Physical Sciences and Mathematics Commons^™

Well-Posedness Properties In Variational Analysis And Its Applications, Wei Ouyang

Wayne State University Dissertations

This dissertation focuses on the study and applications of some significant properties in well-posedness and sensitivity analysis, among which the notions of uniform metric regularity , higher-order metric subregularity and its strong subregularity counterpart play an essential role in modern variational analysis. We derived verifiable sufficient conditions and necessary conditions for those notions in terms of appropriate generalized differential as well as geometric constructions of variational analysis. Concrete examples are provided to illustrate the behavior and compare the results. Optimality conditions of parametric variational systems (PVS) under equilibrium constraints are also investigated via the terms of coderivatives. We derived necessary …

Finite-Difference Approximations And Optimal Control Of Differential Inclusions, Yuan Tian

Wayne State University Dissertations

This dissertation concerns the study of the generalized Bolza type problem for dynamic systems governed by constrained differential inclusions. We develop finite-discrete approximations of differential inclusions by using the implicit Euler scheme and the Runge-Kutta scheme for approximating time derivatives, while an appropriate well-posedness of such approximations is justified. Our principal result establishes the uniform approximation of strong local minimizers for the continuous-time Bolza problem by optimal solutions to the corresponding discretized finite-difference systems by the strengthen $W^{1,2}$-norm approximation of this type in the case ``intermediate" (between strong and weak minimizers) local minimizers under additional assumptions. Especially the implicitly discrete …

Calculator Usage In Secondary Level Classrooms: The Ongoing Debate, Nicole Plummer

Honors College Theses

With technology becoming more prevalent every day, it is imperative that students gain enough experience with different technological tools in order to be successful in the “real-world”. This thesis will discuss the debate and overall support for an increased usage of calculators as tools in the secondary level classroom. When the idea of calculators in the classroom first came to life, many educators were very apprehensive and quite hesitant of this change. Unfortunately, more than 40 years later, there is still hesitation for their usage; and rightfully so. While there are plenty of advantages of calculator use in the classroom, …

Adaptive Stochastic Systems: Estimation, Filtering, And Noise Attenuation, Araz Ryan Hashemi

Wayne State University Dissertations

This dissertation investigates problems arising in identification and control of stochastic systems. When the parameters determining the underlying systems are unknown and/or time varying, estimation and adaptive filter- ing are invoked to to identify parameters or to track time-varying systems. We begin by considering linear systems whose coefficients evolve as a slowly- varying Markov Chain. We propose three families of constant step-size (or gain size) algorithms for estimating and tracking the coefficient parameter: Least-Mean Squares (LMS), Sign-Regressor (SR), and Sign-Error (SE) algorithms.

The analysis is carried out in a multi-scale framework considering the relative size of the gain (rate of …

Properties Of Nonlinear Randomly Switching Dynamic Systems: Mean-Field Models And Feedback Controls For Stabilization, Guangliang Zhao

Wayne State University Dissertations

This dissertation concerns the properties of nonlinear dynamic systems hybrid with Markov switching. It contains two parts. The first part focus on the mean-field models with state-dependent regime switching, and the second part focus on the system regularization and stabilization using feedback control. Throughout this dissertation, Markov switching processes are used to describe the randomness caused by discrete events, like sudden environment change or other uncertainty.

In Chapter 2, the mean-field models we studied are formulated by nonlinear stochastic differential equations hybrid with state-dependent regime switching. It originates from the phase transition problem in statistical physics. The mean-field term is …

Moser-Trudinger And Adams Type Inequalities And Their Applications, Nguyen Lam

Wayne State University Dissertations

In this dissertation, we study some variants of the Moser-Trudinger inequalities and Adams inequalities. The proofs of these inequalities relied crucially on the symmetrization arguments in the literature. By proposing new arguments and approaches, we develop successfully the critical versions of these well-known inequalities in many different settings where the rearrangement arguments may not be existed. As applications of our results, we also study in this dissertation the elliptic equations that contain the exponential nonlinearities.

On Switching Diffusions: The Feynman-Kac Formula And Near-Optimal Controls, Nicholas Baran

Wayne State University Dissertations

We consider diffusions in two different contexts. First, we consider the so-called Feynman-Kac formula(s) for switching diffusions. These formulas provide stochastic representations for solutions of certain weakly coupled elliptical systems of partial differential equations. The formulas are verified for the boundary value problem, the initial value problem, and the initial boundary value problem. Second, we show the existence of near-optimal controls for a system driven by wideband noise in the presence of regime-switching. Using a relaxed control formulation, together with weak convergence methods, we show that given a stochastic optimal control problem, one may find a control that is near-optimal. …

Stability And Controls For Stochastic Dynamic Systems, Zhixin (Harriet) Yang

Wayne State University Dissertations

This dissertation focuses on stability analysis and optimal controls for stochastic dynamic systems. It encompasses two parts. One part of our work gives an in-depth study of stability of linear jump diffusion, linear Markovian jump diffusion, multi-dimensional jump diffusion and

regime-switching jump diffusion together with the associated numerical solutions. The other part of our work is controls for stochastic dynamic systems, to be specific, we concentrated on mean variance types of control under different formulations. We obtained the nearly optimal

mean-variance controls under both two-time-scale and hidden Markov chain formulations and convergence for each case is achieved.

