These topics lie at the heart of many exciting areas of contemporary science and engineering - communication, signal processing, data mining, machine learning, pattern recognition, computational neuroscience, bioinformatics, and cryptography. This textbook introduces theory in tandem with applications. Information theory is taught alongside practical communication systems, such as arithmetic coding for data compression and sparse-graph codes for error-correction. A toolbox of inference techniques, including message-passing algorithms, Monte Carlo methods, and variational approximations, are developed alongside applications of these tools to clustering, convolutional codes, independent component analysis, and neural networks. The final part of the book describes the state of the art in error-correcting codes, including low-density parity-check codes, turbo codes, and digital fountain codes -- the twenty-first century standards for satellite communications, disk drives, and data broadcast. Richly illustrated, filled with worked examples and over 400 exercises, some with detailed solutions, David MacKay's groundbreaking book is ideal for self-learning and for undergraduate or graduate courses. Interludes on crosswords, evolution, and sex provide entertainment along the way. In sum, this is a textbook on information, communication, and coding for a new generation of students, and an unparalleled entry point into these subjects for professionals in areas as diverse as computational biology, financial engineering, and machine learning.

    This is the currently used textbook for "Probabilistic Systems Analysis," an introductory probability course at the Massachusetts Institute of Technology, attended by a large number of undergraduate and graduate students. The book covers the fundamentals of probability theory (probabilistic models, discrete and continuous random variables, multiple random variables, and limit theorems), which are typically part of a first course on the subject. It also contains, a number of more advanced topics, from which an instructor can choose to match the goals of a particular course. These topics include transforms, sums of random variables, least squares estimation, the bivariate normal distribution, and a fairly detailed introduction to Bernoulli, Poisson, and Markov processes. The book strikes a balance between simplicity in exposition and sophistication in analytical reasoning. Some of the more mathematically rigorous analysis has been just intuitively explained in the text, but is developed in detail (at the level of advanced calculus) in the numerous solved theoretical problems. The book has been widely adopted for classroom use in introductory probability courses within the USA and abroad.

    

    The authors have grouped problems so as to illustrate and highlight a number of important techniques and have provided enlightening solutions in all cases. As well as this there are essays on topics that are not only beautiful but also useful. The essays are diverse and enlivened by fresh, non-standard ideas. This book not only teaches techniques but gives a flavour of their past, present and possible future implications. It is a collection of miniature mathematical works in the fullest sense.

    This book offers a full range of exercises, a precise and conceptual presentation, and a new media package designed specifically to meet the needs of today's readers. The exercises gradually increase in difficulty, helping readers learn to generalize and apply the concepts. The refined table of contents introduces the exponential, logarithmic, and trigonometric functions in Chapter 7 of the text.KEY TOPICS Functions, Limits and Continuity, Differentiation, Applications of Derivatives, Integration, Applications of Definite Integrals, Integrals and Transcendental Functions, Techniques of Integration, Further Applications of Integration, Conic Sections and Polar Coordinates, Infinite Sequences and Series, Vectors and the Geometry of Space, Vector-Valued Functions and Motion in Space, Partial Derivatives, Multiple Integrals, Integration in Vector Fields.MARKET For all readers interested in Calculus.

    This volume contains detailed solutions, sometimes multiple solutions, for all the problems, and some solutions offer additional twists for further thought . . . "--CHOICE102 Combinatorial Problems consists of carefully selected problems that have been used in the training and testing of the USA International Mathematical Olympiad (IMO) team. The text provides in-depth enrichment in the important areas of combinatorics by systematically reorganizing and enhancing problem-solving tactics and strategies. The book gradually builds combinatorial skills and techniques and not only broadens the student's view of mathematics, but is also excellent for training teachers.

    Bertrand Russell (1872-1970) Most natural optimization problems, including those arising in important application areas, are NP-hard. Therefore, under the widely believed con jecture that P -=/= NP, their exact solution is prohibitively time consuming. Charting the landscape of approximability of these problems, via polynomial time algorithms, therefore becomes a compelling subject of scientific inquiry in computer science and mathematics. This book presents the theory of ap proximation algorithms as it stands today. It is reasonable to expect the picture to change with time. This book is divided into three parts. In Part I we cover combinato rial algorithms for a number of important problems, using a wide variety of algorithm design techniques. The latter may give Part I a non-cohesive appearance. However, this is to be expected - nature is very rich, and we cannot expect a few tricks to help solve the diverse collection of NP-hard problems. Indeed, in this part, we have purposely refrained from tightly cat egorizing algorithmic techniques so as not to trivialize matters. Instead, we have attempted to capture, as accurately as possible, the individual character of each problem, and point out connections between problems and algorithms for solving them."

