Excursions in Modern Mathematics

Part 1. The Mathematics of Social Choice



1. The Mathematics of Voting: The Paradox of Democracy

1.1 Preference Ballots and Preference Schedules

1.2 The Plurality Method

1.3 The Borda Count Method

1.4 The Plurality-with-Elimination Method (Instant Runoff Voting)

1.5 The Method of Piecewise Comparisons

1.6 Rankings

Profile: Kenneth J. Arrow

Key Concepts

Exercises

Projects and Papers

References and Further Readings



2. The Mathematics of Power: Weighted Voting

2.1 An Introduction to Weighted Voting

2.2 The Banzhaf Power Index

2.3 Applications of the Banzhaf Power Index

2.4 The Shapely-Shubik Power Index

2.5 Applications of the Shapely-Shubik Power Index

Profile: Lloyd S. Shapely

Key Concepts

Exercises

Projects and Papers

References and Further Readings



3. The Mathematics of Sharing: Fair-Division Games

3.1 Fair-Division Games

3.2 Two Players: The Divider-Chooser Method

3.3 The Lone-Divider Method

3.4 The Lone-Chooser Method

3.5 The Last-Diminisher Method

3.6 The Method of Sealed Bids

3.7 The Method of Markers

Profile: Hugo Steinhaus

Key Concepts

Exercises

Projects and Papers

References and Further Readings



4. The Mathematics of Apportionment: Making the Rounds

4.1 Apportionment Problems

4.2 Hamilton's Method and the Quota Rule

4.3 The Alabama and Other Paradoxes

4.4 Jefferson's Method

4.5 Adams's Method

4.6 Webster's Method

Historical Note: A Brief History of Apportionment in the United States

Key Concepts

Exercises

Projects and Papers

References and Further Readings



Mini-Excursion 1: Apportionment Today



Part 2. Management Science



5. The Mathematics of Getting Around: Euler Paths and Circuits

5.1 Euler Circuit Problems

5.2 What is a Graph?

5.3 Graph Concepts and Terminology

5.4 Graph Models

5.5 Euler's Theorems

5.6 Fleury's Algorithm

5.7 Eulerizing Graphs

Profile: Leonard Euler

Key Concepts

Exercises

Projects and Papers

References and Further Readings



6. The Mathematics of Touring: The Traveling Salesman Problem

6.1 Hamilton Circuits and Hamilton Paths

6.2 Complete Graphs

6.3 Traveling Salesman Problems

6.4 Simple Strategies for Solving TSPs

6.5 The Brute-Force and Nearest-Neighbor Algorithms

6.6 Approximate Algorithms

6.7 The Repetitive Nearest-Neighbor Algorithm

6.8 The Cheapest Link Algorithm

Profile: Sir William Rowan Hamilton

Key Concepts

Exercises

Projects and Papers

References and Further Readings



7. The Mathematics of Networks: The Cost of Being Connected

7.1 Trees

7.2 Spanning Trees

7.3 Kruskal's Algorithm

7.4 The Shortest Network Connecting Three Points

7.5 Shortest Networks for Four or More Points

Profile: Evangelista Torricelli

Key Concepts

Exercises

Projects and Papers

References and Further Readings



8. The Mathematics of Scheduling: Chasing the Critical Path

8.1 The Basic Elements of Scheduling

8.2 Directed Graphs (Digraphs)

8.3 Scheduling with Priority Lists

8.4 The Decreasing-Time Algorithm

8.5 Critical Paths

8.6 The Critical-Path Algorithm

8.7 Scheduling with Independent Tasks

Profile: Ronald L. Graham

Key Concepts

Exercises

Projects and Papers

References and Further Readings



Mini-Excursion 2: A Touch of Color



Part 3. Growth And Symmetry



9. The Mathematics of Spiral Growth: Fibonacci Numbers and the Golden Ratio

9.1 Fibonacci's Rabbits

9.2 Fibonacci Numbers

9.3 The Golden Ratio

9.4 Gnomons

9.5 Spiral Growth in Nature

Profile: Leonardo Fibonacci

Key Concepts

Exercises

Projects and Papers

References and Further Readings



10. The Mathematics of Money: Spending it, Saving It, and Growing It

10.1 Percentages

10.2 Simple Interest

10.3 Compound Interest

10.4 Geometric Sequences

10.5 Deferred Annuities: Planned Savings for the Future

Key Concepts

Exercises

Projects and Papers

References and Further Readings



11. The Mathematics of Symmetry: Beyond Reflection

11.1 Rigid Motions

11.2 Reflections

11.3 Rotations

11.4 Translations

11.5 Glide Reflections

11.6 Symmetry as a Rigid Motion

11.7 Patterns

Profile: Sir Roger Penrose

Key Concepts

Exercises

Projects and Papers

References and Further Readings



12. The Geometry of Fractal Shapes: Naturally Irregular

12.1 The Koch Snowflake

12.2 The Sierpinski Gasket

12.3 The Chaos Game

12.4 The Twisted Sierpinski Gasket

12.5 The Mandelbrot Set

Profile: Benoit Mandelbrot

Key Concepts

Exercises

Projects and Papers

References and Further Readings



Mini-Excursion 3: The Mathematics of Population Growth: There is Strength in Numbers



Part 4. Statistics



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

13.1 The Population

13.2 Sampling

13.3 Random Sampling

13.4 Sampling: Terminology and Key Concepts

13.5 The Capture-Recapture Method

13.6 Clinical Studies

Profile: George Gallup

Key Concepts

Exercises

Projects and Papers

References and Further Readings



14. Descriptive Statistics: Graphing and Summarizing Data

14.1 Graphical Descriptions of Data

14.2 Variables

14.3 Numerical Summaries of Data

14.4 Measures of Spread

Profile: W. Edwards Deming

Key Concepts

Exercises

Projects and Papers

References and Further Readings



15. Chances, Probabilities, and Odds: Measuring Uncertainty

15.1 Random Experiments and Sample Spaces

15.2 Counting Outcomes in Sample Spaces

15.3 Permutations and Combinations

15.4 Probability Spaces

15.5 Equiprobable Spaces

15.6 Odds

Profile: Persi Diaconis

Key Concepts

Exercises

Projects and Papers

References and Further Readings



16. The Mathematics of Normal Distributions: The Call of the Bell

16.1 Approximately Normal Distributions of Data

16.2 Normal Curves and Normal Distributions

16.3 Standardizing Normal Data

16.4 The 68-95-99.7 Rule

16.5 Normal Curves as Models of Real-Life Data Sets

16.6 Distributions of Random Events

16.7 Statistical Inference

Profile: Carl Friedrich Gauss

Key Concepts

Exercises

Projects and Papers

References and Further Readings



Mini-Excursion 4: The Mathematics of Managing Risk