Department of Mathematical Sciences

One of the central issues in stability analysis for control systems is how robust a stability property is when external disturbances are presented. This is even more critical when a system is affected by time delay. Systems affected by time delays are ubiquitous in applications. Time delays add more challenges to the task of stability analysis, mainly due to the fact that the state space of a delay system is not a finite-dimensional Euclidean space anymore, but rather an infinite dimensional space of continuous functions defined on the delay interval. In this work, we investigate robust output stability properties for nonlinear systems affected by time delays and external disturbances. Frequently in applications, the requirement of stability properties imposed on the full set of state variables can be too strenuous or even unrealistic. This motivates one to consider robust output stability properties which are related to partial stability analysis in the classic literature.
We start by formulating several notions on integral input-to-output stability and illustrate how these notions are related. We then continue to develop Lyapunov-Krasovskii type of results for such stability properties. As in the other context of Lyapunov stability analysis such as global asymptotic stability and input-to-state stability, a Lyapunov-Krasovskii functional is required to have a decay rate proportional to the magnitudes of the state variables or output variables on the whole delayed interval. This is a difficult feature when trying to construct a Lyapunov-Krasovskii functional. For this issue, we turn our efforts to Lyapunov-Krasovskii functional with a decay rate depending only on the current values of state variables or output variables. Our results lead to a type of Lyapunov-Krasovskii functionals that are more flexible regarding the decay rate, thereby leading to more efficient results for applications.

In this dissertation, we develop analogues to various notion in rigidity theory in the case of angular constraints. Initially, we were interested in angle dependencies. However, by defining a chromatic graph determined by the point-line incidence or angle graph of a given angularity, we were able to determine an assortment of properties. We provide a conjectured angular rigidity matroid involving count matroids, tranversal matroids, and traditional rigidity matroids. We consider realizations of chromatic graphs in R2 as well as C similar to the work in [3]. We extend the notions of pure conditions and infinitesimal motions using the chromatic rigidity matrix by applying techniques from algebra geometric as well as classical geometric results, such as Thales’ theorem. Some realizations I computed inspired curiosity in the space of realizations of angle-constrained graphs. We generate uniformly random sets of angle constraints to consider the space of realizations given these angle sets. We provide some results for the maximum number of possible realizations for some chromatic graphs on four vertices. We conclude with some directions for further research to develop our notions of angle-rigid graphs and their properties.

In 1994 when Peter Shor released his namesake algorithm for factoring and solving the discrete logarithm problem he changed cryptography forever. Many of the state-of-the-art cryptosystems for internet and other computerized communications will become obsolete with the advent of quantum computers. Two distinct approaches have grown to avoid the downfall of secure communication: quantum cryptography which is based in physics and information theory, and post-quantum cryptography which uses mathematical foundations believed not to be weak against even quantum assisted adversaries. This thesis is the culmination of several studies involving cryptanalysis of schemes in both the quantum and post-quantum paradigms as well as mathematically founded constructions in the post-quantum regime.
The first two chapters of this thesis on background information are intended for the reader to more fully grasp the later chapters. The third chapter shows an attack and ultimate futility of a variety of related quantum authentication schemes. The fourth chapter shows a parametric improvement over other state-of-the-art schemes in lattice based cryptography by utilizing a different cryptographic primitive. The fifth chapter proposes an attack on specific parameters of a specific lattice-based cryptographic primitive. Finally, chapter six presents a construction for a fully homomorphic encryption scheme adapted to allow for privacy enhanced machine learning.

Since the population growth systems may suffer impulsive environmental disturbances such as earthquakes, epidemics, tsunamis, hurricanes, and so on, stochastic differential equations(SDEs) that are driven not only by Brownian motion but also by α-stable Lévy noises are more appropriate to model such statistical behavior of non-Gaussian processes with heavy-tailed distribution, having infinite variance and in some cases infinite first moment. In this dissertation, we study stochastic processes defined as solutions to stochastic logistic differential equations driven by multiplicative α-stable Lévy noise. We mainly focus on one-dimensional stochastic logistic jump-diffusion processes driven by Brownian motion and α-stable Lévy motion. First, we present the stability analysis of the solution of a stochastic logistic growth model with multiplicative α-stable Lévy. We establish the existence of a unique global positive solution of this model under certain conditions. Then, we find the sufficient conditions for the almost sure exponential stability of the trivial solution of the model. Next, we provide parameter estimation for the proposed model. In parameter estimation, we use statistical methods to get an optimal and applicable estimator. We also investigate the consistency and asymptotics of the proposed estimator. We assess the validity of the estimators with a simulation study.

Financial time-series data are noisy, volatile, and nonlinear. The classic statistical linear models may not capture those underlying structures of the data. The rapid advancement in artificial intelligence and machine learning techniques, availability of large-scale data, and increased computational capabilities of a machine opens the door to developing sophisticated deep learning models to capture the nonlinearity and hidden information in the data. Creating a robust model by unlocking the power of a deep neural network and using real-time data is essential in this tech era. This study constructs a new computational framework to uncover the information in the financial time-series data and better inform the related parties. It carries out the comparative analysis of the performance of the deep learning models on stock price prediction with a well-balanced set of factors from fundamental data, macroeconomic data, and technical indicators responsible for stock price movement. We further build a novel computational framework through a merger of recurrent neural networks and random compression for the time-series analysis. The performance of the model is tested on a benchmark anomaly time-series dataset. This new computational framework in a compressed paradigm leads to improved computational efficiency and data privacy. Finally, this study develops a custom trading simulator and an agent-based hybrid model by combining gradient and gradient-free optimization methods. In particular, we explore the use of simulated annealing with stochastic gradient descent. The model trains a population of agents to predict appropriate trading behaviors such as buy, hold, or sell by optimizing the portfolio returns. Experimental results on S&P 500 index show that the proposed model outperforms the baseline models.

