Series: Universitext
2017, XI, 673 p. 17 illus., 4 illus. in color.
Softcover
ISBN 978-3-319-61859-3
* Established textbook
* Continues to lead its readers to some of the hottest topics of contemporary
mathematical research
This established reference work continues to provide its readers with a gateway to some
of the most interesting developments in contemporary geometry. It offers insight into a
wide range of topics, including fundamental concepts of Riemannian geometry, such as
geodesics, connections and curvature; the basic models and tools of geometric analysis,
such as harmonic functions, forms, mappings, eigenvalues, the Dirac operator and the
heat flow method; as well as the most important variational principles of theoretical
physics, such as Yang-Mills, Ginzburg-Landau or the nonlinear sigma model of quantum
field theory. The present volume connects all these topics in a systematic geometric
framework. At the same time, it equips the reader with the working tools of the field and
enables her or him to delve into geometric research.
The 7th edition has been systematically reorganized and updated. Almost no page has
been left unchanged. It also includes new material, for instance on symplectic geometry,
as well as the Bishop-Gromov volume growth theorem which elucidates the geometric
role of Ricci curvature.
From the reviews: gThis book provides a very readable introduction to Riemannian
geometry and geometric analysis... With the vast development of the mathematical
subject of geometric analysis, the present textbook is most welcome.h Mathematical
Reviews
gFor readers familiar with the basics of differential geometry and some acquaintance with
modern analysis, the book is reasonably self-contained c The book succeeds very well
in laying out the foundations of modern Riemannian geometry and geometric analysis.
It introduces a number of key techniques and provides a representative overview of the
field.h Monatshefte fur Mathematik
Series: Probability Theory and Stochastic Modelling, Vol. 87
2017, XVI, 644 p. 26 illus.
Hardcover
ISBN 978-1-4471-7347-2
* Comprehensively covers the theory of random sets
* Explores interdisciplinary connections and applications of the theory
* Extensive and updated bibliography
* Includes latest developments in the field
This monograph, now in a thoroughly revised second edition, offers the latest research
on random sets. It has been extended to include substantial developments achieved
since 2005, some of them motivated by applications of random sets to econometrics and
finance.
The present volume builds on the foundations laid by Matheron and others, including
the vast advances in stochastic geometry, probability theory, set-valued analysis, and
statistical inference. It shows the various interdisciplinary relationships of random set
theory within other parts of mathematics, and at the same time fixes terminology and
notation that often vary in the literature, establishing it as a natural part of modern
probability theory and providing a platform for future development. It is completely
self-contained, systematic and exhaustive, with the full proofs that are necessary to gain
insight.
Aimed at research level, Theory of Random Sets will be an invaluable reference for
probabilists; mathematicians working in convex and integral geometry, set-valued
analysis, capacity and potential theory; mathematical statisticians in spatial statistics and
uncertainty quantification; specialists in mathematical economics, econometrics, decision
theory, and mathematical finance; and electronic and electrical engineers interested in
image analysis.
Series: Oberwolfach Seminars, Vol. 47
1st ed. 2017, Approx. 300 p.
Softcover
ISBN 978-3-319-59914-4
* This book introduces the reader to the necessary technical
background to study C*-algebras arising from actions of groups or semigroups
* The text focuses on recent examples and techniques developed in KTheory
* It contains an introduction to Baum-Connes methods as well as
a concise description of the Mackey-Rieffel-Green machine for crossed products
* Much of the material is available here for the first time in book form
This book gives an account of the necessary background for group algebras and crossed
products for actions of a group or a semigroup on a space and reports on some very
recently developed techniques with applications to particular examples. Much of the
material is available here for the first time in book form. The topics discussed are among
the most classical and intensely studied C*-algebras. They are important for applications
in fields as diverse as the theory of unitary group representations, index theory, the
topology of manifolds or ergodic theory of group actions.
Part of the most basic structural information for such a C*-algebra is contained in its Ktheory.
The determination of the K-groups of C*-algebras constructed from group or
semigroup actions is a particularly challenging problem. Paul Baum and Alain Connes
proposed a formula for the K-theory of the reduced crossed product for a group action
that would permit, in principle, its computation. By work of many hands, the formula
has by now been verified for very large classes of groups and this work has led to the
development of a host of new techniques. An important ingredient is Kasparov's bivariant
K-theory.
More recently, also the C*-algebras generated by the regular representation of a
semigroup as well as the crossed products for actions of semigroups by endomorphisms
have been studied in more detail.
Intriguing examples of actions of such semigroups come from ergodic theory as well as
from algebraic number theory. The computation of the K-theory of the corresponding
crossed products needs new techniques. In cases of interest the K-theory of the algebras
reflects ergodic theoretic or number theoretic properties of the action.
**
Series in Real Analysis: Volume 14
160pp Sep 2017
ISBN: 978-981-3221-96-3 (hardcover)
This book offers to the reader a self-contained treatment and systematic exposition of the real-valued theory of a nonabsolute integral on measure spaces. It is an introductory textbook to Henstock*Kurzweil type integrals defined on abstract spaces. It contains both classical and original results that are accessible to a large class of readers.
