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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.
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Date Issued: 2022
PURL: http://purl.flvc.org/fau/fd/FA00013970
Subject Headings: Dynamical systems, Infectious diseases, Parameter estimation
Format: Document (PDF)

Title: Implementation and comparison of the Golay and first order Reed-Muller codes.
Creator: Shukina, Olga., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: In this project we perform data transmission across noisy channels and recover the message first by using the Golay code, and then by using the first-order Reed- Muller code. The main objective of this thesis is to determine which code among the above two is more efficient for text message transmission by applying the two codes to exactly the same data with the same channel error bit probabilities. We use the comparison of the error-correcting capability and the practical speed of the Golay...
Show moreIn this project we perform data transmission across noisy channels and recover the message first by using the Golay code, and then by using the first-order Reed- Muller code. The main objective of this thesis is to determine which code among the above two is more efficient for text message transmission by applying the two codes to exactly the same data with the same channel error bit probabilities. We use the comparison of the error-correcting capability and the practical speed of the Golay code and the first-order Reed-Muller code to meet our goal.
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Date Issued: 2013
PURL: http://purl.flvc.org/fcla/dt/3362579
Subject Headings: Error-correcting codes (Information theory), Coding theory, Computer algorithms, Digital modulation
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: 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.
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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: Input-to-state stability properties for discrete-time nonlinear systems.
Creator: Gao, Kehan, Florida Atlantic University, Wang, Yuan, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: In this thesis, we study the input-to-state stability (scISS) property and related characterizations for discrete-time nonlinear systems. Variations of scISS property were employed in solving particular control problems. The main contribution of this work is to provide a detailed analysis on the relations among various types of notations related to system stability and show that most scISS results for continuous-time nonlinear system can be extended to discrete-time case.
Date Issued: 1999
PURL: http://purl.flvc.org/fcla/dt/15688
Subject Headings: Discrete-time systems, Nonlinear systems, Stability, Control theory
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.
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Date Issued: 2023
PURL: http://purl.flvc.org/fau/fd/FA00014267
Subject Headings: Nonlinear systems, Time delay systems
Format: Document (PDF)

Title: Kicks and Maps A different Approach to Modeling Biological Systems.
Creator: Ippolito, Stephen Anthony, Naudot, Vincent, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: Modeling a biological systems, is a cyclic process which involves constructing a model from current theory and beliefs and then validating that model against the data. If the data does not match, qualitatively or quantitatively then there may be a problem with either our beliefs or the current theory. At the same time directly finding a model from the existing data would make generalizing results difficult. A considerable difficultly in this process is how to specify the model in the first...
Show moreModeling a biological systems, is a cyclic process which involves constructing a model from current theory and beliefs and then validating that model against the data. If the data does not match, qualitatively or quantitatively then there may be a problem with either our beliefs or the current theory. At the same time directly finding a model from the existing data would make generalizing results difficult. A considerable difficultly in this process is how to specify the model in the first place. There is a need to be practice which accounts for the growing use of mathematical and statistical methods. However, as a systems becomes more complex, standard mathematical approaches may not be sufficient. In the field of ecology, the standard techniques involve discrete maps, and continuous models such as ODE's. The intent of this work is to present the mathematics necessary to study hybrids of these two models, then consider two case studies. In first case we con sider a coral reef with continuous change, except in the presence of hurricanes. The results of the data are compared quantitatively and qualitatively with simulation results. For the second case we consider a model for rabies with a periodic birth pulse. Here the analysis is qualitative as we demonstrate the existence of a strange attractor by looking at the intersections of the stable and unstable manifold for the saddle point generating the attractor. For both cases studies the introduction of a discrete event into a continuous system is done via a Dirac Distribution or Measure.
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Date Issued: 2015
PURL: http://purl.flvc.org/fau/fd/FA00004508, http://purl.flvc.org/fau/fd/FA00004508
Subject Headings: Artificial intellligence -- Biological applications, Biology -- Mathematical models, Computational intelligence, Differential dynamical systems, Nonliner mechanics -- Mathematical models
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.
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Date Issued: 2021
PURL: http://purl.flvc.org/fau/fd/FA00013716
Subject Headings: Matrices, Combinatorial analysis, Graph theory
Format: Document (PDF)

