Included in series
North-Holland Mathematics Studies, 195
Description
The book is devoted to various constructions of sets which are
nonmeasurable with respect to invariant (more generally, quasi-invariant)
measures. Our starting point is the classical Vitali theorem
stating the existence of subsets of the real line which are not
measurable in the Lebesgue sense. This theorem stimulated the
development of the following interesting topics in mathematics: 1.
Paradoxical decompositions of sets in finite-dimensional
Euclidean spaces; 2. The theory of non-real-valued-measurable
cardinals; 3. The theory of invariant (quasi-invariant)
extensions of invariant (quasi-invariant) measures. These topics
are under consideration in the book. The role of nonmeasurable
sets (functions) in point set theory and real analysis is
underlined and various classes of such sets (functions) are
investigated . Among them there are: Vitali sets, Bernstein sets,
Sierpinski sets, nontrivial solutions of the Cauchy functional
equation, absolutely nonmeasurable sets in uncountable groups,
absolutely nonmeasurable additive functions, thick uniform
subsets of the plane, small nonmeasurable sets, absolutely
negligible sets, etc. The importance of properties of
nonmeasurable sets for various aspects of the measure extension
problem is shown. It is also demonstrated that there are close
relationships between the existence of nonmeasurable sets and
some deep questions of axiomatic set theory, infinite
combinatorics, set-theoretical topology, general theory of
commutative groups. Many open attractive problems are formulated
concerning nonmeasurable sets and functions.
Audience
Pure mathematicians and post-graduate students. Especially, those
ones whose research interests lie in set theory, real analysis,
measure theory, general topology , geometry of Euclidean spaces,
group theory.
Contents
The book is devoted to various constructions of sets which are
nonmeasurable with respect to invariant (more generally, quasi-invariant)
measures. Our starting point is the classical Vitali theorem
stating the existence of subsets of the real line which are not
measurable in the Lebesgue sense. This theorem stimulated the
development of the following interesting topics in mathematics: 1.
Paradoxical decompositions of sets in finite-dimensional
Euclidean spaces; 2. The theory of non-real-valued-measurable
cardinals; 3. The theory of invariant (quasi-invariant)
extensions of invariant (quasi-invariant) measures. These topics
are under consideration in the book. The role of nonmeasurable
sets (functions) in point set theory and real analysis is
underlined and various classes of such sets (functions) are
investigated . Among them there are: Vitali sets, Bernstein sets,
Sierpinski sets, nontrivial solutions of the Cauchy functional
equation, absolutely nonmeasurable sets in uncountable groups,
absolutely nonmeasurable additive functions, thick uniform
subsets of the plane, small nonmeasurable sets, absolutely
negligible sets, etc. The importance of properties of
nonmeasurable sets for various aspects of the measure extension
problem is shown. It is also demonstrated that there are close
relationships between the existence of nonmeasurable sets and
some deep questions of axiomatic set theory, infinite
combinatorics, set-theoretical topology, general theory of
commutative groups. Many open attractive problems are formulated
concerning nonmeasurable sets and functions.
Hardbound, ISBN: 0-444-51626-3, 352 pages, publication date: 2004
Proceedings of the International Conference on Functional
Analysis and its Applications dedicated to the 110th Anniversary
of Stefan Banach, May 28-31, 2002, Lviv, Ukraine
Included in series
North-Holland Mathematics Studies, 197
Description
The conference took place in Lviv, Ukraine and was dedicated to a
famous Polish mathematician Stefan Banach ?{ the most outstanding
representative of the Lviv mathematical school. Banach spaces,
introduced by Stefan Banach at the beginning of twentieth
century, are familiar now to every mathematician. The book
contains a short historical article and scientific contributions
of the conference participants, mostly in the areas of functional
analysis, general topology, operator theory and related topics.
Audience
Individual mathematicians working in functional analysis,
operator theory, general topology and also in history of
mathematics.
Contents
The conference took place in Lviv, Ukraine and was dedicated to a
famous Polish mathematician Stefan Banach ?{ the most outstanding
representative of the Lviv mathematical school. Banach spaces,
introduced by Stefan Banach at the beginning of twentieth
century, are familiar now to every mathematician. The book
contains a short historical article and scientific contributions
of the conference participants, mostly in the areas of functional
analysis, general topology, operator theory and related topics.
