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Logic is sometimes called the foundation of mathematics: the logician studies the kinds of reasoning used in the individual steps of a proof. Alonzo Church was a pioneer in the field of mathematical logic, whose contributions to number theory and the theories of algorithms and computability laid the theoretical foundations of computer science. His first Princeton book, The Calculi of Lambda-Conversion (1941), established an invaluable tool that computer scientists still use today. Even beyond the accomplishment of that book, however, his second Princeton book, Introduction to Mathematical Logic, defined its subject for a generation. Originally published in Princeton's Annals of Mathematics Studies series, this book was revised in 1956 and reprinted a third time, in 1996, in the Princeton Landmarks in Mathematics series. Although new results in mathematical logic have been developed and other textbooks have been published, it remains, sixty years later, a basic source for understanding formal logic. Church was one of the principal founders of the Association for Symbolic Logic; he founded the Journal of Symbolic Logic in 1936 and remained an editor until 1979 At his death in 1995, Church was still regarded as the greatest mathematical logician in the world.

In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes.

This classic book is an introduction to dynamic programming, presented by the scientist who coined the term and developed the theory in its early stages. In Dynamic Programming, Richard E. Bellman introduces his groundbreaking theory and furnishes a new and versatile mathematical tool for the treatment of many complex problems, both within and outside of the discipline. The book is written at a moderate mathematical level, requiring only a basic foundation in mathematics, including calculus. The applications formulated and analyzed in such diverse fields as mathematical economics, logistics, scheduling theory, communication theory, and control processes are as relevant today as they were when Bellman first presented them. A new introduction by Stuart Dreyfus reviews Bellman's later work on dynamic programming and identifies important research areas that have profited from the application of Bellman's theory.

In real-world problems related to finance, business, and management, mathematicians and economists frequently encounter optimization problems. In this classic book, George Dantzig looks at a wealth of examples and develops linear programming methods for their solutions. He begins by introducing the basic theory of linear inequalities and describes the powerful simplex method used to solve them. Treatments of the price concept, the transportation problem, and matrix methods are also given, and key mathematical concepts such as the properties of convex sets and linear vector spaces are covered. George Dantzig is properly acclaimed as the "e;father of linear programming."e; Linear programming is a mathematical technique used to optimize a situation. It can be used to minimize traffic congestion or to maximize the scheduling of airline flights. He formulated its basic theoretical model and discovered its underlying computational algorithm, the "e;simplex method,"e; in a pathbreaking memorandum published by the United States Air Force in early 1948. Linear Programming and Extensions provides an extraordinary account of the subsequent development of his subject, including research in mathematical theory, computation, economic analysis, and applications to industrial problems. Dantzig first achieved success as a statistics graduate student at the University of California, Berkeley. One day he arrived for a class after it had begun, and assumed the two problems on the board were assigned for homework. When he handed in the solutions, he apologized to his professor, Jerzy Neyman, for their being late but explained that he had found the problems harder than usual. About six weeks later, Neyman excitedly told Dantzig, "e;I've just written an introduction to one of your papers. Read it so I can send it out right away for publication."e; Dantzig had no idea what he was talking about. He later learned that the "e;homework"e; problems had in fact been two famous unsolved problems in statistics.

Explores fundamental concepts in arithmetic. This book begins with the definitions and properties of algebraic fields. It then discusses the theory of divisibility from an axiomatic viewpoint, rather than by the use of ideals. It also gives an introduction to p-adic numbers and their uses, which are important in modern number theory.

Einstein's general theory of relativity requires a curved space for the description of the physical world. If one wishes to go beyond superficial discussions of the physical relations involved, one needs to set up precise equations for handling curved space. The well-established mathematical technique that accomplishes this is clearly described in this classic book by Nobel Laureate P.A.M. Dirac. Based on a series of lectures given by Dirac at Florida State University, and intended for the advanced undergraduate, General Theory of Relativity comprises thirty-five compact chapters that take the reader point-by-point through the necessary steps for understanding general relativity.

In his classic work of geometry, Euclid focused on the properties of flat surfaces. In the age of exploration, mapmakers such as Mercator had to concern themselves with the properties of spherical surfaces. The study of curved surfaces, or non-Euclidean geometry, flowered in the late nineteenth century, as mathematicians such as Riemann increasingly questioned Euclid's parallel postulate, and by relaxing this constraint derived a wealth of new results. These seemingly abstract properties found immediate application in physics upon Einstein's introduction of the general theory of relativity. In this book, Eisenhart succinctly surveys the key concepts of Riemannian geometry, addressing mathematicians and theoretical physicists alike.

