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 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
 Bayesian approach to an exponential hazard regression model with a change point.
 Creator
 Abraha, Yonas Kidane, Qian, Lianfen, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

This thesis contains two parts. The first part derives the Bayesian estimator of the parameters in a piecewise exponential Cox proportional hazard regression model, with one unknown change point for a right censored survival data. The second part surveys the applications of change point problems to various types of data, such as longterm survival data, longitudinal data and time series data. Furthermore, the proposed method is then used to analyse a real survival data.
 Date Issued
 2014
 PURL
 http://purl.flvc.org/fau/fd/FA00004013
 Subject Headings
 Bayesian statistical decision theory, Mathematical statistics, Multivariate analysis  Data processing
 Format
 Document (PDF)
 Title
 An Algorithmic Approach to Tran Van Trung's Basic Recursive Construction of tDesigns.
 Creator
 Lopez, Oscar A., Magliveras, Spyros S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

It was not until the 20th century that combinatorial design theory was studied as a formal subject. This field has many applications, for example in statistical experimental design, coding theory, authentication codes, and cryptography. Major approaches to the problem of discovering new tdesigns rely on (i) the construction of large sets of t designs, (ii) using prescribed automorphism groups, (iii) recursive construction methods. In 2017 and 2018, Tran Van Trung introduced new recursive...
Show moreIt was not until the 20th century that combinatorial design theory was studied as a formal subject. This field has many applications, for example in statistical experimental design, coding theory, authentication codes, and cryptography. Major approaches to the problem of discovering new tdesigns rely on (i) the construction of large sets of t designs, (ii) using prescribed automorphism groups, (iii) recursive construction methods. In 2017 and 2018, Tran Van Trung introduced new recursive techniques to construct t – (v, k, λ) designs. These methods are of purely combinatorial nature and require using "ingredient" tdesigns or resolutions whose parameters satisfy a system of nonlinear equations. Even after restricting the range of parameters in this new method, the task is computationally intractable. In this work, we enhance Tran Van Trung's "Basic Construction" by a robust and efficient hybrid computational apparatus which enables us to construct hundreds of thousands of new t – (v, k, Λ) designs from previously known ingredient designs. Towards the end of the dissertation we also create a new family of 2resolutions, which will be infinite if there are infinitely many Sophie Germain primes.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013233
 Subject Headings
 Combinatorial designs and configurations, Algorithms, tdesigns
 Format
 Document (PDF)
 Title
 Algorithms in Elliptic Curve Cryptography.
 Creator
 Hutchinson, Aaron, Karabina, Koray, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Elliptic curves have played a large role in modern cryptography. Most notably, the Elliptic Curve Digital Signature Algorithm (ECDSA) and the Elliptic Curve Di eHellman (ECDH) key exchange algorithm are widely used in practice today for their e ciency and small key sizes. More recently, the Supersingular Isogenybased Di eHellman (SIDH) algorithm provides a method of exchanging keys which is conjectured to be secure in the postquantum setting. For ECDSA and ECDH, e cient and secure...
Show moreElliptic curves have played a large role in modern cryptography. Most notably, the Elliptic Curve Digital Signature Algorithm (ECDSA) and the Elliptic Curve Di eHellman (ECDH) key exchange algorithm are widely used in practice today for their e ciency and small key sizes. More recently, the Supersingular Isogenybased Di eHellman (SIDH) algorithm provides a method of exchanging keys which is conjectured to be secure in the postquantum setting. For ECDSA and ECDH, e cient and secure algorithms for scalar multiplication of points are necessary for modern use of these protocols. Likewise, in SIDH it is necessary to be able to compute an isogeny from a given nite subgroup of an elliptic curve in a fast and secure fashion. We therefore nd strong motivation to study and improve the algorithms used in elliptic curve cryptography, and to develop new algorithms to be deployed within these protocols. In this thesis we design and develop dMUL, a multidimensional scalar multiplication algorithm which is uniform in its operations and generalizes the well known 1dimensional Montgomery ladder addition chain and the 2dimensional addition chain due to Dan J. Bernstein. We analyze the construction and derive many optimizations, implement the algorithm in software, and prove many theoretical and practical results. In the nal chapter of the thesis we analyze the operations carried out in the construction of an isogeny from a given subgroup, as performed in SIDH. We detail how to e ciently make use of parallel processing when constructing this isogeny.
