Claudi Alsina's Books

Math Made Visual

Creating Images for Understanding Mathematics

By: Claudi Alsina,Roger B. Nelsen

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Math Made Visual

Creating Images for Understanding Mathematics

By: Claudi Alsina,Roger B. Nelsen

Is it possible to make mathematical drawings that help to understand mathematical ideas, proofs, and arguments? The [Author];s of this book are convinced that the answer is yes and the objective of this book is to show how some visualization techniques may be employed to produce pictures that have both mathematical and pedagogical interest. Mathematical drawings related to proofs have been produced since antiquity in China, Arabia, Greece, and India, but only in the last thirty years has there been a growing interest in so-called ``proofs without words''. Hundreds of these have been published in Mathematics Magazine and The College Mathematics Journal, as well as in other journals, books, and on the internet. Often a person encountering a ``proof without words'' may have the feeling that the pictures involved are the result of a serendipitous discovery or the consequence of an exceptional ingenuity on the part of the picture's creator. In this book, the [Author];s show that behind most of the pictures, ``proving'' mathematical relations are some well-understood methods. As the reader shall see, a given mathematical idea or relation may have many different images that justify it, so that depending on the teaching level or the objectives for producing the pictures, one can choose the best alternative.

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Page Count

173

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1614441006

ISBN 13

9781614441007

Norm Derivatives and Characterizations of Inner Product Spaces

By: Claudi Alsina,Justyna Sikorska,Maria Santos Tom s

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Norm Derivatives and Characterizations of Inner Product Spaces

By: Claudi Alsina,Justyna Sikorska,Maria Santos Tom s

The book provides a comprehensive overview of the characterizations of real normed spaces as inner product spaces based on norm derivatives and generalizations of the most basic geometrical properties of triangles in normed spaces. Since the appearance of Jordanvon Neumann's classical theorem (The Parallelogram Law) in 1935, the field of characterizations of inner product spaces has received a significant amount of attention in various literature texts. Moreover, the techniques arising in the theory of functional equations have shown to be extremely useful in solving key problems in the characterizations of Banach spaces as Hilbert spaces. This book presents, in a clear and detailed style, state-of-the-art methods of characterizing inner product spaces by means of norm derivatives. It brings together results that have been scattered in various publications over the last two decades and includes more new material and techniques for solving functional equations in normed spaces. Thus the book can serve as an advanced undergraduate or graduate text as well as a resource book for researchers working in geometry of Banach (Hilbert) spaces or in the theory of functional equations (and their applications).

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Page Count

199

Publisher

World Scientific

Published Date

ISBN 10

9814287261

ISBN 13

9789814287265

Associative Functions

Triangular Norms and Copulas

By: Claudi Alsina,Berthold Schweizer,Maurice J. Frank

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Associative Functions

Triangular Norms and Copulas

By: Claudi Alsina,Berthold Schweizer,Maurice J. Frank

The functional equation of associativity is the topic of Abel's first contribution to Crelle's Journal. Seventy years later, it was featured as the second part of Hilbert's Fifth Problem, and it was solved under successively weaker hypotheses by Brouwer (1909), Cartan (1930) and Aczel (1949). In 1958, B Schweizer and A Sklar showed that the ?triangular norms? introduced by Menger in his definition of a probabilistic metric space should be associative; and in their book Probabilistic Metric Spaces, they presented the basic properties of such triangular norms and the closely related copulas. Since then, the study of these two classes of functions has been evolving at an ever-increasing pace and the results have been applied in fields such as statistics, information theory, fuzzy set theory, multi-valued and quantum logic, hydrology, and economics, in particular, risk analysis.This book presents the foundations of the subject of associative functions on real intervals. It brings together results that have been widely scattered in the literature and adds much new material. In the process, virtually all the standard techniques for solving functional equations in one and several variables come into play. Thus, the book can serve as an advanced undergraduate or graduate text on functional equations.

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Page Count

253

Publisher

World Scientific

Published Date

ISBN 10

9812566716

ISBN 13

9789812566713

When Less is More

Visualizing Basic Inequalities

By: Claudi Alsina,Roger B. Nelsen

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When Less is More

Visualizing Basic Inequalities

By: Claudi Alsina,Roger B. Nelsen

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

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Page Count

216

Publisher

MAA

Published Date

ISBN 10

0883853426

ISBN 13

9780883853429

A Mathematical Space Odyssey

Solid Geometry in the 21st Century

By: Claudi Alsina,Roger B. Nelsen

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A Mathematical Space Odyssey

