On the occasion of the retirement of Wolfram Pohlers the Institut fur Mathematische Logik und Grundlagenforschung of the University of Munster organized a colloquium and a workshop which took place July 17 - 19, 2008. This event brought together proof theorists from many parts of the world who have been acting as teachers, students and collaborators of Wolfram Pohlers and who have been shaping the field of proof theory over the years. The present volume collects papers by the speakers of the colloquium and workshop; and they produce a documentation of the state of the art of contemporary proof theory.

This stress-free layperson's introduction to the intriguing world of numbers is designed to acquaint the general reader with the elegance and wonder of mathematics. This enjoyable volume gives readers a working knowledge of math's most important concepts, an appreciation of its elegant logical structure, and an understanding of its historical significance in creating our contemporary world.

This book offers a fresh context for the ground-breaking work of the great mathematician Andrei Markov. A distinguished collection of scholars and scientists provide exciting new insights into the signifiance and contemporary applicability of Markov's work. (Mathematics)

This book is a product of love and respect. If that sounds rather odd I initially apologise, but let me explain why I use those words. The original manuscript was of course Freudenthal's, but his colleagues have carried the project through to its conclusion with love for the man, and his ideas, and with a respect developed over years of communal effort. Their invitation to me to write this Preface e- bles me to pay my respects to the great man, although I am probably incurring his wrath for writing a Preface for his book without his permission! I just hope he understands the feelings of all colleagues engaged in this particular project. Hans Freudenthal died on October 13th, 1990 when this book project was well in hand. In fact he wrote to me in April 1988, saying "I am thinking about a new book. I have got the sub-title (China Lectures) though I still lack a title". I was astonished. He had retired in 1975, but of course he kept working. Then in 1985 we had been helping him celebrate his 80th birthday, and although I said in an Editorial Statement in Educational Studies in Mathematics (ESM) at the time "we look forward to him enjoying many more years of non-retirement" I did not expect to see another lengthy manuscript.

A fascinating and insightful collection of papers on the strong links between mathematics and culture. The contributions range from cinema and theatre directors to musicians, architects, historians, physicians, graphic designers and writers. The text highlights the cultural and formative character of mathematics, its educational value, and imaginative dimension. These articles are highly interesting, sometimes amusing, and make excellent starting points for researching the strong connection between scientific and literary culture.

This book is devoted to the topological fixed point theory of multivalued mappings including applications to differential inclusions and mathematical economy. It is the first monograph dealing with the fixed point theory of multivalued mappings in metric ANR spaces. Although the theoretical material was tendentiously selected with respect to applications, the text is self-contained. Current results are presented.

This book is one of the first to attempt a systematic in-depth analysis of assessment in mathematics education in most of its important aspects: it deals with assessment in mathematics education from historical, psychological, sociological, epistmological, ideological, and political perspectives. The book is based on work presented at an invited international ICMI seminar and includes chapters by a team of outstanding and prominent scholars in the field of mathematics education. Based on the observation of an increasing mismatch between the goals and accomplishments of mathematics education and prevalent assessment modes, the book assesses assessment in mathematics education and its effects. In so doing it pays particular attention to the need for and possibilities of assessing a much wider range of abilities than before, including understanding, problem solving and posing, modelling, and creativity. The book will be of particular interest to mathematics educators who are concerned with the role of assessment in mathematics education, especially as regards innovation, and to everybody working within the field of mathematics education and related areas: in R&D, curriculum planning, assessment institutions and agencies, teacher trainers, etc.

One of the most effective ways to stimulate students to enjoy intellectual efforts is the scientific competition. In 1894, the Hungarian Mathematical and Physical Society introduced a mathematical competition for high school students. Among the winners were Lipot Fejer, Alfred Haar, Todor Karman, Marcel Riesz, Gabor Szego, and many others who became world-famous scientists. The success of high school competitions led the Mathematical Society to found a college-level contest, named after Miklos Schweitzer. The problems of the Schweitzer contests are proposed and selected by the most prominent Hungarian mathematicians. This book collects the problems posed in contests between 1962 and 1991, which range from algebra, combinatorics, theory of functions, geometry, measure theory, number theory, operator theory, probability theory, and topology to set theory. Solutions are included. The Schweitzer competition is one of the most unique in the world. Experience shows that this competition helps identify research talents. This collection of problems and solutions in several fields in mathematics can serve as a guide for many undergraduates and young mathematicians. The large variety of research-level problems should interest more mature mathematicians and historians of mathematics as well.

