2003 VIII, 134 p. Softcover
3-540-00236-7
The text contains for the first time in book form the state of
the art of homological methods in Functional Analysis like
characterizations of the vanishing of the derived projective
limit functor or the functors Ext1 (E, F) for Frechet and more
general spaces. The researcher in real and complex Analysis finds
powerful tools to solve surjectivity problems e.g. on spaces of
distributions or to characterize the existence of solution
operators.
The requirements from Homological Algebra are minimized: all one
needs is summarized on a few pages. The answers to several
questions of V.P. Palamodov who invented homological methods in
Analysis also show the limits of the program.
Keywords: Functional analysis, derived functors, homological
methods, locally convex spaces
Series: Lecture Notes in Mathematics. Volume. 1810
2003 Approx. 295 p. 95 illus. Hardcover
0-387-95584-4
The aim of this book is NOT to cover discrete mathematics in
depth. Rather, it discusses a number of selected results and
methods, mostly from the areas of combinatorics and graph theory,
along with some elementary number theory and combinatorial
geometry. The authors develop most topics to the extent that they
can describe the discrete mathematics behind an important
application of mathematics such as discrete optimization
problems, the Law of Large Numbers, cryptography, and coding to
name a few. Another feature that is not covered in other discrete
mathematics books is the use of ESTIMATES (How many digits does
100! have? or Which is larger: 2100 or 100!?). There are
questions posed in the text and problems at the end of each
chapter with solutions for many of them at the end of the book.
The book is based on a course taught for several years by two of
the authors at Yale University.
Contents:
Preface.- Let us count!- Combinatorial tools.- Binomial
Coefficients and Pascal's Triangle.- Fibonacci numbers.-
Combinatorial probability.- Integers, divisors, and primes.-
Graphs.- Trees.- Finding the optimum.- Matchings in graphs.-
Combinatorics in geometry.- Euler's formula.- Coloring maps and
graphs.- Finite geometries, codes, Latin squares, and other
pretty creatures.- A glimpse of complexity and cryptography.-
Answers to exercises.- Index.
Series: Undergraduate Texts in Mathematics.
2003 XII, 254 p. 35 illus. Hardcover
0-387-95587-9
Solutions of equations in integers is the central problem of
number theory and focus of this book. Studying number theory
through its algebraic structure shows that the necessary
algebraic concepts--rings and ideals--have no better motivation
than number theory. The book is based on short courses that the
author gave in recent years and it covers most of the results
found in standard number theory courses. The amount of material
is suitable for a one-semester course, with some variation
possible through omission of the optional starred sections. The
author has tried to avoid the ad hoc proofs in favor of unifying
ideas that work in many situations. There are exercises at the
end of almost every section, so that each new idea or proof
receives immediate reinforcement.
Contents:
Preface.- Natural numbers and integers.- The Euclidean algorithm.-
Congruence arithmetic.- The RSA cryptosystem.- The Pell equation.-
The Gaussian Integers.- Quadratic integers.- The four square
theorem.- Quadratic reciprocity.- Rings.- Ideals.- Prime ideals.-
Bibliography.- Index.
Series: Undergraduate Texts in Mathematics.
2003 Approx. 410 p. 111 illus. Hardcover
0-387-95552-6
This book presents methods of solving problems in classical
combinatorics, combinatorial number theory, and combinatorial
geometry. It can be seen as a continuation of the successful book
"Equations and Inequalities" by the same authors.
However, it can be read independently or used as a textbook in
its own right. The authors' aim is to familiarize the reader with
methods for solving problems in elementary mathematics,
accessible to beginning university and advanced high-school
students. They emphasize basic algebraic operations and other
technical skills that are reinforced in numerous examples and
exercises. Answers to all exercises can be found at the end of
the book. The book is intended as a text for a problem-solving
course at the first- or second-year university level, as a text
for enrichment classes for talented high-school students, or for
mathematics competition training.
