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- Title
- Decay for time-dependent Schroedinger equations.
- Creator
- Zhou, Zhen, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
We study the decay in time of solutions of Schrodinger equations of the type du/du=idelta u+iV(t)u, establishing that for small potentials and initial data in L1 the solution u satisfies sup[u(x,t)](x element of R)
Show moreWe study the decay in time of solutions of Schrodinger equations of the type du/du=idelta u+iV(t)u, establishing that for small potentials and initial data in L1 the solution u satisfies sup[u(x,t)](x element of R)Show less - Date Issued
- 1996
- PURL
- http://purl.flvc.org/fcla/dt/12463
- Subject Headings
- Mathematics
- Format
- Document (PDF)
- Title
- Detection of multiple change-points in hazard models.
- Creator
- Zhang, Wei, Qian, Lianfen, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
Change-point detection in hazard rate function is an important research topic in survival analysis. In this dissertation, we firstly review existing methods for single change-point detection in piecewise exponential hazard model. Then we consider the problem of estimating the change point in the presence of right censoring and long-term survivors while using Kaplan-Meier estimator for the susceptible proportion. The maximum likelihood estimators are shown to be consistent. Taking one step...
Show moreChange-point detection in hazard rate function is an important research topic in survival analysis. In this dissertation, we firstly review existing methods for single change-point detection in piecewise exponential hazard model. Then we consider the problem of estimating the change point in the presence of right censoring and long-term survivors while using Kaplan-Meier estimator for the susceptible proportion. The maximum likelihood estimators are shown to be consistent. Taking one step further, we propose an counting process based and least squares based change-point detection algorithm. For single change-point case, consistency results are obtained. We then consider the detection of multiple change-points in the presence of long-term survivors via maximum likelihood based and counting process based method. Last but not least, we use a weighted least squares based and counting process based method for detection of multiple change-points with long-term survivors and covariates. For multiple change-points detection, simulation studies show good performances of our estimators under various parameters settings for both methods. All methods are applied to real data analyses.
Show less - Date Issued
- 2014
- PURL
- http://purl.flvc.org/fau/fd/FA00004173
- Subject Headings
- Problem solving--Data processing., Process control--Statistical methods., Point processes., Mathematical statistics., Failure time data analysis--Data processing., Survival analysis (Biometry)--Data processing.
- Format
- Document (PDF)
- Title
- Detecting essential genes in microarray dataset with unequal number of gene probes.
- Creator
- Zhang, Wei, Qian, Lianfen, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
Microarray technology is a powerful approach for genomic research, which allows the monitoring of expressing profiles for tens of thousands genes in parallel and is already producing huge amounts of data. This thesis is motivated by a special microarray dataset for the bacteria Yersinia Pestis. It contains more than four thousands genes and each gene has different number of observations. The main purpose of this thesis is to detect essentially functional genes. Gene level adjusted multiple t...
Show moreMicroarray technology is a powerful approach for genomic research, which allows the monitoring of expressing profiles for tens of thousands genes in parallel and is already producing huge amounts of data. This thesis is motivated by a special microarray dataset for the bacteria Yersinia Pestis. It contains more than four thousands genes and each gene has different number of observations. The main purpose of this thesis is to detect essentially functional genes. Gene level adjusted multiple t‐test is proposed to handle the problem of unequal number of observations. Furthermore, a comparation study of our method with two other existing methods (Behrens‐Fisher method and Hotelling t‐square method) are presented. The comparison results show that our proposed methods is the best for identifying essential genes.
Show less - Date Issued
- 2012
- PURL
- http://purl.flvc.org/fau/fd/FA00004261
- 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
- Stability analysis for singularly perturbed systems with time-delays.
- Creator
- Yang, Yang, Wang, Yuan, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
Singularly perturbed systems with or without delays commonly appear in mathematical modeling of physical and chemical processes, engineering applications, and increasingly, in mathematical biology. There has been intensive work for singularly perturbed systems, yet most of the work so far focused on systems without delays. In this thesis, we provide a new set of tools for the stability analysis for singularly perturbed control systems with time delays.
