Qualitative Computing: A Computational Journey into Nonlinearity

Contents......Page 12
Pour mes enfants, petits et grands......Page 8
Preface......Page 10
1. Introduction to Qualitative Computing......Page 18
1.1.1 Numeracy is not ubiquitous......Page 19
1.1.2 √2 : An irrational consequence of nonlinearity......Page 20
1.1.3 Zero: Thinking the unthinkable......Page 21
1.1.5 Infinity: Decoding divergent series......Page 22
1.2.1 Classical analysis......Page 23
1.2.4 Hypercomplex numbers of dimension 2k, k ≥ 2\n\n(1843 1912)......Page 24
1.3.1 A paradigm shift......Page 25
1.3.3 The eclipse of the art of computing......Page 26
1.3.4 The rise of numerical linear algebra......Page 28
1.4.1 Hypercomputation in Dickson algebras......Page 31
1.4.2 Homotopic Deviation in associative linear algebra over C......Page 33
1.4.3 Understanding why and explaining how......Page 34
1.4.4 Qualitative Computing......Page 35
2. Hypercomputation in Dickson Algebras......Page 38
2.1.1 Groups, rings and fields......Page 39
2.2.1 The doubling process of Dickson (1912)......Page 40
2.2.3 The k basic generators for Ak, k ≥ 1......Page 42
2.2.4 Productive coupling of linear subspaces in Ak, k ≥ 4......Page 43
2.3.1 The partition Ak = R1 ⊕ Ak, k ≥ 1......Page 44
2.3.2 The commutator for k ≥ 2......Page 45
2.3.3 The associator for k ≥ 3......Page 47
2.3.5 The alternator for k ≥ 4......Page 49
2.3.6 The normalisatrix function for k ≥ 4......Page 52
2.3.7 The subalgebra σx generated by x ∈ A, x  0......Page 54
2.4.1 Definition......Page 56
2.5 The partition Ak = C1 ⊕ Dk, k ≥ 2......Page 57
2.5.2 Algebraic computation in Dk, k ≥ 2......Page 58
2.5.3 The map La for a ∈ Dk......Page 60
2.5.4 The complex scalar product La,Lb F* for a ∈ Dk......Page 61
2.6.2 Colinearity of X and Y in  Ak, k ≥ 4......Page 62
2.6.3 Characterization of alternativity for vectors in\n\nAk, k ≥ 4......Page 64
2.6.4 Alternative subspaces in Ak, k ≥ 4......Page 66
2.7.1 Definitions......Page 68
2.7.3 Octonionic structures......Page 70
2.8 The power map in Ak\\{0}......Page 72
2.8.3 The power map  n : S(Ak+1) . S(Ak+1) restricted \nto a subspace Sm, 2k = m = 2k+1 – 2......Page 73
2.9.1 Motivation......Page 74
2.9.2 The real exponential function......Page 75
2.9.4.1 ex in Ak, k ≥ 2......Page 76
2.9.4.2 The exponential map is onto A k, k ≥ 1......Page 77
2.9.4.3 The exponential of a product x × u in Ak, k ≥ 2......Page 78
2.9.5 When does [eX, eY ] = 0 for X, Y ∈ Ak, k ≥ 2?......Page 79
2.9.6.1 The purely metric approach in  Ak, k ≥ 2......Page 80
2.9.6.3 Right-angled triangles......Page 82
2.9.7.1 Preliminaries......Page 84
2.9.7.2 The formula (2.9.11) under the light of trigonomety......Page 86
2.9.7.3 The particular case t = 1......Page 87
2.9.7.5 An epistemological perspective......Page 88
2.9.8 Summary......Page 89
2.9.9 The real zeros of the ζ function......Page 91
2.10.1 FTA in C......Page 94
2.10.3 Polynomials in the variable x ∈ Ak, k ≥ 2, with \nreal coefficients......Page 95
2.10.4 A topological extension in Ak, k ≥ 2......Page 96
2.11.1 The imaginary units under trigonometric analysis......Page 97
2.12 Bibliographical notes......Page 98
3.1 The multiplication tables in An, n ≥ 0......Page 100
3.1.2.1 Connection with inductive computation......Page 101
3.2 The algorithmic computation of the standard multiplication table Mn......Page 102
3.3.1 The index correspondence Dn......Page 104
3.3.2 The sign matrix Sn associated with ei × ej → ±ek......Page 105
3.4.2 The sign matrices......Page 107
3.5.1 The level m-expansion for z ∈ Ak, 0 < m < k......Page 108
