View All Items

Title: LONESUM MATRICES AND ACYCLIC ORIENTATIONS: ENUMERATION AND ASYMPTOTICS.
Creator: Khera, Jessica, Lundberg, Erik, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: An acyclic orientation of a graph is an assignment of a direction to each edge in a way that does not form any directed cycles. Acyclic orientations of a complete bipartite graph are in bijection with a class of matrices called lonesum matrices, which can be uniquely reconstructed from their row and column sums. We utilize this connection and other properties of lonesum matrices to determine an analytic form of the generating function for the length of the longest path in an acyclic...
Show moreAn acyclic orientation of a graph is an assignment of a direction to each edge in a way that does not form any directed cycles. Acyclic orientations of a complete bipartite graph are in bijection with a class of matrices called lonesum matrices, which can be uniquely reconstructed from their row and column sums. We utilize this connection and other properties of lonesum matrices to determine an analytic form of the generating function for the length of the longest path in an acyclic orientation on a complete bipartite graph, and then study the distribution of the length of the longest path when the acyclic orientation is random. We use methods of analytic combinatorics, including analytic combinatorics in several variables (ACSV), to determine asymptotics for lonesum matrices and other related classes.
Show less
Date Issued: 2021
PURL: http://purl.flvc.org/fau/fd/FA00013716
Subject Headings: Matrices, Combinatorial analysis, Graph theory
Format: Document (PDF)

Title: FORMATION, EVOLUTION, AND BREAKDOWN OF INVARIANT TORI IN DISSIPATIVE SYSTEMS: FROM VISUALIZATION TO COMPUTER ASSISTED PROOFS.
Creator: Fleurantin, Emmanuel, Mireles-James, Jason, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: 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...
Show moreThe 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.
Show less
Date Issued: 2021
PURL: http://purl.flvc.org/fau/fd/FA00013812
Subject Headings: Invariants, Manifolds (Mathematics), Dynamical systems
Format: Document (PDF)

Title: FINANCIAL TIME-SERIES ANALYSIS WITH DEEP NEURAL NETWORKS.
Creator: Rimal, Binod, Hahn, William Edward, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: 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...
Show moreFinancial 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.
Show less
Date Issued: 2022
PURL: http://purl.flvc.org/fau/fd/FA00014009
Subject Headings: Neural networks (Computer science), Deep learning (Machine learning), Time-series analysis, Stocks, Simulated annealing (Mathematics)
Format: Document (PDF)

Title: STABILITY ANALYSIS AND PARAMETER ESTIMATION OF A STOCHASTIC LOGISTIC GROWTH MODEL WITH MULTIPLICATIVE α-STABLE LÉVY NOISE.
Creator: Bhusal, Bikram, Long, Hongwei, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: 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...
Show moreSince 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.
Show less
Date Issued: 2022
PURL: http://purl.flvc.org/fau/fd/FA00014043
Subject Headings: Parameter estimation, Stochastic processes, Lévy processes
Format: Document (PDF)

Title: OPTIMAL PORTFOLIO FOR THE INFORMED INVESTOR IN MISPRICED LEVY MARKET WITH STOCHASTIC VOLATILITY AND POWER UTILITY.
Creator: Zephirin, Duval, Long, Hongwei, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: 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...
Show moreWe 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.
Show less
Date Issued: 2022
PURL: http://purl.flvc.org/fau/fd/FA00014040
Subject Headings: Investments, Portfolio, Lévy processes, Utility functions
Format: Document (PDF)

Title: SELECTED TOPICS IN QUANTUM AND POST-QUANTUM CRYPTOGRAPHY.
Creator: Johnson, Floyd, Bai, Shi, Steinwandt, Rainer, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: 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...
Show moreIn 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.
Show less
Date Issued: 2022
PURL: http://purl.flvc.org/fau/fd/FA00014088
Subject Headings: Quantum cryptography, Cryptography, Homomorphisms (Mathematics), Lattices (Mathematics)
Format: Document (PDF)

