Current Search: Department of Mathematical Sciences (x)
View All Items
Pages
 Title
 ALGORITHMS IN LATTICEBASED CRYPTANALYSIS.
 Creator
 Miller, Shaun, Bai, Shi, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

An adversary armed with a quantum computer has algorithms[66, 33, 34] at their disposal, which are capable of breaking our current methods of encryption. Even with the birth of postquantum cryptography[52, 62, 61], some of best cryptanalytic algorithms are still quantum [45, 8]. This thesis contains several experiments on the efficacy of lattice reduction algorithms, BKZ and LLL. In particular, the difficulty of solving Learning With Errors is assessed by reducing the problem to an instance...
Show moreAn adversary armed with a quantum computer has algorithms[66, 33, 34] at their disposal, which are capable of breaking our current methods of encryption. Even with the birth of postquantum cryptography[52, 62, 61], some of best cryptanalytic algorithms are still quantum [45, 8]. This thesis contains several experiments on the efficacy of lattice reduction algorithms, BKZ and LLL. In particular, the difficulty of solving Learning With Errors is assessed by reducing the problem to an instance of the Unique Shortest Vector Problem. The results are used to predict the behavior these algorithms may have on actual cryptographic schemes with security based on hard lattice problems. Lattice reduction algorithms require several floatingpoint operations including multiplication. In this thesis, I consider the resource requirements of a quantum circuit designed to simulate floatingpoint multiplication with high precision.
Show less  Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013543
 Subject Headings
 Cryptanalysis, Cryptography, Algorithms, Lattices, Quantum computing
 Format
 Document (PDF)
 Title
 ANNIHILATORS AND A + B RINGS.
 Creator
 Epstein, Alexandra Nicole, Klingler, Lee, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

A + B rings are constructed from a ring A and nonempty set of prime ideals of A. Initially, these rings were created to provide examples of reduced rings which satisfy certain annihilator conditions. We describe precisely when A + B rings have these properties, based on the ring A and set of prime ideals of A. We continue by giving necessary and su cient conditions for A + B rings to have various other properties. We also consider annihilators in the context of frames of ideals of reduced rings.
 Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013588
 Subject Headings
 Rings (Algebra)
 Format
 Document (PDF)
 Title
 Graph labeling and nonseparating trees.
 Creator
 Gottipati, Chenchu B., Locke, Stephen C., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

This dissertation studies two independent problems, one is about graph labeling and the other problem is related to connectivity condition in a simple graph. Graph labeling is a rapidly developing area of research in graph theory, having connections with a variety of applicationoriented areas such as VLSI optimization, data structures and data representation. Furthermore, the connectivity conditions in a simple graphs may help us to study the new aspects of ad hoc networks, social networks...
Show moreThis dissertation studies two independent problems, one is about graph labeling and the other problem is related to connectivity condition in a simple graph. Graph labeling is a rapidly developing area of research in graph theory, having connections with a variety of applicationoriented areas such as VLSI optimization, data structures and data representation. Furthermore, the connectivity conditions in a simple graphs may help us to study the new aspects of ad hoc networks, social networks and web graphs. In chapter 2, we study path systems, reduced path systems and how to construct a super edgegraceful tree with any number of edges using path systems. First, we give an algorithm to reduce a labeled path system to a smaller labeled path system of a different type. First, we investigate the cases (m, k) = (3; 5) and (m, k) = (4; 7), where m is the number of paths and 2k is the length of each path, and then we give a generalization for any k, m = 3 and m = 4. We also describe a procedure to construct a superedgegraceful tree with any number of edges.
Show less  Date Issued
 2014
 PURL
 http://purl.flvc.org/fau/fd/FA00004289, http://purl.flvc.org/fau/fd/FA00004289
 Subject Headings
 Computational complexity, Computer graphics, Graph theory, Integrated circuits  Very large scale integration, Mathematical optimization
 Format
 Document (PDF)
 Title
 Modeling and simulating interest rates via timedependent 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 RendlemanBartter model, the Vasicek model, and the CoxIngersollRoss (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 RendlemanBartter model, the Vasicek model, and the CoxIngersollRoss (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 HullWhite onefactor model, which does provide this fit. After calibrating, simulating, and comparing these four models, we find that the HullWhite model gives the best fit to our data sets.
