QUALITATIVE COMPUTING

A Computational Journey Into Nonlinearity

By: Françoise Chatelin

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Hardcover | 19 June 2012

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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 ?2. The celebrated results of Goedel 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.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory Applied Mathematics Nonlinear Science Optimisation Linear Programming Booktopia Publisher Services

World Scientific Publishing

Pour mes enfants, petits et grands	p. vii
Preface	p. ix
Introduction to Qualitative Computing	p. 1
The art of computing before the 20th century	p. 2
The unending evolution of logic due to complexification	p. 6
The 20th century	p. 8
Back to the art of computing	p. 14
Hypercomputation in Dickson Algebras	p. 21
Associativity in algebra	p. 22
Dickson algebras over the real field	p. 23
Properties of the multiplication	p. 27
Left and right multiplication maps	p. 39
The partition A_k = C₁ ⊕D_k, k ≥ 2	p. 40
Alternative vectors in A_k for k ≥ 4	p. 45
Co-alternativity in A_k for k ≥ 4	p. 51
The power map in A_k{0}	p. 55
The exponential function in A_k, k ≥ 0	p. 57
Some extensions of the Fundamental Theorem of Algebra, from A₁ = C to A_k, k ≥ 2	p. 77
Normwise qualification mod 2 in &Chinese character;A_k, k ≥ 2	p. 80
Bibliographical notes	p. 81
Variable Complexity within Noncommutative Dickson Algebras	p. 83
The multiplication tables in A_n, n ≥ 0	p. 83
The algorithmic computation of the standard multiplication table M_n	p. 85
Another algorithmic derivation of M_n, n ≥ 0	p. 87
The right and left multiplication maps	p. 90
Representations of A_k, k ≥ 2 with variable complexity	p. 91
Multiplication in J_k-mA_k	p. 95
The algebra Der (A_k) of derivations for A_k, k ≥ 0	p. 99
Beyond linear derivation	p. 102
The nature of hypercomputation in A_k, k ≥ 0	p. 107
Bibliographical notes	p. 112
Singular Values for the Multiplication Maps	p. 113
Multiplication by a vector x in A_k, k ≥ 0	p. 113
a is not alternative in D_k, k ≥ 4	p. 114
x = + ß1 + t, and ß real, t ∈ D_k, k ≥ 4	p. 120
Complexification of the algebra A_k, k ≥ 3	p. 123
Zerodivisors with two alternative parts in &Chinese character; A_k, k ≥ 3	p. 130
= (a, b) has alternative, orthogonal parts with equal length in &Chinese character; A_k, k ≥ 3	p. 135
The SVD for L_x in A₄	p. 137
Other types of zerodivisors in D_k+1, k ≥ 4	p. 138
Bibliographical notes	p. 149
Computation Beyond Classical Logic	p. 151
Local SVD derivation	p. 153
Pseudo-zerodivisors associated with ∈ _t	p. 158
Local and global SVD analyzed in C₁ for k ≥ 3	p. 159
The measure of a vector a in A_k evolves with k ∈ N	p. 161
Complexification of A_k into A_k+1, k ≥ 2	p. 166
Local SVD for L_l, l = 0, 2, 5, 7	p. 172
About the inductive computation of × from &Chinese character; A_k-1 into D_k, k ≥ 4	p. 180
An epistemological conclusion	p. 187
Bibliographical notes	p. 189
Complexification of the Arithmetic	p. 191
The resurgence of √-1 in A_k, k ≥ 3	p. 191
Self-induction in D_k+1 by a ∈ &Chinese character;A_k, k ≥ 2	p. 194
Complex self-induction by a in D_k, k ≥ 3	p. 196
Spectral analysis of -L²_s for s = (a, a × h) ∈ D_k+1, a ∈ D_k, h ∈ C₁k ≥ 3	p. 198
A geometric viewpoint based on +	p. 203
Monomorphisms from A_m to A_k, 1 ≤ m ≤ 3, k ≥ m	p. 204
Inductive construction of Der	p. 207
An algorithmic evolution of the field R into C by the logistic iteration	p. 220
Other algorithmic evolutions of t from R to C	p. 229
Evolution of u without divergence to ∞	p. 233
An application: The isophasic exponentiation of z in C as a function of the parameter z /	p. 242
Bibliographical notes	p. 244
Homotopic Deviation in Linear Algebra	p. 247
