2003 XIX, 659 pp. Softcover
3-540-01438-1
A gentle introduction to advanced topics
such as parallel
computing, multigrid methods, and special
methods for systems of
PDEs. The goal of all chapters is to ecomputef
solutions to
problems, hence algorithmic and software
issues play a central
role. All software examples use the Diffpack
programming
environment - some experience with Diffpack
is required. There
are also some chapters covering complete
applications, i.e., the
way from a model, expressed as systems of
PDEs, through to
discretization methods, algorithms, software
design,
verification, and computational examples.
Suitable for readers
with a background in basic finite element
and finite difference
methods for partial differential equations.
Contents:
X. Cai, E. Acklam, H. P. Langtangen, A. Tveito:
Parallel
Computing.- X. Cai: Overlapping Domain Decomposition
Methods.- K.-A.
Mardal, G. W. Zumbusch, H. P. Langtangen:
Software Tools for
Multigrid Methods.- K.-A. Mardal, H. P. Langtangen:
Mixed Finite
Elements.- K.-A. Mardal, J. Sundnes, H. P.
Langtangen, A. Tveito:
Systems of PDEs and Block Preconditioning.-
A. Odegard, H. P.
Langtangen, A. Tveito: Object-Oriented Implementation
of Fully
Implicit Methods for Systems of PDEs.- H.
P. Langtangen, H. Osnes:
Stochastic Partial Differential Equations.-
H. P. Langtangen and
K.-A. Mardal: Using Diffpack from Python
Scripts.- X. Cai, A. M.
Bruaset, H. P. Langtangen, G. T. Lines, K.
Samuelsson, W. Shen, A.
Tveito, G. Zumbusch: Performance Modeling
of PDE Solvers.- J.
Sundnes, G.T. Lines, P. Grottum, A. Tveito:
Numerical Methods and
Software for Modeling the Electrical Activity
in the Human Heart.-
O. Skavhaug, B. F. Nielsen, A. Tveito: Mathematical
Models of
Financial Derivatives.- O. Skavhaug, B. F.
Nielsen, A. Tveito:
Numerical Methods for Financial Derivatives.-
T. Thorvaldsen, H.
P. Langtangen, H. Osnes: Finite Element Modeling
of Elastic
Structures.- K. M. Okstad, T. Kvamsdal: Simulation
of Aluminum
Extrusion.-
A. Kjeldstad, H. P. Langtangen, J. Skogseid,
K. Bjorlykke:
Simulation of Deformations, Fluid Flow and
HeatTransfer in
Sedimentary Basins
Series: Lecture Notes in Computational Science
and Engineering.
Vol.. 33
2004 Approx. 157 p. 99 illus. Softcover
3-540-40448-1
Like all of Vladimir Arnold's books, this
book is full of
geometric insight. Arnold illustrates every
principle with a
figure. This book aims to cover the most
basic parts of the
subject and confines itself largely to the
Cauchy and Neumann
problems for the classical linear equations
of mathematical
physics, especially Laplace's equation and
the wave equation,
although the heat equation and the Korteweg-de
Vries equation are
also discussed. Physical intuition is emphasized.
A large number
of problems are sprinkled throughout the
book, and a full set of
problems from examinations given in Moscow
are included at the
end. Some of these problems are quite challenging!
What makes the book unique is Arnold's particular
talent at
holding a topic up for examination from a
new and fresh
perspective. He likes to blow away the fog
of generality that
obscures so much mathematical writing and
reveal the essentially
simple intuitive ideas underlying the subject.
No other
mathematical writer does this quite so well
as Arnold.
Keywords: partial differential equations
Contents:
Preface to the Second Russian Edition.- 1.
The General Theory to
one First-Order Equation.- 2. The General
Theory to one First-Order
Equation (Continued).- 3. Huygense Principle
in the Theory of
Wave Propagation.- 4. The Vibrating String
(dfAlembertfs
Method).- 5. The Fourier Method (for the
Vibrating String).- 6.
The Theory of Oscillations. The Variational
Principle.- 7. The
Theory of Oscillations. The Variational Principle
(Continued).- 8.
