Spectrum - Böcker

This is an introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matrices. While apparent that the conjecture must be true, the proof was elusive. Researchers became drawn to this problem, making connections to aspects of invariant theory, to symmetric functions, to hypergeometric and basic hypergeometric series, and, finally, to the six-vertex model of statistical mechanics. All these threads are brought together in Zeilberger's 1996 proof of the original conjecture. The book is accessible to anyone with a knowledge of linear algebra. Students will learn what mathematicians actually do in an interesting and new area of mathematics, and even researchers in combinatorics will find something new here.

Flatland, Edwin Abbott's story of a two-dimensional universe, as told by one of its inhabitants who is introduced to the mysteries of three-dimensional space, has enjoyed an enduring popularity from the time of its publication in 1884. This fully annotated edition enables the modern-day reader to understand and appreciate the many 'dimensions' of this classic satire. Mathematical notes and illustrations enhance the usefulness of Flatland as an elementary introduction to higher-dimensional geometry. Historical notes show connections to late-Victorian England and to classical Greece. Citations from Abbott's other writings as well as the works of Plato and Aristotle serve to interpret the text. Commentary on language and literary style includes numerous definitions of obscure words. An appendix gives a comprehensive account of the life and work of Flatland's remarkable author.

Flatland, Edwin Abbott's story of a two-dimensional universe, as told by one of its inhabitants who is introduced to the mysteries of three-dimensional space, has enjoyed an enduring popularity from the time of its publication in 1884. This fully annotated edition enables the modern-day reader to understand and appreciate the many 'dimensions' of this classic satire. Mathematical notes and illustrations enhance the usefulness of Flatland as an elementary introduction to higher-dimensional geometry. Historical notes show connections to late-Victorian England and to classical Greece. Citations from Abbott's other writings as well as the works of Plato and Aristotle serve to interpret the text. Commentary on language and literary style includes numerous definitions of obscure words. An appendix gives a comprehensive account of the life and work of Flatland's remarkable author.

Changing the way students learn calculus at New Mexico State University. In the Spring of 1988, Marcus Cohen, Edward D. Gaughan, Arthur Knoebel, Douglas S. Kurtz, and David Penegelley began work on a student project approach to calculus. For the next two years, most of their waking hours (and some of their dreams) would be devoted to writing projects for their students and discovering how to make the use of projects in calculus classes not only successful, but practical as well. A grant from the National Science Foundation made it possible for this experiment to go forward on a large scale. The enthusiasm of the original group of five faculty was contagious, and soon other members of the department were also writing and using projects in their calculus classes. At the present time, about 80% of the calculus students at New Mexico State University are doing projects in their Calculus courses. Teachers can use their methods in teaching their own calculus courses.Student Research Projects in Calculus provides teachers with over 100 projects ready to assign to students in single and multivariable calculus. The authors have designed these projects with one goal in mind: to get students to think for themselves. Each project is a multistep, take-home problem, allowing students to work both individually and in groups. The projects resemble mini-research problems. Most of them require creative thought, and all of them engage the student's analytic and intuitive faculties. the projects often build from a specific example to the general case, and weave together ideas from many parts of the calculus. Project statements are clearly stated and contain a minimum of mathematical symbols. Students must draw their own diagrams, decide for themselves what the problem is about, and what toolsfrom the calculus they will use to solve it. This approach elicits from students an amazing level of sincere questioning, energetic research, dogged persistence, and conscientious communication. Each project has accompanying notes to the instructor, reporting students' experiences. The notes contain helpful information on prerequisites, list the main topics the project explores, and suggests helpful hints. The authors have also provided several introductory chapters to help instructors use projects successfully in their classes and begin to create their own.

