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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Number Theory and Its History
by Oystein Ore
Part of the Dover Books on Mathematics series
A prominent mathematician presents the principal ideas and methods of number theory within a historical and cultural framework. Oystein Ore's fascinating, accessible treatment requires only a basic knowledge of algebra. Topics include prime numbers, the Aliquot parts, linear indeterminate problems, congruences, Euler's theorem, classical construction problems, and many other subjects.
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Geometry of Submanifolds
by Bang-Yen Chen
Part of the Dover Books on Mathematics series
The first two chapters of this frequently cited reference provide background material in Riemannian geometry and the theory of submanifolds. Subsequent chapters explore minimal submanifolds, submanifolds with parallel mean curvature vector, conformally flat manifolds, and umbilical manifolds. The final chapter discusses geometric inequalities of submanifolds, results in Morse theory and their applications, and total mean curvature of a submanifold.
Suitable for graduate students and mathematicians in the area of classical and modern differential geometries, the treatment is largely self-contained. Problems sets conclude each chapter, and an extensive bibliography provides background for students wishing to conduct further research in this area. This new edition includes the author's corrections.
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Theoretical Numerical Analysis
An Introduction to Advanced Techniques
by Peter Linz
Part of the Dover Books on Mathematics series
This concise text introduces numerical analysis as a practical, problem-solving discipline. The three-part presentation begins with the fundamentals of functional analysis and approximation theory. Part II outlines the major results of theoretical numerical analysis, reviewing product integration, approximate expansion methods, the minimization of functions, and related topics. Part III considers specific subjects that illustrate the power and usefulness of theoretical analysis.
Ideal as a text for a one-year graduate course, the book also offers engineers and scientists experienced in numerical computing a simple introduction to the major ideas of modern numerical analysis. Some practical experience with computational mathematics and the ability to relate this experience to new concepts is assumed. Otherwise, no background beyond advanced calculus is presupposed. Moreover, the ideas of functional analysis used throughout the text are introduced and developed only to the extent they are needed.
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The Variational Theory of Geodesics
by M. M. Postnikov
Part of the Dover Books on Mathematics series
Riemannian geometry is a fundamental area of modern mathematics and is important to the study of relativity. Within the larger context of Riemannian mathematics, the active subdiscipline of geodesics (shortest paths) in Riemannian spaces is of particular significance. This compact and self-contained text by a noted theorist presents the essentials of modern differential geometry as well as basic tools for the study of Morse theory. The advanced treatment emphasizes analytical rather than topological aspects of Morse theory and requires a solid background in calculus.
Suitable for advanced undergraduates and graduate students of mathematics, the text opens with a chapter on smooth manifolds, followed by a consideration of spaces of affine connection. Subsequent chapters explore Riemannian spaces and offer an extensive treatment of the variational properties of geodesics and auxiliary theorems and matters.
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Ordinary Differential Equations and Stability Theory
An Introduction
by David A. Sanchez
Part of the Dover Books on Mathematics series
This brief modern introduction to the subject of ordinary differential equations emphasizes stability theory. Concisely and lucidly expressed, it is intended as a supplementary text for advanced undergraduates or beginning graduate students who have completed a first course in ordinary differential equations.
The author begins by developing the notions of a fundamental system of solutions, the Wronskian, and the corresponding fundamental matrix. Subsequent chapters explore the linear equation with constant coefficients, stability theory for autonomous and nonautonomous systems, and the problems of the existence and uniqueness of solutions and related topics. Problems at the end of each chapter and two Appendixes on special topics enrich the text.
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Toward Human-Level Artificial Intelligence
Representation and Computation of Meaning in Natural Language
by Philip C. Jackson
Part of the Dover Books on Mathematics series
How can human-level artificial intelligence be achieved? What are the potential consequences? This book describes a research approach toward achieving human-level AI, combining a doctoral thesis and research papers by the author.
