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All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. My goal is to determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive white noise. That is achieved by computing the local eigenfunctions of the Fokker-Planck evolution operator in linearized neighborhoods of the periodic orbits of the corresponding deterministic system, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. The method applies in principle to both continuous- and discrete-time dynamical systems. Numerical tests of such optimal partitions on unimodal maps support my hypothesis.

A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (non-standard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks. The approach tries to mimic the well known linear situation. The original equation is smoothly transformed into a very specific non-linear equation that is treated as a singular perturbation of the original equation. The transformation provides a classification to all 2-dimensional systems with initial conditions close to a saddle point of the flow generated by the drift vector field. The proof then proceeds by estimates that propagate the small noise nature of the system through the non-linearity. Some proofs are based on geometrical arguments and stochastic pathwise expansions in noise intensity series.

An explicit solution is given for the value of a risk neutral firm with stochastic revenue facing the possibility of bankruptcy. The analysis is conducted in continuous time. Uncertainty is modeled using an Ito process and bankruptcy is modeled as an absorbing boundary. The analysis yields an ordinary differential equation with a closed form solution. The value function is used to calculate the firm's demand for high interest rate loans, showing a positive demand at interest rates which appear intuitively to be excessive. A value function is also derived for a risk neutral lender advancing funds to the firm. The borrowing and lending value functions are then used to examine various aspects of lender-borrower transactions under different bargaining structures. In a competitive lending market, the model shows that credit rationing occurs inevitably. In a monopoly lending market, the lender sets interest rates and maximum loan levels which reduce the borrower to zero profit. When a second borrower is introduced, the lender must allocate limited funds between two borrowers. A lender is shown to squeeze the smaller "riskier" borrower out of the market when the lender's overall credit constraint is tight. Under each bargaining structure, the model is also used to examine changes in the respective "salvage" recoveries of the lender and borrower on bankruptcy. Accepted:

Gene expression is random owing to the low copy numbers of molecules in a living cell and the best way to study it is by use of a stochastic method, specifically the chemical master equation. The method is used here to derive analytically the invariant probability distributions, and expressions for the moments and noise strength for a simple gene model without feedback. Sensitivity analysis, emphasizing particularly the dependence of the probability distributions, the moments, and noise strength is carried out using Metabolic Control Analysis, which uses control coefficients that measure the response of observables when parameters change. Bifurcation analysis is also carried out. The results show that the number of mRNA molecules follows a hypergeometric probability distribution, and that noise decreases as the number of these molecules increases. Metabolic Control Analysis was successfully extended to genetic control mechanisms, with the obtained control coefficients satisfying a summation theorem. The system undergoes stochastic bifurcations as parameters change. / xii, 86 leaves : ill. ; 29 cm

This thesis analyzes the nonlinear dynamics of a drill-string including uncertainty modeling. A drill-string is a slender flexible structure that rotates and digs into the rock in search of oil. A mathematical-mechanical model is developed for this structure including fluid-structure interaction, impact, geometrical nonlinearities and bit-rock interaction. After the derivation of the equations of motion, the system is discretized by means of the finite element method and a computer code is developed for the numerical computations using the software MATLAB. The normal modes of the dynamical system in the prestressed configuration are used to construct a reduced order model for the system. To take into account uncertainties, the nonparametric probabilistic approach, which is able to take into account both system-parameter and model uncertainties, is used. The probability density functions related to the random variables are constructed using the maximum entropy principle and the stochastic response of the system is calculated using the Monte Carlo method. A novel approach to take into account model uncertainties in a nonlinear constitutive equation (bit-rock interaction model) is developed using the nonparametric probabilistic approach. To identify the probabilistic model of the bit-rock interaction model, the maximum likelihood method together with a statistical reduction in the frequency domain (using the Principal Component Analysis) is applied. Finally, a robust optimization problem is performed to find the operational parameters of the system that maximizes its performance, respecting the integrity limits of the system, such as fatigue and instability

No description available.

The purpose of this thesis is to present data pertinent to and propose conclusions regarding the coordination of radiobiologic effectiveness (RBE) and linear energy transfer (LET). RBE is a quantity relating the effectiveness of different radiations in causing cell death. LET is a measure of the rate of energy transferred to material by an ionizing particle. This relationship of these values varies for different particles. The reason for this is still inconclusive. The petitioner has made use of a toolkit for Geant4, known as Geant4-DNA, to perform track-structure analysis on a chromosome model. Geant4 is an object-oriented program for the "simulation of the passage of particles through matter" developed by CERN that makes use of Monte Carlo methods and is expanded by Geant4-DNA to handle low-energy electron physics as well as physic-chemical effects. The chromosome model, in this case, has been developed by the petitioner as a nucleus with a basic, uniform distribution of chromatin. Radiation damage to DNA, in the form of aberrations, lesions and strand breaks, can be coordinated to energy deposited or number of ionizations occurring in the target (in this case DNA or chromatin fiber). Certain threshold values have been established as indicate of different types of DNA damage. The ultimate goal of this work is to score these clusters of events against the threshold values to determine the severity of DNA damage. The final comparison of the results for different particles will provide for a better understanding of the RBE-LET relationships by improving the understanding of the underlying nanodosimetric qualities.

The purpose of this thesis is to develop some new algorithms based on optimization techniques for solving some problems in some areas of telecommunications and the Internet. There are two main parts to this thesis. In the first part we discuss optimization based stochastic and queueing models in telecommunications network corrective maintenance. In the second part we develop optimization based clustering (OBC) algorithms for network evolution and multicast routing. The most typical scenario encountered during mathematical optimization modelling in telecommunications, for example, is to minimize the cost of establishment and maintenance of the networks subject to the performance constraints of the networks and the reliability constraints of the networks as well. Most of these optimization problems are global optimization, that is, they have many local minima and most of these local minima do not provide any useful information for solving these problems. Therefore, the development of effective methods for solving such global optimization problems is important. To run the telecommunications networks with cost-effective network maintenance,we need to establish a practical maintenance model and optimize it. In the first part of the thesis, we solve a known stochastic programming maintenance optimization model with a direct method and then develop some new models. After that we introduce queue programming models in telecommunications network maintenance optimization. The ideas of profit, loss, and penalty will help telecommunications companies have a good view of their maintenance policies and help them improve their service. In the second part of this thesis we propose the use of optimization based clustering (OBC) algorithms to determine level-constrained hierarchical trees for network evolution and multicast routing. This problem is formulated as an optimization problem with a non-smooth, non-convex objective function. Different algorithms are examined for solving this problem. Results of numerical experiments using some artifiicial and real-world databases are reported. / Doctor of Philosophy