Eric Temple Bell (1883-1960) was a distinguished mathematician and a best selling popularizer of mathematics. His Men of Mathematics, still in print after almost sixty years, inspired scores of young readers to become mathematicians. Under the name of John Taine, he also published science fiction novels (among them The Time Stream, Before the Dawn, and The Crystal Horde) that served to broaden the subject matter of that genre during its early years.In The Search for E. T. Bell, Constance Reid has given us a compelling account of this complicated, difficult man who never divulged to anyone, not even to his wife and son, the story of his early life and family background. Her book is thus more of a mystery than a traditional biography. It begins with the discovery of an unexpected inscription in an English churchyard and a series of cryptic notations in a boy's schoolbook. Then comes an inadvertent revelation, by Bell himself, in a respected mathematical journal. You will have to read the book to learn the rest.

Long known as a mathematical storyteller, Howard Eves writes his personal reminiscences, mostly mathematical, some not. The cast of characters includes Albert Einstein, Norbert Wiener, Julian Lowell Coolidge, Maurice Frechet, Nathan Altshiller-Court, G. H. Hardy, and many other interesting figures whom he encountered in a long and active life in mathematics. In Mathematical Reminiscences, we read of Eves' famous mathematical museum, with its lock of Einstein's hair, Hardy's scarf, Veblen's bright yellow lead pencil, Wiener's hat, and much, much, more.

Translated from the original German by Peter Hilton and Jean Pedersen. The 99 points of intersection presented here were collected during a year-long search for surprising concurrence of lines. For each example we find compelling evidence for the sometimes startling fact that in a geometric figure three straight lines, or sometimes circles, pass through one and the same point. Of course, we are familiar with some examples of this from basic elementary geometry - the intersection of medians, altitudes, angle bisectors, and perpendicular bisectors of sides of a triangle. Here there are many more examples - some for figures other than triangles, some where even more than three straight lines pass through a common point. The main part of the book presents 99 points of intersection purely visually. They are developed in a sequence of figures, many without caption or verbal commentary. In addition the book contains general thoughts on and examples of the points of intersection, as well as some typical methods of proving their existence. Many of the examples shown in the book were inspired by questions and suggestions made by students and high-school teachers. Several of those examples have not only a geometrical, but also an intriguing aesthetic, aspect. The book addresses high-school students and students at the undergraduate level as well as their teachers, but will appeal to anyone interested in geometry.

This book is an English translation of a collection of papers in Russian from a conference held in Moscow and St. Petersburg in 1983, 200 years after Euler's death. Two of the Russian papers in the collection are themselves translations from the German. The present English translation appears in 2007, 300 years after his birth. We speak of the ""Age of Euler"". A justification of this term is provided by a list of scientific terms connected with Euler s name and his many contributions to pure mathematics, well-known in the mathematical community and, in part, covered in this volume. What makes this collection remarkable, though, is the extensive treatment of Euler's contributions outside pure mathematics-to astronomy, celestial mechanics, ballistics, history of science, instruments and technology, physics, mechanics, hydromechanics, mechanics of elastic systems, variational principles of mechanics, and a mathematical theory of music. In addition to this survey of his contributions to science, we find here material otherwise very difficult to find detailed accounts of Euler's family life and the careers pursued by his children and grandchildren. Readers otherwise well-informed about Euler and his work will find here much to enhance their appreciation of this extraordinary scientist and human being. For significant and factual knowledge of Euler's contributions to science past and present, this book is easily the best available choice.

This book picks up the history of mathematics from where Sherlock Holmes in Babylon left it. The forty articles of Who Gave You the Epsilon? continue the story of the development of mathematics into the nineteenth and twentieth centuries. The articles have all been published in the Mathematical Association of America journals and are in many cases written by distinguished mathematicians such as G. H. Hardy and B. van der Waerden. The articles are arranged thematically to show the development of analysis, geometry, algebra and number theory through this period of time. Each chapter is preceded by a foreword, giving the historical background and setting and the scene, and is followed by an afterword, reporting on advances in our historical knowledge and understanding since the articles first appeared. This book is ideal for anyone wanting to explore the history of mathematics.