Solid Geometry in the 21st Century

By: Claudi Alsina,Roger B. Nelsen

Solid geometry is the traditional name for what we call today the geometry of three-dimensional Euclidean space. Courses in solid geometry have largely disappeared from American high schools and colleges. The authors are convinced that a mathematical exploration of three-dimensional geometry merits some attention in today’s curriculum. A Mathematical Space Odyssey: Solid Geometry in the 21st Century is devoted to presenting techniques for proving a variety of mathematical results in three-dimensional space, techniques that may improve one’s ability to think visually. Special attention is given to the classical icons of solid geometry (prisms, pyramids, platonic solids, cones, cylinders, and spheres) and many new and classical results: Cavalieri’s principle, Commandino’s theorem, de Gua’s theorem, Prince Rupert’s cube, the Menger sponge, the Schwarz lantern, Euler’s rotation theorem, the Loomis-Whitney inequality, Pythagorean theorems in three dimensions, etc. The authors devote a chapter to each of the following basic techniques for exploring space and proving theorems: enumeration, representation, dissection, plane sections, intersection, iteration, motion, projection, and folding and unfolding. In addition to many figures illustrating theorems and their proofs, a selection of photographs of three-dimensional works of art and architecture are included. Each chapter includes a selection of Challenges for the reader to explore further properties and applications. It concludes with solutions to all the Challenges in the book, references, and a complete index. Readers should be familiar with high school algebra, plane and analytic geometry, and trigonometry. While brief appearances of calculus do occur, no knowledge of calculus is necessary to enjoy this book.

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Page Count

288

Publisher

The Mathematical Association of America

Published Date

ISBN 10

0883853582

ISBN 13

9780883853580

Icons of Mathematics: An Exploration of Twenty Key Images

By: Claudi Alsina ,Roger B. Nelsen

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Icons of Mathematics: An Exploration of Twenty Key Images

By: Claudi Alsina ,Roger B. Nelsen

The authors present twenty icons of mathematics, that is, geometrical shapes such as the right triangle, the Venn diagram, and the yang and yin symbol and explore mathematical results associated with them. As with their previous books (Charming Proofs, When Less is More, Math Made Visual) proofs are visual whenever possible. The results require no more than high-school mathematics to appreciate and many of them will be new even to experienced readers. Besides theorems and proofs, the book contains many illustrations and it gives connections of the icons to the world outside of mathematics. There are also problems at the end of each chapter, with solutions provided in an appendix. The book could be used by students in courses in problem solving, mathematical reasoning, or mathematics for the liberal arts. It could also be read with pleasure by professional mathematicians, as it was by the members of the Dolciani editorial board, who unanimously recommend its publication.

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Page Count

347

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470456168

ISBN 13

9781470456160

A Panoply of Polygons

By: Claudi Alsina,Roger B. Nelsen

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A Panoply of Polygons

By: Claudi Alsina,Roger B. Nelsen

A Panoply of Polygons presents and organizes hundreds of beautiful, surprising and intriguing results about polygons with more than four sides. (A Cornucopia of Quadrilaterals, a previous volume by the same authors, thoroughly explored the properties of four-sided polygons.) This panoply consists of eight chapters, one dedicated to polygonal basics, the next ones dedicated to pentagons, hexagons, heptagons, octagons and many-sided polygons. Then miscellaneous classes of polygons are explored (e.g., lattice, rectilinear, zonogons, cyclic, tangential) and the final chapter presents polygonal numbers (figurate numbers based on polygons). Applications, real-life examples, and uses in art and architecture complement the presentation where many proofs with a visual nature are included. A Panoply of Polygons can be used as a supplement to a high school or college geometry course. It can also be used as a source for group projects or extra-credit assignments. It will appeal, and be accessible to, anyone with an interest in plane geometry. Claudi Alsina and Roger Nelsen are, jointly and individually, the authors of thirteen previous MAA/AMS books. Those books, and this one, celebrate and illuminate the power of visualization in learning, teaching, and creating mathematics.

