Series: Publications of the Scuola Normale Superiore, Vol. 13
Subseries: CRM Series
2012, 2012, Approx. 200 p.
Softcover, ISBN 978-88-7642-426-7
This book contains lecture notes of a series of courses on the regularity theory of partial differential equations and variational problems, held in Pisa and Parma in the years 2009 and 2010. The contributors, Nicola Fusco, Tristan Riviere and Reiner Schatzle, provide three updated and extensive introductions to various aspects of modern Regularity Theory concerning: mathematical modelling of thin films and related free discontinuity problems, analysis of conformally invariant variational problems via conservation laws, and the analysis of the Willmore functional. Each contribution begins with a very comprehensive introduction, and is aimed to take the reader from the introductory aspects of the subject to the most recent developments of the theory.
Content Level ≫ Research
Keywords ≫ Willmore functional - conservation laws - free discontinuity problem - regularity theory
Related subjects ≫ Scuola Normale Superiore
Ernst Kuwert and Reiner Schatzle, The Willmore functional.- Tristan Riviere, The Role of Conservation Laws in the Analysis of Conformally Invariant Problems.- B. De Maria and N. Fusco, Equilibrium configurations of epitaxially strained elastic films.
Series: Monografie Matematyczne, Vol. 73
2012, 2012, X, 380 p.
Hardcover, ISBN 978-3-0348-0366-3
Due: March 31, 2012
First monograph on the mathematical theory of shape optimization for compressible Navier-Stokes equations
Clear explanation of the state-of-the-art developments in a mathematical language that will attract mathematicians to open questions in this important field
New concepts and results are presented
The book presents the modern state of the art in the mathematical theory of compressible Navier-Stokes equations, with a particular emphasis on the applications to aerodynamics. The topics covered include: modeling of compressible viscous flows; modern mathematical theory of nonhomogeneous boundary value problems for viscous gas dynamics equations; applications to optimal shape design in aerodynamics; kinetic theory for equations with oscillating data; new approach to the boundary value problems for transport equations. The monograph offers a comprehensive and self-contained introduction to recent mathematical tools designed to handle the problems arising in the theory.
Content Level ≫ Research
Keywords ≫ computational methods - fluid dynamics - modeling
Related subjects ≫ Dynamical Systems & Differential Equations
Series: Springer Monographs in Mathematics
XXV, 148 p. 47 illus.
Hardcover, ISBN 978-3-642-25633-2
Due: April 30, 2012
Includes numerous problems and exercises which provide a deep insight in the subject and allow to conduct independent research in this topic
Contains many pictures which visualize involved theory
Description of effective computational algorithms for higher genus algebraic curves provides wide opportunities for numerical experiments in mathematics and theoretical physics
The problems of conditional optimization of the uniform (or C-) norm for polynomials and rational functions arise in various branches of science and technology. Their numerical solution is notoriously difficult in case of high degree functions. The book develops the classical Chebyshev's approach which gives analytical representation for the solution in terms of Riemann surfaces. The techniques born in the remote (at the first glance) branches of mathematics such as complex analysis, Riemann surfaces and Teichmuller theory, foliations, braids, topology are applied to approximation problems.
The key feature of this book is the usage of beautiful ideas of contemporary mathematics for the solution of applied problems and their effective numerical realization. This is one of the few books where the computational aspects of the higher genus Riemann surfaces are illuminated. Effective work with the moduli spaces of algebraic curves provides wide opportunities for numerical experiments in mathematics and theoretical physics.?
1 Least deviation problems.- 2 Chebyshev representation of polynomials.- 3 Representations for the moduli space.- 4 Cell decomposition of the moduli space.- 5 Abel’s equations.- 6 Computations in moduli spaces.- 7 The problem of the optimal stability polynomial.- Conclusion.- References.
Series: SpringerBriefs in Mathematics
2012, 2012, VIII, 103 p. 21 illus.
Softcover, ISBN 978-1-4614-3118-3
Due: April 30, 2012
Rainbow connections are natural combinatorial measures that are used in applications to secure the transfer of classified information between agencies in communication networks. Rainbow Connections of Graphs covers this new and emerging topic in graph theory and brings together a majority of the results that deal with the concept of rainbow connections, first introduced by Chartrand et al. in 2006.
The authors begin with an introduction to rainbow connectedness, rainbow coloring, and rainbow connection number. The work is organized into the following categories, computation of the exact values of the rainbow connection numbers for some special graphs, algorithms and complexity analysis, upper bounds in terms of other graph parameters, rainbow connection for dense and sparse graphs, for some graph classes and graph products, rainbow k-connectivity and k-rainbow index, and, rainbow vertex-connection number.
Rainbow Connections of Graphs appeals to researchers and graduate students in the field of graph theory. Conjectures, open problems and questions are given throughout the text with the hope for motivating young graph theorists and graduate students to do further study in this subject.
1. Introduction (Motivation and definitions, Terminology and notations).- 2. (Strong) Rainbow connection number(Basic results, Upper bounds for rainbow connection number, For some graph classes, For dense and sparse graphs, For graph operations, An upper bound for strong rainbow connection number).- 3. Rainbow k-connectivity.- 4. k-rainbow index.- 5. Rainbow vertex-connection number.- 6. Algorithms and computational complexity.- References.
Series: Monografie Matematyczne, Vol. 72
2012, 2012, XII, 386 p.
Hardcover, ISBN 978-3-0348-0369-4
Due: April 30, 2012
Aims at a detailed explanation of Burkholder's method: presents, for most estimates, the steps leading to the discovery of the corresponding special functions
Uses diverse analytic and probabilistic methods to solve the corresponding boundary value problems
Presents a unified up-to-date treatment, illustrated on a variety of examples of different type, difficulty and complexity
Material is completely self-contained
This monograph is a presentation of a unified approach to a certain class of semimartingale inequalities, which can be regarded as probabilistic extensions of classical estimates for conjugate harmonic functions on the unit disc. The approach, which has its roots in the seminal works of Burkholder in the 80s, enables to deduce a given inequality for semimartingales from the existence of a certain special function with some convex-type properties. Remarkably, an appropriate application of the method leads to the sharp version of the estimate under investigation, which is particularly important for applications. These include the theory of quasiregular mappings (with deep implications to the geometric function theory); the boundedness of two-dimensional Hilbert transform and a more general class of Fourier multipliers; the theory of rank-one convex and quasiconvex functions; and more.
The book is divided into a few separate parts. In the introductory chapter we present motivation for the results and relate them to some classical problems in harmonic analysis. The next part contains a general description of the method, which is applied in subsequent chapters to the study of sharp estimates for discrete-time martingales; discrete-time sub- and supermartingales; continuous time processes; the square and maximal functions. Each chapter contains additional bibliographical notes included for reference.?
Preface.- Chapter 1. Introduction.- Chapter 2. Burkholder’s method.- Chapter 3. Martingale inequalities in discrete time.- Chapter 4. Sub- and supermartingale inequalities in discrete time.- Chapter 5. Inequalities in continuous time.- Chapter 6. Inequalities for orthogonal semimartingales.- Chapter 7. Maximal inequalities.- Chapter 8. Square function inequalities.- Appendix.- Bibliography.