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Excursions in Modern Mathematics

by
Edition:
6th
ISBN13:

9780131873636

ISBN10:
0131873636
Format:
Hardcover
Pub. Date:
1/1/2007
Publisher(s):
Prentice Hall
List Price: $121.90

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Summary

For undergraduate courses in Liberal Arts Mathematics, Quantitative Literacy, and General Education. This very successful liberal arts mathematics textbook is a collection of "excursions" into the real-world applications of modern mathematics. The excursions are organized into four independent parts: 1) The Mathematics of Social Choice, 2) Management Science, 3) Growth and Symmetry, and 4) Statistics. Each part consists of four chapters plus a mini-excursion (new feature in 6/e). The book is written in an informal, very readable style, with pedagogical features that make the material both interesting and clear. The presentation is centered on an assortment of real-world examples and applications specifically chosen to illustrate the usefulness, relevance, and beauty of liberal arts mathematics.

Table of Contents

Preface xv
part 1 The Mathematics of Social Choice
The Mathematics of Voting
2(46)
The Paradoxes of Democracy
Preference Ballots and Preference Schedules
4(2)
The Plurality Method
6(4)
The Borda Count Method
10(2)
The Plurality-with-Elimination Method (Instant Runoff Voting)
12(5)
The Method of Pairwise Comparisons (Copeland's Method)
17(6)
Rankings
23(25)
Conclusion: Elections, Fairness, and Arrow's Impossibility Theorem
28(1)
Profile: Kenneth J. Arrow
29(1)
Key Concepts
30(1)
Exercises
30(11)
Projects and Papers
41(1)
Appendix 1: Breaking Ties
42(1)
Appendix 2: A Sampler of Elections in the Real World
43(3)
References and Further Readings
46(2)
Weighted Voting Systems
48(36)
The Power Game
Weighted Voting Systems
50(3)
The Banzhaf Power Index
53(8)
Applications of Banzhaf Power
61(2)
The Shapley-Shubik Power Index
63(5)
Applications of Shapley-Shubik Power
68(16)
Conclusion
70(1)
Profile: Lloyd S. Shapley
71(1)
Key Concepts
72(1)
Exercises
72(7)
Projects and Papers
79(1)
Appendix: Power in the Electoral College
80(2)
References and Further Readings
82(2)
Fair Division
84(44)
The Mathematics of Sharing
Fair-Division Games
86(2)
Two Players: The Divider-Chooser Method
88(1)
The Lone-Divider Method
89(6)
The Lone-Chooser Method
95(3)
The Last-Diminisher Method
98(5)
The Method of Sealed Bids
103(3)
The Method of Markers
106(22)
Conclusion
109(1)
Profile: Hugo Steinhaus
110(1)
Key Concepts
111(1)
Exercises
111(15)
Projects and Papers
126(1)
References and Further Readings
127(1)
The Mathematics of Apportionment
128(32)
Making the Rounds
Apportionment Problems
129(5)
Hamilton's Method and the Quota Rule
134(2)
The Alabama and Other Paradoxes
136(5)
Jefferson's Method
141(3)
Adams's Method
144(1)
Webster's Method
145(15)
Conclusion
147(2)
Historical Note: A Brief History of Apportionment in the United States
149(1)
Key Concepts
150(1)
Exercises
150(5)
Projects and Papers
155(2)
References and Further Readings
157(3)
part 2 Management Science
Euler Circuits
160(36)
The Circuit Comes to Town
Euler Circuit Problems
162(3)
Graphs
165(3)
Graph Concepts and Terminology
168(2)
Graph Models
170(1)
Euler's Theorems
171(3)
Fleury's Algorithm
174(5)
Eulerizing Graphs
179(17)
Conclusion
183(1)
Profile: Leonhard Euler
184(1)
Key Concepts
185(1)
Exercises
185(9)
Projects and Papers
194(1)
References and Further Readings
194(2)
The Traveling Salesman Problem
196(38)
Hamilton Joins the Circuit
Hamilton Circuits and Hamilton Paths
199(1)
Complete Graphs
200(4)
Traveling Salesman Problems
204(3)
Simple Strategies for Solving TSPs
207(3)
The Brute-Force and Nearest-Neighbor Algorithms
210(2)
Approximate Algorithms
212(1)
The Repetitive Nearest-Neighbor Algorithm
213(1)
The Cheapest-Link Algorithm
214(20)
Conclusion
218(2)
Profile: Sir William Rowan Hamilton
220(1)
Key Concepts
221(1)
Exercises
221(9)
Projects and Papers
230(2)
References and Further Readings
232(2)
The Mathematics of Networks
234(40)
The Cost of Being Connected
