Peter D. Lax – författare

This revised edition of a classic book, which established scattering theory as an important and fruitful area of research, reflects the wealth of new results discovered in the intervening years. This new, revised edition should continue to inspire researchers to expand the application of the original ideas proposed by the authors.

Includes sections on the spectral resolution and spectral representation of self adjoint operators, invariant subspaces, strongly continuous one-parameter semigroups, the index of operators, the trace formula of Lidskii, the Fredholm determinant, and more.Assumes prior knowledge of Naive set theory, linear algebra, point set topology, basic complex variable, and real variables.Includes an appendix on the Riesz representation theorem.

This set featuresLinear Algebra and Its Applications, Second Edition (978-0-471-75156-4)Linear Algebra and Its Applications, Second Edition presents linear algebra as the theory and practice of linear spaces and linear maps with a unique focus on the analytical aspects as well as the numerous applications of the subject. In addition to thorough coverage of linear equations, matrices, vector spaces, game theory, and numerical analysis, the Second Edition features student-friendly additions that enhance the book's accessibility, including expanded topical coverage in the early chapters, additional exercises, and solutions to selected problems.Beginning chapters are devoted to the abstract structure of finite dimensional vector spaces, and subsequent chapters address convexity and the duality theorem as well as describe the basics of normed linear spaces and linear maps between normed spaces.Further updates and revisions have been included to reflect the most up-to-date coverage of the topic, including: The QR algorithm for finding the eigenvalues of a self-adjoint matrixThe Householder algorithm for turning self-adjoint matrices into tridiagonal formThe compactness of the unit ball as a criterion of finite dimensionality of a normed linear spaceAdditionally, eight new appendices have been added and cover topics such as: the Fast Fourier Transform; the spectral radius theorem; the Lorentz group; the compactness criterion for finite dimensionality; the characterization of commentators; proof of Liapunov's stability criterion; the construction of the Jordan Canonical form of matrices; and Carl Pearcy's elegant proof of Halmos' conjecture about the numerical range of matrices.Clear, concise, and superbly organized, Linear Algebra and Its Applications, Second Edition serves as an excellent text for advanced undergraduate- and graduate-level courses in linear algebra. Its comprehensive treatment of the subject also makes it an ideal reference or self-study for industry professionals.and Functional Analysis (978-0-471-55604-6) both by Peter D. Lax.

The application by Fadeev and Pavlov of the Lax-Phillips scattering theory to the automorphic wave equation led Professors Lax and Phillips to reexamine this development within the framework of their theory. This volume sets forth the results of that work in the form of new or more straightforward treatments of the spectral theory of the Laplace-Beltrami operator over fundamental domains of finite area; the meromorphic character over the whole complex plane of the Eisenstein series; and the Selberg trace formula. CONTENTS: 1. Introduction. 2. An abstract scattering theory. 3. A modified theory for second order equations with an indefinite energy form. 4. The Laplace-Beltrami operator for the modular group. 5. The automorphic wave equation. 6. Incoming and outgoing subspaces for the automorphic wave equations. 7. The scattering matrix for the automorphic wave equation. 8. The general case. 9. The Selberg trace formula.

The theory of hyperbolic equations is a large subject, and its applications are many: fluid dynamics and aerodynamics, the theory of elasticity, optics, electromagnetic waves, direct and inverse scattering, and the general theory of relativity. This book is an introduction to most facets of the theory and is an ideal text for a second-year graduate course on the subject. The first part deals with the basic theory: the relation of hyperbolicity to the finite propagation of signals, the concept and role of characteristic surfaces and rays, energy, and energy inequalities. The structure of solutions of equations with constant coefficients is explored with the help of the Fourier and Radon transforms. The existence of solutions of equations with variable coefficients with prescribed initial values is proved using energy inequalities. The propagation of singularities is studied with the help of progressing waves. The second part describes finite difference approximations of hyperbolic equations, presents a streamlined version of the Lax-Phillips scattering theory, and covers basic concepts and results for hyperbolic systems of conservation laws, an active research area today. Four brief appendices sketch topics that are important or amusing, such as Huygens' principle and a theory of mixed initial and boundary value problems. A fifth appendix by Cathleen Morawetz describes a nonstandard energy identity and its uses. Information for our distributors: Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University.

