Applied algebraic dynamics:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
de Gruyter
2009
|
| Series: | De Gruyter expositions in mathematics
49 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XXIV, 533 S. 25 cm |
| ISBN: | 9783110203004 |
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Titel: Applied algebraic dynamics
Autor: Anashin, Vladimir
Jahr: 2009
Contents
Preface
vn
1 Algebraic and number-theoretic background. 1
1.1 Facts from number theory. 1
1.1.1 Some useful equalities and congruences. 1
1.1.2 Mobius and Euler functions, Legendre symbol. 3
1.1.3 Distribution of prime numbers. 5
1.2 Basic notions and facts from algebra. 6
1.2.1 Universal algebras. 6
1.2.2 Groups. 9
1.2.3 Rings. 14
1.3 Fields . 17
1.3.1 Finite fields . 17
1.3.2 Non-Archimedean fields. 19
1-4 p-adic numbers. 19
1.4.1 Canonical expansion of p-adic numbers . 22
1.4.2 Tree-like structure ofthe/J-adic numbers. 24
1-5 Ultrametric spaces. 24
1.6 The Haar measure. 26
1 -7 Non-Archimedean rings, m-adic numbers . 28
1.8 Extensions of the field of p-adic numbers . 29
1.8.1 Finite extensions of Qp. 29
1.8.2 The algebraic closure of Qp. 32
1.8.3 Complex / -adic numbers. 33
1.8.4 Krasner's lemma. 33
1 The Commutative Non-Archimedean Dynamics 35
2 Dynamics on algebraic structures . 37
2.1 Basic notions of dynamics. 37
2.1.1 Ergodicity and uniform distribution of sequences. 37
xx Contents
2.2 Dynamics on finite algebraic structures. 39
2.2.1 Hereditary dynamical properties and compatibility. 39
2.2.2 Ergodic polynomial transformations on finite Abelian groups
with operators. 41
2.2.3 Ergodic polynomial transformations on finite commutative
rings. 42
3 p-adic analysis. 48
3.1 Analysis in complete non-Archimedean fields. 48
3.2 Analytic functions. 51
3.3 Hensel's lemma. 52
3.4 Roots of unity. 54
3.5 Non-Archimedean normed spaces . 56
3.6 Multidimensional analysis. 57
3.7 The differentiability modulo pk. 58
3.8 Compatible functions on Zp . 62
3.8.1 Compatibility is equivalent to 1-Lipschitz. 63
3.8.2 Compatibility and differentiability . 66
3.9 Mahler expansion. 75
3.9.1 Identities modulo pk. 76
3.9.2 Mahler expansions of compatible functions. 78
3.10 Special classes of locally analytic functions . 80
3.10.1 Class £. 80
3.10.2 Class £ . 83
3.10.3 Class A . 87
4 p-adic ergodic theory. 90
4.1 Discrete dynamical systems. 90
4.2 Periodic points and their character. 90
4.3 Monomial dynamics. 93
4.3.1 Topologically transitive and minimality. 94
4.3.2 Unique ergodicity. 96
4.4 Measure-preserving and ergodic isometries on Z*. 98
4.4.1 Measure-preserving isometries . 100
4.4.2 1-Lipschitz measure-preserving functions. 102
4.4.3 1-Lipschitz ergodic functions. 105
4.5 Ergodic 1-Lipschitz transformations on Zp. 106
4.5.1 Ergodicity of affine mappings. 106
4.5.2 Ergodicity and measure-preservation in terms of coordinate
functions. 108
4.5.3 Ergodicity and measure-preservation in terms of Mahler ex-
pansion . Ill
Contents xxi
4.6 Measure-preservation and ergodicity of uniformly differentiable func-
tions on Zp . 119
4.6.1 Conditions for measure-preservation. 119
4.6.2 No uniformly differentiable 1 -Lipschitz ergodic transforma-
tions on 7Lnp, n 2. 122
4.6.3 Differentiable ergodic transformations on %p. 125
4.6.4 Measure-preservation and ergodicity of A~, £-, and ^-func-
tions . 132
4.7 Ergodic 1-Lipschitz transformations on p-a ic spheres . 148
4.7.1 1-Lipschitz ergodic transformations on spheres. 148
4.7.2 Ergodicity of .S-functions and of analytic functions . 151
4.7.3 Ergodicity of perturbed monomial mappings. 153
4.7.4 Ergodicity of =A-functions on spheres. 155
4.8 Concluding remarks to p-adic ergodic theory . 156
4.8.1 Continuous p-adic dynamics . 156
4.8.2 Non-minimal dynamics. Non-compatible dynamics. Mixing 159
5 Asymptotic distribution of cycles . 162