This book is the second volume based on lectures for pre-college students given by prominent mathematicians in the Bay Area Mathematical Adventures (BAMA). This book reflects the flavor of the BAMA lectures and the excitement they have generated among the high school and middle school students in the Silicon Valley. The topics cover a wide range of mathematical subjects each treated by a leading proponent of the subject at levels designed to challenge and attract students whose mathematical interests are just beginning. In addition, the treatments given here will intrigue and enchant a more mature mathematician. It is hoped that the publication of these lectures will expose students outside of the San Francisco Bay Area to interesting mathematical topics and treatments outside of their normal experience in the classroom. Mathematical educators are encouraged to offer the students in their own localities similar opportunities to come into contact with exciting adventures in mathematics.

This book is about beautiful mathematical concepts and creations. Mathematical ideas have an aesthetic appeal that can be appreciated by those who have the time and dedication to investigate. Mathematical topics are presented in the categories of words, images, formulas, theorems, proofs, solutions, and unsolved problems. Readers will investigate exciting mathematical topics ranging from complex numbers to arithmetic progressions, from Alcuin's sequence to the zeta function, and from hypercubes to infinity squared. Do you know that a lemniscate curve is the circular inversion of a hyperbola? That Sierpinski's triangle has fractal dimension 1.585....? That a regular septagon can be constructed with straightedge, compass, and an angle trisector? Do you know how to prove Lagrange s theorem that every positive integer is the sum of four squares? Can you find the first three digits of the millionth Fibonacci number? Discover the keys to these and many other mathematical problems. In each case, the mathematics is compelling, elegant, simple, and beautiful. Who should read this book? There is something new for any mathematically-minded person. High school and college students will find motivation for their mathematical studies. Professional mathematicians will find fresh examples of mathematical beauty to pass along to others. Within each chapter, the topics require progressively more prerequisite knowledge. An appendix gives background definitions and theorems, while another gives challenging exercises (with solutions).

This book is about the concept of mathematical maturity. Mathematical maturity is central to a mathematics education. The goal of a mathematics education is to transform the student from someone who treats mathematical ideas empirically and intuitively to someone who treats mathematical ideas analytically and can control and manipulate them effectively.Put more directly, a mathematically mature person is one who can read, analyze, and evaluate proofs. And, most significantly, he/she is one who can create proofs. For this is what modern mathematics is all about: coming up with new ideas and validating them with proofs.The book provides background, data, and analysis for understanding the concept of mathematical maturity. It turns the idea of mathematical maturity from a topic for coffee-room conversation to a topic for analysis and serious consideration.

Beginning with one of the most remarkable ecological collapses of recent times, that of the passenger pigeon, this book surveys collapse processes across the entire spectrum of the natural and man-made world. The reader is taken through extreme weather events, crashing financial markets and companies, the chaotic nature of Earth's orbit, revolutionary political change, the spread of disease and many other fascinating cases. The author's key observation is the consistent occurrence of one or more of six fundamental dynamics ('six sources of collapse'), all of which can be best described and investigated using fundamental mathematical concepts which include low probability events, group dynamics, evolutionary games, instability, nonlinearity and network effects. The exposition assumes a minimal mathematical background, but even professional mathematicians are likely to get rich insights from the range of examples. This book will also be of interest to readers in business and finance, engineering, ecology and political science.

C. Edward Sandifer has been studying Euler for decades and is one of the world's leading experts on his work. This book is the second collection of Sandifer's 'How Euler Did It' columns. Each is a jewel of historical and mathematical exposition. The sum total of years of work and study of the most prolific mathematician in history, this volume will leave the reader marveling at Euler's clever inventiveness, which Sandifer explicates and puts in context.

Founded in 1961 and still thriving today, the recreational magazine Pythagoras has given generations of high school students in the Netherlands a perspective on many branches of mathematics that are not taught in schools. This book is a selection of the best and most inspiring articles from Pythagoras magazine. It contains a mix of problems - some easy, some challenging, always original - as well as playful introductions to subjects in algebra, geometry, topology, number theory, and more. Concepts like sudoku and the magic square are given a whole new dimension. Articles on Escher's 'impossible objects', a recurring subject over the years, are part of a special section on mathematics and art. Smart high school students and anyone else with a sharp and inquisitive mind will find in this book a treasure trove rich enough to keep one's mind engaged for many weeks and months.

