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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Regular Polytopes
by H. S. M. Coxeter
Part of the Dover Books on Mathematics series
Foremost book available on polytopes, incorporating ancient Greek and most modern work done on them. Beginning with polygons and polyhedrons, the book moves on to multi-dimensional polytopes in a way that anyone with a basic knowledge of geometry and trigonometry can easily understand. Definitions of symbols. Eight tables plus many diagrams and examples. 1963 edition.
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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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Pascal's Arithmetical Triangle
The Story of a Mathematical Idea
by A.W.F. Edwards
Part of the Dover Books on Mathematics series
This survey explores the history of the arithmetical triangle, from its roots in Pythagorean arithmetic, Hindu combinatorics, and Arabic algebra to its influence on Newton and Leibniz as well as modern-day mathematicians.
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Worked Examples in Mathematics for Scientists and Engineers
by G. Stephenson
Part of the Dover Books on Mathematics series
This rich collection of fully worked problems in many areas of mathematics covers all the important subjects students are likely to encounter in their courses, from introductory to final-year undergraduate classes. Because lecture courses tend to focus on theory rather than examples, these exercises offer a valuable complement to classroom teachings, promoting the understanding of mathematical techniques and helping students prepare for exams. They will prove useful to undergraduates in mathematics; students in engineering, physics, and chemistry; and postgraduate scientists looking for a way to refresh their skills in specific topics.
The problems can supplement lecture notes and any conventional text. Starting with functions, inequalities, limits, differentiation, and integration, topics encompass integral inequalities, power series and convergence, complex variables, hyperbolic function, vector and matrix algebra, Laplace transforms, Fourier series, vector calculus, and many other subjects.
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Statistical Independence in Probability, Analysis and Number Theory
by Mark Kac
Part of the Dover Books on Mathematics series
This concise monograph in probability by Mark Kac, a well-known mathematician, presumes a familiarity with Lebesgue's theory of measure and integration, the elementary theory of Fourier integrals, and the rudiments of number theory. Readers may then follow Dr. Kac's attempt "to rescue statistical independence from the fate of abstract oblivion by showing how in its simplest form it arises in various contexts cutting across different mathematical disciplines." The treatment begins with an examination of a formula of Vieta that extends to the notion of statistical independence. Subsequent chapters explore laws of large numbers and Émile Borel's concept of normal numbers; the normal law, as expressed by Abraham de Moivre and Andrey Markov's method; and number theoretic functions as well as the normal law in number theory. The final chapter ranges in scope from kinetic theory to continued fractions. All five chapters are enhanced by problems.
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Modern Multidimensional Calculus
by Marshall Evans Munroe
Part of the Dover Books on Mathematics series
A second-year calculus text, this volume is devoted primarily to topics in multidimensional analysis. Concepts and methods are emphasized, and rigorous proofs are sometimes replaced by relevant discussion and explanation. Because of the author's conviction that the differential provides a most elegant and useful tool, especially in a multidimensional setting, the notion of the differential is used extensively and matrix methods are stressed in the study of linear transformations.
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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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Nonlinear Potential Theory of Degenerate Elliptic Equations
by Juha Heinonen
Part of the Dover Books on Mathematics series
A self-contained treatment appropriate for advanced undergraduate and graduate students, this volume offers a detailed development of the necessary background for its survey of the nonlinear potential theory of superharmonic functions.Starting with the theory of weighted Sobolev spaces, the text advances to the theory of weighted variational capacity. Succeeding chapters investigate solutions and supersolutions of equations, with emphasis on refined Sobolev spaces, variational integrals, and harmonic functions. Chapter 7 defines superharmonic functions via the comparison principle, and chapters 8 through 14 form the core of the nonlinear potential theory of superharmonic functions. Topics include balayage; Perron's method, barriers, and resolutivity; polar sets; harmonic measure; fine topology; harmonic morphisms; and quasiregular mappings. The book concludes with explorations of axiomatic nonlinear potential theory and helpful appendixes.
