Allaire, G., Ecole Polytechnique, Paliseau, France

Shape Optimization by the Homogenization Method

2001. Approx. 465 pp. 54 figs. Hardcover
0-387-95298-5

This book provides an introduction to the theory and numerical developments of the homogenization method. It's main features are: a comprehensive presentation of homogenization theory; an introduction to the theory of two-phase composite materials; a detailed treatment of structural optimization by using homogenization; a complete discussion of the resulting numerical algorithms with many documented test problems. It will be of interest to researchers, engineers, and advanced graduate students in applied mathematics, mechanical engineering, and structural optimization.

Contents: Homogenization.- The mathematical modeling of composites.- Optimal design in conductivity.- Optimal design in elasticity.- Numerical algorithms.

Series: Applied Mathematical Sciences. VOL. 146

Saxe, K., Macalester College, St. Paul, MN, USA

Beginning Functional Analysis

2001. Approx. 210 pp. 25 figs. Hardcover
0-387-95224-1

The unifying approach of functional analysis is to view functions as points in some abstract vector space and the differential and integral operators relating these points as linear transformations on these spaces. The author presents the basics of functional analysis with attention paid to both expository style and technical detail, while getting to interesting results as quickly as possible. The book is accessible to students who have completed first courses in linear algebra and real analysis. Topics are developed in their historical context, with bits of pertinent history - including biographies - appearing throughout the text. The book offers suggestions and references for further study, and many exercises.

Karen Saxe is Associate Professor of Mathematics at Macalester College in St. Paul, Minnesota. She received her Ph.D. from the University of Oregon. Before joining the faculty at Macalester, she held a two-year FIPSE post-doctoral position at St. Olaf College in Northfield, Minnesota. She currently serves on the editorial board of the MAA's College Mathematics Journal. This is her first book.

Keywords: Functional analysis

Contents: Metric Spaces, Normed Spaces, Inner Product Spaces.- The Topology of Metric Spaces.- Measure and Integration.- Fourier Analysis in Hilbert Space.- An Introduction to Abstract Linear Operator Theory.- Further Topics.- Appendix A: Complex Numbers.- Appendix B: Basic Set Theory.- Appendix C: Biographies.

Series: Undergraduate Texts in Mathematics.

Pugh, C.C., University of California, Berkeley, CA, USA

Real Mathematical Analysis

2002. Approx. 455 pp. 133 figs. Hardcover
0-387-95297-7

Was plane geometry your favourite math course in high school? Did you like proving theorems? Are you sick of memorising integrals? If so, real analysis could be your cup of tea. In contrast to calculus and elementary algebra, it involves neither formula manipulation nor applications to other fields of science. None. It is Pure Mathematics, and it is sure to appeal to the budding pure mathematician. In this new introduction to undergraduate real analysis the author takes a different approach from past studies of the subject, by stressing the importance of pictures in mathematics and hard problems. The exposition is informal and relaxed, with many helpful asides, examples and occasional comments from mathematicians like Dieudonne, Littlewood and Osserman. The author has taught the subject many times over the last 35 years at Berkeley and this book is based on the honours version of this course. The book contains an excellent selection of more than 500 exercises.

Keywords: Real Analysis, Real Mathematical Analysis

Contents: Real Numbers * A Taste of Topology * Functions of a Real Variable *
Function Spaces * Multivariable Calculus * Lebesgue Theory * Index

Series: Undergraduate Texts in Mathematics.

Kozlov, V.V., Moscow State University, Moscow, Russia

Dynamical Systems X
General Theory of Vortices

2002. Approx. 190 pp. 22 figs. Hardcover
3-540-42207-2

This book contains a mathematical exposition of analogies between classical (Hamiltonian) mechanics, geometrical optics, and hydrodynamics. This theory highlights several general mathematical ideas that appeared in Hamiltonian mechanics, optics and hydrodynamics under different names. In addition, some interesting applications of general vortex theory are discussed in the book such as applications in numerical methods, stability theory, and the theory of exact integration of equations of dynamics. The investigation of families of trajectories of Hamiltonian systems can be reduced to problems of multidimensional ideal fluid dynamics. For example, the well-known Hamilton-Jacobi method corresponds to the case of potential flows.
The book will be of great interest to researchers and postgraduate students interested in mathematical physics, mechanics, and the theory of differential equations.