Complex Proofs of Real Theorems is an extended meditation on Hadamard's famous dictum, ""The shortest and best way between two truths of the real domain often passes through the imaginary one.'' Directed at an audience acquainted with analysis at the first year graduate level, it aims at illustrating how complex variables can be used to provide quick and efficient proofs of a wide variety of important results in such areas of analysis as approximation theory, operator theory, harmonic analysis, and complex dynamics. Topics discussed include weighted approximation on the line, Muntz's theorem, Toeplitz operators, Beurling's theorem on the invariant spaces of the shift operator, prediction theory, the Riesz convexity theorem, the Paley-Wiener theorem, the Titchmarsh convolution theorem, the Gleason-Kahane-Zelazko theorem, and the Fatou-Julia-Baker theorem. The discussion begins with the world's shortest proof of the fundamental theorem of algebra and concludes with Newman's almost effortless proof of the prime number theorem. Four brief appendices provide all necessary background in complex analysis beyond the standard first year graduate course. Lovers of analysis and beautiful proofs will read and reread this slim volume with pleasure and profit.

This book deals with the mathematical side of the theory of shock waves. The author presents what is known about the existence and uniqueness of generalized solutions of the initial value problem subject to the entropy conditions. The subtle dissipation introduced by the entropy condition is investigated and the slow decay in signal strength it causes is shown.

Includes sections on the spectral resolution and spectral representation of self adjoint operators, invariant subspaces, strongly continuous one-parameter semigroups, the index of operators, the trace formula of Lidskii, the Fredholm determinant, and more. Assumes prior knowledge of Naive set theory, linear algebra, point set topology, basic complex variable, and real variables. Includes an appendix on the Riesz representation theorem.

This set featuresLinear Algebra and Its Applications, Second Edition (978-0-471-75156-4)Linear Algebra and Its Applications, Second Edition presents linear algebra as the theory and practice of linear spaces and linear maps with a unique focus on the analytical aspects as well as the numerous applications of the subject. In addition to thorough coverage of linear equations, matrices, vector spaces, game theory, and numerical analysis, the Second Edition features student-friendly additions that enhance the book''s accessibility, including expanded topical coverage in the early chapters, additional exercises, and solutions to selected problems.Beginning chapters are devoted to the abstract structure of finite dimensional vector spaces, and subsequent chapters address convexity and the duality theorem as well as describe the basics of normed linear spaces and linear maps between normed spaces.Further updates and revisions have been included to reflect the most up-to-date coverage of the topic, including: The QR algorithm for finding the eigenvalues of a self-adjoint matrix The Householder algorithm for turning self-adjoint matrices into tridiagonal form The compactness of the unit ball as a criterion of finite dimensionality of a normed linear space Additionally, eight new appendices have been added and cover topics such as: the Fast Fourier Transform; the spectral radius theorem; the Lorentz group; the compactness criterion for finite dimensionality; the characterization of commentators; proof of Liapunov''s stability criterion; the construction of the Jordan Canonical form of matrices; and Carl Pearcy''s elegant proof of Halmos'' conjecture about the numerical range of matrices.Clear, concise, and superbly organized, Linear Algebra and Its Applications, Second Edition serves as an excellent text for advanced undergraduate- and graduate-level courses in linear algebra. Its comprehensive treatment of the subject also makes it an ideal reference or self-study for industry professionals.and Functional Analysis (978-0-471-55604-6) both by Peter D. Lax.

Burstein, and Lax's Calculus with Applications and Computing offers meaningful explanations of the important theorems of single variable calculus. Written with students in mathematics, the physical sciences, and engineering in mind, and revised with their help, it shows that the themes of calculation, approximation, and modeling are central to mathematics and the main ideas of single variable calculus. This edition brings the innovation of the first edition to a new generation of students. New sections in this book use simple, elementary examples to show that when applying calculus concepts to approximations of  functions, uniform convergence is more natural and easier to use than point-wise convergence. As in the original, this edition includes material that is essential for students in science and engineering, including an elementary introduction to complex numbers and complex-valued functions, applications of calculus to modeling vibrations and population dynamics, and an introduction to probability and information theory.