SHERMAN
Spatial statistics and spatio-temporal data
Covariance functions and directional properties
In
the spatial or spatio-temporal context, specifying the correct
covariance function is fundamental to obtain efficient predictions, and
to understand the underlying physical process of interest. This book
focuses on covariance and variogram functions, their role in prediction,
and appropriate choice of these functions in applications. Both recent
and more established methods are illustrated to assess many common
assumptions on these functions, such as, isotropy, separability,
symmetry, and intrinsic correlation. After an extensive introduction to
spatial methodology, the book details the effects of common covariance
assumptions and addresses methods to assess the appropriateness of such
assumptions for various data structures. Key features: * An extensive
introduction to spatial methodology including a survey of spatial
covariance functions and their use in spatial prediction (kriging) is
given. * Explores methodology for assessing the appropriateness of
assumptions on covariance functions in the spatial, spatio-temporal,
multivariate spatial, and point pattern settings. * Provides
illustrations of all methods based on data and simulation experiments to
demonstrate all methodology and guide to proper usage of all methods. *
Presents a brief survey of spatial and spatio-temporal models,
highlighting the Gaussian case and the binary data setting, along with
the different methodologies for estimation and model fitting for these
two data structures. * Discusses models that allow for anisotropic and
nonseparable behaviour in covariance functions in the spatial,
spatio-temporal and multivariate settings. * Gives an introduction to
point pattern models, including testing for randomness, and fitting
regular and clustered point patterns. The importance and assessment of
isotropy of point patterns is detailed. Statisticians, researchers, and
data analysts working with spatial and space-time data will benefit from
this book as well as will graduate students with a background in basic
statistics following courses in engineering, quantitative ecology or
atmospheric science. BROCKWELL ; DAVIS Time Series Theory and Methods This
paperback edition is a reprint of the 1991 edition. Time Series: Theory
and Methods is a systematic account of linear time series models and
their application to the modeling and prediction of data collected
sequentially in time. The aim is to provide specific techniques for
handling data and at the same time to provide a thorough understanding
of the mathematical basis for the techniques. Both time and frequency
domain methods are discussed, but the book is written in such a way that
either approach could be emphasized. The book is intended to be a text
for graduate students in statistics, mathematics, engineering, and the
natural or social sciences. It contains substantial chapters on
multivariate series and state-space models (including applications of
the Kalman recursions to missing-value problems) and shorter accounts of
special topics including long-range dependence, infinite variance
processes, and nonlinear models. Most of the programs used in the book
are available in the modeling package ITSM2000, the student version of
which can be downloaded from
http://www.stat.colostate.edu/~pjbrock/student06.
JOLLIFFE
Principal component analysis The
first edition of this book was the first comprehensive text written
solely on principal component analysis. The second edition updates and
substantially expands the original version, and is once again the
definitive text on the subject. It includes core material, current
research and a wide range of applications. Its length is nearly double
that of the first edition. DUGGAL ; HO JI Null Curves and Hypersurfaces of Semi-Riemannian Manifolds This
is a first textbook that is entirely focused on the up-to-date
developments of null curves with their applications to science and
engineering. It fills an important gap in a second-level course in
differential geometry, as well as being essential for a core
undergraduate course on Riemannian curves and surfaces. The sequence of
chapters is arranged to provide in-depth understanding of a chapter and
stimulate further interest in the next. The book comprises a large
variety of solved examples and rigorous exercises that range from
elementary to higher levels. This unique volume is self-contained and
unified in presenting: a systematic account of all possible null curves,
their Frenet equations, unique null Cartan curves in Lorentzian
manifolds and their practical problems in science and engineering; and,
the geometric and physical significance of null geodesics, mechanical
systems involving curvature of null curves, simple variation problems
and the interrelation of null curves with hypersurfaces. KLAFTER ; SOKOLOV First steps in random walks
from tools to applications The
name "random walk" for a problem of a displacement of a point in a
sequence of independent random steps was coined by Karl Pearson in 1905
in a question posed to readers of "Nature". The same year, a similar
problem was formulated by Albert Einstein in one of his Annus Mirabilis
works. Even earlier such a problem was posed by Louis Bachelier in his
thesis devoted to the theory of financial speculations in 1900. Nowadays
the theory of random walks has proved useful in physics and chemistry
(diffusion, reactions, mixing in flows), economics, biology (from animal
spread to motion of subcellular structures) and in many other
disciplines. The random walk approach serves not only as a model of
simple diffusion but of many complex sub- and super-diffusive transport
processes as well. This book discusses the main variants of random walks
and gives the most important mathematical tools for their theoretical
description. ZELIKIN
Control theory and optimization I
Homogeneous spaces and the Riccati equation in the calculus of variations
The
only monograph on the topic, this book concerns geometric methods in
the theory of differential equations with quadratic right-hand sides,
closely related to the calculus of variations and optimal control
theory. Based on the author's lectures, the book is addressed to
undergraduate and graduate students, and scientific researchers. ONISHCHICK ; VINBERG Lie groups and lie algebras II
A
systematic survey of all the basic results on the theory of discrete
subgroups of Lie groups, presented in a convenient form for users. The
book makes the theory accessible to a wide audience, and will be a
standard reference for many years to come.
