Dover Books on Mathematics
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Challenging Mathematical Problems With Elementary Solutions, Volume I
by A. M. Yaglom
Part 1 of the Dover Books on Mathematics series
Volume I of a two-part series, this book features a broad spectrum of 100 challenging problems related to probability theory and combinatorial analysis. The problems, most of which can be solved with elementary mathematics, range from relatively simple to extremely difficult. Suitable for students, teachers, and any lover of mathematics. Complete solutions.
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Nonnegative Matrices and Applicable Topics in Linear Algebra
by Alexander Graham
Part of the Dover Books on Mathematics series
Nonnegative matrices is an increasingly important subject in economics, control theory, numerical analysis, Markov chains, and other areas. This concise treatment is directed toward undergraduates who lack specialized knowledge at the postgraduate level of mathematics and related fields, such as mathematical economics and operations research.
An Introductory Survey encompasses some aspects of matrix theory and its applications and other relevant topics in linear algebra, including certain facets of graph theory. Subsequent chapters cover various points of the theory of normal matrices, comprising unitary and Hermitian matrices, and the properties of positive definite matrices. An exploration of the main topic, nonnegative matrices, is followed by a discussion of M-matrices. The final chapter examines stochastic, genetic, and economic models. The important concepts are illustrated by simple worked examples. Problems appear at the conclusion of most chapters, with solutions at the end of the book.
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Splines and Variational Methods
by P. M. Prenter
Part of the Dover Books on Mathematics series
One of the clearest available introductions to variational methods, this text requires only a minimal background in calculus and linear algebra. Its self-contained treatment explains the application of theoretic notions to the kinds of physical problems that engineers regularly encounter. The text's first half concerns approximation theoretic notions, exploring the theory and computation of one- and two-dimensional polynomial and other spline functions. Later chapters examine variational methods in the solution of operator equations, focusing on boundary value problems in one and two dimensions. Additional topics include least squares and other Galerkin methods. Many helpful definitions, examples, and exercises appear throughout the book. A classic reference in spline theory, this volume will benefit experts as well as students of engineering and mathematics.
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Theory of Lie Groups
by Claude Chevalley
Part of the Dover Books on Mathematics series
Chevalley's most important contribution to mathematics is certainly his work on group theory. . . . [Theory of Lie Groups] was the first systematic exposition of the foundations of Lie group theory consistently adopting the global viewpoint, based on the notion of analytic manifold. This book remained the basic reference on Lie groups for at least two decades." - Bulletin of the American Mathematical Society Suitable for advanced undergraduate and graduate students of mathematics, this enduringly relevant text introduces the main basic principles that govern the theory of Lie groups. The treatment opens with an overview of the classical linear groups and of topological groups, focusing on the theory of covering spaces and groups, which is developed independently from the theory of paths. Succeeding chapters contain an examination of the theory of analytic manifolds as well as a combination of the notions of topological group and manifold that defines analytic and Lie groups. An exposition of the differential calculus of Cartan follows and concludes with an exploration of compact Lie groups and their representations.
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Linear Algebra and Matrix Theory
by Robert R. Stoll
Part of the Dover Books on Mathematics series
One of the best available works on matrix theory in the context of modern algebra, this text bridges the gap between ordinary undergraduate studies and completely abstract mathematics. 1952 edition.
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Introduction to Linear Algebra and Differential Equations
by John W. Dettman
Part of the Dover Books on Mathematics series
Excellent introductory text for students with one year of calculus. Topics include complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions and boundary-value problems. Includes 48 black-and-white illustrations. Exercises with solutions. Index.
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Set Theory: The Structure of Arithmetic
by Norman T. Hamilton
Part of the Dover Books on Mathematics series
This text is formulated on the fundamental idea that much of mathematics, including the classical number systems, can best be based on set theory. Beginning with a discussion of the rudiments of set theory, authors Norman T. Hamilton and Joseph Landin lead readers through a construction of the natural number system, discussing the integers and the rational numbers, and concluding with an in-depth examination of the real numbers. Drawn from lecture notes for a course intended primarily for high school mathematics teachers, this volume was designed to answer the question, "What is a number?" and to provide a foundation for the study of abstract algebra, elementary Euclidean geometry, and analysis. Upon completion of this treatment - which is suitable for high school mathematics teachers and advanced high school students - readers should be well prepared for introductory courses in abstract algebra and real variables.