Properties of Harmonic Functions.- 9. The
Fundamental Solution
for the Laplacian. Potentials.- 10. The Double
Layer Potential.-
11. Spherical Functions. Maxwellfs Theorem.
The Removable
Singularities Theorem.- 12. Boundary Value
Problems for Laplacefs
Equation. Theory of Linear Equations and
Systems.- A. The
Topological Content of Maxwellfs Theorem
on the Multifield
Representation of Spherical Functions.- B.
Problems.
Series: Universitext.
With a Survey by Richard Sharp: "Periodic
Orbits of
Hyperbolic Flows"
2004 VII, 138 Hardcover
3-540-40121-0
The seminal 1970 Moscow thesis of Grigoriy
A. Margulis, published
for the first time. Entitled "On Some
Aspects of the Theory
of Anosov Systems", it uses ergodic
theoretic techniques to
study the distribution of periodic orbits
of Anosov flows. The
thesis introduces the "Margulis measure"
and uses it to
obtain a precise asymptotic formula for counting
periodic orbits.
This has an immediate application to counting
closed geodesics on
negatively curved manifolds. The thesis also
contains asymptotic
formulas for the number of lattice points
on universal coverings
of compact manifolds of negative curvature.
The thesis is complemented by a survey by
Richard Sharp,
discussing more recent developments in the
theory of periodic
orbits for hyperbolic flows, including the
results obtained in
the light of Dolgopyat's breakthroughs on
bounding transfer
operators and rates of mixing.
Keywords: Anosov Flows, Hyperbolic Flows,
Lebesgue measures,
Dynamical Systems, Periodic Orbits, Riemannian
Geometry
Contents:
G. Margulis: On Some Aspects of the Theory
of Anosov Systems:
1. Some Preliminaries on Anosov Flows.- 2.
Behaviour of Lebesgue
Measures on Leaves of ~mathfrak{S}^{l+1}
under the Action of
Anosov Flows.- 3. Construction of Special
Measures on Leaves of
~mathfrak{S}^{l+1}, mathfrak{S}^{k+1}, mathfrak{S}^l
and
~mathfrak{S}^k.- 4. Construction of a Special
Measure on Wn and
the properties of the flow {Tt} with this
Measure.- 5. Ergodic
Properties of.~mathfrak{S}^k.- 6. Asymptotics
of the Number of
Periodic Trajectories.- 7. Some Asymptotical
Properties of the
Anosov Systems.- Appendix. References.
R. Sharp: Periodic Orbits of Hyperbolic Flows:
0. Introduction.- 1. Definition and Results.-
2. Zeta Functions.-
3. Subshifts of Finite Type and Suspended
Flows.- 4. Ruelle
Transfer Operators.- 5. Extending Zeta Funktions.-
6. Meromorphic
Extensions.- 7. Bounds on the Zeta Function
and Exponential Error
Terms.- 8. Polynomial Error Terms.- 9. Equidistribution
Results.-
10. Finite Group Extensions.- 11. Counting
with Homological
Constraints.- 12. Lalley's Theorem.- 13.
Lattice Point Counting.-
14. Manifolds of Non-Positive Curvature.-
Appendix A: Symbolic
Dynamics.- Appendix B: Livsic Theorems: Cohomology
and Periodic
Orbits.
Series: Springer Monographs in Mathematics.
@
3rd ed. 2004 Approx. 248 p. 250 illus., 6
in color. Hardcover
3-540-40460-0
The mathematical heroes of this book are
"perfect proofs":
brilliant ideas, clever connections and wonderful
observations
that bring new insight and surprising perspectives
on basic and
challenging problems from Number Theory,
Geometry, Analysis,
Combinatorics, and Graph Theory. Thirty beautiful
examples are
presented here. They are candidates for The
Book in which God
records the perfect proofs - according to
the late Paul Erdos,
who himself suggested many of the topics
in this collection. The
result is a book which will be fun for everybody
with an interest
in mathematics, requiring only a very modest
(undergraduate)
mathematical background.
