Description
This is the English translation of the book originally published
in Russian. It contains 20 essays, each dealing with a separate
mathematical topic. The topics range from brilliant mathematical
statements with interesting proofs, to simple and effective
methods of problem-solving, to interesting properties of
polynomials, to exceptional points of the triangle. Many of the
topics have a long and interesting history. The author has
lectured on them to students worldwide.
The essays are independent of one another for the most part, and
each presents a vivid mathematical result that led to current
research in number theory, geometry, polynomial algebra, or
topology.
Contents
Conjugate numbers
Rational parametrizations of the circle
Sums of squares of polynomials
Representing numbers as the sum of two squares
Can any knot be unraveled?
Construction of a regular 17-gon
The Markov equation
Integer-valued polynomials
Chebyshev polynomials
Vectors in geometry
The averaging method and geometric inequalities
Intersection points of the diagonals of regular polygons
The chromatic polynomial of a graph
Brocard points
Diophantine equations for polynomials
The Pascal lines
One butterfly and two butterflies theorems
The Van der Waerden theorem on arithmetical progressions
Isogonal conjugate points
Cubic curves related to the triangle
Details:
Series: Mathematical World, Volume: 16
Publication Year: 2000
ISBN: 0-8218-1944-5
Paging: 75 pp.
Binding: Softcover
Edited by:
B. Brent Gordon, University of Oklahoma, Norman, OK, James D.
Lewis, University of Alberta, Edmonton, AB, Canada,
Stefan Muller-Stach, Universitaa Essen, Germany, Shuji Saito,
Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Japan, and
Noriki Yui, Queen's University, Kingston, ON, Canada
Description
The NATO ASI/CRM Summer School at Banff offered a unique, full,
and in-depth account of the topic, ranging from introductory
courses by leading experts to discussions of the latest
developments by all participants. The papers have been organized
into three categories: cohomological methods; Chow groups and
motives; and arithmetic methods.
As a subfield of algebraic geometry, the theory of algebraic
cycles has gone through various interactions with algebraic
$K$-theory, Hodge theory, arithmetic algebraic geometry, number
theory, and topology. These interactions have led to developments
such as a description of Chow groups in terms of algebraic
$K$-theory, the application of the Merkurjev-Suslin theorem to
the arithmetic Abel-Jacobi mapping, progress on the celebrated
conjectures of Hodge, and of Tate, which compute cycles class
groups respectively in terms of Hodge theory or as the invariants
of a Galois group action on ?tale cohomology, the conjectures of
Bloch and Beilinson, which explain the zero or pole of the $L$-function
of a variety and interpret the leading non-zero coefficient of
its Taylor expansion at a critical point, in terms of arithmetic
and geometric invariant of the variety and its cycle class
groups.
The immense recent progress in the theory of algebraic cycles is
based on its many interactions with several other areas of
mathematics. This conference was the first to focus on both
arithmetic and geometric aspects of algebraic cycles. It brought
together leading experts to speak from their various points of
view. A unique opportunity
was created to explore and view the depth and the breadth of the
subject. This volume presents the intriguing results.
Contents
Cohomological methods
S. Abdulali -- Filtrations on the cohomology of abelian varieties
D. Arapura -- Building mixed Hodge structures
R.-O. Buchweitz and H. Flenner -- The Atiyah-Chern character
yields the semiregularity map as well as the infinitesimal
Abel-Jacobi map
J. Dupont, R. Hain, and S. Zucker -- Regulators and
characteristic classes of flat bundles
B. Harris and B. Wang -- Height pairings asymptotics and
Bott-Chern forms
K. Kato and S. Usui -- Logarithmic Hodge structures and
classifying spaces
Chow groups and motives
M. Asakura -- Motives and algebraic de Rham cohomology
J. I. Burgos Gil -- Hermitian vector bundles and characteristic
classes
M. Hanamura -- The mixed motive of a projective variety
C. Pedrini -- Bloch's conjecture and the $K$-theory of projective
surfaces
N. Ramachandran -- From Jacobians to one-motives: Exposition of a
conjecture of Deligne
S. Saito -- Motives, algebraic cycles and Hodge theory
Arithmetic methods
C. F. Doran -- Picard-Fuchs uniformization: Modularity of
the mirror map and mirror-moonshine
E. Z. Goren -- Hilbert modular varieties in positive
characteristic
Y. Goto -- On the N?ron-Severi groups of some $K$3 surfaces
J. van Hamel -- Torsion zero-cycles and the Abel-Jacobi map over
the real numbers
K. Kimura -- A remark on the Griffiths groups of certain product
varieties
J. Nekov?r -- $p$-adic Abel-Jacobi maps and $p$-adic heights
A. Shiho -- Crystalline fundamental groups and $p$-adic Hodge
theory
H. Verrill and N. Yui -- Thompson series, and the mirror maps of
pencils of $K$3 surfaces
Details:
Series: CRM Proceedings & Lecture Notes, Volume: 24
Publication Year: 2000
ISBN: 0-8218-1954-2
Paging: 432 pp.