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Introduction to Probability Theory with Contemporary Applications
by Lester L. Helms
Part of the Dover Books on Mathematics series
This introduction to probability theory transforms a highly abstract subject into a series of coherent concepts. Its extensive discussions and clear examples, written in plain language, expose students to the rules and methods of probability. Suitable for an introductory probability course, this volume requires abstract and conceptual thinking skills and a background in calculus. Topics include classical probability, set theory, axioms, probability functions, random and independent random variables, expected values, and covariance and correlations. Additional subjects include stochastic processes, continuous random variables, expectation and conditional expectation, and continuous parameter Markov processes. Numerous exercises foster the development of problem-solving skills, and all problems feature step-by-step solutions
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Basic Set Theory
by Azriel Levy
Part of the Dover Books on Mathematics series
An advanced-level treatment of the basics of set theory, this text offers students a firm foundation, stopping just short of the areas employing model-theoretic methods. Geared toward upper-level undergraduate and graduate students, it consists of two parts: the first covers pure set theory, including the basic motions, order and well-foundedness, cardinal numbers, the ordinals, and the axiom of choice and some of it consequences; the second deals with applications and advanced topics such as point set topology, real spaces, Boolean algebras, and infinite combinatorics and large cardinals. An appendix comprises useful information on eliminability and conservation theorems, and numerous exercises help students test their grasp of each topic. 1979 edition. 20 figures.
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Foundations of the Theory of Probability
by A. N. Kolmogorov
Part of the Dover Books on Mathematics series
This famous little book was first published in German in 1933 and in Russian a few years later, setting forth the axiomatic foundations of modern probability theory and cementing the author's reputation as a leading authority in the field. The distinguished Russian mathematician A. N. Kolmogorov wrote this foundational text, and it remains important both to students beginning a serious study of the topic and to historians of modern mathematics. Suitable as a text for advanced undergraduates and graduate students in mathematics, the treatment begins with an introduction to the elementary theory of probability and infinite probability fields. Subsequent chapters explore random variables, mathematical expectations, and conditional probabilities and mathematical expectations. The book concludes with a chapter on the law of large numbers, an Appendix on zero-or-one in the theory of probability, and detailed bibliographies.
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Regular Algebra and Finite Machines
by John Horton Conway
Part of the Dover Books on Mathematics series
World-famous mathematician John H. Conway based this classic text on a 1966 course he taught at Cambridge University. Geared toward graduate students of mathematics, it will also prove a valuable guide to researchers and professional mathematicians. His topics cover Moore's theory of experiments, Kleene's theory of regular events and expressions, Kleene algebras, the differential calculus of events, factors and the factor matrix, and the theory of operators. Additional subjects include event classes and operator classes, some regulator algebras, context-free languages, communicative regular algebra, axiomatic questions, the strength of classical axioms, and logical problems. Complete solutions to problems appear at the end.
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Tensor and Vector Analysis
With Applications to Differential Geometry
by C. E. Springer
Part of the Dover Books on Mathematics series
Concise and user-friendly, this college-level text assumes only a knowledge of basic calculus in its elementary and gradual development of tensor theory. The introductory approach bridges the gap between mere manipulation and a genuine understanding of an important aspect of both pure and applied mathematics. Beginning with a consideration of coordinate transformations and mappings, the treatment examines loci in three-space, transformation of coordinates in space and differentiation, tensor algebra and analysis, and vector analysis and algebra. Additional topics include differentiation of vectors and tensors, scalar and vector fields, and integration of vectors. The concluding chapter employs tensor theory to develop the differential equations of geodesics on a surface in several different ways to illustrate further differential geometry.
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Infinite Series
by Isidore Isaac Hirschman
Part of the Dover Books on Mathematics series
This text for advanced undergraduate and graduate students presents a rigorous approach that also emphasizes applications. Encompassing more than the usual amount of material on the problems of computation with series, the treatment offers many applications, including those related to the theory of special functions. Numerous problems appear throughout the book. The first chapter introduces the elementary theory of infinite series, followed by a relatively complete exposition of the basic properties of Taylor series and Fourier series. Additional subjects include series of functions and the applications of uniform convergence; double series, changes in the order of summation, and summability; power series and real analytic functions; and additional topics in Fourier series. The text concludes with an appendix containing material on set and sequence operations and continuous functions.