5.1 Monomial systems in Cp and in finite extensions of Qp. 163
5.2 Number of cycles of x h+ x" in Qp . 166
5.3 Total number of cycles . 169
5.4 Possible values of the number of cycles. 171
5.5 Probability on the set of prime numbers . 172
5.6 Distribution of cycles. 174
5.7 Expectation value and dispersion. 176
5.8 Fuzzy cycles. 180
II The Non-Commutative Non-Archimedean Dynamics 197
6 Basics of polynomial dynamics on groups. 199
6.1 Non-commutative differential calculus. 200
6.2 Bijective polynomials over finite groups. 204
7 Ergodic polynomials over groups with operators. 205
7.1 Basic properties of groups having ergodic polynomials . 206
7.2 Finite solvable groups having ergodic polynomials. 209
7.2.1 The multivariate case . 209
7.2.2 The univariate case: Nilpotent groups. 212
7.2.3 The univariate case: Solvable groups. 217
7.3 Ergodic theory for profinite groups. 232
7.3.1 Metric and measure on a profinite group. 233
xxii Contents
7.3.2 Equations, the non-commutative Hensel's lemma, and mea-
sure-preserving polynomials over profinite groups. 235
7.3.3 Ergodic polynomials over profinite groups. 237
III Applications 243
8 Automata, computers, combinatorics . 245
8.1 Automata functions are continuous. 245
8.2 Computers think 2-adically. 252
8.3 Differentiable instructions and programs. 259
8.4 Latin squares . 262
9 Pseudorandom numbers . 269
9.1 Pseudorandom generator is a dynamical system. 271
9.1.1 What pseudorandom generators are good?. 272
9.1.2 Why /7-adic ergodic theory?. 274
9.2 Congraential generators of the longest period . 275
9.2.1 Types of congruential generators . 277
9.2.2 Periods of congruential generators. 279
10 Stream ciphers. 305
10.1 How secure are congruential generators?. 306
10.2 Wreath products. 309
10.3 Counter-dependent generators. 314
10.3.1 Special output functions. 325
10.4 Generators based on multivariate functions. 328
10.5 Security issues. 334
10.5.1 The number of transitive compatible mappings. 335
10.5.2 Key recovery and intractability. 337
11 Structure of trajectories . 340
11.1 Distribution in Euclidean space. 340
11.1.1 Points falling on hyperplanes. 341
11.1.2 Lacunas . 347
11.2 Properties of coordinate sequences. 358
11.2.1 Linear and 2-adic complexities. 359
11.2.2 Structure of coordinate sequences. 366
11.3 Distribution of fc-tuples. 371
12 p-adic probability theory. 377
12.1 Historical remarks. 377
12.2 Frequency probability theory. 379
Contents xxiii
12.3 Ensemble probability. 385
12.3.1 Ensembles of infinite volumes. 386
12.3.2 The rules for working with p-adic probabilities. 391
12.3.3 Negative probabilities and / -adic ensemble probabilities . . 396
12.4 Measures. 396
12.5 p-adic probability space. 400
12.6 /7-adic probability measures on the space of binary sequences . . . . 402
12.7 Some technical p- dic results. 403
12.8 p-adic tests for randomness. 404
12.9 Some limit theorems. 408
12.10 Recursive enumeration of the set of p-adic tests. 410
12.11 No jf?-adic universal test. 413
13 p-adic valued quantization. 415
13.1 Toward quantum mechanics with p-adic valued wave functions . 415
13.2 Hilbert spaces. 417
13.3 Groups of unitary isometric operators in the /?-adic Hilbert space . . 419
13.4 Axiomatics of quantum mechanics with /?-adic valued wave functions 421
13.5 Gaussian integral and spaces of square integrable functions. 422
13.6 Gaussian representations of position and momentum operators . . . 425
13.7 One parameter groups generated by position and momentum operators 427
13.8 Operator calculus. 427
13.9 Spectrum of/j-adic position operator. 428
13.10 Concluding remarks on/?-adic quantization. 431
14 m-adic modeling in cognitive science and psychology . 433
14.1 On modeling of mental quantities. 434
14.1.1 Representation of mental states by numbers. 434
14.1.2 Encoding by branches of trees. 437
