Experimental mathematics in action

Experimental Mathematics in Action

Volumn , Issue , 2007, Pages 1-322

  Bailey, David H ^a   Borwein, Jonathan M ^a   Calkin, Neil J ^a   Girgensohn, Roland ^a   Luke, D Russell ^a   Moll, Victor H ^a  

^a NONE

Author keywords

[No Author keywords available]

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EID: 85074549991     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1201/9781439864333     Document Type: Book

References (270)

1. A. Barnikov. A Course in Convexity. Amer. Math. Soc., Providence, RI, 2002.
2. A. Edelman and H. Murakami. Polynomial roots from companion matrix eigenvalues. Mathematics of Computation, 64:763-776, 1995.
3. A.G.Ramm. Recovery of the potential from fixed energy scattering data. Inverse Problems, 4:877-886, 1988.
4. A. Garsia. Arithmetic properties of Bernoulli convolutions. Transactions of the American Mathematical Society, 102:409-432, 1962.
5. A. Granville. Pascal's Triangle Interface. http://www.cecm.sfu.ca/organics/papers/ granville/support/pascalform.html. Accessed August 20, 2006.
6. A. Holroyd, T. Liggett, and D. Romik. Integrals, partitions and cellular automata. Trans. Amer. Math. Soc., 356:3349-3368, 2004.
7. A. Kirsch. Characterization of the shape of a scattering obstacle using the spectral data of the far field operator. Inverse Problems, 14:1489-1512, 1998.
8. A. Kirsch. Factorization of the far field operator for the inhomogeneous medium case and an application to inverse scattering theory. Inverse Problems, 15:413-429, 1999.
9. A. Lasota and M. C. Mackey. Chaos, Fractals and Noise: Stochastic Aspects of Dynamics. Springer-Verlag, Heidelberg, 1994.
10. A. M. Legendre. Theorie des Nombres. Firmin Didot Freres, Paris, 1830.
11. A. Nachman. Global uniqueness for a two-dimensional inverse boundary value problem. The Annals of Mathematics, 143(1):71-96, 1996.
12. A. Nachman. Reconstructions from boundary measurements. Ann. Math., 128:531- 576, 1988.
13. A. Odlyzko. The 1022-nd zero of the Riemann zeta function. Dynamical, spectral, and arithmetic zeta functions. Contemp. Math., 290:139-144, 2001.
14. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev. Integrals and Series. Gordon and Breach Science, Heidelberg, 1992.
15. A. Sommerfeld. Optics. Academic Press, New York, 1954.
16. A. V. Sills and D. Zeilberger. Disturbing the Dyson conjecture (in a good way). Experimental Mathematics, 15(2):187-192, 2005.
17. A. Wintner. On convergent Poisson convolutions. American Journal of Mathematics, 57:827-838, 1935.
18. A. van der Poorten. A proof that Euler missed: Apéry's proof of the irrationality of ? (3). Mathematical Intelligencer, 1(4):195-203, 1978
19. B. Berndt. Ramanujan's Notebooks, Part I. Springer-Verlag, New York, 1985.
20. B. Bollobás. Random graphs, volume 73 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, UK, 2nd edition, 2001.
21. B. Caviness and J. R. Johnson, editors. Quantifier elimination and cylindrical algebraic decomposition. Texts and Monographs in Symbolic Computation, Springer, New York, 1998.
22. B. Jessen and A. Wintner. Distribution functions and the Riemann zeta function. Transactions of the American Mathematical Society, 38:48-88, 1935.
23. B. Solomyak. On the random series S±λi (an Erdos problem). Annals of Mathematics, 142:611-625, 1995.
24. B. Spice. CMU's Simon reflects on how computers will continue to shape the world. Pittsburg Post-Gazette, October 16, 2000. http://www.post-gazette.com/regionstate/ 20001016simon2.asp.
25. B. Yandell. The Honors Class. A K Peters, Natick, MA, 2002.
26. C. Ferguson. Helaman Ferguson: Mathematics in Stone and Bronze. Meridian Creative Group, Erie, PA, 1994.
27. C. Koch. Thinking about the conscious mind, a review of John R. Searle's Mind. A Brief Introduction, Oxford University Press, Oxford, UK, 2004. Science, 306:979- 980, November 5, 2004.
28. C. Pomerance. A tale of two sieves. Notices Amer. Math. Soc., 43(12):1473-1485, 1996.
29. C. Tracy. Integrals for Ising susceptibilty expansion. Personal communication.
30. Centre for Constructive and Experimental Mathematics. EZFacePlus. http://oldweb. cecm.sfu.ca/projects/ezface+. Accessed August 20, 2006.
31. Centre for Constructive and Experimental Mathematics. Inverse Symbolic Calculator. http://oldweb.cecm.sfu.ca/projects/ISC. Accessed August 19, 2006.
32. Clay Mathematics Institute. The Riemann Hypothesis. http://www.claymath.org/ millennium/Riemann Hypothesis/, 2006.
33. D. Ashlock and J. Golden. Evolutionary computation and fractal visualization of sequence data. In Gary B. Fogel and David W. Corne, editors, Evolutionary Computation in Bioinformatics. Morgan Kaufmann Publishers, San Francisco, 2002.