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Technical Calculus With Analytic Geometry
by Judith L. Gersting
Part of the Dover Books on Mathematics series
This well-thought-out text, filled with many special features, is designed for a two-semester course in calculus for technology students with a background in college algebra and trigonometry. The author has taken special care to make the book appealing to students by providing motivating examples, facilitating an intuitive understanding of the underlying concepts involved, and by providing much opportunity to gain proficiency in techniques and skills. Initial chapters cover functions and graphs, straight lines and conic sections, new coordinate systems, the derivative, using the derivative, integration and using the integral. The last four chapters focus on derivatives of transcendental functions, patterns for integrations, series expansion of functions, and differential equations. Throughout, the writing style is clear, readable and informal. Examples are abundant and have complete worked solutions. Practice problems appear in the body of the text in each section; these are relatively easy and are intended to be worked by the student as soon as they are encountered. Each new type of example in the text is followed by a practice problem that allows the student to gain immediate reinforcement in applying the problem-solving technique illustrated by the example. Answers to all practice problems are given at the back of the book, many with worked-out solutions. Other learning aids include the division of complex problem-solving processes into a series of step-by-step tasks, numerous exercises at the end of each section and a Status Check at the end of each chapter that helps students review what they have learned. Additional review exercises and a glossary, with definitions and page references, round out the book.
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Intriguing Mathematical Problems
by Oswald Jacoby
Part of the Dover Books on Mathematics series
Amusing and informative in its approach to solving mathematical bafflers, this treasury of theories, games, puzzles and oddities of all kinds, compiled by one of the world's best card players (Jacoby) and an expert in mathematical recreations (Benson) will delight and fascinate math enthusiasts. Although primarily intended to entertain, the wide variety of puzzles ― ranging from facile curiosities to very difficult intellectual exercises ― will challenge you to keep your mind going full steam. Each of the book's five sections ― "Fun with Numbers," "Fun with Letters," "The Odds: Explorations in Probability," "Where Inference and Reasoning Reign" and "The Answers Are Whole Numbers" ― is made up of approximately 30 problems, with solutions grouped at the end of each section. Math buffs will love testing their puzzle-solving skills on such challenging brainteasers as The Enterprising Snail, Mrs. Crabbe and the Bacon, The Fly and the Bicycles, The Lovesick Cockroaches, The Three Prisoners, Girls Should Live in Brooklyn, Who Was Executed?, Creaker vs. Roadhog, The Crossed Ladders, The Ancient Order of the Greens, and many more. Few of these problems require any advanced mathematical knowledge or prowess. You'll find that simply keeping your wits about you and your logical skills honed are all you need to enjoy a delightful and thought-provoking adventure in recreational mathematics. Foreword. 10 illustrations. 14 tables.
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Recursive Analysis
by R. L. Goodstein
Part of the Dover Books on Mathematics series
Recursive analysis develops natural number computations into a framework appropriate for real numbers. This text is based upon primary recursive arithmetic and presents a unique combination of classical analysis and intuitional analysis. Written by a master in the field, it is suitable for graduate students of mathematics and computer science and can be read without a detailed knowledge of recursive arithmetic. Introductory chapters on recursive convergence and recursive and relative continuity are succeeded by explorations of recursive and relative differentiability, the relative integral, and the elementary functions. A final chapter examines transfinite ordinals, and the text concludes with a helpful appendix of topics related to recursive irrationality and transcendence.
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Iterative Solution of Large Linear Systems
by David M. Young
Part of the Dover Books on Mathematics series
This self-contained treatment offers a systematic development of the theory of iterative methods. Its focal point resides in an analysis of the convergence properties of the successive overrelaxation (SOR) method, as applied to a linear system with a consistently ordered matrix. The text explores the convergence properties of the SOR method and related techniques in terms of the spectral radii of the associated matrices as well as in terms of certain matrix norms. Contents include a review of matrix theory and general properties of iterative methods; SOR method and stationary modified SOR method for consistently ordered matrices; nonstationary methods; generalizations of SOR theory and variants of method; second-degree methods, alternating direction-implicit methods, and a comparison of methods. 1971 edition.
