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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 |
MARC
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| 100 | 1 | |a Chaitin-Chatelin, Françoise, |e author. | |
| 245 | 1 | 0 | |a Qualitative computing : |b a computational journey into nonlinearity / |c Françoise Chatelinches. |
| 264 | 1 | |a Singapore ; |a Hackensack, NJ : |b World Scientific, |c [2012] | |
| 264 | 4 | |c ©2012 | |
| 300 | |a 1 online resource (xv, 582 pages) : |b illustrations, portraits | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 340 | |g polychrome. |2 rdacc |0 http://rdaregistry.info/termList/RDAColourContent/1003 | ||
| 347 | |a text file |2 rdaft |0 http://rdaregistry.info/termList/fileType/1002 | ||
| 380 | |a Bibliography | ||
| 504 | |a Includes bibliographical references and index. | ||
| 520 | |a 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 words - nonlinearity and coupling. This book is unique as it proposes truly original solutions: (1) Using hypercomputation in quadratic algebras, as opposed to the traditional use of linear vector spaces in the 20th century; (2) complementing the classical linear logic by the complex logic which expresses the creative potential of the complex plane. The book illustrates how qualitative computing has been the driving force behind the evolution of mathematics since Pythagoras presented the first incompleteness result about the irrationality of [symbol]2. The celebrated results of Godel and Turing are but modern versions of the same idea: the classical logic of Aristotle is too limited to capture the dynamics of nonlinear computation. Mathematics provides us with the missing tool, the organic logic, which is aptly tailored to model the dynamics of nonlinearity. This logic will be the core of the "Mathematics for Life" to be developed during this century | ||
| 505 | 0 | |a 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. | |
| 505 | 8 | |a 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. | |
| 505 | 8 | |a 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. | |
| 505 | 8 | |a 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. | |
| 546 | |a English. | ||
| 590 | |a eBooks on EBSCOhost |b EBSCO eBook Subscription Academic Collection - Worldwide | ||
| 650 | 0 | |a Social sciences |x Data processing. | |
| 650 | 0 | |a Qualitative research |x Data processing. | |
| 650 | 6 | |a Sciences sociales |x Informatique. | |
| 650 | 6 | |a Recherche qualitative |x Informatique. | |
| 650 | 7 | |a COMPUTERS |x Computer Literacy. |2 bisacsh | |
| 650 | 7 | |a COMPUTERS |x Computer Science. |2 bisacsh | |
| 650 | 7 | |a COMPUTERS |x Data Processing. |2 bisacsh | |
| 650 | 7 | |a COMPUTERS |x Hardware |x General. |2 bisacsh | |
| 650 | 7 | |a COMPUTERS |x Information Technology. |2 bisacsh | |
| 650 | 7 | |a COMPUTERS |x Machine Theory. |2 bisacsh | |
| 650 | 7 | |a COMPUTERS |x Reference. |2 bisacsh | |
| 650 | 7 | |a Social sciences |x Data processing |2 fast | |
| 776 | 0 | 8 | |i Print version: |z 9789814322928 |z 981432292X |
| 856 | 4 | 0 | |u https://ebsco.uam.elogim.com/login.aspx?direct=true&scope=site&db=nlebk&AN=478634 |z Texto completo |
| 880 | 0 | |6 505-00/(S |a Contents -- Pour mes enfants, petits et grands -- Preface -- 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 algebras -- 2.2 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 ≥ 1 -- 2.2.4 Productive coupling of linear subspaces in Ak, k ≥ 4 -- 2.2.5 Other inductive multiplicative processes -- 2.3 Properties of the multiplication -- 2.3.1 The partition Ak = R1 ⊕ Ak, k ≥ 1 -- 2.3.2 The commutator for k ≥ 2 -- 2.3.3 The associator for k ≥ 3 -- 2.3.4 The four real division algebras -- 2.3.5 The alternator for k ≥ 4 -- 2.3.6 The normalisatrix function for k ≥ 4 -- 2.3.7 The subalgebra σx generated by x ∈ A, x 0. | |
