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Provability, Computability and Reflection

Provability, Computability and Reflection

Homotopy Theory contains all the published mathematical work of J. H. C. Whitehead, written between 1947 and 1955. This volume considers the study of simple homotopy types, particularly the realization of problem for homotopy types. It describes Whitehead's version of homotopy theory in terms of CW-complexes. This book is composed of 21 chapters and begins with an overview of a theorem to Borsuk and the homotopy type of ANR. The subsequent chapters deal with four-dimensional polyhedral, the homotopy type of a special kind of polyhedron, and the combinatorial homotopy I and II. These topics are followed by reviews of other homotopy types, such as group extensions with homotopy operators, cohomology systems, secondary boundary operator, algebraic homotopy, and the G-dual of a semi-exact couple. The last chapters examine the connected complex homotopy types and the second non-vanishing homotopy groups. This book will be of great value to mathematicians.

Sixth Form Pure Mathematics, Volume 2, provides an introduction to inverse trigonometric functions, hyperbolic and inverse hyperbolic functions, and a range of mathematical methods including the use of determinants, the manipulation of inequalities, the solution of easy differential equations, and the use of approximate numerical methods. Complex numbers are defined and the various ways of representing and manipulating them are considered. Polar coordinates, curvature, an elementary study of lengths of curves and areas of surfaces of revolution, a more mature discussion of two-dimensional coordinate geometry than was possible in Volume 1, and an elementary introduction to the methods of three dimensional coordinate geometry comprise the geometrical content of the book. Throughout, the authors have tried to preserve the concentric style which they used in Volume 1 and the many worked examples and exercises in each chapter are designed or chosen to provide a continuous reminder of the work of the preceding chapters. Except for Pure Geometry, the two volumes cover almost all of the syllabuses for Advanced Pure Mathematics of the nine Examining Boards. This book provides an adequate course for mathematical pupils at Grammar Schools and a useful introductory course for Science and Engineering students in their first year at University or Technical College or engaged in private study.

Mathematical Analysis: A Special Course covers the fundamentals, principles, and theories that make up mathematical analysis. The title first provides an account of set theory, and then proceeds to detailing the elements of the theory of metric and normed linear spaces. Next, the selection covers the calculus of variations, along with the theory of Lebesgue integral. The text also tackles the geometry of Hilbert space and the relation between integration and differentiation. The last chapter of the title talks about the Fourier transform. The book will be of great use to individuals who want to expand and enhance their understanding of mathematical analysis.

The Theory of Splines and Their Applications

Exploring University Mathematics 2 presents the mathematical concept that is fundamental to the treatment of numbers. This book discusses the positive integers as the basis of common knowledge. Organized into seven chapters, this book begins with an overview of the analytical proof of the essentially geometrical isoperimetric problem. This text then explains that the simple models can be constructed, which lead to sixth-form mathematics. Other chapters consider the important physical ideas in kinetic theory. This book discusses as well the graphical method for establishing the general properties of the solution of a differential equation and applied it to the familiar equation for the motion of a simple pendulum when the amplitude of the oscillation is not small. The final chapter deals with certain methods in the theory of differential equations that are of considerable importance and interest. This book is a valuable resource for students, teachers, and research workers.

Outline Course of Pure Mathematics presents a unified treatment of the algebra, geometry, and calculus that are considered fundamental for the foundation of undergraduate mathematics. This book discusses several topics, including elementary treatments of the real number system, simple harmonic motion, Hooke's law, parabolic motion under gravity, sequences and series, polynomials, binomial theorem, and theory of probability. Organized into 23 chapters, this book begins with an overview of the fundamental concepts of differential and integral calculus, which are complementary processes for solving problems of the physical world. This text then explains the concept of the inverse of a function that is a natural complement of the function concept and introduces a convenient notation. Other chapters illustrate the concepts of continuity and discontinuity at the origin. This book discusses as well the significance of logarithm and exponential functions in scientific and technological contexts. This book is a valuable resource for undergraduates and advanced secondary school students.

Sixth Form Pure Mathematics, Volume 1, Second Edition, is the first of a series of volumes on Pure Mathematics and Theoretical Mechanics for Sixth Form students whose aim is entrance into British and Commonwealth Universities or Technical Colleges. A knowledge of Pure Mathematics up to G.C.E. O-level is assumed and the subject is developed by a concentric treatment in which each new topic is used to illustrate ideas already treated. The major topics of Algebra, Calculus, Coordinate Geometry, and Trigonometry are developed together. This volume covers most of the Pure Mathematics required for the single subject Mathematics at Advanced Level. Early and rapid progress in calculus is made at the beginning of this volume in order to facilitate the student's progress along the most satisfactory lines in Pure Mathematics, in Theoretical Mechanics and in Physics. The worked examples are an essential feature of this book and they are followed by routine exercises within the text of each chapter, associated closely with the work on which they are dependent. The exercises at the end of each chapter collectively embody all the topics of that chapter and, where possible, the preceding chapters also.

