Unsolved Problems in Number Theory

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Author: Richard Guy

Publisher: Springer Science & Business Media

ISBN: 1489935851

Category: Mathematics

Page: 287

View: 5156

Second edition sold 2241 copies in N.A. and 1600 ROW. New edition contains 50 percent new material.

Unsolved Problems in Number Theory

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Author: Richard Guy

Publisher: Springer Science & Business Media

ISBN: 1489935851

Category: Mathematics

Page: 287

View: 2392

Second edition sold 2241 copies in N.A. and 1600 ROW. New edition contains 50 percent new material.

Geometry of Convex Sets

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Author: I. E. Leonard,J. E. Lewis

Publisher: John Wiley & Sons

ISBN: 1119022665

Category: Mathematics

Page: 352

View: 6169

A gentle introduction to the geometry of convex sets in n–dimensional space Geometry of Convex Sets begins with basic definitions of the linear concepts of addition and scalar multiplication and then defines the notion of convexity for subsets of n–dimensional space. Many properties of convex sets can be discovered using just the linear structure. However, for more interesting results, it is necessary to discuss the notion of distance about open sets, closed sets, bounded sets, and compact sets. The book illustrates the interplay between these linear and topological concepts, which makes the notion of convexity so appealing. Thoroughly class–tested, the book discusses topology and convexity in the context of normed linear spaces, specifically with a norm topology on an n–dimensional space. Geometry of Convex Sets also features: An introduction to n–dimensional geometry including points; lines; vectors; distance; norms; inner products; orthogonality; convexity; hyperplanes; and linear functionals An introduction to n–dimensional norm topology including interior points and open sets; accumulation points and closed sets; boundary points and closed sets; compact subsets of n–dimensional space; completeness of n–dimensional space; sequences; equivalent norms; distance between sets; and support hyperplanes Basic properties of convex sets; convex hulls; interior and closure of convex sets; closed convex hulls; accessibility lemma; regularity of convex sets; affine hulls; flats or affine subspaces; affine basis theorem; separation theorems; extreme points of convex sets; supporting hyperplanes and extreme points; existence of extreme points; Krein–Milman theorem; polyhedral sets and polytopes; and Birkhoff s theorem on doubly stochastic matrices Discussions on Helly s theorem; the Art Gallery theorem; Vincensini s problem; Hadwiger s theorems; theorems of Radon and Caratheodory; Kirchberger s theorem; Helly–type theorems for circles; covering problems; piercing problems; sets of constant width; Reuleaux triangles; Barbier s theorem; and Borsuk s problem Geometry of Convex Sets is a useful textbook for upper–undergraduate level courses in geometry of convex sets and is essential for graduate level courses in convex analysis. An excellent reference for academics and readers interested in learning the various applications of higher geometry, the book is also appropriate for teachers who would like to convey a better understanding and appreciation of the field to students.

Counting on Frameworks

Mathematics to Aid the Design of Rigid Structures

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Author: Jack E. Graver

Publisher: Cambridge University Press

ISBN: 9780883853313

Category: Mathematics

Page: 180

View: 4211

Consider a scaffolding that is constructed by bolting together rods and beams. The ultimate question is whether the structure is strong enough to support the workers and their equipment. This is the problem that motivates the area of mathematics known as rigidity theory. The purpose of this book is to develop a mathematical model for the rigidity of structures. In fact the author develops three distinct models in which the structure under consideration is modelled as a framework. These models are the degrees of freedom model and two models based on quadratic equations and linear equations respectively. The author shows that all three of these models agree except for a very small class of specially constructed frameworks. This is a theory with significant practical applications and will be of interest to a wide range of people including those studying graph theory or mathematical modelling.

Latin Squares and Their Applications

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Author: A. Donald Keedwell,József Dénes