Title: SOLVING V.I. ARNOLD’S PROBLEM ABOUT ASYMPTOTIC ENUMERATION OF MORSE FUNCTIONS ON THE 2-SPHERE: A COMBINATORIAL AND ANALYTIC APPROACH WITH COMPUTER ASSISTED PROOFS.
Creator: Dhakal, Bishal, Mireles-James, Jason, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: The goal of this dissertation is to estimate the precise asymptotics for the number of geometric equivalence classes of Morse functions on the 2-sphere. Our approach involves utilizing the Lagrange inversion formula, Cauchy’s coefficient formula, and the saddle point method for the asymptotic analysis of contour integrals to analyze the generating function derived by L. Nicolaescu, expressed as the inverse of an elliptic integral. We utilize complex analysis, nonlinear functional analysis in...
Show moreThe goal of this dissertation is to estimate the precise asymptotics for the number of geometric equivalence classes of Morse functions on the 2-sphere. Our approach involves utilizing the Lagrange inversion formula, Cauchy’s coefficient formula, and the saddle point method for the asymptotic analysis of contour integrals to analyze the generating function derived by L. Nicolaescu, expressed as the inverse of an elliptic integral. We utilize complex analysis, nonlinear functional analysis in infinite sequence spaces, and interval arithmetic to write all the necessary MATLAB programs that validate our results. This work answers questions posed by Arnold and Nicolaescu, furthering our understanding of the topological properties of Morse functions on two-dimensional manifolds. It also demonstrates the effectiveness of a computer assisted approach for asymptotic analysis.
Show less
Date Issued: 2023
PURL: http://purl.flvc.org/fau/fd/FA00014264
Subject Headings: Manifolds (Mathematics), Morse theory, Combinatorial analysis
Format: Document (PDF)

Title: SPATIAL ANALYSIS OF NORTH ATLANTIC STORM TRAJECTORIES.
Creator: Lazar, Austin J., Li, Yang, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: Storms in the North Atlantic Ocean are observed on a continual basis yearly. Storm trajectories exhibit random behavior and are costly to society. Data from the National Oceanic and Atmospheric Administration (NOAA) contains every storm’s track from the year 1851 to 2022. Data of each storm’s track can aid in decision making regarding their behavior. In this article, data analysis is performed on historical storm tracks during the years 1966 to 2022, where access to satellite information is...
Show moreStorms in the North Atlantic Ocean are observed on a continual basis yearly. Storm trajectories exhibit random behavior and are costly to society. Data from the National Oceanic and Atmospheric Administration (NOAA) contains every storm’s track from the year 1851 to 2022. Data of each storm’s track can aid in decision making regarding their behavior. In this article, data analysis is performed on historical storm tracks during the years 1966 to 2022, where access to satellite information is available. Analysis on this data will be used to determine if the storms’ trajectory is statistically dependent on other storm’s trajectories at varying distances in space. The proposed model is a spatial statistical model that is fitted on an in-sample data set to determine the spatial relationship for storm trajectories at all pairwise directions or orientations. Afterwards, the model is assessed on an out-of-sample test data set for performance evaluation.
Show less
Date Issued: 2023
PURL: http://purl.flvc.org/fau/fd/FA00014227
Subject Headings: Spatial analysis (Statistics), Storms, North Atlantic Ocean
Format: Document (PDF)

Title: INTEGRAL INPUT-TO-OUTPUT STABILITY ANALYSIS FOR NONLINEAR SYSTEMS WITH TIME DELAYS.
Creator: Nawarathna, R. H. Harsha, Wang, Yuan, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: 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...
Show moreOne 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.
Show less
Date Issued: 2023
PURL: http://purl.flvc.org/fau/fd/FA00014267
Subject Headings: Nonlinear systems, Time delay systems
Format: Document (PDF)

