Excursions in Modern Mathematics

by Tannenbaum, Peter

• ISBN13: 9780135983355
• ISBN10: 0135983355
• Edition: 3rd
• Format: Hardcover
• Copyright: 2004-01-01
• Publisher: Pearson College Div

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.

Table of Contents

Preface	xiii
PART 1 The Mathematics of Social Choice	2	(146)
1. The Mathematics of Voting: The Paradoxes of Democracy	2	(38)
Preference Ballots and Preference Schedules	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 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