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Page Count

281

Publisher

American Mathematical Society

Published Date

ISBN 10

1470471841

ISBN 13

9781470471842

When Less is More

Visualizing Basic Inequalities

By: Claudi Alsina,Roger B. Nelsen

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When Less is More

Visualizing Basic Inequalities

By: Claudi Alsina,Roger B. Nelsen

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

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Page Count

216

Publisher

MAA

Published Date

ISBN 10

0883853426

ISBN 13

9780883853429

Associative Functions

Triangular Norms and Copulas

By: Claudi Alsina,Berthold Schweizer,Maurice J. Frank

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Associative Functions

Triangular Norms and Copulas

By: Claudi Alsina,Berthold Schweizer,Maurice J. Frank

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Page Count

254

Publisher

World Scientific

Published Date

ISBN 10

9812566716

ISBN 13

9789812566713

Charming Proofs

A Journey into Elegant Mathematics

By: Claudi Alsina,Roger B. Nelsen

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Charming Proofs

A Journey into Elegant Mathematics

By: Claudi Alsina,Roger B. Nelsen

Theorems and their proofs lie at the heart of mathematics. In speaking of the purely aesthetic qualities of theorems and proofs, G. H. Hardy wrote that in beautiful proofs 'there is a very high degree of unexpectedness, combined with inevitability and economy.' Charming Proofs present a collection of remarkable proofs in elementary mathematics that are exceptionally elegant, full of ingenuity, and succinct. By means of a surprising argument or a powerful visual representation, the proofs in this collection will invite readers to enjoy the beauty of mathematics, to share their discoveries with others, and to become involved in the process of creating new proofs. Charming Proofs is organized as follows. Following a short introduction about proofs and the process of creating proofs, the authors present, in twelve chapters, a wide and varied selection of proofs they consider charming. Topics include the integers, selected real numbers, points in the plane, triangles, squares and other polygons, curves, inequalities, plane tilings, origami, colorful proofs, threedimensional geometry, etc. At the end of each chapter are some challenges that will draw the reader into the process of creating charming proofs. There are over 130 such challenges. Charming Proofs concludes with solutions to all of the challenges, references, and a complete index. As in the authors' previous books with the MAA (Math Made Visual and When Less Is More), secondary school, college, and university teachers may wish to use some of the charming proofs in their classrooms to introduce their students to mathematical elegance. Some may wish to use the book as a supplement in an introductory course on proofs, mathematical reasoning, or problem solving.

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Page Count

295

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1614442010

ISBN 13

9781614442011

Icons of Mathematics: An Exploration of Twenty Key Images

By: Claudi Alsina ,Roger B. Nelsen

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Icons of Mathematics: An Exploration of Twenty Key Images

By: Claudi Alsina ,Roger B. Nelsen

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Page Count

347

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470456168

ISBN 13

9781470456160

A Mathematical Space Odyssey

Solid Geometry in the 21st Century

By: Claudi Alsina,Roger B. Nelsen

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A Mathematical Space Odyssey

Solid Geometry in the 21st Century

By: Claudi Alsina,Roger B. Nelsen

Solid geometry is the traditional name for what we call today the geometry of three-dimensional Euclidean space. This book presents techniques for proving a variety of geometric results in three dimensions. Special attention is given to prisms, pyramids, platonic solids, cones, cylinders and spheres, as well as many new and classical results. A chapter is devoted to each of the following basic techniques for exploring space and proving theorems: enumeration, representation, dissection, plane sections, intersection, iteration, motion, projection, and folding and unfolding. The book includes a selection of Challenges for each chapter with solutions, references and a complete index. The text is aimed at secondary school and college and university teachers as an introduction to solid geometry, as a supplement in problem solving sessions, as enrichment material in a course on proofs and mathematical reasoning, or in a mathematics course for liberal arts students.--

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Page Count

271

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1614442169

ISBN 13

9781614442165

Math Made Visual

Creating Images for Understanding Mathematics

By: Claudi Alsina,Roger B. Nelsen

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Math Made Visual

Creating Images for Understanding Mathematics

By: Claudi Alsina,Roger B. Nelsen

The object of this book is to show how visualization techniques may be employed to produce pictures that have interest for the creation, communication and teaching of mathematics. Mathematical drawings related to proofs have been produced since antiquity in China, Arabia, Greece and India but only in the last thirty years has there been a growing interest in so-called 'proofs without words.' In this book the authors show that behind most of the pictures 'proving' mathematical relations are some well-understood methods. The first part of the book consists of twenty short chapters, each one describing a method to visualize some mathematical idea (a proof, a concept, an operation,...) and several applications to concrete cases. Following this the book examines general pedagogical considerations concerning the development of visual thinking, practical approaches for making visualizations in the classroom and a discussion of the role that hands-on material plays in this process.

