Coherent Manipulation and Transport of Matter Waves

Mathematical Analysis of Evolution, Information, and Complexity......Page 5
Contents......Page 7
Preface......Page 17
List of Contributors......Page 21
Prologue......Page 25
1.1 Introduction......Page 33
1.2.1 Weyl’s Seminal Work in 1911–1915......Page 34
1.2.2 The Conjecture of Sommerfeld (1910)......Page 37
1.2.3 The Conjecture of Lorentz (1910)......Page 39
1.2.4 Black Body Radiation: From Kirchhoff to Wien’s Law......Page 40
1.2.5 Black Body Radiation: Rayleigh’s Law......Page 42
1.2.6 Black Body Radiation: Planck’s Law and the Classical Limit......Page 44
1.2.7 Black Body Radiation: The Rayleigh–Einstein–Jeans Law......Page 46
1.2.8 From Acoustics to Weyl’s Law and Kac’s Question......Page 50
1.3.1 The Laplacian on the Flat Torus T(2)......Page 51
1.3.2 The Classical Circle Problem of Gauss......Page 52
1.3.3 The Formula of Hardy–Landau–Voronoï......Page 53
1.3.4 The Trace Formula on the Torus T(2) and the Leading Weyl Term......Page 54
1.3.5 Spectral Geometry: Interpretation of the Trace Formula on the Torus T(2) in Terms of Periodic Orbits......Page 56
1.3.6 The Trace of the Heat Kernel on d-Dimensional Tori and Weyl’s Law......Page 57
1.3.7 Going Beyond Weyl’s Law: One can Hear the Periodic Orbits of the Geodesic Flow on the Torus T(2)......Page 59
1.3.8 The Spectral Zeta Function on the Torus T(2)......Page 60
1.3.9 An Explicit Formula for the Remainder Term in Weyl’s Law on the Torus T(2) and for the Circle Problem......Page 61
1.3.10 The Value Distribution of the Remainder Term in the Circle Problem......Page 64
1.3.11 A Conjecture on the Value Distribution of the Remainder Term in Weyl’s Law for Integrable and Chaotic Systems......Page 66
1.4.1 The Laplace–Beltrami Operator on d-Dimensional Compact Riemann Manifolds M(d) and the Pre-Trace Formula......Page 70
1.4.2 The Sum Rule for the Automorphic Eigenfunctions on M(d)......Page 71
1.4.3 Weyl’s Law on M(d) and its Generalization by Carleman......Page 72
1.4.4 The Selberg Trace Formula and Weyl’s Law......Page 74
1.4.5 The Trace of the Heat Kernel on M(2)......Page 76
1.4.6 The Trace of the Resolvent on M(2) and Selberg’s Zeta Function......Page 77
1.4.7 The Functional Equation for Selberg’s Zeta Function Z(s)......Page 80
1.4.8 An Explicit Formula for the Remainder Term in Weyl’s Law on M(2) and the Hilbert–Polya Conjecture on the Riemann Zeros......Page 81
1.4.9 The Prime Number Theorem vs. the Prime Geodesic Theorem on M(2)......Page 83
1.5.1 Weyl’s Law for Robin Boundary Conditions......Page 84
1.5.2 Weyl’s Law for Unbounded Quantum Billiards......Page 85
1.6 A Proof of Weyl’s Formula......Page 86
1.7 Can One Hear the Shape of a Drum?......Page 91
1.8 Does Diffusion Determine the Domain?......Page 95
References......Page 96
2.1 Introduction......Page 105
2.2 The Exponential Ansatz of Magnus......Page 108
2.3 The Feynman–Dyson Series, and More General Perturbation Techniques......Page 110
2.4.1 Regular Points......Page 112
2.4.2 Singularities of the First Kind......Page 113
2.4.3 Singularities of Second Kind......Page 114
2.5.1 Asymptotic Power Series Expansions......Page 116
2.5.3 Asymptotic Existence Theorems......Page 117
2.5.4 k-Summability......Page 118
2.5.5 Multi-Summability......Page 121
2.5.6 Applications to PDE......Page 122
2.5.7 Perturbed Ordinary Differential Equations......Page 123
