Modern Geometries

(Essays in Public Policy)

James R. Smart

Thomson Brooks/Cole | 1973 | 371 páginas | pdf | 10 Mb

Faltam páginas 146, 147, 148, 149, 158, 159, 160, 161.

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Descrição: This comprehensive, best-selling text focuses on the study of many different geometries -- rather than a single geometry -- and is thoroughly modern in its approach. Each chapter is essentially a short course on one aspect of modern geometry, including finite geometries, the geometry of transformations, convexity, advanced Euclidian geometry, inversion, projective geometry, geometric aspects of topology, and non-Euclidean geometries. This edition reflects the recommendations of the COMAP proceedings on Geometry's Future, the NCTM standards, and the Professional Standards for Teaching Mathematics. References to a new companion text, Active Geometry by David A. Thomas encourage students to explore the geometry of motion through the use of computer software. Using Active Geometry at the beginning of various sections allows professors to give students a somewhat more intuitive introduction using current technology before moving on to more abstract concepts and theorems.

Mathematics the Man Made Universe

Sherman K. Stein

W H FREEMAN & CO | 1963 | djvu | 4,49 Mb

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Descrição: Developed from a course for students in a variety of fields, this highly readable volume covers a vast array of subjects, including number theory, topology, set theory, geometry, algebra, and analysis. Starting with questions on weighing, the primes, the fundamental theory of arithmetic, and rationals and irrationals, the text also surveys the representation of numbers, congruence, probability, much more. Several useful appendices, plus answers, comments for selected exercises.

Table of Contents 

Map; Guide; Preface

1. Questions on weighing
Weighing with a two-pan balance and two measures—Problems raised—Their algebraic phrasing

2. The primes
The Greek prime-manufacturing machine—Gaps between primes—Average gap and 1/1 + 1/2 + 1/3 + . . . + 1/N—Twin primes

3. The Fundamental Theorem of Arithmetic
Special natural numbers—Every special number is prime—"Unique factorization" and "every prime is special" compared—Euclidean algorithm—Every prime number is special—The concealed theorem

4. Rationals and Irrationals
The Pythagorean Theorem-—he square root of 2—Natural numbers whose square root is irrational—Rational numbers and repeating decimals

5. Tiling
The rationals and tiling a rectangle with equal squares—Tiles of various shapes—use of algebra—Filling a box with cubes

6. Tiling and electricity
Current—The role of the rationals—Applications to tiling—Isomorphic structures

7. The highway inspector and the salesman
A problem in topology—Routes passing once over each section of highway—Routes passing once through each town

8. Memory Wheels
A problem raised by an ancient word—Overlapping n-tuplets—Solution—History and applications

9. The Representation of numbers
Representing natural numbers—The decimal system (base ten)—Base two—Base three—Representing numbers between 0 and 1—Arithmetic in base three—The Egyptian system—The decimal system and the metric system

10. Congruence
Two integers congruent modulo a natural number—Relation to earlier chapters—Congruence and remainders—Properties of congruence—Casting out nines—Theorems for later use

11. Strange algebras
Miniature algebras—Tables satisfying rules—Commutative and idempotent tables—Associativity and parentheses—Groups

12. Orthogonal tables
Problem of the 36 officers—Some experiments—A conjecture generalized—Its fate—Tournaments—Application to magic squares

13. Chance
Probability—Dice—The multiplication rule—The addition rule—The subtraction rule—Roulette—Expectation—Odds—Baseball—Risk in making decisions

14. The fifteen puzzle
The fifteen puzzle—A problem in switching cords—Even and odd arrangements—Explanation of the Fifteen puzzle—Clockwise and counterclockwise

15. Map coloring
The two-color theorem—Two three-color theorems—The five-color theorem—The four-color conjecture

16. Types of numbers
Equations—Roots—Arithmetic of polynomials—Algebraic and transcendental numbers—Root r and factor X—r—Complex numbers—Complex numbers applied to alternating current—The limits of number systems

17. Construction by straightedge and compass
Bisection of line segment-Bisection of angle-Trisection of line segment—Trisection of 90° angle—Construction of regular pentagon—Impossibility of constructing regular 9-gon and trisecting 60°

18. Infinite sets
A conversation from the year 1638—Sets and one-to-one correspondence—Contrast of the finite with the infinite—Three letters of Cantor—Cantor's Theorem—Existence of transcendentals

19. A general view
The branches of mathematics—Topology and set theory as geometries—The four "shadow" geometries—Combinatorics—Algebra—Analysis—Probability—Types of proof—Cohen's theorem—Truth and proof—Gödel's theorem