CARTER ; KAMADA ; SAITO
Surfaces in 4-Space "Surfaces
in 4-Space", written by leading specialists in the field, discusses
knotted surfaces in 4-dimensional space and surveys many of the known
results in the area. Results on knotted surface diagrams, constructions
of knotted surfaces, classically defined invariants, and new invariants
defined via quandle homology theory are presented. The last chapter
comprises many recent results, and techniques for computation are
presented. New tables of quandles with a few elements and the homology
groups thereof are included. This book contains many new illustrations
of knotted surface diagrams. The reader of the book will become
intimately aware of the subtleties in going from the classical case of
knotted circles in 3-space to this higher dimensional case. As a survey,
the book is a guide book to the extensive literature on knotted
surfaces and will become a useful reference for graduate students and
researchers in mathematics and physics. LAKSHMIBAI ; RAGHAVAN Standard Monomial Theory
Invariant Theoretic Approach Schubert
varieties provide an inductive tool for studying flag varieties. This
book is mainly a detailed account of a particularly interesting instance
of their occurrence: namely, in relation to classical invariant theory.
More precisely, it is about the connection between the first and second
fundamental theorems of classical invariant theory on the one hand and
standard monomial theory for Schubert varieties in certain special flag
varieties on the other.
DERKSEN ; KEMPER
Computational Invariant Theory This
book, the first volume of a subseries on "Invariant Theory and
Algebraic Transformation Groups", provides a comprehensive and
up-to-date overview of the algorithmic aspects of invariant theory.
Numerous illustrative examples and a careful selection of proofs make
the book accessible to non-specialists.
LANDO ; ZVONKIN
Graphs on surfaces and their applications
Graphs
drawn on two-dimensional surfaces have always attracted researchers by
their beauty and by the variety of difficult questions to which they
give rise. The theory of such embedded graphs, which long seemed rather
isolated, has witnessed the appearance of entirely unexpected new
applications in recent decades, ranging from Galois theory to quantum
gravity models, and has become a kind of a focus of a vast field of
research. The book provides an accessible introduction to this new
domain, including such topics as coverings of Riemann surfaces, the
Galois group action on embedded graphs (Grothendieck's theory of
"dessins d'enfants"), the matrix integral method, moduli spaces of
curves, the topology of meromorphic functions, and combinatorial aspects
of Vassiliev's knot invariants and, in an appendix by Don Zagier, the
use of finite group representation theory. The presentation is concrete
throughout, with numerous figures, examples (including computer
calculations) and exercises, and should appeal to both graduate students
and researchers.
ADLER ; TAYLOR
Random fields and geometry This
monograph is devoted to a completely new approach to geometric problems
arising in the study of random fields. The groundbreaking material in
Part III, for which the background is carefully prepared in Parts I and
II, is of both theoretical and practical importance, and striking in the
way in which problems arising in geometry and probability are
beautifully intertwined. "Random Fields and Geometry" will be useful for
probabilists and statisticians, and for theoretical and applied
mathematicians who wish to learn about new relationships between
geometry and probability. It will be helpful for graduate students in a
classroom setting, or for self-study. Finally, this text will serve as a
basic reference for all those interested in the companion volume of the
applications of the theory.
MILSTEIN ; TRETYAKOV
Stochastic Numerics for Mathematical Physics
Stochastic
differential equations have many applications in the natural sciences.
Besides, the employment of probabilistic representations together with
the Monte Carlo technique allows us to reduce solution of
multi-dimensional problems for partial differential equations to
integration of stochastic equations. This approach leads to powerful
computational mathematics that is presented in the treatise. The authors
propose many new special schemes, some published here for the first
time. In the second part of the book they construct numerical methods
for solving complicated problems for partial differential equations
occurring in practical applications, both linear and nonlinear. All the
methods are presented with proofs and hence founded on rigorous
reasoning, thus giving the book textbook potential. An overwhelming
majority of the methods are accompanied by the corresponding numerical
algorithms which are ready for implementation in practice. The book
addresses researchers and graduate students in numerical analysis,
physics, chemistry, and engineering as well as mathematical biology and
financial mathematics.
