Qualitative computing: A computational journey into nonlinearity

Qualitative Computing: A Computational Journey into Nonlinearity

Volumn , Issue , 2012, Pages 1-582

  Chatelin, Françoise ^a  

^a UNIVERSITÉ DE TOULOUSE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATION THEORY; VECTOR SPACES;

CLASSICAL LINEAR LOGIC; CREATIVE POTENTIALS; HIGH TECHNOLOGY INDUSTRIES; HYPERCOMPUTATION; LINEAR VECTOR SPACE; NONLINEAR COMPUTATIONS; QUADRATIC ALGEBRAS; QUALITATIVE COMPUTING;

COMPUTER CIRCUITS;

EID: 84995538154     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/7904     Document Type: Book

References (204)

1. V. N. Agafonov (1968), Normal sequences and finite automata, Soviet Math. Doklady 9, 324-325.
2. M. Ahmadnasab (2007). Homotopic Deviation theory: a qualitative study. Ph. D. thesis, Univ. Toulouse, Cerfacs Rep. TH/PA/07/120.
3. M. Akian, R. Bapat, and S. Gaubert (2004), Perturbation of eigenvalues of matrix pencils and optimal assignment problem, C.R. Acad. Sci. Paris Ser. I339, 103-108.
4. A. Albert (1942), Quadratic forms permitting composition, Annals of Math. 43, 161-177.
5. D. Allcock (1999), Ideals in the integral octaves. J. Alg. 220, 396-400.
6. U. Alon (2007), Simplicity in biology, Nature 446, 20 March.
7. M. R. Antognazza (2009) Leibniz, an intellectual biography, Cambridge Univ. Press, New York.
8. A. Auric (1915) Suite illimitee de nombres premiers. L’intermédiaire des mathématiciens. 22, 252-253.
9. R. Ayoub (1974) Euler and the Zeta function, Amer. Math. Month 81, 1067-1086.
10. G. Bachet de Meziriac (1612), Probiemes Plaisants et Delectables qui se font par les nombres, A. Blanchard, Paris, 1993 (reprint edition).
11. G. Bachet de Meziriac (undated) Elementorum arithmeticorum libri XIII, Latin manuscript, Bibliotheque Institut de France. J. Baez (2002), The octonions. Bull. AMS, 39, 145-205.
12. Y. Bahri and A. Ilahi (2011), A note on Homotopic Deviation versus Perturbation, Lin. Alg. Appl., 436, 2353-2358, 2012.
13. K. Barner (2001) How old did Fermat become?, Int. J. for Hist. and Ethics of Natural Sciences 9, 209-228.
14. F. S. V. Bazan (2004) Private communication, Cerfacs.
15. F. Benford (1938), The law of anomalous numbers, Proc. Amer. Phil. 78, 551-572.
16. A. Berthoz (2009) La simplexite, Odile Jacob, Paris.
17. D. Biss, D. Christensen, D. Dugger, and D. Isaksen (2007) Large annihilators in Cayley-Dickson algebras II, Bol. Soc. Mat. Mex. 13, 269-292.
18. D. Biss, D. Dugger, and D. Isaksen (2008) Large annihilators in Cayley-Dickson algebras, Comm. Alg. 36, 632-664.
19. D. Biss, D. Christensen, D. Dugger, and D. Isaksen (2009) Eigentheory of Cayley-Dickson algebras, Forum Matem 21, 833-852.
20. D. Blair (2000) Inversion theory and conformal mapping, AMS Student Math. Lib., Providence, R. I.
21. E. Borel (1909) Les probabilites denombrables et leurs applications arithmetiques, Rend. Circ. Mat. Palermo 27, 247-271.
22. R. B. Brown (1967) On generalized Cayley-Dickson algebras, Pacif. J. Math. 20, 415-422.
23. J. Bustoz, A. Feldstein, R. Goodman and S. Linnainmaa (1979) Improved trailing digits Estimates applied to optimal computer arithmetic, J. ACM 26, 716-730.
24. G. Cardano (1545), Ars magna or the rules of algebra, Dover Publ., New York (translated from latin) (1968).
25. E. Cartan (1914) Les groupes reels simples finis et continus., Ann. Scient. Ecole Norm. Sup. 31, 255-262.
26. R. Cawagas et al. (2009) The basic subalgebra structure of the Cayley-Dickson algebra of dimension 32, arXiv:0907.2047 v2 [math.RA] 27 July 2009.
