibung unscharfer Beobachtungen stochastischer Grossen vorzustellen und in die Grundlagen der statistischen Analyse solcher Daten einzufuhren. Der praktische Umgang mit den vorgestellten Theorien und Methoden wird dem Leser anhand zahlreicher Ubungsaufgaben nahergebracht."

A comprehensive philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true.

This volume compiles geostatistical and spatial autoregressive data analyses involving georeferenced socioeconomic, natural resources, agricultural, pollution, and epidemiological variables. Benchmark analyses are followed by analyses of readily available data sets, emphasizing parallels between geostatistical and spatial autoregressive findings. Both SAS and SPSS code are presented for implementation purposes. This informative casebook will serve geographers, regional scientists, applied spatial statisticians, and spatial scientists from across disciplines.

Set-Indexed Martingales offers a unique, comprehensive development of a general theory of Martingales indexed by a family of sets. The authors establish-for the first time-an appropriate framework that provides a suitable structure for a theory of Martingales with enough generality to include many interesting examples. Developed from first principles, the theory brings together the theories of Martingales with a directed index set and set-indexed stochastic processes. Part One presents several classical concepts extended to this setting, including: stopping, predictability, Doob-Meyer decompositions, martingale characterizations of the set-indexed Poisson process, and Brownian motion. Part Two addresses convergence of sequences of set-indexed processes and introduces functional convergence for processes whose sample paths live in a Skorokhod-type space and semi-functional convergence for processes whose sample paths may be badly behaved. Completely self-contained, the theoretical aspects of this work are rich and promising. With its many important applications-especially in the theory of spatial statistics and in stochastic geometry- Set Indexed Martingales will undoubtedly generate great interest and inspire further research and development of the theory and applications.