The research approach, called TalaMind, involves developing an AI system that uses a 'natural language of thought' based on the unconstrained syntax of a language such as English; designing the system as a collection of concepts that can create and modify concepts to behave intelligently in an environment; and using methods from cognitive linguistics for multiple levels of mental representation. Proposing a design-inspection alternative to the Turing Test, these pages discuss 'higher-level mentalities' of human intelligence, which include natural language understanding, higher-level forms of learning and reasoning, imagination, and consciousness. Dr. Jackson gives a comprehensive review of other research, addresses theoretical objections to the proposed approach and to achieving human-level AI in principle, and describes a prototype system that illustrates the potential of the approach.
This book discusses economic risks and benefits of AI, considers how to ensure that human-level AI and superintelligence will be beneficial for humanity, and gives reasons why human-level AI may be necessary for humanity's survival and prosperity.
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Summation of Infinitely Small Quantities
by I. P. Natanson
Part of the Dover Books on Mathematics series
Translated and adapted from a popular Russian educational series, this concise book requires only some background in high school algebra and elementary trigonometry. It explores the fundamental concept of the integral calculus: the limit of the sum of an infinitely increasing number of infinitely decreasing quantities. Mastery of this concept enables the solution of geometry and physics problems, and is an excellent introduction to the systematic study of higher mathematics.
Starting with some algebraic formulas, the treatment proceeds to the determination of the pressure of a liquid on a vertical wall and the calculation of the work done in pumping liquid from a container. Subsequent chapters explore finding the volumes of a cone, pyramid, sphere, and other geometric forms and the measurement of the parabola, ellipse, and sinusoid. The text concludes with a selection of practice problems.
For advanced high school students and college undergraduates.
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Sets, Sequences and Mappings
The Basic Concepts of Analysis
by Kenneth Anderson
Part of the Dover Books on Mathematics series
Students progressing to advanced calculus are frequently confounded by the dramatic shift from mechanical to theoretical and from concrete to abstract. This text bridges the gap, offering a systematic development of the real number system and careful treatment of mappings, sequences, limits, continuity, and metric spaces. The first five chapters consist of a systematic development of many of the important properties of the real number system, plus detailed treatment of such concepts as mappings, sequences, limits, and continuity. The sixth and final chapter discusses metric spaces and generalizes many of the earlier concepts and results involving arbitrary metric spaces. An index of axioms and key theorems appears at the end of the book, and more than 300 problems amplify and supplement the material within the text. Geared toward students who have taken several semesters of basic calculus, this volume is an ideal prerequisite for mathematics majors preparing for a two-semester course in advanced calculus.
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The Mathematics of Games of Strategy
by Melvin Dresher
Part of the Dover Books on Mathematics series
A noted research mathematician explores decision making in the absence of perfect information. His clear presentation of the mathematical theory of games of strategy encompasses applications to many fields, including economics, military, business, and operations research. No advanced algebra or non-elementary calculus occurs in most of the proofs.
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Vector Analysis
by Louis Brand
Part of the Dover Books on Mathematics series
This text was designed as a short introductory course to give students the tools of vector algebra and calculus, as well as a brief glimpse into the subjects' manifold applications. 1957 edition. 86 figures.
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Foundations of Stochastic Analysis
by M. M. Rao
Part of the Dover Books on Mathematics series
This volume considers fundamental theories and contrasts the natural interplay between real and abstract methods. No prior knowledge of probability is assumed. Numerous problems, most with hints. 1981 edition.
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Calculus of Variations
Mechanics, Control And Other Applications
by Charles R. MacCluer
Part of the Dover Books on Mathematics series
First truly up-to-date treatment offers a simple introduction to optimal control, linear-quadratic control design, and more. Broad perspective features numerous exercises, hints, outlines, and appendixes, including a practical discussion of MATLAB. 2005 edition.