Trees
236(4)
Spanning Trees
240(2)
Kruskal's Algorithm
242(2)
The Shortest Network Connecting Three Points
244(6)
Shortest Networks for Four or More Points
250(24)
Conclusion
257(1)
Profile: Evangelista Torricelli
258(1)
Key Concepts
258(1)
Exercises
259(9)
Projects and Papers
268(3)
Appendix: The Soap-Bubble Solution
271(1)
References and Further Readings
272(2)
The Mathematics of Scheduling
274(38)
Directed Graphs and Critical Paths
The Basic Elements of Scheduling
276(5)
Directed Graphs (Digraphs)
281(2)
Scheduling with Priority Lists
283(6)
The Decreasing-Time Algorithm
289(2)
Critical Paths
291(3)
The Critical-Path Algorithm
294(2)
Scheduling with Independent Tasks
296(16)
Conclusion
299(1)
Profile: Ronald L. Graham
300(1)
Key Concepts
300(1)
Exercises
301(8)
Projects and Papers
309(1)
References and Further Readings
309(3)
part 3 Growth and Symmetry
Spiral Growth in Nature
312(26)
Fibonacci Numbers and the Golden Ratio
Fibonacci Numbers
313(4)
The Golden Ratio
317(1)
Gnomons
318(6)
Spiral Growth in Nature
324(14)
Conclusion
327(1)
Profile: Leonardo Fibonacci
328(1)
Key Concepts
329(1)
Exercises
329(6)
Projects and Papers
335(1)
References and Further Readings
336(2)
The Mathematics of Population Growth
338(34)
There Is Strength in Numbers
The Dynamics of Population Growth
339(3)
The Linear Growth Model
342(6)
The Exponential Growth Model
348(9)
The Logistic Growth Model
357(15)
Conclusion
363(1)
Profile: Sir Robert May
363(1)
Key Concepts
364(1)
Exercises
364(6)
Projects and Papers
370(1)
References and Further Readings
371(1)
Symmetry
372(38)
Mirror, Mirror Off the Wall...
Rigid Motions
373(2)
Reflections
375(2)
Rotations
377(3)
Translations
380(1)
Glide Reflections
381(2)
Symmetry as a Rigid Motion
383(5)
Patterns
388(22)
Conclusion
393(1)
Profile: Sir Robert Penrose
393(1)
Key Concepts
394(1)
Exercises
394(11)
Projects and Papers
405(1)
Appendix: The 17 Wallpaper Symmetry Types
406(3)
References and Further Readings
409(1)
The Geometry of Fractal Shapes
410(36)
Fractally Speaking
The Koch Snowflake
412(6)
The Sierpinski Gasket
418(3)
The Chaos Game
421(1)
The Twisted Sierpinski Gasket
422(3)
The Mandelbrot Set
425(21)
Conclusion
432(1)
Profile: Benoit Mandelbrot
433(1)
Key Concepts
434(1)
Exercises
434(8)
Projects and Papers
442(1)
References and Further Readings
443(3)
part 4 Statistics
Collecting Statistical Data
446(30)
Censuses, Surveys, and Clinical Studies
The Population
448(4)
Sampling
452(5)
Random Sampling
457(2)
Sampling: Terminology and Key Concepts
459(1)
The Capture-Recapture Method
460(1)
Clinical Studies
461(15)
Conclusion
465(1)
Profile: George Gallup
466(1)
Key Concepts
467(1)
Exercises
467(7)
Projects and Papers
474(1)
References and Further Readings
475(1)
Descriptive Statistics
476(34)
Graphing and Summarizing Data
Graphical Descriptions of Data
477(4)
Variables
481(6)
Numerical Summaries of Data
487(8)
Measures of Spread
495(15)
Conclusion
498(1)
Profile: W. Fdwards Deming
498(1)
Key Concepts
499(1)
Exercises
499(9)
Projects and Papers
508(1)
References and Further Readings
509(1)
Chances, Probabilities, and Odds
510(30)
Measuring Uncertainty
Random Experiments and Sample Spaces
511(3)
Counting Sample Spaces
514(2)
Permutations and Combinations
516(3)
Probability Spaces
519(4)
Equiprobable Spaces
523(3)
Odds
526(14)
Conclusion
528(1)
Profile: Persi Diaconis
529(1)
Key Concepts
530(1)
Exercises
531(6)
Projects and Papers
537(1)
References and Further Readings
538(2)
Normal Distributions
540(27)
Everything Is Back to Normal (Almost)
Approximately Normal Distributions of Data
542(2)
Normal Curves and Normal Distributions
544(2)
Standardizing Normal Data
546(2)
The 68-95-99.7 Rule
548(1)
Normal Curves as Models of Real-Life Data Sets
549(2)
Distributions of Random Events
551(2)
Statistical Inference
553(14)
Conclusion
557(1)
Profile: Carl Friedrich Gauss
557(1)
Key Concepts
558(1)
Exercises
558(7)
Projects and Papers
565(1)
References and Further Readings
566(1)
Answers to Selected Problems 567(42)
Index 609(10)
Photo Credits 619


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