Mathematical World – serie

Throughout the history of mathematics, maximum and minimum problems have played an important role in the evolution of the field. Many beautiful and important problems have appeared in a variety of branches of mathematics and physics, as well as in other fields of sciences. The greatest scientists of the past - Euclid, Archimedes, Heron, the Bernoullis, Newton, and many others - took part in seeking solutions to these concrete problems. The solutions stimulated the development of the theory, and, as a result, techniques were elaborated that made possible the solution of a tremendous variety of problems by a single method. This book presents fifteen 'stories' designed to acquaint readers with the central concepts of the theory of maxima and minima, as well as with its illustrious history.This book is accessible to high school students and would likely be of interest to a wide variety of readers. In Part One, the author familiarizes readers with many concrete problems that lead to discussion of the work of some of the greatest mathematicians of all time. Part Two introduces a method for solving maximum and minimum problems that originated with Lagrange. While the content of this method has varied constantly, its basic conception has endured for over two centuries. The final story is addressed primarily to those who teach mathematics, for it impinges on the question of how and why to teach. Throughout the book, the author strives to show how the analysis of diverse facts gives rise to a general idea, how this idea is transformed, how it is enriched by new content, and how it remains the same in spite of these changes.

This book is an introduction to elementary topology presented in an intuitive way, emphasizing the visual aspect. Examples of nontrivial and often unexpected topological phenomena acquaint the reader with the picturesque world of knots, links, vector fields, and two-dimensional surfaces. The book begins with definitions presented in a tangible and perceptible way, on an everyday level, and progressively makes them more precise and rigorous, eventually reaching the level of fairly sophisticated proofs. This allows meaningful problems to be tackled from the outset. Another unusual trait of this book is that it deals mainly with constructions and maps, rather than with proofs that certain maps and constructions do or do not exist. The numerous illustrations are an essential feature. The book is accessible not only to undergraduates but also to high school students and will interest any reader who has some feeling for the visual elegance of geometry and topology.

'This is a sample of rich Russian mathematical culture written by professional mathematicians with great experience in working with high school students...Problems are on very simple levels, but building to more complex and advanced work...contains solutions to almost all problems; methodological notes for the teacher...developed for a peculiarly Russian institution (the mathematical circle), but easily adapted to American teachers' needs, both inside and outside the classroom' - from the Translator's notes.What kind of book is this? It is a book produced by a remarkable cultural circumstance in the former Soviet Union which fostered the creation of groups of students, teachers, and mathematicians called 'mathematical circles'. The work is predicated on the idea that studying mathematics can generate the same enthusiasm as playing a team sport - without necessarily being competitive. This book is intended for both students and teachers who love mathematics and want to study its various branches beyond the limits of school curriculum. It is also a book of mathematical recreations and, at the same time, a book containing vast theoretical and problem material in main areas of what authors consider to be 'extracurricular mathematics'. The book is based on a unique experience gained by several generations of Russian educators and scholars.

In most mathematics textbooks, the most exciting part of mathematics - the process of invention and discovery - is completely hidden from the reader. The aim of ""Groups and Symmetry"" is to change all that. By means of a series of carefully selected tasks, this book leads readers to discover some real mathematics. There are no formulas to memorize; no procedures to follow. The book is a guide: Its job is to start you in the right direction and to bring you back if you stray too far. Discovery is left to you. Suitable for a one-semester course at the beginning undergraduate level, there are no prerequisites for understanding the text. Any college student interested in discovering the beauty of mathematics will enjoy a course taught from this book. The book has also been used successfully with non science students who want to fulfill a science requirement. Also available from the AMS by David W. Farmer is ""Knots and Surfaces: A Guide to Discovering Mathematics"".

In most mathematics textbooks, the most exciting part of mathematics - the process of invention and discovery - is completely hidden from the reader. The aim of ""Knots and Surfaces"" is to change all that. By means of a series of carefully selected tasks, this book leads readers to discover some real mathematics. There are no formulas to memorize; no procedures to follow. The book is a guide: its job is to start you in the right direction and to bring you back if you stray too far. Discovery is left to you. Suitable for a one-semester course at the beginning undergraduate level, there are no prerequisites for understanding the text. Any college student interested in discovering the beauty of mathematics will enjoy a course taught from this book. The book has also been used successfully with non science students who want to fulfill a science requirement. Also available from the AMS by David W. Farmer is ""Groups and Symmetry: A Guide to Discovering Mathematics"".