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Page Count

202

Publisher

MAA

Published Date

ISBN 10

0883857464

ISBN 13

9780883857465

A Cornucopia of Quadrilaterals

By: Claudi Alsina,Roger B. Nelsen

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A Cornucopia of Quadrilaterals

By: Claudi Alsina,Roger B. Nelsen

A Cornucopia of Quadrilaterals collects and organizes hundreds of beautiful and surprising results about four-sided figures—for example, that the midpoints of the sides of any quadrilateral are the vertices of a parallelogram, or that in a convex quadrilateral (not a parallelogram) the line through the midpoints of the diagonals (the Newton line) is equidistant from opposite vertices, or that, if your quadrilateral has an inscribed circle, its center lies on the Newton line. There are results dating back to Euclid: the side-lengths of a pentagon, a hexagon, and a decagon inscribed in a circle can be assembled into a right triangle (the proof uses a quadrilateral and circumscribing circle); and results dating to Erdős: from any point in a triangle the sum of the distances to the vertices is at least twice as large as the sum of the distances to the sides. The book is suitable for serious study, but it equally rewards the reader who dips in randomly. It contains hundreds of challenging four-sided problems. Instructors of number theory, combinatorics, analysis, and geometry will find examples and problems to enrich their courses. The authors have carefully and skillfully organized the presentation into a variety of themes so the chapters flow seamlessly in a coherent narrative journey through the landscape of quadrilaterals. The authors' exposition is beautifully clear and compelling and is accessible to anyone with a high school background in geometry.

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Page Count

306

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470453126

ISBN 13

9781470453121

Norm Derivatives and Characterizations of Inner Product Spaces

By: Claudi Alsina,Justyna Sikorska,M. Santos Tomas

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Norm Derivatives and Characterizations of Inner Product Spaces

By: Claudi Alsina,Justyna Sikorska,M. Santos Tomas

1. Introduction. 1.1. Historical notes. 1.2. Normed linear spaces. 1.3. Strictly convex normed linear spaces. 1.4. Inner product spaces. 1.5. Orthogonalities in normed linear spaces -- 2. Norm derivatives. 2.1. Norm derivatives : Definition and basic properties. 2.2. Orthogonality relations based on norm derivatives. 2.3. p'[symbol]-orthogonal transformations. 2.4. On the equivalence of two norm derivatives. 2.5. Norm derivatives and projections in normed linear spaces. 2.6. Norm derivatives and Lagrange's identity in normed linear spaces. 2.7. On some extensions of the norm derivatives. 2.8. p-orthogonal additivity -- 3. Norm derivatives and heights. 3.1. Definition and basic properties. 3.2. Characterizations of inner product spaces involving geometrical properties of a height in a triangle. 3.3. Height functions and classical orthogonalities. 3.4. A new orthogonality relation. 3.5. Orthocenters. 3.6. A characterization of inner product spaces involving an isosceles trapezoid property. 3.7. Functional equations of the height transform -- 4. Perpendicular bisectors in Normed spaces. 4.1. Definitions and basic properties. 4.2. A new orthogonality relation. 4.3. Relations between perpendicular bisectors and classical orthogonalities. 4.4. On the radius of the circumscribed circumference of a triangle. 4.5. Circumcenters in a triangle. 4.6. Euler line in real normed space. 4.7. Functional equation of the perpendicular bisector transform -- 5. Bisectrices in real Normed spaces. 5.1. Bisectrices in real normed spaces. 5.2. A new orthogonality relation. 5.3. Functional equation of the bisectrix transform. 5.4. Generalized bisectrices in strictly convex real normed spaces. 5.5. Incenters and generalized bisectrices -- 6. Areas of triangles in Normed spaces. 6.1. Definition of four areas of triangles. 6.2. Classical properties of the areas and characterizations of inner product spaces. 6.3. Equalities between different area functions. 6.4. The area orthogonality.

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Solve jigsaw puzzle

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Page Count

199

Publisher

World Scientific

Published Date

ISBN 10

981428727X

ISBN 13

9789814287272

Book's by Claudi Alsina

Math Made Visual

Math Made Visual

Norm Derivatives and Characterizations of Inner Product Spaces

Norm Derivatives and Characterizations of Inner Product Spaces

Associative Functions

Associative Functions

When Less is More

When Less is More

A Mathematical Space Odyssey

A Mathematical Space Odyssey

Icons of Mathematics: An Exploration of Twenty Key Images

Icons of Mathematics: An Exploration of Twenty Key Images

A Panoply of Polygons

A Panoply of Polygons

When Less is More

When Less is More

Associative Functions

Associative Functions

Charming Proofs

Charming Proofs

Icons of Mathematics: An Exploration of Twenty Key Images

Icons of Mathematics: An Exploration of Twenty Key Images

A Mathematical Space Odyssey

A Mathematical Space Odyssey

Math Made Visual

Math Made Visual

A Cornucopia of Quadrilaterals

A Cornucopia of Quadrilaterals

Norm Derivatives and Characterizations of Inner Product Spaces

Norm Derivatives and Characterizations of Inner Product Spaces

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