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Theory and Application of Liapunov's Direct Method
by Wolfgang Hahn
Part of the Dover Books on Mathematics series
The groundbreaking work of Russian mathematician A. M. Liapunov (1857–1918) on the stability of dynamical systems was overlooked for decades because of political turmoil. During the Cold War, when it was discovered that his method was applicable to the stability of aerospace guidance systems, interest in his research was rekindled. It has remained high ever since.
This monograph on both the theory and applications of Liapunov's direct method reflects the work of a period when the theory had been studied seriously for some time and reached a degree of completeness and sophistication. It remains of interest to applied mathematicians in many areas. Topics include applications of the stability theorems to concrete problems, the converse of the main theorems, Liapunov functions with certain properties of rate of change, the sensitivity of the stability behavior to perturbations, the critical cases, and generalizations of the concept of stability.
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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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Analytic Theory of Continued Fractions
by Hubert Stanley Wall
Part of the Dover Books on Mathematics series
One of the most authoritative and comprehensive books on the subject of continued fractions, this monograph has been widely used by generations of mathematicians and their students. Dr. Hubert Stanley Wall presents a unified theory correlating certain parts and applications of the subject within a larger analytic structure. Prerequisites include a first course in function theory and knowledge of the elementary properties of linear transformations in the complex plane. Some background in number theory, real analysis, and complex analysis may also prove helpful. The two-part treatment begins with an exploration of convergence theory, addressing continued fractions as products of linear fractional transformations, convergence theorems, and the theory of positive definite continued fractions, as well as other topics. The second part, focusing on function theory, covers the theory of equations, matrix theory of continued fractions, bounded analytic functions, and many additional subjects.
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Almost Periodic Functions
by Harald Bohr
Part of the Dover Books on Mathematics series
Mathematician Harald Bohr, motivated by questions about which functions could be represented by a Dirichlet series, devised the theory of almost periodic functions during the 1920s. His groundbreaking work influenced many later mathematicians, who extended the theory in new and diverse directions. In this volume, Bohr focuses on an essential aspect of the theory - the functions of a real variable - in full detail and with complete proofs.The treatment, which is based on Bohr's lectures, starts with an introduction that leads to discussions of purely periodic functions and their Fourier series. The heart of the book, his exploration of the theory of almost periodic functions, is supplemented by two appendixes that cover generalizations of almost periodic functions and almost periodic functions of a complete variable.
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Harmonic Analysis on Homogeneous Spaces
by Nolan R. Wallach
Part of the Dover Books on Mathematics series
This book is suitable for advanced undergraduate and graduate students in mathematics with a strong background in linear algebra and advanced calculus. Early chapters develop representation theory of compact Lie groups with applications to topology, geometry, and analysis, including the Peter-Weyl theorem, the theorem of the highest weight, the character theory, invariant differential operators on homogeneous vector bundles, and Bott's index theorem for such operators. Later chapters study the structure of representation theory and analysis of non-compact semi-simple Lie groups, including the principal series, intertwining operators, asymptotics of matrix coefficients, and an important special case of the Plancherel theorem. Teachers will find this volume useful as either a main text or a supplement to standard one-year courses in Lie groups and Lie algebras. The treatment advances from fairly simple topics to more complex subjects, and exercises appear at the end of each chapter. Eight helpful Appendixes develop aspects of differential geometry, Lie theory, and functional analysis employed in the main text.
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Numerical Hamiltonian Problems
by J.M. Sanz-Serna
Part of the Dover Books on Mathematics series
This advanced text explores a category of mathematical problems that occur frequently in physics and other sciences. Five preliminary chapters make the book accessible to students without extensive background in this area. Topics include Hamiltonian systems, symplecticness, numerical methods, order conditions, and implementation. The heart of the book, chapters 6 through 10, explores simplistic integration, simplistic order conditions, available simplistic methods, numerical experiments, and properties of simplistic integrators. The final four chapters contain material that is more advanced: generating functions, Lie formalism, high-order methods, and extensions. Many numerical examples appear throughout the text.