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Constructive Real Analysis
by Allen A. Goldstein
Part of the Dover Books on Mathematics series
This text introduces students of mathematics, science, and technology to the methods of applied functional analysis and applied convexity. Topics include iterations and fixed points, metric spaces, nonlinear programming, applications to integral equations, and more. 1967 edition.
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Games, Theory and Applications
by L. C. Thomas
Part of the Dover Books on Mathematics series
Accessible and informative, this introduction to game theory explores 2-person zero-sum games, 2-person non-zero sum games, n-person games, and a variety of applications. Numerous exercises with full solutions. Includes 30 illustrations. 1986 edition.
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Elements of the Theory of Markov Processes and Their Applications
by A. T. Bharucha-Reid
Part of the Dover Books on Mathematics series
Graduate-level text and reference in probability, with numerous scientific applications. Nonmeasure-theoretic introduction to theory of Markov processes and to mathematical models based on the theory. Appendixes. Bibliographies. 1960 edition.
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The Convolution Transform
by Isidore Isaac Hirschman
Part of the Dover Books on Mathematics series
The relation between differential operators and integral transforms is the theme of this work. Discusses finite and non-finite kernels, variation diminishing transforms, asymptotic behavior of kernels, real inversion theory, representation theory, the Weierstrass transform, more.
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Invariant Subspaces
by Heydar Radjavi
Part of the Dover Books on Mathematics series
Broad survey focuses on operators on separable Hilbert spaces. Topics include normal operators, analytic functions of operators, shift operators, invariant subspace lattices, compact operators, invariant and hyperinvariant subspaces, more. 1973 edition.
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Differential Calculus and Its Applications
by Michael J. Field
Part of the Dover Books on Mathematics series
Based on undergraduate courses in advanced calculus, the treatment covers a wide range of topics, from soft functional analysis and finite-dimensional linear algebra to differential equations on submanifolds of Euclidean space. 1976 edition.
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Levels of Infinity
Selected Writings On Mathematics And Philosophy
by Hermann Weyl
Part of the Dover Books on Mathematics series
Original anthology features less-technical essays discussing logic, topology, abstract algebra, relativity theory, and the works of David Hilbert. Most have been long unavailable or previously unpublished in book form. 2012 edition.
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Integral, Measure and Derivative
A Unified Approach
by G. E. Shilov
Part of the Dover Books on Mathematics series
This graduate-level textbook and monograph defines the functions of a real variable through consistent use of the Daniell scheme, offering a rare and useful alternative to customary approaches. The treatment can be understood by any reader with a solid background in advanced calculus, and it features many problems with hints and answers. "The exposition is fresh and sophisticated," declared Sci-Tech Book News, "and will engage the interest of accomplished mathematicians." Part one is devoted to the integral, moving from the Reimann integral and step functions to a general theory, and obtaining the "classical" Lebesgue integral in n space. Part two constructs the Lebesgue-Stieltjes integral through the Daniell scheme using the Reimann-Stieltjes integral as the elementary integral. Part three develops theory of measure with the general Daniell scheme, and the final part is devoted to the theory of the derivative.
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Foundations of Geometry
Euclidean, Bolyai-Lobachevskian, and Projective Geometry
by Karol Borsuk
Part of the Dover Books on Mathematics series
In Part One of this comprehensive and frequently cited treatment, the authors develop Euclidean and Bolyai-Lobachevskian geometry on the basis of an axiom system due, in principle, to the work of David Hilbert. Part Two develops projective geometry in much the same way. An Introduction provides background on topological space, analytic geometry, and other relevant topics, and rigorous proofs appear throughout the text. Topics covered by Part One include axioms of incidence and order, axioms of congruence, the axiom of continuity, models of absolute geometry, and Euclidean geometry, culminating in the treatment of Bolyai-Lobachevskian geometry. Part Two examines axioms of incidents and order and the axiom of continuity, concluding with an exploration of models of projective geometry.
