2009; approx. 326 pp; hardcover
ISBN-13: 978-0-8218-4808-1
Expected publication date is December 17, 2009.
If you are acquainted with mathematics at least to the extent of a standard high school curriculum and like it enough to want to learn more, and if, in addition, you are prepared to do some serious work, then you should start studying this book.
An understanding of the material of the book requires neither a developed ability to reason abstractly nor skill in using the refined techniques of mathematical analysis. In each chapter elementary problems are considered, accompanied by theoretical material directly related to them. There are over 300 problems in the book, most of which are intended to be solved by the reader. In those places in the book where it is natural to introduce concepts outside the high school syllabus, the corresponding definitions are given with examples. And in order to bring out the meaning of such concepts clearly, appropriate (but not too many) theorems are proved concerning them.
Unfortunately, what is sometimes studied at school under the name "mathematics" resembles real mathematics not any closer than a plucked flower gathering dust in a herbarium or pressed between the pages of a book resembles that same flower in the meadow besprinkled with dewdrops sparkling in the light of the rising sun.
Introduction
Induction
Combinatorics
The whole numbers
Geometric transformations
Inequalities
Graphs
The pigeonhole principle
Complex numbers and polynomials
Rational approximations
Mathematics and the computer
Instead of a conclusion: teaching how to look for solutions of problems, or fantasy in the manner of Polya
Solutions of the supplementary problems
Proceedings of Symposia in Applied Mathematics, Volume: 67 Parts 1 & 2.
2009; approx. 1022 pp; hardcover
ISBN-13: 978-0-8218-4728-2
Expected publication date is December 29, 2009.
The International Conference on Hyperbolic Problems: Theory, Numerics and Applications, "HYP2008", was held at the University of Maryland from June 9-14, 2008. This was the twelfth meeting in the bi-annual international series of HYP conferences which originated in 1986 at Saint-Etienne, France, and over the last twenty years has become one of the highest quality and most successful conference series in Applied Mathematics.
The articles in this two-part volume are written by leading researchers as well as promising young scientists and cover a diverse range of multi-disciplinary topics addressing theoretical, modeling and computational issues arising under the umbrella of "hyperbolic PDEs".
Part I
List of plenary talks
S. Benzoni-Gavage and J.-F. Coulombel -- Multidimensional shock waves and surface waves
G.-Q. Chen and M. Feldman -- Shock reflection-diffraction phenomena and multidimensional conservation laws
S. Chen -- Study on Mach reflection and Mach configuration
F. Golse -- Nonlinear regularizing effect for conservation laws
S. Jin -- Numerical methods for hyperbolic systems with singular coefficients: Well-balanced scheme, Hamiltonian preservation, and beyond
A. Kiselev -- Some recent results on the critical surface quasi-geostrophic equation: A review
B. Perthame -- Why hyperbolic and kinetic models for cell populations self-organization?
B. Piccoli -- Flows on networks and complicated domains
Invited talks
D. Amadori and A. Corli -- Global solutions for a hyperbolic model of multiphase flow
F. Ancona and A. Marson -- On the convergence rate for the Glimm scheme
W. Bao and F. Y. Lim -- Analysis and computation for the semiclassical limits of the ground and excited states of the Gross-Pitaevskii equation
G.-Q. Chen, M. Slemrod, and D. Wang -- Conservation laws: Transonic flow and differential geometry
C. Christoforou -- A survey on the $L^1$ comparison of entropy weak solutions to Euler equations in the large with respect to physical parameters
P. D'Ancona, D. Foschi, and S. Selberg -- Low regularity solutions of the Maxwell-Dirac system
A. Dedner and R. Klofkorn -- Stabilization for discontinuous Galerkin methods applied to systems of conservation laws
C. De Lellis -- Ill-posedness for bounded admissible solutions of the 2-dimensional $p$-system
D. Donatelli and P. Marcati -- Applications of dispersive estimates to the acoustic pressure waves for incompressible fluid problems
P. G. LeFloch -- Stability in the $L^1$ norm via a linearization method for nonlinear hyperbolic systems
S. Nishibata and M. Suzuki -- A review of semiconductor models: Global solvability and hierarchy