| 880 | 8 | |6 505-00/(S |a 2.4 Left and right multiplication maps -- 2.4.1 Definition -- 2.4.2 The real scalar product La, Lb F -- 2.5 The partition Ak = C1 ⊕ Dk, k ≥ 2 -- 2.5.1 A characterization of C in Ak, k ≥ 4 -- 2.5.2 Algebraic computation in Dk, k ≥ 2 -- 2.5.3 The map La for a ∈ Dk -- 2.5.4 The complex scalar product La, Lb F* for a ∈ Dk -- 2.6 Alternative vectors in Ak for k ≥ 4 -- 2.6.1 Definition -- 2.6.2 Colinearity of X and Y in Ak, k ≥ 4 -- 2.6.3 Characterization of alternativity for vectors in Ak, k ≥ 4 -- 2.6.4 Alternative subspaces in Ak, k ≥ 4 -- 2.6.5 Fully alternative vectors in Ak, k ≥ 4 -- 2.7 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 -- 2.9.4.3 The exponential of a product x × u in Ak, k ≥ 2 -- 2.9.5 When does [eX, eY] = 0 for X, Y ∈ Ak, k ≥ 2-- 2.9.6 The general Euler formula in Ak, k ≥ 2 for the triple (X, Y, εX + ε′Y) -- 2.9.6.1 The purely metric approach in Ak, k ≥ 2 -- 2.9.6.2 Equilateral triangles -- 2.9.6.3 Right-angled triangles -- 2.9.7 X or Y is a zerodivisor in Dk, k ≥ 4 and [X, Y] = 0 -- 2.9.7.1 Preliminaries -- 2.9.7.2 The formula (2.9.11) under the light of trigonomety -- 2.9.7.3 The particular case t = 1 -- 2.9.7.4 The formula (2.9.12) -- 2.9.7.5 An epistemological perspective -- 2.9.8 Summary -- 2.9.9 The real zeros of the ζ function -- 2.10 Some extensions of the Fundamental Theorem of Algebra, from A1 = C to Ak, k ≥ 2. | |
| 880 | 8 | |6 505-00/(S |a 2.10.1 FTA in C -- 2.10.2 FTA in H -- 2.10.3 Polynomials in the variable x ∈ Ak, k ≥ 2, with real coefficients -- 2.10.4 A topological extension in Ak, k ≥ 2 -- 2.11 Normwise qualification mod 2 in Ak, k ≥ 2 -- 2.11.1 The imaginary units under trigonometric analysis -- 2.11.2 The imaginary ball of radius 2π, 0 excepted -- 2.12 Bibliographical notes -- 3. Variable Complexity within Noncommutative Dickson Algebras -- 3.1 The multiplication tables in An, n ≥ 0 -- 3.1.1 Uniqueness of the multiplication tables for n = 0, 1, 2 -- 3.1.2 The many multiplication tables for n = 3 -- 3.1.2.1 Connection with inductive computation -- 3.1.2.2 Connection with projective geometry -- 3.1.2.3 The structure of the group Aut (G) -- 3.2 The algorithmic computation of the standard multiplication table Mn -- 3.3 Another algorithmic derivation of Mn, n ≥ 0 -- 3.3.1 The index correspondence Dn -- 3.3.2 The sign matrix Sn associated with ei × ej → ±ek -- 3.4 The right and left multiplication maps -- 3.4.1 The Dickson index matrices coalesce -- 3.4.2 The sign matrices -- 3.4.3 The structure of multiplication -- 3.5 Representations of Ak, k ≥ 2 with variable complexity -- 3.5.1 The level m-expansion for z ∈ Ak, 0 <m <k -- 3.5.2 Variable complexity within Ak, k ≥ 2 -- 3.5.3 Expressive coupling -- 3.5.4 Multipure subspaces in Ak, k ≥ 2 -- 3.6 Multiplication in k-mAk -- 3.6.1 The product z = x × u is isometric -- 3.6.2 The product z3 = z1 × z2 in k-m -- 3.6.3 An emerging product in k-m for k 3 -- 3.7 The algebra Der (Ak) of derivations for Ak, k ≥ 0 -- 3.7.1 Definition -- 3.7.2 Der (Ak) for k = 2, 3 -- 3.7.3 Der (Ak) for k ≥ 4 -- 3.8 Beyond linear derivation -- 3.8.1 Derivation as linear causality -- 3.8.2 The nonlinear core of Ak -- 3.8.3 Reducibility by derivation in Dickson algebras -- 3.8.4 Real versus complex causality. | |