The MAA was founded in 1915 to serve as a home for The American Mathematical Monthly. The mission of the Association-to advance mathematics, especially at the collegiate level-has, however, always been larger than merely publishing world-class mathematical exposition. MAA members have explored more than just mathematics; we have, as this volume tries to make evident, investigated mathematical connections to pedagogy, history, the arts, technology, literature, every field of intellectual endeavor. Essays, all commissioned for this volume, include exposition by Bob Devaney, Robin Wilson, and Frank Morgan; history from Karen Parshall, Della Dumbaugh, and Bill Dunham; pedagogical discussion from Paul Zorn, Joe Gallian, and Michael Starbird, and cultural commentary from Bonnie Gold, Jon Borwein, and Steve Abbott.This volume contains 35 essays by all-star writers and expositors writing to celebrate an extraordinary century for mathematics-more mathematics has been created and published since 1915 than in all of previous recorded history. We've solved age-old mysteries, created entire new fields of study, and changed our conception of what mathematics is. Many of those stories are told in this volume as the contributors paint a portrait of the broad cultural sweep of mathematics during the MAA's first century. Mathematics is the most thrilling, the most human, area of intellectual inquiry; you will find in this volume compelling proof of that claim.

At the turn of the twentieth century, mathematical scholarship in the United States underwent a stunning transformation. In 1890, no American professor was producing mathematical research worthy of international attention. Graduate students were then advised to pursue their studies abroad. By the start of World War I, the standing of American mathematics had radically changed. George David Birkhoff, Leonard Dickson, and others were turning out cutting edge investigations that attracted notice in the intellectual centers of Europe. Harvard, Chicago, and Princeton maintained graduate programs comparable to those overseas. This book explores the people, timing, and factors behind this rapid advance.Through the mid-nineteenth century, most American colleges followed a classical curriculum that, in mathematics, rarely reached beyond calculus. With no doctoral programs of any sort in the United States until 1860, mathematical scholarship lagged far behind that in Europe. After the Civil War, visionary presidents at Harvard and Johns Hopkins broadened and deepened the opportunities for study. The breakthrough for mathematics began in 1890 with the hiring, in consecutive years, of William F. Osgood and Maxime Bôcher at Harvard and E. H. Moore at Chicago. Each of these young men had studied in Germany where they acquired vital mathematical knowledge and taste. Over the next few years, Osgood, Bôcher, and Moore established their own research programs and introduced new graduate courses. Working with other like-minded individuals through the nascent American Mathematical Society, the infrastructure of meetings and journals were created. In the early twentieth century, Princeton dramatically upgraded its faculty to give the United States the stability of a third mathematics center. The publication by Birkhoff, in 1913, of the solution to a famous conjecture served notice that American mathematics had earned consideration with the European powers of Germany, France, Italy, England, and Russia.

G. H. Hardy (1877-1947) ranks among the great mathematicians of the twentieth century. He did essential research in number theory and analysis, held professorships at Cambridge and Oxford, wrote important textbooks as well as the classic A Mathematician's Apology, and famously collaborated with J. E. Littlewood and Srinivasa Ramanujan. Hardy was a colorful character with remarkable expository skills. This book is a feast of G. H. Hardy's writing. There are selections of his mathematical papers, his book reviews, his tributes to departed colleagues. Some articles are serious, whereas others display a wry sense of humor. And there are recollections by those who knew Hardy, along with biographical and mathematical pieces written explicitly for this collection. Fans of Hardy should find much here to like. And for those unfamiliar with his work, The G. H. Hardy Reader provides an introduction to this extraordinary individual.

