Noise, Oscillators and Algebraic Randomness: From Noise in Communication Systems of Number Theory Lectures of a School Held in Chapelle des Bois, France, April 5–10, 1999

Preface......Page 5
Contents......Page 6
Michel Planat......Page 8
1 Introduction......Page 13
2.1 The pow r of exponentials......Page 14
2.2 Taylor ’s formula and exponential......Page 15
2.3 Leibniz ’s formula......Page 17
2.4 Exponential vs. logarithm......Page 20
2.5 Infinitesimals and exponentials......Page 23
2.6 Differential equations......Page 26
3.1 An algebraic digression: umbral calculus......Page 31
3.2 Binomial sequences of polynomials......Page 33
3.3 Transformation of polynomials......Page 36
3.4 Expansion formulas......Page 38
3.5 Signal transforms......Page 40
3.6 The inverse problem......Page 42
3.7 A probabilistic application......Page 45
3.8 The Bargmann-Segal transform......Page 47
3.9 The quantum harmonic oscillator......Page 50
4.1 Some divergent series......Page 53
4.2 Polynomials of infinite degree and summation of series......Page 57
4.3 The Euler-Riemann zeta function......Page 59
4.4 Sums of powers of numbers......Page 61
4.5 Variation I: Did Euler really fool himself?......Page 65
4.6 Variation II: Infinite products......Page 70
5 Conclusion: From Euler to Feynman......Page 72
References......Page 74
1 Non-ideal Quantum Measurements......Page 75
2.1 Dissipation and fluctuations......Page 76
2.2 Treatment with quantum .elds......Page 77
2.3 Quantum networks......Page 79
3 Fluctuations in Amplifiers......Page 80
4 The Cold Damped Accelerometer......Page 84
References......Page 87
1 Introduction......Page 88
2 The Microscopic Model......Page 93
2.1 Second quantized formalism......Page 94
Holstein –Primakov representation......Page 97
2.2 Dynamics of the model......Page 98
Form factor......Page 101
Macroscopic response function......Page 102
Conductivity......Page 103
2.4 Conclusion......Page 104
Appendix A......Page 105
Invariance of the commutation relations......Page 107
Appendix B......Page 108
Casimir operator......Page 109
References......Page 110
1 Introduction......Page 111
2.1 Ion motion in a pure quadrupole field......Page 112
2.2 Ion motion in experimental conditions......Page 113
2.3 Experimental observations......Page 115
2.4 Simplified motion equation for radial-axial couplings......Page 116
3.1 The radiofrequency domain......Page 120
3.2 The optical domain......Page 121
3.3 The Ca$^+$ at PIIM, Marseille......Page 123
4 Quantum Optics with Laser-Cooled Ions......Page 124
References......Page 126
1 Introduction......Page 129
2 Extensive Air Showers......Page 130
3 Simulation of EAS Events......Page 131
4 Analysis of Fluctuations......Page 134
5 Multifractal Analysis of the 1/f Noise......Page 137
6 Conclusions......Page 138
References......Page 139
1 Introduction......Page 141
2 Periodic SR in Bistable Dynamic Systems......Page 142
3 Periodic SR in Static Nonlinear Systems......Page 145
4 Aperiodic SR in a Nonlinear Information Channel......Page 149
5 Aperiodic SR in Image Transmission......Page 151
6 Outlook......Page 154
References......Page 155
1 Introduction......Page 160
2 Detrended Fluctuation Analysis Techniques......Page 161
3 Multiaffine Analysis Techniques......Page 164
4 Moving Average Techniques......Page 165
5 Sandpile Model for Rupture and Crashes......Page 167
6 (m,k )-Zipf Techniques......Page 169
7 Basics of i-Variability Diagram Techniques......Page 172
References......Page 173
2 Basic Definitions and Principles......Page 176
2.1 The pendulum as a frequency reference......Page 177
2.2 Damped and stationary oscillations......Page 178
2.4 Accuracy......Page 179
3 Electronic Oscillators......Page 180
4 Frequency-Domain Characterization of Frequency Stability......Page 181
4.1 The power law model......Page 182
5 Time-Domain Characterization of Frequency Stability......Page 184
True variance......Page 185
Two-sample variance......Page 186
References......Page 187
1 Introduction......Page 190
2 Background......Page 191
2.1 Double sideband (DSB) representation of noise......Page 193
3 Traditional Methods......Page 194
3.1 Instrument sensitivity......Page 195
3.2 Additional instrument limitations......Page 196
General two port devices......Page 197
Equal DUT pair......Page 198
Discriminator and delay line......Page 199
Narrow tuning range oscillators......Page 200
5 Interferometric Noise Measurement Method......Page 201
5.1 Design strategies......Page 204
5.2 Further remarks......Page 207
6 Correlation Techniques......Page 208
6.2 Noise theory of the double interferometer......Page 209
Noise floor......Page 212
Noise measurement below the thermal floor......Page 213
References......Page 214
1 Introduction......Page 217
2 Mobility Fluctuation 1/f Noise Induced by Lattice Scattering......Page 218