| 880 | 8 | |6 505-00/(S |a 3.8.5 Epistemological principles of hypercomputation -- 3.9 The nature of hypercomputation in Ak, k ≥ 0 -- 3.9.1 A global summary -- 3.9.2 Information derived from space and time -- 3.9.3 Eidetic computation in Ak, k ≥ 5 -- 3.9.4 About Reason -- 3.10 Bibliographical notes -- 4. Singular Values for the Multiplication Maps -- 4.1 Multiplication by a vector x in Ak, k ≥ 0 -- 4.1.1 x is alternative -- 4.1.2 x is not alternative, k ≥ 4 -- 4.2 a is not alternative in Dk, k ≥ 4 -- 4.2.1 The eigenvalues of -L2 a in a -- 4.2.2 The eigenvalue 1 = a 2 = N(a), a in Dk, k ≥ 4 -- 4.2.3 Other eigenvalues in σa -- 4.2.4 Commuting pairs in Dk -- 4.3 x = α + β + t, α and β real, t ∈ Dk, k ≥ 4 -- 4.3.1 The pythagorean rule -- 4.3.2 The algebra generated by the pair (x,), k ≥ 2 -- 4.3.3 When x = a + b, a and b alternative in Ak and a, b = 0 -- 4.4 Complexification of the algebra Ak, k ≥ 3 -- 4.4.1 The 2 × 2 block representation of -L2 -- 4.4.2 a and b are alternative in Ak, k ≥ 3 -- 4.4.3 The multiplicity of N() -- 4.4.4 The spectral information carried by a × b -- 4.5 Zerodivisors with two alternative parts in Ak, k ≥ 3 -- 4.6 = (a, b) has alternative, orthogonal parts with equal length in Ak, k ≥ 3 -- 4.7 The SVD for Lx in A4 -- 4.7.1 SVD for Lt in D4 -- 4.7.2 SVD for Lx in A4 -- 4.8 Other types of zerodivisors in Dk+1, k ≥ 4 -- 4.8.1 Definitions -- 4.8.2 Not necessarily alternative parts in Ak, k ≥ 4 -- 4.8.3 The vector θt = (a, t k) with a ∈ Dk, t ∈ R*, k ≥ 4 -- 4.8.4 = (a, b) with a, b ∈ Dk, k ≥ 3 -- 4.8.5 The vector ψ = (a,) for a not alternative in Dk, k ≥ 4 -- 4.8.6 max dimZer () for ∈ Dk, k ≥ 4 -- 4.8.7 About the growth of Ld/ d in Ak, k ≥ 4 -- 4.9 Bibliographical notes -- 5. Computation Beyond Classical Logic -- 5.1 Local SVD derivation -- 5.1.1 The head-tail split -- 5.1.2 Local derivations in Ak, k ≥ 3. | |
| 880 | 8 | |6 505-00/(S |a 5.1.3 Nonassociativity of addition deriving from local SVD -- 5.2 Pseudo-zerodivisors associated with λ ∈ σt -- 5.2.1 Definition -- 5.2.2 Is the local SVD derivation absurd-- 5.2.3 Resolution of the logical paradox by complexification -- 5.3 Local and global SVD analyzed in C for k ≥ 3 -- 5.3.1 Threefold partition for C -- 5.3.2 Characteristic curves and points in C for a in Ak, k ≥ 3 -- 5.4 The measure of a vector a in Ak evolves with k ∈ N -- 5.4.1 k = 0 to 3 -- 5.4.2 Measuring a vector a = h + t in Ak, k ≥ 3 -- 5.4.3 F3(t) modified by B: a = α1 + (β + t) -- 5.4.4 F3(t) modified by C: a = β + (α1 + t) -- 5.4.5 The dependence on λ ∈ σt of the geometric frame for a = h + t in Ak, k ≥ 4 -- 5.4.6 The localisation of a from the information given by the pythagorean measure √N -- Part II. Complexification of the Algebra -- 5.5 Complexification of Ak into Ak+1, k ≥ 2 -- 5.5.1 Induction into Ak+1 by a in Ak = C ⊕ Dk, k ≥ 2 -- 5.5.2 The eigenvalues of -L2 ˚i', i' = 0, 1 for ß 0 -- 5.5.3 Global singular values for L, l = 0 to 7 when α β 0, k ≥ 2 -- 5.5.4 The case αβ = 0 -- 5.6 Local SVD for L, l = 0, 2, 5, 7 -- 5.6.1 j = 1, αβ 0 -- 5.6.2 The case αβ = 0 -- 5.6.3 Summary for k ≥ 3 -- 5.6.4 Pseudo-zerodivisors in Ak+1, k ≥ 3 -- 5.6.5 Characteristic curves and points for in Ak+1, k ≥ 3 -- 5.6.6 The contextual measures for a ∈ Ak, k ≥ 2 -- 5.7 About the inductive computation of k ×vk from Ak 1 into Dk, k ≥ 4 -- 5.7.1 About × v and × v -- 5.7.2 The metric equivalence to × v = 0 in D4 -- 5.7.3 × v = 0 is not equivalent to × v = 0 in Dk, k ≥ 5 -- 5.7.4 False zeroproducts in Dk, k ≥ 5 -- 5.8 An epistemological conclusion -- 5.8.1 An overview -- 5.8.2 Nonassociativity ⇒ SVD paradoxes -- 5.8.3 Nonassociativity ⇒ learning by experience -- 5.9 Bibliographical notes -- 6. Complexification of the Arithmetic. | |
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