Binding: Softcover
Description
This volume contains the proceedings from the first DIMACS
meeting on discrete mathematical chemistry held at Rutgers
University (New Brunswick, NJ). The contributions reflect the
presentations and spotlight the breadth of current research on
the topic--from the Benzenoid Clar problem to the Wulff-shape of
sphere packings. Much of the volume reflects the combined
mathematical and physical interest in the new molecules,
fullerenes.
This DIMACS conference highlighted the range of opportunities for
fruitful and informed collaboration across the
mathematics-chemistry boundaries. The interdisciplinary nature of
the contributions pays testament to the fact that
"real" chemistry and "real" mathematics do
indeed interact.
Contents
H. Abeledo and G. Atkinson -- The Clar and Fries
problems for benzenoid hydrocarbons are linear programs
S. C. Basak and B. D. Gute -- Use of graph invariants in QMSA and
predictive toxicology
G. Brinkmann -- Isomorphism rejection in structure generation
programs
L. Bytautas, D. J. Klein, M. Randic, and T. Pisanski --
Foldedness in linear polymers: A difference between graphical and
Euclidean distances
G. Caporossi, P. Hansen, and M. Zheng -- Enumeration of fusenes
to $h$ = 20
D. Cvetkovic -- Characterizing properties of some graph
invariants related to electron charges in the H?ckel molecular
orbital theory
O. D. Friedrichs -- Fast embeddings for planar molecular graphs
O. D. Friedrichs and M. Deza -- More icosahedral fulleroids
A. Dietz, C. Fiorio, M. Habib, and C. Lauren?o -- Representation
of stereochemistry using combinatorial maps
B. T. Fan, A. Panaye, J. H. Yao, S. G. Yuan, and J. P. Doucet --
Geometric symmetry and chemical equivalence
P. W. Fowler, P. E. John, and H. Sachs -- (3-6)-cages, hexagonal
toroidal cages, and their spectra
P. W. Fowler, T. Pisanski, A. Graovac, and J. Zerovnik -- A
generalized ring spiral algorithm for coding fullerenes and other
cubic polyhedra
X. Guo and M. Randic -- An efficient algorithm for determining
fixed bonds and normal components in a bipartite graph
R. Hefferlin -- Numerical solutions of the Laplace equation in
chemical spaces
S. T. Hyde and S. Ramsden -- Chemical frameworks and hyperbolic
tilings
A. Kerber, R. Laue, and T. Wieland -- Discrete mathematics for
combinatorial chemistry
R. B. King -- Carbon networks on cubic infinite periodic minimal
surfaces
S. Klavzar -- Applications of isometric embeddings to chemical
graphs
J. Malkevitch -- Geometrical and combinatorial questions about
fullerenes
P. G. Mezey -- Topological methods of molecular shape analysis:
Continuum models and discretization
H. M. Ohlenbusch, N. Rivier, T. Aste, and B. Dubertret -- Random
networks in two dimensions. Simulations and correlations
C. M. Quinn, D. B. Redmond, and P. W. Fowler -- Group and graph
theoretical perspectives on the structures of large icosahedral
cages
M. Randic, X. Guo, and S. Bobst -- Use of path matrices for a
characterization of molecular structures
P. Rowlinson -- Star sets and star complements in finite graphs:
A spectral construction technique
H. Terrones and M. Terrones -- Geometry and energetics of high
genus fullerenes and nanotubes
P. Vismara and C. Lauren?o -- An abstract representation for
molecular graphs
J. M. Wills -- The Wulff-shape of large periodic sphere packings
A. V. Zeigarnik -- On hypercycles and hypercircuits in
hypergraphs
F. Zhang and H. Li -- On maximal energy ordering of acyclic
conjugated molecules
Details:
Series: DIMACS: Series in Discrete Mathematics and
Theoretical Computer Science,
Publication Year: 2000
ISBN: 0-8218-0987-3
Paging: 392 pp.
Binding: Hardcover
Kazuya Kato, University of Tokyo,
Japan, Nobushige Kurokawa, Tokyo
Institute of Technology, Japan,
and Takeshi Saito, University of Tokyo, Japan
Iwanami Series in Modern Mathematics
Description
This is the English translation of the original Japanese
book. In this volume, "Fermat's Dream", core
theories in modern number theory are introduced. Developments are
given in elliptic curves, $p$-adic numbers, the $\zeta$-function,
and the number fields. This work presents an elegant perspective
on the wonder of numbers. Number Theory 2 on class field theory,
and Number Theory 3 on Iwasawa theory and the theory of modular
forms, are forthcoming in the series.
Contents
Introduction
Rational points on elliptic curves
Conics and $p$-adic numbers
$\zeta$
Algebraic number theory
Rudiments on Dedekind domains
Answers to questions
Answers to exercises
Index
Details:
Series: Translations of Mathematical Monographs,
Volume: 186
Subseries: Iwanami Series in Modern Mathematics
Publication Year: 2000
ISBN: 0-8218-0863-X
Paging: 154 pp.