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Finite-Dimensional Vector Spaces
by Paul R. Halmos
Part of the Dover Books on Mathematics series
A fine example of a great mathematician's intellect and mathematical style, this classic on linear algebra is widely cited in the literature. The treatment is an ideal supplement to many traditional linear algebra texts and is accessible to undergraduates with some background in algebra. The theory is systematically developed by the axiomatic method that has, since von Neumann, dominated the general approach to linear functional analysis and that achieves here a high degree of lucidity and clarity. The presentation is never awkward or dry, as it sometimes is in other 'modern' textbooks; it is as unconventional as one has come to expect from the author. The book contains about 350 well-placed and instructive problems, which cover a considerable part of the subject. All in all this is an excellent work, of equally high value for both student and teacher." - Zentralblatt für Mathematik.
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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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Matrix Theory and Applications for Scientists and Engineers
by Alexander Graham
Part of the Dover Books on Mathematics series
A comprehensive text on matrix theory and its applications, this book is intended for a broad range of students in mathematics, engineering, and other areas of science at the university level. Author Alexander Graham avoids a simple catalogue of techniques by exploring the concepts' underlying principles as well as their numerous applications. Many problems elucidate the text, which includes a substantial answer section at the end. The treatment explores matrices, vector spaces, linear transformations, and the rank and determinant of a matrix. Additional topics include linear equations, eigenvectors and eigenvalues, canonical forms and matrix functions, and inverting a matrix. A Solution to Problems Section, References and a Bibliography conclude the treatment.
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Lectures on the Calculus of Variations
by Oskar Bolza
Part of the Dover Books on Mathematics series
This pioneering modern treatise on the calculus of variations studies the evolution of the subject from Euler to Hilbert. The text addresses basic problems with sufficient generality and rigor to offer a sound introduction for serious study. It provides clear definitions of the fundamental concepts, sharp formulations of the problems, and rigorous demonstrations of their solutions. Many examples are solved completely, and systematic references are given for each theorem upon its first appearance. Initial chapters address the first and second variation of the integral, and succeeding chapters cover the sufficient conditions for an extremum of the integral and Weierstrass's theory of the problem in parameter-representation; Kneser's extension of Weierstrass's theory to cover the case of variable end-points; and Weierstrass's theory of the isoperimetric problems. The final chapter presents a thorough proof of Hilbert's existence theorem.
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Theory of Groups of Finite Order
by W. Burnside
Part of the Dover Books on Mathematics series
After introducing permutation notation and defining group, the author discusses the simpler properties of group that are independent of their modes of representation; composition-series of groups; isomorphism of a group with itself; Abelian groups; groups whose orders are the powers of primes; Sylow's theorem; more. 18 illustrations. A classic introduction.
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A Vector Space Approach to Geometry
by Melvin Hausner
Part of the Dover Books on Mathematics series
A fascinating exploration of the correlation between geometry and linear algebra, this text portrays the former as a subject better understood by the use and development of the latter rather than as an independent field. The treatment offers elementary explanations of the role of geometry in other branches of math and science - including physics, analysis, and group theory - as well as its value in understanding probability, determinant theory, and function spaces. Outstanding features of this volume include discussions of systematic geometric motivations in vector space theory and matrix theory; the use of the center of mass in geometry, with an introduction to barycentric coordinates; axiomatic development of determinants in a chapter dealing with area and volume; and a careful consideration of the particle problem. Students and other mathematically inclined readers will find that this inquiry into the interplay between geometry and other areas offers an enriched appreciation of both subjects.
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Topological Transformation Groups
by Deane Montgomery
Part of the Dover Books on Mathematics series
An advanced monograph on the subject of topological transformation groups, this volume summarizes important research conducted during a period of lively activity in this area of mathematics. The book is of particular note because it represents the culmination of research by authors Deane Montgomery and Leo Zippin, undertaken in collaboration with Andrew Gleason of Harvard University, that led to their solution of a well-known mathematical conjecture, Hilbert's Fifth Problem. The treatment begins with an examination of topological spaces and groups and proceeds to locally compact groups and groups with no small subgroups. Subsequent chapters address approximation by Lie groups and transformation groups, concluding with an exploration of compact transformation groups.
