Table of Contents |
pt. I | Abstract Algebra and Applications | |
1. | Algebra and Geometry Through Hamiltonian Systems / Andrei Konyaev | 3 |
1.1. | Introduction | 3 |
1.2. | Atoms and Their Symmetries | 4 |
1.3. | Integer Lattices of Action Variables for "Spherical Pendulum" System | 7 |
1.4. | Billiards in Confocal Quadrics | 10 |
1.5. | Bertrand's Manifolds and Their Properties | 14 |
1.6. | Lie Algebras with Generic Coadjoint Orbits of Dimension Two | 17 |
| References | 19 |
2. | On Hyperbolic Zeta Function of Lattices / N. N. Dobrovolsky | 23 |
2.1. | Introduction | 23 |
2.1.1. | Lattices | 24 |
2.1.2. | Exponential Sums of Lattices | 26 |
2.1.3. | Multidimensional Quadrature Formulas and Hyperbolic Zeta Function of a Grid | 29 |
2.1.4. | Hyperbolic Zeta Function of Lattices | 34 |
2.1.5. | Generalised Hyperbolic Zeta Function of Lattices | 40 |
2.2. | Functional Equation for Hyperbolic Zeta Function of Integer Lattices | 45 |
2.2.1. | Periodized in the Parameter b Hurwitz Zeta Function | 46 |
2.2.2. | Dirichlet Series with Periodical Coefficients | 47 |
2.2.3. | Functional Equation for Hyperbolic Zeta Function of Integer Lattices | 50 |
2.3. | Functional Equation for Hyperbolic Zeta Function of Cartesian Lattices | 52 |
2.4. | On Some Unsolved Problems of the Theory of Hyperbolic Zeta Function of Lattices | 59 |
| References | 60 |
3. | The Distribution of Values of Arithmetic Functions / G. V. Fedorov | 63 |
| References | 66 |
4. | On the One Method of Constructing Digital Control System with Minimal Structure / V. V. Palin | 67 |
4.1. | The Statement of Problem and Some Familiar Results | 67 |
4.2. | Definitions and Some Preliminary Transformations | 68 |
4.3. | The Method to Obtain the Characteristic of Completely Controllable | 69 |
4.4. | Auxiliary Statements | 69 |
4.5. | The Absence of Associated Vectors Case | 70 |
4.6. | The Case of General Position | 71 |
| Reference | 71 |
5. | On Norm Maps and "Universal Norms" of Formal Groups over Integer Rings of Local Fields / Nikolaj M. Glazunov | 73 |
5.1. | Introduction | 73 |
5.2. | Norm Maps | 75 |
5.3. | Results | 78 |
| References | 80 |
6. | Assignment of Factors Levels for Design of Experiments with Resource Constraints / N. A. Oriekhova | 81 |
6.1. | Introduction | 81 |
6.2. | Hansel Method | 82 |
6.3. | Modification | 83 |
6.4. | Example | 85 |
6.5. | Conclusions | 86 |
| References | 86 |
pt. II | Mechanics and Numerical Methods | |
7. | How to Formulate the Initial-Boundary-Value Problem of Elastodynamics in Terms of Stresses? / D. V. Georgievskii | 89 |
7.1. | The Classic Formulation of the Dynamic Problem and Its Peculiarities | 89 |
7.2. | Ignaczak--Nowacki' Formulation | 91 |
7.3. | Konovalov' Formulation | 92 |
7.4. | Pobedria' Formulation | 93 |
7.5. | One More Possible Formulation | 93 |
| References | 95 |
8. | Finite-Difference Method of Solution of the Shallow Water Equations on an Unstructured Mesh / A. V. Drutsa | 97 |
8.1. | Introduction | 97 |
8.2. | Formulation of the Problem | 97 |
8.3. | Mesh and Mesh Operators | 98 |
8.4. | Finite-Dimensional Problem | 100 |
8.5. | Convergence | 101 |
8.6. | Results of Numerical Experiments | 104 |
8.6.1. | Estimation of Convergence Order | 104 |
8.6.2. | Computation of the Real Geographic Domain | 105 |
| References | 113 |
9. | Dynamics of Vortices in Near-wall Flows with Irregular Boundaries / O. V. Homenko | 115 |
9.1. | Introduction | 115 |
9.2. | Model of Standing Vortex | 117 |
9.3. | Standing Vortex in Cross Groove | 119 |
9.4. | Standing Vortex in an Angular Region | 121 |
9.5. | Resonant Properties of Standing Vortices and Their Behavior in Perturbed Flow | 123 |
9.6. | Summary | 128 |
| References | 128 |
10. | Strongly Convergent Algorithms for Variational Inequality Problem Over the Set of Solutions the Equilibrium Problems / Vladimir V. Semenov | 131 |
10.1. | Introduction | 131 |
10.2. | Preliminaries | 134 |
10.3. | Convergence Analysis | 135 |
10.4. | Concluding Remarks | 145 |
| References | 145 |
pt. III | Long-time Forecasting in Multidisciplinary Investigations | |
11. | Multivalued Dynamics of Solutions for Autonomous Operator Differential Equations in Strongest Topologies / Pavlo O. Kasyanov | 149 |
11.1. | Introduction: Statement of the Problem | 149 |
11.2. | Additional Properties of Solutions | 151 |
11.3. | Attractors in Strongest Topologies | 158 |
11.4. | Application | 160 |
11.5. | Conclusions | 161 |
| References | 161 |
12. | Structure of Uniform Global Attractor for General Non-Autonomous Reaction-Diffusion System / Mikhail Z. Zgurovsky | 163 |
