From Edward Lorenz’s discovery of the Butterfly Effect, to Mitchell Feigenbaum’s calculation of a universal constant, to Benoit Mandelbrot’s concept of fractals, which created a new geometry of nature, Gleick’s engaging narrative focuses on the key figures whose genius converged to chart an innovative direction for science. In Chaos, Gleick makes the story of chaos theory not only fascinating but also accessible to beginners, and opens our eyes to a surprising new view of the universe.

    Popular Author Martin Gardner's Time Travel and other Mathematical Bewilderments

    Multi-terminal active devices are thoroughly covered. The text provides general background for computer-aided circuit analysis and pertinent treatment of simple nonlinear phenomena.

    This book explains the graphic design of illustrations from Thurston's world of low-dimensional geometry and topology. It presents the principles of linear and aerial perspective from the viewpoint of projective geometry.

    Included are a comprehensive presentation and analysis of algorithms of time-dependent phenomena plus beam, plate, and shell theories derived directly from three-dimensional elasticity theory. Solution guide available upon request.

    The neighbors of a strictly 24 dimensional odd unimodular lattice can be found as follows. If a norm -4 vector v E II . corresponds to the sum 25 1 of a strictly 24 dimensional odd unimodular lattice A and a !-dimensional lattice, then there are exactly two nonn-0 vectors of ll25,1 having inner product -2 with v, and these nann 0 vectors correspond to the two even neighbors of A. The enumeration of the odd 24-dimensional lattices. Figure 17.1 shows the neighborhood graph for the Niemeier lattices, which has a node for each Niemeier lattice. If A and B are neighboring Niemeier lattices, there are three integral lattices containing A n B, namely A, B, and an odd unimodular lattice C (cf. [Kne4]). An edge is drawn between nodes A and B in Fig. 17.1 for each strictly 24-dimensional unimodular lattice arising in this way. Thus there is a one-to-one correspondence between the strictly 24-dimensional odd unimodular lattices and the edges of our neighborhood graph. The 156 lattices are shown in Table 17 .I. Figure I 7. I also shows the corresponding graphs for dimensions 8 and 16.

    Problems that have seemed insuperable in previous approaches take on a totally new character in the context of superstring theory, and some of them have been overcome. Interest in the subject has greatly increased following a succession of exciting recent developments. This two-volume book attempts to meet the need for a systematic exposition of superstring theory and its applications accessible to as wide an audience as possible.

    

    This volume studies all aspects of the theory, starting from first principles and probing the function's own challenging theory, with the famous and still unsolved Riemann hypothesis at its heart. The second edition has been revised to include descriptions of work done in the last forty years and is updated with many additional references; it will provide stimulating reading for postgraduates and workers in analytic number theory and classical analysis.

    The mathematics used is rigorous, but not overwhelming, while the authors carefully model physical situations, emphasizing feedback among a beginning model, physical experiments, mathematical predictions, and the subsequent refinement and reevaluation of the physical model itself. Chapter 1 begins with a discussion of various physical problems and equations that play a central role in applications. The following chapters take up the theory of partial differential equations, including detailed discussions of uniqueness, existence, and continuous dependence questions, as well as techniques for constructing conclusions. Specifically, Chapters 2 through 6 deal with problems in one spatial dimension. Chapter 7 is a detailed introduction to the theory of integral equations; then Chapters 8 through 12 treat problems in more spatial variables. Each chapter begins with a discussion of problems that can be treated by elementary means, such as separation of variables or integral transforms, and which lead to explicit, analytical representations of solutions. The minimal mathematical prerequisites for a good grasp of the material in this book are a course in advanced calculus, or an advanced course in science or engineering, and a basic exposure to matrix methods. Students of mathematics, physics, engineering, and other disciplines will find here an excellent guide to mathematical problem-solving techniques with a broad range of applications. For this edition the authors have provided a new section of Solutions and Hints to selected Problems. Suggestions for further reading complete the text.

    The second part is a comprehensive study of recursively enumerable sets and their degrees.

    Topics include first and second variations of an integral, generalizations, isoperimetrical problems, least action, special relativity, elasticity, more. 1963 edition.

    It contains graded exercise sets, as well as clear explanations of number theory with special emphasis on creating and understanding proofs.

    The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and linear representations. The last chapter discusses the connection between Lie algebras, complex groups and compact groups. The book is intended to guide the reader towards further study.

    These subjects are made accessible in the many concrete examples that illustrate techniques of calculation, and in the treatment of all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appear for the first time in this book.

    Projective geometry is simpler: its constructions require only a ruler. In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity. The first two chapters of this book introduce the important concepts of the subject and provide the logical foundations. The third and fourth chapters introduce the famous theorems of Desargues and Pappus. Chapters 5 and 6 make use of projectivities on a line and plane, repectively. The next three chapters develop a self-contained account of von Staudt's approach to the theory of conics. The modern approach used in that development is exploited in Chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all the theorems nontrivially. The concluding chapters show the connections among projective, Euclidean, and analytic geometry.

