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- 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
- 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
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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
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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
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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
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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
- 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
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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
- 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
- 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
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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
- 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
- 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
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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
- 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
- HOMOCLINIC DYNAMICS IN A SPATIAL RESTRICTED FOUR BODY PROBLEM.
- Creator
- Murray, Maxime, James, Jason Mireles, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
- Abstract/Description
-
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....
Show moreThe 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.
Show less - Date Issued
- 2021
- PURL
- http://purl.flvc.org/fau/fd/FA00013758
- Subject Headings
- Boundary value problems, Invariant manifolds, Applied mathematics
- Format
- Document (PDF)
- 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
- ANNIHILATORS AND A + B RINGS.
- Creator
- Epstein, Alexandra Nicole, Klingler, Lee, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
- Abstract/Description
-
A + B rings are constructed from a ring A and nonempty set of prime ideals of A. Initially, these rings were created to provide examples of reduced rings which satisfy certain annihilator conditions. We describe precisely when A + B rings have these properties, based on the ring A and set of prime ideals of A. We continue by giving necessary and su cient conditions for A + B rings to have various other properties. We also consider annihilators in the context of frames of ideals of reduced rings.
- Date Issued
- 2020
- PURL
- http://purl.flvc.org/fau/fd/FA00013588
- Subject Headings
- Rings (Algebra)
- Format
- Document (PDF)
- Title
- DEVELOPING A DEEP LEARNING PIPELINE TO AUTOMATICALLY ANNOTATE GOLD PARTICLES IN IMMUNOELECTRON MICROSCOPY IMAGES.
- Creator
- Jerez, Diego Alejandro, Hahn, William, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
- Abstract/Description
-
Machine learning has been utilized in bio-imaging in recent years, however as it is relatively new and evolving, some researchers who wish to utilize machine learning tools have limited access because of a lack of programming knowledge. In electron microscopy (EM), immunogold labeling is commonly used to identify the target proteins, however the manual annotation of the gold particles in the images is a time-consuming and laborious process. Conventional image processing tools could provide...
Show moreMachine learning has been utilized in bio-imaging in recent years, however as it is relatively new and evolving, some researchers who wish to utilize machine learning tools have limited access because of a lack of programming knowledge. In electron microscopy (EM), immunogold labeling is commonly used to identify the target proteins, however the manual annotation of the gold particles in the images is a time-consuming and laborious process. Conventional image processing tools could provide semi-automated annotation, but those require that users make manual adjustments for every step of the analysis. To create a new high-throughput image analysis tool for immuno-EM, I developed a deep learning pipeline that was designed to deliver a completely automated annotation of immunogold particles in EM images. The program was made accessible for users without prior programming experience and was also expanded to be used on different types of immuno-EM images.
Show less - Date Issued
- 2020
- PURL
- http://purl.flvc.org/fau/fd/FA00013628
- Subject Headings
- Electron microscopy, Immunogold labeling, Image analysis, Deep learning
- Format
- Document (PDF)
- Title
- PREDICTING TROPICAL CYCLONE INTENSITY FROM GEOSYNCHRONOUS SATELLITE IMAGES USING DEEP NEURAL NETWORKS.
- Creator
- Udumulla, Niranga Mahesh, Motta, Francis, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
- Abstract/Description
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Tropical cyclones are among the most devastating natural disasters for human beings and the natural and manmade assets near to Atlantic basin. Estimating the current and future intensity of these powerful storms is crucial to protect life and property. Many methods and models exist for predicting the evolution of Atlantic basin cyclones, including numerical weather prediction models that simulate the dynamics of the atmosphere which require accurate measurements of the current state of the...
Show moreTropical cyclones are among the most devastating natural disasters for human beings and the natural and manmade assets near to Atlantic basin. Estimating the current and future intensity of these powerful storms is crucial to protect life and property. Many methods and models exist for predicting the evolution of Atlantic basin cyclones, including numerical weather prediction models that simulate the dynamics of the atmosphere which require accurate measurements of the current state of the atmosphere (NHC, 2019). Often these models fail to capture dangerous aspects of storm evolution, such as rapid intensification (RI), in which a storm undergoes a steep increase in intensity over a short time. To improve prediction of these events, scientists have turned to statistical models to predict current and future intensity using readily collected satellite image data (Pradhan, 2018). However, even the current-intensity prediction models have shown limited success in generalizing to unseen data, a result we confirm in this study. Therefore, building models for the estimating the current and future intensity of hurricanes is valuable and challenging. In this study we focus on to estimating cyclone intensity using Geostationary Operational Environmental Satellite images. These images represent five spectral bands covering the visible and infrared spectrum. We have built and compared various types of deep neural models, including convolutional networks based on long short term memory models and convolutional regression models that have been trained to predict the intensity, as measured by maximum sustained wind speed.
Show less - Date Issued
- 2020
- PURL
- http://purl.flvc.org/fau/fd/FA00013626
- Subject Headings
- Tropical cyclones, Cyclones--Tropics--Forecasting, Geosynchronous satellites, Neural networks (Computer science)
- Format
- Document (PDF)
- Title
- INFECTION AGE STRUCTURED VECTOR BORNE DISEASE MODEL WITH DIRECT TRANSMISSION.