This is the translation of the Japanese textbook for the grade 11 course, 'Basic Analysis', which is one of three elective courses offered at this level in Japanese high schools. The book includes a thorough treatment of exponential, logarithmic, and trigonometric functions, progressions, and induction method, as well as an extensive introduction to differential and integral calculus.

This is the translation from the Japanese textbook for the grade 11 course, 'General Mathematics'. It is part of the easier of the three elective courses in mathematics offered at this level and is taken by about 40 per cent of students. The book covers basic notions of probability and statistics, vectors, exponential, logarithmic, and trigonometric functions, and an introduction to differentiation and integration.

This text offers a special account of Indian work in diophantine equations during the 6th through 12th centuries and Italian work on solutions of cubic and biquadratic equations from the 11th through 16th centuries. The volume traces the historical development of algebra and the theory of equations from ancient times to the beginning of modern algebra, outlining some modern themes such as the fundamental theorem of algebra, Clifford algebras, and quarternions. It is geared toward undergraduates who have no background in calculus. For other wonderful titles written by this author see: ""Euler through Time: A New Look at Old Themes"", ""Supersymmetry for Mathematicians: An Introduction"", ""The Mathematical Legacy of Harish-Chandra: A Celebration of Representation Theory and Harmonic Analysis"", and ""The Selected Works of V.S. Varadarajan"".

This volume and ""Kvant Selecta: Algebra and Analysis, II"" (MAWRLD/15) are the first volumes of articles published from 1970 to 1990 in the Russian journal, ""Kvant"". The influence of this magazine on mathematics and physics education in Russia is unmatched. This collection represents the Russian tradition of expository mathematical writing at its best. Articles selected for these two volumes are written by leading Russian mathematicians and expositors. Some articles contain classical mathematical gems still used in university curricula today. Others feature cutting-edge research from the twentieth century.The articles in these books are written so as to present genuine mathematics in a conceptual, entertaining, and accessible way. The volumes are designed to be used by students and teachers who love mathematics and want to study its various aspects, thus deepening and expanding the school curriculum. The first volume is mainly devoted to various topics in number theory, whereas the second volume treats diverse aspects of analysis and algebra.

The mathematical theory of games was first developed as a model for situations of conflict, whether actual or recreational. It gained widespread recognition when it was applied to the theoretical study of economics by von Neumann and Morgenstern in Theory of Games and Economic Behavior in the 1940s. The later bestowal in 1994 of the Nobel Prize in economics on Nash underscores the important role this theory has played in the intellectual life of the twentieth century.This volume is based on courses given by the author at the University of Kansas. The exposition is 'gentle' because it requires only some knowledge of coordinate geometry; linear programming is not used. It is 'mathematical' because it is more concerned with the mathematical solution of games than with their applications. Existing textbooks on the topic tend to focus either on the applications or on the mathematics at a level that makes the works inaccessible to most non-mathematicians. This book nicely fits in between these two alternatives. It discusses examples and completely solves them with tools that require no more than high school algebra. In this text, proofs are provided for both von Neumann's Minimax Theorem and the existence of the Nash Equilibrium in the $2 \times 2$ case. Readers will gain both a sense of the range of applications and a better understanding of the theoretical framework of these two deep mathematical concepts.

This volume and ""Kvant Selecta: Algebra and Analysis, I"" (""MAWRLD/14"") are the first volumes of articles published from 1970 to 1990 in the Russian journal, ""Kvant"". The influence of this magazine on mathematics and physics education in Russia is unmatched. This collection represents the Russian tradition of expository mathematical writing at its best. Articles selected for these two volumes are written by leading Russian mathematicians and expositors. Some articles contain classical mathematical gems still used in university curricula today. Others feature cutting-edge research from the twentieth century.The articles in these books are written so as to present genuine mathematics in a conceptual, entertaining, and accessible way. The volumes are designed to be used by students and teachers who love mathematics and want to study its various aspects, thus deepening and expanding the school curriculum. The first volume is mainly devoted to various topics in number theory, whereas the second volume treats diverse aspects of analysis and algebra.

This is the English translation of the book originally published in Russian. It contains 20 essays, each dealing with a separate mathematical topic. The topics range from brilliant mathematical statements with interesting proofs, to simple and effective methods of problem-solving, to interesting properties of polynomials, to exceptional points of the triangle. Many of the topics have a long and interesting history. The author has lectured on them to students worldwide. The essays are independent of one another for the most part, and each presents a vivid mathematical result that led to current research in number theory, geometry, polynomial algebra, or topology.

