This book presents a hierarchy of macroscopic models for semiconductor devices, studying three classes of models in detail: isentropic drift-diffusion equations, energy-transport models, and quantum hydrodynamic equations. The derivation of each, including physical discussions, is shown. Numerical simulations for modern semiconductor devices are performed, showing the particular features of each. The author develops modern analytical techniques, such as positive solution methods, local energy methods for free-boundary problems and entropy methods.

The object of this NATO Advanced Study Institute was to pre sent a tutorial 'introduction both to the basic physics of recent spectacular advances achieved in the field of metrology and to the determination of fundamental physical constants. When humans began to qualify their description of natural phenomena, metrology, the science of measurement, developed along side geometry and mathematics. However, flam antiquity to modern times, the role of metrology was mostly restricted to the need of commercial, social or scientific transactions of local or at most national scope. Beginning with the Renaissance, and particularly in western Europe during the last century, metrology rapidly developed an international character as a result of growing needs for more accurate measurements and common standards in the emerging indus trial society. Although the concerns of metrology are deeply rooted to fundamental sciences, it was, until recently, perceived by much of the scientific community as mostly custodial in character."

The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.

This title includes a superbly written introduction to the fascinating world of mathematical symbols, their meanings, and their uses. Mention the word mathematics and most people's reaction will be one of general bewilderment. There's something about its language of symbols and equations that many find intimidating. The result is that maths and science books for the general public usually avoid the use of symbols, meaning readers never get to fully appreciate the true power and elegance of mathematics. "Math for the Frightened" takes the opposite approach, gently introducing readers to the main concepts of mathematics and painlessly demonstrating how they are expressed as symbols, why symbols are used, and what can be achieved by doing so. If you've ever been curious about mathematics but afraid of its complexity, this book will help you overcome your fears and begin to appreciate math in all its glory.

In the aftermath of the discoveries in foundations of mathematiC's there was surprisingly little effect on mathematics as a whole. If one looks at stan dard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathemat ical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at "unnatural" interpretations of the membership relation."

Given the continuing cataclysmic shift in the economic landscape in the last few years, librarians have been forced to reevaluate not only the traditional services that they offer but also their continued existence and relevance to their academic institutions. Given the new normal of tighter constraint on personnel and materials budgets, librarians now are compelled to find new ways of offering services and forging new relationships with departments and programs outside the traditional library setting.

This volume highlights a number of projects being implemented in academic libraries including: rethinking the entire concept of a library, redefining physical space for new collaborative uses, adapting entrepreneurial techniques to acquire funding, creating new research tools and improving services, forging new consortial partnerships, allying more closely the mission of the library with that of the institution, and adapting public library programs to academic libraries. By re-examining the purpose of an academic library under continuing financial duress, librarians can ensure that their libraries will continue to have relevance to higher education.

This book was published as a special issue of College & Undergraduate Libraries.

A new era of international migration has been accompanied by increasingly restrictive immigration controls to manage migration to more developed countries. The consequence has been fewer routes to enter and/or stay in countries in a regularised way and as a result, an increase in the numbers of undocumented migrants. In this situation undocumented migrants, especially in relation to immigration controls and internal security have come to occupy an important role on the policy agenda of many nation states. The control and regulation of undocumented migrants has become an increasingly politicised issue. This edited collection brings together cutting edge scholarly research papers to explore undocumented migration at the international, national and individual levels. Starting with an overview of the literature on undocumented migration this book explores some of the key areas of research and policy in this area. This includes the making of undocumented migrants, the journey and processes, experiences of being undocumented at the individual level, collective action and return. This fascinating book explores the many facets of undocumented migration and of being an undocumented migrant in different geographical contexts that include Europe, Southern Africa, Central America and North America. This book was originally published as a special issue of Ethnic and Racial Studies.