Keywords: vortex, Hamilton equations, Hamilton-Jacobi method, hydrodynamics

Contents: Introduction.- 1. Hydrodynamics, Geometrical Optics, and Classical Mechanics.- 2. General Vortex Theory.- 3. Geodesics on Lie Groups with Left-Invariant Metrics.- 4. Vortex Method of Integration of Hamilton Equations.- Appendix 1. Invariants of Vorticity and Second Hydrodynamics.- Appendix 2. Quantum Mechanics and Hydrodynamics.- References.- Index.

Series: Encyclopaedia of Mathematical Sciences. VOL. 67

V.P. Golubyatnikov

Uniqueness Questions in Reconstruction of Multidimensional Objects
from Tomography-Type Projection Data

Inverse and Ill-Posed Problems Series

The first part of this new volume in the Inverse and Ill-Posed Problems Series studies uniqeness questions for recovering the shapes of the convex and more complicated bodies from shapes of their projections onto planes of low dimension. Some stability estimates of the solutions to these inverse problems are given.

The second part deals with inverse problems with projection data directly connected to tomography, in partcular to apparent contours of smooth surfaces, which have practical interpretations such as thin cracks in continuous media which are studied in industrial defectoscopy, caustic surfaces which are studies in wave optics, etc.

New results on reconstruction of smooth surfaces from observations of the wave fronts generated by these surfaces are obtained.

This book will be of interest to researchers in the fields of inverse problems, integral geometry and tomography.

Contents:
CHAPTER 1. INTRODUCTION
Notation and basic definitions
Translation equivalence of projections. Preliminary results
CHAPTER 2. SO(2)-CONGRUENCE OF PROJECTIONS
The case of convex bodies
An attempt to relax the asymmetry conditions
The case of ( n - 2)-visible and ( n - 2)-convex bodies
Stability estimates for recovering the shapes of convex bodies from the shapes of their projections
CHAPTER 3. OTHER GROUPS OF CONGRUENCES OF PROJECTIONS
SO(2)-similarity of projections
SO(3)-congruence of projections
SU(2) and U-congruence of projections
CHAPTER 4. APPARENT CONTOURS AND OTHER TOMOGRAPHY-TYPE PROJECTION DATA
Reconstruction of surfaces from the shapes of their apparent contours and the stationary phase observations
Inversion formulae for integral geometry problems and an algorithm of computerized tomography
An inverse problem for the Hamilton-Jacobi equations
Inverse problems for one class of the tomography-type evolution equation
Bibliography

2000; x+120 pages
ISBN 90-6764-332-7

P.G. Danilaev

Coefficient Inverse Problems for Parabolic Type Equations
and Their Application

Inverse and Ill-Posed Problems Series

Contents:
Preface
ON THE ILL-POSEDNESS OF COEFFICIENT INVERSE PROBLEMS AND THE GENERAL APPROACH TO THE STUDY OF THEM
DETERMINING THE COEFFICIENT FOR THE LOWEST TERM OF EQUATION
Setting of the problem. Determination of the coefficient
The difference quasi-inversion problem
A test example
DETERMINING OF THE COEFFICIENT FOR THE LEADING TERMS OF EQUATION
Statement of the problem
The quasi-inversion problem and an estimate of stability of its solution
Simplification of equation of the quasi-inversion method
Simplification of the quasi-inversion problem
Finding the coefficient
Difference quasi-inversion problem
Numerical solution of the quasi-inversion problem
Results of solution of a test example problem
MODIFICATION OF THE METHOD OF DETERMINING THE COEFFICIENT OF THE LEADING TERMS IN AN EQUATION
Modification method
Defining a coefficient
On deriving the main integro-differential equation
GENERALIZATION OF THE DEVELOPED ALGORITHM FOR SOLVING COEFFICIENT INVERSION PROBLEM
ON APPLICATIONS OF COEFFICIENT INVERSE PROBLEMS IN UNDERGROUND FLUID DYNAMICS
Determining of filtration parameters of exploited non-homogeneous oil-strata
Determining the filtration parameters in the case of non-linear filtration
The quasi-inversion problem for the considered cases
Summary
Bibliography

2001; x+116 pages
ISBN 90-6764-348-3