Description-Table Of Contents |
1. Introduction to qualitative computing. 1.1. The art of computing before the 20th century. 1.2. The unending evolution of logic due to complexification. 1.3. The 20th century. 1.4. Back to the art of computing -- 2. Hypercomputation in Dickson algebras. 2.1. Associativity in algebra. 2.2. Dickson algebras over the real field. 2.3. Properties of the multiplication. 2.4. Left and right multiplication maps. 2.5. The partition A[symbol]. 2.6. Alternative vectors in A[symbol] for K [symbol] 4. 2.7. Co-alternativity in A[symbol] for K [symbol] 4. 2.8. The power map in A[symbol]\{lcub}0{rcub}. 2.9. The exponential function in A[symbol], k[symbol]0. 2.10. Some extensions of the Fundamental Theorem of Algebra, from A [symbol]. 2.11. Normwise qualification mod 2 [symbol]. 2.12. Bibliographical notes -- 3. Variable complexity within noncommutative Dickson algebras. 3.1. The multiplication tables in A[symbol], n [symbol] 0. 3.2. The algorithmic computation of the standard multiplication table M[symbol]. 3.3. Another algorithmic derivation of M[symbol], n [symbol] 0. 3.4. The right and left multiplication maps. 3.5. Representations of A[symbol], k [symbol] 2 with variable complexity. 3.6. Multiplication in [symbol]. 3.7. The algebra Der(A[symbol]) of derivations for A[symbol], k [symbol] 0. 3.8. Beyond linear derivation. 3.9. The nature of hypercomputation in A[symbol], k [symbol] 0. 3.10. Bibliographical notes -- 4. Singular values for the multiplication maps. 4.1. Multiplication by a vector x in A[symbol], k [symbol] 0. 4.2. a is not alternative in [symbol], k [symbol] 4. 4.3. x = [symbol]. 4.4. Complexification of the algebra A[symbol], k [symbol] 3. 4.5. Zerodivisors with two alternative parts in [symbol], k [symbol] 3. 4.6. [symbol] = (a,b) has alternative, orthogonal parts with equal length in [symbol], k [symbol] 3. 4.7. The SVD for L[symbol] in A[symbol]. 4.8. Other types of zerodivisors in [symbol], k [symbol] 4. 4.9. Bibliographical notes -- 5. Computation beyond classical logic. 5.1. Local SVD derivation. 5.2. Pseudo-zerodivisors associated with [symbol]. 5.3. Local and global SVD analyzed in [symbol] for k [symbol] 3. 5.4. The measure of a vector [symbol] in A[symbol] evolves with k [symbol]. 5.5. Complexification of A[symbol] into A[symbol], k [symbol] 2. 5.6. Local SVD for L[symbol] = 0, 2, 5, 7. 5.7. About the inductive computation of [symbol] from [symbol] tino [symbol], k [symbol] 4. 5.8. An epistemological conclusion. 5.9. Bibliographical notes. ; 8 6. Complexification of the arithmetic. 6.1. The resurgence of [symbol] in A[symbol], k [symbol] 3. 6.2. Self-induction in [symbol] by [symbol], k [symbol] 2. 6.3. Complex self-induction by [symbol] in [symbol], k [symbol] 3. 6.4. Spectral analysis of [symbol] for s = (a,a, x h), [symbol], k [symbol] 3. 6.5. A geometric viewpoint based on +. 6.6. Monomorphisms from A[symbol]. 6.7. Inductive construction of Der. 6.8. An algorithmic evolution of the field [symbol] into [symbol] by the logistic iteration. 6.9. Other algorithmic evolutions of t from [symbol] to [symbol]. 6.10. Evolution of u without divergence at [symbol]. 6.11. An application: the isophasic exponentiation of z in [symbol] as a function of the parameter [symbol]. 6.12. Bibliographical notes -- 7. Homotopic deviation in linear algebra. 7.1. An introduction to complex homotopic deviation. 7.2. The algebraic tools. 7.3. The resolvent R(t,z) for z [symbol]. 7.4. The spectral field [symbol]. 7.5. Study of the limit set Lim under (7.4.1). 