    In this book, the authors have gathered together a collection of problems from various topics in number theory that they find beautiful, intriguing, and from a certain point of view instructive.

    Pure Mathematics 1 corresponds to unit P1. It covers quadratics, functions, coordinate geometry, circular measure, trigonometry, vectors, series, differentiation and integration.

    van Heijenoort's internationally acclaimed From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (HUP 1967), makes available in English the two most important works in the growth of modern mathematical logic. Heralded by Leibniz, modern logic had its beginnings in the work of Boole, DeMorgan, and Jevons, but the 1879 publication of Gottlob Frege's Begriffsschrift opened a great epoch in the history of logic with the full-form presentation of the propositional calculus and quantification theory.Frege and Gödel: Two Fundamental Texts in Mathematical Logic begins with this short book, which ushered in the classical age of mathematical logic by outlining the construction of a system of logical symbolism. The volume concludes with Gödel's famous incompleteness paper of 1931, which changed the development of logic and the foundations of mathematics by revealing the intrinsic limitations of formal systems, and brought to an end the classical phase.Mr. van Heijenoort has provided a new introduction which sets the Frege and Gödel pieces in perspective in the development of modern logic and points out difficulties in interpretation. Editorial comments, footnotes, and bibliographic information offer additional explanatory material.

    Provides all the propositions and diagrams without the detailed proofs. Readers can use it to see the scope and structure of Elements, identify exactly what Euclid covers and what he doesn't, and to find the location of remembered propositions.

    

    An extensive list of exercises, ranging in difficulty from "routine" to "worthy of independent publication", is included. In each section, there are also exercises that contain material not explicitly discussed in the text before, so as to provide instructors with extra choices if they want to shift the emphasis of their course.It goes without saying that the text covers the classic areas, i.e. combinatorial choice problems and graph theory. What is unusual, for an undergraduate textbook, is that the author has included a number of more elaborate concepts, such as Ramsey theory, the probabilistic method and -- probably the first of its kind -- pattern avoidance. While the reader can only skim the surface of these areas, the author believes that they are interesting enough to catch the attention of some students. As the goal of the book is to encourage students to learn more combinatorics, every effort has been made to provide them with a not only useful, but also enjoyable and engaging reading.

    This guide outlines basic algebraic equations, formulas, properties & operations."

    In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician. In this new introduction to undergraduate real analysis the author takes a different approach from past studies of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians like Dieudonne, Littlewood and Osserman. The author has taught the subject many times over the last 35 years at Berkeley and this book is based on the honours version of this course. The book contains an excellent selection of more than 500 exercises.

    Pure Mathematics 2 corresponds to units P2 and P3. It covers algebra, logarithmic and exponential functions, trigonometry, differentiation, integration, numerical solution of equations, vectors, differential equations and complex numbers.

    The authors highlight connections to other problems, to the curriculum and to more advanced topics. The best problems contain kernels of sophisticated ideas related to important current research, and yet the problems are accessible to undergraduates. The solutions have been compiled from the American Mathematical Monthly, Mathematics Magazine and past competitors. Multiple solutions enhance the understanding of the audience, explaining techniques that have relevance to more than the problem at hand. In addition, the book contains suggestions for further reading, a hint to each problem, separate from the full solution and background information about the competition. The book will appeal to students, teachers, professors and indeed anyone interested in problem solving as a gateway to a deep understanding of mathematics.

    For the first time, this book uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically and physically common phenomena and advancing to a precise specification of the nature of Categories of Sets. Set theory as the algebra of mappings is introduced and developed as a unifying basis for advanced mathematical subjects such as algebra, geometry, analysis, and combinatorics. The formal study evolves from general axioms that express universal properties of sums, products, mapping sets, and natural number recursion.

    This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.

    The author's purpose is to provide a standard reference to all work that has been published since the mid-1970s and to link this work together in a single conceptual framework and a single notational formalism. A working knowledge of basic general relativity is assumed. The book will be essential reading for researchers and postgraduate students in mathematics, theoretical physics, and astronomy interested in cosmic strings.

    

    Alan Beardon covers the ideas of complex numbers, scalar and vector products, determinants, linear algebra, group theory, permutation groups, symmetry groups, and various aspects of geometry including groups of isometries, rotations, and spherical geometry. The emphasis is on the interaction among these topics. The text is divided into short sections, with exercises at the end of each section.