3.5.2 Variable complexity within Ak, k ≥ 2......Page 109
3.5.3 Expressive coupling......Page 110
3.5.4 Multipure subspaces in Ak, k ≥ 2......Page 111
3.6.2 The product z3 = z1 × z2 in k-m......Page 112
3.6.3 An emerging product in  k–m for k   3......Page 115
3.7.1 Definition......Page 116
3.7.3 Der (Ak) for k ≥ 4......Page 117
3.8.2 The nonlinear core of Ak......Page 119
3.8.3 Reducibility by derivation in Dickson algebras......Page 121
3.8.5 Epistemological principles of hypercomputation......Page 122
3.9.1 A global summary......Page 124
3.9.2 Information derived from space and time......Page 125
3.9.3 Eidetic computation in Ak, k ≥ 5......Page 127
3.9.4 About Reason......Page 128
3.10 Bibliographical notes......Page 129
4.1.1 x is alternative......Page 130
4.2.1 The eigenvalues of –L2 \na in a......Page 131
4.2.2 The eigenvalue 1 =  a 2 = N(a), a in Dk, k ≥ 4......Page 134
4.2.4 Commuting pairs in Dk......Page 136
4.3.1 The pythagorean rule......Page 137
4.3.2 The algebra generated by the pair (x,  ), k ≥ 2......Page 138
4.3.3 When x = a + b, a and b alternative in  Ak and  a, b = 0......Page 139
4.4.1 The 2 × 2 block representation of -L2......Page 140
4.4.2 a and b are alternative in  Ak, k ≥ 3......Page 141
4.4.3 The multiplicity of N( )......Page 144
4.4.4 The spectral information carried by a × b......Page 145
4.5 Zerodivisors with two alternative parts in  Ak, k ≥ 3......Page 147
4.6   = (a, b) has alternative, orthogonal parts with equal \nlength in  Ak, k ≥ 3......Page 152
4.7.2 SVD for Lx in A4......Page 154
4.8.2 Not necessarily alternative parts in  Ak, k ≥ 4......Page 155
4.8.3 The vector θt = (a, t  \nk) with a ∈ Dk, t ∈ R*, k ≥ 4......Page 158
4.8.4   = (a, b) with a, b ∈ Dk, k ≥ 3......Page 159
4.8.5 The vector ψ = (a, ) for a not alternative in Dk, \nk ≥ 4......Page 160
4.8.6 max dimZer ( ) for   ∈ Dk, k ≥ 4......Page 162
4.8.7 About the growth of  Ld/ d  in  Ak, k ≥ 4......Page 164
4.9 Bibliographical notes......Page 166
5. Computation Beyond Classical Logic......Page 168
5.1.1 The head-tail split......Page 170
5.1.2 Local derivations in Ak, k ≥ 3......Page 171
5.1.3 Nonassociativity of addition deriving from local SVD......Page 173
5.2.2 Is the local SVD derivation absurd?......Page 175
5.3.1 Threefold partition for C......Page 176
5.3.2 Characteristic curves and points in C  \nfor a in Ak, \nk ≥ 3......Page 177
5.4.1 k = 0 to 3......Page 178
5.4.2 Measuring a vector a = h + t in Ak, k ≥ 3......Page 179
5.4.4 F3(t) modified by C: a = β  \n+ (α1 + t)......Page 180
5.4.5 The dependence on λ ∈ σt of the geometric frame for\n\na = h + t in Ak, k ≥ 4......Page 182
5.5.1 Induction into Ak+1 by a in Ak = C  \n⊕ Dk, k ≥ 2......Page 183
5.5.2 The eigenvalues of  –L2 \ni\' , i\' = 0, 1 for ß   0......Page 185
5.5.3 Global singular values for L  , l = 0 to 7 when α β  \n0, k ≥ 2......Page 187
5.5.4 The case αβ = 0......Page 188
5.6.1 j = 1, αβ   0......Page 189
5.6.3 Summary for k ≥ 3......Page 191
5.6.4 Pseudo-zerodivisors in  Ak+1, k ≥ 3......Page 192
5.6.5 Characteristic curves and points for  in Ak+1, k ≥ 3......Page 194
5.6.6 The contextual measures for a ∈ Ak, k ≥ 2......Page 196
5.7.1 About   × v and  × v......Page 197
5.7.2 The metric equivalence to  × v = 0 in D4......Page 198
5.7.3   × v = 0 is not equivalent to   × v  = 0 in Dk, \nk ≥ 5......Page 199
5.8.1 An overview......Page 204
5.8.2 Nonassociativity ⇒ SVD paradoxes......Page 205
5.9 Bibliographical notes......Page 206
6.1.1   ∈ Dk+1 is a regular zerodivisor......Page 208
