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Excursions in Modern Mathematics

by: Tannenbaum, Peter

Excursions in Modern Mathematics

by: Tannenbaum, Peter

ISBN 13:
9780135983355
ISBN 10:
0135983355
Edition: 3rd
Format: Hardcover
Copyright: 01/01/2004
Publisher: Pearson College Div
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Summary

This text opens a new world of mathematics to the reader. From Voting and Apportionment to Symmetry and Fractals, this best selling text draws students into a new perspective.

Preface

xiii

PART 1 The Mathematics of Social Choice

(146)

1. The Mathematics of Voting: The Paradoxes of Democracy

(38)

Preference Ballots and Preference Schedules

(2)

The Plurality Method

(3)

The Borda Count Method

(1)

The Plurality-with-Elimination Method

(6)

The Method of Pairwise Comparisons

(5)

Rankings

(5)

Conclusion: Fairness and Arrow's Impossibility Theorem

(1)

Exercises

(7)

Appendix 1: Breaking Ties

(2)

Appendix 2: A Sampler of Elections in the Real World

(3)

References and Further Readings

(1)

2 Weighted Voting Systems: The Power Game

(32)

Weighted Voting Systems

(3)

The Banzhaf Power Index

(6)

Applications of the Banzhaf Power Index

(2)

The Shapley-Shubik Power Index

(6)

Applications of the Shapley-Shubik Power Index

(1)

Conclusion

(1)

Exercises

(9)

Appendix: Power in the Electoral College

(1)

References and Further Readings

(1)

3. Fair Division: The Slice Is Right

(40)

Fair-Division Problems and Fair-Division Schemes

(1)

Two Players: The Divider-Chooser Method

(2)

The Lone-Divider Method

(3)

The Lone-Chooser Method

(3)

The Last-Diminisher Method

(5)

The Method of Sealed Bids

(3)

The Method of Markers

(3)

Conclusion

(1)

Exercises

(15)

References and Further Readings

111

(1)

4. The Mathematics of Apportionment: Making the Rounds

112

(36)

Apportionment Problems

114

(2)

A Little Bit of U. S. History

116

(1)

The Mathematics of Apportionment: Basic Concepts

117

(2)

Hamilton's Method

119

(1)

The Quota Rule

120

(1)

The Alabama Paradox

121

(1)

More Problems with Hamilton's Method

122

(4)

Jefferson's Method

126

(2)

Jefferson's Method and The Quota Rule

128

(1)

Adams' Method

129

(3)

Webster's Method

132

(1)

Conclusion: Balinski and Young's Impossibility Theorem

133

(1)

Exercises

134

(7)

Appendix 1: The Huntington-Hill Method

141

(3)

Appendix 2: A Brief History of Apportionment in the United States

144

(2)

References and Further Readings

146

(2)

PART 2 Management Science

148

(152)

5. Euler Circuits: The Circuit Comes to Town

148

(36)

Routing Problems

150

(4)

Graphs

154

(2)

Graph Concepts and Terminology

156

(2)

Graph Models

158

(3)

Euler's Theorems

161

(3)

Fleury's Algorithm

164

(3)

Eulerizing Graphs

167

(5)

Conclusion

172

(1)

Exercises

173

(10)

References and Further Readings

183

(1)

6. The Traveling-Salesman Problem: Hamilton Joins the Circuit

184

(40)

Hamilton Circuits and Hamilton Paths

188

(1)

Complete Graphs

189

(3)

Traveling-Salesman Problems

192

(2)

Simple Strategies for Solving TSPs

194

(3)

The Brute-Force and Nearest-Neighbor Algorithms

197

(5)

The Repetitive Nearest-Neighbor Algorithm

202

(1)

The Cheapest-Link Algorithm

203

(5)

Conclusion

208

(2)

Exercises

210

(12)

References and Further Readings

222

(2)

7. The Mathematics of Networks: Connections!

224

(36)

Trees

226

(4)

Minimum Spanning Trees

230

(1)

Kruskal's Algorithm

230

(2)

The Shortest Distance Between Three Points

232

(6)

Shortest Networks Linking More than Three Points

238

(5)

Conclusion

243

(2)

Exercises

245

(12)

Appendix: The Soap-Bubble Solution

257

(2)

References and Further Readings

259

(1)

8. The Mathematics of Scheduling: Directed Graphs and Critical Paths

260

(40)

The Basic Elements of Scheduling

262

(6)

Directed Graphs

268

(2)

The Priority List Model for Scheduling

270

(7)