G. H. Hardy (1877-1947) ranks among the great mathematicians of the twentieth century. He did essential research in number theory and analysis, held professorships at Cambridge and Oxford, wrote important textbooks as well as the classic A Mathematician's Apology, and famously collaborated with J. E. Littlewood and Srinivasa Ramanujan. Hardy was a colorful character with remarkable expository skills. This book is a feast of G. H. Hardy's writing. There are selections of his mathematical papers, his book reviews, his tributes to departed colleagues. Some articles are serious, whereas others display a wry sense of humor. And there are recollections by those who knew Hardy, along with biographical and mathematical pieces written explicitly for this collection. Fans of Hardy should find much here to like. And for those unfamiliar with his work, The G. H. Hardy Reader provides an introduction to this extraordinary individual.

Certain constants occupy precise balancing points in the cosmos of number, like habitable planets sprinkled throughout our galaxy at just the right distances from their suns. This book introduces and connects four of these constants (phi, pi, e and i), each of which has recently been the individual subject of historical and mathematical expositions. But here we discuss their properties, as a group, at a level appropriate for an audience armed only with the tools of elementary calculus. This material offers an excellent excuse to display the power of calculus to reveal elegant truths that are not often seen in college classes. These truths are described here via the work of such luminaries as Nilakantha, Liu Hui, Hemachandra, Khayyam, Newton, Wallis, and Euler.

This is the first truly comprehensive and thorough history of the development of mathematics in the United States and Canada. This first volume of a two-volume work takes the reader from the European encounters with North America in the fifteenth century up to the emergence of the United States as a world leader in mathematics in the 1930s.In the story of the Colonial period particular emphasis is given to several prominent Colonial figures--Jefferson, Franklin, and Rittenhouse-and four important early colleges-Quebec, Harvard, Yale, and William & Mary. During the first three-quarters of the nineteenth century, mathematics in North America was largely the occupation of scattered individual pioneers: Bowditch, Farrar, Adrain, B. Peirce. This period is given a fuller treatment here than previously in the literature, including the creation of the first PhD programs and attempts to form organizations and found journals.With the founding of Johns Hopkins University in 1876, the American mathematical research community was finally, and firmly, founded. The programs at Hopkins, Chicago, and Clark are detailed as are the influence of major European mathematicians, including especially Klein, Hilbert, and Sylvester. Extensive histories of early areas of American emphasis are provided, including axiomatics, topology, and group theory. Also included are the early histories of statistics and cryptology in America, laying the foundation for the latter topic's role in abstract algebra in the 1950s. The stories of both the American Mathematical Society and the Mathematical Association of America are presented in detail.David Zitarelli is emeritus Professor of Mathematics at Temple University. A decorated and acclaimed teacher, scholar, and expositor, he is one of the world's leading experts on the development of American mathematics. Author or co-author of over a dozen books, this is his magnum opus--sure to become the leading reference on the topic and essential reading, not just for historians. In clear and compelling prose, Zitarelli spins a tale accessible to experts, generalists, and anyone interested in the history of science in North America.

Calculus Gems, a collection of essays written about mathematicians and mathematics, is a spin-off of two appendices (""Biographical Notes"" and ""Variety of Additional Topics"") found in Simmons' 1985 calculus book. With many additions and some minor adjustments, the material will now be available in a separate softcover volume. The text is suitable as a supplement for a calculus course and/or a history of mathematics course, The overall aim is bound up in the question, ""What is mathematics for?"" and in Simmons' answer, ""To delight the mind and help us understand the world"". The essays are independent of one another, allowing the instructor to pick and choose among them. Part A, ""Brief Lives"", is a biographical history of mathematics from earliest times (Thales, 625-547 BC) through the late 19th century (Weierstrass, 1815-1897) that serves to connect mathematics to the broader intellectual and social history of Western civilization. Part B, ""Memorable Mathematics"", is a collection of interesting topics from number theory, geometry, and science arranged in an order roughly corresponding to the order of most calculus courses. Some of these sections have a few problems for the student to solve. Students can gain perspective on the mathematical experience and learn some mathematics not contained in the usual courses, and instructors can assign student papers and projects based on the essays. The book teaches by example that mathematics is more than computation. Original illustrations of influential mathematicians in history and their inventions accompany the brief biographies and mathematical discussions.