    These methods form a broad, coherent and powerful kernel in combinatorial optimization, with strong links to discrete mathematics, mathematical programming and computer science. In eight parts, various areas are treated, each starting with an elementary introduction to the area, with short, elegant proofs of the principal results, and each evolving to the more advanced methods and results, with full proofs of some of the deepest theorems in the area. Over 4000 references to further research are given, and historical surveys on the basic subjects are presented.

    The book develops the main concepts of auction theory from scratch in a self-contained and theoretically rigorous manner. It explores auctions and competitive bidding as games of incomplete information through detailed examinations of themes central to auction theory.This book complements its superb presentation of auction theory with clear and concise proofs of all results on bidding strategies, efficiency, and revenue maximization. It provides discussions on auction-related subjects, including private value auctions; the Revenue Equivalence Principle; auctions with interdependent values; the Revenue Ranking (Linkage) Principle; mechanism design with interdependent values; bidding rings; multiple object auctions; equilibrium and efficiency with private values; and nonidentical objects.This book is essential reading for graduate students taking courses on auction theory, the economics of information, or the economics of incentives, as well as for any serious student of auctions. It will also appeal to professional economists or business analysts working in contract theory, experimental economics, industrial organization, and microeconomic theory.

    Building on the success of the original, this edition includes new material on random signal processing, a new chapter on spectral estimation, greatly expanded coverage of filter banks and wavelets, and new material on the solution of difference equations. Additional steps in mathematical derivations make them easier to follow, and an important new feature is the do-it-yourself section at the end of each chapter, where readers get hands-on experience of solving practical signal processing problems in a range of MATLAB experiments. With 120 worked examples, 20 case studies, and almost 400 homework exercises, the book is essential reading for anyone taking DSP courses. Its unique blend of theory and real-world practical examples also makes it an ideal reference for practitioners.

    Understanding the principles on which it is based is an important topic that requires a knowledge of both computational complexity and a range of topics in pure mathematics. This book provides that knowledge, combining an informal style with strong proofs of the key results to provide an accessible introduction. It includes many examples and exercises, and is based on a highly successful course developed and taught over many years.

    It will serve as an introduction to welcome and guide newcomers to this exciting field, and an indispensible reference for seasoned practitioners. The book starts by introducing the key features of string theory and the theoretical tools needed to get to grips with D-branes. It then builds up the subject in a logical way, discussing further aspects of string theory and advanced applications as the text progresses.

    

    It is based on numerous courses on combinatorial optimization and specialized topics, mostly at graduate level. This book reviews the fundamentals, covers the classical topics (paths, flows, matching, matroids, NP-completeness, approximation algorithms) in detail, and proceeds to advanced and recent topics, some of which have not appeared in a textbook before. Throughout, it contains complete but concise proofs, and also provides numerousexercises and references.This fifth edition has again been updated, revised, and significantlyextended, with more than 60 new exercises and new material on varioustopics, including Cayley's formula, blocking flows, faster"b"-matching separation, multidimensional knapsack, multicommoditymax-flow min-cut ratio, and sparsest cut. Thus, this book represents the state of the art of combinatorial optimization.

    Fractal geometry also appears in art, music and literature, most often without being consciously included by the artist. Consequently, through this we may uncover connections between the arts and sciences, uncommon for students to see in maths and science classes. This book will appeal to teachers who have wanted to include fractals in their mathematics and science classes, to scientists familiar with fractal geometry who want to teach a course on fractals, and to anyone who thinks general scientific literacy is an issue important enough to warrant new approaches.

    Recent research in child development has emphasized the positive effects of early learning. An early exposure to number concepts will enhance your child's cognitive capacity to learn mathematical concepts. The purpose of this book is to assist you as a parent in teaching your child fundamental number concepts at an early age. Given the importance of early learning in cognitive development, Marshmallow Math begins with number concepts that young children can master before they learn to read and write numbers. The book follows a natural progression of skills that starts with simple counting. Addition, subtraction, multiplication, division, and other mathematical concepts are introduced gradually with each new skill building upon mastered skills. The book also explores other important math concepts including sorting and comparing, telling time, spatial awareness, pattern recognition, geometry, measurement, and reasoning. Marshmallow Math provides many quick, simple, and fun activities for you and your child to do together. The activities are appropriate for children ranging in age from toddler through to primary school. Hands-on learning and mental math are emphasized over written work and traditional exercises. Many of the activities described in the book involve the use of counting objects such as marshmallows, pennies, or jellybeans. Having physical objects to look at, pick-up, and count will help to make abstract concepts more concrete for your child. The unique approach set out in Marshmallow Math will help to ensure that your child truly comprehends fundamental number concepts and masters basic math skills. This will give your child both the ability and the confidence to excel in math.