- Date Issued
- 2015
- PURL
- http://purl.flvc.org/fau/fd/FA00004423, http://purl.flvc.org/fau/fd/FA00004423
- Subject Headings
- Biology -- Mathematical models, Biomathematics, Differentiable dynamical systems, Differential equations, Partial -- Numerical solutions, Global analysis (Mathematics), Lyapunov functions, Nonlinear theories
- Format
- Document (PDF)
- Title
- On Boolean algebras and their role in analysis.
- Creator
- Winkowska-Nowak, Katarzyna, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
The relations between complete and $\sigma$-complete covers of a Boolean algebra are examined. The Dedekind completion of a Boolean algebra is shown to be a quotient of any complete cover. Atoms of a Boolean algebra correspond to atoms of the Dedekind completion hence the Dedekind completion of an atomic Boolean algebra is isomorphic to the power set of the set of all atoms. There exists a correspondence between complete (sigma-complete) homomorphisms and full (sigma-complete) ideals. The...
Show moreThe relations between complete and $\sigma$-complete covers of a Boolean algebra are examined. The Dedekind completion of a Boolean algebra is shown to be a quotient of any complete cover. Atoms of a Boolean algebra correspond to atoms of the Dedekind completion hence the Dedekind completion of an atomic Boolean algebra is isomorphic to the power set of the set of all atoms. There exists a correspondence between complete (sigma-complete) homomorphisms and full (sigma-complete) ideals. The explicit form of the Dedekind completion is given for the Boolean algebra generated by all semiopen subintervals of [0,1) as the atomless, complete Boolean algebra of all regularly closed subsets of [0,1). A compatible topology for a Boolean algebra is a topology for which addition and multiplication are continuous. The properties concerning products, quotients, subspaces and uniform completions of topological Boolean algebras are examined. Compact algebras are isomorphic and homeomorphic with power sets, endowed with the product topology. Measure algebras endowed with the weak* topology are compatible if and only if the underlying measure is purely atomic. A new proof of Stone Representation Theorem for a field of sets is given, providing a tool for establishing relations between Stone representation spaces of algebras, covers, subalgebras and quotients.
Show less - Date Issued
- 1996
- PURL
- http://purl.flvc.org/fcla/dt/12454
- Subject Headings
- Mathematics
- Format
- Document (PDF)
- Title
- Computing topological dynamics from time series.
- Creator
- Wess, Mark., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
The topological entropy of a continuous map quantifies the amount of chaos observed in the map. In this dissertation we present computational methods which enable us to compute topological entropy for given time series data generated from a continuous map with a transitive attractor. A triangulation is constructed in order to approximate the attractor and to construct a multivalued map that approximates the dynamics of the linear interpolant on the triangulation. The methods utilize...
Show moreThe topological entropy of a continuous map quantifies the amount of chaos observed in the map. In this dissertation we present computational methods which enable us to compute topological entropy for given time series data generated from a continuous map with a transitive attractor. A triangulation is constructed in order to approximate the attractor and to construct a multivalued map that approximates the dynamics of the linear interpolant on the triangulation. The methods utilize simplicial homology and in particular the Lefschetz Fixed Point Theorem to establish the existence of periodic orbits for the linear interpolant. A semiconjugacy is formed with a subshift of nite type for which the entropy can be calculated and provides a lower bound for the entropy of the linear interpolant. The dissertation concludes with a discussion of possible applications of this analysis to experimental time series.
Show less - Date Issued
- 2008
- PURL
- http://purl.flvc.org/FAU/186294
- Subject Headings
- Algebraic topology, Graph theory, Fixed point theory, Singularities (Mathematics)
- Format
- Document (PDF)
- Title
- Signature schemes in single and multi-user settings.