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Real Analysis
by Edward James McShane
Part of the Dover Books on Mathematics series
This text offers upper-level undergraduates and graduate students a survey of practical elements of real function theory, general topology, and functional analysis. Beginning with a brief discussion of proof and definition by mathematical induction, it freely uses these notions and techniques. The maximality principle is introduced early but used sparingly; an appendix provides a more thorough treatment. The notion of convergence is stated in basic form and presented initially in a general setting. The Lebesgue-Stieltjes integral is introduced in terms of the ideas of Daniell, measure-theoretic considerations playing only a secondary part. The final chapter, on function spaces and harmonic analysis, is deliberately accelerated. Helpful exercises appear throughout the text.
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Interpolation
by J. F. Steffensen
Part of the Dover Books on Mathematics series
In the mathematical subfield of numerical analysis, interpolation is a procedure that assists in "reading between the lines." Topics include displacement symbols and differences, divided differences, formulas of interpolation, much more. 1950 edition.
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Approximate Calculation of Integrals
by V. I. Krylov
Part of the Dover Books on Mathematics series
This introduction to approximate integration approaches its subject from the viewpoint of functional analysis. The 3-part treatment covers concepts and theorems from the theory of quadrature, calculation of definite integrals, and calculation of indefinite integrals. 1962 edition.
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Introduction to Minimax
by V. F. Dem'yanov
Part of the Dover Books on Mathematics series
This user-friendly text offers a thorough introduction to the part of optimization theory that lies between approximation theory and mathematical programming, both linear and nonlinear. Written by two distinguished mathematicians, the expert treatment covers the essentials, incorporating important background materials, examples, and extensive notes. Geared toward advanced undergraduate and graduate students of mathematical programming, the text explores best approximation by algebraic polynomials in both discrete and continuous cases; the discrete problem, with and without constraints; the generalized problem of nonlinear programming; and the continuous minimax problem. Several appendixes discuss algebraic interpolation, convex sets and functions, and other topics. 1974 edition.
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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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Nonlinear Functional Analysis
by Klaus Deimling
Part of the Dover Books on Mathematics series
This volume offers a survey of the main ideas, concepts, and methods that constitute nonlinear functional analysis. It offers extensive commentary and many examples in addition to an abundance of interesting, challenging exercises. Starting with coverage of the development of the Brower degree and its applications, the text proceeds to examinations of degree mappings for infinite dimensional spaces and surveys of monotone and accretive mappings. Subsequent chapters explore the inverse function theory, the implicit function theory, and Newton's methods as well as fixed-point theory, solutions to cones, and the Galerkin method of studying nonlinear equations. The final chapters address extremal problems-including convexity, Lagrange multipliers, and mini-max theorems-and offer an introduction into bifurcation theory. Suitable for graduate-level mathematics courses, this volume also serves as a reference for professionals.
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Special Functions & Their Applications
by N. N. Lebedev
Part of the Dover Books on Mathematics series
Richard Silverman's new translation makes available to English readers the work of the famous contemporary Russian mathematician N. N. Lebedev. Though extensive treatises on special functions are available, these do not serve the student or the applied mathematician as well as Lebedev's introductory and practically oriented approach. His systematic treatment of the basic theory of the more important special functions and the applications of this theory to specific problems of physics and engineering results in a practical course in the use of special functions for the student and for those concerned with actual mathematical applications or uses. In consideration of the practical nature of the coverage, most space has been devoted to the application of cylinder functions and particularly of spherical harmonics. Lebedev, however, also treats in some detail: the gamma function, the probability integral and related functions, the exponential integral and related functions, orthogonal polynomials with consideration of Legendre, Hermite and Laguerre polynomials (with exceptional treatment of the technique of expanding functions in series of Hermite and Laguerre polynomials), the Airy functions, the hypergeometric functions (making this often slighted area accessible to the theoretical physicist), and parabolic cylinder functions. The arrangement of the material in the separate chapters, to a certain degree, makes the different parts of the book independent of each other. Although a familiarity with complex variable theory is needed, a serious attempt has been made to keep to a minimum the required background in this area. Various useful properties of the special functions which do not appear in the text proper will be found in the problems at the end of the appropriate chapters. This edition closely adheres to the revised Russian edition (Moscow, 1965). Richard Silverman, however, has made the book even more useful to the English reader. The bibliography and references have been slanted toward books available in English or the West European languages, and a number of additional problems have been added to this edition.
