Contemporary Mathematics, Volume: 440
2007; 287 pp; softcover
ISBN-10: 0-8218-4247-1
ISBN-13: 978-0-8218-4247-8
Expected publication date is September 23, 2007.
This volume contains a series of articles on wave phenomena and fluid dynamics, highlighting recent advances in these two areas of mathematics. The collection is based on lectures presented at the conference "Fluids and Waves--Recent Trends in Applied Analysis" and features a rich spectrum of mathematical techniques in analysis and applications to engineering, neuroscience, physics, and biology. The mathematical topics discussed range from partial differential equations, dynamical systems and stochastic processes, to areas of classical analysis.
This volume is intended as an introduction to major topics of interest and state-of-the-art analytical research in wave motion and fluid flows. It is helpful to junior mathematicians to stay abreast of new techniques and recent trends in these areas of mathematics. The articles here also provide a unique scientific basis for recent results and new links between current research themes. In summary, this book is a guide for experts in one field to the issues of the other, and will challenge graduate students to investigate these areas of analysis in further detail.
Readership
Graduate students and research mathematicians interested in mathematical aspects of fluid dynamics.
Table of Contents
K. T. Andrews, S. Anderson, R. S. R. Menike, M. Shillor, R. Swaminathan, and J. Yuzwalk -- Vibrations of a damageable string
G. Avalos and R. Triggiani -- The coupled PDE system arising in fluid/structure interaction. Part I: Explicit semigroup generator and its spectral properties
V. Barbu, Z. Grujic, I. Lasiecka, and A. Tuffaha -- Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model
A. Biswas and D. Swanson -- Gevrey regularity of solutions to the 3D Navier-Stokes equations
P. C. Bressloff -- Stimulus-induced bumps in two-dimensional neural field theory
S. N. Chandler-Wilde and M. Lindner -- Wave problems in unbounded domains: Fredholmness and the finite section method
S. Coombes and M. R. Owen -- Exotic dynamics in a firing rate model of neural tissue with threshold accommodation
M. H. Garzon, D. R. Blain, and M. West -- Embedded models of self-assembly of DNA complexes
M. He -- A parameter property of integrodifferential equations with memory
J. M. Lavine, E. C. Eckstein, and J. A. Goldstein -- Stochastic models with negative friction for intermittent rolling of biological mimetics
J. A. H. Murdock -- Multi-parameter oscillatory connection functions in neural field models
N. Popovic -- Front speeds, cut-offs, and desingularization: A brief case study
J. M. Rodriguez and M. H. Garzon -- Neural networks can learn to approximate autonomous flows
C. M. Schober -- Rogue waves, non-Gaussian statistics and proximity to homoclinic data
H. Schurz -- Nonlinear stochastic wave equations in mathbb{R}^1 with power-law
nonlinearity and additive space-time noise
J. Tolosa -- The method of Lyapunov functions of two variables
J. Zhu -- Analysis of map formation in visual perception
CRM Proceedings & Lecture Notes, Volume: 43
2007; 335 pp; softcover
ISBN-10: 0-8218-4351-6
ISBN-13: 978-0-8218-4351-2
Expected publication date is October 13, 2007.
One of the most active areas in mathematics today is the rapidly emerging new topic of "additive combinatorics". Building on Gowers' use of the Freiman-Ruzsa theorem in harmonic analysis (in particular, his proof of Szemeredi's theorem), Green and Tao famously proved that there are arbitrarily long arithmetic progressions of primes, and Bourgain and his co-authors have given non-trivial estimates for hitherto untouchably short exponential sums. There are further important consequences in group theory and in complexity theory and compelling questions in ergodic theory, discrete geometry and many other disciplines. The basis of the subject is not too difficult: it can be best described as the theory of adding together sets of numbers; in particular, understanding the structure of the two original sets if their sum is small. This book brings together key researchers from all of these different areas, sharing their insights in articles meant to inspire mathematicians coming from all sorts of different backgrounds.
Readership
Undergrads, graduate students, and research mathematicians interested in additive combinatorics.