Title: ANGULAR RIGIDITY THEORY IN PLANAR FRAMEWORKS.
Creator: Urizar, David Ricardo, Rosen, Zvi, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: 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...
Show moreIn 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.
Show less
Date Issued: 2023
PURL: http://purl.flvc.org/fau/fd/FA00014291
Subject Headings: Rigidity (Geometry), Algebraic geometry, Graphs
Format: Document (PDF)

Title: TOPOLOGICAL DATA ANALYSIS FOR DATA SCIENCE: THE DELAUNAY-RIPS COMPLEX, TRIANGULATION STABILITIES, AND PROTEIN STABILITY PREDICTIONS.
Creator: Mishra, Amish, Motta, Francis, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: Topological Data Analysis (TDA) is a relatively new field of research that utilizes topological notions to extract discriminating features from data. Within TDA, persistent homology (PH) is a robust method to compute multi-dimensional geometric and topological features of a dataset. Because these features are often stable under certain perturbations of the underlying data, are often discriminating, and can be used for visualization of structure in high-dimensional data and in statistical and...
Show moreTopological Data Analysis (TDA) is a relatively new field of research that utilizes topological notions to extract discriminating features from data. Within TDA, persistent homology (PH) is a robust method to compute multi-dimensional geometric and topological features of a dataset. Because these features are often stable under certain perturbations of the underlying data, are often discriminating, and can be used for visualization of structure in high-dimensional data and in statistical and machine learning modeling, PH has attracted the interest of researchers across scientific disciplines and in many industry applications. However, computational costs may present challenges to effectively using PH in certain data contexts, and theoretical stability results may not hold in practice. In this dissertation, we develop an algorithm that can reduce the computation burden of computing persistent homology on point cloud data. Naming it Delaunay-Rips (DR), we define, implement, and empirically test this computationally tractable simplicial complex construction for computing persistent homology of Euclidean point cloud data. We demonstrate the practical robustness of DR for persistent homology in comparison with other simplical complexes in machine learning applications such as predicting sleep state from patient heart rate. To justify the theoretical stability of DR, we prove the stability of the Delaunay triangulation of a pointcloud P under perturbations of the points of P. Specifically, we impose a notion of genericity on the points of P to ensure stability. In the final chapter, we contribute to the field of computational biology by taking a data-driven approach to learn topological features of designed proteins from their persistence diagrams. We find correlations between the learned topological features and biochemical features to investigate how protein structure relates to features identified by subject-matter experts. We train several machine learning models to assess the performance of incorporating topological features into training with biochemical features. Using cover-tree differencing via entropy reduction (CDER), we identify distinguishing regions of the persistence diagrams of stable/unstable proteins. More notably, we find statistically significant improvement in classification performance (in terms of average precision score) for certain designed secondary structure topologies.
Show less
Date Issued: 2023
PURL: http://purl.flvc.org/fau/fd/FA00014311
Subject Headings: Data Science, Data Analysis, Topology--Data processing, Protein Stability
Format: Document (PDF)

Title: IDENTIFIABILITY ANALYSIS AND OPTIMAL CONTROL OF INFECTIOUS DISEASES EPIDEMICS AND PARAMETERIZATION METHOD FOR (UN)STABLE MANIFOLDS OF IMPLICITLY DEFINED DYNAMICAL SYSTEMS.
Creator: Neupane Timsina, Archana, Tuncer, Necibe, Mireles James, Jason D., Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: 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 and it is locally asymptotically stable when the invasion...
Show moreThis 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.
Show less
Date Issued: 2022
PURL: http://purl.flvc.org/fau/fd/FA00013970
Subject Headings: Dynamical systems, Infectious diseases, Parameter estimation
Format: Document (PDF)