2.6.1 Floquet–Lyapunov Theorem and Floquet Theory......Page 124
2.6.3 The Whittaker–Hill Formula......Page 125
2.6.5 Applications to PDE......Page 126
References......Page 127
3.1.1 General Relativity and the Standard Model of Particle Physics......Page 131
3.1.2 Alternative Theories of Gravity and Historical Overview......Page 143
3.2.1 Lagrange Density and Models......Page 147
3.2.2 The Field Equations......Page 150
3.2.3 Field Equations After Symmetry Breakdown......Page 151
3.2.4 Outlook......Page 156
References......Page 163
4.1 Introduction......Page 169
4.2 Voltage-Based Models......Page 170
4.3 Changing Paradigm – From Biological Networks of Neurons to Artificial Neural Networks......Page 174
4.4 Numerical Simulation of Neural Networks......Page 175
4.5 Population-Based Simulation of Large Spiking Networks......Page 180
4.6 Synaptic Plasticity and Developing Neural Networks......Page 184
References......Page 185
5.1 Introduction......Page 189
5.2 Biological Background......Page 190
5.3 Aims of Modeling......Page 192
5.4 Modeling Techniques......Page 193
5.5 Modeling GRNs with Boolean Networks......Page 194
5.6 Dynamic Behavior of Large Random Networks......Page 197
5.7 Inference of Gene Regulatory Networks from Real Data......Page 201
5.7.2 Identifying Algorithms......Page 202
5.7.3 Noisy Data and the Data First Approach......Page 203
5.7.4 An Information Theoretical Approach......Page 206
5.8 Conclusion......Page 207
References......Page 209
6.1 Symmetries......Page 213
6.2 Quantum Graphs......Page 217
6.3 Energy Methods for Schrödinger Equations......Page 218
6.4 Symmetries in Quantum Graphs......Page 222
6.5 Schrödinger Equation with Potentials......Page 224
6.6 Concluding Remarks and Open Problems......Page 225
References......Page 227
7.1 Introduction......Page 229
7.2 System Architecture......Page 231
7.3.1.1 Feature Extraction......Page 234
7.3.1.2 Feature Compression......Page 235
7.3.2.2 Fixed-Point Arithmetic......Page 236
7.4 Speech Recognition Systems Based on Associative Memory......Page 237
7.4.1.1 Acoustic Models......Page 238
7.4.2.1 Neural Associative Memories......Page 239
7.4.2.2 The Neural Associative Memory-Based Architecture for Word Recognition......Page 241
7.4.2.3 The Functionality of the Architecture......Page 242
7.5 Words to Semantics Conversion using Associative Memory......Page 243
7.5.1 Spoken Word Memory......Page 244
7.5.2 Language Parser......Page 245
7.5.3 Ambiguities......Page 246
7.6 Sample System/Experimental Results......Page 247
7.7 Conclusion......Page 248
References......Page 249
8.1 Introduction......Page 251
8.2 An Overview of Pattern Recognition......Page 254
8.3 Utterance Classification as a Text-Classification Problem......Page 256
8.4 Utterance Corpus Description......Page 257
8.5 Utterance Preprocessing......Page 258
8.6 Feature Extraction Based on Term Clustering......Page 259
8.6.1 Term Vector of Lexical Co-occurrences......Page 260
8.6.2 Hard Term Clustering......Page 261
8.6.2.1 Disambiguation......Page 262
8.6.4 Pole-Based Overlapping Clustering......Page 263
8.6.5 Utterance Categorization......Page 264
8.7.2 Vector Model with Term Weighting for Utterance Classification......Page 265
8.7.2.2 IDF, RIDF and ISCF Scores......Page 266
8.8.1 Classification with One Labeled Utterance and Feature Extraction......Page 267
8.8.2 Classification Based on F Samples per Category......Page 268
References......Page 271