Index
Part II
Contributed talks
G. Alberti, S. Bianchini, and G. Crippa -- Two-dimensional transport equation with Hamiltonian vector fields
A. C. Alvarez, G. Hime, and D. Marchesin -- Analytic regularization of an inverse problem for a system of conservation laws
P. Antonelli and P. Marcati -- On the finite weak solutions to a system in quantum fluid dynamics
K. C. Assi and M. Laforest -- Accuracy of modeling error estimates for discrete velocity models
A. V. Azevedo, A. P. de Souza, F. Furtado, and D. Marchesin -- The Riemann solution for three-phase flow in a porous medium
J. Balbas and X. Qian -- Non-oscillatory central schemes for 3D hyperbolic conservation laws
J. Benz, A. Meister, and P. A. Zardo -- A conservative, positivity preserving scheme for advection-diffusion-reaction equations in biochemical applications
S. Berres and T. Voitovich -- On the spectrum of a rank two modification of a diagonal matrix for linearized fluxes modelling polydisperse sedimentation
S. Bianchini and L. V. Spinolo -- Invariant manifolds for viscous profiles of a class of mixed hyperbolic-parabolic systems
P. Birken and A. Jameson -- Nonlinear iterative solvers for unsteady Navier-Stokes equations
F. Bouchut, C. Klingenberg, and K. Waagan -- An approximate Riemann solver for ideal MHD based on relaxation
R. Burger, A. Coronel, and M. Sepulveda -- Numerical solution of an inverse problem for a scalar conservation law modelling sedimentation
R. Burger, K. H. Karlsen, and J. D. Towers -- A conservation law with discontinuous flux modelling traffic flow with abruptly changing road surface conditions
M. J. Castro-Diaz, P. G. LeFloch, M. L. Munoz-Ruiz, and C. Pares -- Numerical investigation of finite difference schemes for nonconservative hyperbolic systems
F. Cavalli, G. Naldi, G. Puppo, and M. Semplice -- Relaxed schemes for nonlinear evolutionary PDEs
G. Chapiro, G. Hime, A. Mailybaev, D. Marchesin, and A. P. de Souza -- Global asymptotic effects of the structure of combustion waves in porous media
B. Cheng -- Multiscale dynamics of 2D rotational compressible Euler equations-An analytic approach
I. Christov, I. D. Mishev, and B. Popov -- Finite volume methods on unstructured Voronoi meshes for hyperbolic conservation laws
R. M. Colombo, G. Facchi, G. Maternini, and M. D. Rosini -- On the continuum modeling of crowds
R. M. Colombo and G. Guerra -- Balance laws as quasidifferential equations in metric spaces
O. Delestre, S. Cordier, F. James, and F. Darboux -- Simulation of rain-water overland-flow
C. Donadello -- On the vanishing viscosity approximation in the vectorial case
V. Elling -- Counterexamples to the sonic and detachment criteria
J. T. Frings and S. Noelle -- Well-balanced high order scheme for 2-layer shallow water flows
F. G. Fuchs, A. D. McMurry, and S. Mishra -- High-order finite volume schemes for wave propagation in stratified atmospheres
J. M. Gallardo, M. J. Castro, and C. Pares -- High-order finite volume schemes for shallow water equations with topography and dry areas
M. Garavello and B. Piccoli -- Riemann solvers for conservation laws at a node
H. Haasdonk and M. Ohlberger -- Reduced basis method for explcit finite volume approximations of nonlinear conservation laws
J. Haink -- Error estimate for the local discontinuous Galerkin scheme of a diffusive-dispersive equation with convolution
B. Haspot -- Cauchy problem for capillarity Van der Vaals model
H. Hattori -- Viscous conservation laws with discontinuous initial data
G. Hime and V. Matos -- Parallel computation of large amplitude shocks for a system of conservation laws with small data
H. Holden, N. H. Risebro, and H. Sande -- Convergence of front tracking and the Glimm scheme for a model of the flow of immiscible gases
X. Hu and D. Wang -- Global existence and incompressible limit of weak solutions to the multi-dimensional compressible magnetohydrodynamics
H. K. Jenssen and I. A. Kogan -- Construction of conservative systems
E. A. Johnson and J. A. Rossmanith -- Collisionless magnetic reconnection in a five-moment two-fluid electron-positron plasma
M. Jradeh -- Finite difference scheme for a nonlinear damped wave equation derived from brain modulation
K. H. Karlsen and T. K. Karper -- Convergent finite element methods for compressible barotropic Stokes systems
S. Karni and G. Hernandez-Duenas -- A hybrid scheme for flows in porous media
F. Kemm -- Discrete involutions, resonance, and the divergence problem in MHD