Publisher: Elsevier

ISBN: 0444635580

Category: Mathematics

Page: 455

View: 766

Latin Squares and Their Applications, Second edition offers a long-awaited update and reissue of this seminal account of the subject. The revision retains foundational, original material from the frequently-cited 1974 volume but is completely updated throughout. As with the earlier version, the author hopes to take the reader ‘from the beginnings of the subject to the frontiers of research’. By omitting a few topics which are no longer of current interest, the book expands upon active and emerging areas. Also, the present state of knowledge regarding the 73 then-unsolved problems given at the end of the first edition is discussed and commented upon. In addition, a number of new unsolved problems are proposed. Using an engaging narrative style, this book provides thorough coverage of most parts of the subject, one of the oldest of all discrete mathematical structures and still one of the most relevant. However, in consequence of the huge expansion of the subject in the past 40 years, some topics have had to be omitted in order to keep the book of a reasonable length. Latin squares, or sets of mutually orthogonal latin squares (MOLS), encode the incidence structure of finite geometries; they prescribe the order in which to apply the different treatments in designing an experiment in order to permit effective statistical analysis of the results; they produce optimal density error-correcting codes; they encapsulate the structure of finite groups and of more general algebraic objects known as quasigroups. As regards more recreational aspects of the subject, latin squares provide the most effective and efficient designs for many kinds of games tournaments and they are the templates for Sudoku puzzles. Also, they provide a number of ways of constructing magic squares, both simple magic squares and also ones with additional properties. Retains the organization and updated foundational material from the original edition Explores current and emerging research topics Includes the original 73 ‘Unsolved Problems’ with the current state of knowledge regarding them, as well as new Unsolved Problems for further study

Goodness-of-fit

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Author: Pál Révész,Károly Sarkadi,Pranab Kumar Sen

Publisher: North Holland

ISBN: N.A

Category: Mathematics

Page: 624

View: 6906

Fascinating Mathematical People

Interviews and Memoirs

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Author: Donald J. Albers,Gerald L. Alexanderson

Publisher: Princeton University Press

ISBN: 9781400839551

Category: Mathematics

Page: 352

View: 9338

Fascinating Mathematical People is a collection of informal interviews and memoirs of sixteen prominent members of the mathematical community of the twentieth century, many still active. The candid portraits collected here demonstrate that while these men and women vary widely in terms of their backgrounds, life stories, and worldviews, they all share a deep and abiding sense of wonder about mathematics. Featured here--in their own words--are major research mathematicians whose cutting-edge discoveries have advanced the frontiers of the field, such as Lars Ahlfors, Mary Cartwright, Dusa McDuff, and Atle Selberg. Others are leading mathematicians who have also been highly influential as teachers and mentors, like Tom Apostol and Jean Taylor. Fern Hunt describes what it was like to be among the first black women to earn a PhD in mathematics. Harold Bacon made trips to Alcatraz to help a prisoner learn calculus. Thomas Banchoff, who first became interested in the fourth dimension while reading a Captain Marvel comic, relates his fascinating friendship with Salvador Dalí and their shared passion for art, mathematics, and the profound connection between the two. Other mathematical people found here are Leon Bankoff, who was also a Beverly Hills dentist; Arthur Benjamin, a part-time professional magician; and Joseph Gallian, a legendary mentor of future mathematicians, but also a world-renowned expert on the Beatles. This beautifully illustrated collection includes many photographs never before published, concise introductions by the editors to each person, and a foreword by Philip J. Davis. Some images inside the book are unavailable due to digital copyright restrictions.

Geometry Revealed

A Jacob's Ladder to Modern Higher Geometry

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Author: Marcel Berger

Publisher: Springer Science & Business Media

ISBN: 9783540709978

Category: Mathematics

Page: 831

View: 844

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended... In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations. Marcel Berger is t he author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.

Experimentation in Mathematics

Computational Paths to Discovery

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Author: Jonathan M. Borwein,David H. Bailey,Roland Girgensohn

Publisher: CRC Press

ISBN: 1439864195

Category: Mathematics

Page: 368

View: 692

New mathematical insights and rigorous results are often gained through extensive experimentation using numerical examples or graphical images and analyzing them. Today computer experiments are an integral part of doing mathematics. This allows for a more systematic approach to conducting and replicating experiments. The authors address the role of experimental research in the statement of new hypotheses and the discovery of new results that chart the road to future developments. Following the lead of Mathematics by Experiment: Plausible Reasoning in the 21st Century this book gives numerous additional case studies of experimental mathematics in action, ranging from sequences, series, products, integrals, Fourier series, zeta functions, partitions, primes and polynomials. Some advanced numerical techniques are also presented. To get a taste of the material presented in both books view the condensed version.

An Invitation to Mathematics

From Competitions to Research

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Author: Dierk Schleicher,Malte Lackmann

Publisher: Springer Science & Business Media

ISBN: 9783642195334

Category: Mathematics

Page: 220

View: 2063

This Invitation to Mathematics consists of 14 contributions, many from the world's leading mathematicians, which introduce the readers to exciting aspects of current mathematical research. The contributions are as varied as the personalities of active mathematicians, but together they show mathematics as a rich and lively field of research. The contributions are written for interested students at the age of transition between high school and university who know high school mathematics and perhaps competition mathematics and who want to find out what current research mathematics is about. We hope that it will also be of interest to teachers or more advanced mathematicians who would like to learn about exciting aspects of mathematics outside of their own work or specialization. Together with a team of young ``test readers'', editors and authors have taken great care, through a substantial ``active editing'' process, to make the contributions understandable by the intended readership.

