The last twenty years have been witness to a fundamental shift in the way mathematics is practiced. With the continued advance of computing power and accessibility, the view that real mathematicians don't compute no longer has any traction for a newer generation of mathematicians that can really take advantage of computer-aided research, especially given the scope and availability of modern computational packages such as Maple, Mathematica, and MATLAB. The authors provide a coherent variety of accessible examples of modern mathematics subjects in which intelligent computing plays a significant role.
Author(s): David H. Bailey, Jonathan M. Borwein, Neil J. Calkin, Roland Girgensohn, D. Russell Luke, Victor Moll
Publisher: A.K. Peters
Year: 2007
Language: English
Pages: 317
City: Wellesley, Mass
Tags: Математика;Вычислительная математика;
Title Page......Page 1
TOC......Page 3
Mathematical Knowledge as I View It......Page 9
Introduction......Page 10
Philosophy of Experimental Mathematics......Page 11
Our Experimental Mathodology......Page 18
Eight Roles for Computation......Page 19
Finding Things versus Proving Things......Page 22
Conclusions......Page 31
Last Words......Page 33
Algorithms for Experimental Mathematics, Part One......Page 35
High-Precision Arithmetic......Page 36
Integer Relation Detection......Page 37
The BBP Formula for Pi......Page 39
Bifurcation Points in Chaos Theory......Page 41
Sculpture......Page 44
Euler Sums......Page 46
Quantum Field Theory......Page 48
Definite Integrals and Infinite Series Summations......Page 49
Computation of Multivariate Zeta Constants......Page 51
Ramanujan-Type Elliptic Series......Page 52
Experiments with Ramanujan-Type Series......Page 53
Working with the Series Analytically......Page 56
The Wilf-Zeilberger Algorithm......Page 61
Prime Number Computations......Page 63
Roots of Polynomials......Page 66
Numerical Quadrature......Page 68
Tanh-Sinh Quadrature......Page 71
Practical Considerations for Quadrature......Page 73
Infinite Series Summation......Page 75
Apery-Like Summations......Page 78
Exploration and Discovery in Inverse Scattering......Page 87
The Physical Experiment......Page 88
The Model......Page 89
Weak Formulation......Page 95
The Mathematical Experiment: Qualitative Inverse Scattering......Page 97
Where Is the Scatterer? How Big is It?......Page 98
Is the Scatterer Absorbing?......Page 102
What Is the Shape of the Scatterer?......Page 106
Refining the Information......Page 111
Current Research......Page 113
Nowhere Differentiable Functions......Page 115
Functional Equations......Page 117
Schauder Bases......Page 123
Non-differentiability......Page 125
Bernoulli Convolutions......Page 129
Random Vectors and Factoring Integers: A Case Study......Page 139
Fermat's Method......Page 140
Kraitchik's Improvement......Page 141
Brillhart and Morrison......Page 142
Random Models......Page 143
The Constant Weight Model......Page 144
Bounds......Page 145
Which Model Is Best?......Page 150
Refining the Model......Page 151
Experimental Evidence......Page 157
Conclusions......Page 158
Introduction......Page 163
The Project and its Experimental Nature......Page 165
Families and Individuals......Page 166
An Experimental Derivation of Wallis' Formula......Page 169
A Hyperbolic Example......Page 172
A Formula Hidden in the List......Page 176
Some Experiments on Valuations......Page 178
An Error in the Latest Edition......Page 182
Some Examples Involving the Hurwitz Zeta Function......Page 186
Experimental Mathematics: a Computational Conclusion......Page 191
A Little More History......Page 192
The Perko Pair......Page 193
Fractal Cards and Chaos Games......Page 194
A Preliminary Example: Visualizing DNA Strands......Page 197
What Is a Chaos Game?......Page 198
Examples of Chaos Games......Page 199
More on Visualizing DNA......Page 203
Hilbert's Inequality and Witten's Zeta-function......Page 204
Hilbert's (easier) Inequality......Page 205
Witten -functions......Page 209
The Best Constant......Page 211
Computational Challenge Problems......Page 215
Finding (3,1,3,1)......Page 216
/8 or not?......Page 219
Exercises for Chapter 1......Page 227
Exercises for Chapter 2......Page 233
Exercises for Chapter 3......Page 234
Exercises for Chapter 4......Page 237
Exercises for Chapter 5......Page 244
Exercises for Chapter 6......Page 247
Exercises for Chapter 7......Page 250
Exercises for Chapter 8......Page 258
Additional Exercises......Page 266
Bibliography......Page 299
Index......Page 301