14.1.3 Dynamical system approach, artificial intelligence. 438
14.1.4 Unconscious and conscious dynamics - Freudian approach 439
14.1.5 Neuronal hierarchy. 441
14.2 Mental space. 442
14.3 Dynamical thinking in mental space. 442
14.4 Associations and ideas . 443
14.5 Neuronal realization. 444
14.6 Model of cognitive psychology. 446
14.7 Dynamics of associations and ideas. 447
14.8 Advantages of dynamical processing of associations and ideas . 448
14.9 Transformation of unconscious mental flows into conscious flows . 449
14.10 Hidden forbidden wishes, psychoanalysis . 458
14.10.1 Hysteric reactions. 460
xxiv Contents
14.10.2 Feedback control based on doubtful ideas. 461
14.11 Neuro and mental cybernetic bases for the pleasure and reality prin-
ciples . 462
14.12 Consequences for psychology and neuropsychology. 464
14.13 Consequences for psychoanalysis. 465
14.14 Psycho-robots. 467
15 Neuronal hierarchy behind the ultrametric mental space . 468
15.1 Hierarchic neural pathways. 469
15.2 Model: thinking on neuronal tree. 470
15.2.1 Mental field on the brain. 470
15.2.2 Probabilistic dynamics in the mental space. 475
15.3 Diffusion model of dynamics of statistical mental state . 478
15.3.1 Markovean body — • mind fields. 478
15.3.2 Thinking as m-adic diffusion. 479
15.3.3 Discussion. 481
15.4 Postulates. 485
16 Gene expression from dynamics in the 2-adic space . 487
16.1 Description of model. 488
16.1.1 4-adic representation of nucleotides. 488
16.1.2 DNA-reproduction and 4-adic dynamics . 489
16.2 Genetic space. 490
16.2.1 4-adic encoding of DNA and RNA. 490
16.2.2 2-adic encoding. 491
16.3 Dynamical model for degeneracy of the genetic code . 492
17 Genetic code on the diadic plane. 494
17.1 Vertebral mitochondrial and eucaryotic codes . 495
17.2 Parametrization of the set of codons by the diadic plane.' . 495
17.3 Genetic code on the diadic plane. 498
17.4 Physical-chemical regularity of the genetic code. 501
Bibliography . 503
Notation . 527
Index . 529 |
| any_adam_object | 1 |
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| spelling | Anašin, Vladimir S. Verfasser (DE-588)138538522 aut Applied algebraic dynamics by Vladimir Anashin and Andrei Khrennikov Berlin [u.a.] de Gruyter 2009 XXIV, 533 S. 25 cm txt rdacontent n rdamedia nc rdacarrier De Gruyter expositions in mathematics 49 Dynamisches System (DE-588)4013396-5 gnd rswk-swf Algebraische Struktur (DE-588)4001166-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Algebraische Struktur (DE-588)4001166-5 s DE-604 Khrennikov, Andrei 1958- Verfasser (DE-588)128568410 aut De Gruyter expositions in mathematics 49 (DE-604)BV004069300 49 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3189226&prov=M&dok_var=1&dok_ext=htm Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017735811&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Anašin, Vladimir S. Khrennikov, Andrei 1958- Applied algebraic dynamics De Gruyter expositions in mathematics Dynamisches System (DE-588)4013396-5 gnd Algebraische Struktur (DE-588)4001166-5 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4001166-5 |
| title | Applied algebraic dynamics |
| title_auth | Applied algebraic dynamics |
| title_exact_search | Applied algebraic dynamics |
| title_full | Applied algebraic dynamics by Vladimir Anashin and Andrei Khrennikov |
| title_fullStr | Applied algebraic dynamics by Vladimir Anashin and Andrei Khrennikov |
| title_full_unstemmed | Applied algebraic dynamics by Vladimir Anashin and Andrei Khrennikov |
| title_short | Applied algebraic dynamics |
| title_sort | applied algebraic dynamics |
| topic | Dynamisches System (DE-588)4013396-5 gnd Algebraische Struktur (DE-588)4001166-5 gnd |
| topic_facet | Dynamisches System Algebraische Struktur |
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