Bertoin, J., Universite Pierre et Marie Curie,
Paris, France
Marinelli, F., Universita di Roma Tre, Italy
Peres, Y., Hebrew University, Jerusalem, Israel
Bernard, P., Universite Blaise Pascal, Aubiere, France
(Ed.)
Ecole d'Ete de Probabilites de Saint-Flour XXVII - 1997
1999. IX, 289 pp.
3-540-66593-5
Contents: I Subordinators: Examples and Applications:
Foreword.- Elements on subordinators.- Regenerative property.-
Asymptotic behaviour of last passage times.-
Rates of growth of local time.- Geometric properties of
regenerative sets.- Burgers equation with Brownian initial
velocity.- Random covering.- L?vy processes.- Occupation times of
a linear Brownian motion.-
II Lectures on Glauber Dynamics for Discrete Spin Models:
Introduction.- Gibs Measures of Lattice Spin
Models.- The Glauber Dynamics.- One Phase Region.- Boundary Phase
Transitions.- Phase Coexistence.-
Glauber Dynamics for the Dilute Ising Model.-
III Proability on Trees: An Introductory Climb: Preface.- Basic
Definitions and a Few Highlights.- Galton-Watson
Trees.- General percolation on a connected graph.- The
first-Moment method.- Quasi-independent
Percolation.- The second Moment Method.- Electrical Networks.-
Infinite Networks.- The Method of Random
Paths.- Transience of Percolation Clusters.- Subperiodic Trees.-
The Random Walks RW(lambda) .-
Capacity.-.Intersection-Equivalence.- Reconstruction for the
Ising Model on a Tree,- Unpredictable Paths in Z
and EIT in Z3.- Tree-Indexed Processes.- Recurrence for
Tree-Indexed Markov Chains.- Dynamical
Persolation.- Stochastic Domination Between Trees.- ....
Series: Lecture Notes in Mathematics.VOL. 1717
Eberle, A., University of Bielefeld, Germany
1999. VIII, 262 pp.
3-540-66628-1
This book addresses both probabilists working on diffusion
processes and analysts interested in linear parabolic
partial differential equations with singular coefficients. The
central question discussed is whether a given diffusion
operator, i.e., a second order linear differential operator
without zeroth order term, which is a priori defined on
test functions over some (finite or infinite dimensional) state
space only, uniquely determines a strongly
continuous semigroup on a corresponding weighted L space.
Particular emphasis is placed on phenomena causing
non-uniqueness, as well as on the relation between different
notions of uniqueness appearing in analytic and
probabilistic contexts.
Keywords: semigroup, uniqueness, essentially self-adjoint,
generator, diffusion process
Contents: Uniqueness problems in various contexts.- p uniqueness
in finite dimensions.- Markov uniqueness.-
Probabilistic aspects of p and Markov uniqueness.- First steps in
infinite dimensions.
Series: Lecture Notes in Mathematics.VOL. 1718
Courant, R., New York University, New York, USA
John, F., New York University, New York, USA
Reprint of the 1st ed. New York 1989 1999. XVI, 558 pp. 81
figs.
3-540-66569-2
From the reviews: "These books (Introduction to Calculus and
Analysis Vol. I/II) are very well written. The
mathematics are rigorous but the many examples that are given and
the applications that are treated make the
books extremely readable and the arguments easy to understand.
These books are ideally suited for an
undergraduate calculus course. Each chapter is followed by a
number of interesting exercises. More difficult parts
are marked with an asterisk. There are many illuminating
figures...Of interest to students, mathematicians,
scientists and engineers. Even more than that."
Newsletter on Computational and Applied Mathematics, 1991
"...one of the best textbooks introducing several
generations of mathematicians to higher mathematics. ... This
excellent book is highly recommended both to instructors and
students."
