The Genius of Euler: Reflections on his Life and Work

(Spectrum)

William Dunham (Editor)

The Mathematical Association of America | 2007 | 309 páginas | DjVu (11.8 mb)

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This book celebrates the 300th birthday of Leonhard Euler (1707 1783), one of the brightest stars in the mathematical firmament. The book stands as a testimonial to a mathematician of unsurpassed insight, industry, and ingenuity one who has been rightly called the master of us all. The collected articles, aimed at a mathematically literate audience, address aspects of Euler s life and work, from the biographical to the historical to the mathematical. The oldest of these was written in 1872, and the most recent dates to 2006. Some of the papers focus on Euler and his world, others describe a specific Eulerian achievement, and still others survey a branch of mathematics to which Euler contributed significantly. Along the way, the reader will encounter the Konigsberg bridges, the 36-officers, Euler s constant, and the zeta function. There are papers on Euler s number theory, his calculus of variations, and his polyhedral formula. Of special note are the number and quality of authors represented here. Among the 34 contributors are some of the most illustrious mathematicians and mathematics historians of the past century e.g., Florian Cajori, Carl Boyer, George Polya, Andre Weil, and Paul Erdos. And there are a few poems and a mnemonic just for fun.

Contents
Acknowledgments ix
Preface xi
About the Authors xiii
Part I: Biography and Background
Introduction to Part I 3
Leonhard Euler, B. F. Finkel (1897) 5
Leonard Euler, Supreme Geometer (abridged), C. Truesdell (1972) 13
Euler (abridged), Andre Weil (1984) 43
Frederick the Great on Mathematics and Mathematicians (abridged),
Florian Cajori (1927) 51
The Euler-Diderot Anecdote, B. H. Brown (1942) 57
Ars Expositionis: Euler as Writer and Teacher, G. L. Alexanderson (1983) 61
The Foremost Textbook of Modern Times, C. B. Boyer (1951) 69
Leonhard Euler, 1707-1783, J. J. Burckhardt (1983) 75
Euler's Output, A Historical Note, W W. R. Ball (1924) 89
Discoveries (a poem), Marta Sved and Dave Logothetti (1989) 91
Bell's Conjecture (a poem), J. D. Memory (1997) 93
A Response to "Bell's Conjecture" (a poem),
Charlie Marion and William Dunham (1997) 95
Part II: Mathematics
Introduction to Part 2 99
Euler and Infinite Series, Morris Kline (1983) 101
The Genius of Euler: Reflections on his Life and Work
Euler and the Zeta Function, Raymond Ayoub (1974) 113
Addendum to: "Euler and the Zeta Function, " A. G. Howson (1975) 133
Euler Subdues a Very Obstreperous Series (abridged), E. J. Barbeau (1979) 135
On the History ofEuler's Constant, J. W. L. Glaisher (1872) 147
A Mnemonic for Euler's Constant, Morgan Ward (1931) 153
Euler and Differentials, Anthony P. Ferzola (1994) 155
Leonhard Euler's Integral: A Historical Profile of the
Gamma Function, Philip J. Davis (1959) 167
Change of Variables in Multiple Integrals: Euler to Cartan, Victor J. Katz (1982) 185
Euler's Vision of a General Partial Differential Calculus for a
Generalized Kind of Function, Jesper LUtzen (1983) 197
On the Calculus of Variations and Its Major Influences on the
Mathematics of the First Half of Our Century, Erwin Kreyszig (1994) 209
Some Remarks and Problems in Number Theory Related to the
Work of Euler, Paul Erdos and Underwood Dudley (1983) 215
Euler's Pentagonal Number Theorem, George E. Andrews (1983) 225
Euler and Quadratic Reciprocity, Harold M. Edwards (1983) 233
Euler and the Fundamental Theorem of Algebra, William Dunham (1991) 243
Guessing and Proving, George P6lya (1978) 257
The Truth about Kdnigsberg, Brian Hopkins and Robin J. Wilson (2004) 263
Graeco-Latin Squares and a Mistaken Conjecture of Euler,
Dominic Klyve and Lee Stemkoski (2006) 273
Glossary 289
List of Photos 303
Index 305
About the Editor 309