- Creator
- Villanyi, Viktoria., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
In the first chapters we will give a short introduction to signature schemes in single and multi-user settings. We give the definition of a signature scheme and explain a group of possible attacks on them. In Chapter 6 we give a construction which derives a subliminal-free RSA public key. In the construction we use a computationally binding and unconditionally hiding commitment scheme. To establish a subliminal-free RSA modulus n, we have to construct the secret primes p and q. To prove p and...
Show moreIn the first chapters we will give a short introduction to signature schemes in single and multi-user settings. We give the definition of a signature scheme and explain a group of possible attacks on them. In Chapter 6 we give a construction which derives a subliminal-free RSA public key. In the construction we use a computationally binding and unconditionally hiding commitment scheme. To establish a subliminal-free RSA modulus n, we have to construct the secret primes p and q. To prove p and q are primes we use Lehmann's primality test on the commitments. The chapter is based on the paper, "RSA signature schemes with subliminal-free public key" (Tatra Mountains Mathematical Publications 41 (2008)). In chapter 7 a one-time signature scheme using run-length encoding is presented, which in the random oracle model offers security against chosen-message attacks. For parameters of interest, the proposed scheme enables about 33% faster verification with a comparable signature size than a construction of Merkle and Winternitz. The public key size remains unchanged (1 hash value). The main cost for the faster verification is an increase in the time required for signing messages and for key generation. The chapter is based on the paper "A one-time signature using run-length encoding" (Information Processing Letters Vol. 108, Issue 4, (2008)).
Show less - Date Issued
- 2009
- PURL
- http://purl.flvc.org/FAU/215289
- Subject Headings
- Information technology, Security measures, Cryptography, Coding theory, Data encryption (Computer science), DIgital watermarking
- Format
- Document (PDF)
- Title
- Rings of integer-valued polynomials and derivatives.
- Creator
- Villanueva, Yuri., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
For D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integer-valued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D) - {f e K [X]lf(k) (E) c...
Show moreFor D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integer-valued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D) - {f e K [X]lf(k) (E) c D for all 0 < k < r} , for finite E and fixed integer r > 1. These results rely on the work of Skolem [23] and Brizolis [7], who found ways to characterize ideals of Int (E, D) from the values of their polynomials at points in D. We obtain similar results for E = D in case D is local, Noetherian, one-dimensional, analytically irreducible, with finite residue field.
Show less - Date Issued
- 2012
- PURL
- http://purl.flvc.org/FAU/3356899
- Subject Headings
- Rings of integers, Ideals (Algebra), Polynomials, Arithmetic algebraic geometry, Categories (Mathematics), Commutative algebra
- Format
- Document (PDF)
- Title
- ANGULAR RIGIDITY THEORY IN PLANAR FRAMEWORKS.
- Creator
- Urizar, David Ricardo, Rosen, Zvi, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
- Abstract/Description
-
In this dissertation, we develop analogues to various notion in rigidity theory in the case of angular constraints. Initially, we were interested in angle dependencies. However, by defining a chromatic graph determined by the point-line incidence or angle graph of a given angularity, we were able to determine an assortment of properties. We provide a conjectured angular rigidity matroid involving count matroids, tranversal matroids, and traditional rigidity matroids. We consider realizations...
Show moreIn this dissertation, we develop analogues to various notion in rigidity theory in the case of angular constraints. Initially, we were interested in angle dependencies. However, by defining a chromatic graph determined by the point-line incidence or angle graph of a given angularity, we were able to determine an assortment of properties. We provide a conjectured angular rigidity matroid involving count matroids, tranversal matroids, and traditional rigidity matroids. We consider realizations of chromatic graphs in R2 as well as C similar to the work in [3]. We extend the notions of pure conditions and infinitesimal motions using the chromatic rigidity matrix by applying techniques from algebra geometric as well as classical geometric results, such as Thales’ theorem. Some realizations I computed inspired curiosity in the space of realizations of angle-constrained graphs. We generate uniformly random sets of angle constraints to consider the space of realizations given these angle sets. We provide some results for the maximum number of possible realizations for some chromatic graphs on four vertices. We conclude with some directions for further research to develop our notions of angle-rigid graphs and their properties.