6.1.3 The subalgebra Dk versus C2 k–1–1, k   2......Page 209
6.2.2 Real and complex self-inductions......Page 211
6.2.3 Real self-induction with a alternative in  Ak, k ≥ 3......Page 212
6.3.1 s = (a, b) such that a × b ∈ C  \nfor a, b ∈ Dk......Page 213
6.3.2 The octonionic structure generated by s = (a, a × h)......Page 214
6.4.1 The spectrum of – L2\ns, a   Dk,  a  = 1......Page 215
6.4.2 ψ = (a, ) when Zer (a) = {0}, a ∈ Dk......Page 216
6.4.3 About the complete spectrum σs∪s for s = (a, a×h), \na  = 1, k ≥ 4......Page 217
6.4.5 False zeroproducts and spurious eigenelements......Page 219
6.5.1 The manifold V2( Ak)......Page 220
6.6.1 M(Am,Ak)......Page 221
6.6.3 Subsets of monomorphisms for k ≥ m, 1 ≤ m ≤ 3......Page 222
6.6.5 MQ(3, k + 1)......Page 223
6.7.2 Scalar involution in quadratic algebras over a ring......Page 224
6.7.3 The case \n ～= Z2......Page 225
6.7.5 Inward and outward derivations on A in a complexi- fied context......Page 227
6.7.6 The dependence of Int (A) on Zer (3)......Page 230
6.7.7 The dependence of Der (B) on Zer (2) and Zer (3)......Page 231
6.7.8 The case of the scalar field R......Page 233
6.7.9 Summary......Page 234
6.8.3 The logistic iteration......Page 237
6.8.4 The successive iteration (6.8.2) in closed form for h = 1/2, 1 and -1/2......Page 240
6.8.5 The complex dynamics expressed by (6.8.2)......Page 241
6.8.6 A geometric picture in 2D......Page 243
6.8.8 The corresponding evolution of t under h = |t| in I......Page 244
6.9.2 Two basic examples with unobservable periods......Page 246
6.9.3 Algorithmic complexification with Th......Page 247
6.9.4 Algorithmic complexification with Bh......Page 248
6.10.1 The periodic sine function......Page 250
6.10.2 The sine iteration and its local dynamics for h ∈ I......Page 251
6.10.3.1 h   [–1, –1/ ]......Page 252
6.10.3.2 h  [–2, –1]......Page 254
6.10.4 The global dynamics for h ∈ R......Page 256
6.11.1 The induced evolution of sgn z = z/|z|, z ∈ C*......Page 259
6.11.3 Approximation by successive iterations......Page 260
6.12 Bibliographical notes......Page 261
7. Homotopic Deviation in Linear Algebra......Page 14
7.1 An introduction to complex Homotopic Deviation......Page 264
7.1.2 The synthesis A(∞)......Page 265
7.1.3 The spectrum σ(A(t))......Page 266
7.2.2 The deviation matrix E = UV H......Page 267
7.2.4 The communication matrix Mz for z ∈ re(A)......Page 268
7.2.5 Arithmetic properties of z . VH adj(zI  – A)U......Page 269
7.2.6 A is in companion form......Page 272
7.2.7 Scale invariance of the communication process......Page 275
7.2.8 The frontier set in re(A), for r < n......Page 276
7.2.9 Characterization of F(A,E) for r < n......Page 277
7.2.10 The characteristic polynomial π(t, z) for A(t)......Page 278
7.3.1 Existence of R(t, z), t ∈ C......Page 280
7.3.2 Analyticity for t ∈  \nwhen    0......Page 281
7.3.3 The critical set when F(A,E) = re(A)......Page 282
7.3.4 The spectral quotient for (A,E)......Page 285
7.3.5 Analyticity around 0 and ∞......Page 287
7.3.5.2 The case r = 1......Page 288
7.3.6 Observation points with a unit spectral quotient q(z)\n\nfor r ≥ 2......Page 290
7.3.7 Resolution nodes in U......Page 291
7.3.8 Summary......Page 293
7.4.1 The spectral orbits and rays......Page 294
7.4.2 The eigenvalues of A(t) and E(s)......Page 295
7.4.3 The “eigenvalues” of A(∞)......Page 296
7.5.1 Characterization of Lim when g = m......Page 300
7.5.2.2 Zer (X)   {0} for X ∈ Dk, k ≥ 4......Page 305
7.5.3 The Cauchy interlace theorem revisited......Page 307