Table of Contents
A. Granville -- An introduction to additive combinatorics
J. Solymosi -- Elementary additive combinatorics
A. Balog -- Many additive quadruples
E. Szemeredi, A. Balog, and A. Granville -- An old new proof of Roth's theorem
P. Kurlberg -- Bounds on exponential sums over small multiplicative subgroups
B. Green -- Montreal notes on quadratic Fourier analysis
B. Kra -- Ergodic methods in additive combinatorics
T. Tao -- The ergodic and combinatorial approaches to Szemeredi's theorem
I. Z. Ruzsa -- Cardinality questions about sumsets
E. S. Croot III and V. F. Lev -- Open problems in additive combinatorics
M.-C. Chang -- Some problems related to sum-product theorems
J. Cilleruelo and A. Granville -- Lattice points on circles, squares in arithmetic progressions and sumsets of squares
M. B. Nathanson -- Problems in additive number theory. I
K. Gyarmati, S. Konyagin, and I. Z. Ruzsa -- Double and triple sums modulo a prime
A. A. Glibichuk and S. V. Konyagin -- Additive properties of product sets in fields of prime order
G. Martin and K. O'Bryant -- Many sets have more sums than differences
G. Bhowmik and J.-C. Schlage-Puchta -- Davenport's constant for groups
of the form mathbb{Z}_3oplusmathbb{Z}_3oplusmathbb{Z}_{3d}
S. D. Adhikari, R. Balasubramanian, and P. Rath -- Some combinatorial group invariants and their generalizations with weights
Contemporary Mathematics, Volume: 441
2007; 174 pp; softcover
ISBN-10: 0-8218-3820-2
ISBN-13: 978-0-8218-3820-4
Expected publication date is October 31, 2007.
Hopf algebras have proved to be very interesting structures with deep connections to various areas of mathematics, particularly through quantum groups. Indeed, the study of Hopf algebras, their representations, their generalizations, and the categories related to all these objects has an interdisciplinary nature. It finds methods, relationships, motivations and applications throughout algebra, category theory, topology, geometry, quantum field theory, quantum gravity, and also combinatorics, logic, and theoretical computer science.
This volume portrays the vitality of contemporary research in Hopf algebras. Altogether, the articles in the volume explore essential aspects of Hopf algebras and some of their best-known generalizations by means of a variety of approaches and perspectives. They make use of quite different techniques that are already consolidated in the area of quantum algebra. This volume demonstrates the diversity and richness of its subject. Most of its papers introduce the reader to their respective contexts and structures through very expository preliminary sections.
Readership
Graduate students and research mathematicians interested in Hopf algebras, their applications and generalizations.
Table of Contents
B. Day, E. Panchadcharam, and R. Street -- Lax braidings and the Lax centre
G. Karaali -- Dynamical quantum groups-The super story
Y. Kashina -- Groups of grouplike elements of a semisimple Hopf algebra and its dual
S.-H. Ng and P. Schauenburg -- Higher Frobenius-Schur indicators for pivotal categories
F. Panaite -- Doubles of (quasi) Hopf algebras and some examples of quantum groupoids and vertex groups related to them
P. Schauenburg -- Central braided Hopf algebras
M. D. Staic -- A note on anti-Yetter-Drinfeld modules
M. Takeuchi -- Representations of the Hopf algebra U(n)
Courant Lecture Notes, Volume: 16
2007; 126 pp; softcover
ISBN-10: 0-8218-4085-1
ISBN-13: 978-0-8218-4085-6
Expected publication date is December 7, 2007.
This is a brief introduction to stochastic processes studying certain elementary continuous-time processes. After a description of the Poisson process and related processes with independent increments as well as a brief look at Markov processes with a finite number of jumps, the author proceeds to introduce Brownian motion and to develop stochastic integrals and Ito's theory in the context of one-dimensional diffusion processes. The book ends with a brief survey of the general theory of Markov processes.
The book is based on courses given by the author at the Courant Institute and can be used as a sequel to the author's successful book Probability Theory in this series.
Readership
Graduate students and research mathematicians interested in stochastic processes.
Table of Contents
Introduction
Processes with independent increments
Poisson point processes
Jump Markov processes
Brownian motion
One-dimensional diffusions
General theory of Markov processes
Appendix A. Measures on Polish spaces
Appendix B. Additional remarks
Bibliography
Index
Proceedings of Symposia in Applied Mathematics, Volume: 65
2007; approx. 232 pp; hardcover
ISBN-10: 0-8218-4211-0
ISBN-13: 978-0-8218-4211-9
Expected publication date is November 17, 2007.
The articles of this book are written by leading experts in partial differential equations and their applications, who present overviews here of recent advances in this broad area of mathematics. The formation of shocks in fluids, modern numerical computation of turbulence, the breaking of the Einstein equations in a vacuum, the dynamics of defects in crystals, effects due to entropy in hyperbolic conservation laws, the Navier-Stokes and other limits of the Boltzmann equation, occupancy times for Brownian motion in a two dimensional wedge, and new methods of analyzing and solving integrable systems are some of this volume's subjects. The reader will find an exposition of important advances without a lot of technicalities and with an emphasis on the basic ideas of this field.