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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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Semigroups of Linear Operators and Applications
by Jerome A. Goldstein
Part of the Dover Books on Mathematics series
This advanced monograph of semigroup theory explores semigroups of linear operators and linear Cauchy problems. Suitable for graduate students in mathematics as well as professionals in science and engineering, the treatment begins with an introductory survey of the theory and applications of semigroups of operators. Two main sections follow, one dedicated to semigroups of linear operators, and the other to linear Cauchy problems. Author Jerome A. Goldstein emphasizes motivation and heuristics as well as applications. Each of the two sections concludes with further applications and historical notes. Challenging exercises appear throughout the text, which includes a substantial bibliography. This edition has been updated with supplementary transcripts of five lectures given by the author during a 1989 workshop at Blaubeuren, Germany.
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Infinite Sequences and Series
by Konrad Knopp
Part of the Dover Books on Mathematics series
One of the finest expositors in the field of modern mathematics, Dr. Konrad Knopp here concentrates on a topic that is of particular interest to 20th-century mathematicians and students. He develops the theory of infinite sequences and series from its beginnings to a point where the reader will be in a position to investigate more advanced stages on his own. The foundations of the theory are therefore presented with special care, while the developmental aspects are limited by the scope and purpose of the book. All definitions are clearly stated; all theorems are proved with enough detail to make them readily comprehensible. The author begins with the construction of the system of real and complex numbers, covering such fundamental concepts as sets of numbers and functions of real and complex variables. In the treatment of sequences and series that follows, he covers arbitrary and null sequences; sequences and sets of numbers; convergence and divergence; Cauchy's limit theorem; main tests for sequences; and infinite series. Chapter three deals with main tests for infinite series and operating with convergent series. Chapters four and five explain power series and the development of the theory of convergence, while chapter six treats expansion of the elementary functions. The book concludes with a discussion of numerical and closed evaluation of series.
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The Geometry of Geodesics
by Herbert Busemann
Part of the Dover Books on Mathematics series
A comprehensive approach to qualitative problems in intrinsic differential geometry, this text for upper-level undergraduates and graduate students emphasizes cases in which geodesics possess only local uniqueness properties--and consequently, the relations to the foundations of geometry are decidedly less relevant, and Finsler spaces become the principal subject. This direct approach has produced many new results and has materially generalized many known phenomena. Author Herbert Busemann begins with an explanation of the basic concepts, including compact metric spaces, convergence of point sets, motion and isometry, segments, and geodesics. Subsequent topics include Desarguesian spaces, with discussions of Riemann and Finsler spaces and Beltrami's theorem; perpendiculars and parallels, with examinations of higher-dimensional Minkowskian geometry and the Minkowski plane; and covering spaces, including locally isometric space, the universal covering space, fundamental sets, free homotopy and closed geodesics, and transitive geodesics on surfaces of higher genus. Concluding chapters explore the influence of the sign of the curvature on the geodesics, and homogenous spaces, including those with flat bisectors.
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The General Theory of Dirichlet's Series
by G. H. Hardy
Part of the Dover Books on Mathematics series
This classic work explains the theory and formulas behind Dirichlet's series and offers the first systematic account of Riesz's theory of the summation of series by typical means. Its authors rank among the most distinguished mathematicians of the twentieth century: G. H. Hardy is famous for his achievements in number theory and mathematical analysis, and Marcel Riesz's interests ranged from functional analysis to partial differential equations, mathematical physics, number theory, and algebra. Following an introduction, the authors proceed to a discussion of the elementary theory of the convergence of Dirichlet's series, followed by a look at the formula for the sum of the coefficients of a Dirichlet's series in terms of the order of the function represented by the series. They continue with an examination of the summation of series by typical means and of general arithmetic theorems concerning typical means. After a survey of Abelian and Tauberian theorems and of further developments of the theory of functions represented by Dirichlet's series, the text concludes with an exploration of the multiplication of Dirichlet's series.