Contents:
Preface.- Symbols.- Combinatorics.- Combinatorial Arithmetic.-
Combinatorial Geometry.- Hints and Answers.- Bibliography.- Index.
Series: CMS Books in Mathematics. Volume. 12
ISBN: 0-415-30528-4
Pub Date: 28 FEB 2003
Type: Hardback Book
Extent: 328 pages (Dimensions 246x174 mm)
This volume describes the monotone iterative technique which is
used to obtain monotone approximate solutions that converge to
the solution of nonlinear problems of partial differential
equations of elliptic, parabolic and hyperbolic type. This when
combined with the quasilinearization method offers rapid
convergence of approximation to the solution and this has played
a valuable role in unifying a variety of nonlinear problems. The
monograph is divided into two parts. The first part describes the
general methodology systematically utilizing the classic approach
and the second part exhibits the development of the same basic
ideas via the variational technique. The volumes provides a
useful and timely reference text for applied scientists,
engineers and numerical analysts.
Contents:
1. Elliptic Equations 1.1 Introduction 1.2 Monotone Iterates: A
Preview 1.3 Monotone Iterative Technique 1.4 Generalized
Quasilinearization 1.5 Weakly Coupled Mixed Monotone Systems 1.6
Elliptic Systems in Unbounded Domains 1.7 MIT Systems in
Unbounded Domains 1.8 Notes and Comments 2. Parabolic Equations 2.1
Introduction 2.2 Comparision Theorems 2.3 Monotone Iterative
Technique 2.4 Generalized Quasilinearization 2.5 Monotone Flows
and Mixed Monotone Systems 2.6 GCR for Weakly Coupled Systems 2.7
Stability and Vector Lyapunov Functions 2.8 Notes and Comments 3.
Impulsive Parabolic Equations 3.1 Introduction 3.2 Comparison
Results for IPS 3.3 Coupled Lower and Upper Solutions 3.4
Generalized Quasilinearization 3.5 Population Dynamics with
Impulses 3.6 Notes and Comments 4. Hyperbolic Equations 4.1
Introduction 4.2 VP and Comparison Results 4.3 Monotone Iterative
Technique 4.4 The Method of Generalized Quasilinearization 4.5
Notes and Comments 5. Elliptic Equations 5.1 Introduction 5.2
Comparison Result 5.3 MIT: Semilinear Problems 5.4 MIT:
Quasilinear Problems 5.5 MIT: Degenerate Problems 5.6 GQ:
Semilinear Problems 5.7 GQ: Quasilinear Problem 5.8 GQ:
Degenerate Problems 5.9 Notes and Comments 6. Parabolic Equations
6.1 Introduction 6.2 Monotone Iterative Technique 6.3 Generalized
Quasilinearization 6.4 Nonlocal Problems 6.5 GQ: Nonlocal
Problems 6.6 Quasilinear Problems 6.7 GQ: Quasilinear Problems 6.8
Notes and Comments 7. Hyperbolic Equations 7.1 Introduction 7.2
Notation and Comparison Results 7.3 Monotone Iterative Technique
7.4 Generalized Quasilinearization 7.5 Notes and Comments
Appendicies
Series Information:
Mathematical Analysis and Applications
This monograph deals with problems of dynamical reconstruction
of unknown variable characteristics (distributed or boundary
disturbances, coefficients of operators, etc.) for various
classes of systems with distributed parameters (parabolic and
hyperbolic equations, evolutionary variational inequalities, etc.).
The procedures for solving these problems are based on methods of
the theory of feedback control in combination with methods of the
theory of ill-posed problems and noninear analysis. These
procedures are oriented to real-time work and may be realized in
computers. General constructions are illustrated by numerical
examples.
This monograph is of value and interest to specialists in
optimisation theory, as well as to (postgraduate) students in
mathematical physics.