Show less - Date Issued
- 2023
- PURL
- http://purl.flvc.org/fau/fd/FA00014291
- Subject Headings
- Rigidity (Geometry), Algebraic geometry, Graphs
- Format
- Document (PDF)
- Title
- 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
-
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
- Characterizing the Geometry of a Random Point Cloud.
- Creator
- Tyree, Zachariah, Lundberg, Erik, Long, Hongwei, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
This thesis is composed of three main parts. Each chapter is concerned with characterizing some properties of a random ensemble or stochastic process. The properties of interest and the methods for investigating them di er between chapters. We begin by establishing some asymptotic results regarding zeros of random harmonic mappings, a topic of much interest to mathematicians and astrophysicists alike. We introduce a new model of harmonic polynomials based on the so-called "Weyl ensemble" of...
Show moreThis thesis is composed of three main parts. Each chapter is concerned with characterizing some properties of a random ensemble or stochastic process. The properties of interest and the methods for investigating them di er between chapters. We begin by establishing some asymptotic results regarding zeros of random harmonic mappings, a topic of much interest to mathematicians and astrophysicists alike. We introduce a new model of harmonic polynomials based on the so-called "Weyl ensemble" of random analytic polynomials. Building on the work of Li and Wei [28] we obtain precise asymptotics for the average number of zeros of this model. The primary tools used in this section are the famous Kac-Rice formula as well as classical methods in the asymptotic analysis of integrals such as the Laplace method. Continuing, we characterize several topological properties of this model of harmonic polynomials. In chapter 3 we obtain experimental results concerning the number of connected components of the orientation-reversing region as well as the geometry of the distribution of zeros. The tools used in this section are primarily Monte Carlo estimation and topological data analysis (persistent homology). Simulations in this section are performed within MATLAB with the help of a computational homology software known as Perseus. While the results in this chapter are empirical rather than formal proofs, they lead to several enticing conjectures and open problems. Finally, in chapter 4 we address an industry problem in applied mathematics and machine learning. The analysis in this chapter implements similar techniques to those used in chapter 3. We analyze data obtained by observing CAN tra c. CAN (or Control Area Network) is a network for allowing micro-controllers inside of vehicles to communicate with each other. We propose and demonstrate the e ectiveness of an algorithm for detecting malicious tra c using an approach that discovers and exploits the natural geometry of the CAN surface and its relationship to random walk Markov chains.
Show less - Date Issued
- 2018
- PURL
- http://purl.flvc.org/fau/fd/FA00013118
- Subject Headings
- Stochastic processes, Harmonic functions, Random point cloud
- Format
- Document (PDF)
- Title
- The triangle of reflections.
- Creator
- Torres, Jesus, Yiu, Paul Y., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
This thesis presents some results in triangle geometry discovered using dynamic software, namely, Geometer’s Sketchpad, and confirmed with computations using Mathematica 9.0. Using barycentric coordinates, we study geometric problems associated with the triangle of reflections T of a given triangle T, yielding interesting triangle centers and simple loci such as circles and conics. These lead to some new triangle centers with reasonably simple coordinates, and also new properties of some...
Show moreThis thesis presents some results in triangle geometry discovered using dynamic software, namely, Geometer’s Sketchpad, and confirmed with computations using Mathematica 9.0. Using barycentric coordinates, we study geometric problems associated with the triangle of reflections T of a given triangle T, yielding interesting triangle centers and simple loci such as circles and conics. These lead to some new triangle centers with reasonably simple coordinates, and also new properties of some known, classical centers. Particularly, we show that the Parry reflection point is the common point of two triads of circles, one associated with the tangential triangle, and another with the excentral triangle. More interestingly, we show that a certain rectangular hyperbola through the vertices of T appears as the locus of the perspector of a family of triangles perspective with T, and in a different context as the locus of the orthology center of T with another family of triangles.