STATISTICAL MECHANICS JOSEPH EDWARD MAYER... Associate Professor of Chemistry, Columbia University AND MARIA GOEPPERT MAYER Lecturer in Chemistry, Columbia University NEW YORK JOHN WILEY SONS, INC. LONDON CHAPMAN HALL, LIMITED 1940 PREFACE The rapid increase, in the past few decades, of knowledge concerning the structure of molecules has made the science of statistical mechanics a practical tool for interpreting and correlating experimental data. It is therefore desirable to present this subject in a simple manner in order to make it easily available to scientists whose familiarity with theoretical physics is limited. This book, which grew out of lectures and seminars given to graduate students in chemistry and physics, aims to fulfill this purpose. The development of quantum mechanics has altered both the axio matic foundation and the details of the methods of statistical mechanics. Although the results of a large number of statistical calculations are un affected by the introduction of quantum mechanics, the chemists interest happens to be largely in fields where quantum effects are im portant. Consequently, in our presentation, the laws of statistical mechanics are founded on the concepts of both quantum and classical mechanics. The equivalence of the two methods has been stressed, but the quantum-mechanical language has been favored. We believe that this introduction of quantum statistics at the beginning simplifies rather than puts a burden upon the initial concepts. It is to be emphasized that the simpler ideas of quantum mechanics, which are all that is used, are as widely known as the more abstract theorems of classical mechanics which they replace. Simplicity of presentationrather than brevity and elegance has been our endeavor. However, we have not consciously sacrificed rigor. Care has been taken to make the book suitable for reference by sum marizing and tabulating final equations as well as by an attempt to make individual chapters complete in themselves without too much reference to previous subjects. All the theorems and results of mechanics and quantum mechanics which are used later have been summarized, largely without proof, in Chapter 2. The last section, 2k, on Einstein-Bose and Fermi-Dirac systems, ties up closely with Chapters 5 and 16 only. Chapters 3 and 4 contain the derivation of the fundamental statistical laws on which the book is based. Chapter 10 is prerequisite for Chap ters l 1 tol4. Otherwise, individual subj ects may be taken up in different order. vii viii PREFACE In Chapters 7 to 9 considerable space is devoted to the calculation of thermodynamic functions for perfect gases, which was considered justi fied by the value of the results for the chemist. These chapters may be omitted by readers uninterested in the subject. Chapters 13 and 14 on the imperfect gas and condensation theory, respectively, are somewhat more complicated than the remainder, but are included because of our special interest in the subject. The aim of the book is to give the reader a clear understanding of principles and to prepare him thoroughly for the use of the science and the study of recent papers. Many of the simpler applications are dis cussed in some detail, but in general language without comparison with experiment. The more complicated subjects have been omitted, as have been those for which at present only partial solutions are obtained. This choicehas excluded many of the contemporary developments, especially the interesting work of J. G. Kirkwood, L. Onsager, H. Eyring, and W, F. Giauque. In conclusion we express our gratitude to Professors Max Born, Karl F. Hcrzfeld, and Edward Teller, who have read and criticized several parts of the manuscript. We also thank Dr. Elliot Montroll, who aided in reading proof and who made many helpful suggestions. JOSEPH EDWARD MAYER MARIA GOEPPERT MAYER NEW YORK CITY March 31, 1940 Dedicated to our teachers Gilbert N...

Robust designa "that is, managing design uncertainties such as model uncertainty or parametric uncertaintya "is the often unpleasant issue crucial in much multidisciplinary optimal design work. Recently, there has been enormous practical interest in strategies for applying optimization tools to the development of robust solutions and designs in several areas, including aerodynamics, the integration of sensing (e.g., laser radars, vision-based systems, and millimeter-wave radars) and control, cooperative control with poorly modeled uncertainty, cascading failures in military and civilian applications, multi-mode seekers/sensor fusion, and data association problems and tracking systems. The contributions to this book explore these different strategies. The expression "optimization-directeda in this booka (TM)s title is meant to suggest that the focus is not agonizing over whether optimization strategies identify a true global optimum, but rather whether these strategies make significant design improvements.

Considering the stupendous gain in importance, in the banking and insurance industries since the early 1990 s, of mathematical methodology, especially probabilistic methodology, it was a very natural idea for the French "Academie des Sciences" to propose a series of public lectures, accessible to an educated audience, to promote a wider understanding for some of the fundamental ideas, techniques and new tools of the financial industries.