7.5.4 Lim when g < m......Page 309
7.5.5 Connecting | | and | |......Page 311
7.5.6 No interaction between blocks of different size......Page 312
7.5.7 Comparing   A and......Page 315
7.5.8 Convergence of the mean  \n(t) = (1/m) m \ni=1 λi(t)......Page 316
7.5.9 Summary......Page 318
7.6.1 (7.6.1) ⇒ (A,E) = F(A, E)......Page 321
7.6.3   =   = C......Page 322
7.7.1 Definition......Page 323
7.7.2 The algebraic structure of B......Page 324
7.7.3 Partial contour integrations of R(t, z) around (∞, ξ)......Page 325
7.7.5 Comparing the multiplicities of ξ......Page 326
7.7.6 The coincidence a  =    = r at a critical point......Page 328
7.7.7 At a (not critical) frontier point......Page 329
7.7.9 Epistemological significance......Page 332
7.8.1 Various notions of observability at λ ∈ σ(A)......Page 335
7.8.2 Study of Lim ∩ σ(A)......Page 336
7.8.4 limz→-  (z) det Mz, z ∈ re(A).......Page 337
7.8.5  = deg   0......Page 340
7.8.6 The possibility of local organisation when    = 0......Page 341
7.8.7 The particular case Lim = σ(A)......Page 343
7.8.8.1  = 1......Page 344
7.8.8.2  = 0......Page 345
7.8.8.3 Adding colour for r ≥ 2......Page 346
7.8.9 Summary......Page 347
7.9.2 The subset .D\n\n⊂ D......Page 348
7.9.3 Effective eigenvalues for Mz......Page 349
7.10.1 z ∈ re(A)......Page 350
7.10.4 λ ∈ σi......Page 351
7.11.3 The augmented matrix of order   =   + p......Page 352
7.11.5 The algebraic factorisations of   : C3 → C......Page 356
7.11.6 Two rational representations for  (t, z, u) as a \nproduct of 3 or 4 factors......Page 357
7.12.2 The spectral field t → σ(A(t))......Page 358
7.12.3 The portraits in 3D......Page 360
7.13 Bibliographical notes......Page 363
8. The Discrete and the Continuous......Page 364
8.1.1 The Dickson-Albert doubling process over Z2......Page 365
8.1.3 The partition Bk = Ek ⊕ Ok, k ≥ 1......Page 367
8.1.5 The sequence of ones (11 · · · 1) = e2k in Ek, k ≥ 1......Page 369
8.1.6 The set Zer (a) in Bk, k ≥ 1......Page 370
8.2.1 k = 1: multiplication mod 4......Page 371
8.3.3 Inexact multiplication mod 16 for the first 16 elements in B3......Page 373
8.4 The linear space Cn of binary sequences of length n ≥ 1......Page 374
8.4.2 Total order for n = 1 and 2......Page 375
8.4.4 Left and right measures in base 2......Page 376
8.5.2 The complex order in B1 versus the linear order in C2......Page 377
8.5.3 The partial complex order for k ≥ 2......Page 378
8.6.1 Introduction......Page 379
8.7.1 Hilbert tenth problem......Page 381
8.7.2 Exponential diophantine equations......Page 382
8.7.4 Algorithmic information theory for Turing machines......Page 383
8.8.1 Definition......Page 385
8.8.2 The triangle modulo a prime number......Page 386
8.8.3 The central binomial coefficients (2 n) mod p, n  1......Page 388
8.8.3.3 Other rational connections......Page 390
8.8.4 Introduction to the primes 2 and 3......Page 391
8.9.1 Definition......Page 392
8.9.2 Connection with Mersenne and Fermat numbers......Page 393
8.9.3 The Sierpinski triangle as a binary computer......Page 395
8.9.4 The Sierpinski triangle in plane geometry......Page 396
8.9.5 From intuition to thought through (AT)2......Page 398
8.10.2 The cycles......Page 399
8.11.2 Back to Bachet’s integral weighing......Page 401
8.12.1 The ordinary positional notation......Page 402
8.12.2 The scientific notation......Page 403
8.12.3 Computer arithmetic......Page 404
8.12.4 The dichotomy discrete vs continuous over R+*......Page 405
8.12.5 Generalization to a real base b > 1......Page 407
8.12.6 Qualitative versus quantitative aspects of the measure of a number......Page 409
