2nd ed.2003Approx. 360 pp. 59 figs., with
CD-ROM. Hardcover
3-540-44213-8
This second edition of G. Winkler's successful
book on random
field approaches to image analysis, related
Markov Chain Monte
Carlo methods, and statistical inference
with emphasis on
Bayesian image analysis concentrates more
on general principles
and models and less on details of concrete
applications.
Addressed to students and scientists from
mathematics,
statistics, physics, engineering, and computer
science, it will
serve as an introduction to the mathematical
aspects rather than
a survey. Basically no prior knowledge of
mathematics or
statistics is required.
The second edition is in many parts completely
rewritten and
improved, and most figures are new. The topics
of exact sampling
and global optimization of likelihood functions
have been added.
This second edition comes with a CD-ROM by
F.
Friedrich,containing a host of (live) illustrations
for each
chapter. In an interactive environment, readers
can perform their
own experiments to consolidate the subject.
Keywords: Bayesian statistics, Markov chain
Monte Carlo methods,
random fields, spatial modeling, statistical
image analysis
Contents: I.Bayesian Image Analysis.- Introduction.-
The Bayesian
Paradigm.- Cleaning Dirty Pictures.- Random
Fields.- II. The
Gibbs Sampler and Simulated Annealing.- Markov
Chains: Limit
Theorems.- Gibbsian Sampling and Annealing.-
Cooling Schedules.-
Gibbsian Sampling and Annealing Revisited.-
III. More on Sampling
and Annealing.- Metropolis Algorithms.- Eigenvalues
and Related
Topics.- Parallel Algorithms.- IV. Texture
Analysis.-
Partitioning.- Random Fields and Texture
Models.- Bayesian
Texture Classification.- V. Parameter Estimation.-
Maximum
Likelihood Estimation.- Consistency of Spatial
ML .- Computation
of full ML Estimators.- VI. Supplement.-
A Glance at Neural
Networks.- Three Applications.- VII. Appendix.-
A Simulation of
Random Variables.-B Analytical Tools.- C
Physical Imaging Systems.
Series: Applications of Mathematics. Volume.
27
2003XVIII, 460 p. Hardcover
3-540-44228-6
Linear differential equations form the central
topic of this
volume, with the Galois theory being the
unifying theme.
A large number of aspects are presented:
algebraic theory
especially differential Galois theory, formal
theory,
classification, algorithms to decide solvability
in finite terms,
monodromy and Hilbert's 21th problem, asymptotics
and
summability, the inverse problem and linear
differential
equations in positive characteristic. The
appendices aim to help
the reader with the concepts of algebraic
geometry, linear
algebraic groups, sheaves, and tannakian
categories that are used.
This volume will become a standard reference
for all working in
this area of mathematics at the graduate
level, including
graduate students.
Keywords: Arithmetic, Asymptotics, Computer
Algebra, Direct
Problems, Galois Theory, Global Classification,
Inverse Problems,
Linear Differential Equations, Local Classification
Contents: Algebraic Theory: Picard-Vessiot
Theory.- Differential
Operators and Differential Modules.- Formal
Local Theory.-
Algorithmic Considerations
Analytic Theory: Monodromy, the Riemann-Hilbert
Problem and the
Differential Galois Group.- Differential
Equations on the Complex
Sphere and the Riemann-Hilbert Problem.-
Exact Asymptotics.-
Stokes Phenomenon and Differential Galois
Groups.- Stokes
Matrices and Meromorphic Classification.-
Universal Picard-Vessiot
Rings and Galois Groups.- Inverse Problems.-
Moduli for Singular
Differential Equations.- Positive Characteristic
Appendices: Algebraic Geometry.- Tannakian
Categories.- Sheaves
and Cohomology.- Partial Differential Equations.
Series: Grundlehren der mathematischen Wissenschaften.
Volume.
328
2002VIII, 100 p. Softcover
3-540-44243-X
This monograph contributes to the existence
theory of difference
sets, cyclic irreducible codes and similar
objects. The new
method of field descent for cyclotomic integers
of presribed
absolute value is developed. Applications
include the first
substantial progress towards the Circulant
Hadamard Matrix
Conjecture and Ryser`s conjecture since decades.
It is shown that
there is no Barker sequence of length l with
13<1<4x10^(12).
Finally, a conjecturally complete classification
of all
irreducible cyclic two-weight codes is obtained.
Keywords: Barker sequences, Circulant Hadamard
matrices,
Differences sets, irreducible cyclic codes
Contents: 1. Introduction: The nature of
the problems.- The
combinatorial structures in question.- Group
rings, characters,
Fourier analysis.- Number theoretic tools.-
Algebraic-combinatorial
tools. 2. The field descent: The fixing theorem.-
Prescribed
absolute value.- Bounding the absoute value.-
The modulus
equation and the class group. 3. Exponent
bounds: Self-conjugacy
exponent bounds.- Field descent exponent
bounds. 4. Two-weight
irreducible cyclic bounds: A necessary and
sufficient condition.-
All two-weight irreducible cyclic codes?-
Partial proof of
Conjecture 4.2.4.- Two-intersection sets
and sub-difference sets
Series: Lecture Notes in Mathematics. Volume.
