Mathematik
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Kopriva, David A.
Implementing Spectral Methods for Partial Differential Equations
Algorithms for Scientists and Engineers
This book offers a systematic and self-contained approach to solve partial
differential equations numerically using single and multidomain
spectral methods. It contains detailed algorithms in pseudocode for the
application of spectral approximations to both one and two dimensional
PDEs of mathematical physics describing potentials, transport, and wave
propagation. David Kopriva, a well-known researcher in the field with extensive
practical experience, shows how only a few fundamental algorithms form the
building blocks of any spectral code, even for problems with complex
geometries. The book addresses computational and applications scientists, as
it emphasizes the practical derivation and implementation of spectral methods
over abstract mathematics. It is divided into two parts: First comes a primer on
spectral approximation and the basic algorithms, including FFT algorithms,
Gauss quadrature algorithms, and how to approximate derivatives. The
second part shows how to use those algorithms to solve steady and time
dependent PDEs in one and two space dimensions. Exercises and questions
at the end of each chapter encourage the reader to experiment with
the algorithms.
(Series: Scientific Computation)
2009, Approx. 435 p., Hardcover
ISBN: 978-90-481-2260-8
Elliott, David L.
Bilinear Control Systems
Matrices in Action
This book introduces and surveys bilinear control systems theory from a
mathematical viewpoint. The exposition is at the first-year graduate level.
Topics include an introduction to Lie algebras, matrix groups and semigroups;
the controllability, stabilization, and observability of bilinear systems; nonlinear
systems analytically equivalent to bilinear systems; series representations for
input-outputmappings; and random inputs. To assist the reader there are
exercises; Mathematica scripts; and appendices on matrix analysis,
differentiable maps and manifolds, Lie algebra, and Lie groups.
(Series: Applied Mathematical Sciences)
2009, X, 281 p. 8 illus., 2 in color., Hardcover
ISBN: 978-1-4020-9612-9
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den Hollander, Frank
Random Polymers
École d’Été de Probabilités de Saint-Flour XXXVII – 2007
Polymer chains that interact with themselves and/or with their environment are
fascinating objects, displaying a range of interesting physical and chemical
phenomena. The focus in this monograph is on the mathematical description of
some of these phenomena, with particular emphasis on phase transitions as a
function of interaction parameters, associated critical behavior and space-time
scaling. Topics include: self-repellent polymers, self-attracting polymers,
polymers interacting with interfaces, charged polymers, copolymers near linear
or random selective interfaces, polymers interacting with random substrate and
directed polymers in random environment. Different techniques are exposed,
including the method of local times, large deviations, the lace expansion,
generating functions, the method of excursions, ergodic theory, partial
annealing estimates, coarse-graining techniques and martingales. Thus, this
monograph offers a mathematical panorama of polymer chains, which even
today holds plenty of challenges.
May-09
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(Series: Ecole d'Eté Probabilit.Saint-Flour)
2009, XIII, 258 p. 84 illus., Softcover
ISBN: 978-3-642-00332-5
Juhl, Andreas
Families of Conformally Covariant Differential Operators, Q-Curvature
and Holography
The central object of the book is Q-curvature. This important and subtle scalar
Riemannian curvature quantity was introduced by Tom Branson about 15 year
ago in connection with variational formulas for determinants of conformally
covariant differential operators. The book studies structural properties of Qcurvature from an extrinsic point of view by regarding it as a derived quantity of
certain conformally covariant families of differential operators which are
associated to hypersurfaces. The new approach is at the cutting edge of
central developments in conformal differential geometry in the last two
decades (Fefferman-Graham ambient metrics, spectral theory on PoincaréEinstein spaces, tractor calculus, Verma modules and Cartan geometry). The
theory of conformally covariant families is inspired by the idea of holography in
the AdS/CFT-duality. Among other things, it naturally leads to a holographic
description of Q-curvature. The methods admit generalizations in various
directions.
