Modern Mathematics and Mechanics: Fundamentals, Problems and Challenges

Preface......Page 6
 Editors......Page 8
Contents......Page 10
Contributors......Page 19
Part I Differential Geometry......Page 23
  1.1 Introduction......Page 24
  1.2 Expansion in a System of Functions with Dyadic Supports......Page 26
  1.3 Result for Systems of Translates and Dilates......Page 30
  1.4 Concluding Remarks......Page 31
  References......Page 32
   2.1.1 Hamiltonian Vector Fields......Page 33
   2.1.2 Liouville Equivalence of Hamiltonian Integrable Systems......Page 34
   2.1.3 Fomenko-Zieschang Invariants......Page 35
    2.1.3.1 The Case of Atom A......Page 36
    2.1.3.3 The Numerical Marks r and ε......Page 37
    2.1.3.5 Fomenko-Zieschang Theorem and Realization Problem......Page 38
   2.1.4 Simple Examples of Molecules......Page 39
    2.2.1.1 Fundamental Groups of the Atoms......Page 40
    2.2.1.3 Example: Seifert Manifolds......Page 42
    2.2.2.1 Computation of H1(Q3, Z)......Page 43
    2.2.2.2 Example: Poincare Sphere......Page 44
    2.3.1.1 The Torsion of Atoms......Page 45
    2.3.2.1 The Case of Zero r-Marks......Page 46
   2.4.1 Introduction......Page 47
   2.4.2 Main Results......Page 48
  References......Page 49
  3.1 Introduction......Page 51
  3.2 Circulant Matrices: Definitions and Basic Results......Page 52
  3.3 Basic Notions on Discrete Dynamical Systems......Page 53
   3.3.1 Periodic Orbits and Topological Dynamics......Page 54
   3.3.2 Dynamics of Continuous Interval Maps......Page 56
   3.3.3 Piecewise Monotone Maps: Entropy and Attractors......Page 58
   3.3.4 Computing Topological Entropy......Page 60
   3.3.5 Dynamics in Higher Dimension......Page 61
  3.4 Application to Oligopoly Dynamics......Page 62
   3.4.1 Puu–Norin\'s Oligopoly......Page 65
   3.5.1 Chemical Reactions: Belushov–Zhabotinsky Chemical Reaction......Page 70
   3.5.2 Application to Biological Systems......Page 71
   3.5.3 Mathematical Analysis of the Models......Page 72
  References......Page 74
  4.1 Introduction......Page 77
  4.2 Global Attractors of Abstract Multi-Valued Impulsive Dynamical Systems......Page 79
  4.3 Application to Impulsive Parabolic System......Page 85
  References......Page 97
 5 Fraktal and Differential Properties of the Inversor of Digits of Qs-Representation of Real Number......Page 99
  5.1 Introduction......Page 100
  5.2 Qs-Representation of Real Number......Page 101
  5.3 Inversor of Digits of Qs-Representation for Fractional Part of Real Number......Page 103
  5.4 Differential Properties of the Inversor......Page 104
  5.5 Fractal Properties of the Inversor......Page 110
  5.6 Conclusion......Page 113
  References......Page 114
  6.1 Introduction......Page 116
  6.2 Setting of the Problem......Page 118
  6.3 Main Result......Page 119
   6.3.1 Some Sufficient Conditions for (6.17)......Page 122
    6.3.1.1 The Case of Regularly Varying Coefficients......Page 123
   6.3.2 Sharpness of Theorem 6.1......Page 125
  6.4 Some Examples......Page 126
   6.4.1 Population Growth Model......Page 127
   6.4.2 Rendleman–Bartter Model......Page 129
   6.4.3 Asymptotic Behavior of Solutions of Stochastic Differential Equation (6.7)......Page 130
  References......Page 132
Part II Solid Mechanics......Page 134
  7.1 The Classic Galerkin Representation in Isotropic Elasticity......Page 135
  7.2 Splitting of the System of Displacement Equations in Anisotropic Elasticity......Page 137
  7.3 Transversely Isotropic Medium......Page 138
  References......Page 142
  8.1 Introduction and the Main Result......Page 143
  8.2 The Basic Equations......Page 148