MATHEMATICAL ANALYSIS FOR ECONOMISTS BY R. G. D. ALLEN The general science of mathematics is concerned with the investigation of patterns of connectedness, in abstrac tion from the particular relata and the particular modes of connection. ALFRED NORTH WHITEHEAD, Adventures of Ideas To connect elements in laws according to some logical or mathematical pattern is the ultimate ideal of science. MORRIS R. COHEN, Reason and Nature MAQMILLAN AND CO., LIMITED ST. MARTINS STREET, LONDON 1938 PRINTED IN GREAT I3KITAIN FOREWORD TEUS book, which is based on a series of lectures given at the London School of Economics annually since 1931, aims at providing a course of pure mathematics developed in the directions most useful to students of economics. At each stage the mathematical methods described are used in the elucidation of problems of economic theory. Illustrative examples are added to all chapters and it is hoped that the reader, in solving them, will become familiar with the mathematical tools and with their applications to concrete economic problems. The method of treatment rules out any attempt at a systematic development of mathematical economic theory but the essentials of such a theory are to be found either in the text or in the examples. I hope that the book will be useful to readers of different types. The earlier chapters are intended primarily for the student with no mathematical equipment other than that obtained, possibly many years ago, from a matriculation course. Such a student may need to accustom himself to the application of the elementary methods before proceeding to the more powerful processes described in the later chapters. The more advanced reader may use the earlysections for purposes of revision and pass on quickly to the later work. The experienced mathematical economist may find the book as a whole of service for reference and discover new points in some of the chapters. I have received helpful advice and criticism from many mathe maticians and economists. I am particularly indebted to Professor A. L. Bowley and to Dr. J. Marschak and the book includes numerous modifications made as a result of their suggestions on reading the original manuscript. I am also indebted to Mr. G. J. Nash who has read the proofs and has detected a number of slips in my construction of the examples. R. G. D. ALLEN THE LONDON SCHOOL OF ECONOMICS October, 1937 CONTENTS CHAP. PAGE FOREWORD ----------v A SHORT BIBLIOGRAPHY - ..... xiv THE USE OF GREEK LETTERS IN MATHEMATICAL ANALYSIS - - ...... xvi I. NUMBERS AND VARIABLES -------1 1.1 Introduction ---------1 1.2 Numbers of various types ------3 1.3 The real number system -------6 1.4 Continuous and discontinuous variables ... - 7 1.5 Quantities and their measurement ..... 9 1.0 Units of measurement - - - - - - - 13 1.7 Derived quantities - - - - - - - - 14 1.8 The location of points in space - - - - - 1G 1.9 Va viable points and their co-ordinates 20 EXAMPLES 1 The measurement of quantities graphical methods ---------23 . JpOJ ACTIONS AND THEIR DIAGRAMMATIC REPRESENTATION 28 2.1 Definition and examples of functions 28 2.2 The graphs of functions - - - - - - - 32 2.3 Functions and curves - - - - - - - 3 5 2.4 Classification of functions - - - - - - 38 2.5 Function types - - - - - - - - 41 2.6 The symbolic representation of functions of any form - 45 2.7 The diagrammatic method - - - - - - 48 2.8 The solution ofequations in one variable 50 2.9 Simultaneous equations in two variables 54 EXAMPLES II Functions and graphs the solutionjof equa- tions ......... 57 III. ELEMENTARY ANALYTICAL GEOMETRY 61 3.1 Introduction ......... 61 3.2 The gradient of a straight line ..... 03 3.3 The equation of a straight line - - - 66 viii CONTENTS CHAP. 3.4 The parabola 09 3.5 The rectangular hyperbola - - - - - - 72 3.6 The circle 75 3.7 Curve classes and curve systems . - ... 76 3.8 An economic problem in analytical geometry 80 EXAMPLES III--The straight line curves and curve systems 82 IV...

This book explores the creation of knowledge in applied mathematics. It mainly analyses applications of mathematical theories in several contexts. The author analyses the generation of advanced theories that enable people to understand problems in a scientific way, and proposes cognitive models dealing with the observation of human behaviour and its abstraction into comprehensible mathematical models, as this is a main problem in our modern world. This work is directed at people concerned with understanding cognitive processes when tackling complex problems, as it shows the building of knowledge in the making of scientific approaches to any discipline. Using a cross-disciplinary approach, he focuses on the key issues of theories and technologies applicable in a wide variety of contexts, for example in military organizations, in research and development departments and in general strategic planning, as shown in applied cases in Latin America.