27. B. Chabat (1990), Introduction a l’analyse complexe, Vol. 1, Mir, Moscou.
28. G. J. Chaitin (1977) Algorithmic information theory, IBM J. Res. Dev 21, 350-359.
29. G. J. Chaitin (1987). Algorithmic information theory. Cambridge University Press, Cambridge UK.
30. F. Chaitin-Chatelin (1996) Is finite precision arithmetic useful for Physics?, Journ. of Universal Comp. Science 2, 380-395.
31. F. Chaitin-Chatelin (1997), Is nonnormality a serious computational difficulty in practice?, in The quality of Numerical Software (R. F. Boisvert ed.) pp. 300-314, Chapmann and Hall, CRC Press, LLC, Boca Raton, USA.
32. F. Chaitin-Chatelin (1999), The computing power of Geometry, in Numerical Analysis 1999 (D. F. Griffiths and G. A. Watson, eds.) pp. 83-92 CRC Press LLC, Boca Raton, USA.
33. F. Chaitin-Chatelin(2005), The dynamics of matrix coupling with an application to Krylov methods, in Proceedings of NAA2004, Rousse: Bulgaria, Li Z. et al. (eds), pp. 14-24, Lecture Notes in Computer Science, (volume 3401) Springer Verlag, Berlin.
34. F. Chaitin-Chatelin (2007). Computing beyond classical logic: SVD computation in nonassociative Dickson algebras, in Randomness &Complexity, from Leibniz to Chaitin (C. Calude, ed.) pp. 13-23, World Scientific, Singapore.
35. F. Chaitin-Chatelin, V. Fraysse and T. Braconnier (1995), Computations in the neighbourhood of algebraic singularities, Numer. Funct. Anal. and, Optimization 16, 287-302.
36. F. Chaitin-Chatelin and V. Fraysse (1996). Lectures on Finite Precision Computations. SIAM, Philadelphia, USA.
37. F. Chaitin-Chatelin and S. Gratton (1996) Convergence in finite precision of the successive iteration method under high nonnormality, BIT 36, 455-469.
38. F. Chaitin-Chatelin and S. Gratton (2000) On the condition numbers associated with the polar factorization of a matrix, NLAAp. 7, 337-354.
39. F. Chaitin-Chatelin, A. Harrabi and A. Ilahi (2000), About Holder condition numbers and the stratification diagram for defective eigenvalues, in IMACS Journal in Mathematics of Computation, special issue, (E.Houstis, R. Boisvert, and W. Dyksen eds), 54, pp. 397-402.
40. F. Chaitin-Chatelin, T. Meskauskas (2001), Computation with hypercomplex numbers. J. of Nonlinear Analysis 47, 3391-3400.
41. F. Chaitin-Chatelin, V. Toumazou, and E. Traviesas (2000). Accuracy assessment for eigencomputations: variety of backward errors and pseudospectra. Linear Algebra and its Applications 309, 73-83.
42. F. Chaitin-Chatelin, E. Traviesas, and L. Plantie (2000). Understanding Krylov methods in finite precision, in Proceedings of NAA2000, Rousse: Bulgaria, L. Vulkov, J. Wasviewski, P. Yalamov (eds), pp. 187-197, Lecture Notes in Computer Science, volume 1988, Springer Verlag, Berlin.
43. F. Chaitin-Chatelin and E. Traviesas-Cassan (2005). Qualitative Computing. Chapter 5, in Accuracy and Reliability in Scientific Computing, (B. Einarsson ed.), pp. 77-92, SIAM, Philadelphia, USA.
44. F. Chaitin-Chatelin and M. B. Van Gijzen (2006). Analysis of parameterized quadratic eigenvalue problems in Computational Acoustics with Homotopic Deviation theory. NLAAp. 13, 487-512.
45. F. Chatelin (1983). Spectral Approximation of Linear Operators. Academic Press, New York republ. SIAM (2011).
46. F. Chatelin (1987). Chinese translation of Chatelin (1983), Tianjin Univ. Press, Tianjin, PR of China.
47. F. Chatelin (1988). Valeurs propres de matrices. Masson, Paris.
48. F. Chatelin (1993). Eigenvalues of matrices. Wiley, Chichester republ. SIAM (2012).
49. F. Chatelin (2010a) Fourth printing of the japanese translation of Chatelin (1993). Springer Verlag, Tokyo.
50. F. Chatelin (2010b) Numerical information processing under the global rule expressed by the Euler-Riemann Z function defined in the complex plane, Chaos 20, 037 104, p. 1-6.