In Chapter 2, …

Power Operations In The Kunneth And C_2-Equivariant Adams Spectral Sequences With Applications, Sean Michael Tilson

Wayne State University Dissertations

We construct Power operations in the K"unneth spectral sequence and the $C_2$ equivariant Adams spectral sequence. While the operations in the K"unneth spectral sequence are 0 in $Tor$, they still detect operations in the target of the spectral sequence. We then interpret these computations of the homotopy of relative smash products as being related to obstructions to having $E_infty$ ring maps. The operations in the $C_2$-equivariant Adams spectral sequence are a partial extension of the work of Bruner in cite{HRS} and have applications to motivic homotopy theory.

Full Stability In Optimization, Nghia Tran

Wayne State University Dissertations

The dissertation concerns a systematic study of full stability in general optimization models including its conventional Lipschitzian version as well as the new Holderian one. We derive various characterizations of both Lipschitzian and Holderian full stability in nonsmooth optimization, which are new in finite-dimensional and infinite-dimensional frameworks. The characterizations obtained are given in terms of second-order growth conditions and also via second-order generalized differential constructions of variational analysis. We develop effective applications of our general characterizations of full stability to

parametric variational systems including the well-known generalized equations and variational inequalities. Many relationships of full stability with the conventional notions …

Discrete Littlewood-Paley-Stein Theory And Wolff Potentials On Homogeneous Spaces And Multi-Parameter Hardy Spaces, Yayuan Xiao

Wayne State University Dissertations

This dissertation consists of two parts:

In part I, We establish a new atomic decomposition of the multi-parameter Hardy spaces of homogeneous type and obtain the associated $H^p-L^p$ and $H^p-H^p$ boundedness criterions for singular integral operators. On the other hand, we compare the Wolff and Riesz potentials on spaces of homogenous type, followed by a Hardy-Littlewood-Sobolev type inequality. Then we drive integrability estimates of positive solutions to the Lane-Emden type integral systems on spaces of homogeneous type.

In part II, We establish a $(p,2)$-atomic decomposition of the Hardy space associated with different homogeneities for $0

Structure Borne Noise Analysis Using Helmholtz Equation Least Squares Based Forced Vibro Acoustic Components, Logesh Kumar Natarajan

Wayne State University Dissertations

This dissertation presents a structure-borne noise analysis technology that is focused on providing a cost-effective noise reduction strategy. Structure-borne sound is generated or transmitted through structural vibration; however, only a small portion of the vibration can effectively produce sound and radiate it to the far-field. Therefore, cost-effective noise reduction is reliant on identifying and suppressing the critical vibration components that are directly responsible for an undesired sound. However, current technologies cannot successfully identify these critical vibration components from the point of view of direct contribution to sound radiation and hence cannot guarantee the best cost-effective noise reduction.

The technology developed …

Qualitative Properties Of Solutions Of Fully Nonlinear Equations And Overdetermined Problems, Jiuyi Zhu

Wayne State University Dissertations

In section 2 of part I, We study the maximum principles and radial symmetry for viscosity solutions of fully nonlinear partial differential equations. We

obtain the radial symmetry and monotonicity properties for

nonnegative viscosity solutions of fully nonlinear equations under some asymptotic decay rate at infinity. Our symmetry and monotonicity results also

apply to Hamilton-Jacobi-Bellman or Isaccs equations. A new maximum

principle for viscosity solutions to fully nonlinear elliptic equations is established. In section 3, We establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the …

Two-Time-Scale Systems In Continuous Time With Regime Switching And Their Applications, Yousef Talafha

Wayne State University Dissertations

This dissertation is focuses on near-optimal controls for stochastic differential equation with regime switching. The random switching is presented by a continuous-time Markov chain. We use the idea of relaxed control and mean of martingale formulation to show a weak convergence result.