Mathematics and Modern Art: Proceedings of the First ESMA Conference, held in Paris, July 19-22, 2010

(Springer Proceedings in Mathematics)

Claude Bruter

Springer | 2012 | 222 páginas | PDF | 9,3 Mb

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The link between mathematics and art remains as strong today as it was in the earliest instances of decorative and ritual art. Arts, architecture, music and painting have for a long time been sources of new developments in mathematics, and vice versa. Many great painters have seen no contradiction between artistic and mathematical endeavors, contributing to the progress of both, using mathematical principles to guide their visual creativity, enriching their visual environment with the new objects created by the mathematical science.

Owing to the recent development of the so nice techniques for visualization, while mathematicians can better explore these new mathematical objects, artists can use them to emphasize their intrinsic beauty, and create quite new sceneries. This volume, the content of the first conference of the European Society for Mathematics and the Arts (ESMA), held in Paris in 2010, gives an overview on some significant and beautiful recent works where maths and art, including architecture and music, are interwoven. 

The book includes a wealth of mathematical illustrations from several basic mathematical fields including classical geometry, topology, differential geometry, dynamical systems.  Here, artists and mathematicians alike elucidate the thought processes and the tools used to create their work

Contents

A Mathematician and an Artist. The Story of a Collaboration . . 1

Richard S. Palais

Dimensions, a Math Movie  . . 11

Aurelien Alvarez and Jos Leys

Old and New Mathematical Models: Saving the Heritage of the Institut Henri Poincare . . 17

Francois Apery

An Introduction to the Construction of Some Mathematical Objects. . 29

Claude Paul Bruter

Computer, Mathematics and Art . . . 47

Jean-Franc¸ois Colonna

Structure of Visualization and Symmetry in Iterated Function Systems . . . 53

Jean Constant

M.C. Escher’s Use of the Poincare Models of Hyperbolic Geometry. . 69

Douglas Dunham

Mathematics and Music Boxes  . . 79

Vi Hart

My Mathematical Engravings. . 85

Patrice Jeener

Knots and Links As Form-Generating Structures. . 105

Dmitri Kozlov

Geometry and Art from the Cordovan Proportion. . . 117

Antonia Redondo Buitrago and Encarnacion Reyes Iglesias

Dynamic Surfaces . . 131

Simon Salamon

Pleasing Shapes for Topological Objects . . 153

John M. Sullivan

Rhombopolyclonic Polygonal Rosettes Theory . . 167

Francois Tard

USA Mathematical Olympiads 1972-1986 Problems and Solutions

(Anneli Lax New Mathematical Library) 

Murray Klamkin 

Mathematical Association of America | 1989 | 146 páginas | pdf | 1,7 Mb

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People delight in working on problems "because they are there," for the sheer pleasure of meeting a challenge. This is a book full of such delights. In it, Murray S. Klamkin brings together 75 original USA Mathematical Olympiad (USAMO) problems for yearss 1972-1986, with many improvements, extensions, related exercises, open problems, referneces and solutions, often showing alternative approaches. The problems are coded by subject, and solutions are arranged by subject, e.g., algebra, number theory, solid geometry, etc., as an aid to those interested in a particular field. Included is a Glossary of frequently used terms and theorems and a comprehensive bibliography with items numbered and referred to in brackets in the text. This a collection of problemsand solutions of arresting ingenuit, all accessible to secondary school students.

The USAMO has been taken annually by about 150 of the nation's best high school mathematics students. This exam helps to find and encourage high school students with superior mathematical talent and creativity and is the culmination of a three-tiered competition that begins with the American High School Mathematics Examination (AHSME) taken by over 400, 000 students. The eight winners of the USAMO are canidates for the US team in the International Mathematical Olympiad. Schools are encouraged to join this large and important enterprise. See page x of the preface for further information. this book includes a list of all of the top contestants in the USAMO and their schools.