  8.3 Symmetries of the Basic Equations......Page 149
  8.4 Fundamental Solution......Page 150
  References......Page 153
  9.1 Introduction......Page 154
  9.2 Problem Statement and Numerical Procedure......Page 157
   9.2.1 Hydrodynamic Calculations......Page 158
   9.2.2 Far Acoustic Field......Page 161
   9.2.3 Details of the Numerical Scheme......Page 163
   9.3.1 Natural Flow in Open Cylindrical Cavity......Page 165
   9.3.2 Cavity Flow with Fluid Suction......Page 170
  References......Page 174
  10.1 Introduction......Page 176
  10.2 Problem Statement and Method Description......Page 179
  10.3 Numerical Methodology......Page 181
  10.4 Results......Page 184
   10.4.1 Discretization Details......Page 185
   10.4.2 Vortical Flow Patterns and Frequency Analysis......Page 187
   10.4.3 Forces......Page 190
  10.5 Conclusion......Page 194
  References......Page 195
  11.1 Introduction......Page 197
  11.2 Existence of Strong Solutions......Page 199
   11.3.1 Regularised Problems......Page 200
   11.3.2 Existence of Weak Solutions......Page 201
   11.3.3 Existence of Strong Solutions......Page 203
   11.3.4 Asymptotic Behaviour......Page 204
  References......Page 207
Part III Dynamics of Differential and Difference Equations and Applications......Page 209
  12.1 Introduction......Page 210
  12.2 Formulation of Sequence Spaces with Variable Exponents......Page 212
  12.3 Properties of ρ and |·|p......Page 215
  12.4 Properties of the Space P......Page 221
  12.5 Closing Remarks......Page 228
  References......Page 229
  13.1 Introduction......Page 230
  13.2 Preliminaries......Page 231
  13.3 Application......Page 237
  References......Page 251
 14 Non-Lipschitz Homogeneous Volterra Integral Equations......Page 252
  14.1 Introduction......Page 253
  14.2 Increasing Nonlinear Volterra Operators with Locally Bounded Kernels......Page 255
   14.2.1 Continuous and Increasing Kernels......Page 257
   14.2.2 Continuous Like Increasing Kernels......Page 258
   14.2.3 Continuous Kernels......Page 259
   14.2.4 Locally Bounded Kernels......Page 262
  14.3 Increasing Nonlinear Volterra Operators with Locally Integrable Kernels......Page 263
   14.3.1 Non-locally Bounded and Multiple Solutions......Page 264
   14.3.2 Abel Equations as Limit of Volterra Equations......Page 266
  14.4 Numerical Study......Page 267
   14.4.1 Collocation Methods......Page 269
    14.4.1.2 Existence and Uniqueness of Nontrivial Collocation Solutions......Page 270
    14.4.1.3 Blow-Up Collocation Solutions......Page 272
  References......Page 273
  15.1 Introduction and Motivation......Page 275
  15.2 A Glance to the RVT Technique......Page 277
   15.3.1 The Nonlinear Random Differential Equation for a Falling Body......Page 279
   15.3.2 Bayesian Computation of the Parameters of a Fish Weight Growth Model......Page 281
   15.4.1 Bernoulli Method......Page 286
   15.4.2 Application of the Bernoulli Method to Find a Soliton Solution for the Deterministic Nonlinear Dispersive PDE......Page 287
   15.4.3 Obtaining the Probability Density Function of the Soliton Solution......Page 290
  15.5 Conclusions......Page 293
  References......Page 295
  16.1 Introduction......Page 297
   16.2.1 The Setup......Page 299
   16.2.2 The Hypotheses and the Main Result......Page 301
  16.3 The Transversal Perturbations......Page 303
  16.4 The Averaging Error and the Proof of the Main Result......Page 316
  Appendix......Page 321
  References......Page 324
  17.1 Introduction......Page 326
  17.2 Existence, Uniqueness and Continuity of the Solution......Page 329
  References......Page 345
  18.1 Introduction......Page 347
