Edited by: Mihaela Mihailescu-Suliciu, University of Bucharest, Romania

New Trends in Continuum Mechanics

A publication of the Theta Foundation.
2005; 352 pp; hardcover
ISBN: 973-85432-5-8

The volume contains the proceedings of the international conference New Trends in Continuum Mechanics, held at Constanta, Romania between September 8-12, 2003. The conference was devoted to all domains of continuum mechanics; it also included a minisymposium on homogenization and its applications and a section devoted to numerical methods. The volume contains thirty-seven refereed papers; among the subjects covered, are: theoretical and laboratory modelling, mathematical modelling in biology, oil recovery, contact problems with friction, studies on different materials, non-Newtonian fluids, electrorheological and magnetorheological fluids, thermo-mechanical and electro-mechanical problems in solids.

Table of Contents

L. Badea -- One and two-level multiplicative Schwarz methods for the constrained minimization of non-quadratic convex functionals
N. Bontcheva and B. Petzov -- Phase transformation during metal forming processes
A. Capatina and F. Lebon -- Remarks on the equiliibrium finite element method for frictional contact problems
A. Carabineanu -- The study of the self-propulsion of an oscillatory wing by the integral equations method
Q. Chang and W. Sun -- On convergence of the multigrid method for nonnegative definite systems
Y. D. Chashechkin -- Mathematical and laboratory modelling of stratified flows
D. Cioranescu, A. Damlaminan, and G. Griso -- The Stokes problem in perforated domains by the periodic unfolding method
S. Cleja-Tigoiu -- Material symmetry in finite elasto-plasticity with continuum distributed dislocations
M. Cocou and G. Scarella -- A dynamic unilateral contact problem for a cracked viscoelastic body
C. Conca, J. I. Diaz, A. Linan, and C. Timofte -- Homogenization results for chemical reactive flows through porous media
E. M. Craciun -- Stress in prestressed fiber reinforced composite containing two colinear cracks supposed to be under anti-plane shear loadings
N. D. Cristescu -- Theory of falling cylinder viscometers
A. Dumitrache -- An interactive computing method for stall flutter analysis
C. Faciu and M. Mihailescu-Suliciu -- Shape memory effect: A Maxwellian rate type constitutive approach
C. Grandmont, Y. Maday, and B. Maury -- A multiscale/multimodel approach of the respiration tree
D. Jiroveanu and J. Soler -- Numerical modeling of a bubble breakup behavior in homogeneous and isotropic turbulence
V. Marina -- Reading the possibilities to decode the microstructure characteristics from macroexperience
G. Marinoschi -- On a nonlinear boundary value problem of infiltration in unsaturated media
B. Matei -- Nonlinear multiscales representations for homogenization problems: A case study
M. Negreanu -- Discrete inequalities
G. Pasa -- Secondary oil recovery and variable permeability
E. Perez -- Correcting terms from local problems for vibrating systems with concentrated masses
K. Piechor -- On the hydrodynamic limit of the Enskog equation with weak square-well potential
D. Polisevski and R. Schiltz-Bunoiu -- Heat conduction through a first-order jump interface
M. Popescu -- On the optimal control of bilinear systems
R. Raducanu -- On the mortar finite element method in linear elasticity
S. Sburlan and C. Sburlan -- A coincidence degree for bifurcation problems with applications in mechanics of continua
N. Simion -- Models of heat propagation in solid bodies
O. Simionescu-Panait -- Propagation of attenuated waves in isotropic solids subject to initial electro-mechanical fields
D. Socolescu -- On the Leray problems for the stationary and non-stationary Navier-Stokes equations
R. Stavre -- Boundary control of a non-stationary magnetohydrodynamic flow
P. P. Teodorescu, T. Badea, L. Munteanu, and J. Onisoru -- On the wave propatagion in materials with negative stiffness components
P. P. Teodorescu, L. Munteanu, and V. Chiroiu -- On the wave propagation in a chiral medium
V. Tigoiu and C. Cipu -- Flow of some viscoelastic fluids in a falling cylinder viscometer and the evaluation of shear viscosity
A. Ursescu -- Influence of the electrode ends on the channel flow of electrorheological fluids
A. Ursescu and C. Dascalu -- Thermally conductive elliptic hole in an anisotropic solid
C. Vallee, C. Lerintiu, D. Fortune, M. Ban, and G. de Saxce -- Hill's bipotential

Edited by: Chuu-Lian Terng

Integrable Systems, Geometry, and Topology

AMS/IP Studies in Advanced Mathematics, Volume: 36
2006; 256 pp; softcover
ISBN: 0-8218-4048-7
Expected publication date is May 24, 2006.

