This book shows how the practice of script writing can be used both as a pedagogical approach and as a research tool in mathematics education. It provides an opportunity for script-writers to articulate their mathematical arguments and/or their pedagogical approaches. It further provides researchers with a corpus of narratives that can be analyzed using a variety of theoretical perspectives.Various chapters argue for the use of dialogical method and highlight its benefits and special features. The chapters examine both "low tech" implementations as well as the use of a technological platform, LessonSketch. The chapters present results of and insights from several recent studies, which utilized scripting in mathematics education research and practice.

Mathematica by Example, Fifth Edition is an essential desk reference for the beginning Mathematica user, providing step-by-step instructions on achieving results from this powerful software tool. The book fully accounts for the dramatic changes to functionality and visualization capabilities in the most recent version of Mathematica (10.4). It accommodates the full array of new extensions in the types of data and problems that Mathematica can immediately handle, including cloud services and systems, geographic and geometric computation, dynamic visualization, interactive applications and other improvements. It is an ideal text for scientific students, researchers and aspiring programmers seeking further understanding of Mathematica. Written by seasoned practitioners with a view to practical implementation and problem-solving, the book's pedagogy is delivered clearly and without jargon using representative biological, physical and engineering problems. Code is provided on an ancillary website to support the use of Mathematica across diverse applications.

This book provides a one-stop resource for mathematics educators, policy makers and all who are interested in learning more about the why, what and how of mathematics education in Singapore. The content is organized according to three significant and closely interrelated components: the Singapore mathematics curriculum, mathematics teacher education and professional development, and learners in Singapore mathematics classrooms. Written by leading researchers with an intimate understanding of Singapore mathematics education, this up-to-date book reports the latest trends in Singapore mathematics classrooms, including mathematical modelling and problem solving in the real-world context.

Since its inception in the famous 1936 paper by Birkhoff and von Neumann entitled "The logic of quantum mechanics" quantum logic, i.e. the logical investigation of quantum mechanics, has undergone an enormous development. Various schools of thought and approaches have emerged and there are a variety of technical results.
Quantum logic is a heterogeneous field of research ranging from investigations which may be termed logical in the traditional sense to studies focusing on structures which are on the border between algebra and logic. For the latter structures the term quantum structures is appropriate.
The chapters of this Handbook, which are authored by the most eminent scholars in the field, constitute a comprehensive presentation of the main schools, approaches and results in the field of quantum logic and quantum structures. Much of the material presented is of recent origin representing the frontier of the subject.
The present volume focuses on quantum structures. Among the structures studied extensively in this volume are, just to name a few, Hilbert lattices, D-posets, effect algebras MV algebras, partially ordered Abelian groups and those structures underlying quantum probability.
- Written by eminent scholars in the field of logic
- A comprehensive presentation of the theory, approaches and results in the field of quantum logic
- Volume focuses on quantum structures

A Volume in The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education Series Editor Bharath Sriraman, The University of Montana Beliefs and Mathematics is a Festschrift honoring the contributions of Gunter Torner to mathematics education and mathematics. Mathematics Education as a legitimate area of research emerged from the initiatives of well known mathematicians of the last century such as Felix Klein and Hans Freudenthal. Today there is an increasing schism between researchers in mathematics education and those in mathematics as evidenced in the Math wars in the U.S and other parts of the world. Gunter Torner represents an international voice of reason, well respected and known in both groups, one who has successfully bridged and worked in both domains for three decades. His contributions in the domain of beliefs theory are well known and acknowledged. The articles in this book are written by many prominent researchers in the area of mathematics education, several of whom are editors of leading journals in the field and have been at the helm of cutting edge advances in research and practice.The contents cover a wide spectrum of research, teaching and learning issues that are relevant for anyone interested in mathematics education and its multifaceted nature of research. The book as a whole also conveys the beauty and relevance of mathematics in societies around the world. It is a must read for anyone interested in mathematics education.

AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME XXXI FUNCTIONAL ANALYSIS AND SEMI-GROUPS BY EINAR HILLE PROFESSOR OF MATHEMATICS YALE UNIVERSITY PUBLISHED BY THE AMERICAN MATHEMATICAL SOCIETY 531 WEST 116iH STREET, NEW YORK CITY 1948 To KIRSTI And each man hears as the twilight nears, to the beat of his dying hearty The Devil drum on the darkened pane You did it, but was it Art FOREWORD The analytical theory of semi-groups is a recent addition to the ever-growing list of mathematical disciplines. It was my good fortune to take an early interest in this disci pline and to see it reach maturity. It has been a pleasant association I hail a semi-group when I see one and I seem to see them every where Friends have observed, however, that there are mathematical objects which are not semi-groups. The present book is an elaboration of my Colloquium Lectures delivered before the American Mathematical Society at its August, 1944 meeting at Wellesley College. I wish to thank the Society and its officers for their invitation to present and publish these lectures. The book is divided into three parts plus an appendix. My desire to give a practically self-contained presentation of the theory required the inclusion of an elaborate introduc tion to modern functional analysis with special emphasis on function theory in Banach spaces and algebras. This occupies Part One of the book and the Appendix these portions can be read separately from the rest and may be used as a text in a course on operator theory. It is possible to cover most of the material in these six chapters in two terms. The analytical theory of one-parameter semi-groups occupies Part Two while Part Three deals with theapplications to analysis. The latter include such varied topics as trigonometric series and integrals, summability, fractional integration, stochastic theory, and the problem of Cauchy for partial differential equations. In the general theory the reader will also find an alternate approach to ergodic theory. All semi-groups studied in this treatise are referred to a normed topology semi-groups without topology figure in a few places but no details are given. The task of developing an adequate theory of trans formation semi-groups operating in partially ordered spaces is left to more competent hands. The literature has been covered rather incompletely owing to recent war conditions and to the wide range of topics touched upon, which have made it exceedingly difficult to give the proper credits. This investigation has been supported by grants from the American Philosophical Society and from Yale University which are gratefully acknowledged. On the personal side, it is a great pleasure to express my gratitude to the many friends who have aided me in pre paring this book. J. D. Tamarkin, who read and criticized my early work in the field and who vigorously urged its inclusion in the Colloquium Series is beyond the reach of my grati tude. I am deeply indebted to Nelson Dunford and to Max Zorn who have contributed extensively to the book, the former chiefly to Chapters II, III, V, VIII, IX, and XIV, the latter to Chapters IV, VII, and XXII. Both have given me generously of their time and special experience. Various portions of the manuscript have been critically examined and amended by Warren Ambrose, E. G. Begle, H. Cramdr, J. L. Doob, W. Feller, N. Jacobson, D. S. Miller, II. Pollard, C.E. Rickart, and I. E. Segal. To all helpers, named and un named, I extend my warmest thanks. EINAK HILLE New Haven, Conn., December, 1946 CONVENTIONS Each Part of the book starts with a Summary, each Chapter with an Orientation. The chapters are divided into sections and the sections, except orientations, are grouped into paragraphs. Cross references are normally to sections, rarely to paragraphs. Section 3.10 is the tenth section of Chapter III it belongs to 2 which is referred to as 3.2 when necessary...

Edexcel and A Level Modular Mathematics C3 features: Student-friendly worked examples and solutions, leading up to a wealth of practice questions. Sample exam papers for thorough exam preparation. Regular review sections consolidate learning. Opportunities for stretch and challenge presented throughout the course. 'Escalator section' to step up from GCSE. PLUS Free LiveText CD-ROM, containing Solutionbank and Exam Cafe to support, motivate and inspire students to reach their potential for exam success. Solutionbank contains fully worked solutions with hints and tips for every question in the Student Books. Exam Cafe includes a revision planner and checklist as well as a fully worked examination-style paper with examiner commentary.

These Proceedings contain 22 refereed research and survey articles based on lectures given at the Turku Symposium on Number Theory in Memory of Kustaa Inkeri, held in Turku, Finland, from May 31 to June 4, 1999. The subject of the symposium was number theory in a broad sense with an emphasis on recent advances and modern methods. The topics covered in this volume include various questions in elementary number theory, new developments in classical Diophantine problems - in particular of the Fermat and Catalan type, the ABC-conjecture, arithmetic algebraic geometry, elliptic curves, Diophantine approximations, Abelian fields, exponential sums, sieve methods, box splines, the Riemann zeta-function and other Dirichlet series, and the spectral theory of automorphic functions with its arithmetical applications.

This book presents recent research in the field of interaction between computational intelligence and mathematics, ranging from theory to applications. Computational intelligence, or soft computing consists of various bio-inspired methods, especially fuzzy systems, artificial neural networks, evolutionary and memetic algorithms. These research areas were initiated by professionals in various applied fields, such as engineers, economists, and financial and medical experts. Although computational intelligence offered solutions (at least quasi-optimal solutions) for problems with high complexity, vague and undeterministic features, initially little attention was paid to the mathematical models and analysis of the methods successfully applied. A typical example is the extremely successful Mamdani-algorithm, and its modifications and extensions, applied since the mid-1970s, where the first analysis of the simplest cases, showing why this algorithm was so efficient and stable, was not given until the early 1990s. Since the mid-2000s, the authors have organized international conferences annually to focus on the mathematical methodological issues in connection with computational intelligence approaches. These conferences have attracted a large number of submissions with a wide scope of topics and quality. The editors selected several high-quality papers and approached the authors to submit an essentially extended and improved book chapter based on the lectures.This volume is the first contributed book on the subject.

Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.

Integration is the sixth and last of the books that form the core of the Bourbaki series; it draws abundantly on the preceding five Books, especially General Topology and Topological Vector Spaces, making it a culmination of the core six. The power of the tool thus fashioned is strikingly displayed in Chapter II of the author's Theories Spectrales, an exposition, in a mere 38 pages, of abstract harmonic analysis and the structure of locally compact abelian groups.

The first volume of the English translation comprises Chapters 1-6; the present volume completes the translation with the remaining Chapters 7-9.

Chapters 1-5 received very substantial revisions in a second edition, including changes to some fundamental definitions. Chapters 6-8 are based on the first editions of Chapters 1-5. The English edition has given the author the opportunity to correct misprints, update references, clarify the concordance of Chapter 6 with the second editions of Chapters 1-5, and revise the definition of a key concept in Chapter 6 (measurable equivalence relations)."

EIGENFUNCTION EXPANSIONS ASSOCIATED WITH SECOND-ORDER DIFFERENTIAL EQUATIONS BY E. C. TITCHMARSH FJR. S. SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD OXFORD AT THE CLARENDON PRESS 1946 OXFORD UNIVERSITY PRESS AMEN HOUSE, E. G. 4 LONDON EDINBURGH GLASGOW NEW YORK TORONTO MELBOURNE CAPE TOWN BOMBAY CALCUTTA MADRAS GEOFFREY CUMBERLEGE PUBLISHER TO THE UNIVERSITY PREFACE THE idea of expanding an arbitrary function in terms of the solutions of a second-order differential equation goes back to the time of Sturm and Liouville, more than a hundred years ago. The first satisfactory proofs were constructed by various authors early in the twentieth century. Later, a general theory of the singular cases was given by Weyl, who-based i on the theory of integral equations. An alternative method, proceeding via the general theory of linear operators in Hilbert space, is to be found in the treatise by Stone on this subject. Here I have adopted still another method. Proofs of these expansions by means of contour integration and the calculus of residues were given by Cauchy, and this method has been used by several authors in the ordinary Sturm-Liouville case. It is applied here to the general singular case. It is thus possible to avoid both the theory of integral equations and the general theory of linear operators, though of course we are sometimes doing no more than adapt the latter theory to the particular case considered. The ordinary Sturm-Liouville expansion is now well known. I therefore dismiss it as rapidly as possible, and concentrate on the singular cases, a class which seems to include all the most interesting examples. In order to present a clear-cut theory in a reasonablespace, I have had to reject firmly all generalizations. Many of the arguments used extend quite easily to other cases, such as that of two simultaneous first-order equations. It seems that physicists are interested in some aspects of these questions. If any physicist finds here anything that he wishes to know, I shall indeed be delighted but it is to mathematicians that the book is addressed. I believe in the future of mathematics for physicists, but it seems desirable that a writer on this subject should understand physics as well as mathematics. E. C. T. NEW COLLEGE, OXFOBD, 1946. CONTENTS I. THE STUEM-LIOUVILLE EXPANSION ... 1 II. THE SINGULAB CASE SERIES EXPANSIONS . . 19 III. THE GENERAL SINGULAR CASE . . . .39 IV. EXAMPLES 69 V. THE NATURE OF THE SPECTRUM . . .97 VI. A SPECIAL CONVERGENCE THEOREM . . .118 VII. THE DISTRIBUTION OF THE EIGENVALUES . . 124 VIII. FURTHER APPROXIMATIONS TO JV A . . .135 IX. CONVERGENCE OF THE SERIES EXPANSION UNDER FOUBIER CONDITIONS 148 X. SUMMABILITY OF THE SERIES EXPANSION . . 163 REFERENCES 172 THE STURM-LIOUVILLE EXPANSION 1.1. Introduction. Let L denote a linear operator operating on a function y y x. Consider the equation Ly - AT, 1.1.1 where A is a number. A function which satisfies this equation and also certain boundary conditions e. g. which vanishes at x a and x b is called an eigenfunction. The corresponding value of A is called an eigenvalue. Thus ifi t n x is an eigenfunction corresponding to an eigenvalue n, L x Mx. 1.1.2 The object of this book is to study the operator,72 where q x is a given function of x defined over some given interval a, b. In this case y satisfies the second-order differential equation and tff n x satisfies s A- W0- 1J. 5 If we take this and the corresponding equation with m instead of n, multiply by ift m x 9 n x respectively, and subtract, we obtain Hence b A M - AJ J lUaOiM dx 0 m a- a a if i m x and rl x both vanish at x a and x b or satisfy a more general condition of the same kind. If m A n, it follows that b t m x t n x dx Q. 1-1.6 a 4967 2 THE STURM-LIOUVILLE EXPANSION Chap. I By multiplying if necessary by a constant we can arrange that x dx l. 1.1.7 The functions n x then form a normal orthogonal set...