34. D. Bierens de Haan. Nouvelles tables d'integrales definies. Hafner, New York, 2nd edition, 1957.
35. D. Borwein, J. M. Borwein, and R. Girgensohn. Explicit evaluation of Euler sums. Proceedings of the Edinburgh Mathematical Society, 38:273-294, 1995.
36. D. Colton and A. Kirsch. A simple method for solving inverse scattering problems in the resonance region. Inverse Problems, 12(4):383-93, 1996.
37. D. Colton and L. Päivärinta. Transmission eigenvalues and a problem of Hans Lewy. J. Comp. and Appl. Math., 117:91-104, 2000.
38. D. Colton and P. Monk. A linear sampling method for the detection of leukimia using microwaves. SIAM J. Appl. Math, 58(3):926-941, 1998.
39. D. Colton and P. Monk. A novel method for solving the inverse scattering problem for time-harmonic waves in the resonance region II. SIAM J. Appl. Math., 46:506-23, 1986.
40. D. Colton and P. Monk. A novel method for solving the inverse scattering problem for time-harmonic waves in the resonance region. SIAM J. Appl.Math., 45:1039-53, 1985.
41. D. Colton and R. Kress. Eigenvalues of the far field operator for the Helmholtz equation in an absorbing medium. SIAM J. Appl. Math., 55:1742-1735, 1995.
42. D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scattering Theory. Springer-Verlag, New York, 2nd edition, 1998.
43. D. Colton and R. Kress. On the denseness of Herglotz wave functions and electromagnetic Herglotz pairs in Sobolev spaces. Math. Meth. Appl. Sci, 24:1289-1303,2001.
44. D. Colton, H. Haddar, and M. Piana. The linear sampling method in inverse electromagnetic scattering theory. Inverse Problems, 19:S105-S137, 2003.
45. D. Colton, M. Piana, and R. Potthast. A simple method using Morozov's discrepancy principle for solving inverse scattering problems. Inverse Problems, 13(6):1477-1493, 1997.
46. D. Flannery. The Square Root of Two. Copernicus Books, New York, 2006.
47. D. H. Bailey and D. J. Broadhurst. Parallel integer relation detection: techniques and applications. Mathematics of Computation, 70:1719-1736, 2000.
48. D. H. Bailey and J. M. Borwein. Effective error bounds in Euler-Maclaurin based quadrature schemes. http://crd.lbl.gov/~dhbailey/dhbpapers/em-error.pdf, 2005.
49. D. H. Bailey and J. M. Borwein. Experimental mathematics: examples, methods and implications. Notices Amer. Math. Soc., 52(5):502-514, 2005.
50. D. H. Bailey and J. M. Borwein. Experimental mathematics: recent developments and future outlook. In B. Engquist and W. Schmid, editors, Mathematics Unlimited-2001 and Beyond, volume 1, pp. 51-66. Springer-Verlag, 2000.
51. D. H. Bailey and J. M. Borwein. Highly parallel, high-precision numerical integration. http://crd.lbl.gov/~dhbailey/dhbpapers/quadparallel.pdf, 2005.
52. D. H. Bailey and J. M. Borwein. The Experimental Mathematics Website. http:// www.experimentalmath.info, 2006.
53. D. H. Bailey and R. E. Crandall. Random generators and normal numbers. Experimental Mathematics, 11:527-546, 2003.
54. D. H. Bailey and X. S. Li. A comparison of three high-precision quadrature schemes. http://crd.lbl.gov/~dhbailey/dhbpapers/quadrature.pdf, 2002.
55. D. H. Bailey, J. M. Borwein, V. Kapoor, and E. Weisstein. Ten problems in experimental mathematics. Amer. Math. Monthly, 113:481-509, 2006.
56. D. H. Bailey, J. M. Borwein, and D. M. Bradley. Experimental determination of Apéry-like identities for zeta(2n+2). Experimental Mathematics, 15(3):281-289,2006.
57. D. H. Bailey, J. M. Borwein, and R. E. Crandall. Box integrals. Journal of Computational and Applied Mathematics, to appear, available at http://crd.lbl.gov/~dhbailey/ dhbpapers/boxintegrals.pdf, 2006.
58. D. H. Bailey, J. M. Borwein, and R. E. Crandall. Integrals of the Ising class. J. Phys. A, 39:12271-12302, 2006.
59. D. H. Bailey, J. M. Borwein, and R. E. Crandall. On the Khintchine constant. Mathematics of Computation, 66:417-431, 1997.
60. D. H. Bailey, J. M. Borwein, and R. Girgensohn. Experimental evaluation of Euler sums. Experimental Mathematics, 3:17-30, 1994.
61. D. H. Bailey, P. B. Borwein, and S. Plouffe. On the rapid computation of various polylogarithmic constants. Mathematics of Computation, 66:903-913, 1997.
62. D. H. Bailey, Y. Hida, X. S. Li, and B. Thompson. ARPREC: An Arbitrary Precision Computation Package. http://crd.lbl.gov/~dhbailey/dhbpapers/arprec.pdf, 2002.