There is a tradition in Russia that holds that mathematics can be both challenging and fun. One fine outgrowth of that tradition is the magazine, ""Kvant"", which has been enjoyed by many of the best students since its founding in 1970. The articles in ""Kvant"" assume only a minimal background, that of a good high school student, yet are capable of entertaining mathematicians of almost any level. Sometimes the articles require careful thought or a moment's work with a pencil and paper. However, the industrious reader will be generously rewarded by the elegance and beauty of the subjects.This book is the third collection of articles from ""Kvant"" to be published by the AMS. The volume is devoted mainly to combinatorics and discrete mathematics. Several of the topics are well known: nonrepeating sequences, detecting a counterfeit coin, and linear inequalities in economics, but they are discussed here with the entertaining and engaging style typical of the magazine. The two previous collections treat aspects of algebra and analysis, including connections to number theory and other topics. They were published as Volumes 14 and 15 in the ""Mathematical World"" series. The articles are written so as to present genuine mathematics in a conceptual, entertaining, and accessible way. The books are designed to be used by students and teachers who love mathematics and want to study its various aspects, deepening and expanding upon the school curriculum.

'Every map is a tool, a product of human effort and creativity, that represents some aspects of our world or universe...[This] course was powered by the belief that by exploring the mathematical ideas involved in creating and analyzing maps, students would see how mathematics could help them to understand and explain their world' - from the Preface. ""Portraits of the Earth"" exemplifies the AMS' mission to bring the power and vitality of mathematical thought to the nonexpert. It is designed to teach students to think logically and to analyze the technical information that they so readily encounter every day. Maps are exciting, visual tools that we encounter on a daily basis: from street maps to maps of the world accompanying news stories to geologic maps depicting the underground structure of the earth.This book explores the mathematical ideas involved in creating and analyzing maps, a topic that is rarely discussed in undergraduate courses. It is the first modern book to present the famous problem of mapping the earth in a style that is highly readable and mathematically accessible to most students. Feeman's writing is inviting to the novice, yet also interesting to readers with more mathematical experience. Through the visual context of maps and mapmaking, students will see how contemporary mathematics can help them to understand and explain the world. Topics explored are the shape and size of the earth, basic spherical geometry, and why one can't make a perfect flat map of the planet.The author discusses different attributes that maps can have and determines mathematically how to design maps that have the desired features. The distortions that arise in making world maps are quantitatively analyzed. There is an in-depth discussion on the design of numerous map projections - both historical and contemporary - as well as conformal and equal-area maps. Feeman looks at how basic map designs can be modified to produce maps with any center, and he indicates how to generalize methods to produce maps of arbitrary surfaces of revolution. Also included are end-of-chapter exercises and laboratory projects.Particularly interesting is a chapter that explains how to use MapleR add-on software to make maps from geographic data points. This book would make an excellent text for a basic undergraduate mathematics or geography course and would be especially appealing to the teacher who is interested in exciting visual applications in the classroom. It would also serve nicely as supplementary reading for a course in calculus, linear algebra, or differential geometry. Prerequisites include a solid grasp of trigonometry and basic calculus.

This book will bring the beauty and fun of mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts. The first chapter presents the geometry and topology of surfaces. Among other topics, the authors discuss the Poincare-Hopf theorem on critical points of vector fields on surfaces and the Gauss-Bonnet theorem on the relation between curvature and topology (the Euler characteristic).The second chapter addresses various aspects of the concept of dimension, including the Peano curve and the Poincare approach. Also addressed is the structure of three-dimensional manifolds. In particular, it is proved that the three-dimensional sphere is the union of two doughnuts. This is the first of three volumes originating from a series of lectures given by the authors at Kyoto University (Japan). It is suitable for classroom use for high school mathematics teachers and undergraduate mathematics courses in the sciences and liberal arts. The second volume is available as Volume 20 in the AMS series, ""Mathematical World"". A third volume is forthcoming.

This book brings the beauty and fun of mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts. The first chapter talks about the theory of trigonometric and elliptic functions. It includes subjects such as power series expansions, addition and multiple-angle formulas, and arithmetic-geometric means. The second chapter discusses various aspects of the Poncelet Closure Theorem. This discussion illustrates to the reader the idea of algebraic geometry as a method of studying geometric properties of figures using algebra as a tool. This is the second volume originating from a series of lectures given by the authors at Kyoto University (Japan). It is suitable for classroom use for high school mathematics teachers and for undergraduate mathematics courses in the sciences and liberal arts. The first volume is available as Volume 19 in the AMS series, ""Mathematical World"".

