ISBN: 978-3-031-29178-4
Subject: Mathematics and Statistics
Series Title: Developments in Mathematics
This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations.
Traditionally, the label “Calderon-Zygmund theory” has been applied to a distinguished body of works primarily pertaining to the mapping properties of singular integral operators on Lebesgue spaces, in various geometric settings. Volume IV amounts to a versatile Calderon-Zygmund theory for singular integral operators of layer potential type in open sets with uniformly rectifiable boundaries, considered on a diverse range of function spaces. Novel applications to complex analysis in several variables are also explored here.
ISBN: 978-981-99-1651-1
Subject: Mathematics and Statistics
Planned Publication Date: 2023年6月22日
Series Title: Texts and Readings in Mathematics
This book develops the concepts of fundamental convex analysis and optimization by using advanced calculus and real analysis. Brief accounts of advanced calculus and real analysis are included within the book. The emphasis is on building a geometric intuition for the subject, which is aided further by supporting figures. Two distinguishing features of this book are the use of elementary alternative proofs of many results and an eclectic collection of useful concepts from optimization and convexity often needed by researchers in optimization, game theory, control theory, and mathematical economics. A full chapter on optimization algorithms gives an overview of the field, touching upon many current themes. The book is useful to advanced undergraduate and graduate students as well as researchers in the fields mentioned above and in various engineering disciplines.
ISBN: 978-3-031-30680-8
Subject: Mathematics and Statistics
Planned Publication Date: 2023年6月19日
Series Title: Mathematiques et Applications
De nombreux systemes physiques, mecaniques, financiers et economiques peuvent etre decrits par des modeles mathematiques qui visent a optimiser des fonctions, trouver des equilibres et effectuer des arbitrages. Souvent, la convexite des ensembles et des fonctions ainsi que les conditions de monotonie sur les systemes d'inequations qui regissent ces systemes se presentent naturellement dans les modeles. C'est dans cet esprit que nous avons concu ce livre en mettant l'accent sur une approche geometrique qui privilegie l'intuition par rapport a une approche plus analytique. Les demonstrations des resultats classiques ont ete revues dans cette optique et simplifiees. De nombreux exemples d'applications sont etudies et des exercices sont proposes.
Ce livre s'adresse aux etudiants en master de mathematiques appliquees, ainsi qu'aux doctorants, chercheurs et ingenieurs souhaitant comprendre les fondements de l'analyse convexe et de la theorie des inequations variationnelles monotones.
ISBN: 978-3-031-31148-2
Subject: Mathematics and Statistics
Planned Publication Date: 2023年6月19日
Series Title: Probability Theory and Stochastic Modelling
This book describes extensions of Sudakov's classical result on the concentration of measure phenomenon for weighted sums of dependent random variables. The central topics of the book are weighted sums of random variables and the concentration of their distributions around Gaussian laws. The analysis takes place within the broader context of concentration of measure for functions on high-dimensional spheres. Starting from the usual concentration of Lipschitz functions around their limiting mean, the authors proceed to derive concentration around limiting affine or polynomial functions, aiming towards a theory of higher order concentration based on functional inequalities of log-Sobolev and Poincare type. These results make it possible to derive concentration of higher order for weighted sums of classes of dependent variables.
While the first part of the book discusses the basic notions and results from probability and analysis which are needed for the remainder of the book, the latter parts provide a thorough exposition of concentration, analysis on the sphere, higher order normal approximation and classes of weighted sums of dependent random variables with and without symmetries.
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ISBN: 978-3-031-31342-4
Subject: Mathematics and Statistics
Planned Publication Date: 2023年7月10日
Series Title: UNITEXT, La Matematica per il 3+2
This book is focused on the numerical discretization of ordinary differential equations (ODEs), under several perspectives. The attention is first conveyed to providing accurate numerical solutions of deterministic problems. Then, the presentation moves to a more modern vision of numerical approximation, oriented to reproducing qualitative properties of the continuous problem along the discretized dynamics over long times. The book finally performs some steps in the direction of stochastic differential equations (SDEs), with the intention of offering useful tools to generalize the techniques introduced for the numerical approximation of ODEs to the stochastic case, as well as of presenting numerical issues natively introduced for SDEs.
The book is the result of an intense teaching experience as well as of the research carried out in the last decade by the author. It is both intended for students and instructors: for the students, this book is comprehensive and rather self-contained; for the instructors, there is material for one or more monographic courses on ODEs and related topics. In this respect, the book can be followed in its designed path and includes motivational aspects, historical background, examples and a software programs, implemented in Matlab, that can be useful for the laboratory part of a course on numerical ODEs/SDEs.
The book also contains the portraits of several pioneers in the numerical discretization of differential problems, useful to provide a framework to understand their contributes in the presented fields. Last, but not least, rigor joins readability in the book.
ISBN: 978-3-031-31138-3
Subject: Mathematics and Statistics
Planned Publication Date: 2023年7月14日
Series Title: Operator Theory: Advances and Applications
This volume is dedicated to the memory of Sergey Naboko (1950-2020). In addition to original research contributions covering the vast areas of interest of Sergey Naboko, it includes personal reminiscences and comments on the works and legacy of Sergey Naboko’s scientific achievements. Areas from complex analysis to operator theory, especially, spectral theory, are covered, and the papers will inspire current and future researchers in these areas.
ISBN: 978-3-031-29972-8
Subject: Mathematics and Statistics
Planned Publication Date: 2023年6月5日
Series Title: Progress in Mathematics
In this monograph, for elliptic systems with block structure in the upper half-space and t-independent coefficients, the authors settle the study of boundary value problems by proving compatible well-posedness of Dirichlet, regularity and Neumann problems in optimal ranges of exponents. Prior to this work, only the two-dimensional situation was fully understood. In higher dimensions, partial results for existence in smaller ranges of exponents and for a subclass of such systems had been established. The presented uniqueness results are completely new, and the authors also elucidate optimal ranges for problems with fractional regularity data.
The first part of the monograph, which can be read independently, provides optimal ranges of exponents for functional calculus and adapted Hardy spaces for the associated boundary operator. Methods use and improve, with new results, all the machinery developed over the last two decades to study such problems: the Kato square root estimates and Riesz transforms, Hardy spaces associated to operators, off-diagonal estimates, non-tangential estimates and square functions, and abstract layer potentials to replace fundamental solutions in the absence of local regularity of solutions.
ISBN: 978-3-031-29669-7
Subject: Mathematics and Statistics
Planned Publication Date: 2023年7月1日
Series Title: Lecture Notes in Mathematics
This book provides a comprehensive analysis of the existence of weak solutions of unsteady problems with variable exponents. The central motivation is the weak solvability of the unsteady p(.,.)-Navier?Stokes equations describing the motion of an incompressible electro-rheological fluid. Due to the variable dependence of the power-law index p(.,.) in this system, the classical weak existence analysis based on the pseudo-monotone operator theory in the framework of Bochner?Lebesgue spaces is not applicable. As a substitute for Bochner?Lebesgue spaces, variable Bochner?Lebesgue spaces are introduced and analyzed. In the mathematical framework of this substitute, the theory of pseudo-monotone operators is extended to unsteady problems with variable exponents, leading to the weak solvability of the unsteady p(.,.)-Navier?Stokes equations under general assumptions.
Aimed primarily at graduate readers, the book develops the material step-by-step, starting with the basics of PDE theory and non-linear functional analysis. The concise introductions at the beginning of each chapter, together with illustrative examples, graphics, detailed derivations of all results and a short summary of the functional analytic prerequisites, will ease newcomers into the subject.