In recent years people have seen a rapidly growing market for credit derivatives. Among these traded credit derivatives, a growing interest has been shown on multi-name credit derivatives, whose underlying assets are a pool of defaultable securities. For a multi-name credit derivative, the key is the default dependency structure among the underlying portfolio of reference entities, instead of the individual term structure of default probabilities for each single reference entity as in the case of single-name derivative. So far, however, default dependency modeling is still the most demanding open problem in the pricing of credit derivatives. The research in this dissertation is trying to model the default dependency with aid of perturbation method, which was first proposed by Fouque, Papanicolaou and Sircar (2000) as a powerful tool to pricing options under stochastic volatility. Specifically, after a theoretic result regarding the approximation accuracy of the perturbation method and an application of this method to pricing American options under stochastic volatility by Monte Carlo approach, a multi-dimensional Merton model under stochastic volatility is studied first, and then the multi-dimensional generalization of the first-passage model under stochastic volatility comes next, which is then followed by a copula perturbed from the standard Gaussian copula.

Inhibin is one of several hormones which collectively regulate the human female reproductive endocrine system. In recent years, physiologists have been able to separately assay two forms of inhibin. We begin by discussing the physiology and endocrinology that underlie the menstrual cycle. Then, we fit an existing delay differential equation model of the human menstrual cycle to new data describing average hormone levels of young women throughout the menstrual cycle. Next, we consider the existence and stability of equilibrium and periodic solutions, analyze bifurcations, and perform a sensitivity analysis. We compare and contrast these results to previously published results based on the same model but a different data set. Because the introduction of exogenous hormones, whether pharmacological or environmental, can have significant effects on the menstrual cycle, we model the effects of these external hormones. Next, we introduce an expanded model that more fully accounts for the actions of both forms of inhibin. We optimize the parameters to fit the data, then discuss the equilibrium and periodic solutions, bifurcations, and sensitivity for this new model and parameter set. Next, we use the model to consider the effects of exogenous pharmacological hormones. Finally, we discuss the effects on the menstrual cycle of age-related hormone production changes that occur during the peri-menopausal years. This leads to a second parameter set, with only a few variations from the first one, for which the model approximates the monthly hormonal fluctuations for a woman around age 40. In closing, we discuss future possibilities for this line of research.

The purpose of the research has been to develop a class of new finite-element methods, called immersed-interface finite-element methods, to solve elliptic and elasticity interface problems with homogeneous and non-homogeneous jump conditions. Simple non-body-fitted meshes are used. Single functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface. With such functions, the discontinuities across the interface in the solution and flux are removed; and equivalent elliptic and elasticity interface problems with homogeneous jump conditions are formulated. Special finite-element basis functions are constructed for nodal points near the interface to satisfy the homogeneous jump conditions. Error analysis and numerical tests are presented to demonstrate that such methods have an optimal convergence rate. These methods are designed as an efficient component of the finite-element level-set methodology for fast simulation of interface dynamics that does not require re-meshing. Such simulation has been a powerful numerical approach in understanding material properties, biological processes, and many other important phenomena in science and engineering.

Smart material transducers are utilized in wide range of applications, including nanopositioning, fluid pumps, high accuracy, high speed milling, objects, vibration control and/or suppression, and artificial muscles. They are attractive because the resulting devices are solid-state and often very compact. However, the coupling of field or temperature tomechanical deformation, which makes these materials effective transducers, also introduces hysteresis and time-dependent behaviors that must be accommodated in device designs and models before the full potential of compounds can be realized. In this dissertation, we present highly efficient modeling techniques to characterize hysteresis and constitutive nonlinearities in ferroelectric, ferromagnetic, and shape memory alloy compounds and model inversion techniques which permit subsequent linear control designs.

We focus on the problem of clustering large textual data sets. We present 3 well-known clustering algorithms and suggest enhancements involving dimension reduction. We propose a novel method of algorithm aggregation that allows us to use many clustering algorithms at once to arrive on a single solution. This method helps stave off the inconsistency inherent in most clustering algorithms as they are applied to various data sets. We implement our algorithms on several large benchmark data sets.

Let A be an algebraic system with product a*b between elements a and b in A. It is of interest to compare the solvable length t with other invariants, for instance size, order, or dimension of A. Thus we ask, for a given t what is the smallest n such that there is an A of length t and invariant n. It is this problem that we consider for associative algebras, matrix groups, and Lie algebras. We consider A in each case to be subsets of (strictly) upper triangular n by n matrices. Then the invariant is n. We do these for the associative (Lie) algebras of all strictly upper triangular n by n matrices and for the full n by n upper triangular unipotent groups. The answer for n is the same in all cases. Then we restrict the problem to a fixed number of generators. In particular, using only 3 generators and we get the same results for matrix groups and Lie algebras as for the earlier problem. For associative algebras with 1 generator we also get the same result as the general associative algebra case. Finally we consider Lie algebras with 2 generators and here n is larger than in the general case. We also consider the problem of finding the dimension in the associative algebra, the general, and 3 generator Lie algebra cases.