Show less  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
 On the Study of the Aizawa System.
 Creator
 Fleurantin, Emmanuel, MirelesJames, Jason D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

In this report we study the Aizawa field by first computing a Taylor series expansion for the solution of an initial value problem. We then look for singularities (equilibrium points) of the field and plot the set of solutions which lie in the linear subspace spanned by the eigenvectors. Finally, we use the Parameterization Method to compute one and two dimensional stable and unstable manifolds of equilibria for the system.
 Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00005994
 Subject Headings
 Series, Mathematics, Eigenvectors, Aizawa field
 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 threedimensional vector field possessing an inclinationflip 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 threedimensional vector field possessing an inclinationflip 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.
Show less  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)
 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
 Various Approaches on Parameter Estimation in Mixture and NonMixture Cure Models.
 Creator
 Kutal, Durga Hari, Qian, Lianfen, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Analyzing lifetime data with longterm survivors is an important topic in medical application. Cure models are usually used to analyze survival data with the proportion of cure subjects or longterm survivors. In order to include the propor tion of cure subjects, mixture and nonmixture cure models are considered. In this dissertation, we utilize both maximum likelihood and Bayesian methods to estimate model parameters. Simulation studies are carried out to verify the nite sample per...
Show moreAnalyzing lifetime data with longterm survivors is an important topic in medical application. Cure models are usually used to analyze survival data with the proportion of cure subjects or longterm survivors. In order to include the propor tion of cure subjects, mixture and nonmixture cure models are considered. In this dissertation, we utilize both maximum likelihood and Bayesian methods to estimate model parameters. Simulation studies are carried out to verify the nite sample per formance of the estimation methods. Real data analyses are reported to illustrate the goodnessof t via Fr echet, Weibull and Exponentiated Exponential susceptible distributions. Among the three parametric susceptible distributions, Fr echet is the most promising. Next, we extend the nonmixture cure model to include a change point in a covariate for right censored data. The smoothed likelihood approach is used to address the problem of a loglikelihood function which is not di erentiable with respect to the change point. The simulation study is based on the nonmixture change point cure model with an exponential distribution for the susceptible subjects. The simulation results revealed a convincing performance of the proposed method of estimation.
Show less  Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013083
 Subject Headings
 Survival Analysis., Bayesian statistical decision theory., Parameter estimation., Weibull distribution.
 Format
 Document (PDF)
 Title
 The Circular Restricted Four Body Problem is NonIntegrable: A Computer Assisted Proof.
 Creator
 Kepley, Shane, Kalies, William D., MirelesJames, Jason D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Gravitational Nbody problems are central in classical mathematical physics. Studying their long time behavior raises subtle questions about the interplay between regular and irregular motions and the boundary between integrable and chaotic dynamics. Over the last hundred years, concepts from the qualitative theory of dynamical systems such as stable/unstable manifolds, homoclinic and heteroclinic tangles, KAM theory, and whiskered invariant tori, have come to play an increasingly important...
Show moreGravitational Nbody problems are central in classical mathematical physics. Studying their long time behavior raises subtle questions about the interplay between regular and irregular motions and the boundary between integrable and chaotic dynamics. Over the last hundred years, concepts from the qualitative theory of dynamical systems such as stable/unstable manifolds, homoclinic and heteroclinic tangles, KAM theory, and whiskered invariant tori, have come to play an increasingly important role in the discussion. In the last fty years the study of numerical methods for computing invariant objects has matured into a thriving subdiscipline. This growth is driven at least in part by the needs of the world's space programs. Recent work on validated numerical methods has begun to unify the computational and analytical perspectives, enriching both aspects of the subject. Many of these results use computer assisted proofs, a tool which has become increasingly popular in recent years. This thesis presents a proof that the circular restricted four body problem is nonintegrable. The proof of this result is obtained as an application of more general rigorous numerical methods in nonlinear analysis.