63. D.H.Bailey. High-Precision Software Directory. http://crd.lbl.gov/~dhbailey/ mpdist, 2006.
64. D. H. Bailey. Integer relation detection. Computing in Science and Engineering, 2(1):24-28, 2000.
65. D. H. Lehmer. Computer technology applied to the theory of numbers. In W. J. LeVeque, editor, Studies in Number Theory, pp. 117-151. Prentice-Hall, New York, 1969.
66. D. J. Broadhurst and D. Kreimer. Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Physics Letters, B383:403-412, 1997.
67. D. J. Broadhurst, J. A. Gracey, and D. Kreimer. Beyond the triangle and uniqueness relations: non-zeta counterterms at large N from positive knots. Zeitschrift für Physik C, 75:559-574, 1997.
68. D. J. Broadhurst. Massive 3-loop feynman diagrams reducible to sc* primitives of algebras of the sixth root of unity. http://lanl.arxiv.org/abs/hep-th/9803091, 1998.
69. D. Knuth. American Mathematical Monthly Problem 10832. American Mathematical Monthly, 107(9):863, November 2000.
70. D. Manna and V. Moll. A simple example of a new class of Landen transformations. American Mathematical Montly, 114:232-241, 2007.
71. D. Manna and V. Moll. Rational Landen transformations on R. To appear, Mathematics of Computation, 2007.
72. D. Manna, T. Amdeberham, and V. Moll. The 2-adic valuation of a sequence arising from a rational integral. Preprint, May 2006.
73. D. Overbye. An elusive proof and its elusive prover. The New York Times, 155, August 15, 2006.
74. D. R. Luke and R. Potthast. The no response test: a sampling method for inverse scattering problems. SIAM J. Appl. Math., 63(4):1292-1312, 2003.
75. D. R. Luke, J. V. Burke, and R. G. Lyon. Optical wavefront reconstruction: Theory and numerical methods. SIAM Rev., 44:169-224, 2002.
76. D. R. Luke. Image synthesis for inverse obstacle scattering using the eigenfunction expansion theorem. Computing, 75(2):181-196, 2005.
77. D. R. Luke. Multifrequency inverse obstacle scattering: the point source method and generalized filtered backprojeciton. Math. and Computers in Simulation, 66(4):297-314, 2004.
78. D. Zagier. Values of zeta function and their applications. In Proceedings of the First European Congress of Mathematics, volume 2, pp. 497-512, 1994.
79. D. Zeilberger. Identities in Search of Identity. Theoretical Computer Science, 117:23-38, 1993.
80. D'A. Thompson. On Growth and Form. Dover Publications, New York, 1992.
81. E. J. Borowski and J. M. Borwein. Dictionary of Mathematics, 2nd revised edition. HarperCollins, New York, 2002.
82. E. Kasner. The present problems of geometry. Bulletin of the American Mathematical Society, 11:285, 1905.
83. E. Landau. Handbuch der Lehre von der Verteilung der Primzahlen. 2 Bände. Chelsea Publishing Co., New York, 2nd edition, 1953. With an appendix by Paul T. Bateman.
84. E. R. Canfield, P. Erd?os, and C. Pomerance. On a problem of Oppenheim concerning "factorisatio numerorum". J. Number Theory, 17(1):1-28, 1983.
85. E. Regis. Who got Einstein's office? Addison Wesley, Reading, MA, 1988.
86. E. Simmt and B. Davis. Fractal cards: A space for exploration in geometry and discrete maths. Math. Teacher, 91:102-108, 1998.
87. E.W.Weisstein. Lucky number of Euler. MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/LuckyNumberofEuler.html, 2006.
88. E. Wirsing. Über die Zahlen, deren Primteiler einer gegebenen Menge angehören. Arch. Math., 7:263-272, 1956.
89. Eric Weisstein. Wolfram MathWorld Website. http://mathworld.wolfram.com/, 2006.
90. F. Beukers. A note on the irrationality of ? (2) and ? (3). Bull. London Math. Soc., 11:268-272, 1979.
91. F. Cakoni and D. Colton. On the mathematical basis of the linear sampling method. Georg. Math. J., 10:911-925, 2003.
92. F. De-Jun and Y. Wang. Bernoulli convolutions associated with certain non-Pisot numbers. Advances in Mathematics, 187:173-194, 2004.
93. G. Boese and W. Lutter. Stetige, nirgends differenzierbare Funktionen und nicht rektifizierbare Kurven. Mathematische Semesterberichte, 30:228-254, 1983.
94. G. Boros and V. Moll. A criterion for unimodality. Elec. Jour. Comb., 6:1-6, 1999.
95. G. Boros and V. Moll. An integral hidden in Gradshteyn and Ryzhik. Jour. Comp. Applied Math., 106:361-368, 1999.
96. G. Boros and V. Moll. Irresistible Integrals. Cambridge University Press, New York, 1st edition, 2004.
97. G. Boros and V. Moll. Landen transformation and the integration of rational functions. Math. Comp., 71:649-668, 2001.
