Engler, Antonio J., Prestel, Alexander
Valued Fields
Series: Springer Monographs in Mathematics,
2005, Approx. 190 p., Hardcover
ISBN: 3-540-24221-X
About this book
Absolute values and their completions -like the p-adic number
fields- play an important role in number theory. Krull's
generalization of absolute values to valuations made applications
in other branches of mathematics, such as algebraic geometry,
possible. In valuation theory, the notion of a completion has to
be replaced by that of the so-called Henselization.
In this book, the theory of valuations as well as of
Henselizations is developed. The presentation is based on the
knowledge acquired in a standard graduate course in algebra. The
last chapter presents three applications of the general theory -as
to Artin's Conjecture on the p-adic number fields- that could not
be obtained by the use of absolute values only.
Table of contents
Absolute Values.- Valuations.- Extension of Valuations.-
Henselian Fields.- Structure Theory.- Applications of Valuation
Theory.- Ultraproducts of Valued Fields.- Classification of V-topologies.
Runde, Volker
A Taste of Topology
Series: Universitext
2005, Approx. 200 p., Softcover
ISBN: 0-387-25790-X
About this textbook
If mathematics is a language, then taking a topology course at
the undergraduate level is cramming vocabulary and memorizing
irregular verbs: a necessary, but not always exciting exercise
one has to go through before one can read great works of
literature in the original language.
The present book grew out of notes for an introductory topology
course at the University of Alberta. It provides a concise
introduction to set theoretic topology (and to a tiny little bit
of algebraic topology). It is accessible to undergraduates from
the second year on, but even beginning graduate students
canbenefit from some parts.
Great care has been devoted to the selection of examples that are
not self-serving, but already accessible for students who have a
background in calculus and elementary algebra, but not
necessarily in real or complex analysis.
In some points, the book treats its material differently than
other texts on the subject:
* Baire's theorem is derived from Bourbaki's Mittag-Leffler
theorem;
* nets are used extensively, in particular for an intuitive proof
of Tychonoff's theorem;
* a short and elegant, but little known proof for the Stone-Weierstrass
theorem is given.
Table of contents
Preface.- Introduction.- Set Theory.- Metric Spaces.- Set
Theoretic Topology.- Systems of Continuous Functions.- Basic
Algebraic Topology.- The Classical Mittag-Leffler Theorem Derived
from Bourbakifs.- Failure of the Heine-Borel Theorem in
Infinite-Dimensional Spaces.- The Arzela-Ascoli Theorem.-
References.- List of Symbols.- Index.
Mordukhovich, Boris S.
Variational Analysis and Generalized Differentiation I, Basic
Theory
Series: Grundlehren der mathematischen Wissenschaften, Vol.
330
Volume package: Variational Analysis and Generalized
Differentiation
2005, Approx. 450 p., Hardcover
ISBN: 3-540-25437-4
About this book
Variational analysis is a fruitful area in mathematics that, on
the one hand, deals with the study of optimization and
equilibrium problems and, on the other hand, applies
optimization, perturbation, and approximation ideas to the
analysis of a broad range of problems that may not be of a
variational nature.
This monograph in 2 volumes contains a comprehensive and state-of-the
art study of the basic concepts and principles of variational
analysis and generalized differentiation in both finite-dimensional
and infinite dimensional spaces and presents numerous
applications to problems in the optimization, equilibria,
stability and sensitivity, control theory, economics, mechanics,
etc. The first volume is devoted to the basic theory of
variational analysis and generalized differentiations, while the
second volume contains various applications. Both volumes contain
abundant bibliographies and extensive commentaries.
Table of contents
Generalized Differentiation in Banach Spaces: Generalized Normals
to Nonconvex Sets. Coderivatives of Set-Valued Mappings.
Subdifferentials of Nonsmooth Functions.- Extremal Principle in
Variational Analysis: Set Extremality and Nonconvex Separation.
Extremal Principle in Asplund Spaces. Relations with Variational
Principles. Representations and Characterizations in Asplund
Spaces. Versions of the Extremal Principle in Banach Spaces.-
Full Calculus in Asplund Spaces: Calculus Rules for Normals and
Coderivatives. Subdifferential Calculus and Related Topics. SNC
Calculus for Sets and Mappings.- Lipschitzian Stability and
Sensivity Analysis: Neighborhood Criteria and Exact Bounds.
Pointbased Characterizations. Sensitivity Analysis for Constraint
Systems. Sensitivity Analysis for Variational Systems.-
References.- Glossary of Notation.- Index of Statements
Mordukhovich, Boris S.
Variational Analysis and Generalized Differentiation II,
Applications
Series: Grundlehren der mathematischen Wissenschaften, Vol.
331
Volume package: Variational Analysis and Generalized
Differentiation
Approx. 400 p., 2005, 400 p., Hardcover
ISBN: 3-540-25438-2
About this book
Variational analysis is a fruitful area in mathematics that, on
the one hand, deals with the study of optimization and
equilibrium problems and, on the other hand, applies
optimization, perturbation, and approximation ideas to the
analysis of a broad range of problems that may not be of a
variational nature.