    Includes comprehensive coverage of complex numbers, set theory, mapping, functions, Cauchy-Riemann conditions, power series, Taylor series, Green's theorem, Laurent expansions, singularities, residues, transformations, and numerous scientific applications.

    Just as a pair of sunglasses reduces the glare of white light, permitting only the softer green light to pass, so the DFT may be used to modify a signal to achieve a desired effect. In fact, by analyzing the component frequencies of a signal or any system, the DFT can be used in an astonishing variety of problems. Among the applications of the DFT are digital signal processing, oil and gas exploration, medical imaging, aircraft and spacecraft guidance, and the solution of differential equations of physics and engineering. The DFT: An Owner's Manual for the Discrete Fourier Transform explores both the practical and theoretical aspects of the DFT, one of the most widely used tools in science and engineering.

    Topics covered include the meaning and measurement of risk, general single-period portfolio problems, mean-variance analysis and the Capital Asset Pricing Model, the Arbitrage Pricing Theory, complete markets, multiperiod portfolio problems and the Intertemporal Capital Asset Pricing Model, the Black-Scholes option pricing model and contingent claims analysis, 'risk-neutral' pricing with Martingales, Modigliani-Miller and the capital structure of the firm, interest rates and the term structure, and others.

    It includes 70 new articles, particularly in applied mathematics, expanded explanations and appendices, coverage of recent work, and reorganization of older topics.

    This expanded second edition has added thirteen abstracts, a 1988 Scientific American Article, a transcript of a EUROPALIA 89 lecture, an essay on biology, and an extensive bibliography. Its new larger format makes it easier to read. Chaitin's ideas are a fundamental extension of those of Gödel and Turning and have exploded some basic assumptions of mathematics and thrown new light on the scientific method, epistemology, probability theory, and of course computer science and information theory.

    The book emphasizes both the intellectually stimulating parts of geometry and routine arguments or computations in concrete or classical cases, as well as practical and physical applications. The book also teaches the student fundamental concepts and the difference between important reults and minor technical routines. Altogether, the text presents a coherent high school curriculum for the geometry course. There are many examples and exercises.

    The heroes of the book are renowned figures from early eras, as well some scientists of last century. A unique mixture of mathematics, physics, and history, this volume provides biographical glimpses of scientists and their contributions in the context of the social and political background of their times.

    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.

    Reveals mathematics' great power as an alternative language for understanding things and explores such concepts as logic as a computing tool, digital versus analog processes and communication as information transmission.

    The book is directed towards the reader seeking a broad view of the subject rather than elaborate information about technical details.

    

    The roles of symmetry, bifurcation, and Hamiltonian systems in diverse applications are explored.

    This book is the only source for an extended, concentrated focus on the theory and techniques common to various types of intersection graphs. It is a concise treatment of the aspects of intersection graphs that interconnect many standard concepts and form the foundation of a surprising array of applications to biology, computing, psychology, matrices, and statistics. The authors emphasize the underlying tools and techniques and demonstrate how this approach constitutes a definite theory within graph theory. Some of the applications are not widely known or available in the graph theoretic literature and are presented here for the first time. The book also includes a detailed literature guide for many specialized and related areas, a current bibliography, and more than 100 exercises.

    This updated and expanded volume examines gauge theories and their symmetries with an emphasis on their physical and technical aspects. A new introductory chapter gives a systematic overview to classical field theories and a short discussion of their canonical quantization and the discrete symmetries C, P and T. Coverage provides a brief exposition of perturbation theory, the renormalization program, and the use of the renormalization group equation. It then explores topics of current research interest including chiral symmetry and its breaking, anomalies, and low energy effective Lagrangians and some basics of supersymmetry. A chapter on the basics of the electroweak theory is also new to this edition, as well as an appendix that contains a complete set of Feynman rules for the Standard Model. This volume will be ideal for graduate students and researchers in theoretical physics, condensed matter physics, and applied mathematics.

    This exceptional series provides a solid foundation for academic excellence and success in any assessment.

    How many do you need to end up with a complete triangle? Three will do; so will 6, 10, 15, and... These numbers are called triangular numbers. Ask the same question for a triangular pyramid. Four will do; so will 10, 20, 35, and... the so-called pyramidal numbers. This book looks at the discovery of the multiplicity of properties and uses triangular numbers and their many extensions possess.Although often displayed in a triangular array named after the seventeenth-century French philosopher Blaise Pascal, triangular numbers were known many centuries earlier. In this book A.W.F. Edwards traces the Arithmetical Triangle back to its roots in Pythagorean arithmetic, Hindu combinatorics, and Arabic algebra, and gives an account of the progressive solution of combinatorial problems from the earliest recorded examples to the work of Renaissance and later mathematicians. He shows how Pascal's work—so modern in style—in establishing the properties of the numbers and their application in various fields led to Newton's discovery of the binomial theorem for fractional and negative indices and to Leibniz's discovery of calculus.