<p>Numerical methods, LQR control, an abstract formulation andreduced basis techniques for a system consisting of a thin cylindrical shellwith surface-mounted piezoceramic actuators are investigated.Donnell-Mushtari equations,modified to include Kelvin-Voigt damping and passive patch contributions,are used to model the system dynamics. The voltage-induced piezoceramicpatch contributions, used as input inthe control regime, enter the equations as externalforces and moments. Existence and uniqueness of solutions are demonstratedthrough variational and semigroup formulations of the system equations.The semigroup formulation is also used to establish theoretical controlresults and illustrate convergence of the finite dimensional controlsand Riccati operators.The spatial components of the state arediscretized using a Galerkin expansion resulting in an ordinarydifferential equation that can be readily marched in time by existingordinary differential equationsolvers.Full order approximation methods which employ standard basiselements such as cubic or linear splines result in large matrixdimensions rendering the system computationally expensive for real-timesimulations. To lessen on-line computational requirements, reducedbasis methods employing snapshots of the full order model as basisfunctions are investigated.As a first step in validating the model, a shell with obtainable analyticnatural frequencies and modes was considered. The derived frequenciesand modeswere then compared with numerical approximations using full order basisfunctions. Further testing on the static and dynamic performance of the fullorder model was carried out through the following steps:(i) choose true state solutions, (ii) solve for the forces in theequations that would lead to these known solutions, and (iii) comparenumerical results excited by the derived forces with the true solutions.Reduced order methods employing the Lagrange and theKarhunen-Loève proper orthogonal decomposition (POD)basis functions are implemented on the model. Finally, a statefeedback method was developed and investigated computationallyfor both the full order and reduced ordermodels.<P>

<p>A finite, homogeneous, irreducible Markov chain $\mC$ with transitionprobability matrix possesses a unique stationary distribution vector. The questions one can pose in the area of computation of Markov chains include the following:<br>- How does one compute the stationary distributions? <br>- How accurate is the resulting answer? <br>In this thesis, we try to provide answers to these questions. <br><br>The thesis is divided in two parts. The first part deals with the perturbation theory of finite, homogeneous, irreducible Markov Chains, which is related to the first question above. The purpose of this part is to analyze the sensitivity of the stationarydistribution vector to perturbations in the transition probabilitymatrix. The second part gives answers to the question of computing the stationarydistributions of nearly uncoupled Markov chains (NUMC). <P>

<p>This work describes the development of a mathematical model of ahigh-pressure chemical vapor deposition (HPCVD) reactor and nonlinearfeedback methodologies for control of the growth of thin films in thisreactor. Precise control of the film thickness and composition is highlydesirable, making real-time control of the deposition process veryimportant. The source vapor species transport is modeled by the standardgas dynamics partial differential equations, with species decompositionreactions, reduced down to a small number of ordinary differential equationsthrough use of the proper orthogonal decomposition technique. This systemis coupled with a reduced order model of the reactions on the surfaceinvolved in the source vapor decomposition and film deposition on thesubstrate wafer. Also modeled is the real-time observation technique usedto obtain a partial measurement of the deposition process. The utilization of reduced order models greatly simplifies the mathematicalformulation of the physical process so it can be solved quickly enough to beused for real-time model-based feedback control. This control problem isfairly complicated, however, because the surface reactions render the modelnonlinear. Several control methodologies for nonlinear systems are studiedin this work to determine which performs best on test examples similar tothe HPCVD problem. One chosen method is extended to a tracking control toforce certain film growth properties to follow desired trajectories. Thenonlinear control method is used also in the development of a stateestimator which uses the nonlinear partial observation of the nonlinearsystem to create an estimate of the actual state, which the feedback controlformula then can use to guide the HPCVD system. The nonlinear trackingcontrol and estimator techniques are implemented on the HPCVD model and theresults analyzed as to the effectiveness of the reduced order model andnonlinear control.<P>

<p>This work investigates the characteristics of a steady state flow of granular material,under the influence of gravity, in two and three dimensional hoppers of simple geometry.Simulations of such flows are of particular interest to various industries, such as the foodand mining industries, where the handling of large quantities of granular materials in hop-persand silos is routine. While understanding and simulation of time-dependent phenomenaare the ultimate goals in this field, those phenomena are still poorly understood and thustheir study is beyond the scope of this research. It has been observed that steady flowscan provide reasonable approximations, and the corresponding steady state model has con-sequentlybeen the focus of a great deal of research. Historically, these steady state modelshave been approached using only smooth radial fields, and even today most practical hop-perdesign uses these fields as their basis. Our work represents the first time that qualitynumerical methods have been brought to bear on the model equations in their original form,without assuming smoothness of the resulting fields. Two different, yet related, models forstress/velocity consisting of systems of hyperbolic conservation laws and algebraic relationsare considered and discussed. The radial stress and velocity fields, and the stability of thosefields, are studied briefly with both analytical and numerical results presented. More im-portantly,a Runge-Kutta Discontinuous Galerkin method is implemented and applied tovarious boundary value problems involving perturbed stress and velocity fields arising fromdiscontinuous changes in parameters such as hopper wall angle or hopper wall friction.<P>