A volume in Research in Mathematics Education Series Editor Barbara J. Dougherty, Iowa State University Marketing description: Issues of language in mathematics learning and teaching are important for both practical and theoretical reasons. Addressing issues of language is crucial for improving mathematics learning and teaching for students who are bilingual, multilingual, or learning English. These issues are also relevant to theory: studies that make language visible provide a complex perspective of the role of language in reasoning and learning mathematics. What is the relevant knowledge base to consider when designing research studies that address issues of language in the learning and teaching of mathematics? What scholarly literature is relevant and can contribute to research? In order to address issues of language in mathematics education, researchers need to use theoretical perspectives that integrate current views of mathematics learning and teaching with current views on language, discourse, bilingualism, and second language acquisition. This volume contributes to the development of such integrated approaches to research on language issues in mathematics education by describing theoretical perspectives for framing the study of language issues and methodological issues to consider when designing research studies. The volume provides interdisciplinary reviews of the research literature from four very different perspectives: mathematics education (Moschkovich), Cultural-Historical-Activity Theory (Gutierrez, Sengupta-Irving, & Dieckmann), systemic functional linguistics (Schleppegrell), and assessment (Solano-Flores). This volume offers graduate students and researchers new to the study of language in mathematics education an introduction to resources for conceptualizing, framing, and designing research studies. For those already involved in examining language issues, the volume provides useful and critical reviews of the literature as well as recommendations for moving forward in designing research. Lastly, the volume provides a basis for dialogue across multiple research communities engaged in collaborative work to address these pressing issues.

Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.

Over the last twenty years, Professor Franco Giannessi, a highly respected researcher, has been working on an approach to optimization theory based on image space analysis. His theory has been elaborated by many other researchers in a wealth of papers. Constrained Optimization and Image Space Analysis unites his results and presents optimization theory and variational inequalities in their light.

It presents a new approach to the theory of constrained extremum problems, including Mathematical Programming, Calculus of Variations and Optimal Control Problems. Such an approach unifies the several branches: Optimality Conditions, Duality, Penalizations, Vector Problems, Variational Inequalities and Complementarity Problems. The applications benefit from a unified theory.

A number of monographs of various aspects of complex analysis in several variables have appeared since the first version of this book was published, but none of them uses the analytic techniques based on the solution of the Neumann Problem as the main tool.
The additions made in this third, revised edition place additional stress on results where these methods are particularly important. Thus, a section has been added presenting Ehrenpreis' fundamental principle'' in full. The local arguments in this section are closely related to the proof of the coherence of the sheaf of germs of functions vanishing on an analytic set. Also added is a discussion of the theorem of Siu on the Lelong numbers of plurisubharmonic functions. Since the L2 techniques are essential in the proof and plurisubharmonic functions play such an important role in this book, it seems natural to discuss their main singularities.

A groundbreaking, flexible approach to computer science anddata science The Deitels' Introduction to Python for ComputerScience and Data Science: Learning to Program with AI, Big Data and the Cloudoffers a unique approach to teaching introductory Python programming,appropriate for both computer-science and data-science audiences. Providing themost current coverage of topics and applications, the book is paired withextensive traditional supplements as well as Jupyter Notebooks supplements.Real-world datasets and artificial-intelligence technologies allow students towork on projects making a difference in business, industry, government andacademia. Hundreds of examples, exercises, projects (EEPs) and implementationcase studies give students an engaging, challenging and entertainingintroduction to Python programming and hands-on data science. The book's modular architecture enables instructors toconveniently adapt the text to a wide range of computer-science anddata-science courses offered to audiences drawn from many majors.Computer-science instructors can integrate as much or as little data-scienceand artificial-intelligence topics as they'd like, and data-science instructorscan integrate as much or as little Python as they'd like. The book aligns withthe latest ACM/IEEE CS-and-related computing curriculum initiatives and withthe Data Science Undergraduate Curriculum Proposal sponsored by the NationalScience Foundation.