Readership
Graduate students and research mathematicians interested in partial differential equations.
Table of Contents
S. Klainerman -- Null hypersurfaces with finite curvature flux and a breakdown criterion in general relativity
D. Christodoulou -- The formation of shocks in 3-dimensional fluids
F. A. Grunbaum and C. McGrouther -- Occupation time for two dimensional Brownian motion in a wedge
R. Buckingham, A. Tovbis, S. Venakides, and X. Zhou -- The semiclassical focusing nonlinear Schrodinger equation
Y. Li -- An extension to a classical theorem of Liouville and applications
A. S. Fokas -- From Green to Lax via Fourier
J. Jimenez -- Untangling wall turbulence through direct simulations
L. L. Bonilla and A. Carpio -- Defects, singularities and waves
C. D. Levermore -- Fluid dynamics from Boltzmann equations
F. Golse -- From the Boltzmann equation to the incompressible Navier-Stokes equations
C. M. Dafermos -- Hyperbolic conservation laws with involutions and contingent entropies
Mathematical Surveys and Monographs, Volume: 143
2007; 396 pp; hardcover
ISBN-10: 0-8218-4374-5
ISBN-13: 978-0-8218-4374-1
Expected publication date is November 30, 2007.
The Yangians and twisted Yangians are remarkable associative algebras taking their origins from the work of St. Petersburg's school of mathematical physics in the 1980s. The general definitions were given in subsequent work of Drinfeld and Olshansky, and these algebras have since found numerous applications in and connections with mathematical physics, geometry, representation theory, and combinatorics.
The book is an introduction to the theory of Yangians and twisted Yangians,
with a particular emphasis on the relationship with the classical matrix
Lie algebras. A special algebraic technique, the R-matrix formalism, is
developed and used as the main instrument for describing the structure
of Yangians. A detailed exposition of the highest weight theory and the
classification theorems for finite-dimensional irreducible representations
of these algebras is given.
The Yangian perspective provides a unifying picture of several families of Casimir elements for the classical Lie algebras and relations between these families. The Yangian symmetries play a key role in explicit constructions of all finite-dimensional irreducible representations of the orthogonal and symplectic Lie algebras via weight bases of Gelfand-Tsetlin type.
Readership
Graduate students and research mathematicians interested in representation theory and quantum groups.
Table of Contents
Yangian for mathfrak{gl}_N
Twisted Yangians
Irreducible representations of Y(mathfrak{gl}_N)
Irreducible representations of Y(mathfrak{g}_N)
Gelfand-Tsetlin bases for representations of Y(mathfrak{gl}_N)
Tensor products of evaluation modules for Y(mathfrak{gl}_N)
Casimir elements and Capelli identities
Graduate Studies in Mathematics, Volume: 87
2007; approx. 281 pp; hardcover
ISBN-10: 0-8218-4126-2
ISBN-13: 978-0-8218-4126-6
Expected publication date is December 6, 2007.
This book is aimed to provide an introduction to local cohomology which
takes cognizance of the breadth of its interactions with other areas of
mathematics. It covers topics such as the number of defining equations
of algebraic sets, connectedness properties of algebraic sets, connections
to sheaf cohomology and to de Rham cohomology, Grobner bases in the commutative
setting as well as for D-modules, the Frobenius morphism and characteristic
p methods, finiteness properties of local cohomology modules, semigroup
rings and polyhedral geometry, and hypergeometric systems arising from
semigroups.
The book begins with basic notions in geometry, sheaf theory, and homological algebra leading to the definition and basic properties of local cohomology. Then it develops the theory in a number of different directions, and draws connections with topology, geometry, combinatorics, and algorithmic aspects of the subject.
Readership
Graduate students and research mathematicians interested in theory and applications of local cohomology.
Table of Contents
Basic notions
Cohomology
Resolutions and derived functors
Limits
Gradings, filtrations, and Grobner bases
Complexes from a sequence of ring elements
Local cohomology
Auslander-Buchsbaum formula and global dimension
Depth and cohomological dimension
Cohen-Macaulay rings
Gorenstein rings
Connections with sheaf cohomology
Projective varieties
The Hartshorne-Lichtenbaum vanishing theorem
Connectedness
Polyhedral applications
D-modules
Local duality revisited
De Rham cohomology
Local cohomology over semigroup rings
The Frobenius endomorphism
Curious examples
Algorithmic aspects of local cohomology
Holonomic rank and hypergeometric systems
Injective modules and Matlis duality
Bibliography
Index