It is widely acknowledged that the biggest difficulty in defining a Henstock*Kurzweil integral beyond Euclidean spaces is the definition of a set of measurable sets which will play the role of "intervals" in the abstract setting. In this book the author shows a creative and innovative way of defining "intervals" in measure spaces, and prove many interesting and important results including the well-known Radon*Nikodym theorem.
A Nonabsolute Integral on Measure Spaces:
Preliminaries
Existence of a Division and the H-Integral
Fundamental Properties of the H-Integral
The Absolute H-Integral and the McShane-Type Integrals:
The Absolute H-Integral and the M-Integral
The H-Integral and the Lebesgue Integral
The Davies Inetgral and the Davies-McShane Integral
Further Results of the H-Integral:
A Necessary and Sufficient Condition for H-Integrability
Generalised Absolute Continuity and Equi-integrability
The Controlled Convergence Theorem
The Radon*Nikodym Theorem for the H-integral:
The Main Theorem
Descriptive Definition of H-integral
Henstock Integration in the Euclidean Space
Harnack Extension and Convergence Theorems for the H-Integral:
The H-Integral on Metric Spaces
Harnack Extension
The Category Argument
Another Version of Controlled Convergence Theorem
Graduate students and researchers interested in analysis.
392pp Sep 2017
ISBN: 978-981-3143-33-3 (hardcover)
ISBN: 978-981-3143-70-8 (softcover)
Broadly speaking, there are two general approaches to teaching mathematical modeling: 1) the case study approach, and 2) the method based approach (that teaches mathematical techniques with applications to relevant mathematical models). This text emphasizes instead the scientific issues for modeling different phenomena. For the natural or harvested growth of a fish population, we may be interested in the evolution of the population, whether it reaches a steady state (equilibrium or cycle), stable or unstable with respect to a small perturbation from equilibrium, or whether a small change in the environment would cause a catastrophic change, etc. Each scientific issue requires an appropriate model and a different set of mathematical tools to extract information from the model. Models examined are chosen to help explain or justify empirical observations such as cocktail drug treatments are more effective and regenerations after injuries or illness are fast-tracked (compared to original developments).
Volume I of this three-volume set limits its scope to phenomena and scientific issues that are modeled by ordinary differential equations (ODE). Scientific issues such as signal and wave propagation, diffusion, and shock formation involving spatial dynamics to be modeled by partial differential equations (PDE) will be treated in Vol. II. Scientific issues involving randomness and uncertainty are examined in Vol. III.
The Modeling Cycle
Simple Models for Growth of a Population
Estimation of Model Parameters
Steady State Population and Its Stability
Commercial Fishing, Bifurcation and Tragedy of the Common
Interacting Populations
DNA Mutations
Predator-Prey and Other Types of Nonlinear Interacting Populations
Competing Populations and Bi-Stability
HIV Dynamics and Drug Treatments
Cell Lineage, Feedback and Regeneration
Economic Planning for Harvesting a Population and Optimization
Chlamydia Trachomatis and Optimal Control
Genetic Instability and Carcinogenesis
Undergraduates in mathematical biology, mathematical modeling of dynamical systems, optimization and control, viral dynamics (infectious diseases), oncology.
516pp Jul 2017
ISBN: 978-981-3208-92-6 (hardcover)
A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be decomposed into a Cartesian product of the y geometry and the x geometry * except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. In fact, many basic solutions of the Einstein field equations, including the Schwarzschild solution and the Robertson*Walker models, are warped product manifolds.
The first part of this volume provides a self-contained and accessible introduction to the important subject of pseudo-Riemannian manifolds and submanifolds. The second part presents a detailed and up-to-date account on important results of warped product manifolds, including several important spacetimes such as Robertson*Walker's and Schwarzschild's.
The famous John Nash's embedding theorem published in 1956 implies that every warped product manifold can be realized as a warped product submanifold in a suitable Euclidean space. The study of warped product submanifolds in various important ambient spaces from an extrinsic point of view was initiated by the author around the beginning of this century.
The last part of this volume contains an extensive and comprehensive survey of numerous important results on the geometry of warped product submanifolds done during this century by many geometers.
Riemannian and Pseudo-Riemannian Manifolds
Submanifolds
Warped Product Manifolds
Robertson-Walker Spacetimes and Schwarzschild Solution
Contact Metric Manifolds and Submersions
Kahler and Pseudo-Kahler Manifolds
Slant Submanifolds
Generic Submanifolds of Kahler Manifolds
CR-submanifolds of Kahler Manifolds
Warped Products in Riemannian and Kahler Manifolds
Warped Product Submanifolds of Kahler Manifolds
CR-warped Products in Complex Space Forms
More on CR-warped Products in Complex Space Forms
Â-invariants, Submersions and Warped Products
Warped Products in Nearly Kahler Manifolds
Warped Products in Para-Kahler Manifolds
Warped Products in Sasakian Manifolds
Warped Products in Affine Spaces
Graduate students and researchers interested in warped product manifolds and submanifolds in geometry, mathematical physics and general relativity.