Stochastic processes

This dissertation is concerned primarily with the analytical modeling of a class of electromagnetic composite materials using the concepts of stochastical mixture theory, principles of electromagnetics and neuromimetic considerations. The global behavior of the test composite is ascertained in terms of the constitutive relations of the material parameters (having stochastical attributions) and the intramaterial hierarchy is modeled as massively interconnected, interacting units depicting such systems as mimetics of neural networks. Pertinent research efforts enclave the following specific tasks: (i) Modeling a multi-constituent electromagnetic composite medium in terms of the characteristics of its individual constituents and their spatial (random or orderly) dispositions. (ii) Assessment of nonspherical particulate effects (in terms of the stochastical attributes) on the global response of such composite materials. (iii) Evaluation of interparticle interactions and their implicit effects on the effective electromagnetic properties of the composite media. (iv) Assaying the transitional behavior of the test composites and, (v) modeling electromagnetic composites as neuromimetics correlating their effective material characteristics to the corresponding state-transitional response of a massively interconnected neural network. Results arising from these theoretical considerations are compared with data compiled via experimental studies performed (where feasible) or otherwise correlated with theoretical and/or experimental results available elsewhere in the literature. Specific experimental efforts carried out refer to piezoelectric rubber composites and their application in controlling acoustic beamforming via electrical 'pinch off' (which mimics the inhibitory response in a neuronal cell); as well as exclusive experimental tasks to verify the transitional lossy behavior model developed presently using a set of fast-ion conductor composites and dielectric-plus-conductor mixtures. Lastly, inferential conclusions are presented and discussed with an outline on the scope of extensions to the present work.

This dissertation is concerned with modal analysis of plates with properties which vary over the structures. The uncertain geometric and material parameters are treated as random fields which are discretized over individual regions by using a local averaging technique. These discretized properties are then combined with a random perturbation procedure based upon traditional finite element methods. The result is a stochastic finite element method (SFEM) program for modal analysis of plates. This SFEM method is applied to two problems areas. The first application is to provide a new approach for modal analysis of printed circuit boards wherein the circuit board is modeled as an elastic plate with random spatial variation of its properties. The SFEM program is used to predict the effect of this variation on the natural frequencies and mode shapes of the board. Predicted results are compared with those obtained from modal testing of a circuit board. It is shown that variations between the measured and predicted modal parameters can be accounted for by small random variations in the board properties. This approach offers a simple, realistic, and cost-effective way for prediction of board modal properties. The second application is on vibration control of plates by application of surface viscoelastic damping treatments. Existing works generally treat the geometric and material properties of the damping layer as deterministic parameters, although uncertainties in the values of these parameters are commonplace. No work has been done regarding surface damping treatments with uncertain properties. In this thesis, the modal properties of plates with random spatial variation of the damping layer properties are investigated. The effects of this variation on the system natural frequencies, modal loss factors, and mode shapes are calculated by the SFEM program developed. Results are presented for a cantilever aluminum plate with complete PVC surface damping treatment with uncertain properties. In the SFEM modeling of both PC boards and plates with surface damping treatments, the effects on the system eigenvalues/eigenvectors of the correlation distance of the random property field, the correlation constant between the random fluctuations, and the magnitude of the random property variations, are investigated.

By revisiting the popular framework of depicting neuronal (collective) activities as analogous to Ising's spin-glass theory of interacting magnetic spins, the contradictions that coexist with such an analogy are extracted and discussed. To alleviate such contradictions, an alternative strategy of equating the neuronal interactions to the partially anisotropic nematic phase of disorder pertaining to liquid crystals is proposed. Hence, the extent of anisotropy in the neuronal system, quantified in terms of an order-function, is specified to elucidate the nonlinear squashing action of the input-output relations in a neuronal cell. The relevant approach thereof, is based on Langevin's theory considerations as applied to dipole molecules. Further, in view of the stochastical properties due to the inherent disorder associated with the neuronal assembly, the progression of state-transitions across the interconnected cells is modeled as a momentum flow relevant to particle dynamics. Hence, corresponding wave mechanics attributions of such a collective movement of state-transition activity are described in terms of a probabilistic wave function. Lastly, the stochastical aspects of noise-perturbed neuronal dynamics are studied via Fokker-Planck equation representing the Langevin-type relaxational (nonlinear) process associated with the neuronal states. On each of these topics portraying the stochastical characteristics of the neuronal assembly and its activities, newer and/or more exploratory inferences are made, logical conclusions are enumerated and relevant discussions are presented along with the scope for future research to be pursued.