Title: Low rank transitive representations, primitive extensions, and the collision problem in PSL (2, q).
Creator: Thapa Magar, Krishna B., Magliveras, Spyros S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: Every transitive permutation representation of a finite group is the representation of the group in its action on the cosets of a particular subgroup of the group. The group has a certain rank for each of these representations. We first find almost all rank-3 and rank-4 transitive representations of the projective special linear group P SL(2, q) where q = pm and p is an odd prime. We also determine the rank of P SL (2, p) in terms of p on the cosets of particular given subgroups. We then...
Show moreEvery transitive permutation representation of a finite group is the representation of the group in its action on the cosets of a particular subgroup of the group. The group has a certain rank for each of these representations. We first find almost all rank-3 and rank-4 transitive representations of the projective special linear group P SL(2, q) where q = pm and p is an odd prime. We also determine the rank of P SL (2, p) in terms of p on the cosets of particular given subgroups. We then investigate the construction of rank-3 transitive and primitive extensions of a simple group, such that the extension group formed is also simple. In the latter context we present a new, group theoretic construction of the famous Hoffman-Singleton graph as a rank-3 graph.
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Date Issued: 2015
PURL: http://purl.flvc.org/fau/fd/FA00004471, http://purl.flvc.org/fau/fd/FA00004471
Subject Headings: Combinatorial designs and configurations, Cryptography, Data encryption (Computer science), Finite geometries, Finite groups, Group theory, Permutation groups
Format: Document (PDF)

Title: Maximally Prüfer rings.
Creator: Sharma, Madhav, Klingler, Lee, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: In this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong...
Show moreIn this dissertation, we consider six Prufer-like conditions on acommutative ring R. These conditions form a hierarchy. Being a Prufer ring is not a local property: a Prufer ring may not remain a Prufer ring when localized at a prime or maximal ideal. We introduce a seventh condition based on this fact and extend the hierarchy. All the conditions of the hierarchy become equivalent in the case of a domain, namely a Prufer domain. We also seek the relationship of the hierarchy with strong Prufer rings.
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Date Issued: 2015
PURL: http://purl.flvc.org/fau/fd/FA00004465, http://purl.flvc.org/fau/fd/FA00004465
Subject Headings: Approximation theory, Commutative algebra, Commutative rings, Geometry, Algebraic, Ideals (Algebra), Mathematical analysis, Prüfer rings, Rings (Algebra), Rings of integers
Format: Document (PDF)

Title: Message authentication in an identity-based encryption scheme: 1-Key-Encrypt-Then-MAC.
Creator: Amento, Brittanney Jaclyn, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: We present an Identity-Based Encryption scheme, 1-Key-Encrypt-Then-MAC, in which we are able to verify the authenticity of messages using a MAC. We accomplish this authentication by combining an Identity-Based Encryption scheme given by Boneh and Franklin, with an Identity-Based Non-Interactive Key Distribution given by Paterson and Srinivasan, and attaching a MAC. We prove the scheme is chosen plaintext secure and chosen ciphertext secure, and the MAC is existentially unforgeable.
Date Issued: 2010
PURL: http://purl.flvc.org/FAU/2796050
Subject Headings: Data encryption (Computer science), Public key cryptopgraphy, Public key infrastructure (Computer security)
Format: Document (PDF)

Title: A min/max algorithm for cubic splines over k-partitions.
Creator: Golinko, Eric David, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: The focus of this thesis is to statistically model violent crime rates against population over the years 1960-2009 for the United States. We approach this question as to be of interest since the trend of population for individual states follows different patterns. We propose here a method which employs cubic spline regression modeling. First we introduce a minimum/maximum algorithm that will identify potential knots. Then we employ least squares estimation to find potential regression...
Show moreThe focus of this thesis is to statistically model violent crime rates against population over the years 1960-2009 for the United States. We approach this question as to be of interest since the trend of population for individual states follows different patterns. We propose here a method which employs cubic spline regression modeling. First we introduce a minimum/maximum algorithm that will identify potential knots. Then we employ least squares estimation to find potential regression coefficients based upon the cubic spline model and the knots chosen by the minimum/maximum algorithm. We then utilize the best subsets regression method to aid in model selection in which we find the minimum value of the Bayesian Information Criteria. Finally, we preent the R2adj as a measure of overall goodness of fit of our selected model. We have found among the fifty states and Washington D.C., 42 out of 51 showed an R2adj value that was greater than 90%. We also present an overall model of the United States. Also, we show additional applications our algorithm for data which show a non linear association. It is hoped that our method can serve as a unified model for violent crime rate over future years.
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Date Issued: 2012
PURL: http://purl.flvc.org/FAU/3342107
Subject Headings: Spline theory, Data processing, Bayesian statistical decision theory, Data processing, Neural networks (Computer science), Mathematical statistics, Uncertainty (Information theory), Probabilities, Regression analysis
Format: Document (PDF)