This book brings the beauty and fun of mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts. The first chapter talks about the theory of manifolds. It includes discussion of smoothness, differentiability, and analyticity, the idea of local coordinates and coordinate transformation, and a detailed explanation of the Whitney imbedding theorem (both in weak and in strong form).The second chapter discusses the notion of the area of a figure on the plane and the volume of a solid body in space. It includes the proof of the Bolyai-Gerwien theorem about scissors-congruent polynomials and Dehn's solution of the Third Hilbert Problem. This is the third volume originating from a series of lectures given at Kyoto University (Japan). It is suitable for classroom use for high school mathematics teachers and for undergraduate mathematics courses in the sciences and liberal arts. The first and second volumes are available as Volume 19 and Volume 20 in the AMS series, ""Mathematical World"".

A cipher is a scheme for creating coded messages for the secure exchange of information. Throughout history, many different coding schemes have been devised. One of the oldest and simplest mathematical systems was used by Julius Caesar. This is where ""Mathematical Ciphers"" begins. Building on that simple system, Young moves on to more complicated schemes, ultimately ending with the RSA cipher, which is used to provide security for the Internet. This book is structured differently from most mathematics texts. It does not begin with a mathematical topic, but rather with a cipher. The mathematics is developed as it is needed; the applications motivate the mathematics. As is typical in mathematics textbooks, most chapters end with exercises. Many of these problems are similar to solved examples and are designed to assist the reader in mastering the basic material.A few of the exercises are one-of-a-kind, intended to challenge the interested reader. Implementing encryption schemes is considerably easier with the use of the computer. For all the ciphers introduced in this book, JavaScript programs are available from the Web. In addition to developing various encryption schemes, this book also introduces the reader to number theory. Here, the study of integers and their properties is placed in the exciting and modern context of cryptology. ""Mathematical Ciphers"" can be used as a textbook for an introductory course in mathematics for all majors. The only prerequisite is high school mathematics.

This three-volume set addresses the interplay between topology, functions, geometry, and algebra. Bringing the beauty and fun of mathematics to the classroom, the authors offer serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts. It is suitable for advanced high-school students, graduate students, and researchers. The three-volume set includes A Mathematical Gift I, II, and III.

Crisscross, zigzag, bowtie, devil, angel, or star: which are the longest, the shortest, the strongest, and the weakest lacings? Pondering the mathematics of shoelaces, the author paints a vivid picture of the simple, beautiful, and surprising characterizations of the most common shoelace patterns. The mathematics involved is an attractive mix of combinatorics and elementary calculus. This book will be enjoyed by mathematically minded people for as long as there are shoes to lace. Burkard Polster is a well-known mathematical juggler, magician, origami expert, bubble-master, shoelace charmer, and 'Count von Count' impersonator. His previous books include ""A Geometrical Picture Book"", ""The Mathematics of Juggling"", and ""QED: Beauty in Mathematical Proof"". Want to learn more about knot theory? See ""The Knot Book"" by Colin Adams and ""Knots and Links"" by Dale Rolfsen. To read a review published in the ""Gazette of the Australian Mathematical Society"", click here.

The book is devoted to the properties of conics (plane curves of second degree) that can be formulated and proved using only elementary geometry. Starting with the well-known optical properties of conics, the authors move to less trivial results, both classical and contemporary. In particular, the chapter on projective properties of conics contains a detailed analysis of the polar correspondence, pencils of conics, and the Poncelet theorem. In the chapter on metric properties of conics the authors discuss, in particular, inscribed conics, normals to conics, and the Poncelet theorem for confocal ellipses.

'This text should not be viewed as a comprehensive history of algebra before 1600, but as a basic introduction to the types of problems that illustrate the earliest forms of algebra. It would be particularly useful for an instructor who is looking for examples to help enliven a course on elementary algebra with problems drawn from actual historical texts' - Warren Van Egmond about the French edition for ""MathSciNet"". This book does not aim to give an exhaustive survey of the history of algebra up to early modern times but merely to present some significant steps in solving equations and, wherever applicable, to link these developments to the extension of the number system. Various examples of problems, with their typical solution methods, are analyzed, and sometimes translated completely. Indeed, it is another aim of this book to ease the reader's access to modern editions of old mathematical texts, or even to the original texts; to this end, some of the problems discussed in the text have been reproduced in the appendices in their original language (Greek, Latin, Arabic, Hebrew, French, German, Provencal, and Italian) with explicative notes.

