We prove quasisimilar subdecomposable operators without eigenvalues have equal essential spectra. Therefore, quasisimilar hyponormal operators have equal essential spectra. We obtain some results on the spectral pictures of cyclic hyponormal operators. An algebra homomorphism π from <i>H^∞(G)</i> to <i>L(H)</i> is a unital representation for <i>T</i> if <i>π(1) = I</i> and <i>π(x) = T</i>. It is shown that if the boundary of <i>G</i> has zero area measure, then the unital norm continuous representation for a pure hyponormal operator <i>T</i> is unique and is weak star continuous. It follows that every pure hyponormal contraction is in <i>C.₀</i> Let <i>μ</i> represent a positive, compactly supported Borel measure in the plane, <i>C</i>. For each <i>t</i> in [1, ∞ ), the space <i>P^t(μ)</i> consists of the functions in L^t(μ) that belong to the (norm) closure of the (analytic) polynomials. J. Thomson in [T] has shown that the set of bounded point evaluations, <i>bpe μ</i>, for <i>P^t(μ)</i> is a nonempty simply connected region <i>G</i>. We prove that the measure μ restricted to the boundary of <i>G</i> is absolutely continuous with respect to the harmonic measure on <i>G</i> and the space <i>P²(μ)∩C(spt μ) = A(G),</i> where <i>C(spt μ)</i> denotes the continuous functions on <i>spt μ</i> and <i>A(G)</i> denotes those functions continuous on <i>G &macr;</i> that are analytic on <i>G</i>. We also show that if a function <i>f</i> in <i>P²(μ)</i> is zero a.e. <i>μ</i> in a neighborhood of a point on the boundary, then <i>f</i> has to be the zero function. Using this result, we are able to prove that the essential spectrum of a cyclic, self-dual, subnormal operator is symmetric with respect to the real axis. We obtain a reduction into the structure of a cyclic, irreducible, self-dual, subnormal operator. One may assume, in this inquiry, that the corresponding <i>P²(μ)</i> space has <i>bpe μ = D</i>. Necessary and sufficient conditions for a cyclic, subnormal operator <i>S_μ</i> with <i>bpe μ = D</i> to have a self-dual are obtained under the additional assumption that the measure on the unit circle is log-integrable. / Ph. D.

The goal of this paper is to study the LQG problem for a class of infinite dimensional systems. We investigate the convergence of compensator gains for such systems when standard finite element schemes are used to discretize the problem. We are particularly interested in the analysis of the uniformly exponential stability of the corresponding closed - loop systems resulting from the finite dimensional compensators. A specific multiple component flexible structure is used to focus the analysis and to test problem in numerical simulations. An abstract framework for analysis and approximation of the corresponding dynamics system is developed and used to design finite - dimensional compensators. Linear semigroup theory is used to establish that the systems are well posed and to prove the convergence of generic approximation schemes. Approximate solutions of the optimal regulator and optimal observer are constructed via Galerkin - type approximations. Convergence of the scheme is established and numerical results are presented to illustrate the method / Ph. D.

A collocation technique in non-polynomial spline space is presented to approximate solutions of singular neutral functional differential equations (SNFDEs). Using solution representations and general well-posedness results for SNFDEs convergence of the method is shown for a large class of initial data including the case of discontinuous initial function. Using this technique, an identification problem is solved for a particular SNFDE. The technique is also applied to other different examples. Even for the special case in which the initial data is a discontinuous function the identification problem is successfully solved. / Master of Science

Since the introduction of the Born and Rytov approximations for use in random wave propagation some forty years ago, a controversy has boiled over the regions of validity and relative merits of the methods. Although the methods fail for strong fluctuations and distant path lengths, these two perturbation methods are the only approaches available for weak fluctuations in a random in homogeneous media. The approximations have also been applied to the inverse problem for optical and acoustical tomography. The intent of this thesis is to investigate the work of previous authors and attempt to clarify the distinctions of each method. The conclusion will be reached that neither approximation is necessarily better than the other in general for all applications. A careful consideration of the problem following the points given should point towards the use of one approximation over the other. / Master of Science

