| 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 Ak = C1 ⊕Dk, k ≥ 2 | p. 40 |
| Alternative vectors in Ak for k ≥ 4 | p. 45 |
| Co-alternativity in Ak for k ≥ 4 | p. 51 |
| The power map in Ak{0} | p. 55 |
| The exponential function in Ak, k ≥ 0 | p. 57 |
| Some extensions of the Fundamental Theorem of Algebra, from A1 = C to Ak, k ≥ 2 | p. 77 |
| Normwise qualification mod 2 in &Chinese character;Ak, k ≥ 2 | p. 80 |
| Bibliographical notes | p. 81 |
| Variable Complexity within Noncommutative Dickson Algebras | p. 83 |
| The multiplication tables in An, n ≥ 0 | p. 83 |
| The algorithmic computation of the standard multiplication table Mn | p. 85 |
| Another algorithmic derivation of Mn, n ≥ 0 | p. 87 |
| The right and left multiplication maps | p. 90 |
| Representations of Ak, k ≥ 2 with variable complexity | p. 91 |
| Multiplication in Jk-mAk | p. 95 |
| The algebra Der (Ak) of derivations for Ak, k ≥ 0 | p. 99 |
| Beyond linear derivation | p. 102 |
| The nature of hypercomputation in Ak, k ≥ 0 | p. 107 |
| Bibliographical notes | p. 112 |
| Singular Values for the Multiplication Maps | p. 113 |
| Multiplication by a vector x in Ak, k ≥ 0 | p. 113 |
| a is not alternative in Dk, k ≥ 4 | p. 114 |
| x = + ß1 + t, and ß real, t ∈ Dk, k ≥ 4 | p. 120 |
| Complexification of the algebra Ak, k ≥ 3 | p. 123 |
| Zerodivisors with two alternative parts in &Chinese character; Ak, k ≥ 3 | p. 130 |
| = (a, b) has alternative, orthogonal parts with equal length in &Chinese character; Ak, k ≥ 3 | p. 135 |
| The SVD for Lx in A4 | p. 137 |
| Other types of zerodivisors in Dk+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 Ak evolves with k ∈ N | p. 161 |
| Complexification of Ak into Ak+1, k ≥ 2 | p. 166 |
| Local SVD for Ll, l = 0, 2, 5, 7 | p. 172 |
| About the inductive computation of × from &Chinese character; Ak-1 into Dk, k ≥ 4 | p. 180 |
| An epistemological conclusion | p. 187 |
| Bibliographical notes | p. 189 |
| Complexification of the Arithmetic | p. 191 |
| The resurgence of √-1 in Ak, k ≥ 3 | p. 191 |
| Self-induction in Dk+1 by a ∈ &Chinese character;Ak, k ≥ 2 | p. 194 |
| Complex self-induction by a in Dk, k ≥ 3 | p. 196 |
| Spectral analysis of -L2s for s = (a, a × h) ∈ Dk+1, a ∈ Dk, h ∈ C1k ≥ 3 | p. 198 |
| A geometric viewpoint based on + | p. 203 |
| Monomorphisms from Am to Ak, 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 Pz(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 Bk, k ≥ 0 | p. 348 |
| The multiplication tables for k = 1, 2 | p. 354 |
| Partial emergence of multiplication and mod 2k+1 in Bk, k ≥ 3 | p. 356 |
| The linear space Cn of binary sequences of length n ≥ 1 | p. 357 |
| n = 2k: 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 Rk 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[bt], | |
| 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 ∈ Ak, for k ≥ 3 | p. 525 |
| The angles j = <(a, aj) for j = 1 to 4 | p. 533 |
| About the coincidence of a with one of the aj when = ß ≠ 0 | p. 535 |
| The autonomous evolution of = <(a, aM) as a function of r = N(a) = 1 + N(h) > 1 | p. 538 |
| Computational evolution of t out of Dk, 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 da ≥ 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 |
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