8.13.1 The Newcomb conjecture......Page 410
8.13.2 The Borel law of uniform distribution of b-digits......Page 411
8.14.1 The work of P. Levy (1939)......Page 412
8.14.3 The uniform law has a continuous support (C)......Page 415
8.14.4 The resolution of the paradox over R......Page 416
8.14.6 The epistemological value of a scientific computer versus a Turing machine......Page 417
8.15 Finite precision computation over R......Page 418
8.15.1 The division of the significand into lead and trail......Page 419
8.15.3 The distribution of the digits in the significand......Page 421
8.15.4 Influence of b   2......Page 423
8.16 A dynamical perspective on the natural integers......Page 424
8.16.1 Sharkovski’s theorem......Page 425
8.16.2 The epistemological significance of Sharkovski’s ordering for self-composition......Page 426
8.16.3 The function f depends on a real parameter......Page 428
8.16.4 The fixed-points of Lnr , Lr(x) = rx(1 - x)......Page 429
8.16.7 Summary as |h|......Page 433
8.17 Bibliographical notes......Page 434
9.1.1 The four alternative division algebras......Page 436
9.1.2 The theorem of two squares......Page 437
9.1.3 The theorem of four squares......Page 438
9.1.4 The theorem of eight squares......Page 439
9.2.2 Arithmetic with other complex quadratic integers......Page 440
9.2.3 The associative ring R2 of Hurwitz (1896)......Page 443
9.2.4 The alternative ring R3 of Dickson (1923)......Page 444
9.2.5 All ideals in R3 are principal and 2-sided......Page 446
9.2.6 The root vectors for D......Page 447
9.2.7 A first epistemological pause......Page 450
9.3.2 Arithmetic isometries in H......Page 451
9.4.1 Definition......Page 452
9.4.3 The canonical rates of association......Page 453
9.5.2 Study of c  in D......Page 454
9.5.3 Study of d  in D......Page 456
9.6.1 N = 1: Unification by arithmetic multiplicative closure in D×......Page 457
9.6.2 Norms as sums of 1, 2 or 4 identical squares......Page 458
9.6.4 Arithmetic creativity for k = 2, 3......Page 460
9.7.1 The second cycle (f5, f6, f7)......Page 461
9.7.2 The seed lattice of the second kind......Page 462
9.7.4 Study of d......Page 463
9.8 Conclusion......Page 464
9.9 Bibliographical notes......Page 465
10. The Real and the Complex......Page 466
10.1.2 Perspectives on the notion of inclusion......Page 467
10.2.1 Set theory and classical logic......Page 468
10.2.2 Fuzzy set theory......Page 469
10.2.3 Relativity of the concept of setwise inclusion......Page 470
10.2.4 The evolution of logic under iteration with Lh......Page 471
10.2.5 An epistemological pause......Page 473
10.2.6 Comparison between setwise and algebraic inclusions......Page 474
10.3 Isophasic inclusion inside C by exponentiation......Page 476
10.4.1 Setting the problem......Page 477
10.4.2 The descriptive variable is ρ = |z| > 0......Page 478
10.4.3 The descriptive variable is x = Rz   R......Page 480
10.4.4 Comparison between (10.4.2) and (10.4.4)......Page 481
10.4.4.1   = -e   µ   -1/e and µ > 0......Page 482
10.4.4.2   < -e   -1/e < µ < 0......Page 483
10.4.6 Isometric inclusion inside C......Page 484
Part II. About Complex Signals......Page 487
10.5.3 The Cantor set (or discontinuum) in [0, 1]......Page 488
10.5.4 Spectral analysis in lp(N*)......Page 489
10.6.1 Sequences in lp(Z)......Page 490
10.6.2 The commutative Banach algebra l1(Z)......Page 492
10.7.3 A complex signal......Page 493
10.7.4 The algebraic/transcendental dichotomy in C......Page 494
10.8 The continuous Fourier transform as a cognitive tool......Page 496