Appendix A. Review of arithmetic
A quick tour of the basic ideas of arithmetic

Appendix B. Writing mathematics
Some words of advice and caution

Appendix C. The rudiments of algebra
A review of algebra, which is reduced to eleven rules

Appendix D. Teaching mathematics
Suggestions to prospective and practicing teachers

Appendix E. The geometric and harmonic series
Their properties—Applications of geometric series to probability

Appendix F. Space of any dimension
Definition of space of any dimension

Appendix G. Update
Answers and comments for selected exercises

Index

Conceptions, croyances et représentations en maths, sciences et technos

Louise Lafortune, Colette Deaudelin, Pierre-André Doudin, Daniel Marti

Presses de l'Université du Québec | 2005 | 297 páginas | pdf | 3,74 Mb

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Descrição: En éducation, les conceptions, croyances et les représentations des élèves, des étudiants et étudiantes en formation à l’enseignement, des enseignants et enseignantes et des parents côtoient très souvent les connaissances, le savoir et les pratiques. Comment définir ces perceptions? Quelle est leur influence sur l’apprentissage scolaire et plus particulièrement sur celui des mathématiques, des sciences et des technologies?

À partir des résultats de recherche, cet ouvrage propose des pistes d’interventions multidimensionnelles visant à modifier les croyances aussi bien des spécialistes de l’enseignement que des élèves à l’égard des mathématiques, des sciences et des technologies. Des réflexions sur le rapport au savoir des étudiants et étudiantes en formation à l’enseignement et sur les principes pouvant guider la construction des pratiques professionnelles du personnel scolaire sont également au coeur de cet ouvrage.

Calculator's Cunning: The Art of Quick Reckoning

Karl Menninger

Basic Books | 1964 | djvu | 1,53 Mb

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Descrição: An arithmetical trick is a device for solving a problem more easily and surprisingly than by ordinary methods. While one person will see behind it the broad and wonderfully intricate world of numbers, another will be satisfied by his delight in the trick itself. This book will serve both kinds of student. It will teach the beginner to master a series of useful methods and above all to look at the numbers before doing any calculation, in order to use the tricks appropriate to them; for the advanced student, and for anyone who enjoys numbers, it will afford many pleasant glimpses into the land of numbers, and stimulate him to a voyage of discovery.

Numbers have a certain magic. It is no accident that their wonderful properties have grown into the theory of numbers, 'the queen of mathematics'. Those relations which the schoolboy knows as arithmetical tricks belong to elementary number-theory.

Strength In Numbers: Discovering the Joy and Power of Mathematics in Everyday Life

Sherman K. Stein
Wiley | 1999 | 288 páginas | djvu | 2,33 Mb

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Descrição: What do two goats and a car have to do with making good decisions? Was the golden ratio used to build the Great Pyramid of Khufu? Can it be that some numbers are unmistakably "hot," while others are inherently "cool"? With his infectiously enthusiastic and engaging style, award-winning teacher and author Sherman K. Stein offers a new appreciation for mathematics, from the beauty of its logic ("as inevitable and memorable as a Mozart symphony") to its amazing power and pervasiveness in our lives. Requiring no math knowledge beyond basic arithmetic and high school geometry, Strength in Numbers is an enlightening introduction to all the math you'll ever need.

Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football

Wayne L. Winston

Princeton University Press | 2009 | 376 páginas | pdf | 1,03 Mb

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Descrição: athletics is a remarkably entertaining book that shows readers how to use simple mathematics to analyze a range of statistical and probability-related questions in professional baseball, basketball, and football, and in sports gambling. How does professional baseball evaluate hitters? Is a singles hitter like Wade Boggs more valuable than a power hitter like David Ortiz? Should NFL teams pass or run more often on first downs? Could professional basketball have used statistics to expose the crooked referee Tim Donaghy? Does money buy performance in professional sports?In Mathletics, Wayne Winston describes the mathematical methods that top coaches and managers use to evaluate players and improve team performance, and gives math enthusiasts the practical tools they need to enhance their understanding and enjoyment of their favorite sports--and maybe even gain the outside edge to winning bets. Mathletics blends fun math problems with sports stories of actual games, teams, and players, along with personal anecdotes from Winston's work as a sports consultant. Winston uses easy-to-read tables and illustrations to illuminate the techniques and ideas he presents, and all the necessary math concepts--such as arithmetic, basic statistics and probability, and Monte Carlo simulations--are fully explained in the examples.After reading Mathletics, you will understand why baseball teams should almost never bunt, why football overtime systems are unfair, why points, rebounds, and assists aren't enough to determine who's the NBA's best player--and much, much more.