9.1 Introduction......Page 275
9.2.2 DNA Microarray Technology......Page 276
9.3 Cluster Analysis......Page 277
9.3.1 Clustering Microarray Data......Page 278
9.3.2.1 Hierarchical Clustering......Page 280
9.3.2.2 Partitional Clustering......Page 282
9.3.2.3 Incremental Updates......Page 284
9.3.2.4 Model-Based Clustering......Page 285
9.3.2.5 Spectral Clustering and Other Graph-Based Methods......Page 286
9.3.2.6 Biclustering......Page 287
9.3.3 Cluster Validation......Page 289
9.4 Semi-Supervised Clustering......Page 291
9.4.1 Modeling Background Knowledge......Page 293
9.4.2 Results......Page 294
9.4.3 Challenges......Page 295
9.5 Summary......Page 296
References......Page 298
10.1 Introduction......Page 305
10.1.1 Overall View......Page 306
10.3 Background from Physical Cosmology......Page 307
10.4 Image Formation and Characterization......Page 309
10.4.1.1 Perspective Projection......Page 310
10.4.1.3 Mollweide Projection......Page 311
10.4.2.1 Quantization Property......Page 312
10.4.2.3 Scale Space Property......Page 313
10.4.3.2 Covariance and Correlation......Page 314
10.4.3.3 Joint Histogram......Page 315
10.4.3.5 Fourier Transformation of One Observation of an RF......Page 316
10.4.3.7 Two-Point Correlation Function and Power Spectrum......Page 317
10.4.4 Image Registration......Page 319
10.4.4.1 Data Term......Page 320
10.5 Methods of Image Processing......Page 321
10.5.2.1 Gaussian......Page 322
10.5.2.2 First-Order Derivative......Page 323
10.5.2.4 Gabor Filter......Page 324
10.5.2.5 Gabor Filter Bank......Page 325
10.5.3 Morphological Filtering......Page 326
10.5.4 Extraction of Image Structures......Page 327
10.6 Invariant Features of Images......Page 328
10.6.1.2 Fourier Descriptors......Page 329
10.6.2.1 Stereography......Page 330
10.6.2.2 Topology......Page 331
10.6.3.1 Stochastic Geometry......Page 333
10.6.3.2 Integral Geometry......Page 334
10.6.3.3 Applications......Page 335
10.7 Concluding Remarks......Page 339
References......Page 341
11.1 Introduction......Page 343
11.2 Hypothesis Boosting Problem......Page 344
11.3 Learn......Page 345
11.4 Boosting by Majority......Page 347
11.5 AdaBoost......Page 348
11.5.1 Training Sample Error......Page 349
11.5.2 Generalization Error......Page 350
11.5.3 AdaBoost on Noisy Data......Page 351
11.6 BrownBoost......Page 352
11.7 AdaBoost for Feature Selection......Page 355
11.8 Conclusion......Page 357
References......Page 363
12.1 Introduction and History......Page 365
12.2.1 Applications in Theory......Page 366
12.2.2 Applications in Practice......Page 367
12.2.3 Special Case: Applications in the Field of Information Transmission......Page 368
12.3.1 Notation......Page 371
12.3.2 The Sampling Theorem......Page 372
12.3.3.1 Dirichlet’s Theorem......Page 373
12.3.3.2 A First Attempt of a Proof......Page 374
12.3.3.3 The Efficient Proof......Page 375
12.3.4.1 Tempered Distributions......Page 376
12.3.4.2 Fourier Transformation......Page 377
12.3.4.3 Inversion Theorem......Page 378
12.3.4.4 Examples......Page 379
12.3.4.5 Convolution......Page 380
12.3.4.6 The Conventional Proof......Page 382
12.3.4.7 A Convolution Theorem for a Specific Function......Page 384
References......Page 385
13.2 Introduction to Linear Codes......Page 387
13.3 Introduction to Forward Error Correction......Page 389
13.3.2 Additive White Gaussian Noise Channel, AWGN......Page 390
