Since the publication of "Theory of Games and Economic Behavior" by von Neumann and Morgenstern, the concept of games has played an increasing role in economics. It also plays a role of growing importance in other sciences, including biology, political science, and psychology. Many scientists have made seminal advances and continue to be leaders in the field, including Harsanyi, Shapley, Shubik, and Selten. Professor Robert Aumann, in addition to his important contributions to game theory and economics, made a number of significant contributions to mathematics. This volume provides a collection of essays in mathematical economics and game theory, including cutting-edge research on noncooperative game theory and its foundations, bargaining theory, and general equilibrium theory. Also included is a reprint of Aumann's classic paper, "Acceptable Points in General Cooperative n-Person Games" and of the oft-cited, yet hard to find, paper by Maschler, "The Worth of a Cooperative Enterprise to Each Member". This book illustrates the wide range of applications of mathematics to economics, game theory, and social choice. The volume is dedicated to Professor Robert J. Aumann, Hebrew University, Jerusalem, Israel, for his contributions in mathematics and social sciences.

The Law of Causality and its Limits was the principal philosophical work of the physicist turned philosopher, Philipp Frank. Born in Vienna on March 20, 1884, Frank died in Cambridge, Massachusetts on July 21, 1966. He received his doctorate in 1907 at the University of Vienna in theoretical physics, having studied under Ludwig Boltzmann; his sub sequent research in physics and mathematics was represented by more than 60 scientific papers. Moreover his great success as teacher and expositor was recognized throughout the scientific world with publication of his collaborative Die Differentialgleichungen der Mechanik und Physik, with Richard von Mises, in 1925-27. Frank was responsible for the second volume, on physics, and especially noted for his authoritative article on classical Hamiltonian mechanics and optics. Among his earliest papers were those, beginning in 1908, devoted to special relativity, which together with general relativity and physical cosmology occupied him throughout his life. Already in 1907, Frank published his seminal paper 'Kausalgesetz und Erfahrung' ('Experience and the Law of Causality'), much later collected with a splendid selection of his essays on philosophy of science, in English (1941c and 1949g, in our Bibliography). Joining the first 'Vienna Circle' in the first decade of the 20th century, with Hans Hahn, mathematician, and Otto Neurath, sociologist and economist, and deeply influenced by studies of Ernst Mach's critical conceptual histories of science and by the striking challenge of Poincare and Duhem, Frank continued his epistemological investigations.

Lax and Nirenberg are two of the most distinguished mathematicians of our times. Their work on partial differential equations (PDEs) over the last half-century has dramatically advanced the subject and has profoundly influenced the course of mathematics. A huge part of the development in PDEs during this period has either been through their work, motivated by it or achieved by their postdocs and students. A large number of mathematicians honored these two exceptional scientists in a week-long conference in Venice (June 1996) on the occasion of their 70th birthdays. This volume contains the proceedings of the conference, which focused on the modern theory of nonlinear PDEs and their applications. Among the topics treated are turbulence, kinetic models of a rarefied gas, vortex filaments, dispersive waves, singular limits and blow-up of solutions, conservation laws, Hamiltonian systems, and others.The conference served as a forum for the dissemination of new scientific ideas and discoveries and enhanced scientific communication by bringing together such a large number of scientists working in related fields. The event allowed the international mathematics community to honor two of its outstanding members.

Spinors are used extensively in physics. It is widely accepted that they are more fundamental than tensors, and the easy way to see this is through the results obtained in general relativity theory by using spinors -- results that could not have been obtained by using tensor methods only. The foundation of the concept of spinors is groups; spinors appear as representations of groups. This textbook expounds the relationship between spinors and representations of groups. As is well known, spinors and representations are both widely used in the theory of elementary particles. The authors present the origin of spinors from representation theory, but nevertheless apply the theory of spinors to general relativity theory, and part of the book is devoted to curved space-time applications. Based on lectures given at Ben Gurion University, this textbook is intended for advanced undergraduate and graduate students in physics and mathematics, as well as being a reference for researchers.

