Continuous and Distributed Systems: Theory and Applications

Preface......Page 6
Contents......Page 8
Contributors......Page 14
318052_1_En_1_PartFrontmatter_OnlinePDF......Page 18
 1.1 Introduction......Page 19
 1.2 Atoms and Their Symmetries......Page 20
 1.3 Integer Lattices of Action Variables for ``Spherical Pendulum\'\' System......Page 23
 1.4 Billiards in Confocal Quadrics......Page 26
 1.5 Bertrand\'s Manifolds and Their Properties......Page 30
 1.6 Lie Algebras with Generic Coadjoint Orbits  of Dimension Two......Page 33
 References......Page 35
 2.1 Introduction......Page 38
  2.1.1 Lattices......Page 39
  2.1.2 Exponential Sums of Lattices......Page 41
  2.1.3 Multidimensional Quadrature Formulas and Hyperbolic  Zeta Function of a Net......Page 44
  2.1.4 Hyperbolic Zeta Function of Lattices......Page 49
  2.1.5 Generalised Hyperbolic Zeta Function of Lattices......Page 55
 2.2 Functional Equation for Hyperbolic Zeta Function  of Integer Lattices......Page 60
  2.2.1 Periodized in the Parameter b Hurwitz Zeta Function......Page 61
  2.2.2 Dirichlet Series with Periodic Coefficients......Page 62
  2.2.3 Functional Equation for Hyperbolic Zeta Zunction  of Integer Lattices......Page 65
 2.3 Functional Equation for Hyperbolic Zeta Function  of Cartesian Lattices......Page 67
 2.4 On Some Unsolved Problems of the Theory of Hyperbolic Zeta Function of Lattices......Page 74
 References......Page 75
3 The Distribution of Values of Arithmetic Functions......Page 78
 References......Page 81
 4.1 The Statement of Problem and Some Familiar Results......Page 82
 4.2 Definitions and Some Preliminary Transformations......Page 83
 4.4 Auxiliary Statements......Page 84
 4.5 The Absence of Associated Vectors Case......Page 85
 Reference......Page 86
 5.1 Introduction......Page 87
 5.2 Norm Maps......Page 89
 5.3 Results......Page 92
 References......Page 94
 6.1 Introduction......Page 95
 6.2 Hansel Method......Page 96
 6.3 Modification......Page 97
 6.4 Example......Page 99
 References......Page 100
Part IIMechanics and Numerical Methods......Page 101
 7.1 The Classic Formulation of the Dynamic Problem  and its Peculiarities......Page 102
 7.2 Ignaczak--Nowacki\' Formulation......Page 104
 7.3 Konovalov\' Formulation......Page 105
 7.5 One More Possible Formulation......Page 106
 References......Page 108
 8.2 Formulation of the Problem......Page 109
 8.3 Mesh and Mesh Operators......Page 110
 8.4 Finite-Dimensional Problem......Page 112
 8.5 Convergence......Page 113
  8.6.1 Estimation of Convergence Order......Page 116
  8.6.2 Computation of the Real Geographic Domain......Page 117
 References......Page 125
 9.1 Introduction......Page 126
 9.2 Model of Standing Vortex......Page 128
 9.3 Standing Vortex in Cross Groove......Page 130
 9.4 Standing Vortex in an Angular Region......Page 132
 9.5 Resonant Properties of Standing Vortices  and Their Behavior in Perturbed Flow......Page 134
 References......Page 139
 10.1 Introduction......Page 141
 10.2 Preliminaries......Page 144
 10.3 Convergence Analysis......Page 145
 References......Page 155
Part IIILong-time Forecasting in Multidisciplinary Investigations......Page 157
 11.1 Introduction: Statement of the Problem......Page 158
 11.2 Additional Properties of Solutions......Page 160
 11.3 Attractors in Strongest Topologies......Page 167
 11.4 Application......Page 169
 References......Page 170
 12.1 Introduction......Page 172
 12.2 Setting of the Problem......Page 173
 12.3 Multi-Valued Processes and Uniform Attractors......Page 174
 12.4 Uniform Global Attractor for RD-System......Page 183
 References......Page 189
 13.1 Introduction......Page 190
 13.2 Topological Properties of Strong Solutions......Page 192
 13.3 Proof of Theorem 13.2......Page 193
 13.4 Proof of Theorem 13.1......Page 194
 References......Page 196
 14.1 Introduction......Page 197
 14.2 Statement of the Problem and Spectrum of the Linear Operator......Page 199
 14.3 Sufficient Conditions for the Existence of Inertial Manifolds......Page 201
  14.4.1 New Norm in the Spaces......Page 205
  14.4.2 New Norm in the Spaces......Page 206
  14.4.3 New Norm in the Space......Page 208
  14.4.4 End of the Proof of Theorem 14.3......Page 210
 References......Page 211
 15.1 Introduction......Page 212
 15.2 On Compact Global Attractor for Reaction-Diffusion Inclusion in Unbounded Domain......Page 215
 15.3 Regularity of All Weak Solutions and Their Attractors......Page 224
 References......Page 226
 16.1 Introduction......Page 228
 16.2 Setting of the Problem......Page 229
 16.3 Preliminaries......Page 230
 16.4 Properties of Solutions......Page 232
 16.5 The Existence of a Global Attractor......Page 238
 16.6 Global Attractors for Typically Discontinuous Interaction Functions......Page 239
 References......Page 244
Part IVControl Theory and Decision Making......Page 245
 17.1 Introduction......Page 246
 17.2 Theorem of Existence of Statistical Regularities......Page 248
 17.3 The Proof......Page 251
 17.4 Applications in Decision Theory......Page 252
 References......Page 254
 18.1 Introduction......Page 256
 18.2 Model Description......Page 257
 18.3 Reduction of POMDPs to COMDPs and Optimality Results......Page 261
 18.4 Example......Page 267
 18.5 Conclusions......Page 268
 References......Page 269
 19.1 Introduction......Page 270
 19.2 Notation and Preliminaries......Page 271
 19.3 Radon Measures and Convergence in Variable Spaces......Page 275
 19.4 The Model of Quasistatic Evolution of Damage in an Elastic Material......Page 278
 19.5 Setting of the Optimal Control Problems and Existence Theorem for Optimal Traction......Page 283
 References......Page 291
 20.1 Introduction......Page 292
 20.2 Notation and Preliminaries......Page 294
 20.3 Setting of the Optimal Control Problem......Page 299
 20.4 Compensated Compactness Lemma in Variable Lebesgue and Sobolev Spaces......Page 300
 20.5  Existence of H-Optimal Solutions......Page 301
 20.6 Attainability of H-Optimal Solutions......Page 302
 References......Page 305
 21.1 Introduction......Page 307
 21.2 Setting of the Problem......Page 308
 21.3 Main Results......Page 309
 21.4 Conclusions......Page 315
 References......Page 316
22 Guaranteed Safety Operation of Complex Engineering Systems......Page 317
 22.1 Introduction......Page 318
 22.2 Information Platform of Engineering Diagnostics  of the Complex Object Operation......Page 319
 22.3 Diagnostic of Reanimobile\'s Functioning......Page 325
 22.4 Conclusion......Page 329
 References......Page 330
Appendix A......Page 331
Appendix C......Page 332
Appendix D......Page 334