This monograph in two volumes contains a comprehensive and state-of-the
art study of the basic concepts and principles of variational
analysis and generalized differentiation in both finite-dimensional
and infinite dimensional spaces and presents numerous
applications to problems in the optimization, equilibria,
stability and sensitivity, control theory, economics, mechanics,
etc. The first volume is mainly devoted to the basic theory of
variational analysis and generalized differentiations, while the
second volume contains various applications. Both volumes contain
abundant bibliographies and extensive commentaries.
Table of contents
Constrained Optimization and Equilibria: Necessary Optimality
Conditions in Nondifferentiable Programming. Mathematical
Programs with Equilibrium Constraints. Multiobjective
Optimization. Subextremality and Suboptimality at Linear Rate.-
Optimal Control of Evolution Systems in Banach Spaces: Optimal
Control of Discrete-Time and Continuous-time Evolution Inclusions.
Necessary Optimality Conditions for Differential Inclusions
without Relaxation. Maximum Principle for Continuous-Time Systems
with Smooth Dynamics. Approximate Maximum Principle in Optimal
Control.- Optimal Control of Distributed Systems: Optimization of
Differential-Algebraic Inclusions with Delays. Neumann Boundary
Control of Semilinear Constrained Hyperbolic Equations. Drichelet
Boundary Control of Linear Constrained Hyperbolic Equations.
Minimax Control of Parabolic Systems with Pointwise State
Constraints.- Applications to Economics: Models of Welfare
Economics. Second Welfare Theorem for Nonconvex Economics.
Nonconvex Economics with Ordered Commodity Spaces. Further
Extensions and Public Goods.- References.- Glossary of Notation.-
Index of Statements.
Logan, J. David
A First Course in Differential Equations
Series: Undergraduate Texts in Mathematics,
2005, Approx. 250 p., Hardcover
ISBN: 0-387-25963-5
About this textbook
While the standard sophomore course on elementary differential
equations is typically one semester in length, most of the texts
currently being used for these courses have evolved into calculus-like
presentations that include a large collection of methods and
applications, packaged with state-of-the-art color graphics,
student solution manuals, the latest fonts, marginal notes, and
web-based supplements. All of this adds up to several hundred
pages of text and can be very expensive. Many students do not
have the time or desire to read voluminous texts and explore
internet supplements. Thats what makes the format of this
differential equations book unique. It is a one-semester, brief
treatment of the basic ideas, models, and solution methods. Its
limited coverage places it somewhere between an outline and a
detailed textbook. The author writes concisely, to the point, and
in plain language. Many worked examples and exercises are
included. A student who works through this primer will have the
tools to go to the next level in applying ODEs to problems in
engineering, science, and applied mathematics. It will also give
instructors, who want more concise coverage, an alternative to
existing texts.
This text also encourages students to use a computer algebra
system to solve problems numerically. It can be stated with
certainty that the numerical solution of differential equations
is a central activity in science and engineering, and it is
absolutely necessary to teach students scientific computation as
early as possible. Templates of MATLAB programs that solve
differential equations are given in an appendix. Maple and
Mathematica commands are given as well. The author taught this
material on several ocassions to students who have had a standard
three-semester calculus sequence. It has been well received by
many students who appreciated having a small, definitive parcel
of material to learn. Moreover, this text gives students the
opportunity to start reading mathematics at a slightly higher
level than experienced in pre-calculus and calculus; not every
small detail is included. Therefore the book can be a bridge in
their progress to study more advanced material at the junior-senior
level, where books leave a lot to the reader and are not packaged
with elementary formats.
Table of contents
Preface.- Differential Equations and Models.- Analytic Solutions
and Approximations.-Second-Order Differential Equations.- Laplace
Transforms.- Linear Systems.- Nonlinear Systems.- Appendix A:
References.- Appendix B: A Brief on MATLAB.- Appendix C: Maple
Commands.- Appendix D: Sample Examinations.- Appendix E:
Solutions to Selected Exercises.
Nepomnyashchy, Alexander, Simanovskii, Ilya, Legros, Jean Claude
Interfacial Convection in Multilayer Systems
Series: Springer Monographs in Mathematics,
2005, Approx. 300 p. 178 illus., Hardcover
ISBN: 0-387-22194-8
About this book
The interfacial convection in multilayer systems in a widespread
phenomenon which is of great importance in numerous branches of
technology. This book, for the first time, classifies all the
known types of convective instabilities in such systems, and
discusses their peculiarities. An overview is provided of the
wide variety of steady and oscillatory patterns, waves, and other
dynamic phenomena characteristic for multilayer fluid systems.
Various physical effects, including heat and mass transfer,
thermal and mechanical couplings on the interfaces, interfacial
deformability, and the influence of surfactants on different
types of convective motions are investigated.
Table of contents
Introduction.- Types of convective instabilities in systems with
an interface.- Benard problem in multilayer systems with
undeformable interfaces.- Benard problem in multilayer systems
with deformable interfaces.- Stability of flows.- Outlook.