Hardbound, ISBN: 0-444-51373-6, 342 pages, publication date: 2004
Description
This book would be invaluable for any graduate student in partial
differential equations or in applied mathematics. Each chapter
brings indeed new ideas and new techniques which can be used in
these fields. The different chapters can be read independently
and are of great pedagogical value. The advanced researcher will
find the book the most recent achievements in various fields.
Key features:
Independent chapters
Most recent advances in each fields
High didactic quality
Self contained
Excellence of the contributors
Wide range of topics
Contents
Solutions of quasilinear second order elliptic boundary value
problems via degree theory (C. Bandle, W. Reichel).
Stationary Navier-Stokes problem in a two-dimensional exterior
domain (G. P Galdi).
Qualitative properties of solutions to elliptic problems (W.-M.
Ni).
On some basic aspects of the relationship between the calculus of
variations and differential equations (P. Pedregal).
On a class of singular perturbation problems (I. Shafrir).
Nonlinear spectral problems for degenerate elliptic operators (P.
Takac).
Analytical aspects of Liouville-type equations with singular
sources (G. Tarantello).
Elliptic equations involving measures (L. Ve).
Hardbound, ISBN: 0-444-51126-1, 648 pages, publication date: 2004
Description
This book contains several introductory texts concerning the main
directions in the theory of evolutionary partial differential
equations. The main objective is to present clear, rigorous, and
in depth surveys on the most important aspects of the present
theory.
Contents
Semigroups and evolution equations: Calculus, regularity and
kernel estimates (W. Arendt).
The front tracking method for systems of conservation laws (A.
Bressan).
Current issues on singular and degenerate evolution equations (E.
DiBenedetto, J.M. Urbano, V. Vespri).
Nonlinear hyperbolic-parabolic coupled systems (L. Hsiao, S.
Jiang).
Nonlinear parabolic equations and systems (A. Lunardi).
L1-stability of nonlinear waves in scalar conservation laws (D.
Serre).
Kinetic formulations of parabolic and hyperbolic PDE's: from
theory to numerics (B. Perthame).
Hardbound, ISBN: 0-444-51131-8, 612 pages, publication date: 2004
Description
Parameter Estimation and Inverse Problems will primarlily be a
textbook for advanced advanced undergrad/intro graduate courses.
Class notes have been developed and reside on the WWW for
faciliting use and feedback by colleagues (see http://www.ees.nmt.edu/Geop/Classes/GEOP529.html).
The authors' treatment promotes an understanding of fundamental
and practical issus associated with pararmeter fitting and
inverse problems including basic theory of inverse problems,
stastical issues, computational issues and an understanding of
how to analyze the success and limitations of solutiosn to these
probles. Second the text will be a practical resource for general
students and professional researchers, where well-referenced
techniques and concepts can be readily picked up on a chapter-by-chapter
basis. The text is structured around a course at New Mexico Tech
and is designed to be acessible to typical graduate students in
physical sciences without an extensive mathematical background.
The text will be accompanied by a Website containing Mathwork's
Matlab code corresponding to all examples.
Audience
Astrophysics Applied Mathematics Atmospheric Science Geologiocal
Engineering Geophysics Hydrology Oceanography Students: Every
year the U.S. awards 3,400 PhD in the physcial sciences along
with another 5,300 Engineering PhDs and 1,000 mathematics PhDs (NSF:
http://www.nsf.gov/sbe/srs/nsf02305/). Presumably, the potential
textbook market is approx. 10,000 students/year.
Contents
Preface 1. Introduction 2. Linear Regression 3. Discretizing
Continuous Inverse Problems 4. Rank Deficiency and Ill-Conditioning
5. Tikhonov Regularization 6. Iterative Methods 7. Other
Regularization Techniques 8. Fourier Techniques 9. Nonlinear
Regression 10. Nonlinear Inverse Problems 11. Bayesian Methods
Appendix A: Review of Linear Algebra Appendix B: Review of
Probability and Statistics Appendix C: Glossary of Notation
Bibliography Index
Hardbound, ISBN: 0-12-065604-3, 425 pages