Daniel Alpay, Ben-Gurion University of the Negev, Beer-sheva, Israel
The Schur Algorithm, Reproducing Kernel Spaces and System Theory
From a review of the French edition:
"This excellent survey showing a rich interplay between functional analysis, complex analysis and systems science is very informative and can be highly recommended to functional analysts curious about the systems science impact of their discipline or to theoretically inclined systems scientists, in particular those involved in the realization theory."
-- Zentralblatt fur Mathematik
Description
The class of Schur functions consists of analytic functions on the unit disk that are bounded by $1$. The Schur algorithm associates to any such function a sequence of complex constants, which is much more useful than the Taylor coefficients. There is a generalization to matrix-valued functions and a corresponding algorithm. These generalized Schur functions have important applications to the theory of linear operators, to signal processing and control theory, and to other areas of engineering.
In this book, Alpay looks at matrix-valued Schur functions and their applications from the unifying point of view of spaces with reproducing kernels. This approach is used here to study the relationship between the modeling of time-invariant dissipative linear systems and the theory of linear operators. The inverse scattering problem plays a key role in the exposition. The point of view also allows for a natural way to tackle more general cases, such as nonstationary systems, non-positive metrics, and pairs of commuting nonself-adjoint operators. This is the English translation of a volume originally published in French by the Societe Mathematique de France. Translated by Stephen S. Wilson.
Contents
Introduction
Reproducing kernel spaces
Theory of linear systems
Schur algorithm and inverse scattering problem
Operator models
Interpolation
The indefinite case
The non-stationary case
Riemann surfaces
Conclusion
Bibliography
Index
Details:
Series: SMF/AMS Texts and Monographs, Volume: 5
Publication Year: 2001
ISBN: 0-8218-2155-5
Paging: 150 pp.
Binding: Softcover
Alexander Givental, University of California, Berkeley, CA
Linear Algebra and Differential Equations
Expected publication date is August 12, 2001
Description
This is based on the course, "Linear Algebra and Differential Equations", taught by the author to sophmore students at UC Berkeley.
From the Introduction: "We accept the currently acting syllabus as an outer constraint ... but otherwise we stay rather far from conventional routes.
"In particular, at least half of the time is spent to present the entire agenda of linear algebra and its applications in the $2D$ environment; Gaussian elimination occupies a visible but supporting position; abstract vector spaces intervene only in the review section. Our eye is constantly kept on why?, and very few facts (the fundamental theorem of algebra, the uniqueness and existence theorem for solutions of ordinary differential equations, the Fourier convergence theorem, and the higher-dimensional Jordan normal form theorem) are stated and discussed without proof."
Specific material in the book is organized as follows: Chapter 1 discusses geometry on the plane, including vectors, analytic geometry, linear transformations and matrices, complex numbers, and eigenvalues. Chapter 2 presents differential equations (both ODEs and PDEs), Fourier series, and the Fourier method. Chapter 3 discusses classical problems of linear algebra, matrices and determinants, vectors and linear systems, Gaussian elimination, quadratic forms, eigenvectors, and vector spaces. The book concludes with a sample final exam.
Contents
Geometry on the plane
Differential equations
Linear algebra
Details:
Series: Berkeley Mathematical Lecture Notes Volume: 11
Publication Year: 2001
ISBN: 0-8218-2850-9
Paging: 132 pp.
Binding: Softcover
V. P. Maslov, Moscow State University, Russia,
and G. A. Omel'yanov, Moscow Institute of Electronic Engineering, Russia
Geometric Asymptotics for Nonlinear PDE. I
Expected publication date is August 26, 2001
Description
The study of asymptotic solutions to nonlinear systems of partial differential equations is a very powerful tool in the analysis of such systems and their applications in physics, mechanics, and engineering. In the present book, the authors propose a new powerful method of asymptotic analysis of solutions, which can be successfully applied in the case of the so-called "smoothed shock waves", i.e., nonlinear waves which vary fast in a neighborhood of the front and slowly outside of this neighborhood. The proposed method, based on the study of geometric objects associated to the front, can be viewed as a generalization of the geometric optics (or WKB) method for linear equations. This volume offers to a broad audience a simple and accessible presentation of this new method.
The authors present many examples originating from problems of hydrodynamics, nonlinear optics, plasma physics, mechanics of continuum, and theory of phase transitions (free boundary problems). In the examples, characterized by smoothing of singularities due to dispersion or diffusion, asymptotic solutions in the form of distorted solitons, kinks, breathers, or smoothed shock waves are constructed. By a unified rule, a geometric picture is associated with each physical problem that allows for obtaining tractable asymptotic formulas and provides a geometric interpretation of the physical process. Included are many figures illustrating the various physical effects.
Contents
Introduction
Waves in one-dimensional nonlinear media
Nonlinear waves in multidimensional media
Asymptotic solutions of some pseudodifferential equations and dynamical systems with small dispersion
Problems with a free boundary
Multi-phase asymptotic solutions
Asymptotics of stationary solutions to the Navier-Stokes equations describing stretched vortices
List of equations
Bibliography
Details:
Series: Translations of Mathematical Monographs, Volume: 202
Publication Year: 2001
ISBN: 0-8218-2109-1
Paging: 285 pp.