First published in 1978, this study considers the impact of dissenting voices upon literature, religion and politics in order to reassess the nonconformist contribution to English culture from the eighteenth century through to the twentieth. This historical survey takes into the account the contribution of a wealth of seminal literary figures such as the poets Isaac Watts, Charles Wesley and William Blake; and the novelists Elizabeth Gaskell, George Elliot, Mark Rutherford and D. H. Lawrence. However, far from consigning his study merely to literature, Davie also includes important orators like Robert Hall; scientists like Michael Farraday and Philip Gosse; political activists like Joseph Priestly, and soldiers like Orde Wingate. Unitarians, Sandemanians, Wesleyan Methodists and the Plymouth Brethren are considered, as well as the older denominations.

Many student private pilots don't realize at the start of their course that many hours of study are required on top of the in-class schedule. This book will help those trainee pilots without science backgrounds, or those that need a refresher, to brush up on the necessary theory. It covers subjects that will be encountered many times during the PPL course, such as principles of flight, aircraft general knowledge, flight performance and planning, meteorology, navigation and human factors. The content is organized around two main groups of information, namely core knowledge, concentrating more on the concepts; and a practical toolbox, dedicated to some techniques that will be required during the course.

Many online applications, especially in the financial industries, are running on blockchain technologies in a decentralized manner, without the use of an authoritative entity or a trusted third party. Such systems are only secured by cryptographic protocols and a consensus mechanism. As blockchain-based solutions will continue to revolutionize online applications in a growing digital market in the future, one needs to identify the principal opportunities and potential risks. Hence, it is unavoidable to learn the mathematical and cryptographic procedures behind blockchain technology in order to understand how such systems work and where the weak points are. Cryptographic Primitives in Blockchain Technology provides an introduction to the mathematical and cryptographic concepts behind blockchain technologies and shows how they are applied in blockchain-based systems. This includes an introduction to the general blockchain technology approaches that are used to build the so-called immutable ledgers, which are based on cryptographic signature schemes. As future quantum computers will break some of the current cryptographic primitive approaches, Andreas Bolfing considers their security and presents the current research results that estimate the impact on blockchain-based systems if some of the cryptographic primitive break. Based on the example of Bitcoin, he shows that weak cryptographic primitives pose a possible danger for the ledger, which can be overcome through the use of the so-called post-quantum cryptographic approaches.

The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Clear, plastic straws can be used for counting and making shapes.

In Dutch "WISKOBAS" stands for a particular kind of mathematics in the elementary school (ages 6-12). In tum Wiskobas was one of the depart ments in the IOWO, the Institute for the Development of Mathematics Education. This institute was concerned with the development of material for mathematics education as well as the related research on the possibility of change from the then existing arithmetic instruction to the future mathematics education. The present publication Three Dimensions has three aims: to give a picture of the goals Wiskobas set for future mathematics education, at the same time to show how such goals can be described, and to show the theoretical framework of the Wiskobas curriculum. The problem at hand is not at all simple. What is more, Wiskobas' ideas about mathematics education cannot literally be translated into strings of words. So how can we face the accusation that our objectives are unattain able and the goal itself irrational? In order to avoid this vagueness as much as possible and for the sake of clarity, this book makes continuous use of illustrations of mathematics education. In these examples both the subject-matter and the methods of description of the goals are illustrated as explicitly as possible, while at the same time creating the opportunity to read between the lines. The reader is urged to follow carefully the mathe matical material at the start of each chapter. This advice applies both to the more general education oriented, and to the more mathematical didactical reader."