PCT, Spin and Statistics, and All That is the classic summary of and introduction to the achievements of Axiomatic Quantum Field Theory. This theory gives precise mathematical responses to questions like: What is a quantized field? What are the physically indispensable attributes of a quantized field? Furthermore, Axiomatic Field Theory shows that a number of physically important predictions of quantum field theory are mathematical consequences of the axioms. Here Raymond Streater and Arthur Wightman treat only results that can be rigorously proved, and these are presented in an elegant style that makes them available to a broad range of physics and theoretical mathematics.

Presents an account of two pillars of the techniques of nonlinear dynamics and chaos theory: stable and chaotic behavior. This title discusses cases in which N-body motions are stable, covering topics such as Hamiltonian systems, the (Moser) twist theorem, and aspects of Kolmogorov-Arnold-Moser theory.

In his work on rings of operators in Hilbert space, John von Neumann discovered a new mathematical structure that resembled the lattice system Ln. In characterizing its properties, von Neumann founded the field of continuous geometry. This book, based on von Neumann's lecture notes, begins with the development of the axioms of continuous geometry, dimension theory, and--for the irreducible case--the function D(a). The properties of regular rings are then discussed, and a variety of results are presented for lattices that are continuous geometries, for which irreducibility is not assumed. For students and researchers interested in ring theory or projective geometries, this book is required reading.

Fibre bundles are an integral part of differential geometry. This book begins with an introduction to bundles, including such topics as differentiable manifolds and covering spaces. It then provides brief surveys of advanced topics, such as homotopy theory and cohomology theory, before using them to study further properties of fibre bundles.

Presents a study of network flow problems. This title introduces the models and algorithms that can be used in the fields of transportation systems, manufacturing, inventory planning, image processing, and internet traffic. It is suitable for the researchers working with networks.

This book offers a concise introduction to the angular momentum, one of the most fundamental quantities in all of quantum mechanics. Beginning with the quantization of angular momentum, spin angular momentum, and the orbital angular momentum, the author goes on to discuss the Clebsch-Gordan coefficients for a two-component system. After developing the necessary mathematics, specifically spherical tensors and tensor operators, the author then investigates the 3-j, 6-j, and 9-j symbols. Throughout, the author provides practical applications to atomic, molecular, and nuclear physics. These include partial-wave expansions, the emission and absorption of particles, the proton and electron quadrupole moment, matrix element calculation in practice, and the properties of the symmetrical top molecule.

Provides a clear introduction to one of the important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, this book goes on to examine tangent spaces, oriented manifolds, and vector fields. It discusses concepts such as homotopy, the index number of a map, and the Pontryagin construction.

Available for the first time in paperback, R. Tyrrell Rockafellar's classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar's theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions. This book has firmly established a new and vital area not only for pure mathematics but also for applications to economics and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book. There is also a guide for the reader who may be using the book as an introduction, indicating which parts are essential and which may be skipped on a first reading.

When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied. The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "e;higher order"e; derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "e;functors"e; and of their "e;derived functors."e; This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.

In this renowned volume, Hermann Weyl discusses the symmetric, full linear, orthogonal, and symplectic groups and determines their different invariants and representations. Using basic concepts from algebra, he examines the various properties of the groups. Analysis and topology are used wherever appropriate. The book also covers topics such as matrix algebras, semigroups, commutators, and spinors, which are of great importance in understanding the group-theoretic structure of quantum mechanics. Hermann Weyl was among the greatest mathematicians of the twentieth century. He made fundamental contributions to most branches of mathematics, but he is best remembered as one of the major developers of group theory, a powerful formal method for analyzing abstract and physical systems in which symmetry is present. In The Classical Groups, his most important book, Weyl provided a detailed introduction to the development of group theory, and he did it in a way that motivated and entertained his readers. Departing from most theoretical mathematics books of the time, he introduced historical events and people as well as theorems and proofs. One learned not only about the theory of invariants but also when and where they were originated, and by whom. He once said of his writing, "e;My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful."e; Weyl believed in the overall unity of mathematics and that it should be integrated into other fields. He had serious interest in modern physics, especially quantum mechanics, a field to which The Classical Groups has proved important, as it has to quantum chemistry and other fields. Among the five books Weyl published with Princeton, Algebraic Theory of Numbers inaugurated the Annals of Mathematics Studies book series, a crucial and enduring foundation of Princeton's mathematics list and the most distinguished book series in mathematics.

In this classic of statistical mathematical theory, Harald Cramer joins the two major lines of development in the field: while British and American statisticians were developing the science of statistical inference, French and Russian probabilitists transformed the classical calculus of probability into a rigorous and pure mathematical theory. The result of Cramer's work is a masterly exposition of the mathematical methods of modern statistics that set the standard that others have since sought to follow. For anyone with a working knowledge of undergraduate mathematics the book is self contained. The first part is an introduction to the fundamental concept of a distribution and of integration with respect to a distribution. The second part contains the general theory of random variables and probability distributions while the third is devoted to the theory of sampling, statistical estimation, and tests of significance.