Show less  Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013113
 Subject Headings
 Curves, Elliptic, Cryptography, Algorithms
 Format
 Document (PDF)
 Title
 An Algorithmic Approach to The Lattice Structures of Attractors and Lyapunov functions.
 Creator
 Kasti, Dinesh, Kalies, William D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Ban and Kalies [3] proposed an algorithmic approach to compute attractor repeller pairs and weak Lyapunov functions based on a combinatorial multivalued mapping derived from an underlying dynamical system generated by a continuous map. We propose a more e cient way of computing a Lyapunov function for a Morse decomposition. This combined work with other authors, including Shaun Harker, Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes the process of nding a...
Show moreBan and Kalies [3] proposed an algorithmic approach to compute attractor repeller pairs and weak Lyapunov functions based on a combinatorial multivalued mapping derived from an underlying dynamical system generated by a continuous map. We propose a more e cient way of computing a Lyapunov function for a Morse decomposition. This combined work with other authors, including Shaun Harker, Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes the process of nding a global Lyapunov function for Morse decomposition very e  cient. One of the them is to utilize highly memorye cient data structures: succinct grid data structure and pointer grid data structures. Another technique is to utilize Dijkstra algorithm and Manhattan distance to calculate a distance potential, which is an essential step to compute a Lyapunov function. Finally, another major technique in achieving a signi cant improvement in e ciency is the utilization of the lattice structures of the attractors and attracting neighborhoods, as explained in [32]. The lattice structures have made it possible to let us incorporate only the joinirreducible attractorrepeller pairs in computing a Lyapunov function, rather than having to use all possible attractorrepeller pairs as was originally done in [3]. The distributive lattice structures of attractors and repellers in a dynamical system allow for general algebraic treatment of global gradientlike dynamics. The separation of these algebraic structures from underlying topological structure is the basis for the development of algorithms to manipulate those structures, [32, 31]. There has been much recent work on developing and implementing general compu tational algorithms for global dynamics which are capable of computing attracting neighborhoods e ciently. We describe the lifting of sublattices of attractors, which are computationally less accessible, to lattices of forward invariant sets and attract ing neighborhoods, which are computationally accessible. We provide necessary and su cient conditions for such a lift to exist, in a general setting. We also provide the algorithms to check whether such conditions are met or not and to construct the lift when they met. We illustrate the algorithms with some examples. For this, we have checked and veri ed these algorithms by implementing on some noninvertible dynamical systems including a nonlinear Leslie model.
Show less  Date Issued
 2016
 PURL
 http://purl.flvc.org/fau/fd/FA00004668
 Subject Headings
 Differential equations  Numerical solutions., Differentiable dynamical systems., Algorithms.
 Format
 Document (PDF)
 Title
 A Study on Partially Homomorphic Encryption Schemes.
 Creator
 Mithila, Shifat P., Karabina, Koray, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

High processing time and implementation complexity of the fully homomorphic encryption schemes intrigued cryptographers to extend partially homomorphic encryption schemes to allow homomorphic computation for larger classes of polynomials. In this thesis, we study several public key and partially homomorphic schemes and discuss a recent technique for boosting linearly homomorphic encryption schemes. Further, we implement this boosting technique on CGS linearly homomorphic encryption scheme to...
Show moreHigh processing time and implementation complexity of the fully homomorphic encryption schemes intrigued cryptographers to extend partially homomorphic encryption schemes to allow homomorphic computation for larger classes of polynomials. In this thesis, we study several public key and partially homomorphic schemes and discuss a recent technique for boosting linearly homomorphic encryption schemes. Further, we implement this boosting technique on CGS linearly homomorphic encryption scheme to allow one single multiplication as well as arbitrary number of additions on encrypted plaintexts. We provide MAGMA source codes for the implementation of the CGS scheme along with the boosted CGS scheme.