The Early Mathematics of Leonhard Euler gives an article-by-article description of Leonhard Euler's early mathematical works; the 50 or so mathematical articles he wrote before he left St. Petersburg in 1741 to join the Academy of Frederick the Great in Berlin. These early pieces contain some of Euler's greatest work, the Konigsberg bridge problem, his solution to the Basel problem, and his first proof of the Euler-Fermat theorem. It also presents important results that we seldom realize are due to Euler; that mixed partial derivatives are (usually) equal, our $f(x)$ notation, and the integrating factor in differential equations.The books shows how contributions in diverse fields are related, how number theory relates to series, which, in turn, relate to elliptic integrals and then to differential equations. There are dozens of such strands in this beautiful web of mathematics. At the same time, we see Euler grow in power and sophistication, from a young student when at 18 he published his first work on differential equations (a paper with a serious flaw) to the most celebrated mathematician and scientist of his time. It is a portrait of the world's most exciting mathematics between 1725 and 1741, rich in technical detail, woven with connections within Euler's work and with the work of other mathematicians in other times and places, laced with historical context.

Optical Illusions in Rome is a beautifully written and richly illustrated guide that takes the reader on a tour through ingenious uses of geometry to create illusory impressions of space and grandeur in Italian Renaissance art and architecture in the Eternal City. The book takes us to some of the most striking and historically important uses of optical illusion and includes works of Peruzzi, Borromini, and Pozzo. The artworks are analyzed geometrically and placed in their historical context. The notes on visiting the art described make the volume the perfect companion for a study trip to Rome. A chapter on the principles of perspective geometry and a collection of exercises make the book a wonderful resource for a module on perspective in a geometry or art history course. The mathematical discussion is kept at a level accessible to a reader with a familiarity with high school geometry.Kirsti Andersen is a distinguished historian of mathematics and emerita faculty at Aarhus University. Her previous book, The Geometry of an Art, is widely recognized as the definitive work on the history of the use of perspective in European art. Viktor Blasjo, the translator, is a historian of mathematics on the faculty at Utrecht University. Blasjo has won both the Ford and Polya prizes for expository writing from the Mathematical Association of America.

Lobachevski Illuminated provides an historical introduction to non-Euclidean geometry. Within its pages, readers will be guided step-by-step through a new translation of Lobachevski's groundbreaking book, The Theory of Parallels.Extensive commentary situates Lobachevski's work in its mathematical, historical, and philosophical context, thus granting readers a vision of the mysterious and beautiful world of non-Euclidean geometry as seen through the eyes of one of its discoverers. Although Lobachevski's 170-year-old text is challenging to read on its own, Seth Braver's carefully arranged ""illuminations'' render this classic accessible to any modern reader (student, professional, or layman) undaunted by high school mathematics.

The Ladies' Diary was an annual almanac published in England from 1704 to 1840. It was designed to provide useful information to women; the subtitle reveals the purpose, Containing New Improvements in Arts and Sciences, and Many Entertaining Particulars: Designed for the Use and Diversion of the Fair Sex. It contained meteorological and astronomical information, recipes, health and medical advice, scientific information, and mathematical puzzles and problems. Readers were encouraged to, and did, send solutions and original problems and puzzles of their own for publication in the next year's issue. Frank Swetz, one of the founding Editors of Convergence, the MAA's online journal of the history of mathematics, wondered about the historical and sociological conditions that supported The Ladies' Diary. In this volume he unearths the story of the Diary's creation and of the community of people surrounding it. We learn who the editors were and something about the contributors and readers. Swetz explores the sociological and cultural circumstances that made this unique almanac full of mathematics popular for over a century. The book includes scores of puzzles from the Diary, many in the form of riddles, rebuses, and poems.