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On Riemann's Theory of Algebraic Functions and Their Integrals
A Supplement to the Usual Treatises
by Felix Klein
Part of the Dover Books on Mathematics series
A great researcher, writer, and teacher in an era of tremendous mathematical ferment, Felix Klein (1849–1925) occupies a prominent place in the history of mathematics. His many talents included an ability to express complicated mathematical ideas directly and comprehensively, and this book, a consideration of the investigations in the first part of Riemann's Theory of Abelian Functions, is a prime example of his expository powers. The treatment introduces Riemann's approach to multiple-value functions and the geometrical representation of these functions by what later became known as Riemann surfaces. It further concentrates on the kinds of functions that can be defined on these surfaces, confining the treatment to rational functions and their integrals. The text then demonstrates how Riemann's mathematical ideas about Abelian integrals can be arrived at by thinking in terms of the flow of electric current on surfaces. Klein's primary concern is preserving the sequence of thought and offering intuitive explanations of Riemann's notions, rather than furnishing detailed proofs. Deeply significant in the area of complex functions, this work constitutes one of the best introductions to the origins of topological problems.
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An Introduction to Algebraic Structures
by Joseph Landin
Part of the Dover Books on Mathematics series
As the author notes in the preface, "The purpose of this book is to acquaint a broad spectrum of students with what is today known as 'abstract algebra.'" Written for a one-semester course, this self-contained text includes numerous examples designed to base the definitions and theorems on experience, to illustrate the theory with concrete examples in familiar contexts, and to give the student extensive computational practice. The first three chapters progress in a relatively leisurely fashion and include abundant detail to make them as comprehensible as possible. Chapter One provides a short course in sets and numbers for students lacking those prerequisites, rendering the book largely self-contained. While Chapters Four and Five are more challenging, they are well within the reach of the serious student. The exercises have been carefully chosen for maximum usefulness. Some are formal and manipulative, illustrating the theory and helping to develop computational skills. Others constitute an integral part of the theory, by asking the student to supply proofs or parts of proofs omitted from the text. Still others stretch mathematical imaginations by calling for both conjectures and proofs. Taken together, text and exercises comprise an excellent introduction to the power and elegance of abstract algebra. Now available in this inexpensive edition, the book is accessible to a wide range of students, who will find it an exceptionally valuable resource.
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The Theory of Groups
by Marshall Hall
Part of the Dover Books on Mathematics series
This encyclopedic treatment of the current knowledge of group theory was widely praised upon its 1959 publication for its readability and accessibility. Today this volume remains useful as an unsurpassed resource for learning and reviewing the basics of a fundamental and ever-expanding area of modern mathematics. Suitable for advanced undergraduate mathematics majors and graduate students in math, the treatment is largely self-contained and offers numerous helpful exercises.
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Analytic Functions
by M.A. Evgrafov
Part of the Dover Books on Mathematics series
This highly regarded text is directed toward advanced undergraduates and graduate students in mathematics who are interested in developing a firm foundation in the theory of functions of a complex variable. The treatment departs from traditional presentations in its early development of a rigorous discussion of the theory of multiple-valued analytic functions on the basis of analytic continuation. Thus it offers an early introduction of Riemann surfaces, conformal mapping, and the applications of residue theory. M. A. Evgrafov focuses on aspects of the theory that relate to modern research and assumes an acquaintance with the basics of mathematical analysis derived from a year of advanced calculus.
Starting with an introductory chapter containing the fundamental results concerning limits, continuity, and integrals, the book addresses analytic functions and their properties, multiple-valued analytic functions, singular points and expansion in series, the Laplace transform, harmonic and subharmonic functions, extremal problems and distribution of values, and other subjects. Chapters are largely self-contained, making this volume equally suitable for the classroom or independent study.