Show less  Date Issued
 2017
 PURL
 http://purl.flvc.org/fau/fd/FA00004840, http://purl.flvc.org/fau/fd/FA00004840
 Subject Headings
 Computer networksSecurity measures., Computer security., ComputersAccess controlCode words., Cyberinfrastructure., Computer network architectures., Cryptography., Number theoryData processing.
 Format
 Document (PDF)
 Title
 A Constructive Theory of Ordered Sets and their Completions.
 Creator
 Joseph, Jean S., Richman, Fred, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

The context for the development of this work is constructive mathematics without the axiom of countable choice. By constructive mathematics, we mean mathematics done without the law of excluded middle. Our original goal was to give a list of axioms for the real numbers R by only considering the order on R. We instead develop a theory of ordered sets and their completions and a theory of ordered abelian groups.
 Date Issued
 2018
 PURL
 http://purl.flvc.org/fau/fd/FA00013007
 Subject Headings
 Constructive mathematics, Ordered sets, Abelian groups
 Format
 Document (PDF)
 Title
 CONTRIBUTIONS TO QUANTUMSAFE CRYPTOGRAPHY: HYBRID ENCRYPTION AND REDUCING THE T GATE COST OF AES.
 Creator
 Pham, Hai, Steinwandt, Rainer, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Quantum cryptography offers a wonderful source for current and future research. The idea started in the early 1970s, and it continues to inspire work and development toward a popular goal, largescale communication networks with strong security guarantees, based on quantummechanical properties. Quantum cryptography builds on the idea of exploiting physical properties to establish secure cryptographic operations. A particular quantumbased protocol has gathered interest in recent years for...
Show moreQuantum cryptography offers a wonderful source for current and future research. The idea started in the early 1970s, and it continues to inspire work and development toward a popular goal, largescale communication networks with strong security guarantees, based on quantummechanical properties. Quantum cryptography builds on the idea of exploiting physical properties to establish secure cryptographic operations. A particular quantumbased protocol has gathered interest in recent years for its use of mesoscopic coherent states. The AlphaEta protocol has been designed to exploit properties of coherent states of light to transmit data securely over an optical channel. AlphaEta aims to draw security from the uncertainty of any measurement of the transmitted coherent states due to intrinsic quantum noise. We propose a framework to combine this protocol with classical preprocessing, taking into account errorcorrection for the optical channel and establishing a strong provable security guarantee. Integrating a stateoftheart solution for fast authenticated encryption is straightforward, but in this case the security analysis requires heuristic reasoning.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013339
 Subject Headings
 Cryptography, Quantum computing, Algorithms, Mesoscopic coherent states
 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 socalled "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 socalled "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 KacRice 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 orientationreversing 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 microcontrollers 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
 CHARACTERIZATIONS OF LINEAR ISOMETRIES ON COMPLEX SEQUENCE SPACES.
 Creator
 Babun Codorniu, Omar, Zhang, XiaoDong, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

An operator acting on a Banach space is called an isometry if it preserves the norm of the space. An interesting problem is to determine the form or structure of linear isometries on Banach spaces. This can be done in some instances. This dissertation presents several theorems that provide necessary and sufficient conditions for some linear operators acting on finite and infinite dimensional sequence spaces of complex numbers to be isometries. In all cases, the linear isometries have the form...
Show moreAn operator acting on a Banach space is called an isometry if it preserves the norm of the space. An interesting problem is to determine the form or structure of linear isometries on Banach spaces. This can be done in some instances. This dissertation presents several theorems that provide necessary and sufficient conditions for some linear operators acting on finite and infinite dimensional sequence spaces of complex numbers to be isometries. In all cases, the linear isometries have the form of a permutation of the elements of the sequences in the given space, with each component of each sequence multiplied by a complex number of absolute value 1.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013354
 Subject Headings
 Banach spaces, Isometrics (Mathematics), Matrices, Linear operators, Normed linear spaces
 Format
 Document (PDF)
 Title
 Elliptic curves: identitybased signing and quantum arithmetic.