    For each of them, the authors provide discussion of the main conceptual points, an exposition of many practical calculations of physical qualities and a comparison of these quantitive predictions with experimental results.

    Like its predecessor, it was created with an audience of students working toward a non-science related degree in mind. A non-intimidating, easy-to-understand textbook companion, this new edition has more explanatory graphs and illustrations and double the number of practice problems.First edition of this book has sold more copies than any of the other 70+ books on the subjsect.Twice as many practice problems in the second edition.More college students are now required to take calculus in college than ever before.Author is an award-winning calculus teacher praised for his ability to make this topic fun and approachable.His website, calculus-help.com, reaches thousands of students every month.

    This class-tested 2007 introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.

    It is intended toget the readers interested in the subject and also convey some basic concepts.Many bright students from age 12 upwards are bored with routine class work inMaths and will enjoy these puzzles which will sharpen their logical reasoning.Thus, the subject dreaded by many can indeed be very interesting, absorbing andenjoyable, at least as much as chess or bridge. Fun And Fundamentals Of Mathematics is designed to arousean interest in mathematics amongst the readers, thereby serving as asupplementary text for the 12-18 age group.

    One of the most remarkable interactions between geometry and physics since 1980 has been an application of quantum field theory to topology and differential geometry. This book focuses on a relationship between two-dimensional quantum field theory and three-dimensional topology which has been studied intensively since the discovery of the Jones polynomial in the middle of the 1980s and Witten's invariant for 3-manifolds derived from Chern-Simons gauge theory.

    In fact, they sit at the confluence of analysis, algebra, geometry, and number theory. The first edition of this volume was respected, both as a textbook and as a source for results, ideas, and references. It helped to spark a growing interest in the mathematical community to bring it back into print. In this second edition, Iwaniec treats the spectral theory of automorphic forms as the study of the space $L (H\Gamma)$, where $H$ is the upper half-plane and $\Gamma$ is a discrete subgroup of volume-preserving transformations of $H$. He combines various techniques from analytic number theory. Among the topics discussed are Eisenstein series, estimates for Fourier coefficients of automorphic forms, the theory of Kloosterman sums, the Selberg trace formula, and the theory of small eigenvalues."

    It covers almost all known estimates. The emphasis is on distribution-free properties of the estimates.

    Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own leisurely treatment.

    Such an epoch was, in the history of the Arabs, that of the Caliphs AL mansur, harun AL rashid, and AL mamun, the illustrious contemporaries of charlemagne; to the glory of which era, in the volume now offered to the public, a new monument is endeavoured to be raised.This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

    The first chapter describes the historical development of the classical theory, and the second chapter includes a modern treatment of Runge-Kutta and extrapolation methods. Chapter three begins with the classical theory of multistep methods, and concludes with the theory of general linear methods. The reader will benefit from many illustrations, a historical and didactic approach, and computer programs which help him/her learn to solve all kinds of ordinary differential equations. This new edition has been rewritten and new material has been included.

    Without sacrificing mathematical precision, Anton and Busby focus on the aspects of linear algebra that are most likely to have practical value to the student while not compromising the intrinsic mathematical form of the subject. Throughout Contemporary Linear Algebra , students are encouraged to look at ideas and problems from multiple points of view.

    The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularization (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the fundamental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.

    A central theme of this book is that many instances of pattern formation can be understood within a single framework: symmetry. This book applies symmetry methods to increasingly complex kinds of dynamic behavior: equilibria, period-doubling, time-periodic states, homoclinic and heteroclinic orbits, and chaos. Examples are drawn from both ODEs and PDEs. In each case the type of dynamical behavior being studied is motivated through applications, drawn from a wide variety of scientific disciplines ranging from theoretical physics to evolutionary biology.

    This is especially - but not uniquely- true of Chapters 3, 4, and 6, which may be regarded as advanced versions of the corresponding chapters in Mathematical Reflections. Like its predecessor, the present work consists of nine chapters, each devoted to a lively mathematical topic, and each capable, in principle, of being read independently of the other chapters.' Thus this is not a text which- as is the intention of most standard treatments of mathematical topics - builds systematically on certain common themes as one proceeds 1Mathematical Reflections - In a Room with Many Mirrors, Springer Undergraduate Texts in Math- ematics, 1996; Second Printing 1998. We will refer to this simply as MR. 2There was an exception in MR; Chapter 9 was concerned with our thoughts on the doing and teaching of mathematics at the undergraduate level.