Title: Minimal zero-dimensional extensions.
Creator: Chiorescu, Marcela, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: The structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are...
Show moreThe structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are ideals of R primary for distinct prime ideals of height greater than zero. A subsidiary result is the classification of minimal zero-dimensional extensions of general ZPI-rings.
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Date Issued: 2009
PURL: http://purl.flvc.org/FAU/210447
Subject Headings: Algebra, Abstract, Noetherian rings, Commutative rings, Modules (Algebra), Algebraic number theory
Format: Document (PDF)

Title: THE MINIMUM K-CENTER PROBLEM FOR GRID GRAPH.
Creator: HSUEH, CHI-FU, Florida Atlantic University, Hadlock, Frank O., Hoffman, Frederick, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: A study was made of the problem of locating M facilities on a connected grid graph, so that M is the minimum and so that every demand node on the graph is within given distance K of one of these M facilities. We call this problem briefly the G(N,K,M) problem, with N denoting the total number of demand nodes. An algorithm for solving this problem by using backtrack technique is presented in this thesis. A heuristic algorithm is also present; although the resulting M is not always minimum, it...
Show moreA study was made of the problem of locating M facilities on a connected grid graph, so that M is the minimum and so that every demand node on the graph is within given distance K of one of these M facilities. We call this problem briefly the G(N,K,M) problem, with N denoting the total number of demand nodes. An algorithm for solving this problem by using backtrack technique is presented in this thesis. A heuristic algorithm is also present; although the resulting M is not always minimum, it tends to be near minimum. The advantage over the backtrack algorithm is that the heuristic algorithm operates very quickly. Algorithms represented in this thesis are programmed in the Pascal language for the Univac 1100 computer at Florida Atlantic University, Boca Raton, Florida.
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Date Issued: 1981
PURL: http://purl.flvc.org/fcla/dt/14077
Subject Headings: Graph theory, Algorithms
Format: Document (PDF)

Title: Modeling and simulating interest rates via time-dependent mean reversion.
Creator: Dweck, Andrew Jason, Long, Hongwei, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: The purpose of this thesis is to compare the effectiveness of several interest rate models in fitting the true value of interest rates. Up until 1990, the universally accepted models were the equilibrium models, namely the Rendleman-Bartter model, the Vasicek model, and the Cox-Ingersoll-Ross (CIR) model. While these models were probably considered relatively accurate around the time of their discovery, they do not provide a good fit to the initial term structure of interest rates, making...
Show moreThe purpose of this thesis is to compare the effectiveness of several interest rate models in fitting the true value of interest rates. Up until 1990, the universally accepted models were the equilibrium models, namely the Rendleman-Bartter model, the Vasicek model, and the Cox-Ingersoll-Ross (CIR) model. While these models were probably considered relatively accurate around the time of their discovery, they do not provide a good fit to the initial term structure of interest rates, making them substandard for use by traders in pricing interest rate options. The fourth model we consider is the Hull-White one-factor model, which does provide this fit. After calibrating, simulating, and comparing these four models, we find that the Hull-White model gives the best fit to our data sets.
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Date Issued: 2014
PURL: http://purl.flvc.org/fau/fd/FA00004103, http://purl.flvc.org/fau/fd/FA00004103
Subject Headings: Game theory, Investment analysis, Options (Finance), Recursive functions, Stochastic differential equations
Format: Document (PDF)