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Individual Choice Behavior
A Theoretical Analysis
by R. Duncan Luce
Part of the Dover Books on Mathematics series
This influential treatise presents upper-level undergraduates and graduate students with a mathematical analysis of choice behavior. It begins with the statement of a general axiom upon which the rest of the book rests; the following three chapters, which may be read independently of each other, are devoted to applications of the theory to substantive problems: psychophysics, utility, and learning. Applications to psychophysics include considerations of time- and space-order effects, the Fechnerian assumption, the power law and its relation to discrimination data, interaction of continua, discriminal processes, signal detectability theory, and ranking of stimuli. The next major theme, utility theory, features unusual results that suggest an experiment to test the theory. The final chapters explore learning-related topics, analyzing the stochastic theories of learning as the basic approach-with the exception that distributions of response strengths are assumed to be transformed rather than response probabilities. The author arrives at three classes of learning operators, both linear and nonlinear, and the text concludes with a useful series of appendixes.
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Infinite Series
by James M Hyslop
Part of the Dover Books on Mathematics series
Intended for advanced undergraduates and graduate students, this concise text focuses on the convergence of real series. Definitions of the terms and summaries of those results in analysis that are of special importance in the theory of series are specified at the outset. In the interests of maintaining a succinct presentation, discussion of the question of the upper and lower limits of a function is confined to an outline of those properties with a direct bearing on the convergence of series. The central subject of this text is the convergence of real series, but series with complex terms and real infinite products are also examined as illustrations of the main theme. Infinite integrals appear only in connection with the integral test for convergence. Topics include functions and limits, real sequences and series, series of non-negative terms, general series, series of functions, the multiplication of series, infinite products, and double series. Prerequisites include a familiarity with the principles of elementary analysis.
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Tensor Analysis on Manifolds
by Richard L. Bishop
Part of the Dover Books on Mathematics series
Striking just the right balance between formal and abstract approaches, this text proceeds from generalities to specifics. Topics include function-theoretical and algebraic aspects, manifolds and integration theory, several important structures, and adaptation to classical mechanics. 1980 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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Theory of Functions, Parts I and II
by Konrad Knopp
Part of the Dover Books on Mathematics series
Two volumes of a classic 5-volume work in one handy edition. Part I considers general foundations of the theory of functions; Part II stresses special functions and characteristic, important types of functions, selected from single-valued and multiple-valued classes. Demonstrations are full and proofs given in detail. Introduction. Bibliographies.
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Introductory Numerical Analysis
by Anthony J. Pettofrezzo
Part of the Dover Books on Mathematics series
Geared toward undergraduate mathematics majors, engineering students, and future high school mathematics teachers, this text offers an understanding of the principles involved in numerical analysis. Its main theme is interpolation from the standpoint of finite differences, least squares theory, and harmonic analysis. Additional considerations include the numerical solutions of ordinary differential equations and approximations through Fourier series. Discussions of the relationships between the calculus of finite differences and the calculus of infinitesimals will prove especially important to future teachers of mathematics. More than seventy worked-out illustrative examples are featured; some include solutions by different methods, showing the relative merits of a variety of approaches. Over 280 multipart exercises range from drill problems to those requiring some degree of ingenuity on the part of the student. Answers are provided to problems with numerical solutions. The only prerequisites are a grasp of differential and integral calculus and some familiarity with determinants. An appendix containing definitions and several theorems from elementary determinant theory is included.