 Creator
 Budhathoki, Parshuram, Steinwandt, Rainer, Eisenbarth, Thomas, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Pairingfriendly curves and elliptic curves with a trapdoor for the discrete logarithm problem are versatile tools in the design of cryptographic protocols. We show that curves having both properties enable a deterministic identitybased signing with “short” signatures in the random oracle model. At PKC 2003, Choon and Cheon proposed an identitybased signature scheme along with a provable security reduction. We propose a modification of their scheme with several performance benefits. In...
Show morePairingfriendly curves and elliptic curves with a trapdoor for the discrete logarithm problem are versatile tools in the design of cryptographic protocols. We show that curves having both properties enable a deterministic identitybased signing with “short” signatures in the random oracle model. At PKC 2003, Choon and Cheon proposed an identitybased signature scheme along with a provable security reduction. We propose a modification of their scheme with several performance benefits. In addition to faster signing, for batch signing the signature size can be reduced, and if multiple signatures for the same identity need to be verified, the verification can be accelerated. Neither the signing nor the verification algorithm rely on the availability of a (pseudo)random generator, and we give a provable security reduction in the random oracle model to the (`)Strong DiffieHellman problem. Implementing the group arithmetic is a costcritical task when designing quantum circuits for Shor’s algorithm to solve the discrete logarithm problem. We introduce a tool for the automatic generation of addition circuits for ordinary binary elliptic curves, a prominent platform group for digital signatures. Our Python software generates circuit descriptions that, without increasing the number of qubits or Tdepth, involve less than 39% of the number of Tgates in the best previous construction. The software also optimizes the (CNOT) depth for F2linear operations by means of suitable graph colorings.
Show less  Date Issued
 2014
 PURL
 http://purl.flvc.org/fau/fd/FA00004182, http://purl.flvc.org/fau/fd/FA00004182
 Subject Headings
 Coding theory, Computer network protocols, Computer networks  Security measures, Data encryption (Computer science), Mathematical physics, Number theory  Data processing
 Format
 Document (PDF)
 Title
 Curve shortening in secondorder lagrangian.
 Creator
 Adams, Ronald Edward, Kalies, William D., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

A secondorder Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lowerorder derivatives play a key role in forcing certain types of dynamics. However, the application of...
Show moreA secondorder Lagrangian system is a generalization of a classical mechanical system for which the Lagrangian action depends on the second derivative of the state variable. Recent work has shown that the dynamics of such systems c:an be substantially richer than for classical Lagrangian systems. In particular, topological properties of the planar curves obtained by projection onto the lowerorder derivatives play a key role in forcing certain types of dynamics. However, the application of these techniques requires an analytic restriction on the Lagrangian that it satisfy a twist property. In this dissertation we approach this problem from the point of view of curve shortening in an effort to remove the twist condition. In classical curve shortening a family of curves evolves with a velocity which is normal to the curve and proportional to its curvature. The evolution of curves with decreasing action is more general, and in the first part of this dissertation we develop some results for curve shortening flows which shorten lengths with respect to a Finsler metric rather than a Riemannian metric. The second part of this dissertation focuses on analytic methods to accommodate the fact that the Finsler metric for secondorder Lagrangian system has singularities. We prove the existence of simple periodic solutions for a general class of systems without requiring the twist condition. Further; our results provide a frame work in which to try to further extend the topological forcing theorems to systems without the twist condition.
Show less  Date Issued
 2014
 PURL
 http://purl.flvc.org/fau/fd/FA00004175, http://purl.flvc.org/fau/fd/FA00004175
 Subject Headings
 Critical point theory (Mathematical analysis), Differentiable dynamical systems, Geometry,Differential, Lagrange equations, Lagrangian functions, Mathematical optimization, Surfaces of constant curvature
 Format
 Document (PDF)
 Title
 DETERMINANTS OF WOMEN'S ATTITUDE TOWARDS INTIMATE PARTNER VIOLENCE: EVIDENCE FROM BANGLADESH.