Title: ON THE IMAGE COUNTING PROBLEM FROM GRAVITATIONAL LENSING.
Creator: Perry, Sean, Lundberg, Erik, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: 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...
Show moreDue 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.
Show less
Date Issued: 2022
PURL: http://purl.flvc.org/fau/fd/FA00013889
Subject Headings: Gravitational lense, Gravitational lenses--Mathematics, Persistent homology
Format: Document (PDF)

Title: PRIVACY-PRESERVING TOPOLOGICAL DATA ANALYSIS USING HOMOMORPHIC ENCRYPTION.
Creator: Gold, Dominic, Motta, Francis, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: Computational tools grounded in algebraic topology, known collectively as topological data analysis (TDA), have been used for dimensionality-reduction to preserve salient and discriminating features in data. This faithful but compressed representation of data through TDA’s flagship method, persistent homology (PH), motivates its use to address the complexity, depth, and inefficiency issues present in privacy-preserving, homomorphic encryption (HE)-based machine learning (ML) models, which...
Show moreComputational tools grounded in algebraic topology, known collectively as topological data analysis (TDA), have been used for dimensionality-reduction to preserve salient and discriminating features in data. This faithful but compressed representation of data through TDA’s flagship method, persistent homology (PH), motivates its use to address the complexity, depth, and inefficiency issues present in privacy-preserving, homomorphic encryption (HE)-based machine learning (ML) models, which permit a data provider (often referred to as the Client) to outsource computational tasks on their encrypted data to a computationally-superior but semi-honest party (the Server). This work introduces efforts to adapt the well-established TDA-ML pipeline on encrypted data to realize the benefits TDA can provide to HE’s computational limitations as well as provide HE’s provable security on the sensitive data domains in which TDA has found success in (e.g., sequence, gene expression, imaging). The privacy-protecting technologies which could emerge from this foundational work will lead to direct improvements to the accessibility and equitability of health care systems. ML promises to reduce biases and improve accuracies of diagnoses, and enabling such models to act on sensitive biomedical data without exposing it will improve trustworthiness of these systems.
Show less
Date Issued: 2024
PURL: http://purl.flvc.org/fau/fd/FA00014440
Subject Headings: Data encryption (Computer science), Homomorphisms (Mathematics), Privacy-preserving techniques (Computer science), Machine learning
Format: Document (PDF)

Title: SPACES OF MINIMAL PRIME ELEMENTS OF ALGEBRAIC FRAMES WITHOUT FIP.
Creator: Madinya, Albert Anthony, Bhattacharjee, Papiya, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: The following dissertation investigates algebraic frames and their spaces of minimal prime elements with respect to the Hull-Kernel topology and Inverse topology. Much work by other authors has been done in obtaining internal characterizations in frame-theoretic terms for when these spaces satisfy certain topological properties, but most of what is done is under the auspices of the finite intersection property. In the first half of this dissertation, we shall add to the literature more...
Show moreThe following dissertation investigates algebraic frames and their spaces of minimal prime elements with respect to the Hull-Kernel topology and Inverse topology. Much work by other authors has been done in obtaining internal characterizations in frame-theoretic terms for when these spaces satisfy certain topological properties, but most of what is done is under the auspices of the finite intersection property. In the first half of this dissertation, we shall add to the literature more characterizations in this context, and in the second half we will study general algebraic frames and investigate which, if any, of the known theorems generalize to algebraic frames not necessarily with the FIP. Throughout this investigative journey, we have found that certain ideals and filters of algebraic frames play a pivotal role in determining internal characterizations of the algebraic frames for when interesting topological properties occur in its space of minimal prime elements. In this dissertation, we investigate completely prime filters and compactly generated filters on algebraic frames. We introduce a new concept of subcompact elements and subcompactly generated filters. One of our main results is that the inverse topology on the space of minimal prime elements is compact if and only if every maximal subcompactly generated filter is completely prime. Furthermore, when the space of minimal prime elements is compact, then each minimal prime has what we are calling the compact absoluteness property.
Show less
Date Issued: 2024
PURL: http://purl.flvc.org/fau/fd/FA00014410
Subject Headings: Mathematics, Applied mathematics
Format: Document (PDF)

