Series: Developments in Mathematics, Vol. 22
1st Edition., 2011, VIII, 280 p.
Hardcover, ISBN 978-1-4419-9812-5
Due: July 29, 2011
.This book fills a real gap in the analytical literature. After many years and many results of analytic regularity for partial differential equations, the only access to the technique known as (T^p)_phi has remained embedded in the research papers themselves, making it difficult for a graduate student or a mature mathematician in another discipline to master the technique and use it to advantage. This monograph takes a particularly non-specialist approach, one might even say gentle, to smoothly bring the reader into the heart of the technique and its power, and ultimately to show many of the results it has been instrumental in proving. Another technique developed simultaneously by F. Treves is developed and compared and contrasted to ours.
The techniques developed here are tailored to proving real analytic regularity to solutions of sums of squares of vector fields with symplectic characteristic variety and others, real and complex. The motivation came from the field of several complex variables and the seminal work of J. J. Kohn. It has found application in non-degenerate (strictly pseudo-convex) and degenerate situations alike, linear and non-linear, partial and pseudo-differential equations, real and complex analysis. The technique is utterly elementary, involving powers of vector fields and carefully chosen localizing functions. No knowledge of advanced techniques, such as the FBI transform or the theory of hyperfunctions is required. In fact analyticity is proved using only C^infty techniques.
The book is intended for mathematicians from graduate students up, whether in analysis or not, who are curious which non-elliptic partial differential operators have the property that all solutions must be real analytic. Enough background is provided to prepare the reader with it for a clear understanding of the text, although this is not, and does not need to be, very extensive. In fact, it is very nearly true that if the reader is willing to accept the fact that pointwise bounds on the derivatives of a function are equivalent to bounds on the L^2 norms of its derivatives locally, the book should read easily.
1. What this book is and is not.- 2. Brief Introduction.- 3.Overview of Proofs.- 4. Full Proof for the Heisenberg Group.- 5. Coefficients.- 6. Pseudo-differential Problems.- 7. Sums of Squares and Real Vector Fields.- 8. bar{partial}-Neumann and the Boundary Laplacian.- 9. Symmetric Degeneracies.- 10. Details of the Previous Chapter. -11. Non-symplectic Strategem ahe.- 12. Operators of Kohn Type Which Lose Derivatives.- 13. Non-linear Problems.- 14. Treves' Approach.- 15. Appendix.- Bibliography.
Series: Lecture Notes in Mathematics, Vol. 2022
Subseries: Mathematical Biosciences Subseries
1st Edition., 2011, VIII, 196 p. 27 illus. in color.
Softcover, ISBN 978-3-642-20529-3
Due: July 2011
This volume presents explicit approximations of the quasi-stationary distribution and of the expected time to extinction from the state one and from quasi-stationarity for the stochastic logistic SIS model. The approximations are derived separately in three different parameter regions, and then combined into a uniform approximation across all three regions. Subsequently, the results are used to derive thresholds as functions of the population size N.
Content Level ā Research
Keywords ā 60-J28; 92-D30 - Birth-death processes - Extinction time - Logistic SIS model - Quasi-stationarity - Threshold results
Related subjects ā Life Sciences - Probability Theory and Stochastic Processes
1 Introduction.- 2 Model Formulation.- 3 A Birth-Death Process with Finite State Space and with an Absorbing State at the Origin.- 4 The SIS Model: First Approximations of the Quasi-Stationary Distribution.- 5 Some Approximations Involving the Normal Distribution.- 6 Preparations for the Study of the Stationary Distribution p(1) of the SIS Model.- 7 Approximation of the Stationary Distribution p(1) of the SIS Model.- 8 Preparations for the Study of the Stationary Distribution p(0) of the SIS Model.- 9 Approximation of the Stationary Distribution p(0) of the SIS Model.- 10 Approximation of Some Images UnderY for the SIS Model.- 11 Approximation of the Quasi-Stationary Distribution q of the SIS Model.- 12 Approximation of the Time to Extinction for the SIS Model.- 13 Uniform Approximations for the SIS Model.- 14 Thresholds for the SIS Model.- 15 Concluding Comments.