Wittgenstein wrote this book during 1932-1934 - the period just before he began to dictate the Blue Book. In Part I he discusses the notions of "proposition," "sense," "language," "grammar"; what "saying something" is, what distinguishes signs form random marks or noises. Must we start with "primary" signs which need no explanation? In what sense have we a general concept of proposition or of language? The phrases "family of cases" and "family similarities," which the Investigations use, are here; and comparison brings out what is special in the later development. But although it is close to the Investigations at some points, and to the Philosophische Bemerkungen at others, the Philosophical Grammar is an independent work and discusses much that is not in either of them. It is Wittgenstein's fullest treatment of logic and mathematics in their connection with his later understanding of "proposition," "sign," and "system." In Part II he writes on logical inference and generality - criticizing views of Frege and Russell and earlier views of his own, developing his conception of "law of a series" and of " ... and so on"--Leading to his discussion of mathematics, which fills two fifths of the volume: the ideas of "foundations of mathematics," of cardinal numbers, of mathematical proof, and especially of inductive or recursive proofs (with reference to Skolem), which he treats to a depth and extent beyond anything he said of them elsewhere.

“I am Sonya Vronsky, professor of mathematics at Tel Aviv University, and this is the story of a day in late August. On this remarkable day I kissed a student, pursued a lover, found my father, and left my brother.” So begins A Wall of Light , a novel which chronicles a single day in the life of Sonya, a thirty-two-year-old deaf woman about to break out of her predictable routine. Sonya lives in Tel Aviv with her protective half-brother, Kostya; their household has dwindled from five to two. Anna, their mother, is now in a nursing home and Noah, Kostya’s son, is living in Berlin. Kostya, wracked with guilt for the tragedies that have befallen Sonya, also grapples with the memory of his wife, Iris, a lawyer murdered in the course of a dangerous investigation seventeen years earlier. As we move through Sonya’s day, Noah and Anna narrate their stories as well. Noah’s journal entries cover the years 1980-1993, and Anna’s letters to Andrei, her married lover in Russia, are written in 1957, after Anna has emigrated to Israel to build a new life for herself and her son, Kostya. While Sonya’s story moves rapidly through the events of a single day, Noah and Anna’s voices take the reader back in time, filling in the circumstances that have led Sonya to this pivotal moment. We learn that Sonya has already endured two catastrophes. At age twelve, a medical mishap leaves her deaf, and at eighteen, while studying at university in Beersheba, Sonya is assaulted by two hoodlums. Throughout the novel, Sonya’s experiences, instigated by both human error and human evil, are echoed by the larger, political violence that haunts modern Israel. While Noah’s and Anna’s voices shed light on Sonya’s journey, they also provide insights into the political and cultural fabric of Israel from the mid 1950s to the present. Noah’s journal entries, starting with his tenth birthday and ending shortly after his army service, map his coming of age. We see him wrestling with his sexual identity and first sexual encounters, the fallout from his mother’s leftist politics, and his own conscription to the army. Anna’s secret letters to Andrei offer an outsider’s perspective on the new Israeli state. The remarkable events of Sonya’s day are set in motion when her brother gives her an antihistamine. Overcome with sleepiness, she dismisses her morning class early, asking only one student, Matar, to stay behind. She wants to understand what lies behind his unusual expression. He answers that he has been involved in war crimes, and surprises Sonya by kissing her. Sonya feels that she has been roused from a long slumber and as the novel progresses we see the ways in which her awakened desire shapes her choices. She decides to take a taxi home from the university and impulsively invites the taxi driver inside and seduces him. He complies, but when she tells him she’s deaf, he flees in confusion. Sonya is convinced that she has fallen in love with him, and decides to pursue him. She solicits her brother’s help and sets out to find her lover. Sonya’s search gains in intensity and purpose as she travels to East Jerusalem. There she encounters the walls that prevent Palestinians from moving freely through the West Bank. After an Alice in Wonderland -like journey past numerous obstacles, Sonya finally makes it to her lover’s house. This second encounter leads Sonya to a central revelation: the identity of her father. As this day of awakened desire and dispelled secrets closes, Sonya is able to step out from under the protective wing of her brother into a life that reflects both the ambiguity and uncertainty of contemporary Israel and her own personal possibilities.