We consider a portfolio optimization problem in stochastic volatility jump-diffusion model. The model is a mispriced Lévy market that contains informed and uninformed investors. Contrarily to the uninformed investor, the informed investor knows that a mispricing exists in the market. The stock price follows a jump-diffusion process, the mispricing and volatility are modelled by Ornstein-Uhlenbeck (O-U) process and Cox-Ingersoll-Ross (CIR) process, respectively. We only present results for the informed investor whose goal is to maximize utility from terminal wealth over a finite investment horizon under the power utility function. We employ methods of stochastic calculus namely Hamilton-Jacobi-Bellman equation, instantaneous centralized moments of returns and three-level Crank-Nicolson method. We solve numerically the partial differential equation associated with the optimal portfolio. Under the power utility function, analogous results to those obtain in the jump-diffusion model under logarithmic utility function and deterministic volatility are obtained.

This dissertation is a study about applied dynamical systems on two concentrations. First, on the basis of the growing association between opioid addiction and HIV infection, a compartmental model is developed to study dynamics and optimal control of two epidemics; opioid addiction and HIV infection. We show that the disease-free-equilibrium is locally asymptotically stable when the basic reproduction number R0 = max(Ru0; Rv0) < 1; here Rv0 is the reproduction number of the HIV infection, and Ru0 is the reproduction number of the opioid addiction. The addiction-only boundary equilibrium exists when Ru0 > 1 and it is locally asymptotically stable when the invasion number of the opioid addiction is Ruinv < 1: Similarly, HIV-only boundary equilibrium exists when Rv0 > 1 and it is locally asymptotically stable when the invasion number of the HIV infection is Rvinv < 1. We study structural identifiability of the parameters, estimate parameters employing yearly reported data from Central for Disease Control and Prevention (CDC), and study practical identifiability of estimated parameters. We observe the basic reproduction number R0 using the parameters. Next, we introduce four distinct controls in the model for the sake of control approach, including treatment for addictions, health care education about not sharing syringes, highly active anti-retroviral therapy (HAART), and rehab treatment for opiate addicts who are HIV infected. US population using CDC data, first applying a single control in the model and observing the results, we better understand the influence of individual control. After completing each of the four applications, we apply them together at the same time in the model and compare the outcomes using different control bounds and state variable weights. We conclude the results by presenting several graphs.

Due to the phenomenon of gravitational lensing, light from distant sources may appear as several images. The “image counting problem from gravitational lensing" refers to the question of how many images might occur, given a particular distribution of lensing masses. A common model treats the lensing masses as a finite collection of points situated in a finite collection of planes. The position of the apparent images correspond to the critical points of a real-valued function and also as solutions to a system of complex rational equations. Herein, we give upper bounds for the number of images in a point mass multiplane ensemble with an arbitrary number of masses in an arbitrary number of planes. We give lower bounds on the number of solutions in a closely related problem concerning gravitational equilibria. We use persistence homology to investigate two different stochastic ensembles. Finally we produce a multiplane ensemble, related to the maximal one plane ensemble, that produces a large number of images.

The goal of this work is to study smooth invariant sets using high order approximation schemes. Whenever possible, existence of invariant sets are established using computer-assisted proofs. This provides a new set of tools for mathematically rigorous analysis of the invariant objects. The dissertation focuses on application of these tools to a family of three dimensional dissipative vector fields, derived from the normal form of a cusp-Hopf bifurcation. The vector field displays a Neimark-Sacker bifurcation giving rise to an attracting invariant torus. We examine the torus via parameter continuation from its appearance to its breakdown, scrutinizing its dynamics between these events. We also study the embeddings of the stable/unstable manifolds of the hyperbolic equilibrium solutions over this parameter range. We focus on the role of the invariant manifolds as transport barriers and their participation in global bifurcations. We then study the existence and regularity properties for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations and lay out a constructive method of computer assisted proof which pertains to explicit problems in non-perturbative regimes. We get verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We look at two important cases of rotational and resonant tori. Finally, we study the related problem of approximating two dimensional subcenter manifolds of conservative systems. As an application, we compare two methods for computing the Taylor series expansion of the graph of the subcenter manifold near a saddle-center equilibrium solution of a Hamiltonian system.

The set of transverse homoclinic intersections for a saddle-focus equilibrium in the planar equilateral restricted four body problem admits certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection, or homoclinic web. In this thesis, the planar restricted four body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way, we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale Tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddle-focus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycle-to-cycle homoclinics in the spatial problem are robust with respect to small changes in energy. The method uses high order Fourier-Taylor and Chebyshev series approximations in conjunction with the parameterization method, a general functional analytic framework for invariant manifolds. Tools that admit a natural notion of a-posteriori error analysis. Finally, we develop and implement a validation algorithm which we later use to obtain Theorems confirming the existence of homoclinic dynamics. This approach, known as the Radii polynomial, is a contraction mapping argument which can be applied to both the parameterized manifold and the Chebyshev arcs. When the Theorem applies, it guarantees the existence of a true solution near the approximation and it provides an upper bound on the C0 norm of the truncation error.