J. Kim and R. J. LeVeque -- Two-layer shallow water system and its applications
W. J. Lambert and D. Marchesin -- Asymptotic rarefaction waves for balance laws with stiff sources
R. Liska, R. Loubere, P.-H. Maire, J. Briel, S. Galera, and P. Vachal -- Comparison of staggered and cell-centered Lagrangian and ALE hydrodynamical methods
M. Luka?ova-Medvidova and E. Tadmor -- On the entropy stability of Roe-type finite volume methods
A. Madrane and E. Tadmor -- Entropy stability of Roe-type upwind finite volume methods on unstructured grids
F. Marche and C. Berthon -- A robust high order VFRoe scheme for shallow water equations
S. Mishra and E. Tadmor -- Vorticity preserving schemes using potential-based fluxes for the system wave equation
T. Nakamura and S. Nishibata -- Half space problem for the compressible Navier-Stokes equation
M. Nolte and D. Kroner -- Computing the effective Hamiltonian for a time-dependent Hamiltonian
R. Pan and K. Zhao -- Initial boundary value problems for compressible Euler equations with damping
M. Pelanti and F. Bouchut -- A relaxation method for modeling two-phase shallow granular flows
Y.-J. Peng and J. Ruiz -- Riemann problem for Born-Infeld systems
E. Peterson, M. Shearer, T. P. Witelski, and R. Levy -- Stability of traveling waves in thin liquid films driven by gravity and surfactant
M. V. Popov and S. D. Ustyugov -- Piecewise parabolic method on a local stencil for hyerbolic conservation laws
X. Qian, J. Balbas, A. Bhattacharjee, and H. Yang -- A numerical study of magnetic reconnection: A central scheme for Hall MHD
M. Ricciuto and A. Bollermann -- Accuracy of stabilized residual distribution for shallow water flows including dry beds
O. Rouch and P. Arminjon -- Using the entropy production rate to enhance aritificial compression
O. Rozanova -- Blow up of smooth solutions to the barotropic compressible magnetohydrodynamic equations with finite mass and energy
G. Russo and A. Khe -- High order well balanced schemes for systems of balance laws
V. M. Shelkovich -- Transport of mass, momentum and energy in zero-pressure gas dynamics
W. Shen -- On a model of granular flow
K.-M. Shyue -- A simple unified coordinates method for compressible homogeneous two-phase flows
D. L. Tkachev and A. M. Blokhin -- Courant-Friedrich's hypothesis and stability of the weak shock
R. Touma -- Unstaggered central schemes for MHD and SMHD
Y. Ueda, T. Nakamura, and S. Kawashima -- Stability of planar stationary wave for damped wave equation with nonlinear convection in half space
D. Wright, M. Frank, and A. Klar -- The minimum entropy approximation to the radiative transfer equation
Index
Mathematical Surveys and Monographs, Volume: 158
2009; approx. 445 pp; hardcover
ISBN-13: 978-0-8218-0331-8
Expected publication date is January 16, 2010.
The main subject of this book is applications of methods of scattering theory to differential operators, primarily the Schrodinger operator.
There are two different trends in scattering theory for differential operators. The first one relies on the abstract scattering theory. The second one is almost independent of it. In this approach the abstract theory is replaced by a concrete investigation of the corresponding differential equation. In this book both of these trends are presented. The first half of this book begins with the summary of the main results of the general scattering theory of the previous book by the author, Mathematical Scattering Theory: General Theory, American Mathematical Society, 1992. The next three chapters illustrate basic theorems of abstract scattering theory, presenting, in particular, their applications to scattering theory of perturbations of differential operators with constant coefficients and to the analysis of the trace class method.
In the second half of the book direct methods of scattering theory for differential operators are presented. After considering the one-dimensional case, the author returns to the multi-dimensional problem and discusses various analytical methods and tools appropriate for the analysis of differential operators, including, among others, high- and low-energy asymptotics of the Green function, the scattering matrix, ray and eikonal explansions.
The book is based on graduate courses taught by the author at Saint-Petersburg (Russia) and Rennes (France) Universities and is oriented towards a reader interested in studying deep aspects of scattering theory (for example, a graduate student in mathematical physics).