The Decreasing-Time Algorithm

277

(1)

Critical Paths

278

(4)

The Critical-Path Algorithm

282

(2)

Scheduling with Independent Tasks

284

(3)

Conclusion

287

(1)

Exercises

288

(9)

References and Further Readings

297

(3)

PART 3 Growth and Symmetry

300

(124)

9. Spiral Growth in Nature: Fibonacci Numbers and the Golden Ration

300

(28)

Fibonacci Numbers

302

(3)

The Equation x(2) = x + 1 and the Golden Ratio

305

(3)

Gnomons

308

(7)

Gnomonic Growth

315

(2)

Conclusion

317

(1)

Exercises

318

(8)

References and Further Readings

326

(2)

10. The Mathematics of Population Growth: There Is Strength in Numbers

328

(30)

The Dynamics of Population Growth

330

(3)

The Linear Growth Model

333

(5)

The Exponential Growth Model

338

(7)

The Logistic Growth Model

345

(5)

Conclusion

350

(1)

Exercises

351

(6)

References and Further Readings

357

(1)

Geometric Symmetry

360

(1)

Rigid Motions

361

(1)

Reflections

362

(2)

Rotations

364

(2)

Translations

366

(1)

Glide Reflections

366

(1)

Symmetry Revisited

367

(6)

Patterns

373

(4)

Conclusion

377

(1)

Exercises

378

(10)

Appendix: The Seventeen Wallpaper Pattern Types

388

(3)

References and Further Readings

391

(1)

12. Fractal Geometry: Fractally Speaking

392

(32)

The Koch Snowflake

394

(6)

The Sierpinski Gasket

400

(2)

The Chaos Game

402

(1)

The Twisted Sierpinski Gasket

403

(3)

Symmetry of Scale in Art and Literature

406

(1)

The Mandelbrot Set

407

(6)

Conclusion: Fractals

413

(3)

Exercises

416

(5)

References and Further Readings

421

(3)

PART 4 Statistics

424

(111)

13. Collecting Statistical Data: Censuses, Surveys, and Studies

424

(28)

The Population

426

(2)

Case Study 1: The 1990 U. S. Census

428

(1)

Surveys

429

(1)

Case Study 2: The 1936 Literary Digest Poll

430

(2)

Case Study 3: The 1948 Presidential Election

432

(2)

Random Sampling

434

(2)

Case Study 4: Modern Public Opinion Polls: Stratified Samples

436

(2)

Sampling: Terminology and Key Concepts

438

(2)

Clinical Studies

440

(1)

Case Study 5: The 1954 Salk Polio Vaccine Field Trials

441

(3)

Conclusion

444

(1)

Exercises

445

(6)

References and Further Readings

451

(1)

14. Descriptive Statistics: Graphing and Summarizing Data

452

(32)

Graphical Descriptions of Data

454

(3)

Variables: Quantitative and Qualitative; Continuous and Discrete

457

(5)

Numerical Summaries of Data

462

(9)

Measures of Spread

471

(3)

Conclusion

474

(1)

Exercises

475

(8)

References and Further Readings

483

(1)

15. Chances, Probability, and Odds: Measuring Uncertainty

484

(26)

Random Experiments and Sample Spaces

486

(2)

Counting: The Multiplication Rule

488

(2)

Permutations and Combinations

490

(6)

Probability Spaces

496

(2)

Probability Spaces with Equally Likely Outcomes

498

(4)

Odds

502

(1)

Conclusion

503

(1)

Exercises

504

(5)

References and Further Readings

509

(1)

16. Normal Distributions: Everything Is Back to Normal (Almost)

510

(25)

Approximately Normal Distributions of Data

512

(2)

Normal Curves and Their Properties

514

(4)

Normal Curves as Models of Real-Life Data Sets

518

(1)

Normal Distributions of Random Events

519

(2)

Statistical Inference

521

(5)

Conclusion

526

(1)

Exercises

527

(6)

References and Further Readings

533

(2)

Answers to Selected Problems

535

(44)

Index

579

(8)

Photo Credits

587

Amazon no longer offers textbook rentals. We do!

Excursions in Modern Mathematics

Excursions in Modern Mathematics

9780135983355

0135983355

Summary

Table of Contents

Supplemental Materials

Write a Review

Amazon no longer offers textbook rentals. We do!

Excursions in Modern Mathematics

Excursions in Modern Mathematics

9780135983355

0135983355

Summary

Table of Contents

Supplemental Materials

Write a Review

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