Title: LATTICE SIGNATURES BASED ON MODULE-NTRU.
Creator: Kottal, Sulani Thakshila Baddhe Vidhanalage, Bai, Shi, Karabina, Koray, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: Euclidean lattices have attracted considerable research interest as they can be used to construct efficient cryptographic schemes that are believed to be quantum-resistant. The NTRU problem, introduced by J. Hoffstein, J. Pipher, and J. H. Silverman in 1996 [16], serves as an important average-case computational problem in lattice-based cryptography. Following their pioneer work, the NTRU assumption and its variants have been used widely in modern cryptographic constructions such as...
Show moreEuclidean lattices have attracted considerable research interest as they can be used to construct efficient cryptographic schemes that are believed to be quantum-resistant. The NTRU problem, introduced by J. Hoffstein, J. Pipher, and J. H. Silverman in 1996 [16], serves as an important average-case computational problem in lattice-based cryptography. Following their pioneer work, the NTRU assumption and its variants have been used widely in modern cryptographic constructions such as encryption, signature, etc. Let Rq = Zq[x]/ (xn + 1) be a quotient polynomial ring. The standard NTRU problem asks to recover short polynomials f, g E Rq such that h - g/ f (mod q), given a public key h and the promise that such elements exist. In practice, the degree n is often a power of two. As a generalization of NTRU, the Module-NTRU problems were introduced by Cheon, Kim, Kim, and Son (IACR ePrint 2019/1468), and Chuengsatiansup, Prest, Stehle, Wallet, and Xagawa (ASIACCS '20). In this thesis, we presented two post-quantum Digital Signature Schemes based on the Module-NTRU problem and its variants.
Show less
Date Issued: 2024
PURL: http://purl.flvc.org/fau/fd/FA00014407
Subject Headings: Lattice theory, Cryptography, Public key cryptography, Applied mathematics
Format: Document (PDF)

Title: PARAMETERIZATION OF INVARIANT CIRCLES IN MAPS.
Creator: Blessing, David Charles, James, J. D. James, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: We explore a novel method of approximating contractible invariant circles in maps. The process begins by leveraging improvements on Birkhoff's Ergodic Theorem via Weighted Birkhoff Averages to compute high precision estimates on several Fourier modes. We then set up a Newton-like iteration scheme to further improve the estimation and extend the approximation out to a sufficient number of modes to yield a significant decay in the magnitude of the coefficients of high order. With this...
Show moreWe explore a novel method of approximating contractible invariant circles in maps. The process begins by leveraging improvements on Birkhoff's Ergodic Theorem via Weighted Birkhoff Averages to compute high precision estimates on several Fourier modes. We then set up a Newton-like iteration scheme to further improve the estimation and extend the approximation out to a sufficient number of modes to yield a significant decay in the magnitude of the coefficients of high order. With this approximation in hand, we explore the phase space near the approximate invariant circle with a form numerical continuation where the rotation number is perturbed and the process is repeated. Then, we turn our attention to a completely different problem which can be approached in a similar way to the numerical continuation, finding a Siegel disk boundary in a holomorphic map. Given a holomorphic map which leads to a formally solvable cohomological equation near the origin, we use a numerical continuation style process to approximate an invariant circle in the Siegel disk near the origin. Using an iterative scheme, we then enlarge the invariant circle so that it approximates the boundary of the Siegel disk.
Show less
Date Issued: 2024
PURL: http://purl.flvc.org/fau/fd/FA00014464
Subject Headings: Dynamical systems, Nonlinearity (Mathematics), Numerical analysis, Parameterization
Format: Document (PDF)