The problems are intriguing and the solutions elegant and informative. Students and teachers will enjoy working these challenging problems. Indeed, all hose who are mathematically inclined will find many delights and pleasant challenges in this book.

Contents
Editors' Note Vii
Preface ix
USA Olympiad Problems 1
Solutions of Olympiad Problems 15
Algebra (A) 15
Number Theory (N.T.) 30
Plane Geometry (P.G.) 45
Solid Geometry (S.G.) 55
Geometric Inequalities (G.I.) 66
Inequalities (I) 81
Combinatorics & Probability (C.& P.) 93
Appendix 105
List of Symbols 110
Glossary 111
References 120

Mathematical Cognition

(Current Perspectives on Cognition, Learning, and Instruction)

James Royer

Information Age Publishing | 2003 | 272 páginas | rar - pdf | 1,3 Mb

link (password: matav)

CONTENTS
Introduction
James M. Royer ix
1. The Development of Math Competence in the Preschool and  Early School Years: Cognitive Foundations and Instructional Strategies
Sharon Griffin 1
2. Perspectives on Mathematics Strategy Development
Martha Carr and Hillary Hettinger 33
3. Mathematical Problem Solving
Richard E. Mayer 69
4. Learning Disabilities in Basic Mathematics: Deficits in Memory and Cognition
David C. Geary and Mary K. Hoard 93
5. Relationships Among Basic Computational Automaticity, Working Memory, and Complex Mathematical Problem Solving: What We Know and What We Need to Know
Loel T. Tronsky and James M. Royer 117
6. Mathematics Instruction: Cognitive, Affective and Existential Perspectives
Allan Feldman 147
7. A Brief History of American K-12 Mathematics Education in the 20th Century
David Klein 175
8. Assessment in Mathematics: A Developmental Approach
John Pegg 260

Gödel, Escher, Bach: Una Eterna Trenza Dorada

 Douglas R. Hofstadter

Consejo nacional de ciencia - versão em castelhano | 1979 | 915 páginas | pdf | 44 Mb

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scribd.com

epub - 170 Mb - epubbud.com

versão em inglês

El Autor es Douglas R. Hofstadter y es un libro ganador del Premio Pulitzer. Aunque en la primera edición en español su titulo fue “Gödel, Escher, Bach: una eterna trenza dorada”, posteriormente fue cambiado por el actual.
El libro toma la forma de una interacción entre varias narrativas. Los capítulos principales se alternan con diálogos entre los personajes imaginarios, inspirados por la narración de Lewis Carroll Lo que le dijo la tortuga a Aquiles, que aparece en el libro. En éste, Aquiles y la tortuga discuten una paradoja relativa a los modus ponens. Hofstadter basa los otros diálogos en éste, presentando al cangrejo y a un genio, entre otros. Estas narrativas se sumergen con frecuencia en la autorreferencia y la metaficción.
Un diálogo particularmente significativo en el libro está ingeniosamente escrito en la forma de un canon en espejo, en el cual cada línea antes del punto medio corresponde a una línea idéntica pasado el punto medio, si bien la conversación da una sensación extraña debido al uso de frases comunes que pueden ser usadas como saludos o despedidas (“buen día”) y la colocación de las líneas que, bajo cercana inspección, dobla como una respuesta a una pregunta en la línea siguiente.
El libro fue considerado un tiempo como intraducible, dado que da gran énfasis en los así llamados “retruécanos estructurales”, como en el diálogo del canon en espejo, que se lee exactamente igual, oración por oración, tanto en forma normal como al revés, exceptuando el voilá del primer fragmento que es sustituido por viola en el segundo.
En este libro hay muchos campos tratados y es un libro “denso” que requiere varias relecturas y echar mano de la enciclopedia Internet para comprender ciertas partes, algunos de los campos que toca son:
* Metamatemática* Simetría* Inteligencia artificial* Sistema formal, Teoría de la computabilidad* Paradoja* Zen* Genética* Biología molecular* Lógica, Teoría de números* Tipografía y sintaxis* Cerebro, Mente y Cognición* Sintaxis vs. Semántica* Libre albedrío vs. Determinismo* Holismo vs. reduccionismo* El lenguaje de programación Lisp* Isomorfismos y significado* Capas yuxtapuestas de significado, contrapunto, semiótica, códigos* Autorreferencia, recursión, Bucles extraños* Auto-organización, sensación emergente de identidad: consciente (por Ej. “Yo soy una oración verdadera, y lo que declaro es que no puedo ser comprobado dentro de este sistema al que pertenezco” o “Yo soy verdadero, pero mi verdad transciende este universo”)