Effective information analysis generally boils down to the geometry of the data represented by a graph. Typical applications include social networks, transportation networks, the spread of epidemic disease, brain's neuronal networks, gene data on biological regulatory networks, telecommunication networks, knowledge graph, which are lying on the non-Euclidean graph domain. To describe the geometric structures, graph matrices such as adjacency matrix or graph Laplacian can be employed to reveal latent patterns. This thesis focuses on the theoretical analysis of graph neural networks and the development of methods for specific applications using graph representation. Four methods are proposed, including rational neural networks for jump graph signal estimation, RemezNet for robust attribute prediction in the graph, ICNet for integrated circuit security, and CNF-Net for dynamic circuit deobfuscation. For the first method, a recent important state-of-art method is the graph convolutional networks (GCN) nicely integrate local vertex features and graph topology in the spectral domain. However, current studies suffer from drawbacks: graph CNNs rely on Chebyshev polynomial approximation which results in oscillatory approximation at jump discontinuities since Chebyshev polynomials require degree $Omega$(poly(1/$epsilon$)) to approximate a jump signal such as $|x|$. To reduce complexity, RatioanlNet is proposed to integrate rational function and neural networks for graph node level embeddings. For the second method, we propose a method for function approximation which suffers from several drawbacks: non-robustness and infeasibility issue; neural networks are incapable of extracting analytical representation; there is no study reported to integrate the superiorities of neural network and Remez. This work proposes a novel neural network model to address the above issues. Specifically, our method utilizes the characterizations of Remez to design objective functions. To avoid the infeasibility issue and deal with the non-robustness, a set of constraints are imposed inspired by the equioscillation theorem of best rational approximation. The third method proposes an approach for circuit security. Circuit obfuscation is a recently proposed defense mechanism to protect digital integrated circuits (ICs) from reverse engineering. Estimating the deobfuscation runtime is a challenging task due to the complexity and heterogeneity of graph-structured circuit, and the unknown and sophisticated mechanisms of the attackers for deobfuscation. To address the above-mentioned challenges, this work proposes the first graph-based approach that predicts the deobfuscation runtime based on graph neural networks. The fourth method proposes a representation for dynamic size of circuit graph. By analyzing SAT attack method, a conjunctive normal form (CNF) bipartite graph is utilized to characterize the complexity of this SAT problem. To overcome the difficulty in capturing the dynamic size of the CNF graph, an energy-based kernel is proposed to aggregate dynamic features. / Doctor of Philosophy / Graph data is pervasive throughout most fields, including pandemic spread network, social network, transportation roads, internet, and chemical structure. Therefore, the applications modeled by graph benefit people's everyday life, and graph mining derives insightful opinions from this complex topology. This paper investigates an emerging technique called graph neural newton (GNNs), which is designed for graph data mining. There are two primary goals of this thesis paper: (1) understanding the GNNs in theory, and (2) apply GNNs for unexplored and values real-world scenarios. For the first goal, we investigate spectral theory and approximation theory, and a unified framework is proposed to summarize most GNNs. This direction provides a possibility that existing or newly proposed works can be compared, and the actual process can be measured. Specifically, this result demonstrates that most GNNs are either an approximation for a function of graph adjacency matrix or a function of eigenvalues. Different types of approximations are analyzed in terms of physical meaning, and the advantages and disadvantages are offered. Beyond that, we proposed a new optimization for a highly accurate but low efficient approximation. Evaluation of synthetic data proves its theoretical power, and the tests on two transportation networks show its potentials in real-world graphs. For the second goal, the circuit is selected as a novel application since it is crucial, but there are few works. Specifically, we focus on a security problem, a high-value real-world problem in industry companies such as Nvidia, Apple, AMD, etc. This problem is defined as a circuit graph as apply GNN to learn the representation regarding the prediction target such as attach runtime. Experiment on several benchmark circuits shows its superiority on effectiveness and efficacy compared with competitive baselines. This paper provides exploration in theory and application with GNNs, which shows a promising direction for graph mining tasks. Its potentials also provide a wide range of innovations in graph-based problems.

Function approximation is an important problem. This work presents applications of interpolants to modeling random variables. Specifically, this work studies the prediction of distributions of random variables applied to computer system throughput variability. Existing approximation methods including multivariate adaptive regression splines, support vector regressors, multilayer perceptrons, Shepard variants, and the Delaunay mesh are investigated in the context of computer variability modeling. New methods of approximation using Box splines, Voronoi cells, and Delaunay for interpolating distributions of data with moderately high dimension are presented and compared with existing approaches. Novel theoretical error bounds are constructed for piecewise linear interpolants over functions with a Lipschitz continuous gradient. Finally, a mathematical software that constructs monotone quintic spline interpolants for distribution approximation from data samples is proposed. / Doctor of Philosophy / It is common for scientists to collect data on something they are studying. Often scientists want to create a (predictive) model of that phenomenon based on the data, but the choice of how to model the data is a difficult one to answer. This work proposes methods for modeling data that operate under very few assumptions that are broadly applicable across science. Finally, a software package is proposed that would allow scientists to better understand the true distribution of their data given relatively few observations.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 1977 / Vita. / Includes bibliographical references. / Ph. D. / Ph. D. Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science

In this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong Prufer rings. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015 / FAU Electronic Theses and Dissertations Collection

This thesis focuses on the developing of statistical methodology in reliability and quality engineering, and to assist the decision-makings at enterprise level, process level, and product level. In Chapter II, we propose a multi-level statistical modeling strategy to characterize data from spatial logistics systems. The model can support business decisions at different levels. The information available from higher hierarchies is incorporated into the multi-level model as constraint functions for lower hierarchies. The key contributions include proposing the top-down multi-level spatial models which improve the estimation accuracy at lower levels; applying the spatial smoothing techniques to solve facility location problems in logistics. In Chapter III, we propose methods for modeling system service reliability in a supply chain, which may be disrupted by uncertain contingent events. This chapter applies an approximation technique for developing first-cut reliability analysis models. The approximation relies on multi-level spatial models to characterize patterns of store locations and demands. The key contributions in this chapter are to bring statistical spatial modeling techniques to approximate store location and demand data, and to build system reliability models entertaining various scenarios of DC location designs and DC capacity constraints. Chapter IV investigates the power law process, which has proved to be a useful tool in characterizing the failure process of repairable systems. This chapter presents a procedure for detecting and estimating a mixture of conforming and nonconforming systems. The key contributions in this chapter are to investigate the property of parameter estimation in mixture repair processes, and to propose an effective way to screen out nonconforming products. The key contributions in Chapter V are to propose a new method to analyze heavily censored accelerated life testing data, and to study the asymptotic properties. This approach flexibly and rigorously incorporates distribution assumptions and regression structures into estimating equations in a nonparametric estimation framework. Derivations of asymptotic properties of the proposed method provide an opportunity to compare its estimation quality to commonly used parametric MLE methods in the situation of mis-specified regression models.