13.3.3 Maximum A Posteriori (MAP) and Maximum Likelihood (ML) Decoding......Page 391
13.4 Algebraic–Geometric Codes......Page 392
13.5 Computation of Riemann–Roch Spaces......Page 395
13.6 Decoding up to Half the Minimum Distance......Page 397
13.7 Interpolation-Based Decoding......Page 400
13.8 Power Decoding of Low Rate Reed–Solomon Codes......Page 404
13.9 Interpolation-Based Soft-Decision Decoding......Page 405
13.10 Soft-Decision Decoding with the Dorsch Algorithm......Page 408
References......Page 409
14.2 Introduction......Page 411
14.3 The Basic Unit: Analytical Study of the Dipole......Page 415
14.3.1 Well-Posedness Results......Page 416
14.3.2 Linearization......Page 417
14.4 The Basic Unit: Numerical Analysis of the Dipole......Page 420
14.5 Model of a Recurrent Network......Page 421
14.6 Discussion and Conclusions......Page 423
References......Page 424
15.1.1 Central Ideas......Page 427
15.1.2 Outline of the Article......Page 428
15.2 How to Factor Numbers......Page 430
15.2.1 Prime Numbers, a Primality Test and a Naive Approach to Factorization......Page 431
15.2.3 Problems with this Algorithm and Probability of Success......Page 432
15.3.1 Encoding in Quantum Systems......Page 433
15.3.2 Mapping of Periodicity......Page 435
15.3.4 Phase State......Page 436
15.3.5 Subtleties......Page 437
15.4.1 Scattering Atoms off a Standing Wave......Page 438
15.4.2 Method of Stationary Phase......Page 439
15.4.3 Interference in Phase Space......Page 440
15.5 Exponential Growth of Hilbert Space as a Resource of Exponential Speedup......Page 441
15.6 Conclusions......Page 442
15.A.1 Basic Idea......Page 443
15.A.3 Modular Exponentiation......Page 444
15.A.3.2 Periodicity......Page 445
15.B.1 Residues Represented as a Matrix......Page 446
15.B.3 Period of Function from Chinese Remainder Theorem......Page 448
15.C.1 Definition......Page 449
15.C.3 A Compact Formula for φ......Page 450
15.E Primitive Root......Page 452
15.E.2 Periods for Prime Numbers......Page 453
15.F.1 Expression for the Period......Page 455
15.F.2 Analysis of Different Cases......Page 456
15.G Elements of Atom Optics......Page 457
15.G.2 Momentum Distribution......Page 458
15.G.3 Discreteness of Momentum due to Interference......Page 459
15.H Factorization with a Gauss Sum due to its Periodicity......Page 461
References......Page 462
16.1 Introduction......Page 465
16.2 Factorization of Integers: Classical Algorithms......Page 466
16.3 Graph Isomorphism: Classical Algorithms......Page 467
16.4 Quantum Algorithms for Integer Factorization......Page 468
16.4.1 The Quantum Fourier Transform and Period Finding......Page 469
16.4.2 Generalization of the Period-Finding Algorithm......Page 471
16.5.1 The Hidden-Subgroup Problem and Graph Isomorphism......Page 475
16.5.2 The Quantum Query Model and Graph Isomorphism......Page 476
16.5.3 Quantum Walks and the Fix-Automorphism Problem......Page 477
16.6 Reductions of Integer Factorization and Graph Isomorphism to Ring Isomorphism......Page 479
16.6.1 Factoring Integers and Finding Ring Automorphisms......Page 480
16.6.2 Graph Isomorphism and Ring Isomorphism......Page 481
References......Page 483
17.1 Introduction......Page 487
17.1.1 Recursion for the Expected Number of Comparisons......Page 489
17.2 An Upper Bound......Page 490
17.3 A Lower Bound......Page 491
17.4 The δ-Random Source......Page 494
Further Reading......Page 496
Index......Page 497