The first chapter is devoted to the study of stochastic Li´enard equations with random switching. The motivation of our study stems from modeling of complex systems in which both continuous dynamics and discrete events are present. The continuous component is a solution of a stochastic Li´enard equation and the discrete component is a Markov chain with a finite …

Dg And Hdg Methods For Curved Structures, Li Fan

Wayne State University Dissertations

We introduce and analyze discontinuous Galerkin methods

for a Naghdi type arch model. We prove that, when the numerical traces are properly chosen, the methods display optimal convergence uniformly with respect to the thickness of the arch. These methods are thus free from membrane and shear locking.

We also prove that, when polynomials of degree $k$ are used,

{\em all} the numerical traces superconverge with a rate of order

h ^2k+1.

Based on the superconvergent phenomenon and we show how to

post-process them in an element-by-element fashion

to obtain a far better approximation. Indeed, we prove that,

if polynomials …

Nodal Geometry Of Eigenfunctions On Smooth Manifolds And Hardy-Littlewood-Sobolev Inequalities On The Heisenberg Group, Xiaolong Han

Wayne State University Dissertations

Part I: Let (M,g) be a n dimensional smooth, compact, and connected Riemannian manifold without boundary, consider the partial differential equation on M:

-Δu=Λu,

in which Δ is the Laplace-Beltrami operator. That is, u is an eigenfunction with eigenvalue Λ. We analyze the asymptotic behavior of eigenfunctions as Λ go to ∞ (i.e., limit of high energy states) in terms of the following aspects.

(1) Local and global properties of eigenfunctions, including several crucial estimates for further investigation.

(2) Write the nodal set of u as N={u=0}, estimate the size of N using Hausdorff measure. Particularly, surrounding the conjecture that …

Stabilization And Classification Of Poincare Duality Embeddings, John Whitson Peter

Wayne State University Dissertations

We define a space E(K,X) of Poincare Duality embeddings and show that such spaces admit a highly connected stabilization map.

This serves as a tool for classifying Poincare Duality embeddings in terms of the homotopy types of their complements. In

particular, a Poincare embedding gives rise to a fiberwise duality

map in the category of retractive spaces over X. We use this construction to obtain a highly connected classification map with target a moduli space of unstable complements for Poincare embeddings. As consequences, we obtain stabilization and classication results for

smooth embeddings.

Large Deviations Of Stochastic Systems And Applications, Qi He

Wayne State University Dissertations

This dissertation focuses on large deviations of stochastic systems with applications to optimal control and system identification. It encompasses analysis of two-time-scale Markov processes and system identification with regular and quantized data. First, we develops large deviations principles for systems driven by continuous-time Markov chains with twotime scales and related optimal control problems. A distinct feature of our setup is that the Markov chain under consideration is time dependent or inhomogeneous. The use of two time-scale formulation stems from the effort of reducing computational complexity in a wide variety of applications in control, optimization, and systems theory. Starting with a …

Consensus-Type Stochastic Approximation Algorithms, Yu Sun

Wayne State University Dissertations

This work is concerned with asymptotic properties of consensus-type algorithms for networked systems whose topologies switch randomly. The regime-switching process is modeled as a discrete-time Markov chain with a nite state space. The consensus control is achieved by designing stochastic approximation algorithms. In the setup, the regime-switching process (the Markov chain) contains a rate parameter

"Ε> 0 in the transition probability matrix that characterizes how frequently the topology switches. On the other hand, the consensus control algorithm uses a step-size Μ that denes how fast the network states are updated. Depending on their relative values, three distinct scenarios emerge. Under …

Finitely Presented Modules Over The Steenrod Algebra In Sage, Michael J. Catanzaro

Wayne State University Theses

No abstract provided.

Variational Analysis And Optimal Control Of The Sweeping Process, Hoang Dinh Nguyen

Wayne State University Dissertations

We formulate and study an optimal control problem for the sweeping(Moreau) process, where control functions enter the moving sweeping

set. To the best of our knowledge, this is the first study in the literature devoted to optimal control of the sweeping process. We first establish an existence theorem of optimal solutions and then derive necessary optimality conditions for this optimal control problem of a new type, where the dynamics is governed by discontinuous differential inclusions with variable right-hand sides. Our approach to necessary optimality conditions is based on the method of discrete approximations and advanced tools of variational analysis and …

Moduli Spaces And Cw Structures Arising From Morse Theory, Lizhen Qin

Wayne State University Dissertations

In this dissertation, we study the moduli spaces and CW Structures arising from Morse theory.

Suppose M is a smooth manifold and f is a Morse function on it. We consider the negative gradient flow of f. Suppose the flow satisfies transversality. This naturally defines the moduli spaces of flow lines and gives a stratication of M by its unstable manifolds. The gluing of broken flow lines can also be constructed.