(Series: Progress in Mathematics)
2009, Approx. 505 p., Hardcover
ISBN: 978-3-7643-9899-6
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Kowol, Gerhard
Projektive Geometrie und Cayley-Klein Geometrien der Ebene
In diesem Buch wird am Beispiel der ebenen reellen und komplexen
projektiven Geometrie und der davon abgeleiteten Cayley-Klein-Geometrien
versucht aufzuzeigen, dass das Mathematisieren eine weit über das
Fachspezifische hinausgehende Bedeutung hat - sowohl in
erkenntnistheoretischer Hinsicht als auch in Bezug auf Anwendungen.
Ersteres wird durch den anschaulich-synthetischen Zugang, der im Laufe der
Darstellung durch den analytischen ergänzt wird, belegt und durch
philosophische und mathematikhistorische Erörterungen untermauert; letzteres
erstreckt sich auch auf wenig bekannte Anwendungen innerhalb der Botanik,
Kristallografie, Mechanik und Psychologie. Des weiteren werden bislang kaum
bzw. nicht in Buchform dargestellte Themen behandelt wie: Natürliche
Geometrie von J. Hjelmslev, Beweis des Parallelenaxioms nach P. Lorenzen
(konstruktive euklidische Geometrie), Imaginärtheorie nach L. Locher-Ernst,
Wegkurven und Wegflächen, Koordinatisierung der Cayley-Klein-Ebenen. Das
Buch ist soweit wie möglich elementar gehalten; nur eine Vertrautheit mit
mathematischer Argumentation sowie Grundkenntnisse der euklidischen
Geometrie werden vorausgesetzt.
May-09
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May-09
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May-09
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2009, Etwa 450 S., Hardcover
ISBN: 978-3-7643-9901-6
Kravchenko, Vladislav V.
Applied Pseudoanalytic Function Theory
Pseudoanalytic function theory generalizes and preserves many crucial
features of complex analytic function theory. The Cauchy-Riemann system is
replaced by a much more general first-order system with variable coefficients
which turns out to be closely related to important equations of mathematical
physics. This relation supplies powerful tools for studying and solving
Schrödinger, Dirac, Maxwell, Klein-Gordon and other equations with the aid of
complex-analytic methods.
The book is dedicated to these recent developments in pseudoanalytic function
theory and their applications as well as to multidimensional generalizations.
It is directed to undergraduates, graduate students and researchers interested
in complex-analytic methods, solution techniques for equations of
mathematical physics, partial and ordinary differential equations.
(Series: Frontiers in Mathematics)
2009, Approx. 195 p., Softcover
ISBN: 978-3-0346-0003-3
Ding, Jiu; Zhou, Aihui
Statistical Properties of Deterministic Systems
Part of Tsinghua University Texts, "Statistical Properties of Deterministic
Systems" discusses the fundamental theory and computational methods of the
statistical properties of deterministic discrete dynamical systems. After
introducing some basic results from ergodic theory, two problems related to the
dynamical system are studied: first the existence of absolute continuous
invariant measures, and then their computation. They correspond to the
functional analysis and numerical analysis of the Frobenius-Perron operator
associated with the dynamical system.
The book can be used as a text for graduate students in applied mathematics
and in computational mathematics; it can also serve as a reference book for
researchers in the physical sciences, life sciences, and engineering.
Dr. Jiu Ding is a professor at the Department of Mathematics of the University
of Southern Mississippi; Dr. Aihui Zhou is a professor at the Academy of
Mathematics and Systems Science of the Chinese Academy of Sciences.
(Series: Tsinghua University Texts)
2009, Approx. 270 p. 4 illus., Hardcover
ISBN: 978-3-540-85366-4
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Hackbusch, Wolfgang
Hierarchische Matrizen
Algorithmen und Analysis
Bei der Diskretisierung von Randwertaufgaben und Integralgleichungen
entstehen große, eventuell auch voll besetzte Matrizen. Es wird eine neuartige
Methode dargestellt, die es erstmals erlaubt, derartige Matrizen nicht nur
effizient zu speichern, sondern auch alle Matrixoperationen einschließlich der
Matrixinversion bzw. der Dreieckszerlegung approximative durchzuführen.