    It is based around a number of difficult old problems that live at the interface of analysis and number theory. Some of these problems are the following: The Integer Chebyshev Problem. Find a nonzero polynomial of degree n with integer eoeffieients that has smallest possible supremum norm on the unit interval. Littlewood's Problem. Find a polynomial of degree n with eoeffieients in the set { + 1, -I} that has smallest possible supremum norm on the unit disko The Prouhet-Tarry-Escott Problem. Find a polynomial with integer co- effieients that is divisible by (z - l)n and has smallest possible 1 norm. (That 1 is, the sum of the absolute values of the eoeffieients is minimal.) Lehmer's Problem. Show that any monie polynomial p, p(O) i- 0, with in- teger coefficients that is irreducible and that is not a cyclotomic polynomial has Mahler measure at least 1.1762 .... All of the above problems are at least forty years old; all are presumably very hard, certainly none are completely solved; and alllend themselves to extensive computational explorations. The techniques for tackling these problems are various and include proba- bilistic methods, combinatorial methods, "the circle method," and Diophantine and analytic techniques. Computationally, the main tool is the LLL algorithm for finding small vectors in a lattice. The book is intended as an introduction to a diverse collection of techniques.

    This carefully written book can be read by any student who knows some topology, providing a useful method to quickly learn this novel homotopy-theoretic point of view of algebraic topology.

    It describes relations with standard cohomology theory and provides complete proofs. Coverage also presents basic concepts and results of homotopical algebra. This second edition contains numerous corrections.

    It contains a fascinating collection of articles by some of the top names in the field, such as Elwyn Berlekamp and John Conway, plus other researchers in mathematics and computer science, together with some top game players. The articles run the gamut from new theoretical approaches (infinite games, generalizations of game values, 2-player cellular automata, Alpha-Beta pruning under partial orders) to the very latest in some of the hottest games (Amazons, Chomp, Dot-and-Boxes, Go, Chess, Hex). Many of these advances reflect the interplay of the computer science and the mathematics. The book ends with an updated bibliography by A. Fraenkel and an updated and annotated list of combinatorial game theory problems by R. K. Guy.

    These techniques are explained in the context of matrix groups as a principal example.

    Volume I is a suitable text for advanced undergraduates and beginning graduate students. Volume II requires a much higher level of mathematical maturity, with contents ranging from binary quadratic forms to rational number theory. Problems and hints for solutions. 1956 edition. Supplementary Reading. List of Symbols. Index.

    This volume provides an accessible introduction to the basic elements of algebraic codes and discusses their use in a variety of applications. The author describes a range of important coding techniques, including Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. Throughout the book, mathematical theory is illustrated by reference to many practical examples. The book is written for graduate students of electrical and computer engineering and practicing engineers whose work involves communications or signal processing.

    This book deals with the most basic algorithms in the area. Most of them can be viewed as “algorithmic jewels” and deserve reader-friendly presentation. One of the main aims of the book is to present several of the most celebrated algorithms in a simple way by omitting obscuring details and separating algorithmic structure from combinatorial theoretical background. The book reflects the relationships between applications of text-algorithmic techniques and the classification of algorithms according to the measures of complexity considered. The text can be viewed as a parade of algorithms in which the main purpose is to discuss the foundations of the algorithms and their interconnections. One can partition the algorithmic problems discussed into practical and theoretical problems. Certainly, string matching and data compression are in the former class, while most problems related to symmetries and repetitions in texts are in the latter. However, all the problems are interesting from an algorithmic point of view and enable the reader to appreciate the importance of combinatorics on words as a tool in the design of efficient text algorithms.In most textbooks on algorithms and data structures, the presentation of efficient algorithms on words is quite short as compared to issues in graph theory, sorting, searching, and some other areas. At the same time, there are many presentations of interesting algorithms on words accessible only in journals and in a form directed mainly at specialists. This book fills the gap in the book literature on algorithms on words, and brings together the many results presently dispersed in the masses of journal articles. The presentation is reader-friendly; many examples and about two hundred figures illustrate nicely the behaviour of otherwise very complex algorithms.

    It includes a thorough treatment of the local theory using the tools of commutative algebra, an extensive development of sheaf theory and the theory of coherent analytic and algebraic sheaves, proofs of the main vanishing theorems for these categories of sheaves, and a complete proof of the finite dimensionality of the cohomology of coherent sheaves on compact varieties. The vanishing theorems have a wide variety of applications and these are covered in detail.

    Homotopy theory arises from choosing a class of maps, called weak equivalences, and then passing to the homotopy category by localizing with respect to the weak equivalences - by creating a new category in which the weak equivalences are isomorphisms.