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Tables of Contents for Laboratories in Mathematical Experimentation
Chapter/Section Title
Page #
Page Count
Preface
vii
8
Introduction
xv
1 Iteration of Linear Functions
1
15
1.1 Introduction
1
1
1.2 What is iteration?
2
1
1.3 The mathematical ideas
3
4
1.4 Questions to explore
7
4
1.5 Discussion
11
2
1.6 Bibliography
13
1
Computer programs
14
2
2 Cyclic Difference Sets
16
14
2.1 Introduction
16
1
2.2 Arithmetic modulo 15
17
2
2.3 Cyclic difference sets modulo m
19
2
2.4 Questions to explore
21
3
2.5 Discussion
24
1
Computer programs
25
5
3 The Euclidean Algorithm
30
24
3.1 Introduction
30
1
3.2 The algorithm
31
4
3.3 Questions and discussion
35
5
3.4 Linear Diophantine Equations
40
5
3.5 Additional topic
45
1
Computer programs
46
8
4 Prime Numbers
54
25
4.1 Introduction
54
1
4.2 Listing prime numbers
55
5
4.3 Functions generating primes
60
4
4.4 Distribution of primes
64
3
4.5 Further reading
67
1
Computer programs
67
12
5 The Coloring of Graphs
79
15
5.1 Introduction
79
2
5.2 Introduction to the mathematical ideas
81
9
5.3 Questions to explore
90
3
5.4 Bibliography
93
1
6 Randomized Response Surveys
94
25
6.1 Introduction
94
1
6.2 Asking sensitive questions
95
1
6.3 Background
96
1
6.4 Questions to explore
97
15
Computer programs
112
7
7 Polyhedra
119
4
7.1 Introduction
119
1
7.2 Questions and discussion
120
2
7.3 Additional topic
122
1
8 The p-adic Numbers
123
19
8.1 Introduction
123
2
8.2 Absolute values on Q
125
3
8.3 The real numbers
128
3
8.4 The p-adic numbers
131
11
9 Parametric Curve Representation
142
17
9.1 Introduction
142
1
9.2 Symmetries and closed curves
143
6
9.3 Questions to explore
149
3
9.4 Polar representation of curves
152
3
9.5 Additional ideas to explore
155
1
Computer programs
155
4
10 Numerical Integration
159
22
10.1 Introduction
159
1
10.2 Standard numerical methods
160
3
10.3 Automating the standard methods
163
3
10.4 Questions to explore
166
3
10.5 Monte Carlo methods
169
3
10.6 Higher dimensions
172
3
Computer programs
175
6
11 Sequences and Series
181
22
11.1 Introduction
181
1
11.2 The mathematical ideas
182
3
11.3 The harmonic series
185
6
11.4 The natural logarithm
191
4
11.5 Euler's constant
195
3
11.6 Additional exercises and questions
198
2
Computer programs
200
3
12 Experiments in Periodicity
203
16
12.1 Introduction
203
2
12.2 Area accumulation using CALCWIN
205
7
12.3 A new type of function
212
2
12.4 Antiderivatives of periodic functions
214
1
12.5 Finding the periodic antiderivative
215
2
12.6 Further investigation
217
2
13 Iteration to Solve Equations
219
7
13.1 Introduction
219
3
13.2 Improving convergence
222
1
13.3 Questions to explore
223
1
Computer programs
224
2
14 Iteration of Quadratic Functions
226
17
14.1 Introduction
226
1
14.2 Some theory
226
1
14.3 Iterating f(x) = ax(1 - x)
227
3
14.4 The Feigenbaum diagram
230
1
14.5 Examining chaos
231
3
14.6 The tent and sawtooth functions
234
1
14.7 Conjugacy
235
1
14.8 Iterating other functions
236
1
14.9 Listening to chaos
236
1
14.10 Bibliography
237
1
Computer programs
237
6
15 Iterated Linear Maps in the Plane
243
16
15.1 Introduction
243
1
15.2 Multiplying matrices
244
3
15.3 An example to start
247
5
15.4 Questions to explore
252
1
15.5 Discussion
253
2
Computer programs
255
4
16 Euclidean Algorithm for Complex Integers
259
16
16.1 Introduction
259
1
16.2 Complex integers
260
8
16.3 Questions and discussion
268
3
Computer programs
271
4
Index
275