7.6. About the limit and frontier points in re(A). 7.7. The mutation matrix B[symbol] at [symbol]. 7.8. The observation point is the eigenvalue [symbol](A). 7.9. Algorithmic complexification of the homotopy parameter t, [symbol] < 1. 7.10. The family of pencils P[symbol](t) = [symbol], where the parameter z varies in [symbol]. 7.11. About contextual algebraic computation. 7.12. Visualization tools. 7.13. Bibliographical notes -- 8. The discrete and the continuous. 8.1. The self-conjugate binary algebras B[symbol], k [symbol] 0. 8.2. The multiplication tables for k = 1, 2. 8.3. Partial emergence of multiplication mod 2[symbol] in B[symbol], k [symbol] 3. 8.4. The linear space C[symbol] of binary sequences of length n [symbol] 1. 8.5. n = 2[symbol]: an alternative complex order. 8.6. The base b-expansion of n, b [symbol] 2. 8.7. Mechanical uncomputability. 8.8. The arithmetic triangle. 8.9. The arithmetic triangle mod 2. 8.10. The triangle mod 3. 8.11. Connections between 2 and 3. 8.12. Two digital representations of real numbers. 8.13. The Borel-Newcomb paradox for real numbers. 8.14. Sum of random variables computed modulo 1. 8.15. Finite precision computation over [symbol]. 8.16. A dynamical perspective on the natural integers. 8.17. Bibliographical notes -- 9. Arithmetic in the four Dickson division algebras. 9.1. A review of the three theorems of squares. 9.2. The rings R[symbol] of hypercomplex integers, k [symbol] 3. 9.3. Isometries in 3 and 4 dimensions. 9.4. The rate of association in G. 9.5. The first cycle (f1,f2,f3). 9.6. A second epistemological pause. 9.7. The last three canonical vectors f5 to f7. 9.8. Conclusion. 9.9. Bibliographical notes. ; 8 10. The real and the complex. 10.1. About the relativity stemming from an algorithmic quantification of a quality. 10.2. Setwise inclusion in [symbol]. 10.3. Isophasic inclusion inside [symbol] by exponentiation. 10.4. Metric inclusion inside [symbol] under exponentiation. 10.5. The Cantor space {lcub}0,1{rcub}[symbol]. 10.6. Doubly infinite sequences. 10.7. Evolution from R[symbol] to C[symbol]. 10.8. The continuous Fourier transform as a cognitive tool. 10.9. The scalar product [symbol]. 10.10. Bibliographical notes -- 11. The organic logic of hypercomputation. 11.1. About the representations of complex integers. 11.2. The inductive points of [symbol] with norm n [symbol] 2. 11.3. An algorithm for organic arithmetic in [symbol][b]. 11.4. Comparison between |z| and |vis(z)| for z [symbol]. 11.5. The rings [symbol] = n [symbol] 2 for hyperarithmetic. 11.6. The synthetic power of [symbol] stemming from [symbol]. 11.7. The organic logic for hypercomputation. 11.8. The organic measure set for the source vector a [symbol], for k [symbol] 3. 11.9. The angles [symbol] = [symbol] for j = 1 to 4. 11.10. About the coincidence of a with one of the a[symbol] when |[symbol]| = |[symbol]| [symbol] 0. 11.11. The autonomous evolution of [symbol] as a function of r = N(a) = 1 + N(h) [symbol] 1. 11.12. Computational evolution of t out of [symbol], k [symbol] 3. 11.13. Autonomous evolution based on the spectral information in [symbol]. 11.14. Bibliographical notes -- 12. The organic intelligence in numbers. 12.1. About the zeros of the [symbol] function. 12.2. Algebraic depth and p = [symbol]. 12.3. The two families of complex zeros for [symbol] in the light of hypercomputation. 12.4. The algebras with d[symbol] [symbol] 2 are sources of common sense. 12.5. The algebraic reductions with p = 1/2. 12.6. Thinking in 1 or 2 dimensions: thought or intuition. 12.7. A review of hypercomputation. |