98. G.Boros and V. Moll. The double square root, Jacobi polynomials and Ramanujan's master theorem. Jour. Comp. Applied Math., 130:337-344, 2001.
99. G. Boros, V. Moll, and J. Shallit. The 2-adic valuation of the coefficients of a polynomial. Scientia, 7:37-50, 2001.
100. G.E. Collins. Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In G. Goos and J. Hartmanis, editors, Automata Theory and Formal Languages 2nd GI Conference Kaiserslautern, May 20-23, 1975, Lecture Notes in Computer Science 33, pp. 134-183, Springer-Verlag, Berlin, 1975.
101. G. Eskin. The inverse scattering problem in two dimensions at fixed frequency. Comm. in Part. Diff. Eq., 26J. :1055-1090, 2001.
102. G. Faber. Über stetige Funktionen I. Mathematische Annalen, 66:81-93, 1908.
103. G. Faber. Über stetige Funktionen II. Mathematische Annalen, 69:372-443, 1910.
104. G.Forsythe. Pitfalls in computation, or why a math book isn't enough. Amer. Math. Monthly, 77:931-956, 1970.
105. G. Freud. Über trigonometrische Approximation und Fouriersche Reihen. Mathematische Zeitschrift, 78:252-262, 1962.
106. G. H. Hardy and S. Ramanujan. The normal number of prime factors of a number n. Quarterly Journal of Mathematics, 48:76-92, 1917.
107. G. H. Hardy, J. E. Littlewood, and G. Pólya. Hilbert's double series theorem; On Hilbert's inequality. In Inequalities, pp. 226-227; pp. 308-309. Cambridge University Press, Cambridge, UK, 2nd edition, 1988.
108. G. H. Hardy. Prolegomena to a chapter on inequalities. In Collected Papers, volume 2, pp. 471-489. Oxford University Press, Oxford, UK, 1967.
109. G. H. Hardy. Weierstrass's non-differentiable function. Transactions of the American Mathematical Society, 17:301-325, 1916.
110. G. J. Chaitin. G J Chaitin Home Page. http://www.umcs.maine.edu/~chaitin. Accessed August 20, 2006.
111. G. J. Chaitin. Thoughts on the Riemann hypothesis. Math. Intelligencer, 26(1):4-7, 2004.
112. G. J. O. Jameson. Hilbert's inequality and related results, available at http://www. maths.lancs.ac.uk/~jameson/hilbert.pdf, 2006.
113. G. Kolada. In Math, computers don't lie. Or do they? New York Times, April 6, 2004.
114. G. Lakoff and R. E. Nunez. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books, New York, 2001.
115. G. Polya. Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving (Combined Edition). Wiley & Sons, New York, 1981.
116. G. V. Balakin, V. F. Kolchin, and V. I. Khokhlov. Hypercycles in a random hypergraph. Diskret. Mat., 3(3):102-108, 1991.
117. G. de Rham. Sur certaines équations fonctionnelles. In École Polytechnique de l'Université de Lausanne, Centennaire 1853-1953, pp. 95-97. École Polytechnique de l'Université de Lausanne, 1953.
118. G. de Rham. Sur une courbe plane. Journal de Mathématiques Pures et Appliquées, 35:25-42, 1956.
119. H. Cohn. A short proof of the continued fraction expansion of e. Amer. Math. Monthly, 113:57-61, 2006.
120. H.H. Chan. Some new identities involving p, 1/p and 1/p2. Preprint, 2006.
121. H.J. Bremermann. The Encyclopaedia of Ignorance, chapter Complexity and Transcomputability, pp. 167-174. Pocket Books, New York, 1977.
122. H. M. Edwards. Essays in Constructive Mathematics. Springer, New York, 2005.
123. H. Montgomery. Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis. Amer. Math. Soc., Providence, RI, 1994. CBMS Regional Conference Series in Mathematics.
124. H. P. McKean and V. Moll. Elliptic Curves: Function Theory, Geometry, Arithmetic. Cambridge University Press, New York, NY, 1997.
125. H. Poincaré. L'intuition et la logique en mathématiques. In La Valeur de La Science, Chapter 1. Flammarion, Paris, 1905.
126. H. Poincaré. Mathematical invention. In Raymond Ayoub, editor, Musings of the Masters, pp. 20-30. MAA, 2004.
127. H. R. P. Ferguson, D. H. Bailey, and S. Arno. Analysis of PSLQ, An Integer Relation Finding Algorithm. Mathematics of Computation, 68:351-369, 1999.
128. H. W. Lenstra Jr. Primality testing with cyclotomic rings. Preprint, 2002.
129. H. Zimmer. Computational Problems, Methods, and Results in Algebraic Number Theory. Springer-Verlag, New York, 1972.
130. H.Wilf. Computer programs from the book A=B, and related programs. http://www. cis.upenn.edu/~wilf/progs.html. Accessed August 20, 2006.
131. I. Aliev. Siegel's lemma and sum-distinct sets. Preprint, 2005.
132. I. Damgard, P. Landrock, and C. Pomerance. Average case error estimates for the strong probable prime test. Mathematics of Computation, 61:177-194, 1993.