This thesis is concerned with nonlinear dynamical systems subject to random or combined random and deterministic excitations. To this end, a systematic procedure is first developed to obtain the exact stationary probability density for the response of a nonlinear system under both additive and multiplicative excitations of Gaussian white noises. This procedure is applicable to a class of systems called the class of generalized stationary potential. The basic idea is to separate the circulatory probability flow from the noncirculatory flow, thus obtaining two sets of equations for the probability potential. It is shown that previously published exact solutions are special cases of this class. For those nonlinear systems not belonging to the class of generalized stationary potential, an approximate solution technique is developed on the basis of weighted residuals. The original system is replaced by the closest system belonging to the class of generalized stationary potential, in the sense that the statistically weighted residuals are zero for some suitably selected weighting functions. The consistency of the approximation technique is proved in terms of certain statistical moments. The above exact and approximate solution techniques are extended to two types of nonlinear systems: one subjected to non-Gaussian impulsive noise excitations and another subjected to combined harmonic and broad-band random excitations. Approximation procedures are devised to obtain stationary probabilistic solutions for these two types of problems. Monte Carlo simulations are performed to substantiate the accuracy of the approximate solution procedures.

A model for earthquake ground motion is developed in this dissertation using principles of geophysics and stochastics. The earth is idealized as being composed of horizontally stratified layers, with uniform physical properties for each layer. The seismic source is assumed to be the result of shear dislocation propagating on a fault line, which is further discretized into a series of point sources at equal intervals. The fundamental problem of the ground motion in a layered medium due to a point source at a given source location is first considered. The governing equations of three-dimensional wave motion in a uniform layer are presented and solved in both Cartesian and cylindrical coordinates. Wave propagation in a multi-layered medium is then analyzed in detail, in which the wave scattering matrices are introduced so that stability and accuracy in numerical calculation can be guaranteed. A detailed review of the mechanism of seismic point source is also provided. Based on the fundamental solution for a point source, an earthquake model is constructed by superposing the solutions associated with a series of point sources along a line which are activated sequentially at random times. Statistical characteristics of earthquake ground motion is then obtained by applying a generalized version of the random-pulse-train theory and its evolutionary spectral representation. Finally the effects of uneven interface on the earthquake ground motion is also analyzed using a first-order perturbation approach.

This thesis proposes a sensor approach for quantifying the hydrodynamic performance of Ocean Current Turbines (OCT), and investigates the influence of sensor-specific noise and sampling rates on calculated turbine performance. Numerical models of the selected sensors are developed, and then utilized to add stochastic measurement error to numerically-generated, non-stochastic OCT data. Numerically-generated current velocity and turbine performance measurements are used to quantify the relative influence of sensor-specific error and sampling limitations on sensor measurements and calculated OCT performance results. The study shows that the addition of sensor error alters the variance and mean of OCT performance metric data by roughly 7.1% and 0.24%, respectively, for four evaluated operating conditions. It is shown that sensor error results in a mean, maximum and minimum performance metric to Signal to Noise Ration (SNR) of 48.6% and 6.2%, respectively.

Submerged turbines which harvest energy from ocean currents are an important potential energy resource, but their harsh and remote environment demands an automated system for machine condition monitoring and prognostic health monitoring (MCM/PHM). For building MCM/PHM models, vibration sensor data is among the most useful (because it can show abnormal behavior which has yet to cause damage) and the most challenging (because due to its waveform nature, frequency bands must be extracted from the signal). To perform the necessary analysis of the vibration signals, which may arrive rapidly in the form of data streams, we develop three new wavelet-based transforms (the Streaming Wavelet Transform, Short-Time Wavelet Packet Decomposition, and Streaming Wavelet Packet Decomposition) and propose modifications to the existing Short-TIme Wavelet Transform. ... The proposed algorithms also create and select frequency-band features which focus on the areas of the signal most important to MCM/PHM, producing only the information necessary for building models (or removing all unnecessary information) so models can run on less powerful hardware. Finally, we demonstrate models which can work in multiple environmental conditions. ... Our results show that many of the transforms give similar results in terms of performance, but their different properties as to time complexity, ability to operate in a fully streaming fashion, and number of generated features may make some more appropriate than others in particular applications, such as when streaming data or hardware limitations are extremely important (e.g., ocean turbine MCM/PHM).

