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A renowned mathematician who considers himself both applied and
theoretical in his approach, Peter Lax has spent most of his
professional career at NYU, making significant contributions to
both mathematics and computing. He has written several important
published works and has received numerous honors including the
National Medal of Science, the Lester R. Ford Award, the Chauvenet
Prize, the Semmelweis Medal, the Wiener Prize, and the Wolf Prize.
Several students he has mentored have become leaders in their
fields.
Two volumes span the years from 1952 up until 1999, and cover
many varying topics, from functional analysis, partial differential
equations, and numerical methods to conservation laws, integrable
systems and scattering theory. After each paper, or collection of
papers, is a commentary placing the paper in context and where
relevant discussing more recent developments. Many of the papers in
these volumes have become classics and should be read by any
serious student of these topics. In terms of insight, depth, and
breadth, Lax has few equals. The reader of this selecta will
quickly appreciate his brilliance as well as his masterful touch.
Having this collection of papers in one place allows one to follow
the evolution of his ideas and mathematical interests and to
appreciate how many of these papers initiated topics that developed
lives of their own.
A renowned mathematician who considers himself both applied and
theoretical in his approach, Peter Lax has spent most of his
professional career at NYU, making significant contributions to
both mathematics and computing. He has written several important
published works and has received numerous honors including the
National Medal of Science, the Lester R. Ford Award, the Chauvenet
Prize, the Semmelweis Medal, the Wiener Prize, and the Wolf Prize.
Several students he has mentored have become leaders in their
fields.
Two volumes span the years from 1952 up until 1999, and cover
many varying topics, from functional analysis, partial differential
equations, and numerical methods to conservation laws, integrable
systems andscattering theory.After each paper, or collection of
papers, is a commentary placing the paper in context and where
relevant discussing more recent developments.Many of the papers in
these volumes have become classics and should be read by any
serious student of these topics.In terms of insight, depth, and
breadth, Lax has few equals.The reader of this selecta will quickly
appreciate his brilliance as well as his masterful touch.Having
this collection of papers in one place allows one to follow the
evolution of his ideas and mathematical interests and to appreciate
how many of these papers initiated topics that developed lives of
their own."
This new edition of 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."
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.
These lecture notes of the courses presented at the first CIME
session 1994 by leading scientists present the state of the art in
recent mathematical methods in Nonlinear Wave Propagation.
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.
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.
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.
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.
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