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Based on courses given at Eoetvoes Lorand University (Hungary) over the past 30 years, this introductory textbook develops the central concepts of the analysis of functions of one variable - systematically, with many examples and illustrations, and in a manner that builds upon, and sharpens, the student's mathematical intuition.

The famous problems of squaring the circle, doubling the cube and trisecting an angle captured the imagination of both professional and amateur mathematicians for over two thousand years. Despite the enormous effort and ingenious attempts by these men and women, the problems would not yield to purely geometrical methods. It was only the development. of abstract algebra in the nineteenth century which enabled mathematicians to arrive at the surprising conclusion that these constructions are not possible. In this book we develop enough abstract algebra to prove that these constructions are impossible. Our approach introduces all the relevant concepts about fields in a way which is more concrete than usual and which avoids the use of quotient structures (and even of the Euclidean algorithm for finding the greatest common divisor of two polynomials). Having the geometrical questions as a specific goal provides motivation for the introduction of the algebraic concepts and we have found that students respond very favourably. We have used this text to teach second-year students at La Trobe University over a period of many years, each time refining the material in the light of student performance.

This much-anticipated textbook illuminates the field of discrete mathematics with examples, theory, and applications of the discrete volume of a polytope. It weaves a unifying thread through basic yet deep ideas in discrete geometry, combinatorics, and number theory.

Unlike other texts in differential geometry, this book develops the architecture necessary to introduce symplectic and contact geometry alongside its Riemannian cousin. Emphasizes the consequences of a definition and the use of examples and constructions.

Payoffs include: concise picture proofs of the Monotone and Dominated Convergence Theorems, a one-line/one-picture proof of Fubini's theorem from Cavalieri's Principle, and, in many cases, the ability to see an integral result from measure theory.

This text plugs a gap in the standard curriculum by linking set theory with analysis. It features a distinctive, detailed treatment of the real numbers system, and combines an introduction to set theory with exposition of the essence of analysis.

This book offers a collection of historical essays detailing a large variety of mathematical disciplines and issues; it's accessible to a broad audience. This third edition includes new chapters on simple groups and new sections on alternating groups and the Poincare conjecture. Many more exercises have been added as well as commentary that helps place the exercises in context.

This book develops the theory of multivariable analysis, building on the single variable foundations established in the companion volume, Real Analysis: Foundations and Functions of One Variable.

This textbook is designed for a one year course covering the fundamentals of partial differential equations, aimed towards advanced undergraduates and beginning graduate students. It features examples and exercises connected to real world problems.

The authors' novel approach to some interesting mathematical concepts - not normally taught in other courses - places them in a historical and philosophical setting.

The third edition of this concise, popular textbook on elementary differential equations gives instructors an alternative to the many voluminous texts on the market.

Outlines an elementary, one semester course, which exposes students to both the process of rigor, and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. This book focuses on questions which give analysis its inherent fascination.

Covers various basic topics in calculus of several variables, including vectors, curves, functions of several variables, gradient, tangent plane, maxima and minima, potential functions, curve integrals, Green's theorem, multiple integrals, surface integrals, Stokes' theorem, and the inverse mapping theorem and its consequences.

"This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory.

This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics. It includes exercises and examples at the end of each section.

This is a short text in linear algebra, intended for a one-term course. He then starts with a discussion of linear equations, matrices and Gaussian elimination, and proceeds to discuss vector spaces, linear maps, scalar products, determinants, and eigenvalues.

Written to accompany a one- or two-semester course, this text combines rigor and wit to cover a plethora of topics from integers to uncountable sets. It teaches methods such as axiom, theorem, and proof through the mathematics rather than in abstract isolation.

The text was written with four pedagogical goals in mind: offer a variety of topics in one course, get to the main themes and tools as efficiently as possible, show the relationships between the different topics, and include recent results to convince students that mathematics is a living discipline.

Widely used graphics clarify both concrete and abstract concepts, helping students visualize the proofs of many results. Freely accessible solutions to every-other-odd exercise are posted to the book's Springer website.

Written by one of the foremost experts in the field, Algebraic Combinatorics is a unique undergraduate textbook that will prepare the next generation of pure and applied mathematicians. The combination of the author¿s extensive knowledge of combinatorics and classical and practical tools from algebra will inspire motivated students to delve deeply into the fascinating interplay between algebra and combinatorics. Readers will be able to apply their newfound understanding to mathematical, engineering, and business models. Prerequisites include a basic knowledge of linear algebra over a field, existence of finite fields, and rudiments of group theory. The topics in each chapter build on one another and include extensive problem sets as well as hints to selected exercises. Key topics include walks on graphs, cubes and the Radon transform, the Matrix-Tree Theorem, de Bruijn sequences, the Erd¿s¿Moser conjecture, electrical networks, the Sperner property, shellability of simplicialcomplexes and face rings. There are also three appendices on purely enumerative aspects of combinatorics related to the chapter material: the RSK algorithm, plane partitions, and the enumeration of labeled trees.The new edition contains a bit more content than intended for a one-semester advanced undergraduate course in algebraic combinatorics, enumerative combinatorics, or graph theory. Instructors may pick and choose chapters/sections for course inclusion and students can immerse themselves in exploring additional gems once the course has ended. A chapter on combinatorial commutative algebra (Chapter 12) is the heart of added material in this new edition. The author gives substantial application without requisites needed for algebraic topology and homological algebra. A sprinkling of additional exercises and a new section (13.8) involving commutative algebra, have been added.From reviews of the first edition:¿This gentle book provides the perfect stepping-stone up. The various chapters treat diverse topics ¿ . Stanley¿s emphasis on ¿gems¿ unites all this ¿he chooses his material to excite students and draw them into further study. ¿ Summing Up: Highly recommended. Upper-division undergraduates and above.¿ ¿D. V. Feldman, Choice, Vol. 51(8), April, 2014

Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface. In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book.

Every mathematician agrees that every mathematician must know some set theory; The purpose of the book is to tell the beginning student of advanced mathematics the basic set theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism.

This textbook invites the reader to develop a holistic grounding in mathematical finance, where concepts and intuition play as important a role as powerful mathematical tools.

The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).The book may serve as a first or second course in undergraduate abstract algebra and with some supplementation perhaps, for beginning graduate level courses in algebraic geometry or computational algebra.

The fourth edition adds material related to mathematical finance as well as expansions on stable laws and martingales.From the reviews: "Almost thirty years after its first edition, this charming book continues to be an excellent text for teaching and for self study."

An introduction to complex analysis for students with some knowledge of complex numbers from high school. Topics studied include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces, with emphasis placed on the three geometries: spherical, euclidean, and hyperbolic.

The unifying approach of functional analysis is to view functions as points in abstract vector space and the differential and integral operators as linear transformations on these spaces.