Show less  Date Issued
 2017
 PURL
 http://purl.flvc.org/fau/fd/FA00004997
 Subject Headings
 Dissertations, Academic  Florida Atlantic University, Mathematical physics., Invariants., Dynamical systems
 Format
 Document (PDF)
 Title
 The Covering Numbers of Some Finite Simple Groups.
 Creator
 Epstein, Michael, Magliveras, Spyros S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups in a minimal cover of G. Here we determine the covering numbers of the projective special unitary groups U3(q) for q ≤ 5, and give upper and lower bounds for the covering number of U3(q) when...
Show moreA finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups in a minimal cover of G. Here we determine the covering numbers of the projective special unitary groups U3(q) for q ≤ 5, and give upper and lower bounds for the covering number of U3(q) when q > 5. We also determine the covering number of the McLaughlin sporadic simple group, and verify previously known results on the covering numbers of the HigmanSims and Held groups.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013203
 Subject Headings
 Finite simple groups, Covering numbers
 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.
Show less  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
 Stability analysis for singularly perturbed systems with timedelays.
 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
 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
 QuantumResistant Key Agreement and Key Encapsulation.
 Creator
 Robinson, Angela, Steinwandt, Rainer, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

We explore quantumresistant key establishment and hybrid encryption. We nd that while the discrete logarithm problem is e ciently solved by a quantum computer using Shor's algorithm, some instances are insecure even using classical computers. The discrete logarithm problem based on a symmetric group Sn is e  ciently solved in polynomial time. We design a PUFbased 4round group key establishment protocol, adjusting the model to include a physical channel capable of PUF transmission, and...
Show moreWe explore quantumresistant key establishment and hybrid encryption. We nd that while the discrete logarithm problem is e ciently solved by a quantum computer using Shor's algorithm, some instances are insecure even using classical computers. The discrete logarithm problem based on a symmetric group Sn is e  ciently solved in polynomial time. We design a PUFbased 4round group key establishment protocol, adjusting the model to include a physical channel capable of PUF transmission, and modify adversarial capabilities with respect to the PUFs. The result is a novel group key establishment protocol which avoids computational hardness assumptions and achieves key secrecy. We contribute a hybrid encryption scheme by combining a key encapsulation mechanism (KEM) with a symmetric key encryption scheme by using two hash functions. We require only oneway security in the quantum random oracle model (QROM) of the KEM and onetime security of the symmetric encryption scheme in the QROM. We show that this hybrid scheme is INDCCA secure in the QROM. We rely on a powerful theorem by Unruh that provides an upper bound on indistinguishability between the output of a random oracle and a random string, when the oracle can be accessed in quantum superposition. Our result contributes to the available INDCCA secure encryption schemes in a setting where quantum computers are under adversarial control. Finally, we develop a framework and describe biometric visual cryptographic schemes generically under our framework. We formalize several security notions and de nitions including sheet indistinguishability, perfect indistinguishability, index recovery, perfect index privacy, and perfect resistance against false authentication. We also propose new and generic strategies for attacking eBVC schemes such as new distinguishing attack, new index recovery, and new authentication attack. Our quantitative analysis veri es the practical impact of our framework and o ers concrete upper bounds on the security of eBVC.
Show less  Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013023
 Subject Headings
 Quantum computing, Data encryption (Computer science), Cryptography
 Format
 Document (PDF)
 Title
 Quantum Circuits for Cryptanalysis.
 Creator
 Amento, Brittanney Jaclyn, Steinwandt, Rainer, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Finite elds of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these elds can have a signi cant impact on the resource requirements for quantum arithmetic. In particular, we show how the Gaussian normal basis representations and \ghostbit basis" representations can be used to implement inverters with a quantum circuit of depth O(mlog(m)). To the best of our knowledge, this is the rst construction with...