Heralds of Science: As Represented by Two Hundred Epochal Books and Pamphlets is a comprehensive guide to the most significant scientific publications in history. Written by Bern Dibner, a renowned collector of rare scientific books and manuscripts, this book features 200 seminal works that have shaped the course of scientific progress. The book is divided into six sections, each covering a different area of science: astronomy, mathematics, physics, chemistry, medicine, and technology. Each section includes a brief introduction to the field, followed by a selection of books and pamphlets that have had a significant impact on that area of study. The book is beautifully illustrated with full-color photographs of the featured publications, as well as portraits of the scientists who wrote them. The text is written in an accessible style, making it easy for readers of all levels of scientific knowledge to understand and appreciate the importance of these works. Heralds of Science is a must-read for anyone interested in the history of science and the key figures and publications that have shaped our understanding of the world around us. It is a valuable resource for students, researchers, and anyone with a passion for science and its impact on society.This is a new release of the original 1955 edition.This scarce antiquarian book is a facsimile reprint of the old original and may contain some imperfections such as library marks and notations. Because we believe this work is culturally important, we have made it available as part of our commitment for protecting, preserving, and promoting the world's literature in affordable, high quality, modern editions, that are true to their original work.

The transformation of mathematics from its ancient Greek practice to its development in the medieval Arab-speaking world is approached by focusing on a single problem proposed by Archimedes and the many solutions offered. From a practice of mathematics based on the localized solution (originating in the polemical practices of early Greek science), we see a transition to a practice of mathematics based on the systematic approach (grounded in the deuteronomic practices of Late Antiquity and the Middle Ages). A radically new interpretation is accordingly offered of the historical trajectory of pre-modern mathematics.

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

Written by one of the most respected figures in American linguistics, this book develops an approach to the analysis of language on a mathematical model. Harris presents a formal theory of language structure, in which syntax is characterized as an orderly system of departure from random combinings of sounds, words, and all the elements of language. He argues that the combining of words in a sentence constitutes a mathematical object, and that each departure from randomness is a contribution to the structure and meaning of a sentence. Discussing the differences in the structure and content of language, mathematics, and music, Harris shows that the use of language in a science constitutes a distinguishable sub-language. Remarkable and compelling, Harris's magnum opus will be considered the classical analysis of the structuring of information and development of language.

A Rigorous And Lively Introduction To The Theory Of Ordinary Differential Equations. The Approach Is Essentially Geometric And Extremely Appealing To The Intuition. The Careful And Lucid Way In Which This Book Is Written Puts It Well Within The Grasp Of The Senior Mathematics Student.

This book is about computational methods based on operator splitting. It consists of twenty-three chapters written by recognized splitting method contributors and practitioners, and covers a vast spectrum of topics and application areas, including computational mechanics, computational physics, image processing, wireless communication, nonlinear optics, and finance. Therefore, the book presents very versatile aspects of splitting methods and their applications, motivating the cross-fertilization of ideas.

The application by Fadeev and Pavlov of the Lax-Phillips scattering theory to the automorphic wave equation led Professors Lax and Phillips to reexamine this development within the framework of their theory. This volume sets forth the results of that work in the form of new or more straightforward treatments of the spectral theory of the Laplace-Beltrami operator over fundamental domains of finite area; the meromorphic character over the whole complex plane of the Eisenstein series; and the Selberg trace formula. CONTENTS: 1. Introduction. 2. An abstract scattering theory. 3. A modified theory for second order equations with an indefinite energy form. 4. The Laplace-Beltrami operator for the modular group. 5. The automorphic wave equation. 6. Incoming and outgoing subspaces for the automorphic wave equations. 7. The scattering matrix for the automorphic wave equation. 8. The general case. 9. The Selberg trace formula.

Lyapunov methods and equations of parabolic type.- Multiple solutions of nonlinear partial differential equations.- Fading memory and functional-differential equations.- Singular perturbation by a quasilinear operator.- Remarks on branching from multiple eigenvalues.- Asymptotic analysis of a class of nonlinear integral equations.- Remarks about bifurcation and stability of quasi-periodic solutions which bifurcate from periodic solutions of the Navier Stokes equations.- Bifurcation of periodic solutions into invariant tori: The work of Ruelle and Takens.- Ergodic theory and statistical mechanics of non-equilibrium processes.- Mathematical problems in theoretical biology.- On predator-prey equations simulating an immune response.- Bifurcation theory for gradient systems.- Six lectures on the transition to instability.- Groundwater flow as a singular perturbation problem and remarks about numerical methods.- Nonlinear problems in nuclear reactor analysis.- Some non-linear problems in statistical mechanics and biology.- Stability properties and periodic behavior of controlled biochemical systems.