Acta Scientiarum Mathematicarum, 1991
Keywords: Differential equations Integral calculus calculus of
variations, curves and surfaces
Contents: Functions of Several Variables and Their Derivatives:
Points and Points Sets in the Plane and in
Space; Functions of Several Independent Variables; Continuity;
The Partial Derivatives of a Function; The
Differential of a Function and Its Geometrical Meaning; Functions
of Functions (Compound Functions) and the
Introduction of New Independent Variables; The mean Value Theorem
and Taylor's Theorem for Functions of
Several Variables; Integrals of a Function Depending on a
Parameter; Differentials and Line Integrals; The
Fundamental Theorem on Integrability of Linear Differential
Forms; Appendix.-
Vectors, Matrices, Linear Transformations: Operatios with
Vectors; Matrices and Linear Transformations;
Determinants; Geometrical Interpretation of Determinants; Vector
Notions in Analysis.-
Developments and Applications of the Differential Calculus:
Implicit Functions; Curves and Surfaces in Implicit
Form; Systems of Functions, Transformations, and Mappings;
Applications; Families of Curves, Families of
Surfaces, and Their Envelopes; Alternating Differential Forms;
Maxima and Minima; Appendix.-
Multiple Integrals: Areas in the Plane; Double Integrals;
Integrals over Regions in three and more Dimensions;
Space Differentiation. Mass and Density; Reduction of the
Multiple Integral to Repeated Single Integrals;
Transformation of Multiple Integrals; Improper Multiple
Integrals; Geometrical Applications; Physical Applications;
Multiple Integrals in Curvilinear Coordinates; Volumes and
Surface Areas in Any Number of Dimensions; Improper
Single Integrals as Functions of a Parameter; The Fourier
Integral; The Eulerian Integrals (Gamma Function);
Appendix
Series: Classics in Mathematics.
Duistermaat, J.J., Utrecht University, Utrecht,
The Netherlands
Kolk, J.A.C., Utrecht University, Utrecht, The Netherlands
1999. VIII, 334 pp.
3-540-15293-8
This book is a (post)graduate textbook on Lie groups and Lie
algebras. Its aim is to give a broad introduction to
the field with an emphasis on using differential-geometrical
methods, in the spirit of Lie himself. The structure of
compact Lie groups is analyzed in terms of the action of the
group on itself by conjugation. The book culminates
in the classification of the representations of compact Lie
groups and in their realization as sections of
holomorphic line bundles over flag manifolds. The relations with
algebraic and analytic models are also discussed.
A review of the required background material is provided in
appendices.
Keywords: Lie groups Lie algebras Representations of groups
Representations of algebras Group actions
Series: Universitext.
Fokkink, W., Centrum voor Wiskunde en Informatica, Amsterdam, The Netherlands
1999. VIII, 163 pp. 11 figs.
3-540-66579-X
Automated and semi-automated manipulation of so-called labelled
transition systems has become an important
means in discovering flaws in software and hardware systems.
Process algebra has been developed to express such
labelled transition systems algebraically, which enhances the
ways of manipulation by means of equational logic and
term rewriting.
The theory of process algebra has developed rapidly over the last
twenty years, and verification tools have been
developed on the basis of process algebra, often in cooperation
with techniques related to model checking. This
textbook gives a thorough introduction into the basics of process
algebra and its applications.
Keywords: process algebra, concurrency, algebraic specification,
operational semantics, term rewriting,
verification
Series: Texts in Theoretical Computer Science. An EATCS Series.
O'Meara, T.O., University of Notre Dame, IN, USA
Reprint of the 1st ed. Berlin Heidelberg New York 1963. Corr.
3rd printing 1973 1999. XIV, 344 pp. 10 figs.
3-540-66564-1
From the reviews: "O'Meara treats his subject from this
point of view (of the interaction with algebraic groups).
He does not attempt an encyclopedic coverage ...nor does he
strive to take the reader to the frontiers of
knowledge... . Instead he has given a clear account from first
principles and his book is a useful introduction to
the modern viewpoint and literature. In fact it presupposes only
undergraduate algebra (up to Galois theory
inclusive)... The book is lucidly written and can be warmly
recommended.
J.W.S. Cassels, The Mathematical Gazette, 1965
"Anyone who has heard O'Meara lecture will recognize in
every page of this book the crispness and lucidity of
the author's style;... The organization and selection of material
is superb... deserves high praise as an excellent
example of that too-rare type of mathematical exposition
combining conciseness with clarity...