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Algebraic Theory of Numbers
Translated from the French by Allan J. Silberger
by Pierre Samuel
Part of the Dover Books on Mathematics series
Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics-algebraic geometry, in particular. This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Galois theory, Noetherian rings and modules, and rings of fractions. It covers the basics, starting with the divisibility theory in principal ideal domains and ending with the unit theorem, finiteness of the class number, and the more elementary theorems of Hilbert ramification theory. Numerous examples, applications, and exercises appear throughout the text.
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The Science of Measurement
A Historical Survey
by Herbert Arthur Klein
Part of the Dover Books on Mathematics series
Witty, imaginative coverage of metrology-concepts of weight, length, volume, temperature, time, nuclear radiation, thermal power, light, pressure, much more. Nontechnical.
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The Advanced Geometry of Plane Curves and Their Applications
by C. Zwikker
Part of the Dover Books on Mathematics series
This study of many important curves, their geometrical properties, and their applications features material not customarily treated in texts on synthetic or analytic Euclidean geometry. Its wide coverage, which includes both algebraic and transcendental curves, extends to unusual properties of familiar curves along with the nature of lesser known curves.Informative discussions of the line, circle, parabola, ellipse, and hyperbola presuppose only the most elementary facts. The less common curves - cissoid, strophoid, spirals, the leminscate, cycloid, epicycloid, cardioid, and many others - receive introductions that explain both their basic and advanced properties. Derived curves-the involute, evolute, pedal curve, envelope, and orthogonal trajectories-are also examined, with definitions of their important applications. These range through the fields of optics, electric circuit design, hydraulics, hydrodynamics, classical mechanics, electromagnetism, crystallography, gear design, road engineering, orbits of subatomic particles, and similar areas in physics and engineering. The author represents the points of the curves by complex numbers, rather than the real Cartesian coordinates, an approach that permits simple, direct, and elegant proofs.
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The Riemann Zeta-Function
Theory and Applications
by Aleksandar Ivic
Part of the Dover Books on Mathematics series
This extensive survey presents a comprehensive and coherent account of Riemann zeta-function theory and applications. Starting with elementary theory, it examines exponential integrals and exponential sums, the Voronoi summation formula, the approximate functional equation, the fourth power moment, the zero-free region, mean value estimates over short intervals, higher power moments, and omega results. Additional topics include zeros on the critical line, zero-density estimates, the distribution of primes, the Dirichlet divisor problem and various other divisor problems, and Atkinson's formula for the mean square. End-of-chapter notes supply the history of each chapter's topic and allude to related results not covered by the book. 1985 edition.
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Numbers
Their History and Meaning
by Graham Flegg
Part of the Dover Books on Mathematics series
Extremely readable, jargon-free book for general readers traces the evolution of counting systems, from the primitive techniques of antiquity to computers. Text examines the earliest endeavors to count and record numbers, initial attempts to solve problems by using equations, and origins of infinite cardinal arithmetic.
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Introduction to Bessel Functions
by Frank Bowman
Part of the Dover Books on Mathematics series
A full, clear introduction to the properties and applications of Bessel functions, this self-contained text is equally useful for the classroom or for independent study. Topics include Bessel functions of zero order, modified Bessel functions, definite integrals, asymptotic expansions, and Bessel functions of any real order. More than 200 problems throughout.
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The Qualitative Theory of Ordinary Differential Equations
An Introduction
by Fred Brauer
Part of the Dover Books on Mathematics series
This highly regarded text presents a self-contained introduction to some important aspects of modern qualitative theory for ordinary differential equations. It is accessible to any student of physical sciences, mathematics or engineering who has a good knowledge of calculus and of the elements of linear algebra. In addition, algebraic results are stated as needed; the less familiar ones are proved either in the text or in appendixes. The topics covered in the first three chapters are the standard theorems concerning linear systems, existence and uniqueness of solutions, and dependence on parameters. The next three chapters, the heart of the book, deal with stability theory and some applications, such as oscillation phenomena, self-excited oscillations and the regulator problem of Lurie. One of the special features of this work is its abundance of exercises-routine computations, completions of mathematical arguments, extensions of theorems and applications to physical problems. Moreover, they are found in the body of the text where they naturally occur, offering students substantial aid in understanding the ideas and concepts discussed. The level is intended for students ranging from juniors to first-year graduate students in mathematics, physics or engineering; however, the book is also ideal for a one-semester undergraduate course in ordinary differential equations, or for engineers in need of a course in state space methods.
