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Excursions in Modern Mathematics

ISBN: 9780135983355 | 0135983355
Edition: 5th
Format: Hardcover
Publisher: Pearson College Div
Pub. Date: 1/1/2004

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SummaryTable of Contents
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.
Prefacexiii
PART 1 The Mathematics of Social Choice2(146)
1. The Mathematics of Voting: The Paradoxes of Democracy
2(38)
Preference ... MORE
4(2)
The Plurality Method
6(3)
The Borda Count Method
9(1)
The Plurality-with-Elimination Method
10(6)
The Method of Pairwise Comparisons
16(5)
Rankings
21(5)
Conclusion: Fairness and Arrow's Impossibility Theorem
26(1)
Exercises
27(7)
Appendix 1: Breaking Ties
34(2)
Appendix 2: A Sampler of Elections in the Real World
36(3)
References and Further Readings
39(1)
2 Weighted Voting Systems: The Power Game
40(32)
Weighted Voting Systems
42(3)
The Banzhaf Power Index
45(6)
Applications of the Banzhaf Power Index
51(2)
The Shapley-Shubik Power Index
53(6)
Applications of the Shapley-Shubik Power Index
59(1)
Conclusion
60(1)
Exercises
61(9)
Appendix: Power in the Electoral College
70(1)
References and Further Readings
71(1)
3. Fair Division: The Slice Is Right
72(40)
Fair-Division Problems and Fair-Division Schemes
75(1)
Two Players: The Divider-Chooser Method
76(2)
The Lone-Divider Method
78(3)
The Lone-Chooser Method
81(3)
The Last-Diminisher Method
84(5)
The Method of Sealed Bids
89(3)
The Method of Markers
92(3)
Conclusion
95(1)
Exercises
96(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 Science148(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 Symmetry300(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 Statistics424(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 Problems535(44)
Index579(8)
Photo Credits587

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