Sacred Geometry: Philosophy & Practice

 Robert Lawlor

Thames & Hudson | 1982 | 112 páginas | pdf | 17 Mb

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Art and Imagination: These large-format, gloriously-illustrated paperbacks cover Eastern and Western religion and philosophy, including myth and magic, alchemy and astrology. The distinguished authors bring a wealth of knowledge, visionary thinking and accessible writing to each intriguing subjecArt and Imagination: These large-format, gloriously-illustrated paperbacks cover Eastern and Western religion and philosophy, including myth and magic, alchemy and astrology. The distinguished authors bring a wealth of knowledge, visionary thinking and accessible writing to each intriguing subjec

An introduction to the geometry which, as modern science now confirms, underlies the structure of the universe.

The thinkers of ancient Egypt, Greece and India recognized that numbers governed much of what they saw in their world and hence provided an approach to its divine creator. Robert Lawlor sets out the system that determines the dimension and the form of both man-made and natural structures, from Gothic cathedrals to flowers, from music to the human body. By also involving the reader in practical experiments, he leads with ease from simple principles to a grasp of the logarithmic spiral, the Golden Proportion, the squaring of the circle and other ubiquitous ratios and proportions. 

Contents
Introduction 4
The Practice of Geometry 7
Sacred Geometry : Metaphor of Universal Order 16
The Primal Act: The Division of Unity 23
Workbook 1 : The Square Cut by its Diagonal; square root 2 25-27
Workbook 2: The square root 3 and the Vesica Piscis 32-33
Workbook 3: The square root 5 36-37
Alternation
Workbook 4: Alternation 4W1
Proportiorl andthe Golden Section
Workbook 5: The Golden Proportion 48-52
Gnomonic Expan~iona nd the Creation of Spirals
Workbook 6: Gnomonic spirals 67-70
The Squaring of the Circle
Workbook 7: Squaring the circle 7479
Mediation : Geometry becomes Music
Workbook 8: Geometry and Music 83-85
Anthropos I
The Genesis of Cosmic Volumes
Workbook 9: The Platonic Solids 98-1 02
Bibliography

Sources of Illustrations

A Guide to Complex Variables

(Dolciani Mathematical Expositions)

Steven G. Krantz

Mathematical Association of America | 2008 |202 páginas | 1 Mb

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versão draft -pdf - link direto

This is a book about complex variables that gives the reader a quick and accessible introduction to the key topics. While the coverage is not comprehensive, it certainly gives the reader a solid grounding in this fundamental area. There are many figures and examples to illustrate the principal ideas, and the exposition is lively and inviting. An undergraduate wanting to have a first look at this subject or a graduate student preparing for the qualifying exams, will find this book to be a useful resource.


In addition to important ideas from the Cauchy theory, the book also include sthe Riemann mapping theorem, harmonic functions, the argument principle, general conformal mapping and dozens of other central topics.
Readers will find this book to be a useful companion to more exhaustive texts in the field. It is a valuable resource for mathematicians and non-mathematicians alike.

Table of Contents
Preface
1. The complex plane
2. Complex line integrals
3. Applications of the Cauchy theory
4. Isolated singularities and Laurent series
5. The argument principle
6. The geometric theory of holomorphic functions
7. Harmonic functions
8. Infinite series and products
9. Analytic continuation.