Show moreFinite elds of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these elds can have a signi cant impact on the resource requirements for quantum arithmetic. In particular, we show how the Gaussian normal basis representations and \ghostbit basis" representations can be used to implement inverters with a quantum circuit of depth O(mlog(m)). To the best of our knowledge, this is the rst construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the ItohTsujii algorithm which exploits the property that, in a normal basis representation, squaring corresponds to a permutation of the coe cients. We give resource estimates for the resulting quantum circuit for inversion over binary elds F2m based on an elementary gate set that is useful for faulttolerant implementation. Elliptic curves over nite elds F2m play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with a ne or projective coordinates. In this thesis we show that changing the curve representation allows a substantial reduction in the number of Tgates needed to implement the curve arithmetic. As a tool, we present a quantum circuit for computing multiplicative inverses in F2m in depth O(mlogm) using a polynomial basis representation, which may be of independent interest. Finally, we change our focus from the design of circuits which aim at attacking computational assumptions on asymmetric cryptographic algorithms to the design of a circuit attacking a symmetric cryptographic algorithm. We consider a block cipher, SERPENT, and our design of a quantum circuit implementing this cipher to be used for a key attack using Grover's algorithm as in [18]. This quantum circuit is essential for understanding the complexity of Grover's algorithm.
Show less  Date Issued
 2016
 PURL
 http://purl.flvc.org/fau/fd/FA00004662, http://purl.flvc.org/fau/fd/FA00004662
 Subject Headings
 Artificial intelligence, Computer networks, Cryptography, Data encryption (Computer science), Finite fields (Algebra), Quantum theory
 Format
 Document (PDF)
 Title
 Quantum Circuits for Symmetric Cryptanalysis.
 Creator
 Langenberg, Brandon Wade, Steinwandt, Rainer, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Quantum computers and quantum computing is a reality of the near feature. Companies such as Google and IBM have already declared they have built a quantum computer and tend to increase their size and capacity moving forward. Quantum computers have the ability to be exponentially more powerful than classical computers today. With this power modeling behavior of atoms or chemical reactions in unusual conditions, improving weather forecasts and traffic conditions become possible. Also, their...
Show moreQuantum computers and quantum computing is a reality of the near feature. Companies such as Google and IBM have already declared they have built a quantum computer and tend to increase their size and capacity moving forward. Quantum computers have the ability to be exponentially more powerful than classical computers today. With this power modeling behavior of atoms or chemical reactions in unusual conditions, improving weather forecasts and traffic conditions become possible. Also, their ability to exponentially speed up some computations makes the security of todays data and items a major concern and interest. In the area of cryptography, some encryption schemes (such as RSA) are already deemed broken by the onset of quantum computing. Some encryption algorithms have already been created to be quantum secure and still more are being created each day. While these algorithms in use today are considered quantumsafe not much is known of what a quantum attack would look like on these algorithms. Specifically, this paper discusses how many quantum bits, quantum gates and even the depth of these gates that would be needed for such an attack. The research below was completed to shed light on these areas and offer some concrete numbers of such an attack.
Show less  Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013010
 Subject Headings
 Quantum computing, Cryptography, Cryptanalysis, Data encryption (Computer science), Computer algorithms
 Format
 Document (PDF)
 Title
 Output Stability Analysis for Nonlinear Systems with Time Delays.
 Creator
 Gallolu Kankanamalage, Hasala Senpathy, Wang, Yuan, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Systems with time delays have a broad range of applications not only in control systems but also in many other disciplines such as mathematical biology, financial economics, etc. The time delays cause more complex behaviours of the systems. It requires more sophisticated analysis due to the infinite dimensional structure of the space spaces. In this thesis we investigate stability properties associated with output functions of delay systems. Our primary target is the equivalent Lyapunov...