133. I. Daubechies and J. C. Lagarias. Two-scale difference equations. I. Existence and global regularity of solutions. SIAM Journal of Mathematical Analysis, 22:1338-1410, 1991.
134. I. Daubechies and J. C. Lagarias. Two-scale difference equations. II. Local regularity, infinite products of matrices, and fractals. SIAM Journal of Mathematical Analysis, 23:1031-1079, 1992.
135. I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math., 41:909-996, 1988.
136. I. Daubechies. Ten Lectures on Wavelets. SIAM, Philadelphia, 1992.
137. I. Doust, M. D. Hirschhorn, and J. Ho. Trigonometric identities, linear algebra, and computer algebra. Amer. Math. Monthly, 112(2):155-164, 2005.
138. I. E. Pritsker and R. S. Varga. The Szeg?o curve, zero distribution and weighted approximation. Trans. Amer. Math. Soc., 349:4085-4105, 1997.
139. I. Kotsireas and K. Karamanos. Exact computation of the bifurcation point b4 of the logistic map and the Bailey-Broadhurst conjectures. International Journal of Bifurcation and Chaos, 14:2417-2423, 2004.
140. I. Lakatos. Proofs and Refutations. Cambridge University Press, Cambridge, UK,1976.
141. I. Nemes, M. Petkovsek, H. Wilf, and D. Zeilberger. How to do MONTHLY problems with your computer. Amer. Math. Monthly, 104:505-519, 1997.
142. I. Pinelis. On l'Hopital type rules for monotonicity. Jour. Inequal. Pure Appl. Math., 7(2), Article 40 (electronic), 2006. Available at http://jipam.vu.edu.au/article.php? sid=657.
143. I. S. Gradshteyn and I.i M. Rhyzik. Table of Integrals, Series, and Products. Edited by A. Jeffrey and D. Zwillinger. Academic Press, New York, 6th edition, 2000.
144. I. Vardi. Integrals, an introduction to analytic number theory. Amer. Math. Mon., 95:308-315, 1988.
145. J. Alvarez, M. Amadis, G. Boros, D. Karp, V. Moll, and L. Rosales. An extension of a criterion for unimodality. Elec. Jour. Comb., 8:1-7, 2001.
146. J. Gerver. More on the differentiability of the Riemann function. American Journal of Mathematics, 93:33-41, 1971.
147. J. Gerver. The differentiability of the Riemann function at certain rational multiples of p. American Journal of Mathematics, 92:33-55, 1970.
148. J. Grabiner. Newton, Maclaurin, and the authority of Mathematics. Amer. Math. Monthly, VOL#:841-852, December 2004.
149. J. Guillera. About a new kind of Ramanujan-type series. Experimental Mathematics, 12(4):507-510, 2003.
150. J. Guillera. Generators of some Ramanujan formulas. The Ramanujan Journal, to appear.
151. J. Guillera. Some Binomial Series Obtained by the WZ-Method. Advances in Applied Mathematics, 29:599-603, 2002.
152. J. K. Brown. Solid tools for visualizing science. Science, VOl#: 1136-37, November 19, 2004.
153. J. Little. On the zeroes of two families of polynomials arising from certain rational integrals. Rocky Mountain Journal, 35(4):1205-1215, 2005.
154. J. M. Borwein and A. S. Lewis. Convex Analysis and Nonlinear Optimization. CMS Books. Springer-Verlag, New York, 2000.
155. J. M. Borwein and D. Broadhurst. Determination of rational Dirichlet-zeta invariants of hyperbolic manifolds and Feynman knots and links. http://arxiv.org/hep-th/ 9811173, 1998.
156. J. M. Borwein and D. H. Bailey. Mathematics by Experiment: Plausible Reasoning in the 21st century. A K Peters Ltd, Natick, MA, 2003.
157. J. M. Borwein and D. M. Bradley. Empirically determined Apéry-like formulae for ? (4n+3). Experimental Mathematics, 6(3):181-194, 1997.
158. J. M. Borwein and D. M. Bradley. On two fundamental identities for Euler sums. Preprint, February 2005.
159. J. M. Borwein and D. M. Bradley. Searching symbolically for Apéry-like formulae for values of the Riemann zeta function. ACM SIGSAM Bulletin of Algebraic and Symbolic Manipulation, 30(2):2-7, 1996
160. J. M. Borwein and D. M. Bradley. Thirty two Goldbach variations. Int. J. Num. Theory, 21:65-103, 2006.
161. J. M. Borwein and I. J. Zucker. Elliptic integral evaluation of the gamma function at rational values of small denominator. IMA Journal on Numerical Analysis, 12:519- 526, 1992.
162. J. M. Borwein and M. Chamberland. Integer powers of arcsin. D-DRIVE preprint 288, submitted April 2005.
163. J. M. Borwein and P. B. Borwein. Challenges for mathematical computing. Computing in Science & Engineering, 3:48-53, 2001.
164. J. M. Borwein and P. B. Borwein. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. CMS Series of Monographs and Advanced books in Mathematics, John Wiley, Hoboken, NJ, 1987.