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Adventures in Mathematical Reasoning
by Sherman Stein
Part of the Dover Books on Mathematics series
Equally appealing to beginners and to the mathematically adept, this book bridges the humanities and sciences to explore applications behind computers, cell phones, measurement of astronomical distance, cell growth, and other areas. Eight fascinating examples show how understanding certain topics in advanced mathematics requires nothing more than arithmetic and common sense. Each chapter begins with a question about strings consisting of nothing more than two letters, and every such question raises intriguing problems to be explored and solved. Author Sherman Stein proceeds at a measured pace that permits readers to move through the chapters in a leisurely fashion, omitting none of the steps. His approach makes complex subjects, from topology to set theory to probability, both accessible and exciting.
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Initial Boundary Value Problems in Mathematical Physics
by Rolf Leis
Part of the Dover Books on Mathematics series
Based on the author's lectures at the University of Bonn in 1983–84, this book introduces classical scattering theory and the time-dependent theory of linear equations in mathematical physics. Topics include proof of the existence of wave operators, some special equations of mathematical physics, exterior boundary value problems, radiation conditions, and limiting absorption principles.
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The Green Book of Mathematical Problems
by Kenneth Hardy
Part of the Dover Books on Mathematics series
Rich selection of 100 practice problems - with hints and solutions - for students preparing for the William Lowell Putnam and other undergraduate-level mathematical competitions. Features real numbers, differential equations, integrals, polynomials, sets, other topics. Hours of stimulating challenge for math buffs at varying degrees of proficiency. References.
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Taxicab Geometry
An Adventure in Non-Euclidean Geometry
by Eugene F. Krause
Part of the Dover Books on Mathematics series
Fascinating, accessible introduction to unusual mathematical system in which distance is not measured by straight lines. Illustrated topics include applications to urban geography and comparisons to Euclidean geometry. Selected answers to problems.
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Spheroidal Wave Functions
by Carson Flammer
Part of the Dover Books on Mathematics series
Intended to facilitate the use and calculation of spheroidal wave functions, this applications-oriented text features a detailed and unified account of the properties of these functions. Addressed to applied mathematicians, mathematical physicists, and mathematical engineers, it presents tables that provide a convenient means for handling wave problems in spheroidal coordinates. Topics include separation of the scalar wave equation in spheroidal coordinates, angle and radial functions, integral representations and relations, and expansions in spherical Bessel function products. Additional subjects include recurrence relations of Whittaker type, asymptotic expansions for large values of c, and vector wave functions. The text concludes with an appendix, references, and tables of numerical values.
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The Early Mathematical Manuscripts of Leibniz
by G. W. Leibniz
Part of the Dover Books on Mathematics series
The manuscripts and correspondence of Leibniz possess a special interest: they are invaluable as aids to the study of their author's part in the invention and development of the infinitesimal calculus. In addition, the main ideas behind Leibniz's philosophical theories lay here, in his mathematical work. This volume consists of two sections. The first part features Leibniz's own accounts of his work, and the second section comprises critical and historical notes and essays. An informative Introduction leads to the "postscript" to Leibniz's 1703 letter to James Bernoulli, his "Historia et Origio Calculi Differentialis," and manuscripts of the period 1673-77. Essays by the distinguished scholar C. I. Gerhardt follow--Leibniz in London and Leibniz and Pascal, along with additional letters and manuscripts by Leibniz.