Keywords: analysis, combinatorics, geometry,
number theory,
proofs
From the Reviews of the First Edition:
" ... This is a wonderful book that
can be recommended to
anybody who is in any way connected to mathematics.
Those who
have ever experienced the beauty of mathematics
will experience
the chill again. For those who have never
experienced that, this
book is just the right one to start."
Acta Scientiarum
Mathematicarum, 1999, Vol. 65, 769-770
"... Inside PFTB (Proofs from The Book)
is indeed a glimpse
of mathematical heaven, where clever insights
and beautiful ideas
combine in astonishing and glorious ways.
There is vast wealth
within its pages, one gem after another.
Some of the proofs are
classics, but many are new and brilliant
proofs of classical
results. ...Aigner and Ziegler do not claim
to have presented the
definitive collection of great mathematics.
In their brief
introduction they write: "We have no
definition or
characterization of what constitutes a proof
from THE BOOK: all
we offer is the examples that we have selected,
hoping that our
readers will share our enthusiasm about brilliant
ideas, clever
insights and wonderful observations."
I do. ... "
Notices of the American Mathematical Society,
August 1999
"... This book is a pleasure to hold
and to look at: ample
margins, nice photos, instructive pictures,
and beautiful
drawings ... It is a pleasure to read as
well: the style is clear
and entertaining, the level is close to elementary,
the necessary
background is given separately, and the proofs
are brilliant.
Moreover, the exposition makes them transparent.
..." London
Mathematical Society Newsletter, January
1999
"... Clearly this second edition is
dangerously suited to
infect the reader with the enthusiasm of
the authors." J.Elstrodt
(Munster), Zentralblatt fur Mathematik 0978.00002
2003 Approx. 490 p. Hardcover
1-85233-759-1
Statistical Inference for Ergodic Diffusion
Processes encompasses
a wealth of results from over ten years of
mathematical
literature. It provides a comprehensive overview
of existing
techniques, and presents - for the first
time in book form - many
new techniques and approaches. An elementary
introduction to the
field at the start of the book introduces
a class of examples -
both non-standard and classical - that reappear
as the
investigation progresses to illustrate the
merits and demerits of
the procedures. The statements of the problems
are in the spirit
of classical mathematical statistics, and
special attention is
paid to asymptotically efficient procedures.
Today, diffusion
processes are widely used in applied problems
in fields such as
physics, mechanics and, in particular, financial
mathematics.
This book provides a state-of-the-art reference
that will prove
invaluable to researchers, and graduate and
postgraduate
students, in areas such as financial mathematics,
economics,
physics, mechanics and the biomedical sciences.
Keywords: Statistical inference, Statistical
processes, Diffusion
processes
Contents: Preface.- Introduction.- 1. Diffusion
Processes and
Statistical Problems.- 2. Parameter Estimation.-
3. Special
Models.- 4. Nonparametric Estimation.- 5.
Hypothesis Testing.-
Historical Remarks.- References.- Index.
Series: Springer Series in Statistics.
2004 Approx. 220 p. 19 illus. Hardcover
0-387-40248-9
Written in the spirit of Liboff's acclaimed
text on Quantum
Mechanics, this introduction to group theory
offers an
exceptionally clear presentation with a good
sense of what to
explain, which examples are most appropriate,
and when to give a
counter-example. Group Theory Primer is an
ideal introductory
text for undergraduates in physics, engineering,
materials
science, and chemistry. It should also provide
a good background
for those students who go on to use group
theory in such
applications as nuclear and particle physics.
Liboff covers the
standard topics, but in a way that allows
students to see the
physical implications of the defined concept.
Among the many
introductions to group theory pitched at
the undergraduate level,
few can match this text for the logic and
lucidity of its
presentation.
Contents: Groups and Subgroups.- Classes
and Platonic Solids.-
Matrices, Irreps and the Great Orthogonality
Theorem.- Quantum
Mechanics, The Full Rotation Group, and Young
Diagrams.- Space
Groups, Brillouin Zone and Group of k.- Atoms
in Crystals and
Correlation Diagrams.- Elements of Abstract
Algebra and the
Galois Group.
Series: Undergraduate Texts in Contemporary
Physics.