10.8.1 The Fourier integral......Page 497
10.8.2 The spectral analysis of TF......Page 498
10.8.4 The decidability of elementary geometry......Page 500
10.9.1 Properties of the real scalar product  (f) for f......Page 501
10.9.2 The epistemological role of   and   applied to f......Page 503
10.9.4 Application to a complex signal......Page 506
10.9.5 The signal h = zf for f......Page 507
10.10 Bibliographical notes......Page 510
11. The Organic Logic of Hypercomputation......Page 512
11.1.1 The quadratic diophantine equation for b,......Page 514
11.1.2 Complex integers in the basis {1, b}......Page 515
11.1.3 The organic notation when n  2......Page 516
11.1.4 The organic integers for n   2......Page 518
11.1.5 Back (ward) analysis......Page 521
11.2.1 Description of the inductive points of norm n ≥ 2......Page 524
11.2.2 The scalar product σ =  b, ξ......Page 527
11.3.1 The complex vision map for n ≥ 1......Page 528
11.3.2 About the arithmetic algorithm......Page 530
11.3.4 A study of |vis(z)| for |z| bounded from below......Page 531
11.4.1 Analysis at z   0 in X0......Page 532
11.5 The rings Z[bt], |bt|2 = n ≥ 2 for hyperarithmetic......Page 533
11.5.2 Change of basis bt   1   – bt......Page 534
11.5.3 The two fundamental rings Z[i] and Z[j]......Page 535
11.5.5 The arithmetic monad......Page 536
11.6.1 The emergence of π by exponentiation in H......Page 537
11.6.3 Complex analysis C → C: holomorphy......Page 538
11.7.3 The multiplicative interpretation......Page 539
11.7.5 A sociological remark......Page 540
11.8.2 Comparison between SV (a) and CM(a)......Page 542
11.8.3 The formula D makes the local geometry appear one dimensional......Page 543
11.8.4 Local metric information for a in Ak, k ≥ 3......Page 544
11.8.5 A homogeneous formulation when   > 0......Page 547
\n11.9 The angles θj =  (a, aj) for j = 1 to 4......Page 550
11.10 About the coincidence of a with one of the aj when |α| = \n|β|   0......Page 552
11.10.3 A qualitative analysis......Page 553
11.10.5 Summary......Page 554
11.11.1 An emergent law for the evolution of θj , j = 2, 4......Page 555
11.11.3 The limit N(h) → ∞......Page 556
11.12.1 The transformation t  → dy = y  \n+ t by homotopy, \ny ∈ R......Page 557
11.12.2 The transformation t  → ex = t + x by homotopy, \nx ∈ R......Page 558
11.12.3 The 2D-evolution t  → a = x + y + t, x, y ∈ R......Page 559
11.13.1 Pure imaginary evolution along   from t......Page 560
11.13.3 The general evolution in Ak......Page 561
11.14 Bibliographical notes......Page 562
12.1.1 The η function for s ∈ C......Page 564
12.2.1 Introduction......Page 565
12.2.3 Ak = Am * Ak–m, 0   m   k for k   2......Page 566
12.3.2 The zeros with real part in ]0, 1[, within Ak, k ≥ 1......Page 567
12.4.1 The emergence of sense by integration of   – 1......Page 568
12.4.3 The commutative fields R and C in the light of hypercomputation......Page 569
12.4.4 Turing machines vs hypercomputation......Page 570
12.5 The algebraic reductions with p = 1/2......Page 571
12.5.4 The algebras with zerodivisors......Page 572
12.6.1 Thinking based on R vs C......Page 573
12.6.2 Synchronicity vs randomness in C......Page 574
12.6.3 The shadows of randomness......Page 576
12.6.5 Eastern vs Western philosophies......Page 577
12.6.6 The evolution of philosophy in Europe......Page 579
12.7.1 Looking for meaning......Page 580
12.7.2 Looking backward......Page 581
12.7.3 Looking forward......Page 582
12.7.4 Looking ahead......Page 583
Bibliography......Page 584
Index......Page 594