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Anwendungen findet diese Technik nicht nur bei der Lösung großer
Gleichungssysteme, sondern auch bei Matrixgleichungen und der Berechnung
von Matrixfunktionen.
2009, XX, 451 S. 65 Abb., Hardcover
ISBN: 978-3-642-00221-2
Lunardi, Alessandra
Interpolation theory
This book contains the lecture notes of a basic course in interpolation theory.
In the mathematical literature there are many good books on the subject, but
none of them is very elementary, and in many cases the basic principles are
hidden below great generality. In this second edition, the principles of
interpolation theory are illustrated aiming at simplification rather than at
generality. The abstract theory is reduced as far as possible, and many
examples and applications are given, especially to operator theory and to
regularity in partial differential equations. Moreover the treatment is selfcontained, the only prerequisite being the knowledge of basic functional
analysis.
(Series: Lecture Notes (Scuola Normale Superiore))
2nd ed., 2009, Approx. 190 p., Softcover
ISBN: 978-88-7642-342-0
Krantz, Steven G.
Explorations in Harmonic Analysis
With Applications to Complex Function Theory and the Heisenberg
Group
This self-contained text provides an introduction to modern harmonic analysis
in the context in which it is actually applied, in particular, through complex
function theory and partial differential equations. It takes the novice
mathematical reader from the rudiments of harmonic analysis (Fourier series)
to the Fourier transform, pseudodifferential operators, and finally to Heisenberg
analysis.
Within the textbook, the new ideas on the Heisenberg group are applied to the
study of estimates for both the Szegö and Poisson–Szegö integrals on the unit
ball in complex space. Thus the main theme of the book is also tied into
complex analysis of several variables. With a rigorous but well-paced
exposition, this text provides all the necessary background in singular and
fractional integrals, as well as Hardy spaces and the function theory of several
complex variables, needed to understand Heisenberg analysis.
Explorations in Harmonic Analysis is ideal for graduate students in
mathematics, physics, and engineering. Prerequisites include a fundamental
background in real and complex analysis and some exposure to functional
analysis.
(Series: Applied and Numerical Harmonic Analysis)
2009, XIV, 366 p. 26 illus., Hardcover
ISBN: 978-0-8176-4668-4
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Milmeister, Gérard
The Rubato Composer Music Software
Component-Based Implementation of a Functorial Concept Architecture
Both modern mathematical music theory and computer science are strongly
influenced by the theory of categories and functors. One outcome of this
research is the data format of denotators, which is based on set-valued
presheaves over the category of modules and diaffine homomorphisms. The
functorial approach of denotators deals with generalized points in the form of
arrows and allows the construction of a universal concept architecture. This
architecture is ideal for handling all aspects of music, especially for the
analysis and composition of highly abstract musical works.
This book presents an introduction to the theory of module categories and the
theory of denotators, as well as the design of a software system, called Rubato
Composer, which is an implementation of the category-theoretic concept
framework. The application is written in portable Java and relies on plug-in
components, so-called rubettes, which may be combined in data flow networks
for the generation and manipulation of denotators.
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The Rubato Composer system is open to arbitrary extension and is freely
available under the GPL license. It allows the developer to build specialized
rubettes for tasks that are of interest to composers, who in turn combine them
to create music. It equally serves music theorists, who use them to extract
information from and manipulate musical structures. They may even develop
new theories by experimenting with the many parameters that are at their
disposal thanks to the increased flexibility of the functorial concept
architecture.
Two contributed chapters by Guerino Mazzola and Florian Thalmann illustrate
the application of the theory as well as the software in the development of
compositional tools and the creation of a musical work with the help of the
Rubato framework.
(Series: Computational Music Science)
2009, XX, 301 p. 143 illus., Hardcover
ISBN: 978-3-642-00147-5
Rohde, Jan Christian
Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication
The main goal of this book is the construction of families of Calabi-Yau 3manifolds with dense sets of complex multiplication fibers. The new families
are determined by combining and generalizing two methods.