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The Umbral Calculus
by Steven Roman
Part of the Dover Books on Mathematics series
Geared toward upper-level undergraduates and graduate students, this elementary introduction to classical umbral calculus requires only an acquaintance with the basic notions of algebra and a bit of applied mathematics (such as differential equations) to help put the theory in mathematical perspective.
The text focuses on classical umbral calculus, which dates back to the 1850s and continues to receive the attention of modern mathematicians. Subjects include Sheffer sequences and operators and their adjoints, with numerous examples of associated and other sequences. Related topics encompass the connection constants problem and duplication formulas, the Lagrange inversion formula, operational formulas, inverse relations, and binomial convolution. The final chapter offers a glimpse of the newer and less well-established forms of umbral calculus.
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Lectures on the Mathematical Method in Analytical Economics
by Jacob T. Schwartz
Part of the Dover Books on Mathematics series
An early but still useful and frequently cited contribution to the science of mathematical economics, this volume is geared toward graduate students in the field. Prerequisites include familiarity with the basic theory of matrices and linear transformations and with elementary calculus. Author Jacob T. Schwartz begins his treatment with an exploration of the Leontief input-output model, which forms a general framework for subsequent material. An introductory treatment of price theory in the Leontief model is followed by an examination of the business-cycle theory, following ideas pioneered by Lloyd Metzler and John Maynard Keynes. In the final section, Schwartz applies the teachings of previous chapters to a critique of the general equilibrium approach devised by Léon Walras as the theory of supply and demand, and he synthesizes the notions of Walras and Keynes.
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Infinite Abelian Groups
by Irving Kaplansky
Part of the Dover Books on Mathematics series
In the Introduction to this concise monograph, the author states his two main goals: first, "to make the theory of infinite abelian groups available in a convenient form to the mathematical public; second, to help students acquire some of the techniques used in modern infinite algebra." Suitable for advanced undergraduates and graduate students in mathematics, the text requires no extensive background beyond the rudiments of group theory. Starting with examples of abelian groups, the treatment explores torsion groups, Zorn's lemma, divisible groups, pure subgroups, groups of bounded order, and direct sums of cyclic groups. Subsequent chapters examine Ulm's theorem, modules and linear transformations, Banach spaces, valuation rings, torsion-free and complete modules, algebraic compactness, characteristic submodules, and the ring of endomorphisms. Many exercises appear throughout the book, along with a guide to the literature and a detailed bibliography.
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Numerical Methods for Two-Point Boundary-Value Problems
by Herbert B. Keller
Part of the Dover Books on Mathematics series
Elementary yet rigorous, this concise treatment explores practical numerical methods for solving very general two-point boundary-value problems. The approach is directed toward students with a knowledge of advanced calculus and basic numerical analysis as well as some background in ordinary differential equations and linear algebra. After an introductory chapter that covers some of the basic prerequisites, the text studies three techniques in detail: initial value or "shooting" methods, finite difference methods, and integral equations methods. Sturm-Liouville eigenvalue problems are treated with all three techniques, and shooting is applied to generalized or nonlinear eigenvalue problems. Several other areas of numerical analysis are introduced throughout the study. The treatment concludes with more than 100 problems that augment and clarify the text, and several research papers appear in the Appendixes.
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Math Through the Ages
A Gentle History for Teachers and Others
by William P. Berlinghoff
Part of the Dover Books on Mathematics series
Designed for students just beginning their study of the discipline, this concise introductory history of mathematics is supplemented by brief but in-depth sketches of the more important individual topics. Covering such subjects as algebra symbols, negative numbers, the metric system, quadratic equations, and much more, this widely adopted work invites and encourages further study of mathematics.