Show less - Date Issued
- 2014
- PURL
- http://purl.flvc.org/fau/fd/FA00004167, http://purl.flvc.org/fau/fd/FA00004167
- Subject Headings
- Geometer's Sketchpad, Geometry -- Study and teaching, Geometry, Hyperbolic, Mathematics -- Computer network resources, Problem solving
- Format
- Document (PDF)
- Title
- Stability analysis for nonlinear systems with time-delays.
- Creator
- Tiwari, Shanaz, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
In this work, we investigate input-to-state stability (ISS) and other related stability properties for control systems with time-delays. To overcome the complexity caused by the presence of the delays, we adopt a Razumikhin approach. The underlying idea of this approach is to treat the delayed variables as system uncertainties. The advantage of this approach is that one works in the more familiar territory of stability analysis for delay-free systems in the context of ISS instead of carrying...
Show moreIn this work, we investigate input-to-state stability (ISS) and other related stability properties for control systems with time-delays. To overcome the complexity caused by the presence of the delays, we adopt a Razumikhin approach. The underlying idea of this approach is to treat the delayed variables as system uncertainties. The advantage of this approach is that one works in the more familiar territory of stability analysis for delay-free systems in the context of ISS instead of carrying out stability analysis on systems of functional differential equations. Our first step is to provide criteria on ISS and input-to-input stability properties based on the Razumikhin approach. We then turn our attention to large-scale interconnected systems. It has been well recognized that the small-gain theory is a powerful tool for stability analysis of interconnected systems. Using the Razumikhin approach, we develop small-gain theorems for interconnected systems consisting of two or more subs ystems with time-delays present either in the interconnection channels or within the subsystems themselves. As an interesting application, we apply our results to an existing model for hematopoesis, a blood cell production process,and improve the previous results derived by linear methods. Another important stability notion in the framework of ISS is the integral ISS (iISS) property. This is a weaker property than ISS, so it supplies to a larger class of systems. As in the case of ISS, we provide Razumikhin criteria on iISS for systems with delays. An example is presented to illustrate that though very useful in practice, the Razumikhin approach only provides sufficient conditions, not equivalent conditions. Finally, we address stability of time-varying systems with delays in the framework of ISS., In particular, we consider Lyapunov-Razumikhin functions whose decay rates are affected by time-varying functions that can be zero or even negative on some sets of non-zero measure. Our motivation is that it is often less demanding to find or construct such a Lyapunov function than one with a uniform decay rate. We also extend our small-gain theorems to the time-varying case by treating the time-varying system as an auxiliary time-invariant system.
Show less - Date Issued
- 2012
- PURL
- http://purl.flvc.org/FAU/3352880
- Subject Headings
- Nonlinear systems, Simulation methods, Control theory, Stability, Mathematical models, Mathematical optimization
- Format
- Document (PDF)
- Title
- Random Harmonic Polynomials.
- Creator
- Thomack, Andrew, Lundberg, Erik, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
The study of random polynomials and in particular the number and behavior of zeros of random polynomials have been well studied, where the rst signi cant progress was made by Kac, nding an integral formula for the expected number of zeros of real zeros of polynomials with real coe cients. This formula as well as adaptations of the formula to complex polynomials and random elds show an interesting dependency of the number and distribution of zeros on the particular method of randomization....
Show moreThe study of random polynomials and in particular the number and behavior of zeros of random polynomials have been well studied, where the rst signi cant progress was made by Kac, nding an integral formula for the expected number of zeros of real zeros of polynomials with real coe cients. This formula as well as adaptations of the formula to complex polynomials and random elds show an interesting dependency of the number and distribution of zeros on the particular method of randomization. Three prevalent models of signi cant study are the Kostlan model, the Weyl model, and the naive model in which the coe cients of the polynomial are standard Gaussian random variables. A harmonic polynomial is a complex function of the form h(z) = p(z) + q(z) where p and q are complex analytic polynomials. Li and Wei adapted the Kac integral formula for the expected number of zeros to study random harmonic polynomials and take particular interest in their interpretation of the Kostlan model. In this thesis we nd asymptotic results for the number of zeros of random harmonic polynomials under both the Weyl model and the naive model as the degree of the harmonic polynomial increases. We compare the ndings to the Kostlan model as well as to the analytic analogs of each model. We end by establishing results which lead to open questions and conjectures about random harmonic polynomials. We ask and partially answer the question, \When does the number and behavior of the zeros of a random harmonic polynomial asymptotically emulate the same model of random complex analytic polynomial as the degree increases?" We also inspect the variance of the number of zeros of random harmonic polynomials, motivating the work by the question of whether the distribution of the number of zeros concentrates near its as the degree of the harmonic polynomial increases.