3 Phonon Fine Structure in the 1/f Noise of Semiconductor Devices......Page 219
4 Surface and Bulk Phonons in the 1/f Noise of Metals......Page 220
5 1/f Noise Induced by Surface and Bulk Atomic Motion......Page 221
6 Physical Significance of the 1/f Noise Parameter......Page 222
7.1 Metals......Page 223
7.2 Semiconductors......Page 225
8 Conclusion......Page 228
References......Page 229
1 Introduction......Page 233
2 Conventional Quantum 1/f Effect......Page 234
3 Derivation of the Coherent Quantum 1/f Noise Effect......Page 237
4 Sufficient Criterion for Fundamental 1/f Noise......Page 239
5 Application to QED: Quantum 1/f Effect as a Special Case......Page 242
6 Derivation of the Conventional Quantum 1/f Noise Effect in Second Quantization......Page 243
7 Physical Derivation of Coherent Quantum 1/f Noise Effect......Page 248
8 Derivation of Mobility Quantum 1/f Noise in $n^+ - p$ Diodes......Page 251
9 Quantum 1/f Noise in SQUIDS......Page 254
10 Quantum 1/f Noise in Bulk Acoustic Wave and SAW Quartz Resonators......Page 255
11 A Different Approach to 1/f Noise from Frequency Mixing Experiments......Page 260
12 Discussion......Page 262
References......Page 264
1.1 Theoretical background......Page 266
1.2 Experiments......Page 270
2.1 The frequency of beat signals from diophantine approximation......Page 275
2.2 The amplitude of beat signals and the Franel–Landau shift......Page 279
2.3 Diophantine signal processing and 1/f frequency fluctuations......Page 282
2.4 The Riemann zeta function and the Riemann hypothesis and physics......Page 285
References......Page 287
1 Introduction......Page 289
2 Experimental and Computed Data......Page 290
3.1 False nearest neighbour percentage......Page 292
3.2 Correlation dimension......Page 294
4.1 Experimental time series......Page 295
4.2 Computed time series......Page 298
5 Discussion and Conclusion......Page 302
References......Page 303
1 Introduction......Page 306
2 Continued Fractions and Action of $F_2$ on $Q$......Page 307
2.1 Continued fractions and $F^+_2$......Page 308
2.2 Geometry of continued fractions......Page 310
3 Arithmetic Resolution Space......Page 312
3.1 An analogue of the dynamical devil’s staircase in number theory......Page 315
4.1 On the boundary of locking resonances......Page 316
4.2 Dynamics of transient resonance zones......Page 317
4.3 Accumulation of locking resonance zones......Page 318
4.4 Approximation property......Page 319
4.5 Length of locking resonance zones......Page 320
5 Truncated Resolution Space......Page 322
References......Page 323
1 Hamiltonian Chaos and the Standard Map......Page 325
2 The Critical Constants......Page 327
3 Complex Analytic Maps......Page 328
4 Continued Fractions and the Brjuno Function......Page 331
5 The Brjuno Series and Diophantine conditions......Page 334
6 The Brjuno Operator......Page 335
7 Application to Hölder–continuous Functions......Page 337
8 The Complexification of the Brjuno Function......Page 339
References......Page 342
1 A Few “Principles ”......Page 344
2.1 Block complexity......Page 345
Morphisms and finite automata......Page 346
Some links between these algorithms......Page 348
The Thue-Morse sequence revisited......Page 349
Real numbers with automatic continued fraction expansion......Page 350
3.2 Topological entropy......Page 351
The Wiener spectrum......Page 352
References......Page 353
1 Introduction......Page 356
2 Symbolic Dynamics at the Feigenbaum Points......Page 357
4 Entropy Analysis and Complexity at a Feigenbaum Point......Page 359
5 Other Coarse–Gr ined Statistical Properties at the Feigenbaum Points......Page 362
6 Replacements and Morphisms......Page 363
7 Digital Approach,Transcendence and Non-Normality......Page 364
8 On the Feigenbaum Constants $\\delta$ and $\\alpha$......Page 365
9 The “Standard ”Conjecture of Chaos......Page 366
10 Conclusion......Page 367
References......Page 368
1 Transcendental Values of Böttcher Functions......Page 371
2 Lehmer’s Problem and the Entropy of Algebraic Dynamical Systems......Page 373
3 Canonical Heights and Dynamical Systems......Page 375
References......Page 376
1 Introduction......Page 378
2 Notation......Page 379
3 Results......Page 380
4 An Algorithm......Page 384
5 An Example......Page 385
References......Page 386
1 The Modular Function and Elliptic Curves......Page 387
2 Complex Multiplication......Page 391
3 Schneider’s Theorem......Page 393
4 Generalisations......Page 394
References......Page 396
1 Introduction......Page 397
2 The Case of the Classical Markoff Theory......Page 399
3 More General Diophantine Equations......Page 402
4 Solving the Generalized Equations......Page 406
5 Application to the Analysis of the Markoff Spectrum......Page 408
6 A Link with the Representation of Free Groups......Page 410
References......Page 415