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The Penrose Transform
Its Interaction With Representation Theory
by Robert J. Baston
Part of the Dover Books on Mathematics series
In recent decades twistor theory has become an important focus for students of mathematical physics. Central to twistor theory is the geometrical transform known as the Penrose transform, named for its groundbreaking developer. Geared toward students of physics and mathematics, this advanced text explores the Penrose transform and presupposes no background in twistor theory and a minimal familiarity with representation theory. An introductory chapter sketches the development of the Penrose transform, followed by reviews of Lie algebras and flag manifolds, representation theory and homogeneous vector bundles, and the Weyl group and the Bott-Borel-Weil theorem. Succeeding chapters explore the Penrose transform in terms of the Bernstein-Gelfand-Gelfand resolution, followed by worked examples, constructions of unitary representations, and module structures on cohomology. The treatment concludes with a review of constructions and suggests further avenues for research.
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Lectures on Ergodic Theory
by Paul R. Halmos
Part of the Dover Books on Mathematics series
This concise classic by Paul R. Halmos, a well-known master of mathematical exposition, has served as a basic introduction to aspects of ergodic theory since its first publication in 1956. Suitable for advanced undergraduates and graduate students in mathematics, the treatment covers recurrence, mean and pointwise convergence, ergodic theorem, measure algebras, and automorphisms of compact groups. Additional topics include weak topology and approximation, uniform topology and approximation, invariant measures, unsolved problems, and other subjects.
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Elements of the Theory of Functions
by Konrad Knopp
Part of the Dover Books on Mathematics series
This well-known book provides a clear and concise review of general function theory via complex variables. Suitable for undergraduate math majors, the treatment explores only those topics that are simplest but are also most important for the development of the theory. Prerequisites include a knowledge of the foundations of real analysis and of the elements of analytic geometry. The text begins with an introduction to the system of complex numbers and their operations. Then the concept of sets of numbers, the limit concept, and closely related matters are extended to complex quantities. Final chapters examine the elementary functions, including rational and linear functions, exponential and trigonometric functions, and several others as well as their inverses, including the logarithm and the cyclometric functions. Numerous examples clarify the essential ideas, and proofs are expressed in a direct manner without sacrifice of completeness or rigor.
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A Modern View of Geometry
by Leonard M. Blumenthal
Part of the Dover Books on Mathematics series
Elegant and original, this exposition explores the foundations and development of both Euclidean and non-Euclidean geometry, particularly the postulational geometry of planes. Emphasis is placed upon the coordination of affine and projective planes as well as the basic unity of algebra and geometry. Geared toward undergraduate and graduate students, the treatment begins with a brief but engaging sketch of the historical background of Euclidean geometry and an elementary summary of set theory and propositional calculus. Subsequent chapters explore coordinates in an affine plane, including those with Desargues and Pappus properties, and coordinatizing projective planes. The final two chapters contain detailed developments of simple sets of postulates for the Euclidean and non-Euclidean planes.
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Lectures on Integral Equations
by Harold Widom
Part of the Dover Books on Mathematics series
This concise and classic volume presents the main results of integral equation theory as consequences of the theory of operators on Banach and Hilbert spaces. In addition, it offers a brief account of Fredholm's original approach. The self-contained treatment requires only some familiarity with elementary real variable theory, including the elements of Lebesgue integration, and is suitable for advanced undergraduates and graduate students of mathematics. Other material discusses applications to second order linear differential equations, and a final chapter uses Fourier integral techniques to investigate certain singular integral equations of interest for physical applications as well as for their own sake. A helpful index concludes the text.
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Groups, Rings, Modules
by Maurice Auslander
Part of the Dover Books on Mathematics series
This classic monograph is geared toward advanced undergraduates and graduate students. The treatment presupposes some familiarity with sets, groups, rings, and vector spaces. The four-part approach begins with examinations of sets and maps, monoids and groups, categories, and rings. The second part explores unique factorization domains, general module theory, semisimple rings and modules, and Artinian rings. Part three's topics include localization and tensor products, principal ideal domains, and applications of fundamental theorem. The fourth and final part covers algebraic field extensions and Dedekind domains. Exercises are provided at the end of each chapter.
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The Calculus Primer
by William L. Schaaf
Part of the Dover Books on Mathematics series
Comprehensive but concise, this introduction to differential and integral calculus covers all the topics usually included in a first course. The straightforward development places less emphasis on mathematical rigor, and the informal manner of presentation sets students at ease. Many carefully worked-out examples illuminate the text, in addition to numerous diagrams, problems, and answers. Bearing the needs of beginners constantly in mind, the treatment covers all the basic concepts of calculus: functions, derivatives, differentiation of algebraic and transcendental functions, partial differentiation, indeterminate forms, general and special methods of integration, the definite integral, partial integration, and other fundamentals. Ample exercises permit students to test their grasp of subjects before moving forward, making this volume appropriate not only for classroom use but also for review and home study.