DEBNATH
The legacy of Leonhard Euler This
book primarily serves as a historical research monograph on the
biographical sketch and career of Leonhard Euler and his major
contributions to numerous areas in the mathematical and physical
sciences. It contains fourteen chapters describing Euler's works on
number theory, algebra, geometry, trigonometry, differential and
integral calculus, analysis, infinite series and infinite products,
ordinary and elliptic integrals and special functions, ordinary and
partial differential equations, calculus of variations, graph theory and
topology, mechanics and ballistic research, elasticity and fluid
mechanics, physics and astronomy, probability and statistics. The book
is written to provide a definitive impression of Euler's personal and
professional life as well as of the range, power, and depth of his
unique contributions. This tricentennial tribute commemorates Euler the
great man and Euler the universal mathematician of all time. Based on
the author's historically motivated method of teaching, special
attention is given to demonstrate that Euler's work had served as the
basis of research and developments of mathematical and physical sciences
for the last 300 years. An attempt is also made to examine his
research and its relation to current mathematics and science. Based on a
series of Euler's extraordinary contributions, the historical
development of many different subjects of mathematical sciences is
traced with a linking commentary so that it puts the reader at the
forefront of current research. CHARPENTIER ; GHYS ; LESNE The scientific legacy of Poincaré Henri
Poincare (1854-1912) was one of the greatest scientists of his time,
perhaps the last one to have mastered and expanded almost all areas in
mathematics and theoretical physics. He created new mathematical
branches, such as algebraic topology, dynamical systems, and automorphic
functions, and he opened the way to complex analysis with several
variables and to the modern approach to asymptotic expansions. He
revolutionized celestial mechanics, discovering deterministic chaos. In
physics, he is one of the fathers of special relativity, and his work in
the philosophy of sciences is illuminating. For this book, about twenty
world experts were asked to present one part of Poincare's
extraordinary work. Each chapter treats one theme, presenting Poincare's
approach, and achievements, along with examples of recent applications
and some current prospects. Their contributions emphasize the power and
modernity of the work of Poincare, an inexhaustible source of
inspiration for researchers, as illustrated by the Fields Medal awarded
in 2006 to Grigori perelman for his proof of the Poincare conjecture
stated a century before. This book can be read by anyone with a
master's (even a bachelor's) degree in mathematics, or physics, or more
generally by anyone who likes mathematical and physical ideas. Rather
than presenting detailed proofs, the main ideas are explained, and a
bibliography is provided for those who wish to understand the technical
details.
BOBENKO ; KLEIN
Computational
approach to Riemann surfaces This
volume offers a well-structured overview of existent computational
approaches to Riemann surfaces and those currently in development. The
authors of the contributions represent the groups providing publically
available numerical codes in this field. Thus this volume illustrates
which software tools are available and how they can be used in practice.
In addition examples for solutions to partial differential equations
and in surface theory are presented. The intended audience of this book
is twofold. It can be used as a textbook for a graduate course in
numerics of Riemann surfaces, in which case the standard undergraduate
background, i.e., calculus and linear algebra, is required. In
particular, no knowledge of the theory of Riemann surfaces is expected;
the necessary background in this theory is contained in the Introduction
chapter. At the same time, this book is also intended for specialists
in geometry and mathematical physics applying the theory of Riemann
surfaces in their research. It is the first book on numerics of Riemann
surfaces that reflects the progress made in this field during the last
decade, and it contains original results. There are a growing number of
applications that involve the evaluation of concrete characteristics of
models analytically described in terms of Riemann surfaces. Many
problem settings and computations in this volume are motivated by such
concrete applications in geometry and mathematical physics. BRENDLE
Ricci flow and the sphere theorem
"In
1982, R. Hamilton introduced a nonlinear evolution equation for
Riemannian metrics with the aim of finding canonical metrics on
manifolds. This evolution equation is known as the Ricci flow, and it
has since been used widely and with great success, most notably in
Perelman's solution of the Poincare conjecture. Furthermore, various
convergence theorems have been established. This book provides a concise
introduction to the subject as well as a comprehensive account of the
convergence theory for the Ricci flow. The proofs rely mostly on maximum
principle arguments. Special emphasis is placed on preserved curvature
conditions, such as positive isotropic curvature. One of the major
consequences of this theory is the Differentiable Sphere Theorem: a
compact Riemannian manifold, whose sectional curvatures all lie in the
interval (1,4], is diffeomorphic to a spherical space form. This
question has a long history, dating back to a seminal paper by H. E.