51. F. Chatelin (2011a), A computational journey into the mind, Nat. Comput., Special Issue: UC2010 (J. Timmis and K. Morita guest eds.) pp. 1-13, doi:10.1007/s11047-011-9269-6; 11, 67-79 (2012).
52. F. Chatelin (2011b), Polymorphic Information Processing in weaving computation: An approach through cloth geometry, Cerfacs Tech. Rep. TR/PA/11/27.
53. F. Chatelin (2011c), On relating computation, Cerfacs Tech. Rep. TR/PA/11/37.
54. F. Chatelin (2011d), About weaving computation in 2 and 3 real dimensions, Cerfacs Tech. Rep. TR/PA/11/119.
55. F. Chatelin (2012), Computation in Mind, chapter in Philosophy of Mind (N. Gotsiridze-Columbus ed.), Nova Science publ., Hauppage NY.
56. F. Chatelin and T. Braconnier (1993), About the qualitative computation of Jordan forms, ZAMM. J. Angew. Math. Mech. 74 (2), pp. 105-113.
57. F. Chatelin and M. Ahmadnasab (2009). Some contributions of Homotopic Deviation to the theory of matrix pencils, in Proceedings of NAA 2008, Lozenetz, Bulgaria, S. Margenov et al. (eds), pp. 13-19, Lecture Notes in Computer Science volume 5434, Springer Verlag, Berlin.
58. K. Ciesielski and Z. Pogoda (2008), On ordering the natural numbers or the Sharkovski theorem, Am. Math. Month. 115, 159-165.
59. J. B. Conrey (2003) The Riemann hypothesis, Notices AMS 50, 341-353.
60. J. H. Conway and R. K. Guy (1996), The Book of Numbers. Springer-Verlag, New York.
61. J. H. Conway and D. A. Smith (2003), On Quaternions and Octonions. A. K. Peters, Natick.
62. H.S.M. Coxeter (1946) Integral Cayley numbers, Duke Math. J. 13, 561-578.
63. M.J. Crowe (1994) History of vector analysis, Dover Publ. Inc., New York.
64. T. Dantzig (1954) Number. The language of science, 4th edition, revised and augmented, The Free Press, Macmillan Publ., New York.
65. M. Davis, H. Putman, J. Robinson (1961) The decision problem for exponential diophantine equations, Annals of Math. 74, 425-436.
66. C. F. Degen (1818) Adumbratio demonstrationis theorematis arithmetici maxime universalis, Mémoires Acad. impériale des Sc. de Saint-Petersbourg 8, 207-219.
67. J. W. Demmel (1987) On condition numbers and the distance to the nearest ill-posed problem, Numer. Math. 51, 251-289.
68. L. E. Dickson (1912). Linear Algebras, Trans. AMS 13, 59-73.
69. L. E. Dickson (1919), On quaternions and their generalization and the history of the 8 square theorem, Annals of Math. 20, 155-171, 1919.
70. L. E. Dickson (1923), A new simple theory of hypercomplex integers. J. de Math. Pures et Appl. 2, 281-326.
71. L. E. Dickson (2005) History of the theory of numbers, Vol. 1 (Divisibility and primality) Dover, Mineola, NY. (republication of the 1919 edition).
72. J. Dieudonne (1987) Pour l’honneur de l’esprit humain. Les mathematiques d’aujourd’hui, Hachette, Paris.
73. H. L. Dreyfus (1979) What computers can’t do. The limits of Artificial Intelligence, Revised edition, Harper Colophon Books, New York.
74. J. Duda (2008a) Complex base numeral systems, ArXiv: 0712.1309 v3 [math. DS] 23, Feb. 2008.
75. J. Duda (2008b) Number systems using a complex base, The Wolfram Demonstration Project, http://demonstration.wolfram.com.
76. P. Eakin and A. Sathaye (1990) On automorphisms and derivations of Cayley-Dickson algebras J. Algebra 129, 263-278.
77. U. Eco (1995) The search for the perfect language (The making of Europe), Blackwell Publ., Oxford, UK.
78. H. M. Edwards (2001) Riemann’s zeta function, Dover Public., Mineola.
79. S. Eilenberg and I. Niven (1944) Fundamental theorem of algebra for quaternions, Bull. AMS 50, 246-248.
80. L. Euler (1749) Remarques sur un beau rapport entre les series des puissances tant directes qu’inverses, Mémoires de l’Acad. Sc. Berlin 17, 83-106, 1768.