Firstly, the method of E. Viehweg and K. Zuo, who have constructed a
deformation of the Fermat quintic with a dense set of CM fibers by a tower of
cyclic coverings. Using this method, new families of K3 surfaces with dense
sets of CM fibers and involutions are obtained.
Secondly, the construction method of the Borcea-Voisin mirror family, which in
the case of the author's examples yields families of Calabi-Yau 3-manifolds
with dense sets of CM fibers, is also utilized. Moreover fibers with complex
multiplication of these new families are also determined.
This book was written for young mathematicians, physicists and also for
experts who are interested in complex multiplication and varieties with complex
multiplication. The reader is introduced to generic Mumford-Tate groups and
Shimura data, which are among the main tools used here. The generic
Mumford-Tate groups of families of cyclic covers of the projective line are
computed for a broad range of examples.
(Series: Lecture Notes in Mathematics)
2009, IX, 228 p., Softcover
ISBN: 978-3-642-00638-8
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Romanovski, Valery G.; Shafer, Douglas S.
The Center and Cyclicity Problems
A Computational Algebra Approach
In the last three decades, advances in methods for investigating polynomial
ideals and their varieties have provided new possibilities for approaching two
long-standing problems in the theory of differential equations: the Poincaré
center problem and the cyclicity problem (the problem of bifurcation of limit
cycles from singular trajectories).
Using a computational algebra approach, this work addresses the center and
cyclicity problems as behaviors of dynamical systems and families of
polynomial systems. The text first lays the groundwork for computational
algebra and gives the main properties of ideals in polynomial rings and their
affine varieties; this is followed by a discussion regarding the theory of normal
forms and stability of differential equations. The center and cyclicity problems
are then explored in detail.
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The book contains numerous examples, pseudocode displays of all the
computational algorithms, historical notes, nearly two hundred exercises, and
an extensive bibliography. Completely self-contained, it is thus suitable mainly
as a textbook for a graduate course in the subject but also as a reference for
researchers.
2009, XII, 330 p. 4 illus., Softcover
ISBN: 978-0-8176-4726-1
Giaquinta, Mariano; Modica, Giuseppe
Mathematical Analysis
An Introduction to Functions of Several Variables
This self-contained work is an introductory presentation of basic ideas,
structures, and results of differential and integral calculus for functions of
several variables.
The wide range of topics covered include: differential calculus of several
variables, including differential calculus of Banach spaces, the relevant results
of Lebesgue integration theory, differential forms on curves, a general
introduction to holomorphic functions, including singularities and residues,
surfaces and level sets, and systems and stability of ordinary differential
equations. An appendix highlights important mathematicians and other
scientists whose contributions have made a great impact on the development
of theories in analysis.
Mathematical Analysis: An Introduction to Functions of Several Variables
motivates the study of the analysis of several variables with examples,
observations, exercises, and illustrations. It may be used in the classroom
setting or for self-study by advanced undergraduate and graduate students
and as a valuable reference for researchers in mathematics, physics, and
engineering.
Other books recently published by the authors include: Mathematical Analysis:
Functions of One Variable, Mathematical Analysis: Approximation and Discrete
Processes, and Mathematical Analysis: Linear and Metric Structures and
Continuity, all of which provide the reader with a strong foundation in modernday analysis.
2009, XII, 348 p. 105 illus., Hardcover
ISBN: 978-0-8176-4509-0
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Andrews, George E.; Berndt, Bruce C.
Ramanujan's Lost Notebook
Part II
This volume is the second of approximately four volumes that the authors plan
to write on Ramanujan’s lost notebook, which is broadly interpreted to include
all material published in The Lost Notebook and Other Unpublished Papers in
1988. The primary topics addressed in the authors’ second volume on the lost
notebook are q-series, Eisenstein series, and theta functions. Most of the
entries on q-series are located in the heart of the original lost notebook, while
the entries on Eisenstein series are either scattered in the lost notebook or are
found in letters that Ramanujan wrote to G.H. Hardy from nursing homes.
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2009, XII, 420 p. 8 illus., Hardcover
ISBN: 978-0-387-77765-8
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