These lectures were given at the "Academie des Sciences" in Paris by internationally renowned experts in mathematical finance, and later written up for this volume which develops, in simple yet rigorous terms, some challenging topics such as risk measures, the notion of arbitrage, dynamic models involving fundamental stochastic processes like Brownian motion and Levy processes.

The Ariadne s thread leads the reader from Louis Bachelier s thesis 1900 to the famous Black-Scholes formula of 1973 and to most recent work close to Malliavin s stochastic calculus of variations. The book also features a description of the trainings of French financial analysts which will help them to become experts in these fast evolving mathematical techniques."

This book will serve as a valuable source of information about triangulations for the graduate student and researcher. With emphasis on computational issues, it presents the basic theory necessary to construct and manipulate triangulations. In particular, the book gives a tour through the theory behind the Delaunay triangulation, including algorithms and software issues. It also discusses various data structures used for the representation of triangulations.

The KK-theory of Kasparov is now approximately twelve years old; its power, utility and importance have been amply demonstrated. Nonethe less, it remains a forbiddingly difficult topic with which to work and learn. There are many reasons for this. For one thing, KK-theory spans several traditionally disparate mathematical regimes. For another, the literature is scattered and difficult to penetrate. Many of the major papers require the reader to supply the details of the arguments based on only a rough outline of proofs. Finally, the subject itself has come to consist of a number of difficult segments, each of which demands prolonged and intensive study. is to deal with some of these difficul Our goal in writing this book ties and make it possible for the reader to "get started" with the theory. We have not attempted to produce a comprehensive treatise on all aspects of KK-theory; the subject seems too vital to submit to such a treatment at this point. What seemed more important to us was a timely presen tation of the very basic elements of the theory, the functoriality of the KK-groups, and the Kasparov product."

This book offers a new conceptual framework for reflecting on the role of information and communication technology in mathematics education. Discussion focuses on how computers, writing and oral discourse transform education at an epistemological as well as a political level. Building on examples, research and theory, the authors propose that knowledge is not constructed solely by humans, but by collectives of humans and technologies of intelligence.

For many years physics and mathematics have had a fruitful influence on one another. Classical mechanics and celestial mechanics have produced very deep problems whose solutions have enhanced mathematics. On the other hand, mathematics itself has found interesting theories which then (sometimes after many years) have been reflected in physics, confirming the thesis that nothing is more practical than a good theory. The same is true for the younger physical discipline -of quantum mechanics. In the 1930s two events, not at all random, became: The mathematical back grounds of both quantum mechanics and probability theory. In 1936, G. Birkhoff and J. von Neumann published their historical paper "The logic of quantum mechanics," in which a quantum logic was suggested. The mathematical foundations of quantum mechanics remains an outstanding problem of mathematics, physics, logic and philosophy even today. The theory of quantum logics is a major stream in this axiomatical knowledge river, where L(H), the system of all closed subspaces of a Hilbert space H, due to J. von Neumann, plays an important role. When A.M. Gleason published his solution to G. Mackey's problem showing that any state (= probability measure) corresponds to a density operator, he probably did not anticipate that his solution would become a cornerstone of ax iomati cal theory of quantum mechanics nor that it would provide many interesting applications to mathematics."

When the 50th anniversary of the birth of Information Theory was celebrated at the 1998 IEEE International Symposium on Informa tion Theory in Boston, there was a great deal of reflection on the the year 1993 as a critical year. As the years pass and more perspec tive is gained, it is a fairly safe bet that we will view 1993 as the year when the "early years" of error control coding came to an end. This was the year in which Berrou, Glavieux and Thitimajshima pre sented "Near Shannon Limit Error-Correcting Coding and Decoding: Turbo Codes" at the International Conference on Communications in Geneva. In their presentation, Berrou et al. claimed that a combi nation of parallel concatenation and iterative decoding can provide reliable communications at a signal to noise ratio that is within a few tenths of a dB of the Shannon limit. Nearly fifty years of striving to achieve the promise of Shannon's noisy channel coding theorem had come to an end. The implications of this result were immediately apparent to all -coding gains on the order of 10 dB could be used to dramatically extend the range of communication receivers, increase data rates and services, or substantially reduce transmitter power levels. The 1993 ICC paper set in motion several research efforts that have permanently changed the way we look at error control coding."