The articles in this volume are based on lectures from a program on integrable systems and differential geometry held at Taiwan's National Center for Theoretical Sciences. As is well-known, for many soliton equations, the solutions have interpretations as differential geometric objects, and thereby techniques of soliton equations have been successfully applied to the study of geometric problems.

The article by Burstall gives a beautiful exposition on isothermic surfaces and their relations to integrable systems, and the two articles by Guest give an introduction to quantum cohomology, carry out explicit computations of the quantum cohomology of flag manifolds and Hirzebruch surfaces, and give a survey of Givental's quantum differential equations. The article by Heintze, Liu, and Olmos is on the theory of isoparametric submanifolds in an arbitrary Riemannian manifold, which is related to the n-wave equation when the ambient manifold is Euclidean. Mukai-Hidano and Ohnita present a survey on the moduli space of Yang-Mills-Higgs equations on Riemann surfaces. The article by Terng and Uhlenbeck explains the gauge equivalence of the matrix non-linear Schrodinger equation, the Schrodinger flow on Grassmanian, and the Heisenberg Feromagnetic model.

The book provides an introduction to integrable systems and their relation to differential geometry. It is suitable for advanced graduate students and research mathematicians.

Table of Contents

F. E. Burstall -- Isothermic surfaces: Conformal geometry, Clifford algebras and integrable systems
M. A. Guest -- Introduction to homological geometry: part I
M. A. Guest -- Introduction to homological geometry: part II
E. Heintze, X. Liu, and C. Olmos -- Isoparametric submanifolds and a Chevalley-type restriction theorem
M. Mukai-Hidano and Y. Ohnita -- Gauge-Theoretic approach to harmonic maps and subspaces in moduli spaces
C.-L. Terng and K. Uhlenbeck -- Schrodinger flows on Grassmannians

Edited by: James Carlson, Clay Mathematics Institute, Cambridge, MA, Arthur Jaffe, Harvard University, Cambridge, MA, and Andrew Wiles, Institute for Advanced Study, Princeton, NJ

The Millennium Prize Problems

2006; approx. 169 pp; hardcover
ISBN: 0-8218-3679-X

Guided by the premise that solving some of the world's most important mathematical problems will advance the field, this book offers a fascinating look at the seven unsolved Millennium Prize problems. This work takes the unprecedented approach of describing these important and difficult problems at the professional level.

In announcing the seven problems and a US7 million prize fund in 2000, the Clay Mathematics Institute emphasized that mathematics still constitutes an open frontier with important unsolved problems. The descriptions in this book serve the Institute's mission to "further the beauty, power and universality of mathematical thinking."

Separate chapters are devoted to each of the seven problems: the Birch and Swinnerton-Dyer Conjecture, the Hodge Conjecture, the Navier-Stokes Equation, the P versus NP Problem, the Poincare Conjecture, the Riemann Hypothesis, and Quantum Yang-Mills Theory.

An essay by Jeremy Gray, a well-known expert in the history of mathematics, outlines the history of prize problems in mathematics and shows how some of mathematics' most important discoveries were first revealed in papers submitted for prizes. Numerous photographs of mathematicians who shaped mathematics as it is known today give the text a broad historical appeal. Anyone interested in mathematicians' continued efforts to solve important problems will be fascinated with this text, which places into context the historical dimension of important achievements.

