Mathematics of Random Phenomena: Random Vibrations of Mechanical Structures

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes.

Table of Contents:
A)Theory.- I. Mathematical Prerequisites.- I.1. Linear Algebra.- I.2. Transposition.- I.3. Functional Analysis.- I.4. Prerequisites from the Theory of Measure and Integration.- I.5. Sobolev Spaces.- II. Principles of Probability Theory.- II.1. Principles of Probability Theory.- II.2. Characteristic Functions. Moments.- II.3. Common Probability Laws.- II.4. Relation Between Probability Theory and Statistics.- II.5. Effective Description of Probability Laws.- III. Stochastic Processes and Random Fields.- III.1. Classical Stochastic Processes.- III.2. Processes of Second Order.- III.3. Transformations of Stochastic Processes.- III.4. Integral Representation of Classical Stochastic Processes of Second Order.- III.5. Conditional Expectations and Applications to Processes.- III.6. The Markovian Property.- III.7. Statistics on Random Functions.- IV. Linear Filtering and Spectral Analysis.- IV.1. Linear Transformations of Signals.- IV.2. Convolution Filters.- IV.3. Finite-Dimensional Linear Oscillators.- IV.4. Linear Integral Transformations of M.S. Continuous Second-Order Stochastic Processes.- IV.5. Convolution Filtering of M.S. Continuous Second-Order Stationary Processes.- IV.6. Stochastic Linear Oscillators.- V. Statistics on Sample Paths.- V.1. Introduction.- V.2. Processes with Absolutely Continuous Sample Paths.- V.3. Polygonal Approximation of Sample Paths.- V.4. Mean Number of Crossings for Almost All Levels.- V.5. Mean Number of Crossings.- V.6. Mean Number of Up crossings (Downcrossings).- V.7. Relationship Between the Mean Number of Upcrossings and the Study of Absolute Maxima on an Interval.- V.8. Mean Number of Local Maxima and Minima.- B)Applications.- VI. Dynamic Response of Structures Subjected to Wind Loads.- VI.1. Brief Review of Fluid Mechanics.- VI.2. Probabilistic Description of the Atmospheric Turbulent Boundary Layer.- VI.3. Along-Wind Dynamic Response to Flexible Structures.- VI.4. Experimental Verification of the Model>.- VII. Probabilistic Approach to the Effects of Earthquakes on Structures.- VII.1. General Facts.- VII.2. Probabilistic Modelling of Strong-Motion Earthquakes.- VII.3. Responses of Systems with Linear Behavior.- VII.4. Response of Structures with Nonlinear Behavior.- VII.5. Numerical Resolution of the Fokker-Planck Equation.- VIII. Probabilistics Models of Sea Wave and Applications.- VIII.1. Statement of the Problem and Hypotheses.- VIII.2. Prerequisites on the Deterministic Wave.- VIII.3. Probabilistic Model of Short-Term Sea States.- VIII.4. Application to the Study of the Response of a Certain Type of Offshore Structures.- IX. Mechanical Fatigue under Random Loading.- IX.1. General Facts Concerning Fatigue.- IX.2. Various Notions of Cycles. Counting Methods.- IX.3. Fatigue under Constant-Cycle Loading.- IX.4. Fatigue under Loading with Constant Cycles on Intervals.- IX.5.Fatigue under Random Loading.- C)Applications.- X. Measures on Vector Spaces and Stochastic Modelling.- X.1. Locally Convex Spaces.- X.2. Duality for Locally Convex Spaces.- X.3. Cylindrical Models and Models of Random Elements.- X.4. Study of Certain ?-Algebras.- X.5. Constructing Models.- X.6. Linear Processes of Order p..- X.7. Stochastic Hilbert Space. Reproducing Hilbert Space.- X.8. Weak Integrals.- XI. Generalized Stochastic Processes.- XI.1. Random Vector Distributions.- XI.2. Linear Transformations of Generalized Processes.- XI.3. Spectral Theory of Generalized Processes.- XI.4. Linear Equations.- XI.5. Linear Oscillators with Arbitrarily Many Degrees of Freedom.- XI.6. Stochastic Hilbert Spaces and Regularity of Generalized Processes.- XII. Markov Realization of Gaussian Processes.- XII.1. Physically Realizable Processes.- XII.2. Markov Realization of Processes with Spectral Density of Rational Type.- XII.3. Approximate Markov Realization of Physically Realizable Processes.- XII.4. Filtering Techniques for Random Fields.- XII.5. Application of Random Field Filtering to Approximate Markovianization.- XIII. Stochastic Differential Equations.- XIII.1. Stochastic Integrals.- XIII.2. Indefinite Stochastic Integrals and Ito’s Differential Calculus.- XIII.3. Existence and Uniqueness of Solutions in the Uniform Lipschitz Case.- XIII.4. Existence and Uniqueness of Solutions in other Cases.- XIII.5. The Fokker-Planck Equation.- XIII.6. Law of the First Exit from an Open Set.- XIII.7. Markovianization of Some Stochastic Initial-Value Problems.- XIV. Random Oscillations with Intermittent Constraints.- XIV.1. Deterministic Multivalued Differential Equations.- XIV.2. Deterministic Oscillations with Intermittent Constraints.- XIV.3. Multivalued Stochastic Differential Equations.- XIV.4. The Diffusion Operator of (MSDE).- XIV.5. Diffusion Equation for the Probability Distributions for (MSDE) in the Product Situation.- XIV.6. Multivalued Equations and Stochastic Differential Equations with Boundary Conditions.- XIV.7. Generalized FPE for a (MSDE) of Product type.- XV. Linear Oscillators Subjected to Squared Gaussian Processes.- XV.1. Existence Theorem for Reproducing Hilbert Spaces.- XV.2. Fourier Transform of Images of Gaussian Measures under Certain Nonlinear Mappings.- XV.3. Proof of the Topological Property XV.2.4.c.- XV.4. Example of a Nonlinear Transformation of a Gaussian Measure.- XV.5. An Approximation Procedure.- Additional References.