  18.2 The Bass–Pyke Theorem......Page 348
  18.3 Uniform Law of Large Numbers for Random Signed Measures......Page 350
  18.4 Proof of Theorem 18.2......Page 351
  18.5 Homogeneous Random Fields and Stationary Measures......Page 353
  18.6 Stochastic Integrals......Page 358
  18.7 Concluding Remarks......Page 360
  References......Page 361
 19 On Comparison Results for Neutral Stochastic Differential Equations of Reaction-Diffusion Type in L2( Rd)......Page 363
  19.1 Introduction......Page 364
  19.2 Problem Definition......Page 365
  19.3 Preliminaries......Page 369
   19.3.1 Comparison Theorem for Finite-Dimensional Case......Page 370
   19.3.2 Approximation Properties......Page 372
  19.4 Proof of Theorem 19.1......Page 373
  References......Page 407
  20.1 Introduction......Page 408
   20.2.1 Internal Practical Stability......Page 411
   20.2.2 Weak Internal Practical Stability......Page 413
   20.2.4 External Practical Stability......Page 414
  20.3 Maximum Set of Initial Conditions: Linear Case......Page 415
  20.4 Internal Practical Stabilization......Page 418
  References......Page 420
Part IV Modern Methods of Optimization and Control Sciences for Continuum Mechanics......Page 422
  21.1 Introduction......Page 423
  21.2 Proof of Theorem 21.1......Page 425
  21.3 Examples of Applications......Page 428
  References......Page 432
  22.1 Introduction......Page 434
  22.2 Problem Statement......Page 436
  22.3 Suppression of Constrained Disturbances in Impulse Processes with Multirate Sampling Based on Invariant Ellipsoids Method......Page 438
  22.4 Design of Multirate Impulse Processes Control Systems for Stabilization of CM Nodes......Page 442
  22.5 Example of Human Resources Management in IT Company Based on CM Weights Increments with Multirate Sampling......Page 444
  22.6 Conclusion......Page 449
  References......Page 452
  23.1 Introduction......Page 453
  23.2 On Consistency of Optimal Control Problem (23.2)–(23.5)......Page 456
  23.3 On Approximating Optimal Control Problems and Their Previous Analysis......Page 460
  23.4 Asymptotic Analysis of Approximating OCP (23.26)–(23.29)......Page 467
  23.5 Optimality Conditions for Approximating OCP (23.26)–(23.29)......Page 481
  References......Page 487
  24.1 Introduction......Page 489
  24.2 Setting of the Problem......Page 490
  24.3 Main Results......Page 491
  References......Page 499
  25.1 Introduction......Page 500
  25.2 The Problem Statement......Page 501
   25.3.1 Unbounded Control......Page 504
   25.3.2 Bounded Control......Page 510
  References......Page 511
  26.1 Introduction......Page 512
  26.2 Differential-Difference Games of Approach with Commutative Matrices......Page 515
  26.3 Differential-Difference Games of Approach with Pure Time Delay......Page 524
  References......Page 530
  27.1 Introduction......Page 532
  27.2 Formulation of the Problem......Page 533
   27.3.2 Extension of a Fuzzy Preference Relation to the Class of Fuzzy Sets......Page 536
  27.4 Fuzzy Set of Non-dominated Alternatives......Page 537
  27.5 Effective Maximizing Alternatives......Page 543
  27.6 Conclusion......Page 545
  References......Page 546
 28 Using Wavelet Techniques to Approximate the Subjacent Risk of Death......Page 548
  28.1 Introduction......Page 549
  28.2 The Biometric Model......Page 550
  28.3 Wavelet Graduation......Page 553
   28.3.1 Wavelet Graduation Problems......Page 554
   28.3.2 Combining Bootstrap and Wavelet Graduation......Page 555
  28.4 Validation and Applications......Page 557
   28.4.1 Application to Real Data......Page 561
  28.5 Conclusions......Page 562
  References......Page 563