An introduction to complex Homotopic Deviation	p. 248
The algebraic tools	p. 250
The resolvent R(t, z) for z ∈ re(A), t ∈ C	p. 263
The spectral field t → (A(t)), t ∈ C	p. 277
Study of the limit set Lim under (7.4.1)	p. 283
About the limit and frontier points in re(A)	p. 304
The mutation matrix B at ∈ F(A, E) ≠ {re(A),0}	p. 306
The observation point is the eigenvalue ∈ (A)	p. 318
Algorithmic complexification of the homotopy parameter t,	p. 331
The family of pencils P_z(t) = (A - zI) + tE, where the parameter z varies in C	p. 333
About contextual algebraic computation	p. 335
Visualization tools	p. 341
Bibliographical notes	p. 346
The Discrete and the Continuous	p. 347
The self-conjugate binary algebras B_k, k ≥ 0	p. 348
The multiplication tables for k = 1, 2	p. 354
Partial emergence of multiplication and mod 2^k+1 in B_k, k ≥ 3	p. 356
The linear space C_n of binary sequences of length n ≥ 1	p. 357
n = 2_k: An alternative complex order	p. 360
The base b-expansion of n, b ≥ 2	p. 362
Mechanical uncomputability	p. 364
The arithmetic triangle	p. 368
The arithmetic triangle mod 2	p. 375
The triangle mod 3	p. 382
Connections between 2 and 3	p. 384
Two digital representations of real numbers	p. 385
The Borel-Newcomb paradox for real numbers	p. 393
Sum of random variables computed modulo 1	p. 395
Finite precision computation over R	p. 401
A dynamical perspective on the natural integers	p. 407
Bibliographical notes	p. 417
Arithmetic in the Four Dickson Division Algebras	p. 419
A review of the three theorems of squares	p. 419
The rings R_k of hypercomplex integers, k ≤ 3	p. 423
Isometries in 3 and 4 dimensions	p. 434
The rate of association in G	p. 435
The first cycle (f₁, f₂, f₃)	p. 437
A second epistemological pause	p. 440
The last three canonical vectors f₅ to f₇	p. 444
Conclusion	p. 447
Bibliographical notes	p. 448
The Real and the Complex	p. 449
About the relativity stemming from an algorithmic quantification of a quality	p. 450
Setwise inclusion in R	p. 451
Isophasic inclusion inside C by exponentiation	p. 459
Metric inclusion inside C under exponentiation	p. 460
The Cantor space {0, 1}^N	p. 471
Doubly infinite sequences	p. 473
Evolution from R⁺_* to C*	p. 476
The continuous Fourier transform as a cognitive tool	p. 479
The scalar product <∂_tf, _tf>	p. 484
Bibliographical notes	p. 493
The Organic Logic of Hypercomputation	p. 495
About the representations of complex integers	p. 497
The inductive points of C with norm n ≥ 2	p. 507
An algorithm for organic arithmetic in Z[b]	p. 511
Comparison between z and vis (z) for z ∈ Z[b]	p. 515
The rings Z[b_t],
The synthetic power of C stemming from Z	p. 520
The organic logic for hypercomputation	p. 522
The organic measure set for the source vector a ∈ A_k, for k ≥ 3	p. 525
The angles _j = <(a, a_j) for j = 1 to 4	p. 533
About the coincidence of a with one of the a_j when = ß ≠ 0	p. 535
The autonomous evolution of = <(a, a^M) as a function of r = N(a) = 1 + N(h) > 1	p. 538
Computational evolution of t out of D_k, k ≥ 3	p. 540
Autonomous evolution based on the spectral information in _t	p. 543
Bibliographical notes	p. 545
The Organic Intelligence in Numbers	p. 547
About the zeros of the function	p. 547
Algebraic depth and p = Rs	p. 548
The two families of complex zeros for in the light of hypercomputation	p. 550
The algebras with d_a ≥ 2 are sources of common sense	p. 551
The algebraic reductions with p = 1/2	p. 554
Thinking in 1 or 2 dimensions: Thought or intuition	p. 556
A review of hypercomputation	p. 563
Bibliography	p. 567
Index	p. 577
Table of Contents provided by Ingram. All Rights Reserved.

QUALITATIVE COMPUTING

A Computational Journey Into Nonlinearity

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