Binding: Hardcover
Edited by: Luca Capogna and Loredana Lanzani, University of Arkansas, Fayetteville, AR
Harmonic Analysis and Boundary Value Problems
Description
This volume presents research and expository articles by the participants of the 25th Arkansas Spring Lecture Series on "Recent Progress in the Study of Harmonic Measure from a Geometric and Analytic Point of View" held at the University of Arkansas (Fayetteville). Papers in this volume provide clear and concise presentations of many problems that are at the forefront of harmonic analysis and partial differential equations.
The following topics are featured: the solution of the Kato conjecture, the "two bricks" problem, new results on Cauchy integrals on non-smooth curves, the Neumann problem for sub-Laplacians, and a new general approach to both divergence and nondivergence second order parabolic equations based on growth theorems. The articles in this volume offer both students and researchers a comprehensive volume of current results in the field.
Contents
J. D. Sykes and R. M. Brown -- The mixed boundary problem in $L^p$ and Hardy spaces for Laplace's equation on a Lipschitz domain
D. Danielli, N. Garofalo, and D.-M. Nhieu -- Sub-elliptic Besov spaces and the characterization of traces on lower dimensional manifolds
S. Hofmann -- The solution of the Kato problem
M. B. Korey -- A decomposition of functions with vanishing mean oscillation
D. Mitrea and M. Mitrea -- General second order, strongly elliptic systems in low dimensional nonsmooth manifolds
E. Ferretti and M. V. Safonov -- Growth theorems and Harnack inequality for second order parabolic equations
Z. Shen -- Absolute continuity of generalized periodic Schrodinger operators
G. C. Verchota -- The use of Rellich identities on certain nongraph boundaries
J. Verdera -- $L^2$ boundedness of the Cauchy integral and Menger curvature
Details:
Series: Contemporary Mathematics, Volume: 277
Publication Year: 2001
ISBN: 0-8218-2745-6
Paging: 158 pp.
Binding: Softcover
Edited by: Eric Todd Quinto, Tufts University, Medford, MA, Leon Ehrenpreis, Temple University, Philadelphia, PA, Adel Faridani, Oregon State University, Corvallis, OR, Fulton Gonzalez, Tufts University, Medford, MA, and Eric Grinberg, Temple University, Philadelphia, PA
Radon Transforms and Tomography
Expected publication date is August 25, 2001
Description
One of the most exciting features of the fields of Radon transforms and tomography is the strong relationship between high-level pure mathematics and applications to areas such as medical imaging and industrial nondestructive evaluation. The proceedings featured in this volume bring together fundamental research articles in the major areas of Radon transforms and tomography.
This volume includes expository papers that are of special interest to beginners as well as advanced researchers. Topics include local tomography and wavelets, Lambda tomography and related methods, tomographic methods in RADAR, ultrasound, Radon transforms and differential equations, and the Pompeiu problem.
The major themes in Radon transforms and tomography are represented among the research articles. Pure mathematical themes include vector tomography, microlocal analysis, twistor theory, Lie theory, wavelets, harmonic analysis, and distribution theory. The applied articles employ high-quality pure mathematics to solve important practical problems. Effective scanning geometries are developed and tested for a NASA wind tunnel. Algorithms for limited electromagnetic tomographic data and for impedance imaging are developed and tested. Range theorems are proposed to diagnose problems with tomography scanners. Principles are given for the design of X-ray tomography reconstruction algorithms, and numerical examples are provided.
This volume offers readers a comprehensive source of fundamental research useful to both beginners and advanced researchers in the fields.
Contents
Expository papers
C. A. Berenstein -- Local tomography and related problems
M. Cheney -- Tomography problems arising in synthetic aperture radar
A. Faridani, K. A. Buglione, P. Huabsomboon, O. D. Iancu, and J. McGrath -- Introduction to local tomography
F. Natterer -- Algorithms in ultrasound tomography
E. T. Quinto -- Radon transforms, differential equations, and microlocal analysis
L. Zalcman -- Supplementary bibliography to "A bibliographic survey of the Pompeiu problem"
Research papers
T. Bailey and M. Eastwood -- Twistor results for integral transforms
J. Boman -- Injectivity for a weighted vectorial Radon transform
O. Dorn, E. L. Miller, and C. M. Rappaport -- Shape reconstruction in 2D from limited-view multifrequency electromagnetic data
L. Ehrenpreis -- Three problems at Mount Holyoke
F. B. Gonzalez -- A Paley-Wiener theorem for central functions on compact Lie groups
I. Pesenson and E. L. Grinberg -- Inversion of the spherical Radon transform by a Poisson type formula
S. H. Izen and T. J. Bencic -- Application of the Radon transform to calibration of the NASA-Glenn icing research wind tunnel
A. Katsevich -- Range theorems for the Radon transform and its dual
S. K. Patch -- Moment conditions $\emph{indirectly}$ improve image quality
A. Rieder -- Principles of reconstruction filter design in 2D-computerized tomography
B. Rubin and D. Ryabogin -- The $k$-dimensional Radon transform on the $n$-sphere and related wavelet transforms
S. Siltanen, J. L. Mueller, and D. Isaacson -- Reconstruction of high contrast 2-D conductivities by the algorithm of A. Nachman
L. B. Vertgeim -- Integral geometry problem with incomplete data for tensor fields in a complex space
Details:
Series: Contemporary Mathematics, Volume: 278
Publication Year: 2001
ISBN: 0-8218-2135-0
Paging: 261 pp.
Binding: Softcover