81. A. Feldstein and R. Goodman (1976) Convergence estimates for the distribution of trailing digits, J. ACM 23, 287-297.
82. W. Feller (1971) An Introduction to Probability Theory and its Applications, 2nd edition, Vol. 2, John Wiley and Sons, UK.
83. P. Fermat (1999), Oeuvres de Pierre de Fermat. I. La theorie des nombres. Librairie Albert Blanchard, Paris.
84. G. Ferraro (2002) Convergence and formal manipulation of series from the origin of calculus to about 1730, Annals of Science 59, 179-199.
85. N. J. Fine (1947) Binomial coefficients modulo a prime, Amer. Math. Month. 54, 589-592.
86. L. R. Ford (1925) On the closeness of approach of complex rational fractions to a complex irrational number, Trans. Amer. Math. Soc. 27, 146-154.
87. J. B. J. Fourier (1807) Memoire sur la propagation de la chaleur dans les corps solides, Acad. Sciences, Institut National, Paris.
88. F. R. Gantmacher (1960). The theory of matrices. Vols. I and II, Chelsea Publishing Company, New York.
89. A. Gardiner (2002) Understanding infinity: the mathematics of infinite processes. Dover Public., Mineola.
90. L. Gilder (2008), The age of entanglement, Alfred A. Knopf, New York.
91. J. Gray (2008) Plato’s ghost. The modernist transformation of mathematics, Princeton Univer. Press, Princeton and Oxford.
92. K. Godel (1972), Some remarks on the undecidability results, pp. 305-306 in Collected works II, Publications 1938-1974, Oxford Univ. Press, Oxford (1990)
93. G. Godefroy (1997) L’aventure des nombres. Editions Odile Jacob, Paris.
94. J. Guckenheimer (1980) One dimensional dynamics, Ann. NY Acad. Sci. 357, 343-347.
95. J. Hadamard (1945), An essay on the Psychology of Invention in the Mathematical Field. Princeton U. Press, New York.
96. R. W. Hamilton (1848) Note respecting the researches of John T. Graves, Esq. Trans, Royal Irish Acad. 21, 338-341.
97. R. W. Hamming (1970) On the distribution of numbers, Bell System Tech. J. 49, 1609-1625.
98. R. W. Hamming (1993) Digital filters, 3rd edition, Dover, Mineola, NY.
99. G. H. Hardy (1914), Sur les zeros de la fonction Z(s) de Riemann, C. R. Acad. Sc. Paris pp. 1012-1014.
100. G. H. Hardy and E. M. Wright (1979), An Introduction to the Theory of Numbers, fifth edition. Oxford Science Publ., Clarendon Press, Oxford.
101. F. Hausdorff (1919) Dimension und ausseres Mass, Math. Ann. 79, 157-179.
102. C. Hermite (1864) Sur un nouveau developpement en serie des fonctions, C. R. Acad. Sc. Paris 58, 93-100, 266-273.
103. R. A. Horn and C. Johnson (1985). Matrix analysis, Cambridge Univ. Press, Cambridge, UK.
104. A. Hurwitz (1919), Vorlesungen über die Zahlentheorie der Quaternio-nen. Julius Springer, Berlin.
105. A. Ilahi (1998) Validation du calcul sur ordinateur: application de la théorie des singularités. Ph. D. Thesis, Univ. Toulouse, Cerfacs Tech. Rep. TH/PA/98/31. A. Ilahi (2009) Private communication (e-mail).
106. N. Jacobson (1939) Cayley numbers and normal simple Lie algebras of type G, Duke Math. J. 5, 775-783.
107. N. Jacobson (1958) Composition algebras and their automorphisms, Rend. Circ. Mat. Palermo 7(Serie 2), 55-80.
108. J. Jacques (1992), La molécule et son double Hachette, Paris.
109. M. V. Jakobson (1981) Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Commun. Math. Phys. 81, 39-88.
110. J. P. Jones and Y. V. Matijasevich (1984) Register machine proof of the theorem on exponential diophantine representation of enumerable sets, J. Symbolic Logic 49, 818-829.