 Creator
 Khan, Md Tareq Ferdous, Qian, Lianfen, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

This thesis uses Bangladesh Demographic and Health Survey 2014 data to identify the important determinants due to which women justification towards intimate partner violence (IPV) varies. Statistical analyses reveal that among the individuallevel independent variables age at first marriage, respondent's education, decision score, religion, NGO membership, access to information, husband's education, normalized wealth score, and division indicator have significant effects on the women's...
Show moreThis thesis uses Bangladesh Demographic and Health Survey 2014 data to identify the important determinants due to which women justification towards intimate partner violence (IPV) varies. Statistical analyses reveal that among the individuallevel independent variables age at first marriage, respondent's education, decision score, religion, NGO membership, access to information, husband's education, normalized wealth score, and division indicator have significant effects on the women's attitude towards IPV. It shows that other than religion, NGO membership, and division indicator, the higher the value of the variable, the lower the likelihood of justifying IPV. However, being a Muslim, NGO member, and resident of other divisions, women are found more tolerant of IPV from their respective counterparts. Among the three communitylevel variables, only the mean decision score is found significant in lowering the likelihood. The thesis concludes with some policy recommendations and a proposal for future research.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013325
 Subject Headings
 Intimate partner violence, Bangladesh, Women
 Format
 Document (PDF)
 Title
 DEVELOPING A DEEP LEARNING PIPELINE TO AUTOMATICALLY ANNOTATE GOLD PARTICLES IN IMMUNOELECTRON MICROSCOPY IMAGES.
 Creator
 Jerez, Diego Alejandro, Hahn, William, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Machine learning has been utilized in bioimaging in recent years, however as it is relatively new and evolving, some researchers who wish to utilize machine learning tools have limited access because of a lack of programming knowledge. In electron microscopy (EM), immunogold labeling is commonly used to identify the target proteins, however the manual annotation of the gold particles in the images is a timeconsuming and laborious process. Conventional image processing tools could provide...
Show moreMachine learning has been utilized in bioimaging in recent years, however as it is relatively new and evolving, some researchers who wish to utilize machine learning tools have limited access because of a lack of programming knowledge. In electron microscopy (EM), immunogold labeling is commonly used to identify the target proteins, however the manual annotation of the gold particles in the images is a timeconsuming and laborious process. Conventional image processing tools could provide semiautomated annotation, but those require that users make manual adjustments for every step of the analysis. To create a new highthroughput image analysis tool for immunoEM, I developed a deep learning pipeline that was designed to deliver a completely automated annotation of immunogold particles in EM images. The program was made accessible for users without prior programming experience and was also expanded to be used on different types of immunoEM images.
Show less  Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013628
 Subject Headings
 Electron microscopy, Immunogold labeling, Image analysis, Deep learning
 Format
 Document (PDF)
 Title
 Distinguishability of Public Keys and Experimental Validation: The McEliece PublicKeyed Cryptosystem.
 Creator
 Pham, Hai, Steinwandt, Rainer, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

As quantum computers continue to develop, they pose a threat to cryptography since many popular cryptosystems will be rendered vulnerable. This is because the security of most currently used asymmetric systems requires the computational hardness of the integer factorization problem, the discrete logarithm or the elliptic curve discrete logarithm problem. However, there are still some cryptosystems that resist quantum computing. We will look at codebased cryptography in general and the...