165. J. M. Borwein and Q. J. Zhu. Techniques of Variational Analysis. CMS Books. Springer-Verlag, New York, 2005.
166. J. M. Borwein and R. Corless. Emerging tools for experimental mathematics. Amer. Math. Monthly, 106:889-909, 1999.
167. J. M. Borwein and R. Crandall. On the Ramanujan AGM fraction. Part II: the complex-parameter case. Experimental Mathematics, 13:287-296, 2004.
168. J. M. Borwein and R. Girgensohn. Addition theorems and binary expansions. Canadian Journal of Mathematics, 47:262-273, 1995.
169. J. M. Borwein and T. S. Stanway. Knowledge and community in mathematics. The Mathematical Intelligencer, 2005.
170. J. M. Borwein, D. H. Bailey, and R. Girgensohn. Experimentation in Mathematics: Computational Paths to Discovery. A K Peters Ltd, Natick, MA, 2003.
171. J. M. Borwein, D. J. Broadhurst, and J. Kamnitzer. Central binomial sums, multiple Clausen values, and zeta values. Experimental Mathematics, 10:25-41, 2001.
172. J. M. Borwein, D. M. Bradley, D. J. Broadhurst, and P. Lison?ek. Special values of multidimensional polylogarithms. Transactions of the American Mathematical Society, 353:907-941, 2001.
173. J. M. Borwein, D. M. Bradley, and R. E. Crandall. Computational strategies for the Riemann zeta function. Journal of Computational and Applied Mathematics, 121:247-296, 2000.
174. J. M. Borwein, I. J. Zucker, and J. Boersma. The evaluation of character Euler double sums. To appear in Ramanujan Journal, 15, 2008. In press, D-DRIVE preprint 260.
175. J. M. Borwein, P. B. Borwein, and D. A. Bailey. Ramanujan, modular equations and pi or how to compute a billion digits of pi. Amer. Math. Monthly, 96:201-219, 1989.
176. J. M. Borwein, R. Crandall, and G. Fee. On the Ramanujan AGM fraction. part I: the real-parameter case. Experimental Mathematics, 13:275-286, 2004.
177. J. M. Borwein. Jonathan Borwein's Homepage. http://users.cs.dal.ca/~jborwein, 2006.
178. J. M. Borwein. The 100 digit challenge: an extended review. Math Intelligencer, 27(4):40-48, 2005.
179. J. M. Steele. The Cauchy-Schwarz Master Class. MAA Problem Books Series. Cambridge University Press, Cambridge, UK, 2004.
180. J. P. Kahane. Lacunary Taylor and Fourier series. Bulletin of the American Mathematical Society, 70:199-213, 1964.
181. J. Schauder. Zur Theorie stetiger Abbildungen in Funktionalräumen. Mathematische Zeitschrift, 26:47-65, 1927.
182. J. Sondow. Jonathan Sondow's Home Page. http://home.earthlink.net/~jsondow/, 2006.
183. J.-L. Marichal and M. J. Mossinghoff. Slices, slabs, and sections of the unit hypercube. Preprint, available at http://xxx.lanl.gov/pdf/math.MG/0607715, 2006.
184. Ji Chung-Gang. A simple proof of a curious congruence by Zhao. Proceedings of the American Mathematical Society, 133:3469-3472, 2005.
185. Johann Wolfgang Goethe. Maxims and Reflections. Penguin Classics, London, 1998. Elizabeth Stopp, trans.
186. Jun Wu. On the Lévy constant for quadratic irrationals. Proceedings of the American Mathematical Society, 134(6):1631-1634, 2006. Electronically published on December 15, 2005.
187. K. Devlin. Mathematics: The Science of Patterns. Owl Books, New York, 1996.
188. K. E. Atkinson. An Introduction to Numerical Analysis. John Wiley and Sons, Hoboken, NJ, 1989.
189. K. Gödel. Some basic theorems on the foundations. In Collected Works, Vol. III. Unpublished essays and lectures. Oxford University Press, New York, 1995.
190. K. Weierstraß. Über continuirliche Functionen eines reellen Arguments, die für keinenWerth des letzteren einen bestimmten Differentialquotienten besitzen. (Gelesen in der Königl. Akademie derWissenschaften am 18. Juli 1872). In Mathematische Werke von Karl Weierstraß, vol. 2, pp. 71-74. Mayer & Müller, Berlin, 1895.
191. K.Wegschaider. Computer generated proofs of binomial multi-sum identities. Master Thesis, RISC-Linz, May 1997.
192. L. Comtet. Advanced Combinatorics: The Art of Finite and Infinite Expansion. D. Reidel Publishing, Dordrecht, Holland, 1974.
193. L. Lewin. Dilogarithms and Associated Functions. Macdonald, London, 1958.
194. L. Lewin. Polylogarithms and Associated Functions. North Holland, New York,1981.
195. L. Monier. Evaluation and comparison of two efficient probabilistic primality testing algorithms. Theoretical Computer Science, 12:97-108, 1980.
196. M. A. Morrison and John J. Brillhart. A method of factoring and the factorization of F7. Math. Comp., 29:183-205, 1975. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday.