Show less - Date Issued
- 2017
- PURL
- http://purl.flvc.org/fau/fd/FA00004986
- Subject Headings
- Dissertations, Academic -- Florida Atlantic University, Random polynomials., Functions., Polynomials.
- 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.
Show less - 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
- 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.
Show less - 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
- Revisiting leisure activities and the risk of dementia in the elderly with special focus on dancing.
- Creator
- Stevens, Carrie., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
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Data was provided by researchers of the Einstein Aging Study (EAS) of the Albert Einstein College of Medicine, Yeshiva University whom statistically analyzed data from the Bronx Aging Study cohort, concluding that participation in cognitive leisure activities and one physical activity, dancing, were associated with a reduced risk of dementia [1]. We explore data from a second (the EAS) cohort, utilizing Cox Proportional-Hazards and extended Cox regression [13]. Cognitive leisure activities in...
Show moreData was provided by researchers of the Einstein Aging Study (EAS) of the Albert Einstein College of Medicine, Yeshiva University whom statistically analyzed data from the Bronx Aging Study cohort, concluding that participation in cognitive leisure activities and one physical activity, dancing, were associated with a reduced risk of dementia [1]. We explore data from a second (the EAS) cohort, utilizing Cox Proportional-Hazards and extended Cox regression [13]. Cognitive leisure activities in general, and particularly doing crossword puzzles, reading books, watching television, and emailing are associated with a reduced risk of dementia. Doing aerobics, learning computer programming, babysitting, dancing, jogging singing, and weight training are associated with an increased risk of dementia. Participation in cognitive leisure activities in general, and reading books in particular, remains highly significant even after adjustment for well-known risk factors [14] such as: age, cognitive status, depression, medical illnesses, gender, ethnicity, education and economic status.
Show less - Date Issued
- 2011
- PURL
- http://purl.flvc.org/FAU/3334097
- Subject Headings
- Aging, Psychological aspects, Older people, Health and hygiene, Forecasting, Older people, Mental health, Forecasting, Alzheimer's disease
- 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
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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.
Show less - 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
- The existence of minimal logarithmic signatures for classical groups.
- Creator
- Singhi, Nikhil., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
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A logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a...
Show moreA logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a group G of order Qk j=1 pj mj , the length of any LS for G satises `() Pk j=1mjpj . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS). The MLS conjecture states that every finite simple group has an MLS. If the conjecture is true then every finite group will have an MLS. The conjecture was shown to be true by a number of researchers for a few classes of finite simple groups. However, the problem is still wide open. This dissertation addresses the MLS conjecture for the classical simple groups. In particular, it is shown that MLS's exist for the symplectic groups Sp2n(q), the orthogonal groups O 2n(q0) and the corresponding simple groups PSp2n(q) and 2n(q0) for all n 2 N, prime power q and even prime power q0. The existence of an MLS is also shown for all unitary groups GUn(q) for all odd n and q = 2s under the assumption that an MLS exists for GUn 1(q). The methods used are very general and algorithmic in nature and may be useful for studying all nite simple groups of Lie type and possibly also the sporadic groups. The blocks of logarithmic signatures constructed in this dissertation have cyclic structure and provide a sort of cyclic decomposition for these classical groups.
Show less - Date Issued
- 2011
- PURL
- http://purl.flvc.org/FAU/3172943
- Subject Headings
- Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups
- Format
- Document (PDF)