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Special Matrices and Their Applications in Numerical Mathematics
by Miroslav Fiedler
Part of the Dover Books on Mathematics series
This revised and corrected second edition of a classic book on special matrices provides researchers in numerical linear algebra and students of general computational mathematics with an essential reference. Author Miroslav Fiedler, a Professor at the Institute of Computer Science of the Academy of Sciences of the Czech Republic, Prague, begins with definitions of basic concepts of the theory of matrices and fundamental theorems. In subsequent chapters, he explores symmetric and Hermitian matrices, the mutual connections between graphs and matrices, and the theory of entrywise nonnegative matrices. After introducing M-matrices, or matrices of class K, Professor Fiedler discusses important properties of tensor products of matrices and compound matrices and describes the matricial representation of polynomials. He further defines band matrices and norms of vectors and matrices. The final five chapters treat selected numerical methods for solving problems from the field of linear algebra, using the concepts and results explained in the preceding chapters.
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Shape Theory
Categorical Methods of Approximation
by J. M. Cordier
Part of the Dover Books on Mathematics series
This in-depth treatment uses shape theory as a "case study" to illustrate situations common to many areas of mathematics, including the use of archetypal models as a basis for systems of approximations. It offers students a unified and consolidated presentation of extensive research from category theory, shape theory, and the study of topological algebras. A short introduction to geometric shape explains specifics of the construction of the shape category and relates it to an abstract definition of shape theory. Upon returning to the geometric base, the text considers simplical complexes and numerable covers, in addition to Morita's form of shape theory. Subsequent chapters explore Bénabou's theory of distributors, the theory of exact squares, Kan extensions, the notion of a stable object, and stability in an Abelian context. The text concludes with a brief description of derived functors of the limit functor theory-the concept that leads to movability and strong movability of systems-and illustrations of the equivalence of strong movability and stability in many contexts.
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Introductory Complex Analysis
by Richard A. Silverman
Part of the Dover Books on Mathematics series
A shorter version of A. I. Markushevich's masterly three-volume Theory of Functions of a Complex Variable, this edition is appropriate for advanced undergraduate and graduate courses in complex analysis. Numerous worked-out examples and more than 300 problems, some with hints and answers, make it suitable for independent study. 1967 edition.
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The Gamma Function
by Emil Artin
Part of the Dover Books on Mathematics series
This brief monograph on the gamma function was designed by the author to fill what he perceived as a gap in the literature of mathematics, which often treated the gamma function in a manner he described as both sketchy and overly complicated. Author Emil Artin, one of the twentieth century's leading mathematicians, wrote in his Preface to this book, "I feel that this monograph will help to show that the gamma function can be thought of as one of the elementary functions, and that all of its basic properties can be established using elementary methods of the calculus." Generations of teachers and students have benefitted from Artin's masterly arguments and precise results. Suitable for advanced undergraduates and graduate students of mathematics, his treatment examines functions, the Euler integrals and the Gauss formula, large values of x and the multiplication formula, the connection with sin x, applications to definite integrals, and other subjects.
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Introduction to Real Analysis
by Michael J. Schramm
Part of the Dover Books on Mathematics series
This text forms a bridge between courses in calculus and real analysis. It focuses on the construction of mathematical proofs as well as their final content. Suitable for upper-level undergraduates and graduate students of real analysis, it also provides a vital reference book for advanced courses in mathematics. The four-part treatment begins with an introduction to basic logical structures and techniques of proof, including discussions of the cardinality concept and the algebraic and order structures of the real and rational number systems. Part Two presents in-depth examinations of the completeness of the real number system and its topological structure. Part Three reviews and extends the previous explorations of the real number system, and the final part features a selection of topics in real function theory. Numerous and varied exercises range from articulating the steps omitted from examples and observing mechanical results at work to the completion of partial proofs within the text.