197. M. Agrawal, N. Kayal, and N. Saxena. PRIMES is in P. Annals of Mathematics, 160:781-793, 2004.
198. M. Chamberland and V. Moll. Dynamics of the degree six Landen transformation. Discrete and Dynamical Systems, 2006. Preprint.
199. M. D. Hirschhorn. A difficult limit. Australian Math. Gazette, 31:125-127, 2004.
200. M. Dummett. Frege: Philosophy of Language. Harvard University Press, Cambridge, MA, 1973.
201. M. F. Barnsley. Fractal functions and interpolation. Constructive Approximation, 2:303-329, 1986.
202. M. G. Beumer. Some special integrals. Amer. Math. Monthly, 68:645-647, 1961
203. M. Hata. Improvement in the irrationality measures of p and p2. Proceedings of the Japanese Academy, Ser. A Mathematical Sciences, 68:283-286, 1992.
204. M. Hata. On Weierstrass's non-differentiable function. Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique, 307:119-123, 1988.
205. M. Hata. Singularities of the Weierstrass type functions. Journal d'Analyse Mathématique, 51:62-90, 1988.
206. M. Hoffman. References on multiple zeta values and Euler sums. http://www.usna. edu/Users/math/meh/biblio.html, 2006.
207. M. Ikehata. Enclosing a polygonal cavity in a two-dimensional bounded domain from Cauchy data. Inverse Problems, 15:1231-1241, 1999.
208. M. Kauers and P. Paule. A computer proof of Moll's log-concavity conjecture. Preprint, May 2006.
209. M. Kauers. SumCracker-A package for manipulating symbolic sums and related objects. Technical report 2005-21, SFB F013, Johannes Kepler Universität, 2005 (submitted).
210. M. Koecher. Letter (German). Mathematics Intelligencer, 2(2):62-64, 1979/80.
211. M. Petkovsek, H. Wilf, and D. Zeilberger. A=B. A K Peters, Ltd., Natick, MA, 1996.
212. M. Planck. Scientific Autobiography and Other Papers. Philosophical Library, New York, 1949. Trans. F. Gaynor.
213. M. Rabin. Probabilistic algorithms for testing primality. Journal of Number Theory, 12:128-138, 1980.
214. M. Reed and B. Simon. Methods of Modern Mathematical Physics, volume III. Scattering Theory. Academic Press, New York, 1979.
215. M. Trott. Mathematica Guidebooks. http://www.mathematicaguidebooks.org/ downloads/N 2 01 Evaluated.nb. v2006.
216. N. J. Calkin. Dependent sets of constant weight binary vectors. Combin. Probab. Comput., 6(3):263-271, 1997.
217. Neil Sloane. The On-Line Encyclopedia of Integer Sequences. http://www.research. att.com/~njas/sequences/index.html, 2006.
218. Numerical Algorithms Group. Numerical Algorithms Group Website. http://www. nag.co.uk, 2006.
219. O. Espinosa and V. Moll. On some definite integrals involving the Hurwitz zeta function. Part 1. The Ramanujan Journal, 6:159-188, 2002.
220. O. Espinosa and V. Moll. On some definite integrals involving the Hurwitz zeta function. Part 2. The Ramanujan Journal, 6:449-468, 2002.
221. O. Espinosa and V. Moll. The evaluation of tonheim double sums. Part 1. Journal of Number Theory, 116:200-229, 2006.
222. P. Borwein. An Efficient Algorithm for the Riemann Zeta Function. http://www. cecm.sfu.ca/preprints/1995pp.html, 1995.
223. P. C. Berry, A. D. Falkoff, and K. E. Iverson. Using the computer to compute: a direct but neglected approach to teaching Mathematics. Technical Report 320-2988, IBM New York Scientific Center, May 1970.
224. P. Erdös and M. Kac. The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math., 62:738-742, 1940.
225. P. Ernest. Social Constructivism As a Philosophy of Mathematics. State University of New York Press, Albany, NY, 1998.
226. P. Monk. Finite Element Methods for Maxwell's Equations. Clarendon Press, Oxford, 2003.
227. P. Preuss. Mathematics: The Statue in the Stone. Berkeley Lab Research Review, 23:12-13, Fall 2000.
228. P. Winkler. Mathematical Puzzles: A Connosseur's Collection. A K Peters, Natick, MA, 2004.
229. P. du Bois-Reymond. Versuch einer Classification der willkürlichen Functionen reeller Argumente nach ihren Aenderungen in den kleinsten Intervallen. Journal für die reine und angewandte Mathematik (Crelle's Journal), 79:21-37, 1875.
230. Proof and beauty. The Economist, 375(8240), April 2, 2005.
231. R. C. Lewontin. In the beginning was the word. Science (Human Genome Issue), 291:1263-1264, February 16, 2001.
232. R. D. Mauldin and K. Simon. The equivalence of some Bernoulli convolutions to Lebesgue measure. Proceedings of the American Mathematical Society, 126:2733-2736, 1998.