Title: ANALYSIS OF CRYPTOGRAPHIC EFFICIENCY: ELLIPTIC CURVE SCALAR MULTIPLICATION AND CONSTANT-TIME POLYNOMIAL INVERSION IN POST-QUANTUM CRYPTOGRAPHY.
Creator: Dutta, Abhraneel, Persichetti, Edoardo, Karabina, Koray, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: An efficient scalar multiplication algorithm is vital for elliptic curve cryptosystems. The first part of this dissertation focuses on a scalar multiplication algorithm based on scalar recodings resistant to timing attacks. The algorithm utilizes two recoding methods: Recode, which generalizes the non-zero signed all-bit set recoding, and Align, which generalizes the sign aligned columns recoding. For an ℓ-bit scalar split into d subscalars, our algorithm has a computational cost of ⌈⌈ℓ logk...
Show moreAn efficient scalar multiplication algorithm is vital for elliptic curve cryptosystems. The first part of this dissertation focuses on a scalar multiplication algorithm based on scalar recodings resistant to timing attacks. The algorithm utilizes two recoding methods: Recode, which generalizes the non-zero signed all-bit set recoding, and Align, which generalizes the sign aligned columns recoding. For an ℓ-bit scalar split into d subscalars, our algorithm has a computational cost of ⌈⌈ℓ logk(2)⌉/d⌉ point additions and k-scalar multiplications and a storage cost of kd−1(k − 1) – 1 points on E. The “split and comb” method further optimizes computational and storage complexity. We find the best setting to be with a fixed base point on a Twisted Edwards curve using a mix of projective and extended coordinates, with k = 2 generally offering the best performance. However, k = 3 may be better in certain applications. The second part of this dissertation is dedicated to constant-time polynomial inversion algorithms in Post-Quantum Cryptography (PQC). The computation of the inverse of a polynomial over a quotient ring or finite field is crucial for key generation in post-quantum cryptosystems like NTRU, BIKE, and LEDACrypt. Efficient algorithms must run in constant time to prevent side-channel attacks. We examine constant-time algorithms based on Fermat’s Little Theorem and the Extended GCD Algorithm, providing detailed time complexity analysis. We find that the constant-time Extended GCD inversion algorithm is more efficient, performing fewer field multiplications. Additionally, we explore other exponentiation algorithms similar to the Itoh-Tsuji inversion method, which optimizes polynomial multiplications in the BIKE/LEDACrypt setup. Recent results on hardware implementations are also discussed.
Show less
Date Issued: 2024
PURL: http://purl.flvc.org/fau/fd/FA00014492
Subject Headings: Cryptography, Curves, Elliptic, Polynomials
Format: Document (PDF)

Title: AN INVESTIGATION ON PRACTICAL ASPECTS OF POST-QUANTUM CRYPTOGRAPHY.
Creator: Karagoz, Emrah, Persichetti, Edoardo, Karabina, Koray, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: The security of the current public-key cryptographic schemes, based on integer factorization and discrete logarithm problems, is expected to be totally broken with the development of quantum computers utilizing Shor’s algorithm. As a result, The National Institute of Standards and Technology (NIST) initiated the Post-Quantum Cryptography (PQC) standardization process in 2016, inviting researchers to submit candidate algorithms that are both resistant to quantum attacks and efficient for real...
Show moreThe security of the current public-key cryptographic schemes, based on integer factorization and discrete logarithm problems, is expected to be totally broken with the development of quantum computers utilizing Shor’s algorithm. As a result, The National Institute of Standards and Technology (NIST) initiated the Post-Quantum Cryptography (PQC) standardization process in 2016, inviting researchers to submit candidate algorithms that are both resistant to quantum attacks and efficient for real world applications. Researchers have since studied various aspects of the candidate algorithms, such as their security against quantum attacks and efficient implementation on different platforms. In this thesis, we investigate the practical aspects of Post-Quantum Cryptography and contribute to several topics. First, we focus on the knapsack problem and its security under classical and quantum attacks. Second, we improve the secure biometric template generation algorithm NTT-Sec, proposing an enhanced version, NTT-Sec-R, and providing an in-depth design and security analysis. Third, we work on optimizing implementations of the post-quantum secure signature scheme LESS and polynomial inversion algorithms for code-based schemes. Finally, we analyze a proposed countermeasure for the exposure model of SIKE, the isogeny-based scheme that is a candidate in NIST’s Round 4.
Show less
Date Issued: 2024
PURL: http://purl.flvc.org/fau/fd/FA00014549
Subject Headings: Cryptography, Knapsack problem (Mathematics), Computer science
Format: Document (PDF)