TABLE OF CONTENTS:

Preface xi
Acknowledgments xiii
List of Abbreviations xv

Part I. Baseball 1

Chapter 1: Baseball's Pythagorean Theorem 3
Chapter 2: Who Had a Better Year, Nomar Garciaparra or Ichiro Suzuki? 11
The Runs- Created Approach
Chapter 3: Evaluating Hitters by Linear Weights 17
Chapter 4: Evaluating Hitters by Monte Carlo Simulation 30
Chapter 5: Evaluating Baseball Pitchers and Forecasting Future Pitcher Performance 41
Chapter 6: Baseball Decision- Making 52
Chapter 7: Evaluating Fielders 64
Sabermetrics' Last Frontier
Chapter 8: Player Win Averages 71
Chapter 9: The Value of Replacement Players 79
Evaluating Trades and Fair Salary
Chapter 10: Park Factors 84
Chapter 11: Streakiness in Sports 87
Chapter 12: The Platoon Effect 102
Chapter 13: Was Tony Perez a Great Clutch Hitter? 106
Chapter 14: Pitch Count and Pitcher Effectiveness 110
Chapter 15: Would Ted Williams Hit .406 Today? 113
Chapter 16: Was Joe DiMaggio's 56- Game Hitting Streak the Greatest Sports Record of All Time? 116
Chapter 17: Major League Equivalents 123

Part II. Football 125

Chapter 18: What Makes NFL Teams Win? 127
Chapter 19: Who's Better, Tom Brady or Peyton Manning? 132
Chapter 20: Football States and Values 138
Chapter 21: Football Decision- Making 101 143
Chapter 22: A State and Value Analysis of the 2006 Super Bowl 151
Champion Colts
Chapter 23: If Passing Is Better Than Running, Why Don't 158
Teams Always Pass?
Chapter 24: Should We Go for a One- Point or Two- Point Conversion? 165
Chapter 25: To Give Up the Ball Is Better Than to Receive 172
The Case of College Football Overtime
Chapter 26: Why Is the NFL's Overtime System Fatally Flawed? 175
Chapter 27: How Valuable Are High Draft Picks in the NFL? 180

Part III. Basketball 185

Chapter 28: Basketball Statistics 101 187
The Four- Factor Model
Chapter 29: Linear Weights for Evaluating NBA Players 195
Chapter 30: Adjusted_/_Player Ratings 202
Chapter 31: NBA Lineup Analysis 224
Chapter 32: Analyzing Team and Individual Matchups 228
Chapter 33: NBA Players' Salaries and the Draft 233
Chapter 34: Are NBA Officials Prejudiced? 237
Chapter 35: Are College Basketball Games Fixed? 242
Chapter 36: Did Tim Donaghy Fix NBA Games? 244
Chapter 37: End- Game Basketball Strategy 248

Part IV. Playing with Money, and Other Topics for Serious Sports Fans 253

Chapter 38: Sports Gambling 101 255
Chapter 39: Freakonomics Meets the Bookmaker 262
Chapter 40: Rating Sports Teams 266
Chapter 41: Which League Has Greater Parity, The NFL or the NBA? 283
Chapter 42: The Ratings Percentage Index (RPI) 287
Chapter 43: From Point Ratings to Probabilities 290
Chapter 44: Optimal Money Management 298
The Kelly Growth Criteria
Chapter 45: Ranking Great Sports Collapses 303
Chapter 46: Can Money Buy Success? 311
Chapter 47: Does Joey Crawford Hate the Spurs? 319
Chapter 48: Does Fatigue Make Cowards of Us All? 321
The Case of NBA Back- to- Back Games and NFL Bye Weeks
Chapter 49: Can the Bowl Championship Series Be Saved? 324
Chapter 50: Comparing Players from Different Eras 331
Chapter 51: Conclusions 335

Index of Databases 341
Annotated Bibliography 343
Index 353

Guide to Mathematical Modelling

(Mathematical guides)
Dilwyn Edwards, Mike Hamson

Palgrave Macmillan | 2001 | 240 páginas | pdf | 3,4 Mb

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Descrição: A basic introduction to mathematical modelling, this book encourages the reader to participate in the investigation of a wide variety of modelling examples. These are carefully paced so that readers can identify and develop the skills which are required for successful modelling. The examples also promote an appreciation of the enormous range of problems to which mathematical modelling skills can usefully be applied.

Contents

What is Modelling ?
Getting Started
Modelling Methodology
Modelling Skills
Using Difference Equations
Using Differential Equations
Using Random Numbers
Using data
Example Models
Report Writing and Presentation