This volume consists of chapters written by eminent scientists and engineers from the international community and present significant advances in several theories, methods and applications of an interdisciplinary research. These contributions focus on both old and recent developments of Global Optimization Theory, Convex Analysis, Calculus of Variations, Discrete Mathematics and Geometry, as well as several applications to a large variety of concrete problems, including applications of computersto the study of smoothness and analyticity of functions, applications to epidemiological diffusion, networks, mathematical models of elastic and piezoelectric fields, optimal algorithms, stability of neutral type vector functional differential equations, sampling and rational interpolation for non-band-limited signals, recurrent neural network for convex optimization problems and experimental design.

The book also contains some review works, which could prove particularly useful for a broader audience of readers in Mathematical and Engineering subjects and especially to graduate students who search for the latest information."

Originally published in 1986, this work examines how key figures such as Garfinkel, Sacks and Cicourel have revolutionised thinking about how sociology's presuppositions about 'being social' are grounded. Yet until the appearance of this book there were no clear and authoritative introductions to the main thinkers in the field or their work. In assessing the critical reception of Ethnomethodology, Sharrock and Anderson argue persuasively that much is wide of the mark - as they say, the real argument has yet to begin.

This book is a product of the BACOMET group, a group of educators-mainly educators of prospective teachers of mathematics-who first came together in 1980 to engage in study, discussion, and mutual reflection on issues in mathematics education. BACOMET is an acronym for BAsic Components of Mathematics Education for Teachers. The group was formed after a series of meetings in 1978-1979 between Geoffrey Howson, Michael Otte, and the late Bent Christiansen. In the ensuing years, BACOMET initiated several projects that resulted in published works. The present book is the main product of the BACOMET project entitled Meaning and Communication in Mathematics Education. This theme was chosen because of the growing recognition internationally that teachers of mathematics must deal with questions of meaning, sense making, and communication if their students are to be proficient learners and users of mathematics. The participants in this project were the following: Nicolas Balacheff (Grenoble, France) Maria Bartolini Bussi (Modena, Italy) Rolf Biehler (Bielefeld, Germany) Robert Davis (New Brunswick, NJ, USA) Willibald Dorfler (Klagenfurt, Austria) Tommy Dreyfus (Holon, Israel) Joel Hillel (Montreal, Canada) Geoffrey Howson (Southampton, England) Celia Hoyles-Director (London, England) Jeremy Kilpatrick-Director (Athens, GA, USA) Christine Keitel (Berlin, Germany) Colette Laborde (Grenoble, France) Michael Otte (Bielefeld, Germany) Kenneth Ruthven (Cambridge, England) Anna Sierpinska (Montreal, Canada) Ole Skovsmose-Director (Aalborg, Denmark) Conversations about directions the project might take began in May 1993 at a NATO Advanced Research Workshop of the previous BACOMET project in VIII PREFACE

1. 1 A paradigm About one hundred years ago, Maurice Couette, a French physicist, de signed an apparatus consisting of two coaxial cylinders, the space between the cylinders being filled with a viscous fluid and the outer cylinder being rotated at angular velocity O2. The purpose of this experiment was, follow ing an idea of the Austrian physicist Max Margules, to deduce the viscosity of the fluid from measurements of the torque exerted by the fluid on the inner cylinder (the fluid is assumed to adhere to the walls of the cylinders). At least when O is not too large, the fluid flow is nearly laminar and 2 the method of Couette is valuable because the torque is then proportional to 110, where II is the kinematic viscosity of the fluid. If, however, O is 2 2 increased to a very large value, the flow becomes eventually turbulent. A few years later, Arnulph Mallock designed a similar apparatus but allowed the inner cylinder to rotate with angular velocity 01, while O2 = o. The surprise was that the laminar flow, now known as the Couette flow, was not observable when 0 exceeded a certain "low" critical value Ole, even 1 though, as we shall see in Chapter II, it is a solution of the model equations for any values of 0 and O ."

This book studies the situation over discrete Abelian groups with wide range applications. It covers classical functional equations, difference and differential equations, polynomial ideals, digital filtering and polynomial hypergroups, giving unified treatment of several different problems. There is no other comprehensive work in this field. The book will be of interest to graduate students, research workers in harmonic analysis, spectral analysis, functional equations and hypergroups.