Show moreSystems with time delays have a broad range of applications not only in control systems but also in many other disciplines such as mathematical biology, financial economics, etc. The time delays cause more complex behaviours of the systems. It requires more sophisticated analysis due to the infinite dimensional structure of the space spaces. In this thesis we investigate stability properties associated with output functions of delay systems. Our primary target is the equivalent Lyapunov characterization of inputtooutput stability (ios). A main approach used in this work is the Lyapuno Krasovskii functional method. The Lyapunov characterization of the so called outputLagrange stability is technically the backbone of this work, as it induces a Lyapunov description for all the other output stability properties, in particular for ios. In the study, we consider two types of output functions. The first type is defined in between Banach spaces, whereas the second type is defined between Euclidean spaces. The Lyapunov characterization for the first type of output maps provides equivalence between the stability properties and the existence of the LyapunovKrasovskii functionals. On the other hand, as a special case of the first type, the second type output renders flexible Lyapunov descriptions that are more efficient in applications. In the special case when the output variables represent the complete collection of the state variables, our Lyapunov work lead to Lyapunov characterizations of iss, complementing the current iss theory with some novel results. We also aim at understanding how output stability are affected by the initial data and the external signals. Since the output variables are in general not a full collection of the state variables, the overshoots and decay properties may be affected in different ways by the initial data of either the state variables or just only the output variables. Accordingly, there are different ways of defining notions on output stability, making them mathematically precisely. After presenting the definitions, we explore the connections of these notions. Understanding the relation among the notions is not only mathematically necessary, it also provides guidelines in system control and design.
Show less  Date Issued
 2017
 PURL
 http://purl.flvc.org/fau/fd/FA00004935, http://purl.flvc.org/fau/fd/FA00004935
 Subject Headings
 Nonlinear systems., Time delay systems., Multiagent systems., Adaptive control systems., Chaotic behavior in systems.
 Format
 Document (PDF)
 Title
 PARAMETER ESTIMATION FOR GEOMETRIC L EVY PROCESSES WITH STOCHASTIC VOLATILITY.
 Creator
 Chhetri, Sher B., Long, Hongwei, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

In finance, various stochastic models have been used to describe the price movements of financial instruments. After Merton's [38] seminal work, several jump diffusion models for option pricing and risk management have been proposed. In this dissertation, we add alphastable Levy motion to the process related to dynamics of logreturns in the BlackScholes model where the volatility is assumed to be constant. We use the sample characteristic function approach in order to study parameter...
Show moreIn finance, various stochastic models have been used to describe the price movements of financial instruments. After Merton's [38] seminal work, several jump diffusion models for option pricing and risk management have been proposed. In this dissertation, we add alphastable Levy motion to the process related to dynamics of logreturns in the BlackScholes model where the volatility is assumed to be constant. We use the sample characteristic function approach in order to study parameter estimation for discretely observed stochastic differential equations driven by Levy noises. We also discuss the consistency and asymptotic properties of the proposed estimators. Simulation results of the model are also presented to show the validity of the estimators. We then propose a new model where the volatility is not a constant. We consider generalized alphastable geometric Levy processes where the stochastic volatility follows the CoxIngersollRoss (CIR) model in Cox et al. [9]. A number of methods have been proposed for estimating parameters for stable laws. However, a complication arises in estimation of the parameters in our model because of the presence of the unobservable stochastic volatility. To combat this complication we use the sample characteristic function method proposed by Press [48] and the conditional least squares method as mentioned in Overbeck and Ryden [47] to estimate all the parameters. We then discuss the consistency and asymptotic properties of the proposed estimators and establish a Central Limit Theorem. We perform simulations to assess the validity of the estimators. We also present several tables to show the comparison of estimators using different choices of arguments ui's. We conclude that all the estimators converge as expected regardless of the choice of ui's.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013294
 Subject Headings
 Stochastic models, Lévy processes, Parameter estimation, Finance, Simulations
 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 currentintensity 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, CyclonesTropicsForecasting, Geosynchronous satellites, Neural networks (Computer science)
 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 rank3 and rank4 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 rank3 and rank4 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 rank3 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 HoffmanSingleton graph as a rank3 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)