1797
2003XVI, 492 p. Hardcover
3-540-44133-6
Global class field theory is a major achievement
of algebraic
number theory, based on the functorial properties
of the
reciprocity map and the existence theorem.
The author works out
the consequences and the practical use of
these results by giving
detailed studies and illustrations of classical
subjects (classes,
ideles, ray class fields, symbols, reciprocity
laws, Hasse's
principles, the Grunwald-Wang theorem, Hilbert's
towers,...). He
also proves some new or less-known results
(reflection theorem,
structure of the abelian closure of a number
field) and lays
emphasis on the invariant (/cal T) p, of
abelian p-ramification,
which is related to important Galois cohomology
properties and p-adic
conjectures. This book, intermediary between
the classical
literature published in the sixties and the
recent computational
literature, gives much material in an elementary
way, and is
suitable for students, researchers, and all
who are fascinated by
this theory.
Keywords: Abelian closure, Class field theory,
idele groups,
number fields, reciprocity laws
Contents: Preface Introduction to Global
Class Field Theory
Chapter I: Basic Tools and Notations 1) Places
of a number field
2) Embeddings of a Number Field in its Completions
3) Number and
Ideal Groups 4) Idele Groups - Generalized
Class Groups 5)
Reduced Ideles - Topological Aspects 6) Kummer
Extensions Chapter
II: Reciprocity Maps - Existence Theorems
1) The Local
Reciprocity Map - Local Class Field Theory
2) Idele Groups in an
Extension L/K 3) Global Class Field Theory:
Idelic Version 4)
Global Class Field Theory: Class Group Version
5) Ray Class
Fields 6) The Hasse Principle - For Norms
- For Powers 7) Symbols
Over Number Fields - Hilbert and Regular
Kernels Chapter III:
Abelian Extensions with Restricted Ramification
- Abelian Closure
1) Generalities on H(T)/H and its Subextensions
2) Computation of
A(T) := Gal(H(T)/K) and T(T) := tor(A(T))
3) Study of the
compositum of the Zp-extensions - The p-adic
Conjecture 4)
Structure Theorems for the Abelian Closure
of K 5) Explicit
Computations in Incomplete p-Ramification
6) The Radical of the
Maximal Elementary Subextension of the compositum
of the Zp-extensions
Chapter IV: Invariant Classes Formulas in
p-ramification - Genus
Theory 1) Reduction to the Case of p-Ramification
2) Injectivity
of the Transfer Map: A(K,p) to A(L,p) 3)
Determination of
invariant classes of A(L,p) and T(L,p) -
p-Rationality 4) Genus
Theory with Ramification and Decomposition
Chapter V: Cyclic
Extensions with Prescribed Ramification 1)
Study of an Example 2)
Construction of a Governing Field 3) Conclusion
and Perspectives
Appendix: Arithmetical Interpretation of
the second cohomology
group of G(T,S) over Zp 1) A General Approach
by Class Field
Theory 2) Complete p-Ramification Without
Finite Decomposition 3)
The General Case - Infinitesimal Knot Groups
Bibliography Index
of Notations
Series: Springer Monographs in Mathematics.
2003Approx. 400 p. With CD-ROM. Hardcover
3-540-43826-2
This book contains a series of research papers
on subjects
related to the work of Niels Henrik Abel,
written by some of the
foremost specialists in their fields.
Some of the authors have been specifically
invited to present
papers, discussing the influence of Abel
in a mathematical-historical
context. Others have submitted papers presented
at the Abel
Bicentennial Conference, Oslo June 3-8, 2002.
The idea behind the
book has been to produce a text covering
a substantial part of
the legacy of Abel, as perceived at the beginning
of the 21st
century.
Keywords: algebraic geometry, complex analysis,
history of
mathematics, non-communtative geometry
Contents: From the Contents: History of Mathematics:
A. Stubhaug,
Ch. Houzel, F. Catanese, and B. Mazur, together
with research
papers by D. Lazard, G. Frei, N. Schappacher
on Algebraic
Equations/Arithmetics, by G. Faltings, C.
Ciliberto, A.
Brigaglia, S. Kleiman, W. Fulton, and T.
Ekedahl on Algebraic
Geometry, by P.Griffiths, M. Green, Y.-T.
Siu, and J.-E. Bjorck
on Complex Analysis, by G. Henkin, M. Passare,
and S. Gindikin on
Differential equations, and by Yu.Manin.
Moreover the plenary speakers at the conference
have all been
invited to submit papers. This includes A.
Connes, G. Faltings, W.
Fulton, H.W.Lenstra, K.A. Ribet, D. Sullivan,
A. Wiles, M.
Kontsevich, and D. Zagier.