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Partial Differential Equations
This is the second edition of the now definitive text on partial differential equations (PDE). It offers a comprehensive survey of modern techniques in the theoretical study of PDE with particular emphasis on nonlinear equations. Its wide scope and clear exposition make it a great text for a graduate course in PDE. For this edition, the author has made numerous changes, including a new chapter on nonlinear wave equations, more than 80 new exercises, several new sections, a significantly expanded bibliography. About the First Edition: I have used this book for both regular PDE and topics courses. It has a wonderful combination of insight and technical detail...Evans' book is evidence of his m...
An Introduction to Stochastic Differential Equations
These notes provide a concise introduction to stochastic differential equations and their application to the study of financial markets and as a basis for modeling diverse physical phenomena. They are accessible to non-specialists and make a valuable addition to the collection of texts on the topic. --Srinivasa Varadhan, New York University This is a handy and very useful text for studying stochastic differential equations. There is enough mathematical detail so that the reader can benefit from this introduction with only a basic background in mathematical analysis and probability. --George Papanicolaou, Stanford University This book covers the most important elementary facts regarding stoch...
Measure Theory and Fine Properties of Functions
This book provides a detailed examination of the central assertions of measure theory in n-dimensional Euclidean space and emphasizes the roles of Hausdorff measure and the capacity in characterizing the fine properties of sets and functions. Topics covered include a quick review of abstract measure theory, theorems and differentiation in Mn, lower Hausdorff measures, area and coarea formulas for Lipschitz mappings and related change-of-variable formulas, and Sobolev functions and functions of bounded variation. The text provides complete proofs of many key results omitted from other books, including Besicovitch's Covering Theorem, Rademacher's Theorem (on the differentiability a.e. of Lipschitz functions), the Area and Coarea Formulas, the precise structure of Sobolev and BV functions, the precise structure of sets of finite perimeter, and Alexandro's Theorem (on the twice differentiability a.e. of convex functions). Topics are carefully selected and the proofs succinct, but complete, which makes this book ideal reading for applied mathematicians and graduate students in applied mathematics.
Chance, Strategy, and Choice
Games and elections are fundamental activities in society with applications in economics, political science, and sociology. These topics offer familiar, current, and lively subjects for a course in mathematics. This classroom-tested textbook, primarily intended for a general education course in game theory at the freshman or sophomore level, provides an elementary treatment of games and elections. Starting with basics such as gambling, zero-sum and combinatorial games, Nash equilibria, social dilemmas, and fairness and impossibility theorems for elections, the text then goes further into the theory with accessible proofs of advanced topics such as the Sprague-Grundy theorem and Arrow's impossibility theorem. * Uses an integrative approach to probability, game, and social choice theory * Provides a gentle introduction to the logic of mathematical proof, thus equipping readers with the necessary tools for further mathematical studies * Contains numerous exercises and examples of varying levels of difficulty * Requires only a high school mathematical background.
Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem
In this volume, the authors demonstrate under some assumptions on $f $, $f $ that a solution to the classical Monge-Kantorovich problem of optimally rearranging the measure $\mu{ }=f dx$ onto $\mu =f dy$ can be constructed by studying the $p$-Laplacian equation $- \roman{div}(\vert DU_p\vert p-2}Du_p)=f -f $ in the limit as $p\rightarrow\infty$. The idea is to show $u_p\rightarrow u$, where $u$ satisfies $\vert Du\vert\leq 1, -\roman{div}(aDu)=f -f $ for some density $a\geq0$, and then to build a flow by solving a nonautonomous ODE involving $a, Du, f $ and $f $
Partial Differential Equations
Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.
Mathematics and 21st Century Biology
The exponentially increasing amounts of biological data along with comparable advances in computing power are making possible the construction of quantitative, predictive biological systems models. This development could revolutionize those biology-based fields of science. To assist this transformation, the U.S. Department of Energy asked the National Research Council to recommend mathematical research activities to enable more effective use of the large amounts of existing genomic information and the structural and functional genomic information being created. The resulting study is a broad, scientifically based view of the opportunities lying at the mathematical science and biology interface. The book provides a review of past successes, an examination of opportunities at the various levels of biological systemsâ€" from molecules to ecosystemsâ€"an analysis of cross-cutting themes, and a set of recommendations to advance the mathematics-biology connection that are applicable to all agencies funding research in this area.
Basic Research in Information Science and Technology for Air Force Needs
The U.S. Air Force is developing new force capabilities appropriate to an emerging array of threats. It is clear that advances in information science and technology (IS&T) are essential for most of these new capabilities. As a consequence, the Air Force is finding it necessary to refocus its IS&T basic research program to provide stronger support for reaching these goals. To assist this effort, the AFOSR asked the NRC for a study to create a vision and plan for the IS&T-related programs within the Office's Mathematics and Space Science Directorate. This report provides an assessment of basic research needs for Air Force systems and communications, software, information management and integration, and human interactions with IS&T systems. The report also offers a set of priorities for basic IS&T research, and an analysis of funding mechanisms its support.
Can We Trust the Bible on the Historical Jesus?
This book features a learned and fascinating debate between two great Bible scholars about the New Testament as a reliable source on the historical Jesus. Bart Ehrman, an agnostic New Testament scholar, debates Craig Evans, an evangelical New Testament scholar, about the historical Jesus and what constitutes "history." Their interaction includes such compelling questions as: What are sound methods of historical investigation? What are reliable criteria for determining the authenticity of an ancient text? What roles do reason and inference play? And, of course, interpretation? Readers of this debate—regardless of their interpretive inclinations and biases—are sure to find some confirmatio...
Defense Modeling, Simulation, and Analysis
Modeling, simulation, and analysis (MS&A) is a crucial tool for military affairs. MS&A is one of the announced pillars of a strategy for transforming the U.S. military. Yet changes in the enterprise of MS&A have not kept pace with the new demands arising from rapid changes in DOD processes and missions or with the rapid changes in the technology available to meet those demands. To help address those concerns, DOD asked the NRC to identify shortcomings in current practice of MS&A and suggest where and how they should be resolved. This report provides an assessment of the changing mission of DOD and environment in which it must operate, an identification of high-level opportunities for MS&A research to address the expanded mission, approaches for improving the interface between MS&A practitioners and decision makers, a discussion of training and continuing education of MS&A practitioners, and an examination of the need for coordinated military science research to support MS&A.