The Last Recreations

Hydras, Eggs, and Other Mathematical Mystifications

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Author: Martin Gardner

Publisher: Springer Science & Business Media

ISBN: 0387258272

Category: Mathematics

Page: 392

View: 5611

Of all of Martin Gardners writings, none gained him a wider audience or was more central to his reputation than his Mathematical Recreations column in Scientific American - which virtually defined the genre of popular mathematics writing for a generation. Flatland, Hydras and Eggs: Mathematical Mystifications is the final collection of these columns, covering the period roughly from 1979 to Gardners retirement in 1986. As always in his published collections, Gardner includes letters commenting on the ideas presented in his articles. These columns show him at the top of his form and should not be missed by anyone with an interest in mathematics.

Euler's Gem

The Polyhedron Formula and the Birth of Topology

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Author: David S. Richeson

Publisher: Princeton University Press

ISBN: 0691154570

Category: Mathematics

Page: 317

View: 5433

Leonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child. Euler's Gem tells the illuminating story of this indispensable mathematical idea. From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V-E+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map. Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, Euler's Gem will fascinate every mathematics enthusiast.

Euler

The Master of Us All

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Author: William Dunham

Publisher: MAA

ISBN: 9780883853283

Category: Mathematics

Page: 185

View: 8278

Leonhard Euler was one of the most prolific mathematicians that have ever lived. This book examines the huge scope of mathematical areas explored and developed by Euler, which includes number theory, combinatorics, geometry, complex variables and many more. The information known to Euler over 300 years ago is discussed, and many of his advances are reconstructed. Readers will be left in no doubt about the brilliance and pervasive influence of Euler's work.

When Less is More

Visualizing Basic Inequalities

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Author: Claudi Alsina,Roger B. Nelsen

Publisher: MAA

ISBN: 9780883853429

Category: Mathematics

Page: 181

View: 6941

Introduces the richness and variety of inequalities in mathematics using illustration and visualisation.

Research Problems in Discrete Geometry

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Author: Peter Brass,William O. J. Moser,János Pach

Publisher: Springer Science & Business Media

ISBN: 0387238158

Category: Mathematics

Page: 499

View: 1069

This book is a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and the most important partial results related to these problems.

Famous Puzzles of Great Mathematicians

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Author: Miodrag Petkovi_

Publisher: American Mathematical Soc.

ISBN: 0821848143

Category: Mathematics

Page: 325

View: 3725

This entertaining book presents a collection of 180 famous mathematical puzzles and intriguing elementary problems that great mathematicians have posed, discussed, and/or solved. The selected problems do not require advanced mathematics, making this book accessible to a variety of readers. Mathematical recreations offer a rich playground for both amateur and professional mathematicians. Believing that creative stimuli and aesthetic considerations are closely related, great mathematicians from ancient times to the present have always taken an interest in puzzles and diversions. The goal of this book is to show that famous mathematicians have all communicated brilliant ideas, methodological approaches, and absolute genius in mathematical thoughts by using recreational mathematics as a framework. Concise biographies of many mathematicians mentioned in the text are also included. The majority of the mathematical problems presented in this book originated in number theory, graph theory, optimization, and probability. Others are based on combinatorial and chess problems, while still others are geometrical and arithmetical puzzles. This book is intended to be both entertaining as well as an introduction to various intriguing mathematical topics and ideas. Certainly, many stories and famous puzzles can be very useful to prepare classroom lectures, to inspire and amuse students, and to instill affection for mathematics.

Unsolved Problems in Geometry

Unsolved Problems in Intuitive Mathematics

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Author: Hallard T. Croft,Kenneth Falconer,Richard K. Guy

Publisher: Springer Science & Business Media

ISBN: 1461209633

Category: Mathematics

Page: 199

View: 4571

Mathematicians and non-mathematicians alike have long been fascinated by geometrical problems, particularly those that are intuitive in the sense of being easy to state, perhaps with the aid of a simple diagram. Each section in the book describes a problem or a group of related problems. Usually the problems are capable of generalization of variation in many directions. The book can be appreciated at many levels and is intended for everyone from amateurs to research mathematicians.