12.1. | Introduction | 163 |
12.2. | Setting of the Problem | 164 |
12.3. | Multivalued Processes and Uniform Attractors | 165 |
12.4. | Uniform Global Attractor for RD-System | 174 |
| References | 180 |
13. | Topological Properties of Strong Solutions for the 3D Navier-Stokes Equations / Nina V. Zadoianchuk | 181 |
13.1. | Introduction | 181 |
13.2. | Topological Properties of Strong Solutions | 183 |
13.3. | Proof of Theorem 13.2 | 184 |
13.4. | Proof of Theorem 13.1 | 185 |
| References | 187 |
14. | Inertial Manifolds and Spectral Gap Properties for Wave Equations with Weak and Strong Dissipation / Natalia Chalkina | 189 |
14.1. | Introduction | 189 |
14.2. | Statement of the Problem and Spectrum of the Linear Operator | 191 |
14.3. | Sufficient Conditions for the Existence of Inertial Manifolds | 193 |
14.4. | Proof of Theorem 14.3 | 197 |
14.4.1. | New Norm in the Spaces Hk, k = 1, ... k1 | 197 |
14.4.2. | New Norm in the Spaces Hk, k = k1 + 1, ..., k2 | 198 |
14.4.3. | New Norm in the Space H[∞] | 200 |
14.4.4. | End of the Proof of Theorem 14.3 | 202 |
| References | 203 |
15. | On Regularity of All Weak Solutions and Their Attractors for Reaction-Diffusion Inclusion in Unbounded Domain / Pavlo O. Kasyanov | 205 |
15.1. | Introduction | 205 |
15.2. | On Compact Global Attractor for Reaction-Diffusion Inclusion in Unbounded Domain | 208 |
15.3. | Regularity of All Weak Solutions and Their Attractors | 217 |
| References | 219 |
16. | On Global Attractors for Autonomous Damped Wave Equation with Discontinuous Nonlinearity / Liliia S. Paliichuk | 221 |
16.1. | Introduction | 221 |
16.2. | Setting of the Problem | 222 |
16.3. | Preliminaries | 223 |
16.4. | Properties of Solutions | 225 |
16.5. | The Existence of a Global Attractor | 231 |
16.6. | Global Attractors for Typically Discontinuous Interaction Functions | 232 |
| References | 237 |
pt. IV | Control Theory and Decision Making | |
17. | On the Regularities ofMass Random Phenomena / Valery A. Labkovsky | 241 |
17.1. | Introduction | 241 |
17.2. | Theorem of Existence of Statistical Regularities | 243 |
17.3. | The Proof | 246 |
17.4. | Applications in Decision Theory | 247 |
17.5. | Concluding Remarks | 249 |
| References | 249 |
18. | Optimality Conditions for Partially Observable Markov Decision Processes / Mikhail Z. Zgurovsky | 251 |
18.1. | Introduction | 251 |
18.2. | Model Description | 252 |
18.3. | Reduction of POMDPs to COMDPs and Optimality Results | 256 |
18.4. | Example | 262 |
18.5. | Conclusions | 263 |
| References | 264 |
19. | On Existence of Optimal Solutions to Boundary Control Problem for an Elastic Body with Quasistatic Evolution of Damage / Gunter Leugering | 265 |
19.1. | Introduction | 265 |
19.2. | Notation and Preliminaries | 266 |
19.3. | Radon Measures and Convergence in Variable Spaces | 270 |
19.4. | The Model of Quasistatic Evolution of Damage in an Elastic Material | 273 |
19.5. | Setting of the Optimal Control Problems and Existence Theorem for Optimal Traction | 278 |
| References | 286 |
20. | On Existence and Attainability of Solutions to Optimal Control Problems in Coefficients for Degenerate Variational Inequalities of Monotone Type / Olga P. Kupenko | 287 |
20.1. | Introduction | 287 |
20.2. | Notation and Preliminaries | 289 |
20.3. | Setting of the Optimal Control Problem | 294 |
20.4. | Compensated Compactness Lemma in Variable Lebesgue and Sobolev Spaces | 295 |
20.5. | Existence of H-Optimal Solutions | 296 |
20.6. | Attainability of H-Optimal Solutions | 297 |
| References | 300 |
21. | Distributed Optimal Control in One Non-Self-Adjoint Boundary Value Problem / O. K. Mazur | 303 |
21.1. | Introduction | 303 |
21.2. | Setting of the Problem | 304 |
21.3. | Main Results | 305 |
21.4. | Conclusions | 311 |
| References | 312 |
22. | Guaranteed Safety Operation of Complex Engineering Systems / Andrii M. Raduk | 313 |
22.1. | Introduction | 314 |
22.2. | Information Platform of Engineering Diagnostics of the Complex Object Operation | 315 |
22.3. | Diagnostic of Reanimobile's Functioning | 321 |
22.4. | Conclusion | 325 |
| References | 326 |
Appendix A | To the Arithmetics of the Bose--Maslov Condensate Statistics | 327 |
Appendix B | Numerical Algorithms for Multiphase Flows and Applications | 329 |
Appendix C | Singular Trajectories of the First Order in Problems with Multidimensional Control Lying in a Polyhedron | 331 |
Appendix D | The Guaranteed Result Principle in Decision Problems | 333 |