Title: Multivariate finite operator calculus applied to counting ballot paths containing patterns [electronic resource].
Creator: Sullivan, Shaun, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: Counting lattice paths where the number of occurrences of a given pattern is monitored requires a careful analysis of the pattern. Not the length, but the characteristics of the pattern are responsible for the difficulties in finding explicit solutions. Certain features, like overlap and difference in number of ! and " steps determine the recursion formula. In the case of ballot paths, that is paths the stay weakly above the line y = x, the solutions to the recursions are typically polynomial...
Show moreCounting lattice paths where the number of occurrences of a given pattern is monitored requires a careful analysis of the pattern. Not the length, but the characteristics of the pattern are responsible for the difficulties in finding explicit solutions. Certain features, like overlap and difference in number of ! and " steps determine the recursion formula. In the case of ballot paths, that is paths the stay weakly above the line y = x, the solutions to the recursions are typically polynomial sequences. The objects of Finite Operator Calculus are polynomial sequences, thus the theory can be used to solve the recursions. The theory of Finite Operator Calculus is strengthened and extended to the multivariate setting in order to obtain solutions, and to prepare for future applications.
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Date Issued: 2011
PURL: http://purl.flvc.org/FAU/3174076
Subject Headings: Combinatorial probabilities, Lattice paths, Combinatorial enumeration problems, Generating functions
Format: Document (PDF)

Title: Negligible Variation, Change of Variables, and a Smooth Analog of the Hobby-Rice Theorem.
Creator: Rutherfoord, Vermont Charles, Sagher, Yoram, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: This dissertation concerns two topics in analysis. The rst section is an exposition of the Henstock-Kurzweil integral leading to a necessary and su cient condition for the change of variables formula to hold, with implications for the change of variables formula for the Lebesgue integral. As a corollary, a necessary and suf- cient condition for the Fundamental Theorem of Calculus to hold for the HK integral is obtained. The second section concerns a challenge raised in a paper by O. Lazarev...
Show moreThis dissertation concerns two topics in analysis. The rst section is an exposition of the Henstock-Kurzweil integral leading to a necessary and su cient condition for the change of variables formula to hold, with implications for the change of variables formula for the Lebesgue integral. As a corollary, a necessary and suf- cient condition for the Fundamental Theorem of Calculus to hold for the HK integral is obtained. The second section concerns a challenge raised in a paper by O. Lazarev and E. H. Lieb, where they proved that, given f1….,fn ∈ L1 ([0,1] ; C), there exists a smooth function φ that takes values on the unit circle and annihilates span {f1...., fn}. We give an alternative proof of that fact that also shows the W1,1 norm of φ can be bounded by 5πn + 1. Answering a question raised by Lazarev and Lieb, we show that if p > 1 then there is no bound for the W1,p norm of any such multiplier in terms of the norms of f1...., fn.
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Date Issued: 2016
PURL: http://purl.flvc.org/fau/fd/FA00004627
Subject Headings: Mathematical analysis., Measure theory., Henstock-Kurzweil integral.
Format: Document (PDF)

Title: New Geometric Large Sets.
Creator: Hurley, Michael Robert, Magliveras, Spyros S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: Let V be an n-dimensional vector space over the field of q elements. By a geometric t-[q^n, k, λ] design we mean a collection D of k-dimensional subspaces of V, called blocks, such that every t-dimensional subspace T of V appears in exactly λ blocks in D. A large set, LS [N] [t, k, q^n], of geometric designs is a collection on N disjoint t-[q^n, k, λ] designs that partitions [V K], the collection of k-dimensional subspaces of V. In this work we construct non-isomorphic large sets using...
Show moreLet V be an n-dimensional vector space over the field of q elements. By a geometric t-[q^n, k, λ] design we mean a collection D of k-dimensional subspaces of V, called blocks, such that every t-dimensional subspace T of V appears in exactly λ blocks in D. A large set, LS [N] [t, k, q^n], of geometric designs is a collection on N disjoint t-[q^n, k, λ] designs that partitions [V K], the collection of k-dimensional subspaces of V. In this work we construct non-isomorphic large sets using methods based on incidence structures known as the Kramer-Mesner matrices. These structures are induced by particular group actions on the collection of subspaces of the vector space V. Subsequently, we discuss and use computational techniques for solving certain linear problems of the form AX = B, where A is a large integral matrix and X is a {0,1} solution. These techniques involve (i) lattice basis-reduction, including variants of the LLL algorithm, and (ii) linear programming. Inspiration came from the 2013 work of Braun, Kohnert, Ostergard, and Wassermann, [17], who produced the first nontrivial large set of geometric designs with t ≥ 2. Bal Khadka and Michael Epstein provided the know-how for using the LLL and linear programming algorithms that we implemented to construct the large sets.
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Date Issued: 2016
PURL: http://purl.flvc.org/fau/fd/FA00004732, http://purl.flvc.org/fau/fd/FA00004732
Subject Headings: Group theory., Finite groups., Factorial experiment designs., Irregularities of distribution (Number theory), Combinatorial analysis.
Format: Document (PDF)