2002; viii+270 pages
ISBN 90-6764-369-6
Contents
Introduction
PROBLEMS OF DYNAMICAL MODELING IN ABSTRACT SYSTEMS
Reconstruction of inputs in dynamical systems. The method of
auxiliary controlled models
Method of Stabilization of Lyapunov Functionals
Reconstruction of distributed controls in abstract systems. The
case of instantaneous restrictions imposed on controls
APPROXIMATION OF INPUTS IN PARABOLIC EQUATIONS
Reconstruction of distributed controls in nonlinear parabolic
equations
Approximation of boundary controls in Neumann conditions and of
coefficients of the elliptic operator
Reconstruction of intensities of point sources through results of
sensory measurements
Approximation of boundary controls. The case of Dirichlet
boundary conditions
APPROXIMATION OF INPUTS IN PARABOLIC VARIATIONAL INEQUALITIES
Dynamical discrepancy method
The role of a priori information in the problem of control
reconstruction. The method of smoothing functional
Applications to inverse problems of mathematical physics and
mechanics
FINITE-DIMENSIONAL APPROXIMATION OF INPUTS. EXAMPLES
Reconstruction of boundary controls. The dynamical discrepancy
method
Reconstruction of distributed control. The method of smoothing
functional
Reconstruction of coefficients of the elliptic operators and
boundary controls
The results of computer modeling
Bibliography
M.M. Lavrentiev is the author of many fundamental scientific
results in many directions of mathematics and its applications,
such as differential equations, inverse and ill-posed problems,
tomography, numerical and applied mathematics. His results in the
theory of inverse problems for differential equations and in
tomography are well known all over the world.
To honour him on the occasion of his 70 birthday renowned
scientists in this field of mathematics, both from East and West,
have contributed to this special collection of papers on ill-posed
and inverse problems, which will be of interest to anyone working
in this field.
2002; xii+468 pages
ISBN 90-6764-362-9
Contents
Representations of functions of many complex variables and
inverse problems for kinetic equations
Yu.E. Anikonov
Uniqueness in determining piecewise analytic coefficients in
hyperbolic equations
Yu.E. Anikonov, J. Cheng and M. Yamamoto
Direct and inverse problems for evolution integro-differential
equations of the first-order in time
J.S. Azamatov and A. Lorenzi
How to see waves under the Earth surface (the BC-method for
geophysicists)
M.I. Belishev
Global theorem of uniqueness of solution to inverse coefficient
problem for a quasilinear hyperbolic equation
A.M. Denisov
Identification of parameters in polymer crystallization,
semiconductor models and elasticity via iterative regularization
methods
H.W. Engl
The tomato salad problem in spherical stereology
R. Gorenflo
Two methods in inverse problem and extraction formulae
M. Ikehata
Identification of the unknown potential in the nonstationary
Schrodinger equation
A.D. Iskenderov
Iterative methods of solving inverse problems for hyperbolic
equations
S.I. Kabanikhin
Carleman estimates and inverse problems: Uniqueness and
convexification of multiextremal objective functions
M.V. Klibanov
Convergence analysis of a Landweber-Kaczmarz method for solving
nonlinear ill-posed problems
R. Kowar and O. Scherzer
A sampling method for an inverse boundary value problem for
harmonic vector fields
R. Kress
Approaching a partial differential equation of mixed elliptic-hyperbolic
type
R. Magnanini and G. Talenti
Complex geometrical optics solutions and pseudoanalytic matrices
G. Nakamura and G. Uhlmann
Numerical solution of inverse evolution problems via the
nonlinear Levitan equation
F. Natterer
An inverse problem for a parabolic equation with final
overdetermination
A.I. Prilepko and D.S. Tkachenko
Uniqueness theorems for an inverse problem related to local
heterogeneities and data on a piece of a plane
V.G. Romanov
On ill-posed problems and Professor Lavrentiev
P.C. Sabatier
Regularization and iterative approximation for linear ill-posed
problems in the space of functions of bounded variation
V.V. Vasin
A posteriori error estimation for ill-posed problems on some
sourcewise represented or compact sets
A.G. Yagola and V.N. Titarenko
Multidimensional inverse problems for hyperbolic equations with
point sources
V.G. Yakhno