Piaget—A Practical Consideration deals with the general theories and work of Jean Piaget, with an account of a short follow up study of his work on the development of the concept of geometry. Piaget's theory concerning the general pattern of intellectual development in children is discussed, along with his study of children from two English primary schools in differing environments. This book is comprised of seven chapters and opens with a brief background on the life of Piaget and his approach to his work. The next chapter summarizes Piaget's general theories, including his concept of intelligence and its development; his studies of the progress made by children towards higher levels of equilibrium; and the Piagetian stages of development. The reader is then introduced to Piaget's experiments and their results on the development of the thought, language, and moral judgment of the child. The final chapter considers some recorded criticisms of Piaget's methods and maturation system. This monograph will be a useful resource for educators and psychologists.

Introduction to Modern Mathematics focuses on the operations, principles, and methodologies involved in modern mathematics. The monograph first tackles the algebra of sets, natural numbers, and functions. Discussions focus on groups of transformations, composition of functions, an axiomatic approach to natural numbers, intersection of sets, axioms of the algebra of sets, fields of sets, prepositional functions of one variable, and difference of sets. The text then takes a look at generalized unions and intersections of sets, Cartesian products of sets, and equivalence relations. The book ponders on powers of sets, ordered sets, and linearly ordered sets. Topics include isomorphism of linearly ordered sets, dense linear ordering, maximal and minimal elements, quasi-ordering relations, inequalities for cardinal numbers, sets of the power of the continuum, and Cantor's theorem. The manuscript then examines elementary concepts of abstract algebras, functional calculus and its applications in mathematical proofs, and propositional calculus and its applications in mathematical proofs. The publication is a valuable reference for mathematicians and researchers interested in modern mathematics.

Introductory College Mathematics: With Linear Algebra and Finite Mathematics is an introduction to college mathematics, with emphasis on linear algebra and finite mathematics. It aims to provide a working knowledge of basic functions (polynomial, rational, exponential, logarithmic, and trigonometric); graphing techniques and the numerical aspects and applications of functions; two- and three-dimensional vector methods; the fundamental ideas of linear algebra; and complex numbers, elementary combinatorics, the binomial theorem, and mathematical induction. Comprised of 15 chapters, this book begins with a discussion on functions and graphs, paying particular attention to quantities measured in the real number system. The next chapter deals with linear and quadratic functions as well as some of their applications. Tips on graphing are offered. Subsequent chapters focus on polynomial functions, along with graphs of factored polynomials; rational functions; exponential and logarithm functions; and trigonometric functions. Identities and inverse functions, vectors and matrices, and trigonometry are also explored, together with complex numbers, linear transformations, and the geometry of space. The book concludes by considering finite mathematics, with particular reference to mathematical induction and the binomial theorem. This monograph will be a useful resource for undergraduate students of mathematics and algebra.

In the decade before 1900, the Italian mathematician Giuseppe Peano was one of the most original and influential pioneers of modern mathematical logic. He made significant contributions to the development of the foundations of mathematics and the axiomatic method (for example, his postulates for the natural numbers), dimension theory (including the space-filling curve), measure theory, vector analysis, differential equations, and the rigorization of analysis. Several of Peano's works have been translated into other languages; here for the first time is a generous selection of works translated into English. Fifteen articles, one booklet, and parts of two books and one monograph, published between 1883 and 1921, chosen with the interests of mathematicians and logicians in mind, are included. Each selection is preceded by an introductory note. The volume also contains a biographical sketch, a chronological list of Peano's publications (larger by one fifth than any previously published list), and a bibliography on the life and work of Peano. This selection will appeal especially to historians of mathematics and logic, but also to those mathematicians and logicians who wish to know more about how their subject came to be.

An international symposium held to honor Alfred Tarski; co-sponsored by the University of California, Berkeley, the Association for Symbolic Logic [and] the International Union for History and Philosophy of Science--Division of Logic, Methodology, and Philosophy of Science.

Kunihiko Kodaira's influence in mathematics has been fundamental and international, and his efforts have helped lay the foundations of modern complex analysis. These three volumes contain Kodaira's written contributions, published in a large number of journals and books between 1937 and 1971. The volumes cover chronologically the major periods of Kodaira's mathematical concentration and reflect his collaboration with other prominent theoreticians.Thus they begin with early works that discuss the application of Hilbert space methods to differential equations, and the use of elementary solutions to prove regularity theorems for strongly elliptic systems of partial differential equations.Originally published in 1975.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Kunihiko Kodaira's influence in mathematics has been fundamental and international, and his efforts have helped lay the foundations of modern complex analysis. These three volumes contain Kodaira's written contributions, published in a large number of journals and books between 1937 and 1971. The volumes cover chronologically the major periods of Kodaira's mathematical concentration and reflect his collaboration with other prominent theoreticians. It was in the second period that Kodaira did his fundamental work on harmonic integrals. The third period is chiefly characterized by the application of harmonic integrals and of the theory of sheaves to algebraic geometry and to complex manifolds.Originally published in 1975.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.