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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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Tensors, Differential Forms, and Variational Principles
by David Lovelock
Part of the Dover Books on Mathematics series
Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms, interaction between the concept of invariance and the calculus of variations. Emphasis is on analytical techniques, with large number of problems, from routine manipulative exercises to technically difficult assignments.
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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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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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Operational Calculus in Two Variables and Its Applications
by V. A. Ditkin
Part of the Dover Books on Mathematics series
A concise monograph by two Russian experts provides an account of the operational calculus in two variables based on the two-dimensional Laplace transform. Suitable for advanced undergraduates and graduate students in mathematics, the treatment requires some familiarity with operational calculus in one variable. Part One of the two-part approach presents the fundamental theory in two chapters, examining the two-dimensional Laplace transform and offering basic definitions and theorems of the operational calculus in two variables and its applications. Part Two presents tables of formulae for various categories of functions, including rational and irrational functions; exponential and logarithmic functions; cylinder, integral, and confluent hypergeometric functions; and other areas.
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Algebraic Equations
An Introduction to the Theories of Lagrange and Galois
by Edgar Dehn
Part of the Dover Books on Mathematics series
Meticulous and complete, this presentation of Galois' theory of algebraic equations is geared toward upper-level undergraduate and graduate students. The theories of both Lagrange and Galois are developed in logical rather than historical form. And they are given a more thorough exposition than is customary. For this reason, and also because the author concentrates on concrete applications of algebraic theory, Algebraic Equations is an excellent supplementary text, offering students a concrete introduction to the abstract principles of Galois theory. Of further value are the many numerical examples throughout the book, which appear with complete solutions. A third of the text focuses on the basic ideas of algebraic theory, giving detailed explanations of integral functions, permutations, and groups. in addition to a very clear exposition of the symmetric group and its functions. A study of the theory of Lagrange follows. Using Lagrange's solvent as a basis for the solution of the general quadratic, cubic, and biquadratic equations. After a discussion of various groups (including isomorphic, transitive, and Abelian groups), a detailed study of Galois theory covers the properties of the Galoisian function, resolvent. and group, the general equation, reductions of the group, natural irrationality. and other features. The book concludes with the application of Galoisian theory to the solution of such special equations as Abelian, cyclic, metacyclic, and quintic equations.
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The Schwarz Lemma
by Sean Dineen
Part of the Dover Books on Mathematics series
The Schwarz lemma is among the simplest results in complex analysis that capture the rigidity of holomorphic functions. This self-contained volume provides a thorough overview of the subject; it assumes no knowledge of intrinsic metrics and aims for the main results, introducing notation, secondary concepts, and techniques as necessary. Suitable for advanced undergraduates and graduate students of mathematics, the two-part treatment covers basic theory and applications. Starting with an exploration of the subject in terms of holomorphic and subharmonic functions, the treatment proves a Schwarz lemma for plurisubharmonic functions and discusses the basic properties of the Poincaré distance and the Schwarz-Pick systems of pseudodistances. Additional topics include hyperbolic manifolds, special domains, pseudometrics defined using the (complex) Green function, holomorphic curvature, and the algebraic metric of Harris. The second part explores fixed point theorems and the analytic Radon-Nikodym property.
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The Absolute Differential Calculus (Calculus of Tensors)
by Tullio Levi-Civita
Part of the Dover Books on Mathematics series
Written by a towering figure of twentieth-century mathematics, this classic examines the mathematical background necessary for a grasp of relativity theory. Tullio Levi-Civita provides a thorough treatment of the introductory theories that form the basis for discussions of fundamental quadratic forms and absolute differential calculus, and he further explores physical applications. Part one opens with considerations of functional determinants and matrices, advancing to systems of total differential equations, linear partial differential equations, algebraic foundations, and a geometrical introduction to theory. The second part addresses covariant differentiation, curvature-related Riemann's symbols and properties, differential quadratic forms of classes zero and one, and intrinsic geometry. The final section focuses on physical applications, covering gravitational equations and general relativity.