Show moreAs quantum computers continue to develop, they pose a threat to cryptography since many popular cryptosystems will be rendered vulnerable. This is because the security of most currently used asymmetric systems requires the computational hardness of the integer factorization problem, the discrete logarithm or the elliptic curve discrete logarithm problem. However, there are still some cryptosystems that resist quantum computing. We will look at codebased cryptography in general and the McEliece cryptosystem specifically. Our goal is to understand the structure behind the McEliece scheme, including the encryption and decryption processes, and what some advantages and disadvantages are that the system has to offer. In addition, using the results from Courtois, Finiasz, and Sendrier's paper in 2001, we will discuss a digital signature scheme based on the McEliece cryptosystem. We analyze one classical algebraic attack against the security analysis of the system based on the distinguishing problem whether the public key of the McEliece scheme is generated from a generating matrix of a binary Goppa code or a random binary matrix. The idea of the attack involves solving an algebraic system of equations and we examine the dimension of the solution space of the linearized system of equations. With the assistance from a paper in 2010 by Faugere, GauthierUmana, Otmani, Perret, Tillich, we will see the parameters needed for the intractability of the distinguishing problem.
Show less  Date Issued
 2015
 PURL
 http://purl.flvc.org/fau/fd/FA00004535, http://purl.flvc.org/fau/fd/FA00004535
 Subject Headings
 Coding theory, Combinatorial analysis, Data encryption (Computer science), Data transmission systems  Security measures, Information theory, McEliece, Robert J.  Influence, Public key cryptography
 Format
 Document (PDF)
 Title
 Detection of multiple changepoints in hazard models.
 Creator
 Zhang, Wei, Qian, Lianfen, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Changepoint detection in hazard rate function is an important research topic in survival analysis. In this dissertation, we firstly review existing methods for single changepoint detection in piecewise exponential hazard model. Then we consider the problem of estimating the change point in the presence of right censoring and longterm survivors while using KaplanMeier estimator for the susceptible proportion. The maximum likelihood estimators are shown to be consistent. Taking one step...
Show moreChangepoint detection in hazard rate function is an important research topic in survival analysis. In this dissertation, we firstly review existing methods for single changepoint detection in piecewise exponential hazard model. Then we consider the problem of estimating the change point in the presence of right censoring and longterm survivors while using KaplanMeier 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 changepoint detection algorithm. For single changepoint case, consistency results are obtained. We then consider the detection of multiple changepoints in the presence of longterm 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 changepoints with longterm survivors and covariates. For multiple changepoints 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 solvingData processing., Process controlStatistical methods., Point processes., Mathematical statistics., Failure time data analysisData processing., Survival analysis (Biometry)Data processing.
 Format
 Document (PDF)
 Title
 General monotonicity, interpolation of operators, and applications.
 Creator
 Grigoriev, Stepan M., Sagher, Yoram, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Assume that {φn} is an orthonormal uniformly bounded (ONB) sequence of complexvalued functions de ned on a measure space (Ω,Σ,µ), and f ∈ L1(Ω,Σ,µ). Let be the Fourier coefficients of f with respect to {φn} . R.E.A.C. Paley proved a theorem connecting the Lpnorm of f with a related norm of the sequence {cn}. Hardy and Littlewood subsequently proved that Paley’s result is best possible within its context. Their results were generalized by Dikarev, Macaev, Askey, Wainger, Sagher, and later by...
Show moreAssume that {φn} is an orthonormal uniformly bounded (ONB) sequence of complexvalued functions de ned on a measure space (Ω,Σ,µ), and f ∈ L1(Ω,Σ,µ). Let be the Fourier coefficients of f with respect to {φn} . R.E.A.C. Paley proved a theorem connecting the Lpnorm of f with a related norm of the sequence {cn}. Hardy and Littlewood subsequently proved that Paley’s result is best possible within its context. Their results were generalized by Dikarev, Macaev, Askey, Wainger, Sagher, and later by Tikhonov, Li yand, Booton and others.The present work continues the generalization of these results.
Show less  Date Issued
 2014
 PURL
 http://purl.flvc.org/fau/fd/FA00004290, http://purl.flvc.org/fau/fd/FA00004290
 Subject Headings
 Combinatorial optimization, Differential dynamical systems, Functions of complex variables, Inequalities (Mathematics), Nonsmooth optimization
 Format
 Document (PDF)
 Title
 HOMOCLINIC DYNAMICS IN A SPATIAL RESTRICTED FOUR BODY PROBLEM.
