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A computational journey into the mind. (English) Zbl 1251.30054
Summary: The first half of this paper describes some salient features of hypercomputation in Dickson algebras. Such quadratic algebras form an appropriate framework for nonlinear computations which does not limit a priori the computational power of multiplication. They underlie paradoxical mathematics whose potential interest to analyse some computational aspects of the human mind which resist the classical approach is presented. In its last part, the paper offers new glimpses on the organic logic for hypercomputation by developing a fresh look at plane geometry in relation with the \(\zeta\) function.

MSC:
30G35 Functions of hypercomplex variables and generalized variables
17A30 Nonassociative algebras satisfying other identities
03F60 Constructive and recursive analysis
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[1] Atkins R et al (2010) A characterization of hyperbolic iterated function systems. Topol Proc 36:189–211 · Zbl 1196.28011
[2] Baez J (2001) The octonions. Bull AMS 39:145–205 · Zbl 1026.17001
[3] Barnsley M (1988) Fractals everywhere. Academic Press, San Diego · Zbl 0691.58001
[4] Berry G (2009) Penser, modéliser et maitriser le calcul informatique. Collège de France/Fayard, Paris
[5] Berthoz A (2009) La simplexité. Odile Jacob, Paris
[6] Blair D (2000) Inversion theory and conformal mapping. Student Mathematical Library, Providence · Zbl 0956.30001
[7] Calude C, Chatelin F (2010) A dialogue about Qualitative Computing. Bull EATCS 101:29–41
[8] Calude C et al (2010) The complexity of proving chaoticity and the Church–Turing thesis. Chaos 20:037103 · Zbl 1311.68080
[9] Chaitin-Chatelin F Computing beyond classical logic: SVD computation in nonassociative Dickson algebras. In: Calude C (eds) Randomness & complexity, from Leibniz to Chaitin. pp 13–23 World Scientific, Singapore, (2007) · Zbl 1135.03341
[10] Chatelin F (1993) Eigenvalues of matrices. Wiley, Chichester · Zbl 0807.15016
[11] Chatelin F (2010) Numerical information processing under the global rule expressed by the Euler–Riemann {\(\zeta\)} function defined in the complex plane. Chaos 20:037104 · Zbl 1311.11090
[12] Chatelin F (2011) Qualitative computing, a computational journey into nonlinearity. World Scientific, Singapore · Zbl 1251.03001
[13] Conway J, Smith D (2003) On quaternions and octonions. A. K. Peters, Natick · Zbl 1098.17001
[14] Dickson L (1912) Linear algebras. Trans AMS 13:59–73 · JFM 43.0162.09
[15] Dickson L (1923) A new simple theory of hypercomplex integers. J Math Pures Appl 2:281–326 · JFM 49.0089.01
[16] Douillet P (2009) Pencils of cycles in the triangle plane. Working Paper ENSAIT, University of Lille
[17] Eakin P, Sathaye A (1990) On automorphisms and derivations of Cayley–Dickson algebras. J Algebra 129:263–278 · Zbl 0699.17022
[18] Gutzwiller MC (1990) Chaos in classical and quantum mechanics. Springer, Berlin · Zbl 0727.70029
[19] Hurwitz A (1919) Vorlesungen über die Zahlentheorie der Quaternionen. Julius Springer, Berlin · JFM 47.0106.01
[20] Jordan C (1874) Mémoire sur les formes bilinéaires. J Math Pures Appl 19(Série 2):35–54 · JFM 06.0070.02
[21] Klein E (2005) Chronos: how time shapes the Universe. Avalon Publishing Group, New York
[22] Mahler K (1943) On ideals in the Cayley–Dickson algebra. Proc R Irish Acad 48:123–133 · Zbl 0061.05301
[23] Maxwell JC (1891) A treatise on electricity and magnetism, 3rd edn in 2 vols (republ. 1954). Dover, New York
[24] Pérez Velasco PP, de Lara J (2009) Matrix graph grammars and monotone complex logics. ArXiv:0902.0850 v1 [cs. DM]
[25] Pfieffer R, van Hook C (1993) Circles, vectors and linear algebra. Math Mag 66:75–86 · Zbl 0785.51005
[26] Plouffe S (1998) The computation of certain numbers using a ruler and compass. J Integer Seq 1:article 98.1.3 · Zbl 1010.11071
[27] Poincaré H (1898) On the foundations of geometry. Monist 9:1–43 · JFM 29.0167.01
[28] Poincaré H (1902) La Science et l’Hypothèse. Flammarion, Paris · JFM 34.0080.12
[29] Poon C-S, Young DL (2006) Non associative learning as gated neural integrator and differentiator in stimulus-response pathways. Behav Brain Funct 2:29
[30] Rumely R (1986) Arithmetic over the ring of algebraic integers. J Reine Angew Math 368:127–133 · Zbl 0581.14014
[31] Schafer R (1954) On algebras formed by the Cayley–Dickson process. Am J Math 76:435–446 · Zbl 0059.02901
[32] Schafer R (1966) An introduction to nonassociative algebras. Academic Press, New York · Zbl 0145.25601
[33] Ungar AA (1988) The Thomas rotation formulation underlying a nonassociative group structure for relativistic velocities. Appl Math Lett 1:403–405 · Zbl 0706.20047
[34] Ungar AA (2001) Beyond the Einstein addition law and its gyroscopic Thomas precession. Kluwer, Dordrecht · Zbl 0972.83002
[35] Valaas L (2006) Triangles in hyperbolic geometry, e-manuscript, Senior project. Whitman College
[36] Varičak V (1910) Anwendung der Lobatchefskjschen Geometrie in der Relativtheorie. Phys Z 10:826–829
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