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 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, MirelesJames, 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 computerassisted 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 cuspHopf 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 computerassisted 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 cuspHopf bifurcation. The vector field displays a NeimarkSacker 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 nonperturbative 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 saddlecenter 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 TIMESERIES 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 timeseries 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 largescale 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 timeseries 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 largescale 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 realtime data is essential in this tech era. This study constructs a new computational framework to uncover the information in the financial timeseries 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 wellbalanced 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 timeseries analysis. The performance of the model is tested on a benchmark anomaly timeseries 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 agentbased hybrid model by combining gradient and gradientfree 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), Timeseries analysis, Stocks, Simulated annealing (Mathematics)
 Format
 Document (PDF)
 Title
 STABILITY ANALYSIS AND PARAMETER ESTIMATION OF A STOCHASTIC LOGISTIC GROWTH MODEL WITH MULTIPLICATIVE αSTABLE 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 αstable Lévy noises are more appropriate to model such statistical behavior of nonGaussian processes with heavytailed 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 αstable Lévy noises are more appropriate to model such statistical behavior of nonGaussian processes with heavytailed 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 αstable Lévy noise. We mainly focus on onedimensional stochastic logistic jumpdiffusion processes driven by Brownian motion and αstable Lévy motion. First, we present the stability analysis of the solution of a stochastic logistic growth model with multiplicative αstable 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 jumpdiffusion 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 jumpdiffusion process, the mispricing and volatility are modelled by OrnsteinUhlenbeck (OU) process and CoxIngersollRoss (CIR) process, respectively. We only present results for the...
Show moreWe consider a portfolio optimization problem in stochastic volatility jumpdiffusion 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 jumpdiffusion process, the mispricing and volatility are modelled by OrnsteinUhlenbeck (OU) process and CoxIngersollRoss (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 HamiltonJacobiBellman equation, instantaneous centralized moments of returns and threelevel CrankNicolson 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 jumpdiffusion 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 POSTQUANTUM 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 stateoftheart 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 postquantum 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 stateoftheart 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 postquantum 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 postquantum paradigms as well as mathematically founded constructions in the postquantum 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 stateoftheart schemes in lattice based cryptography by utilizing a different cryptographic primitive. The fifth chapter proposes an attack on specific parameters of a specific latticebased 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 2SPHERE: A COMBINATORIAL AND ANALYTIC APPROACH WITH COMPUTER ASSISTED PROOFS.
 Creator
 Dhakal, Bishal, MirelesJames, 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 2sphere. 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 2sphere. 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 twodimensional 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 insample data set to determine the spatial relationship for storm trajectories at all pairwise directions or orientations. Afterwards, the model is assessed on an outofsample 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 INPUTTOOUTPUT 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 finitedimensional 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 finitedimensional 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 inputtooutput stability and illustrate how these notions are related. We then continue to develop LyapunovKrasovskii type of results for such stability properties. As in the other context of Lyapunov stability analysis such as global asymptotic stability and inputtostate stability, a LyapunovKrasovskii 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 LyapunovKrasovskii functional. For this issue, we turn our efforts to LyapunovKrasovskii functional with a decay rate depending only on the current values of state variables or output variables. Our results lead to a type of LyapunovKrasovskii 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 pointline 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 pointline 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 angleconstrained 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 anglerigid 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 DELAUNAYRIPS 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 multidimensional 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 highdimensional 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 multidimensional 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 highdimensional 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 DelaunayRips (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 datadriven 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 subjectmatter experts. We train several machine learning models to assess the performance of incorporating topological features into training with biochemical features. Using covertree 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, TopologyData 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 diseasefreeequilibrium 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 diseasefreeequilibrium 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 addictiononly boundary equilibrium exists when Ru0 > 1 and it is locally asymptotically stable when the invasion number of the opioid addiction is Ruinv < 1: Similarly, HIVonly 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 antiretroviral 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 realvalued 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 realvalued 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 lensesMathematics, Persistent homology
 Format
 Document (PDF)
 Title
 Dynamics of low and high pathogenic avian influenza in wild and domestic bird populations.
 Creator
 Tuncer, Necibe, Torres, Juan, Martcheva, Maia, Barfield, Michael, Holt, Robert D.
 Date Issued
 20160114
 PURL
 http://purl.flvc.org/fau/fd/FAUIR000194
 Format
 Citation