111. C. Jordan (1874) Memoire sur les formes bilineaires, J. Math. Pures Appl. 19(Serie 2), 35-54.
112. C. G. Jung and W. Pauli (1955), The interpretation of Nature and the Psyche, Random House, New York.
113. R. Kaplan (1999) The nothing that is. Oxford University Press, Oxford, UK. T. Kato (1984) Perturbation Theory for Linear Operators, Second revised edition, Springer-Verlag, Berlin.
114. S.H. Khalil and Paul Yiu (1997) The Cayley-Dickson algebras, a theorem of A. Hurwitz, and quaternions, Bull. Soc. Sciences et Lettres, Lodz 47, 117-169.
115. I. Kleiner (1998) From numbers to rings: the early history of ring theory, Elem. Math. 53, 18-35.
116. M. Kline (1983) Euler and infinite series, Math. Mag. 56, 307-314.
117. D. Knuth (1981) The art of computer programming, Vol. 2 (2nd edition), Addison-Wesley, Reading, US.
118. B. Kosko (1990) Fuzziness vs. Probability, Int. J. General Systems 17, 211-240. S. G. Kranz (2008) Complex variables. Chapman &Hall, CRC, Boca Raton.
119. E. Kummer (1852) Über die Ergänzungssatze zu den allgemeinen Re- ciprocitätsgesetzen, J. Reine Angew. Math. 44, 93-146.
120. J. L. Lagrange (1770) Demonstration d’un theorème d’Arithmetique, Nouveaux Mémoires de l’Acad. royale Sc. et Belles-Lettres de Berlin, annee 1770, 189-201.
121. J. Lambek (1995) If Hamilton had prevailed: quaternions in physics, Math. Intell. 17, 7-15.
122. P. J. C. Lamont (1963a) Arithmetics in Cayley’s algebra, Glasgow Mathem. J. 6, 99-106.
123. P. J. C. Lamont (1963b) Ideals in Cayley’s algebra, Indag. Math. 25, 394-400. K. Lamotke (ed.) (1998), Les nombres, Vuibert, Paris.
124. E. Landau (1906-1907) Euler und die Funtionalgleichung der Riemannschen Zetafunktion, Bibl. Math. 7, 69-79.
125. J. Langou (2009) Translation and modern interpretation of Laplace’s Theorie analytique des probabilites, pages 505-512, 516-520. arXiv:0907.4695v1, 27 July 2009.
126. P. S. Laplace (1820) Sur l’application du calcul des probabilites a la philosophie naturelle; in Theorie analytique des probabilites, Premier Supplement, pp. 505-512, 516-520, 3eme edition, Mme Vve Courcier, Paris.
127. D. Leborgne (1991) Calcul differentiel complexe “Que sais-je?”, PUF, Paris. P. Levy (1939) L’addition des variables aleatoires sur une circonference, Bull.Soc. Math. France 67, 1-41.
128. P. Levy (1954) Theorie de l’addition des variables aleatoires, 2eme edition, Gauthier-Villars, Paris (reprint edition 2003, Gabay, Paris).
129. A. M. Legendre (1830) Theorie des nombres, 3eme edition, Firmin-Didot, Paris.
130. T. Y. Li and J. A. Yorke (1975) Period three implies chaos, Amer. Math. Month. 82, 985-992.
131. V. B. Lidskii (1965). Perturbation theory of non-conjugate operators. U.S.S.R Comput. Math. Phys. 1, 73-85.
132. E. N. Lorenz (1980) Noisy periodicity and reverse bifurcation, Annals N. Y. Acad. Sc. 357, 282-291.
133. E. Lucas (1878a) Theoremes d’Arithmetique, Atti Reale Acad. Scienze Torino 13, 271-284.
134. E. Lucas (1878b) Sur les congruences des nombres euleriens et les coefficients differentiels des fonctions trigonometriques suivant un module premier, Bull. Soc. Math. France 6, 49-54.
135. E. Lucas (1891) Theorie des nombres Gauthier-Villars, Paris (reprint edition 1991, Gabay, Paris).
136. U. Lumiste (2007) Tarski’s system of geometry and betweenness geometry with the group of movements, Proc. Estonian Acad. Sc. Phys. Math. 56, 252-263.
137. K. Mahler (1943) On ideals in the Cayley-Dickson algebra Proc. Roy. Irish Acad. 48, 123-133.
138. B. Mandelbrot (1980) Fractal aspects of the iteration of z ^ Az(1-z) for complex A and z, Annals NY Acad. Sc. 357, 249-259.