Many people intuitively sense that there is a connection between mathematics and music. If nothing else, both involve counting. There is, of course, much more to the association. David Wright's book is an investigation of the interrelationships between mathematics and music, reviewing the needed background concepts in each subject as they are encountered. Along the way, readers will augment their understanding of both mathematics and music. The text explores the common foundations of the two subjects, which are developed side by side. Musical and mathematical notions are brought together, such as scales and modular arithmetic, intervals and logarithms, tone and trigonometry, and timbre and harmonic analysis. When possible, discussions of musical and mathematical notions are directly interwoven. Occasionally the discourse dwells for a while on one subject and not the other, but eventually the connection is established, making this an integrative treatment of the two subjects. The book is a text for a freshman level college course suitable for musically inclined or mathematically inclined students, with the intent of breaking down any apprehension that either group might have for the other subject. Exercises are given at the end of each chapter. The mathematical prerequisites are a high-school level familiarity with algebra, trigonometry, functions, and graphs. Musically, the student should have had some exposure to musical staffs, standard clefs, and key signatures, though all of these are explained in the text.

How quickly can you compute the remainder when dividing $109837^{97}$ by 120143? Why would you even want to compute this? And what does this have to do with cryptography? Modern cryptography lies at the intersection of mathematics and computer sciences, involving number theory, algebra, computational complexity, fast algorithms, and even quantum mechanics. Many people think of codes in terms of spies, but in the information age, highly mathematical codes are used every day by almost everyone, whether at the bank ATM, at the grocery checkout, or at the keyboard when you access your email or purchase products online. This book provides a historical and mathematical tour of cryptography, from classical ciphers to quantum cryptography. The authors introduce just enough mathematics to explore modern encryption methods, with nothing more than basic algebra and some elementary number theory being necessary. Complete expositions are given of the classical ciphers and the attacks on them, along with a detailed description of the famous Enigma system. The public-key system RSA is described, including a complete mathematical proof that it works. Numerous related topics are covered, such as efficiencies of algorithms, detecting and correcting errors, primality testing and digital signatures. The topics and exposition are carefully chosen to highlight mathematical thinking and problem solving. Each chapter ends with a collection of problems, ranging from straightforward applications to more challenging problems that introduce advanced topics. Unlike many books in the field, this book is aimed at a general liberal arts student, but without losing mathematical completeness.

The theory of fixed points finds its roots in the work of Poincare, Brouwer, and Sperner and makes extensive use of such topological notions as continuity, compactness, homotopy, and the degree of a mapping. Fixed point theorems have numerous applications in mathematics; most of the theorems ensuring the existence of solutions for differential, integral, operator, or other equations can be reduced to fixed point theorems. In addition, these theorems are used in such areas as mathematical economics and game theory. This book presents a readable exposition of fixed point theory. The author focuses on the problem of whether a closed interval, square, disk, or sphere has the fixed point property.Another aim of the book is to show how fixed point theory uses combinatorial ideas related to decomposition (triangulation) of figures into distinct parts called faces (simplexes), which adjoin each other in a regular fashion. All necessary background concepts - such as continuity, compactness, degree of a map, and so on - are explained, making the book accessible even to students at the high school level. In addition, the book contains exercises and descriptions of applications. Readers will appreciate this book for its lucid presentation of this fundamental mathematical topic.

This unique book presents simple mathematical models of various aspects of sports with applications to sports training and competitions. Requiring only a background in precalculus, it would be suitable as a textbook for courses in mathematical modeling and operations research at the high school or college level Coaches and those who participate in sports will find it interesting as well. The lively writing style and wide range of topics make this book especially appealing.

The Mathematics of Voting and Elections: A Hands-On Approach, Second Edition, is an inquiry-based approach to the mathematics of politics and social choice. The aim of the book is to give readers who might not normally choose to engage with mathematics recreationally the chance to discover some interesting mathematical ideas from within a familiar context, and to see the applicability of mathematics to real-world situations. Through this process, readers should improve their critical thinking and problem solving skills, as well as broaden their views of what mathematics really is and how it can be used in unexpected ways. The book was written specifically for non-mathematical audiences and requires virtually no mathematical prerequisites beyond basic arithmetic. At the same time, the questions included are designed to challenge both mathematical and non-mathematical audiences alike. More than giving the right answers, this book asks the right questions.The book is fun to read, with examples that are not just thought-provoking, but also entertaining. It is written in a style that is casual without being condescending. But the discovery-based approach of the book also forces readers to play an active role in their learning, which should lead to a sense of ownership of the main ideas in the book. And while the book provides answers to some of the important questions in the field of mathematical voting theory, it also leads readers to discover new questions and ways to approach them.In addition to make small improvements in all the chapters, this second edition contains several new chapters. Of particular interest might be Chapter 12 which covers a host of topics related to gerrymandering.