Table of Contents

J. Gray -- A history of prizes in mathematics
A. Wiles -- The Birch and Swinnerton-Dyer conjecture
P. Deligne -- The Hodge conjecture
C. L. Fefferman -- Existence and smoothness of the Navier-Stokes equation
J. Milnor -- The Poincare conjecture
S. Cook -- The P versus NP problem
E. Bombieri -- The Riemann hypothesis
A. Jaffe and E. Witten -- Quantum Yang-Mills theory
Rules for the millennium prizes
Authors' biographies

J. L. Berggren, Simon Fraser University, Burnaby, BC, Canada, and R. S. D. Thomas, University of Manitoba, Winnipeg, Manitoba, Canada

Euclid's Phaenomena:
A Translation and Study of a Hellenistic Treatise in Spherical Astronomy

History of Mathematics Volume: 29
2006; 132 pp; softcover
ISBN: 0-8218-4072-X

The book contains a translation and study of Euclid's Phaenomena, a work which once formed part of the mathematical training of astronomers from Central Asia to Western Europe. Included is an introduction that sets Euclid's geometry of the celestial sphere, and its application to the astronomy of his day, into its historical context for readers not already familiar with it. So no knowledge of astronomy or advanced mathematics is necessary for an understanding of the work. The book shows mathematical astronomy shortly before the invention of trigonometry, which allowed the calculation of exact results and the subsequent composition of Ptolemy's Almagest.

The Phaenomena itself begins with an introduction (possibly not by Euclid) followed by eighteen propositions set out in geometrical style about how arcs of the zodiacal circle move across the sky. The astronomical application is to the small arc of that circle occupied by the Sun, but the Sun is not mentioned. This work and the (roughly) contemporaneous treatises of Autolycus and Aristarchos form a corpus of the oldest extant works on mathematical astronomy. Together with Euclid's Optics one has the beginnings of the history of science as an application of mathematics.

Table of Contents

Introduction
Euclid's presuppositions
Notes on the translation
Sigla
Euclid's Phaenomena translated with commentary
Euclid's Phaenomena
English glossary of selected technical terms and phrases
Greek glossary of selected technical terms and phrases
Bibliography
Index of names
Index of subjects
Index of subjects (Greek)

Edited by: Hermann Brunner and Xiao-Qiang Zhao, Memorial University of Newfoundland, St. John's, NL, Canada, and Xingfu Zou, University of Western Ontario, London, ON, Canada

Nonlinear Dynamics and Evolution Equations

Fields Institute Communications Volume: 48
2006; 311 pp; hardcover
ISBN: 0-8218-3721-4
List Price: US109

The papers in this volume reflect a broad spectrum of current research activities on the theory and applications of nonlinear dynamics and evolution equations. They are based on lectures given during the International Conference on Nonlinear Dynamics and Evolution Equations at Memorial University of Newfoundland, St. John's, NL, Canada, July 6-10, 2004. This volume contains thirteen invited and refereed papers. Nine of these are survey papers, introducing the reader to, and describing the current state of the art in major areas of dynamical systems, ordinary, functional and partial differential equations, and applications of such equations in the mathematical modelling of various biological and physical phenomena. These papers are complemented by four research papers that examine particular problems in the theory and applications of dynamical systems.

Table of Contents

J. Arino and P. van den Driessche -- Disease spread in metapopulations
P. W. Bates -- On some nonlocal evolution equations arising in materials science
W. Craig -- Invariant tori for Hamiltonian PDE
N. Dancer -- Stable and not too unstable solutions on R^n for small diffusion
Y. Du and J. Shi -- Some recent results on diffusive predator-prey models in spatially heterogeneous environment
S. A. Gourley and J. Wu -- Delayed non-local diffusive systems in biological invasion and disease spread
J. Jiang -- Asymptotic behavior for systems comparable to quasimonotone systems
T. Krisztin -- C^1-smoothness of center manifolds for differential equations with state-dependent delay
C. Rousseau -- Normal forms for germs of analytic families of planar vector fields unfolding a generic saddle-node or resonant saddle
R. Saghin and Z. Xia -- Generic properties of symplectic diffeomorphisms
B. D. Sleeman -- Mathematical aspects of modelling tumour angiogenesis
G. S. K. Wolkowicz -- Interpretation of the generalized asymmetric May-Leonard model of three species competition as a food web in a chemostat
Y. Yi and X. Zhang -- On exact Poisson structures