In this dissertation we will present a stochastic optimization algorithm and use it and other mathematical techniques to tackle problems arising in medicinal chemistry. In Chapter 1, we present some background about stochastic optimization and the Accelerated Random Search (ARS) algorithm. We then present a novel improvement of the ARS algorithm, DIrected Accelerated Random Search (DARS), motivated by some theoretical results, and demonstrate through numerical results that it improves upon ARS. In Chapter 2, we use DARS and other methods to address issues arising from the use of mixture-based combinatorial libraries in drug discovery. In particular, we look at models associated with the biological activity of these mixtures and use them to answer questions about sensitivity and robustness, and also present a novel method for determining the integrity of the synthesis. Finally, in Chapter 3 we present an in-depth analysis of some statistical and mathematical techniques in combinatorial chemistry, including a novel probabilistic approach to using structural similarity to predict the activity landscape.

The present work uses statistical mechanics tools to investigate the dynamics of markets, prices, trades and wealth distribution. We studied the evolution of market dynamics in different stages of historical development by analyzing commodity prices from two distinct periods : ancient Babylon, and medieval and early modern England. We find that the first-digit distributrions of both Babylon and England commodity prices follow Benford's Law, indicating that the data represent empirical observations typically arising from a free market. Further, we find that the normalized prices of both Babylon and England agricultural commodities are characterized by stretched exponential distributions, and exhibit persistent correlations of a power law type over long periods of up to several centuries, in contrast to contemporary markets. Our findings suggest that similar market interactions may underlie the dynamics of ancient agricultural commodity prices, and that these interactions may remain stable across centuries. To further investigate the dynamics of markets, we present the analogy between transfers of money between individuals and the transfer of energy through particle collisions by means of the kinetic theory of gases. We introduce a theoretical framework of how micro rules of trading lead to the emergence of income and wealth distribution. Particularly, we study the effects of different types of distribution of savings/investments among individuals in a society and different welfare/subsidies redistribution policies. Results show that while considering savings propensities, the models approach empirical distributions of wealth quite well. The effect of redistribution better captures specific features of the distributions which earlier models failed to do. Moreover, the models still preserve the exponential decay observed in empirical income distributions reported by tax data and surveys.

Wireless devices in wireless networks are powered typically by small batteries that are not replaceable nor recharged in a convenient way. To prolong the operating lifetime of networks, energy efficiency is indicated as a critical issue and energy-efficient resource allocation designs have been extensively developed. We investigated energy-efficient schemes that prolong network operating lifetime in wireless sensor networks and in wireless relay networks. In Chapter 2, the energy-efficient resource allocation that minimizes a general cost function of average user powers for small- or medium-scale wireless sensor networks, where the simple time-division multiple-access (TDMA) is adopted as the multiple access scheme. A class of Ç-fair cost-functions is derived to balance the tradeoff between efficiency and fairness in energy-efficient designs. Based on such cost functions, optimal channel-adaptive resource allocation schemes are developed for both single-hop and multi-hop TDMA sensor networks. In Chapter 3, optimal power control methods to balance the tradeoff between energy efficiency and fairness for wireless cooperative networks are developed. It is important to maximize power efficiency by minimizing power consumption for a given quality of service, such as the data rate; it is also equally important to evenly or fairly distribute power consumption to all nodes to maximize the network life. The optimal power control policy proposed is derived in a quasi-closed form by solving a convex optimization problem with a properly chosen cost-function. To further optimize a wireless relay network performance, an orthogonal frequency division multiplexing (OFDM) based multi-user wireless relay network is considered in Chapter 4.