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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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Induction in Geometry
by L.I. Golovina
Part of the Dover Books on Mathematics series
Induction in Geometry discusses the application of the method of mathematical induction to the solution of geometric problems, some of which are quite intricate. The book contains 37 examples with detailed solutions and 40 for which only brief hints are provided. Most of the material requires only a background in high school algebra and plane geometry; chapter six assumes some knowledge of solid geometry, and the text occasionally employs formulas from trigonometry. Chapters are self-contained, so readers may omit those for which they are unprepared.
To provide additional background, this volume incorporates the concise text, The Method of Mathematical Induction. This approach introduces this technique of mathematical proof via many examples from algebra, geometry, and trigonometry, and in greater detail than standard texts. A background in high school algebra will largely suffice; later problems require some knowledge of trigonometry. The combination of solved problems within the text and those left for readers to work on, with solutions provided at the end, makes this volume especially practical for independent study.
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Lapses in Mathematical Reasoning
by V. M. Bradis
Part of the Dover Books on Mathematics series
Designed as a methodDesigned as a method for teaching correct mathematical thinking to high school students, this book contains a brilliantly constructed series of what the authors call "lapses," erroneous statements that are part of a larger mathematical argument. These lapses lead to sophism or mathematical absurdities. The ingenious idea behind this technique is to lead the student deliberately toward a clearly false conclusion. The teacher and student then go back and analyze the lapse as a way to correct the problem. The authors begin by focusing on exercises in refuting erroneous mathematical arguments and their classification. The remaining chapters discuss examples of false arguments in arithmetic, algebra, geometry, trigonometry, and approximate computations. Ideally, students will come to the correct insights and conclusions on their own; however, each argument is followed by a detailed analysis of the false reasoning. Stimulating and unique, this book is an intriguing and enjoyable way to teach students critical mathematical reasoning skills.
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Rings of Continuous Functions
by Leonard Gillman
Part of the Dover Books on Mathematics series
Designed as a text as well as a treatise for active mathematicians, this volume begins with an unusual notice: "The book is addressed to those who know the meaning of each word in the title." As such, it constituted the first systematic account of the theory of rings of continuous functions, and it has retained its secure position as the basic graduate-level book in this area. The authors focus on characterizing the maximal ideals and classifying their residue class fields. Problems concerning extending continuous functions from a subspace to the entire space play a fundamental role in the study, and these problems are discussed in extensive detail. A thorough treatment of the Stone-Čech compactification is supplemented with a number of related topics: the Ulam measure problem, the theory of uniform spaces, and a small but significant portion of dimension theory. Hundreds of problems of varying difficulty appear throughout the text, providing additional details, describing counterexamples, and outlining new topics.
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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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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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Lectures on Measure and Integration
by Harold Widom
Part of the Dover Books on Mathematics series
These well-known and concise lecture notes present the fundamentals of the Lebesgue theory of integration and an introduction to some of the theory's applications. Suitable for advanced undergraduates and graduate students of mathematics, the treatment also covers topics of interest to practicing analysts. Author Harold Widom emphasizes the construction and properties of measures in general and Lebesgue measure in particular as well as the definition of the integral and its main properties. The notes contain chapters on the Lebesgue spaces and their duals, differentiation of measures in Euclidean space, and the application of integration theory to Fourier series.
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Invitation to Geometry
by Z. A. Melzak
Part of the Dover Books on Mathematics series
Intended for students of many different backgrounds with only a modest knowledge of mathematics, this text features self-contained chapters that can be adapted to several types of geometry courses. Only a slight acquaintance with mathematics beyond the high-school level is necessary, including some familiarity with calculus and linear algebra. This text's introductions to several branches of geometry feature topics and treatments based on memorability and relevance. The author emphasizes connections with calculus and simple mechanics, focusing on developing students' grasp of spatial relationships. Subjects include classical Euclidean material, polygonal and circle isoperimetry, conics and Pascal's theorem, geometrical optimization, geometry and trigonometry on a sphere, graphs, convexity, and elements of differential geometry of curves. Additional material may be conveniently introduced in several places, and each chapter concludes with exercises of varying degrees of difficulty.