This Seminar began in Moscow in November 1943 and has continued without interruption up to the present. We are happy that with this vol ume, Birkhiiuser has begun to publish papers of talks from the Seminar. It was, unfortunately, difficult to organize their publication before 1990. Since 1990, most of the talks have taken place at Rutgers University in New Brunswick, New Jersey. Parallel seminars were also held in Moscow, and during July, 1992, at IRES in Bures-sur-Yvette, France. Speakers were invited to submit papers in their own style, and to elaborate on what they discussed in the Seminar. We hope that readers will find the diversity of styles appealing, and recognize that to some extent this reflects the diversity of styles in a mathematical society. The principal aim was to have interesting talks, even if the topic was not especially popular at the time. The papers listed in the Table of Contents reflect some of the rich variety of ideas presented in the Seminar. Not all the speakers submit ted papers. Among the interesting talks that influenced the seminar in an important way, let us mention, for example, that of R. Langlands on per colation theory and those of J. Conway and J. McKay on sporadic groups. In addition, there were many extemporaneous talks as well as short discus sions."

Intended for first- or second-year undergraduates, this introduction to discrete mathematics covers the usual topics of such a course, but applies constructivist principles that promote - indeed, require - active participation by the student. Working with the programming language ISETL, whose syntax is close to that of standard mathematical language, the student constructs the concepts in her or his mind as a result of constructing them on the computer in the syntax of ISETL. This dramatically different approach allows students to attempt to discover concepts in a "Socratic" dialog with the computer. The discussion avoids the formal "definition-theorem" approach and promotes active involvement by the reader by its questioning style. An instructor using this text can expect a lively class whose students develop a deep conceptual understanding rather than simply manipulative skills. Topics covered in this book include: the propositional calculus, operations on sets, basic counting methods, predicate calculus, relations, graphs, functions, and mathematical induction.

Contents -
Preface -
1. REVIEW OF BASIC MATERIAL - FUNCTIONS, INEQUALITIES - 2. DIFFERENTIAL CALCULUS - 3. INTEGRATION - 4. FUNCTIONS OF MANY VARIABLES; PARTIAL DIFFERENTIATION - 5. VECTORS - 6. SERIES, TAYLOR-MACLAURIN SERIES - 7. COMPLEX NUMBERS - 8. ORTHOGONAL FUNCTIONS AND FOURIER SERIES - 9. DETERMINANTS - 10. MATRICES - 11. DIFFERENTIAL EQUATIONS - 12. PARTIAL DIFFERENTIAL EQUATIONS - 13. NUMERICAL METHODS - 14. ELEMENTARY STATISTICS AND ERROR ANALYSIS - Problems for Solution - Bibliography - Answers to Problems - Index

A Volume in International Perspectives on Mathematics Education - Cognition, Equity & Society Series Editor Bharath Sriraman, The University of Montana and Lyn English, Queensland University of Technology This volume represents a serious attempt to understand what it is that structures the pedagogical experience. In that attempt there are two main objectives. One is a theoretical interest that involves examining the issue of the subjectivity of the teacher and exploring how intersubjective negotiations shape the production of classroom practice. A second objective is to apply these understandings to the production of mathematical knowledge and to the construction of identities in actual mathematics classrooms. To that end the book will contain substantial essays that draw on postmodern philosophies of the social to explore theory's relationship with the practice of mathematics pedagogy. Unpacking Pedagogy takes new ideas seriously and engages readers in theory development. Groundbreaking in content, the book investigates how our thinking about classroom practice in general, and mathematics teaching (and learning), in particular, might be transformed. As a key resource for interrogating and understanding classroom life, the book's sophisticated analyses allow readers to build new knowledge about mathematics pedagogy. In turn, that new knowledge will provide them with the tools to engage more actively in educational criticism and to play a role in educational change.

Exam Board: Edexcel Level: AS/A-level Subject: Mathematics First Teaching: September 2017 First Exam: June 2018 Endorsed for Edexcel Help students to develop their knowledge and apply their reasoning to mathematical problems with worked examples, stimulating activities and assessment support tailored to the 2017 Edexcel specification. The content benefits from the expertise of subject specialist Keith Pledger and the support of MEI (Mathematics in Education and Industry). -Prepare students for assessment with skills-building activities, worked examples and practice questions tailored to the changed criteria. -Develop a fuller understanding of mathematical concepts with real world examples that help build connections between topics and develop mathematical modelling skills. -Cement understanding of problem-solving, proof and modelling with dedicated sections on these key areas. -Confidently teach the new statistics requirements with five dedicated statistics chapters and questions around the use of large data sets. -Cover the use of technology in Mathematics with a variety of questions based around the use of spreadsheets, graphing software and graphing calculators. -Provide clear paths of progression that combine pure and applied maths into a coherent whole.