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The Magic of Numbers
by Eric Temple Bell
Part of the Dover Books on Mathematics series
From one of the foremost interpreters for lay readers of the history and meaning of mathematics: a stimulating account of the origins of mathematical thought and the development of numerical theory. It probes the work of Pythagoras, Galileo, Berkeley, Einstein, and others, exploring how "number magic" has influenced religion, philosophy, science, and mathematics
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Rotations, Quaternions, and Double Groups
by Simon L. Altmann
Part of the Dover Books on Mathematics series
This self-contained text presents a consistent description of the geometric and quaternionic treatment of rotation operators, employing methods that lead to a rigorous formulation and offering complete solutions to many illustrative problems. Geared toward upper-level undergraduates and graduate students, the book begins with chapters covering the fundamentals of symmetries, matrices, and groups, and it presents a primer on rotations and rotation matrices. Subsequent chapters explore rotations and angular momentum, tensor bases, the bilinear transformation, projective representations, and the geometry, topology, and algebra of rotations. Some familiarity with the basics of group theory is assumed, but the text assists students in developing the requisite mathematical tools as necessary.
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Introduction to Vector and Tensor Analysis
by Robert C. Wrede
Part of the Dover Books on Mathematics series
A broad introductory treatment, this volume examines general Cartesian coordinates, the cross product, Einstein's special theory of relativity, bases in general coordinate systems, maxima and minima of functions of two variables, line integrals, integral theorems, fundamental notions in n-space, Riemannian geometry, algebraic properties of the curvature tensor, and more. 1963 edition.
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Introduction to the Theory of Sets
by Joseph Breuer
Part of the Dover Books on Mathematics series
Set theory permeates much of contemporary mathematical thought. This text for undergraduates offers a natural introduction, developing the subject through observations of the physical world. Its progressive development leads from concrete finite sets to cardinal numbers, infinite cardinals, and ordinals. Although set theory begins in the intuitive and the concrete, it ascends to a very high degree of abstraction. All that is necessary to its grasp, declares author Joseph Breuer, is patience. Breuer illustrates the grounding of finite sets in arithmetic, permutations, and combinations, which provides the terminology and symbolism for further study. Discussions of general theory lead to a study of ordered sets, concluding with a look at the paradoxes of set theory and the nature of formalism and intuitionalism. Answers to exercises incorporated throughout the text appear at the end, along with an appendix featuring glossaries and other helpful information.
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Introduction to Mathematical Thinking
The Formation of Concepts in Modern Mathematics
by Friedrich Waismann
Part of the Dover Books on Mathematics series
This enlightening survey of mathematical concept formation holds a natural appeal to philosophically minded readers, and no formal training in mathematics is necessary to appreciate its clear exposition of mathematic fundamentals. Rather than a system of theorems with completely developed proofs or examples of applications, readers will encounter a coherent presentation of mathematical ideas that begins with the natural numbers and basic laws of arithmetic and progresses to the problems of the real-number continuum and concepts of the calculus. Contents include examinations of the various types of numbers and a criticism of the extension of numbers; arithmetic, geometry, and the rigorous construction of the theory of integers; the rational numbers, the foundation of the arithmetic of natural numbers, and the rigorous construction of elementary arithmetic. Advanced topics encompass the principle of complete induction; the limit and point of accumulation; operating with sequences and differential quotient; remarkable curves; real numbers and ultrareal numbers; and complex and hypercomplex numbers. In issues of mathematical philosophy, the author explores basic theoretical differences that have been a source of debate among the most prominent scholars and on which contemporary mathematicians remain divided. 27 figures. Index.
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Intuitive Concepts in Elementary Topology
by B. H. Arnold
Part of the Dover Books on Mathematics series
Classroom-tested and much-cited, this concise text offers a valuable and instructive introduction for undergraduates to the basic concepts of topology. It takes an intuitive rather than an axiomatic viewpoint, and can serve as a supplement as well as a primary text. A few selected topics allow students to acquire a feeling for the types of results and the methods of proof in mathematics, including mathematical induction. Subsequent problems deal with networks and maps, provide practice in recognizing topological equivalence of figures, examine a proof of the Jordan curve theorem for the special case of a polygon, and introduce set theory. The concluding chapters examine transformations, connectedness, compactness, and completeness. The text is well illustrated with figures and diagrams.