- Creator
- Giri, Sunil, Tuncer, Necibe, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
- Abstract/Description
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Mathematical modeling is a powerful tool to study and analyze the disease dynamics prevalent in the community. This thesis studies the dynamics of two time since infection structured vector borne models with direct transmission. We have included disease induced death rate in the first model to form the second model. The aim of this thesis is to analyze whether these two models have same or different disease dynamics. An explicit expression for the reproduction number denoted by R0 is derived....
Show moreMathematical modeling is a powerful tool to study and analyze the disease dynamics prevalent in the community. This thesis studies the dynamics of two time since infection structured vector borne models with direct transmission. We have included disease induced death rate in the first model to form the second model. The aim of this thesis is to analyze whether these two models have same or different disease dynamics. An explicit expression for the reproduction number denoted by R0 is derived. Dynamical analysis reveals the forward bifurcation in the first model. That is when the threshold value R0 < 1, disease free-equilibrium is stable locally implying that if there is small perturbation of the system, then after some time, the system will return to the disease free equilibrium. When R0 > 1 the unique endemic equilibrium is locally asymptotically stable. For the second model, analysis of the existence and stability of equilibria reveals the existence of backward bifurcation i.e. where the disease free equilibrium coexists with the endemic equilibrium when the reproduction number R02 is less than unity. This aspect shows that in order to control vector borne disease, it is not sufficient to have reproduction number less than unity although necessary. Thus, the infection can persist in the population even if the reproduction number is less than unity. Numerical simulation is presented to see the bifurcation behaviour in the model. By taking the reproduction number as the bifurcation parameter, we find the system undergoes backward bifurcation at R02 = 1. Thus, the model has backward bifurcation and have two positive endemic equilibrium when R02 < 1 and unique positive endemic equilibrium whenever R02 > 1. Stability analysis shows that disease free equilibrium is locally asymptotically stable when R02 < 1 and unstable when R02 > 1. When R02 < 1, lower endemic equilibrium in backward bifurcation is locally unstable.
Show less - Date Issued
- 2020
- PURL
- http://purl.flvc.org/fau/fd/FA00013552
- Subject Headings
- Vector Borne Diseases, Mathematical models, Simulations, Dynamics--Mathematical models
- Format
- Document (PDF)
- Title
- ALGORITHMS IN LATTICE-BASED CRYPTANALYSIS.
- Creator
- Miller, Shaun, Bai, Shi, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
- Abstract/Description
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An adversary armed with a quantum computer has algorithms[66, 33, 34] at their disposal, which are capable of breaking our current methods of encryption. Even with the birth of post-quantum cryptography[52, 62, 61], some of best cryptanalytic algorithms are still quantum [45, 8]. This thesis contains several experiments on the efficacy of lattice reduction algorithms, BKZ and LLL. In particular, the difficulty of solving Learning With Errors is assessed by reducing the problem to an instance...
Show moreAn adversary armed with a quantum computer has algorithms[66, 33, 34] at their disposal, which are capable of breaking our current methods of encryption. Even with the birth of post-quantum cryptography[52, 62, 61], some of best cryptanalytic algorithms are still quantum [45, 8]. This thesis contains several experiments on the efficacy of lattice reduction algorithms, BKZ and LLL. In particular, the difficulty of solving Learning With Errors is assessed by reducing the problem to an instance of the Unique Shortest Vector Problem. The results are used to predict the behavior these algorithms may have on actual cryptographic schemes with security based on hard lattice problems. Lattice reduction algorithms require several floating-point operations including multiplication. In this thesis, I consider the resource requirements of a quantum circuit designed to simulate floating-point multiplication with high precision.
Show less - Date Issued
- 2020
- PURL
- http://purl.flvc.org/fau/fd/FA00013543
- Subject Headings
- Cryptanalysis, Cryptography, Algorithms, Lattices, Quantum computing
- Format
- Document (PDF)
- Title
- THE CHANGE POINT PROBLEM FOR TWO CLASSES OF STOCHASTIC PROCESSES.
- Creator
- Ball, Cory, Long, Hongwei, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
- Abstract/Description
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The change point problem is a problem where a process changes regimes because a parameter changes at a point in time called the change point. The objective of this problem is to estimate the change point and each of the parameters of the stochastic process. In this thesis, we examine the change point problem for two classes of stochastic processes. First, we consider the volatility change point problem for stochastic diffusion processes driven by Brownian motions. Then, we consider the drift...
Show moreThe change point problem is a problem where a process changes regimes because a parameter changes at a point in time called the change point. The objective of this problem is to estimate the change point and each of the parameters of the stochastic process. In this thesis, we examine the change point problem for two classes of stochastic processes. First, we consider the volatility change point problem for stochastic diffusion processes driven by Brownian motions. Then, we consider the drift change point problem for Ornstein-Uhlenbeck processes driven by _-stable Levy motions. In each problem, we establish the consistency of the estimators, determine asymptotic behavior for the changing parameters, and finally, we perform simulation studies to computationally assess the convergence of parameters.
Show less - Date Issued
- 2020
- PURL
- http://purl.flvc.org/fau/fd/FA00013462
- Subject Headings
- Stochastic processes, Change-point problems, Brownian motion processes, Ornstein-Uhlenbeck process, Computer simulation
- Format
- Document (PDF)