233. R. E. Crandall and C. Pomerance. Prime Numbers: A Computational Perspective. Springer-Verlag, Heidelberg, 2001.
234. R. E. Crandall and J. P. Buhler. On the evaluation of Euler sums. Experimental Mathematics, 3:275-284, 1994.
235. R. E. Crandall and J. S. Papadopoulos. On the implementation of the AKS-class primality tests. Technical report, Advanced Computation Group, Apple Computer, 2003.
236. R. Girgensohn. Functional equations and nowhere differentiable functions. Aequationes Mathematicae, 46:243-256, 1993.
237. R. Girgensohn. Nowhere differentiable solutions of a system of functional equations. Aequationes Mathematicae, 47:89-99, 1994.
238. R. Graham, D. Knuth, and O. Patashnik. Concrete Mathematics. Addison Wesley, Boston, 2nd edition, 1994.
239. R. Hersh. Fresh breezes in the philosophy of mathematics. Amer. Math. Monthly, 102:589-594, August 1995.
240. R. Hersh. What is Mathematics Really? Oxford University Press, Oxford, UK,1999.
241. R. Kershner and A. Wintner. On symmetric Bernoulli convolutions. American Journal of Mathematics, 57:541-548, 1935.
242. R. Kress. Linear Integral Equations, volume 82 of Applied Mathematical Sciences. Springer Verlag, New York, 2nd edition, 1999.
243. R. L. Devaney, N. Goodmann, and A. Vajiac. Fractalina. http://math.bu.edu/DYSYS/ applets/fractalina.html. Accessed August 20, 2006.
244. R. Larson, B. Edwards, and D. Heyd. Calculus of a Single Variable. Houghton Mifflin Company, Boston-New York, 6th edition, 1998.
245. R. Leis. Initial Boundary Value Problems in Mathematical Physics. B.G. Teubner, Stuttgart, 1986.
246. R. Novikov. Multidimensional inverse spectral problems for the equation -f + (v(x)-eu(x))f. Trans. Func. Anal. and its Appl., 22:263-272, 1988.
247. R. Novikov. The inverse scattering problem on a fixed energy for the twodimensional Schrödinger operator. J. Funct. Anal., 1992:409-463, 1992.
248. R. Potthast, J. Sylvester, and S. Kusiak. A 'range test' for determining scatterers with unknown physical properties. Inverse Problems, 19(3):533-547, 2003.
249. R. Potthast. Point Sources and Multipoles in Inverse Scattering Theory. Chapman & Hall, London, 2001.
250. R. Preston. The Mountains of Pi. New Yorker, March 2, 1992.
251. R. Salem. A remarkable class of algebraic integers, proof of a conjecture by Vijayaraghavan. Duke Mathematical Journal, 11:103-108, 1944.
252. Random House Webster's Unabridged Dictionary. Random House, New York, 1999.
253. S. M. Zemyan. On the zeroes of the nth partial sum of the exponential series. American Mathematical Monthly, 110(10):891-909, 2005,.
254. S. R. Finch. Mathematical Constants. Cambridge University Press, Cambridge, UK, 2003.
255. S. Robinson. New method said to solve key problem in Math. New York Times, August 8, 2002.
256. S. Wagon, F. Bornemann, D. Laurie, and J. Waldvogel. The SIAM 100 Digit Challenge: A Study in High-Accuracy Numerical Computing. SIAM, Philadelphia, 2004.
257. S. Wagon. Fourteen proofs of a result about tiling a rectangle. American Mathematical Monthly, 94(7):601-617, 1987.
258. T. Arens. Why linear sampling works. Inverse Problems, 20:163-173, 2004.
259. T. Chow. What is a closed-form number? Amer. Math. Monthly, 106:440-448, 1999.
260. T. Hales. A proof of the Kepler conjecture. Ann. Math., 162:1065-1185, 2005.
261. T. J. Osler. A proof of the continued fraction expansion of e-1/m. Amer. Math. Monthly, 113:62-65, 2006.
262. T. S. Kuhn. The Structure of Scientific Revolutions. University of Chicago Press, Chicago, 3rd edition, 1996.
263. T. Takagi. A simple example of the continuous function without derivative. Proceedings of the Physico-Mathematical Society of Japan (Ser. II), 1:176-177, 1903.
264. T. Tymoczko. Computer, proofs and mathematics: a philosophical investigation of the four-color problem. Math. Magazine, 53:131-138, 1980.
265. T. Tymoczko. The four-color problem and its philosophical significance. J. Philos., 76:57-83, 1979.
266. The GNU Project. GMP. http://swox.com/gmp/, 2006.
267. V. F. Kolchin and V. I. Khokhlov. On the number of cycles in a random nonequiprobable graph. Diskret. Mat., 2(3):137-145, 1990.
268. VeriSign announces Valentine's Day-related online shopping activity. http://www.verisign.com/verisign-inc/news-and-events/news-archive/ us-news-2005/page 028504.html, February 2005.
269. W. Grobner and N. Hofreiter. Integraltafel, volume 1. Springer-Verlag, Vienna, 1st edition, 1949.
270. W. Grobner and N. Hofreiter. Integraltafel, volume 2. Springer-Verlag, Vienna, 1st edition, 1950.