R. Jacobowitz, Bulletin of the AMS, 1965
Keywords: Quadratic Forms Arithmetic Theory of Fields Arithmetic
Theory of Rings
"The exposition follows the tradition of the lectures of
Emil Artin who enjoyed developing a subject from first
principles and devoted much research to finding the simplest
proofs at every stage." - American Mathematical
Monthly
Contents: Prerequisites ad Notation Part One: Arithmetic Theory
of Fields
I Valuated Fields
Valuations
Archimedean Valuations
Non-Archimedean valuations
Prolongation of a complete valuation to a finite extension
Prolongation of any valuation to a finite separable extension
Discrete valuations
II Dedekind Theory of Ideals Dedekind axioms for S
Ideal theory
Extension fields
III Fields of Number Theory
Rational global fields
Local fields
Global fields
Part Two: Abstract Theory of Quadratic Forms
VI Quadratic Forms and the Orthogonal Group
Forms, matrices and spaces
Quadratic spaces
Special subgroups of On(V)
V The Algebras of Quadratic Forms
Tensor products
Wedderburn's theorem on central simple algebras
Extending the field of scalars
The clifford algebra
The spinor norm
Special subgroups of On(V)
Quaternion algebras
The Hasse algebra
VI The Equivalence of Quadratic Forms
Complete archimedean fields
Finite fields
Local fields
Global notation
Squares and norms in global fields
Quadratic forms over global fields
VII Hilbert's Reciprocity Law
Proof of the reciprocity law
Existence of forms with prescribed local behavior
The quadratic reciprocity law
Part Four: Arithmetic Theory of Quadratic Forms over Rings
VIII Quadratic Forms over Dedekind Domains
Abstract lattices
Lattices in quadratic spaces
IX Integral Theory of Quadratic Forms over Local Fields
Generalities
Classification of lattices over non-dyadic fields
Classification of Lattices over dyadic fields
Effective determination of the invariants
Special subgroups of On(V)
X Integral Theory of Quadratic Forms over Global Fields
Elementary properties of the orthogonal group over arithmetic
fields
The genus and the spinor genus
Finiteness of class number
The class and the spinor genus in the indefinite case
The indecomposable splitting of a definite lattice
Definite unimodular lattices over the rational integers
Bibliography
Index Bibliography
Index
Series: Classics in Mathematics.
Seroul, R., Universite Louis Pasteur, Strasbourg, France
1999. VIII, 395 pp.
3-540-66422-X
The aim of this book is to teach mathematics students how to
program using their knowledge of mathematics. For
this they require only to know how to construct a proof. The
entire book's emphasis is on "how to think" when
programming. Three methods for constructing an algorithm or a
program are used: a) manipulation and enrichment
of existing code; b) use of recurrent sequences; c) deferral of
code writing, in order to deal with one difficulty at
a time. Many theorems are mathematically proved and programmed.
The last chapter explains how a compiler
works and shows how to compile "by hand" little (but
not trivial--even recursive) programs. The book is intended
for anyone who thinks mathematically and wants to program and
play with mathematics.
Keywords: Programming, sequences, code transformation, top-down
programming
Contents: Programming Proverbs.- Review of Arithmetic.- An
Algorithmic Description Language.- How to
Create an Algorithm.- Algorithms and Classical Constructions.-
The Pascal Language.- How to Write a Program.-
The Integers.- The Comlex Numbers.- Polynomials.- Matrices.-
Recursion.- Elements of Compiler Theory.
Series: Universitext.
Neukirch, J.
Schmidt, A., University of Heidelberg, Germany
Wingberg, K., University of Heidelberg, Germany
1999. XV, 699 pp.
3-540-66671-0
Galois modules over local and global fields form the main subject
of this monograph, which can serve both as a
textbook for students, and as a reference book for the working
mathematician, on cohomological topics in number
theory. The first part provides the necessary algebraic
background. The arithmetic part deals with Galois groups
of local and global fields: local Tate duality, the structure of
the absolute Galois group of a local field, extensions
of global fields with restricted ramification, cohomology of the
id?le and the id?le class groups, Poitou-Tate
duality for finitely generated Galois modules, the Hasse
principle, the theorem of Grundwald-Wang, Leopoldt's
conjecture, Riemann's existence theorem for number fields,
embedding problems, the theorems of Iwasawa and of
Safarevic on solvable groups as Galois groups over global fields,
Iwasawa theory of local and global number fields,
and the characterization of number fields by their absolute
Galois groups.
Keywords: Algebraic number fields Cohomology theory Galois groups
Contents: I Algebraic Theory: Cohomology of Profinite Groups.-
Some Homological Algebra.- Duality Properties
of Profinite Groups.- Free Products of Profinite Groups.- Iwasawa
Modules
II Arithmetic Theory: Galois Cohomology.- Cohomology of Local
Fields.- Cohomology of Global Fields.- The
Absolute Galois Group of a Global Field.- Restricted
Ramification.- Iwasawa Theory of Number Fields; Anabelian
Geometry.- Literature.- Index.
Series: Grundlehren der mathematischen Wissenschaften.BD. 323