Rauch in 1951, and it was resolved in 2007 by the author and Richard
Schoen."
TIMASHEV
Homogeneous spaces and
equivariant embeddings Homogeneous
spaces of linear algebraic groups lie at the crossroads of algebraic
geometry, theory of algebraic groups, classical projective and
enumerative geometry, harmonic analysis, and representation theory. By
standard reasons of algebraic geometry, in order to solve various
problems on a homogeneous space it is natural and helpful to compactify
it keeping track of the group action, i.e. to consider equivariant
completions or, more generally, open embeddings of a given homogeneous
space. Such equivariant embeddings are the subject of this book. We
focus on classification of equivariant embeddings in terms of certain
data of "combinatorial" nature (the Luna--Vust theory) and description
of various geometric and representation-theoretic properties of these
varieties in terms of these data. The class of spherical varieties,
intensively studied during the last three decades, is of special
interest in the scope of this book. Spherical varieties include many
classical examples, such as Grassmannians, flag varieties, and varieties
of quadrics, as well as well-known toric varieties. We tried to cover
all most important issues, including a substantial progress obtained in
the theory of spherical varieties and around it quite recently.
SEGUINS PAZZI
Invitation aux formes quadratiques
Une somme extraordinaire sur un
chapitre trop souvent ignoré, cet ouvrage sur les formes quadratiques
livre au brillant
taupin, à l'agrégatif tout comme au mathématicien confirmé un choix impressionnant de sujets et de thèmes en relation avec ce
domaine capital. Démarrant avec les fondements, dans un cadre rigoureux et précis, l'auteur nous guide, juste après les
théorèmes de classification, vers les premières applicationsgéométriques des formes quadratiques que sont l'étude des
coniques et des quadriques. Il nous offre au passage une véritable introduction à la géométrie affine et projective et une
incursion inattendue du côté de la géométrie différentielle, avec le lemme de Morse et la notion de courbure. L'auteur
s'applique
ensuite à présenter, pour la première fois en France comme à
l'étranger, la théorie des formes quadratiques rationnelles dans une
approche relativement élémentaire et progressive, où les nombreux
exemples et les applications ne manquent pas.
C'est l'occasion aussi d'une introduction aux corps p-adiques et aux théorèmes reliant les passages local/global.
L'étude algébrique couvre évidemment les théorèmes de Witt, les formes de Pfister, les algèbres de Clifford et l'examen des
groupes
orthogonaux et spinoriels, tous aussi chers aux géomètres qu'aux
physiciens théoriciens. L'ouvrage se termine sur une étude approfondie
du cas de la caractéristique 2, souvent ignoré ou escamoté dans les
livres sur le sujet.
L'ouvrage s'accompagne de magnifiques dessins, de plus de neuf cents exercices et problèmes, ainsi que d'un index extrê-
mement fourni, le tout dans un style et une finition particulièrement soignés.
SAINT-GERVAIS Uniformisation des surfaces de Riemann retour sur un théorème centenaire
En 1907, Paul Koebe et Henri
Poincaré démontraient presque simultanément le théorème
duniformisation : Toute surface de Riemann simplement connexe est
isomorphe au plan, au disque ou à la sphère.
Il a fallu tout un
siècle avant doser énoncer ce théorème et den donner une
démonstration convaincante, grâce aux travaux de Gauss, Riemann,
Schwarz, Klein, Poincaré et Koebe (entre autres). Ce livre propose
quelques points de vue sur la maturation de ce théorème. Lévolution du
théorème duniformisation sest faite en parallèle avec lapparition de
la géométrie algébrique, la création de lanalyse complexe, les
premiers balbutiements de lanalyse fonctionnelle, avec le foisonnement
de la théorie des équations différentielles linéaires et la naissance
de la topologie. Le théorème duniformisation est lun des fils
conducteurs du XIXe siècle mathématique.
Il ne sagit pas ici de
décrire lhistoire dun théorème mais de revenir sur des preuves
anciennes, de les lire avec des yeux de mathématiciens modernes, de
sinterroger sur la validité de ces preuves et dessayer de compléter
celles-ci en respectant autant que possible les connaissances de
lépoque, voire, si cela savère nécessaire, en utilisant des outils
mathématiques
modernes qui nétaient pas à la disposition de leurs auteurs.
Ce
livre sera utile aux mathématiciens daujourdhui qui souhaitent jeter
un regard sur lhistoire de leur discipline. Il pourra également
permettre à des étudiants de niveau master daccéder à ces concepts si
importants de la recherche contemporaine en utilisant une voie
inhabituelle.