Basic notation
Introduction
Basic concepts
Smooth theory. The Schrodinger operator
Smooth theory. General differential operators
Scattering for perturbations of trace class type
Scattering on the half-line
One-dimensional scattering
The limiting absorption principle (LAP), the radiation conditions and the expansion theorem
High- and lower-energy asymptotics
The scattering matrix (SM) and the scattering cross section
The spectral shift function and trace formulas
The Schrodinger operator with a long-range potential
The LAP and radiation estimates revisited
Review of the literature
Bibliography
Index
History of Mathematics, Volume: 36
2009; approx. 396 pp; hardcover
ISBN-13: 978-0-8218-4718-3
Expected publication date is February 4, 2010.
Henri Poincare (1854-1912) was one of the greatest scientists of his time, perhaps the last one to have mastered and expanded almost all areas in mathematics and theoretical physics. He created new mathematical branches, such as algebraic topology, dynamical systems, and automorphic functions, and he opened the way to complex analysis with several variables and to the modern approach to asymptotic expansions. He revolutionized celestial mechanics, discovering deterministic chaos. In physics, he is one of the fathers of special relativity, and his work in the philosophy of sciences is illuminating.
For this book, about twenty world experts were asked to present one part of Poincare's extraordinary work. Each chapter treats one theme, presenting Poincare's approach, and achievements, along with examples of recent applications and some current prospects. Their contributions emphasize the power and modernity of the work of Poincare, an inexhaustible source of inspiration for researchers, as illustrated by the Fields Medal awarded in 2006 to Grigori Perelman for his proof of the Poincare conjecture stated a century before.
This book can be read by anyone with a master's (even a bachelor's) degree in mathematics, or physics, or more generally by anyone who likes mathematical and physical ideas. Rather than presenting detailed proofs, the main ideas are explained, and a bibliography is provided for those who wish to understand the technical details.
E. Ghys -- Poincare and his disk
N. Bergeron -- Differential equations with algebraic coefficients over arithmetic manifolds
E. Kowalski -- Poincare and analytic number theory
J.-P. Francoise -- The theory of limit cycles
D. Cerveau -- Singular points of differential equations: On a theorem of Poincare
M. Nauenberg -- Periodic orbits of the three body problem: Early history, contributions of Hill and Poincare, and some recent developments
N. Anantharaman -- On the existence of closed geodesics
F. Beguin -- Poincare's memoir for the Prize of King Oscar II: Celestial harmony entangled in homoclinic intersections
E. Ghys -- Variations on Poincare's recurrence theorem
G. Boffetta, G. Lacorata, and A. Vulpiani -- Low-dimensional chaos and asymptotic time behavior in the mechanics of fluids
A. Yger -- The concept of "residue" after Poincare: Cutting across all of mathematics
L. Bessieres, G. Besson, and M. Boileau -- The proof of the Poincare conjecture, according to Perelman
J. Mawhin -- Henri Poincare and the partial differential equations of mathematical physics
P. Cartier -- Poincare's calculus of probabilities
M. M. France -- Poincare and geometric probability
P.-P. Grivel -- Poincare and Lie's third theorem
M. Le Bellac -- The Poincare group
Y. Pomeau -- Henri Poincare as an applied mathematician
G. Heinzmann -- Henri Poincare and his thoughts on the philosophy of science
Graduate Studies in Mathematics, Volume: 109
2010; 265 pp; hardcover
ISBN-13: 978-0-8218-4083-2
Expected publication date is January 14, 2010.
This book presents elementary models of transport in continuous media and a corresponding body of mathematical technique. Physical topics include convection and diffusion as the simplest models of transport; local conservation laws with sources as the general framework of continuum mechanics; ideal fluid as the simplest model of a medium with mass; momentum and energy transport; and finally, free surface waves, in particular, shallow water theory.
There is a strong emphasis on dimensional analysis and scaling. Some topics, such as physical similarity and similarity solutions, are traditional. In addition, there are reductions based on scaling, such as incompressible flow as a limit of compressible flow, and shallow water theory derived asymptotically from the full equations of free surface waves. More and deeper examples are presented as problems, including a series of problems that model a tsunami approaching the shore.
The problems form an embedded subtext to the book. Each problem is followed by a detailed solution emphasizing process and craftsmanship. The problems express the practice of applied mathematics as the examination and re-examination of simple but essential ideas in many interrelated examples.
Transport processes: the basic prototypes
Convection
Diffusion
Local conservation laws
Superposition
Superposition of point source solutions
$\delta$-functions
Scaling-based reductions in basic fluid mechanics
Ideal fluid mechanics
Free surface waves
Solution of the shallow water equations
Bibliography
Index