From a review of the first edition: Beautifully written and well organized ... indispensable for those interested in certain areas of mathematical physics ... for the expert and beginner alike. The author deserves to be congratulated both for his work in unifying a subject and for showing workers in the field new directions for future development. --Zentralblatt MATH This is a second edition of a well-known book on the theory of trace ideals in the algebra of operators in a Hilbert space. Because of the theory's many different applications, the book was widely used and much in demand. For this second edition, the author has added four chapters on the closely related theory of rank one perturbations of self-adjoint operators. He has also included a comprehensive index and an addendum describing some developments since the original notes were published. This book continues to be a vital source of information for those interested in the theory of trace ideals and in its applications to various areas of mathematical physics.

Mental Chronometry (MC) comprises a variety of techniques for measuring the speed with which the brain processes information. First developed in mid-1800, MC was subsequently eclipsed by more complex and practically useful types of psychometric tests stemming from Alfred Binet. This class of mental tests, however, has no true metric relating the test scores to any specific properties of the brain per se. The scores merely represent an ordinal scale, only ranking individuals according to their overall performance on a variety of complex mental tasks. The resulting scores represent no more than ranks rather than being a true metrical scale of any specific dimension of brain function. Such an ordinal scale, which merely ranks individuals in some defined population, possesses no true scale properties, possessing neither a true zero or equal intervals throughout the scale. This deficiency obstructs the development of a true natural science of mental ability. The present burgeoning interest in understanding individual differences in mental abilities in terms of the natural sciences, biology and the brain sciences in particular, demands direct measures that functionally link brain and behavior. One such natural ratio scale is time itself - the time it takes the brain to perform some elementary cognitive task, measured in milliseconds. After more than 25 years researching MC, Jensen here presents results on an absolute scale showing times for intake of visual and auditory information, for accessing short-term and long-term memory, and other cognitive skills, as a function of age, at yearly intervals from 3 to 80 years. The possible uses of MC in neurological diagnosis and the monitoring of drug effects on cognition, the chronometric study of special time-sensitive talents such as musical performance, and presents a theory of general intelligence, or g, as a function of the rate of oscillation of neural action potentials as

Logic is of course a general resource for reasoning at large. But in the first half of the twentieth century, it developed particularity with a view to mathematical applications, and the field of mathematical logic came into being and flourished. In the second half of the century, much the same happened with regard to philosophical applications. Hence philosophical logic. The deliberations of this book cover a varied but interrelated array of key issues in the field. They address the representation of information in linguistic formulation, and modes of cogent demonstration in logic, mathematics, and empirical investigation, as well as the role of logic in philosophical deliberations. Overall, the book seeks to demonstrate and illustrate the utility of logic as a productive resource for rational inquiry at large.

The word apory stems from the Greek aporia, meaning impasse or perplexing difficulty. In Aporetics, Nicholas Rescher defines an apory as a group of individually plausible but collectively incompatible theses. Rescher examines historic, formulaic, and systematic apories and couples these with aporetic theory from other authors to form this original and comprehensive survey. Citing thinkers from the pre-Socratics through Spinoza, Hegel, and Nicolai Hartmann, he builds a framework for coping with the complexities of divergent theses, and shows in detail how aporetic analysis can be applied to a variety of fields including philosophy, mathematics, linguistics, logic, and intellectual history. Rescher's in-depth examination reveals how aporetic inconsistency can be managed through a plausibility analysis that breaks the chain of inconsistency at its weakest link by deploying right-of-way precedence based on considerations of cognitive centrality. Thus while involvement with cognitive conflicts and inconsistencies are pervasive in human thought, aporetic analysis can provide an effective means of damage control.

This volume is number five in the 11-volume Handbook of the History of Logic. It covers the first 50 years of the development of mathematical logic in the 20th century, and concentrates on the achievements of the great names of the period--Russell, Post, Gödel, Tarski, Church, and the like. This was the period in which mathematical logic gave mature expression to its four main parts: set theory, model theory, proof theory and recursion theory. Collectively, this work ranks as one of the greatest achievements of our intellectual history. Written by leading researchers in the field, both this volume and the Handbook as a whole are definitive reference tools for senior undergraduates, graduate students and researchers in the history of logic, the history of philosophy, and any discipline, such as mathematics, computer science, and artificial intelligence, for whom the historical background of his or her work is a salient consideration. • The entire range of modal logic is covered• Serves as a singular contribution to the intellectual history of the 20th century• Contains the latest scholarly discoveries and interpretative insights