Qualitative computing : a computational journey into nonlinearity /
High technology industries are in desperate need for adequate tools to assess the validity of simulations produced by ever faster computers for perennial unstable problems. In order to meet these industrial expectations, applied mathematicians are facing a formidable challenge summarized by these wo...
Clasificación: | Libro Electrónico |
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Autor principal: | |
Formato: | Electrónico eBook |
Idioma: | Inglés |
Publicado: |
Singapore ; Hackensack, NJ :
World Scientific,
[2012]
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Temas: | |
Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Pour mes enfants, petits et grands; Preface; Contents; 7. Homotopic Deviation in Linear Algebra; 1. Introduction to Qualitative Computing; 1.1 The art of computing before the 20th century; 1.1.1 Numeracy is not ubiquitous; 1.1.2 √2 : An irrational consequence of nonlinearity; 1.1.3 Zero: Thinking the unthinkable; 1.1.4 v1-: A complex consequence of nonlinearity; 1.1.5 Infinity: Decoding divergent series; 1.2 The unending evolution of logic due to complexification; 1.2.1 Classical analysis; 1.2.2 The creative role of zero; 1.2.3 The evolutive pressure of paradoxes on logic.
- 1.2.4 Hypercomplex numbers of dimension 2k, k ≥ 2(1843 1912)1.3 The 20th century; 1.3.1 A paradigm shift; 1.3.2 Fixing the laws of logic a priori; 1.3.3 The eclipse of the art of computing; 1.3.4 The rise of numerical linear algebra; 1.3.5 Contemporary experimental sciences; 1.4 Back to the art of computing; 1.4.1 Hypercomputation in Dickson algebras; 1.4.2 Homotopic Deviation in associative linear algebra over C; 1.4.3 Understanding why and explaining how; 1.4.4 Qualitative Computing; 2. Hypercomputation in Dickson Algebras; 2.1 Associativity in algebra; 2.1.1 Groups, rings and fields.
- 2.1.2 Real algebras2.2 Dickson algebras over the real field; 2.2.1 The doubling process of Dickson (1912); 2.2.2 "Complexification" of Ak-1: Ak = Ak-1×1 Ak-1×, k 1; 2.2.3 The k basic generators for Ak, k e"1; 2.2.4 Productive coupling of linear subspaces in Ak, k e"4; 2.2.5 Other inductive multiplicative processes; 2.3 Properties of the multiplication; 2.3.1 The partition Ak = R1 ₅"Ak, k e"1; 2.3.2 The commutator for k e"2; 2.3.3 The associator for k e"3; 2.3.4 The four real division algebras; 2.3.5 The alternator for k e"4; 2.3.6 The normalisatrix function for k e"4.
- 2.6.5 Fully alternative vectors in Ak, k ≥ 42.7 Co-alternativity in Ak for k ≥ 4; 2.7.1 Definitions; 2.7.2 Quaternionic structures; 2.7.3 Octonionic structures; 2.8 The power map in Ak\{0}; 2.8.1 Preliminaries; 2.8.2 Definition; 2.8.3 The power map n : S(Ak+1) . S(Ak+1) restricted to a subspace Sm, 2k = m = 2k+1
- 2; 2.9 The exponential function in Ak, k ≥ 0; 2.9.1 Motivation; 2.9.2 The real exponential function; 2.9.3 The complex exponential function; 2.9.4 The hypercomplex exponential in Ak, k ≥ 2; 2.9.4.1 ex in Ak, k ≥ 2; 2.9.4.2 The exponential map is onto A k, k ≥ 1.