The authors consider operators of the form in a bounded domain of where are nonsmooth Hörmander's vector fields of step such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution for and provide growth estimates for and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that also possesses second derivatives, and they deduce the local solvability of , constructing, by means of , a solution to with Hölder continuous . The authors also prove estimates on this solution.
The authors consider operators of the form in a bounded domain of where are nonsmooth Hörmander's vector fields of step such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution for and provide growth estimates for and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that also possesses second derivatives, and they deduce the local solvability of , constructing, by means of , a solution to with Hölder continuous . The authors also prove estimates on this solution.
Hörmander's operators are an important class of linear elliptic-parabolic degenerate partial differential operators with smooth coefficients, which have been intensively studied since the late 1960s and are still an active field of research. This text provides the reader with a general overview of the field, with its motivations and problems, some of its fundamental results, and some recent lines of development.
Hörmander operators are a class of linear second order partial differential operators with nonnegative characteristic form and smooth coefficients, which are usually degenerate elliptic-parabolic, but nevertheless hypoelliptic, that is highly regularizing. The study of these operators began with the 1967 fundamental paper by Lars Hörmander and is intimately connected to the geometry of vector fields.Motivations for the study of Hörmander operators come for instance from Kolmogorov-Fokker-Planck equations arising from modeling physical systems governed by stochastic equations and the geometric theory of several complex variables. The aim of this book is to give a systematic exposition of a relevant part of the theory of Hörmander operators and vector fields, together with the necessary background and prerequisites.The book is intended for self-study, or as a reference book, and can be useful to both younger and senior researchers, already working in this area or aiming to approach it.
Mental disorders arise from neural and psychological mechanisms that have been built and shaped by natural selection across our evolutionary history. Looking at psychopathology through the lens of evolution is the only way to understand the deeper nature of mental disorders and turn a mass of behavioral, genetic, and neurobiological findings into a coherent, theoretically grounded discipline. The rise of evolutionary psychopathology is part of an exciting scientific movement in psychology and medicine -- a movement that is fundamentally transforming the way we think about health and disease. Evolutionary Psychopathology takes steps toward a unified approach to psychopathology, using the concepts of life history theory -- a biological account of how individual differences in development, physiology and behavior arise from tradeoffs in survival and reproduction -- to build an integrative framework for mental disorders. This book reviews existing evolutionary models of specific conditions and connects them in a broader perspective, with the goal of explaining the large-scale patterns of risk and comorbidity that characterize psychopathology. Using the life history framework allows for a seamless integration of mental disorders with normative individual differences in personality and cognition, and offers new conceptual tools for the analysis of developmental, genetic, and neurobiological data. The concepts presented in Evolutionary Psychopathology are used to derive a new taxonomy of mental disorders, the Fast-Slow-Defense (FSD) model. The FSD model is the first classification system explicitly based on evolutionary concepts, a biologically grounded alternative to transdiagnostic models. The book reviews a wide range of common mental disorders, discusses their classification in the FSD model, and identifies functional subtypes within existing diagnostic categories.
This book covers technologies, applications, tools, languages, procedures, advantages, and disadvantages of reconfigurable supercomputing using Field Programmable Gate Arrays (FPGAs). The target audience is the community of users of High Performance Computers (HPC) who may benefit from porting their applications into a reconfigurable environment. As such, this book is intended to guide the HPC user through the many algorithmic considerations, hardware alternatives, usability issues, programming languages, and design tools that need to be understood before embarking on the creation of reconfigurable parallel codes. We hope to show that FPGA acceleration, based on the exploitation of the data parallelism, pipelining and concurrency remains promising in view of the diminishing improvements in traditional processor and system design. Table of Contents: FPGA Technology / Reconfigurable Supercomputing / Algorithmic Considerations / FPGA Programming Languages / Case Study: Sorting / Alternative Technologies and Concluding Remarks
This book is dedicated to preparing prospective college students for the study of mathematics. It can be used at the end of high school or during the first year of college, for personal study or for introductory courses. It aims to set a meeting between two relatives who rarely speak to each other: the Mathematics of Beauty, which shows up in some popular books and films, and the Mathematics of Toil, which is widely known. Toil can be overcome through an appropriate method of work. Beauty will be found in the achievement of a way of thinking. The first part concerns the mathematical language: the expressions “for all”, “there exists”, “implies”, “is false”, ...; what is a proof by contradiction; how to use indices, sums, induction. The second part tackles specific difficulties: to study a definition, to understand an idea and apply it, to fix a slightly wrong argument, to discuss suggestions, to explain a proof. The third part presents customary techniques and points of view in college mathematics. The reader can choose one of three difficulty levels (A, B, C).
Proceedings of the Second International Symposium on the Gerasimov-Drell-Hearn Sum Rule and the Spin Structure of the Nucleon : Genova, Italy, 3-6 July, 2002
Proceedings of the Second International Symposium on the Gerasimov-Drell-Hearn Sum Rule and the Spin Structure of the Nucleon : Genova, Italy, 3-6 July, 2002
Containing the proceedings of the GDH 2002 symposium, this is a review of results on the nucleon spin structure and related sum rules using real and virtual photons. Theoretical developments and high precision data from different laboratories are presented and discussed. The work offers a comprehensive picture of the nucleon spin studies from the perturbative domain down to the resonance and low momentum transfer region.
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