$New methods for estimating fractal dimensions of coastlines$

Title: New methods for estimating fractal dimensions of coastlines.
Creator: Klotzbach, Jonathan David, Florida Atlantic University, Voss, Richard, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: A coastline is an example of a statistically self-similar fractal. A standard characterization walks a ruler of fixed size along the coast and estimates fractal dimension from the power-law relationship between length and ruler size. Multiple intersection can lead to ambiguity in choosing the next step. The standard method always chooses the first intersection along the curve. Variations were developed to choose intersections which highlight geographic properties. The land method measures...
Show moreA coastline is an example of a statistically self-similar fractal. A standard characterization walks a ruler of fixed size along the coast and estimates fractal dimension from the power-law relationship between length and ruler size. Multiple intersection can lead to ambiguity in choosing the next step. The standard method always chooses the first intersection along the curve. Variations were developed to choose intersections which highlight geographic properties. The land method measures accessibility to the coast from land at each size while the water method probes water access. Measurements on sections of the East and West Coasts of the United States typically showed the land length exceeding water. Jumps in water length as step size decreased were often caused by narrow rivers or bays which have few corresponding land features. Simple recursive constructions were inadequate to model this asymmetry.
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Date Issued: 1998
PURL: http://purl.flvc.org/fcla/dt/15545
Subject Headings: Fractals, Boundaries, Coasts
Format: Document (PDF)

Title: New Results in Group Theoretic Cryptology.
Creator: Sramka, Michal, Florida Atlantic University, Magliveras, Spyros S., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: With the publication of Shor's quantum algorithm for solving discrete logarithms in finite cyclic groups, a need for new cryptographic primitives arose; namely, for more secure primitives that would prevail in the post-quantum era. The aim of this dissertation is to exploit some hard problems arising from group theory for use in cryptography. Over the years, there have been many such proposals. We first look at two recently proposed schemes based on some form of a generalization of the...
Show moreWith the publication of Shor's quantum algorithm for solving discrete logarithms in finite cyclic groups, a need for new cryptographic primitives arose; namely, for more secure primitives that would prevail in the post-quantum era. The aim of this dissertation is to exploit some hard problems arising from group theory for use in cryptography. Over the years, there have been many such proposals. We first look at two recently proposed schemes based on some form of a generalization of the discrete logari thm problem (DLP), identify their weaknesses, and cryptanalyze them. By applying the exper tise gained from the above cryptanalyses, we define our own generalization of the DLP to arbitrary finite groups. We show that such a definition leads to the design of signature schemes and pseudo-random number generators with provable security under a security assumption based on a group theoretic problem. In particular, our security assumption is based on the hardness of factorizing elements of the projective special linear group over a finite field in some representations. We construct a one-way function based on this group theoretic assumption and provide a security proof.
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Date Issued: 2006
PURL: http://purl.flvc.org/fau/fd/FA00000878
Subject Headings: Group theory, Mathematical statistics, Cryptography, Combinatorial designs and configurations, Data encryption (Computer science), Coding theory
Format: Document (PDF)

Title: Nonlinear Phenomena from a Reinjected Horseshoe.
Creator: Fontaine, Marcus, Kalies, William D., Naudot, Vincent, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: A geometric model of a reinjected cuspidal horseshoe is constructed, that resembles the standard horseshoe, but where the set of points that escape are now reinjected and contribute to richer dynamics. We show it is observed in the unfolding of a three-dimensional vector field possessing an inclination-flip homoclinic orbit with a resonant hyperbolic equilibrium. We use techniques from classical dynamical systems theory and rigorous computational symbolic dynamics with algebraic topology to...
Show moreA geometric model of a reinjected cuspidal horseshoe is constructed, that resembles the standard horseshoe, but where the set of points that escape are now reinjected and contribute to richer dynamics. We show it is observed in the unfolding of a three-dimensional vector field possessing an inclination-flip homoclinic orbit with a resonant hyperbolic equilibrium. We use techniques from classical dynamical systems theory and rigorous computational symbolic dynamics with algebraic topology to show that for suitable parameters the flow contains a strange attractor.
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Date Issued: 2016
PURL: http://purl.flvc.org/fau/fd/FA00004591
Subject Headings: Nonlinear theories., Computational dynamics., Attractors (Mathematics), Chaotic behavior in systems., Mathematical physics.
Format: Document (PDF)

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