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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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Introduction to Partial Differential Equations and Hilbert Space Methods
by Karl E. Gustafson
Part of the Dover Books on Mathematics series
Easy-to-use text examines principal method of solving partial differential equations, 1st-order systems, computation methods, and much more. Over 600 exercises, with answers for many. Ideal for a 1-semester or full-year course.
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Abstract Sets and Finite Ordinals
An Introduction to the Study of Set Theory
by G. B. Keene
Part of the Dover Books on Mathematics series
This text unites the logical and philosophical aspects of set theory in a manner intelligible both to mathematicians without training in formal logic and to logicians without a mathematical background. It combines an elementary level of treatment with the highest possible degree of logical rigor and precision. Starting with an explanation of all the basic logical terms and related operations, the text progresses through a stage-by-stage elaboration that proves the fundamental theorems of finite sets. It focuses on the Bernays theory of finite classes and finite sets, exploring the system's basis and development, including Stage I and Stage II theorems, the theory of finite ordinals, and the theory of finite classes and finite sets. This volume represents an excellent text for undergraduates studying intermediate or advanced logic as well as a fine reference for professional mathematicians.
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The Elements of Non-Euclidean Geometry
by D. M.Y. Sommerville
Part of the Dover Books on Mathematics series
This volume became the standard text in the field almost immediately upon its original publication. Renowned for its lucid yet meticulous exposition, it can be appreciated by anyone familiar with high school algebra and geometry. Its arrangement follows the traditional pattern of plane and solid geometry, in which theorems are deduced from axioms and postulates. In this manner, students can follow the development of non-Euclidean geometry in strictly logical order, from a fundamental analysis of the concept of parallelism to such advanced topics as inversion and transformations. Topics include elementary hyperbolic geometry; elliptic geometry; analytic non-Euclidean geometry; representations of non-Euclidean geometry in Euclidean space; and space curvature and the philosophical implications of non-Euclidean geometry. Additional subjects encompass the theory of the radical axes, homothetic centers, and systems of circles; inversion, equations of transformation, and groups of motions; and the classification of conics. Although geared toward undergraduate students, this text treats such important and difficult topics as the relation between parataxy and parallelism, the absolute measure, the pseudosphere, Gauss' proof of the defect-area theorem, geodesic representation, and other advanced subjects. In addition, its 136 problems offer practice in using the forms and methods developed in the text.
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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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Basic Concepts in Modern Mathematics
by John Edward Hafstrom
Part of the Dover Books on Mathematics series
An in-depth survey of some of the most readily applicable essentials of modern mathematics, this concise volume is geared toward undergraduates of all backgrounds as well as future math majors. By focusing on relatively few fundamental concepts, the text delves deeply enough into each subject to challenge students and to offer practical applications. The opening chapter introduces the program of study and discusses how numbers developed. Subsequent chapters explore the natural numbers; sets, variables, and statement forms; mappings and operations; groups; relations and partitions; integers; and rational and real numbers. Prerequisites include high school courses in elementary algebra and plane geometry.
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Numerical Methods
by Germund Dahlquist
Part of the Dover Books on Mathematics series
Practical text strikes fine balance between students' requirements for theoretical treatment and needs of practitioners, with best methods for large- and small-scale computing. Prerequisites are minimal (calculus, linear algebra, and preferably some acquaintance with computer programming). Text includes many worked examples, problems, and an extensive bibliography.
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Naive Set Theory
by Paul R. Halmos
Part of the Dover Books on Mathematics series
This classic by one of the twentieth century's most prominent mathematicians offers a concise introduction to set theory. Suitable for advanced undergraduates and graduate students in mathematics, it employs the language and notation of informal mathematics. There are very few displayed theorems; most of the facts are stated in simple terms, followed by a sketch of the proof. Only a few exercises are designated as such since the book itself is an ongoing series of exercises with hints. The treatment covers the basic concepts of set theory, cardinal numbers, transfinite methods, and a good deal more in 25 brief chapters.