Binding: Softcover
Description
Model theoretic algebra has witnessed remarkable progress in the
last few years. It has found profound applications in other areas
of mathematics, notably in algebraic geometry and in singularity
theory.
Since Wilkie's results on the o-minimality of the expansion of
the reals by the exponential function, and most recently even by
all Pfaffian functions, the study of o-minimal expansions of the
reals has become a fascinating topic. The quest for analogies
between the semi-algebraic case and the o-minimal case has set a
direction to this research.
Through the Artin-Schreier Theory of real closed fields, the
structure of the non-archimedean models in the semi-algebraic
case is well understood. For the o-minimal case, so far there has
been no systematic study of the non-archimedean models. The goal
of this monograph is to serve this purpose.
The author presents a detailed description of the non-archimedean
models of the elementary theory of certain o-minimal expansions
of the reals in which the exponential function is definable. The
example of exponential Hardy fields is worked out with particular
emphasis. The basic tool is valuation theory, and a sufficient
amount of
background material on orderings and valuations is presented for
the convenience of the reader.
Contents
Preliminaries on valued and ordered modules
Non-archimedean exponential fields
Valuation theoretic interpretation of the growth and Taylor
axioms
The exponential rank
Construction of exponential fields
Models for the elementary theory of the reals with restricted
analytic functions and exponentiation
Exponential Hardy fields
The model theory of contraction groups
Bibliography
Index
List of notation
Details:
Series: Fields Institute Monographs, Volume: 12
Publication Year: 2000
ISBN: 0-8218-0943-1
Paging: 166 pp.
Binding: Hardcover
A publication of Societe Mathematique de
France.
Description
This volume presents the geometric properties of $n$-manifolds
($n\geq 2$) with geometric structures modeled on $(\mathbf {R}
P^n, \textnormal{PGL}(n+1, \mathbf {R}))$, i.e., $n$-manifolds
with projectively flat torsion-free affine connections. The
author begins by defining the notion of $i$-convexity of such
manifolds (due to Carriere) for integers $i$, $1 \leq i \leq
n-1$, which are generalizations of convexity. Given a real
projective $n$-manifold $M$, the failure of an $(n-1)$-convexity
of $M implies the existence of a certain geometric object, an
$n$-crescent, in the completion $\check M$ of the universal cover
$\tilde M$ of $M$. This further implies the existence of a
particular type of affine submanifold in $M$ and gives a natural
decomposition of $M$ into simpler real projective manifolds, some
of which are
$(n-1)$-convex and others which are affine, more specifically
concave affine. It is useful to have such a decomposition
particularly in dimension three. The result will later aid in
studying the geometric and topological properties of radiant
affine $3$-manifolds leading to their classification. The author
obtains a consequence for affine Lie groups.
Contents
Part I. An introduction to real projective
structures
Introduction
Convex subsets of the real projective sphere
Convex subsets in the Kuiper completions
Part II. $(n-1)$-convexity and decomposition
$(n-1)$-convexity and $n$-crescents
The transversal intersection of $n$-crescents
Hemispheric $n$-crescents and two faced submanifolds
Bihedral $n$-crescents and two-faced submanifolds
The preservation of crescents after decomposing and splitting
The construction of concave affine manifolds
Splitting and decomposing manifolds
Left-invariant real projective structures on Lie groups
Part III. Appendices
A. Two miscellaneous theorems
B. Shrinking and expanding $n$-balls by projective maps
Frequently used symbols
Bibliography
Index
Details:
Series: Memoires de la Societe Math?matique de France, ISSN:
0249-633X
Number: 78
Publication Year: 1999
ISBN: 2-85629-079-5
Paging: 102 pp.
Binding: Softcover
Description
The book treats free probability theory, which has been
extensively developed since the early eighties. The emphasis is
put on entropy and the random matrix model approach. It is a
unique presentation demonstrating the extensive interrelation
between the topics. Wigner's theorem and its broad
generalizations, such as asymptotic freeness of independent
matrices, are explained in detail. Consistent throughout is the
parallelism between the normal and semicircle laws. The authors
present Voiculescu's multivariate free entropy theory with full
proofs and extend the results to unitary operators. Some
applications to operator algebras are also given.
The book is the first essentially full-scale presentation on free
probability theory and includes improvements of results and
proofs in current literature. The combinatorial aspects of the
specialized topics are emphasized; many examples are given. The
book would be a suitable text for graduate courses in free
probability theory.
Contents
Overview
Probability laws and noncommutative random variables
The free relation
Analytic function theory and infinitely divisible laws
Random matrices and asymptotically free relation
Large deviations for random matrices
Free entropy of noncommutative random variables
Relation to operator algebras
Bibliography
Index
Details:
Series: Mathematical Surveys and Monographs,
Publication Year: 2000
ISBN: 0-8218-2081-8
Paging: approximately 386 pp.
Binding: Hardcover