We prove that, under certain assumptions, these moduli spaces can be compactified and the compactified spaces are smooth manifolds with corners. Moreover, these compactified manifolds satisfy certain orientation formulas. …

Spectral Collocation Method For Compact Integral Operators, Can Huang

Wayne State University Dissertations

We propose and analyze a spectral collocation method for integral

equations with compact kernels, e.g. piecewise smooth kernels and

weakly singular kernels of the form $\frac{1}{|t-s|^\mu}, \;

0<\mu<1. $ We prove that 1) for integral equations, the convergence

rate depends on the smoothness of true solutions $y(t)$. If $y(t)$

satisfies condition (R): $\|y^{(k)}\|_{L^\infty[0,T]}\leq

ck!R^{-k}$}, we obtain a geometric rate of convergence; if $y(t)$

satisfies condition (M): $\|y^{(k)}\|_{L^{\infty}[0,T]}\leq cM^k $,

we obtain supergeometric rate of convergence for both Volterra

equations and Fredholm equations and related integro differential

equations; 2) for eigenvalue problems, the convergence rate depends

on the smoothness of eigenfunctions. The same convergence rate for

the largest modulus eigenvalue approximation …

Genus 0, 1, 2 Actions Of Some Almost Simple Groups Of Lie Rank 2, Xianfen Kong

Wayne State University Dissertations

Please see the paper.

Thanks.

Numerical Methods For Problems Arising In Risk Management And Insurance, Zhuo Jin

Wayne State University Dissertations

In this dissertation we investigate numerical methods for problems annuity purchasing and dividend optimization arising in risk management and insurance. We consider the models with Markov regime-switching process. The regime-switching model contains both continuous and discrete components in their evolution and is referred to as a hybrid system. The discrete events are used to model the random factors that cannot formulated by differential equations. The switching process between regimes is modulated as a finite state Markov chain.

As is widely recognized, this regime-switching model appears to be more versatile and more realistic. However, because of the regime switching and the …

Asymptotic Expansions And Stability Of Hybrid Systems With Two-Time Scales, Dung Tien Nguyen

Wayne State University Dissertations

In this dissertation, we consider solutions of hybrid systems in which both continuous dynamics and discrete events coexists. One

of the main ingredients of our models is the two-time-scale formulation. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations for switching diffusion in two classes of models, namely, fast switching systems and fast diffusion systems. To prove the validity of the asymptotic expansions, uniform error bounds are obtained.

In the second part of the dissertation, a singular linear system is considered. Again a two-time-scale formulation is used. Under suitable conditions, the system has …

New Variational Principles With Applications To Optimization Theory And Algorithms, Hung Minh Phan

Wayne State University Dissertations

In this dissertation we investigate some applications of variational analysis in optimization theory and algorithms. In the first part we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, under the name of tangential extremal principles and rated extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. These developments are in the core geometric theory of variational analysis. Our study includes the basic theory and applications to problems of semi-infinite programming …

Methods Of Variational Analysis In Pessimistic Bilevel Programming, Samarathunga M. Dassanayaka

Wayne State University Dissertations

Bilevel programming problems are of growing interest both from theoretical and practical points of view. These models are used in various applications, such as economic planning, network design, and so on. The purpose of this dissertation is to study the pessimistic (or strong) version of bilevel programming problems in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data) and can be treated by using appropriate tools of modern variational analysis and generalized differentiation developed by B. Mordukhovich.

This dissertation begins with analyzing pessimistic bilevel programs, formulation of the problems, literature review, practical application, existence of …

Variational Analysis In Parametric Optimization, Yen Nhi Nguyen Thi

Wayne State University Dissertations

The dissertation is devoted to the development of variational analysis and generalized differentiation in infinite dimensions. We derive new calculus rules for both first-order partial subdifferentials and second-order partial subdifferentials in the framework of general Banach spaces as well as more developed rules in the framework of Asplund spaces. This calculus is applied in the study of sensitivity analysis for solution maps to the parameterized generalized equations in Asplund spaces, where both bases and fields are parameter-dependent multifunctions. We analyze the parametric sensitivity of either stationary points or stationary point multiplier multifunctions associated with parameterized optimization problems under consideration. The …

Asymptotic Properties Of Markov Modulated Sequences With Fast And Slow Time Scales, Son Luu Nguyen

Wayne State University Dissertations

In this dissertation we investigate asymptotic properties of Markov modulated random processes having two-time scales. The model contains a number of mixing sequences modulated by a switching process that is a discrete-time Markov chain. The motivation of our study stems from applications in manufacturing systems, communication networks, and economic systems, in which regime-switching models are used.

This thesis focuses on asymptotic properties of the Markov modulated processes under suitable scaling. Our main effort focuses on obtaining weak convergence and strong approximation results.