Series: Lecture Notes in Mathematics, Vol. 2024
1st Edition., 2011, X, 204 p.
Softcover, ISBN 978-3-642-21136-2
Due: July 6, 2011
This volume introduces a systematic approach to the solution of some mathematical problems that arise in the study of the hyperbolic-parabolic systems of equations that govern the motions of thermodynamic fluids. It is intended for a wide audience of theoretical and applied mathematicians with an interest in compressible flow, capillarity theory, and control theory.
The focus is particularly on recent results concerning nonlinear asymptotic stability, which are independent of assumptions about the smallness of the initial data. Of particular interest is the loss of control that sometimes results when steady flows of compressible fluids are upset by large disturbances. The main ideas are illustrated in the context of three different physical problems:
(i) A barotropic viscous gas in a fixed domain with compact boundary. The domain may be either an exterior domain or a bounded domain, and the boundary may be either impermeable or porous.
(ii) An isothermal viscous gas in a domain with free boundaries.
(iii) A heat-conducting, viscous polytropic gas.
1 Topics in Fluid Mechanics.- 2 Topics in Stability.- 3 Barotropic Fluids with Rigid Boundary.- 4 Isothermal Fluids with Free Boundaries.- 5 Polytropic Fluids with Rigid Boundary.
Ecole dfEte de Probabilites de Saint-Flour XL 2010
Series: Lecture Notes in Mathematics, Vol. 2025
Subseries: Ecole d'Ete de Probabilites de Saint-Flour
1st Edition., 2011, Approx. 140 p. 15 illus.
Softcover, ISBN 978-3-642-21155-3
Due: July 7, 2011
Understanding the effect of disorder on critical phenomena is a central issue in statistical mechanics. In probabilistic terms: what happens if we perturb a system exhibiting a phase transition by introducing a random environment? The physics community has approached this very broad question by aiming at general criteria that tell whether or not the addition of disorder changes the critical properties of a model: some of the predictions are truly striking and mathematically challenging. We approach this domain of ideas by focusing on a specific class of models, the "pinning models," for which a series of recent mathematical works has essentially put all the main predictions of the physics community on firm footing; in some cases, mathematicians have even gone beyond, settling a number of controversial issues. But the purpose of these notes, beyond treating the pinning models in full detail, is also to convey the gist, or at least the flavor, of the "overall picture," which is, in many respects, unfamiliar territory for mathematicians.
1 Introduction.- 2 Homogeneous pinning systems: a class of exactly solved models.- 3 Introduction to disordered pinning models.- 4 Irrelevant disorder estimates.- 5 Relevant disorder estimates: the smoothing phenomenon.- 6 Critical point shift: the fractional moment method.- 7 The coarse graining procedure.- 8 Path properties.
Series: Selected Works in Probability and Statistics
1st Edition., 2011, X, 1211 p. 1 illus.
Hardcover, ISBN 978-1-4419-9674-9
Due: July 29, 2011
This volume is dedicated to the memory of the late Oded Schramm (1961-2008), distinguished statistician. Throughout his life, Oded made profound and beautiful contributions to mathematics that will have a lasting influence.
In these two volumes, Editors Itai Benjamini and Olle Haggstrom have collected some of his papers, supplemented with three survey papers by Steffen Rohde, Haggstrom and Cristophe Garban that further elucidate his work. The papers within are a representative collection that shows the breadth, depth, enthusiasm and clarity of his work, with sections on Geometry, Noise Sensitivity, Random Walks and Graph Limits, Percolation, and finally Schramm-Loewner Evolution. An introduction by the Editors and a comprehensive bibliography of Odedfs publications complete the volume. The book will be of especial interest to researchers in probability and statistics, and in the history of these subjects.
Content Level ā Research
Related subjects ā Statistical Theory and Methods