Title: TOPOLOGICAL MACHINE LEARNING WITH UNREDUCED PERSISTENCE DIAGRAMS.
Creator: Abreu, Nicole Juliana, Motta, Francis, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
Abstract/Description: A common topological data analysis approach used in the experimental sciences involves creating machine learning pipelines that incorporate discriminating topological features derived from persistent homology (PH) of data samples, encoded in persistence diagrams (PDs) and associated topological feature vectors. Often the most computationally demanding step is computing PH through an algorithmic process known as boundary matrix reduction. In this work, we introduce several methods to generate...
Show moreA common topological data analysis approach used in the experimental sciences involves creating machine learning pipelines that incorporate discriminating topological features derived from persistent homology (PH) of data samples, encoded in persistence diagrams (PDs) and associated topological feature vectors. Often the most computationally demanding step is computing PH through an algorithmic process known as boundary matrix reduction. In this work, we introduce several methods to generate topological feature vectors from unreduced boundary matrices. We compared the performance of classifiers trained on vectorizations of unreduced PDs to vectorizations of fully-reduced PDs across several benchmark ML datasets. We discovered that models trained on PDs built from unreduced diagrams can perform on par and even outperform those trained on full-reduced diagrams. This observation suggests that machine learning pipelines which incorporate topology-based features may benefit in terms of computational cost and performance by utilizing information contained in unreduced boundary matrices.
Show less
Date Issued: 2024
PURL: http://purl.flvc.org/fau/fd/FA00014518
Subject Headings: Machine learning, Topology, Data sets
Format: Document (PDF)

FAU Theses and Dissertations (133)	+ -
FAU College Collections (8)	+ -

Cryptography (17)	+ -
Data encryption (Computer science) (16)	+ -
Mathematics (10)	+ -
Mathematical optimization (7)	+ -
Algorithms (6)	+ -

Coding theory (6)	+ -
Data processing (6)	+ -
Mathematical models (6)	+ -
Combinatorial designs and configurations (5)	+ -
Stochastic processes (5)	+ -
Combinatorial analysis (4)	+ -
Combinatorial group theory (4)	+ -
Dynamical systems (4)	+ -
Finite groups (4)	+ -
Graph theory (4)	+ -
Mathematical physics (4)	+ -
Mathematical statistics (4)	+ -
Nonlinear systems (4)	+ -
Quantum computing (4)	+ -
Abelian groups (3)	+ -
Algebra, Abstract (3)	+ -
Algebraic number theory (3)	+ -
Applied mathematics (3)	+ -
Chaotic behavior in systems (3)	+ -
Commutative rings (3)	+ -
Computer algorithms (3)	+ -
Computer security (3)	+ -
Control theory (3)	+ -
Curves, Algebraic (3)	+ -
Differential equations, Partial (3)	+ -

Electronic Thesis or Dissertation (133)	+ -
Program Review (7)	+ -
article (1)	+ -

Charles E. Schmidt College of Science (140)	+ -
Department of Mathematical Sciences (140)	+ -
Florida Atlantic University (99)	+ -
Magliveras, Spyros S. (8)	+ -
Long, Hongwei (7)	+ -

Search In

Pages

Pages

Sort Results By:

Narrow results by:

Collection

Topical Subject

Genre

Type of Resource

Language

Creator

Format

Owner