139. J. Marchant (2008), Decoding the heavens William Heinemann, London.
140. J. C. Maxwell (1870) On the application of the ideas of the calculus of quaternions to electromagnetic phenomena, in Vol. 2, Maxwell’s Scientific Papers (Sir W. Niven, ed.), pp. 570-576, Dover Publ. New York (1890).
141. J. C. Maxwell (1871) Remarks on the mathematical classification of physical quantities, Proc. London Math. Soc. 3, 224-233.
142. S. Mazzanti (2007) Exponential diophantine representation of relations: two simple constructions, Tech. Rep. DADI/WP-1307, Univ. IUAV di Venezia. K. McCrimmon(1985a) Nonassociative algebras with scalar involution, Pacif. J. Math 116, 85-109.
143. K. Mc Crimmon(1985b) Derivations and Cayley derivations of generalized Cayley-Dickson algebras, Pacif. J. Math. 117, 163-182.
144. M. Minsky (1961). Recursive unsolvability of Post’s problem of “tag” and other topics in the theory of Turing machines, Annals of Math. 72, 437-455.
145. R. Montgomery (1991) How much does the rigid body rotate? A Berry’s phase from the 18th century, Amer. J. Phys. 59, 394-398.
146. G. Moreno (1998), The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Matem. Mex. 4, 13-27.
147. G. Moreno (2001), The higher dimensional Cayley-Dickson algebras, Manuscript, CINVESTAV del IPN, Mexico, Mex.
148. G. Moreno (2003), The exponential map on the Cayley-Dickson algebras, Manuscript, CINVESTAV del IPN, Mexico, Mex. Also ArXiv: math/0405424v1 [math. QA] 21 May 2004.
149. G. Moreno (2004), Hopf construction map in higher dimensions, Bol. Soc. Mat. Mex. 10, 383-398.
150. G. Moreno (2005), Constructing zerodivisors in higher dimensional Cayley-Dickson algebras. ArXiv: math/051257 v1 [math.RA], 22 Dec. 2005.
151. G. Moreno (2006a), Alternative elements in the Cayley-Dickson algebras, in Topics in Mathematical Physics, Relativity and Cosmology in honor of Jerzy Plebansky (H. Garcia-Compean et al. eds.) pp. 333-346. World Scientific, Hackensack, NJ.
152. G. Moreno (2006b), Monomorphisms between Cayley-Dickson algebras, Chapter 22 in Nonassociative algebras and applications (L. V. Sabinin et al. eds.) pp. 281-289. CRC Press, Taylor and Francis.
153. G. Moreno and G. Morales (2004), Multiplication on Cayley-Dickson algebras, Manuscript, CINVESTAV del IPN, Mexico, Mex.
154. J. Moro, J. Burke, and M. Overton (1997). On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure. SIAM J. Matrix Analysis and Applications 18, 793-817.
155. H. Nagashima and Y. Baba(1999) Introduction to chaos. IOP Publ., Bristol, UK.
156. J. Needham (1977), La science chinoise et l’occident, Seuil, Points Sciences, Paris.
157. S. Newcomb (1881) Note on the frequency of use of different digits in natural numbers, Amer. J. Math 4, 39-40.
158. E. M. Oblow (1988) Supertracks, supertrack functions and chaos in the quadratic map, Phys. Letters A 128, 406-412.
159. V. Ovsienko and S. Tabachnikov (2009) What is the Schwarzian derivative, Notices of the AMS 56, 34-36.
160. B. N. Parlett (1968) Global convergence of the basic QR algorithm on Hessenberg matrices, Math. Comp. 22, 803-817.
161. B. N. Parlett (1980) The symmetric eigenvalue problem, Prentice Hall, Englewood Cliffs, NJ.
162. L. Pasteur (1848), Sur les relations qui peuvent exister entre la forme cristalline, la composition chimique et le sens de la polarisation rotatoire Ann. Chim. Phys. 24, pp. 441-459.
163. R. Penrose (1989) The emperor’s new mind: concerning computers, minds and physics, Oxford Univ. Press, Oxford.
164. A. Pixton (2009), Alternators in the Cayley Dickson algebras, Forum Matem. 21, 853-869.
165. K. Plofker (2008), Mathematics in India, Princeton U. Press, Princeton, NJ.
166. H. Poincare (1896) Calcul des probabilites, George Carre, Paris.
167. H. Poincaré (1905), La valeur de la Science, Flammarion, Paris, republished in 1970.
168. H. Poincaré (1907) Le hasard, Revue du Mois 3, 257-276, republished as Chapter 4 in Science et Methode (1908) Flammarion, Paris.