    Acquaints the specialist in relativity theory with some global techniques for the treatment of space-times and will provide the pure mathematician with a way into the subject of general relativity.

    The questions that receive the most attention are those that deal with discrete optimization problems for points in Euclidean space, such as the minimum spanning tree, the traveling-salesman tour, and minimal-length matchings. Still, there are several nongeometric optimization problems that receive full treatment, and these include the problems of the longest common subsequence and the longest increasing subsequence. The philosophy that guides the exposition is that analysis of concrete problems is the most effective way to explain even the most general methods or abstract principles.

    The topics of the first edition have been enhanced with additional discussion, new numerical experiments, and improved figures. New topics in the second edition include nonlinear equations, Neumann boundary conditions, variable mesh and variable coefficient problems, anisotropic problems, algebraic multigrid (AMG), adaptive methods, and finite elements. This introductory book is ideally suited as a companion textbook for graduate numerical analysis courses. It is written for computational mathematicians, engineers, and other scientists interested in learning about multigrid.

    Ranging from relatively simple to extremely difficult, their topics include points and lines, topology, convex polygons, theory of primes, nondecimal counting, and other subjects. Suitable for students, teachers, and any lover of mathematics. Complete solutions.

    At the present time there is a great revival of interest in these topics not only for their own sake but also because of their applications to so many areas of mathematical research from group theory and number theory to topology and differential equations. In this book the authors give elementary accounts of many aspects of classical complex function theory including M�bius transformations, elliptic functions, Riemann surfaces, Fuchsian groups and modular functions. A distinctive feature of their presentation is the way in which they have incorporated into the text many interesting topics from other branches of mathematics. This book is based on lectures given to advanced undergraduates and is well-suited as a textbook for a second course in complex function theory. Professionals will also find it valuable as a straightforward introduction to a subject which is finding widespread application throughout mathematics.

    Problems that have seemed insuperable in previous approaches take on a totally new character in the context of superstring theory, and some of them have been overcome. Interest in the subject has greatly increased following a succession of exciting recent developments. This two-volume book attempts to meet the need for a systematic exposition of superstring theory and its applications accessible to as wide an audience as possible.

    In the context of Riemannian manifolds, an independent mathematical theory has developed around them. Recently, it has produced several striking results, which have been of great interest also to physicists. This Ergebnisse volume is the first book which presents an up-to-date overview of the state of the art in this field. "Einstein Manifold"s is a successful attempt to organize the abundant literature, with emphasis on examples. Parts of it can be used separately as introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.

    The completely revised Second Edition introduces subspace methods, methods that utilize frequency domain data, and these key non-linear black box methods: neural networks, wavelet transforms, neuro-fuzzy modeling and hinging hyperplanes.KEY TOPICS: Leader in the field Lennart Ljung introduces systems and models, time-invariant linear systems, time-varying and nonlinear systems. He presents several approaches to system identification, including nonparametric time- and frequency-domain methods; parameter estimation; convergence and consistency; asymptotic distribution of parameter estimates; linear regressions, iterative search and recursive estimation. He also presents detailed coverage of key issues that can make or break system identification projects: defining objectives, designing experiments, selecting criteria, and controlling the bias distribution of transfer-function estimates.MARKET: For all engineering and control systems professionals, faculty and students.

    It is no coincidence that the fundamental science of numbers has come to be known as the "Queen of Mathematics." Indeed some of the most complex conventions of the mathematical mind have evolved from the study of basic problems of number theory.Andr� Weil, one of the outstanding contributors to number theory, has written an historical exposition of this subject; his study examines texts that span roughly thirty-six centuries of arithmetical work -- from an Old Babylonian tablet, datable to the time of Hammurapi to Legendre's Essai sur la Th�orie des Nombres (1798). Motivated by a desire to present the substance of his field to the educated reader, Weil employs an historical approach in the analysis of problems and evolving methods of number theory and their significance within mathematics. In the course of his study Weil accompanies the reader into the workshops of four major authors of modern number theory (Fermat, Euler, Lagrange and Legendre) and there he conducts a detailed and critical examination of their work. Enriched by a broad coverage of intellectual history, Number Theory represents a major contribution to the understanding of our cultural heritage.

    Advanced Engineering Mathematics features a greater number of examples and problems and is fine-tuned throughout to improve the clear flow of ideas. The computer plays a more prominent role than ever in generating computer graphics used to display concepts and problem sets, incorporating the use of leading software packages. Computational assistance, exercises and projects have been included to encourage students to make use of these computational tools. The content is organized into eight parts and covers a wide spectrum of topics including Ordinary Differential Equations, Vectors and Linear Algebra, Systems of Differential Equations and Qualitative Methods, Vector Analysis, Fourier Analysis, Orthogonal Expansions, and Wavelets, Partial Differential Equations, Complex Analysis, and Probability and Statistics.