 Creator
 Murray, Maxime, James, Jason Mireles, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

The set of transverse homoclinic intersections for a saddlefocus equilibrium in the planar equilateral restricted four body problem admits certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection, or homoclinic web. In this thesis, the planar restricted four body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view....
Show moreThe set of transverse homoclinic intersections for a saddlefocus equilibrium in the planar equilateral restricted four body problem admits certain simple homoclinic orbits which form the skeleton of the complete homoclinic intersection, or homoclinic web. In this thesis, the planar restricted four body problem is viewed as an invariant subsystem of the spatial problem, and the influence of this planar homoclinic skeleton on the spatial dynamics is studied from a numerical point of view. Starting from the vertical Lyapunov families emanating from saddle focus equilibria, we compute the stable/unstable manifolds of these spatial periodic orbits and look for intersections between these manifolds near the fundamental planar homoclinics. In this way, we are able to continue all of the basic planar homoclinic motions into the spatial problem as homoclinics for appropriate vertical Lyapunov orbits which, by the Smale Tangle theorem, suggest the existence of chaotic motions in the spatial problem. While the saddlefocus equilibrium solutions in the planar problems occur only at a discrete set of energy levels, the cycletocycle homoclinics in the spatial problem are robust with respect to small changes in energy. The method uses high order FourierTaylor and Chebyshev series approximations in conjunction with the parameterization method, a general functional analytic framework for invariant manifolds. Tools that admit a natural notion of aposteriori error analysis. Finally, we develop and implement a validation algorithm which we later use to obtain Theorems confirming the existence of homoclinic dynamics. This approach, known as the Radii polynomial, is a contraction mapping argument which can be applied to both the parameterized manifold and the Chebyshev arcs. When the Theorem applies, it guarantees the existence of a true solution near the approximation and it provides an upper bound on the C0 norm of the truncation error.
Show less  Date Issued
 2021
 PURL
 http://purl.flvc.org/fau/fd/FA00013758
 Subject Headings
 Boundary value problems, Invariant manifolds, Applied mathematics
 Format
 Document (PDF)
 Title
 HLOCAL RINGS.
 Creator
 Omairi, Akeel, Klingler, Lee, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any Rmodule which decomposes into a _nite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains and in a 2011 paper Ay and Klingler obtain similar results for Noetherian reduced rings. We characterize the UDI property for Noetherian...
Show moreWe say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any Rmodule which decomposes into a _nite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains and in a 2011 paper Ay and Klingler obtain similar results for Noetherian reduced rings. We characterize the UDI property for Noetherian rings in general.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013336
 Subject Headings
 Noetherian rings, Prüfer rings, Local rings
 Format
 Document (PDF)
 Title
 ACCURATE HIGH ORDER COMPUTATION OF INVARIANT MANIFOLDS FOR LONG PERIODIC ORBITS OF MAPS AND EQUILIBRIUM STATES OF PDE.
 Creator
 Gonzalez, Jorge L., MirelesJames, Jason, Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

The study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future...
Show moreThe study of the long time behavior of nonlinear systems is not effortless, but it is very rewarding. The computation of invariant objects, in particular manifolds provide the scientist with the ability to make predictions at the frontiers of science. However, due to the presence of strong nonlinearities in many important applications, understanding the propagation of errors becomes necessary in order to quantify the reliability of these predictions, and to build sound foundations for future discoveries. This dissertation develops methods for the accurate computation of highorder polynomial approximations of stable/unstable manifolds attached to long periodic orbits in discrete time dynamical systems. For this purpose a multiple shooting scheme is applied to invariance equations for the manifolds obtained using the Parameterization Method developed by Xavier Cabre, Ernest Fontich and Rafael De La Llave in [CFdlL03a, CFdlL03b, CFdlL05].
Show less  Date Issued
 2020
 PURL
 http://purl.flvc.org/fau/fd/FA00013468
 Subject Headings
 Invariant manifolds, Nonlinear systems, Diffeomorphisms, Parabolic partial differential equations, Differential equations, Partial
 Format
 Document (PDF)