169. H. Poincare (1908a) L’avenir des mathematiques, Rev. gen. Sciences pures et app. 19, 930-939.
170. H. Poincare (1908b), L’invention mathematiques, Bull. Inst. Gen. Psychol., 8, 175-187.
171. B. Poonen (2008) Undecidability in Number Theory, Notices AMS 55, 344-350.
172. V. Puiseux (1850). Recherches sur les fonctions algebriques. J. Math. Pures Appl. 51, 465-581.
173. A. Robert (1992) Exemples de Discret-Continu en Analyse, in Le Labyrinthe du Continu (J. M. Salanskis, H. Sinaceur eds.), pp. 279-295, SpringerVerlag France, Paris.
174. A. Robert (2000) A Course in p-adic Analysis, Springer Verlag, New York.
175. E. Robson (2008) Mathematics in Ancient Iraq, Princeton U. Press, Princeton, NJ.
176. R. Rumely (1986) Arithmetic over the ring of algebraic integers, J. Reine Angew.Math. 368, 127-133.
177. R. D. Schafer (1954), On algebras formed by the Cayley-Dickson process, Amer.J. Math. 76, 435-446.
178. R. D. Schafer (1966), An introduction to Nonassociative Algebras, Academic Press, New York.
179. I. Schur (1909) On the characteristic roots of a linear substitution with an application to the theory of integral equations, Math. Ann. 66, 488-510.
180. I. Schur (1917-1918) Über Potenzreichen, die in Innern des Einheitskreises beschurankt sind I-II, J. Reine Angew. Math. 147, 205-232, 148, 122-145.
181. A. N. Sharkovski (1964) Coexistence of cycles of a continuous map of a line into itself, Ukr. Math. J. 16, 161-171.
182. J. Sherman and W. J. Morrison (1950) Adjustment of an inverse matrix corresponding to change in the elements of a given matrix, Ann. Math. Statist. 21, 124-127.
183. M. Shishikura (1998) The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Annals of Math. 147, 225-267.
184. n-ons: New kinds of numbers, Tech. Rep., http://math.temple.edu/~wds/homepage/works.html
185. G. W. Stewart (1993) On the early history of the Singular Value Decomposition, SIAM Rev. 35, 551-566.
186. o. 40, University Park and London.
187. A. Tarski and S. Givant (1999) Tarski’s system of Geometry, Bull. Symb. Logic 5, 175-212.
188. G. Tenenbaum and M. Mendes-France (1997) Les nombres premiers, “Que sais-je?”, PUF, Paris.
189. o. 382, 690-691.
190. P. Turner (1982) The distribution of leading significant digits IMA J. Num. Anal. 2, 407-412.
191. A. Ungar (2008), Analytic hyperbolic geometry and Albert Einstein’s Special Theory of Relativity World Scientific, Singapore.
192. M. B. Van Gijzen (2002) Private communication, Cerfacs.
193. M. L. Wantzel (1837) Recherches sur les moyens de reconnaitre si un probleme de Geometrie peut se résoudre par la règle et le compas, J. Mathém. Pures et Appl. 2, 336-372.
194. A. Weil (1984) Number Theory. An approach through history, Birkhauser, Boston, USA.
195. N. Wiener (1958) The Fourier integral and certain of its applications, Dover Publ, New York.
196. J. H. Wilkinson (1988), The algebraic eigenvalue problem, Oxford University Press, Oxford, UK.
197. C. R. Woese (2004), A new biology for a new century, Microbiol. and Molec.Biology Reviews, 68, 173-186.
198. C. R. Woese (2007), Biology’s next revolution, Nature, 445, 25 January.
199. S. Wolfram (1984) Geometry of binomial coefficients, Amer. Math. Month. 91, 566-571.
200. S. Wolfram (2002) A new Kind of Science, Wolfram Media, Champaign, Il.
201. P. Yale (1966) Automorphisms of the complex numbers, Mathem. Magazine 39, 135-141.
202. P. Yiu (2006) Private communication (e-mail).
203. J. Ziv and A. Lempel (1978) Compression of individual sequences via variable-rate coding, IEEE Trans. Inf. Theor. 24, 530-536.
204. M. A. Zorn (1935) The automorphisms of Cayley’s non-associative algebra, Proc. Nat. Acad. Sci. 21, 355-358.