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 Title
 20092010 Program Review Mathematics.
 Creator
 Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Florida Atlantic University Departmental Dashboard Indicators. Department program reviews for Charles E. Schmidt College of Science, Florida Atlantic University.
 Date Issued
 20092010
 PURL
 http://purl.flvc.org/fau/fd/FA00007676
 Format
 Document (PDF)
 Title
 20102011 Program Review Mathematics.
 Creator
 Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Florida Atlantic University Departmental Dashboard Indicators. Department program reviews for Charles E. Schmidt College of Science, Florida Atlantic University.
 Date Issued
 20102011
 PURL
 http://purl.flvc.org/fau/fd/FA00007683
 Format
 Document (PDF)
 Title
 20122013 Program Review Mathematics.
 Creator
 Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Florida Atlantic University Departmental Dashboard Indicators. Department program reviews for Charles E. Schmidt College of Science, Florida Atlantic University.
 Date Issued
 20122013
 PURL
 http://purl.flvc.org/fau/fd/FA00007690
 Format
 Document (PDF)
 Title
 20132014 Program Review Mathematics.
 Creator
 Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Florida Atlantic University Departmental Dashboard Indicators. Department program reviews for Charles E. Schmidt College of Science, Florida Atlantic University.
 Date Issued
 20132014
 PURL
 http://purl.flvc.org/fau/fd/FA00007697
 Format
 Document (PDF)
 Title
 20142015 Program Review Mathematics.
 Creator
 Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Florida Atlantic University Departmental Dashboard Indicators. Department program reviews for Charles E. Schmidt College of Science, Florida Atlantic University.
 Date Issued
 20142015
 PURL
 http://purl.flvc.org/fau/fd/FA00007704
 Format
 Document (PDF)
 Title
 20152016 Program Review Mathematics.
 Creator
 Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Florida Atlantic University Departmental Dashboard Indicators. Department program reviews for Charles E. Schmidt College of Science, Florida Atlantic University.
 Date Issued
 20152016
 PURL
 http://purl.flvc.org/fau/fd/FA00007711
 Format
 Document (PDF)
 Title
 20162017 Program Review Mathematics.
 Creator
 Florida Atlantic University, Department of Mathematical Sciences, Charles E. Schmidt College of Science
 Abstract/Description

Florida Atlantic University Departmental Dashboard Indicators. Department program reviews for Charles E. Schmidt College of Science, Florida Atlantic University.
 Date Issued
 20162017
 PURL
 http://purl.flvc.org/fau/fd/FA00007718
 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
 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
 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)
 Title
 Algebraic and combinatorial aspects of group factorizations.
 Creator
 Bozovic, Vladimir., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

The aim of this work is to investigate some algebraic and combinatorial aspects of group factorizations. The main contribution of this dissertation is a set of new results regarding factorization of groups, with emphasis on the nonabelian case. We introduce a novel technique for factorization of groups, the socalled free mappings, a powerful tool for factorization of a wide class of abelian and nonabelian groups. By applying a certain group action on the blocks of a factorization, a number...
Show moreThe aim of this work is to investigate some algebraic and combinatorial aspects of group factorizations. The main contribution of this dissertation is a set of new results regarding factorization of groups, with emphasis on the nonabelian case. We introduce a novel technique for factorization of groups, the socalled free mappings, a powerful tool for factorization of a wide class of abelian and nonabelian groups. By applying a certain group action on the blocks of a factorization, a number of combinatorial and computational problems were noted and studied. In particular, we analyze the case of the group Aut(Zn) acting on blocks of factorization of Zn. We present new theoretical facts that reveal the numerical structure of the stabilizer of a set in Zn, under the action of Aut(Zn). New algorithms for finding the stabilizer of a set and checking whether two sets belong to the same orbit are proposed.
Show less  Date Issued
 2008
 PURL
 http://purl.flvc.org/FAU/107805
 Subject Headings
 Physical measurements, Mapping (Mathematics), Combinatorial enumeration problems, Algebra, Abstract
 Format
 Document (PDF)
 Title
 An algebraic attack on block ciphers.
 Creator
 Matheis, Kenneth., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

The aim of this work is to investigate an algebraic attack on block ciphers called Multiple Right Hand Sides (MRHS). MRHS models a block cipher as a system of n matrix equations Si := Aix = [Li], where each Li can be expressed as a set of its columns bi1, . . . , bisi . The set of solutions Ti of Si is dened as the union of the solutions of Aix = bij , and the set of solutions of the system S1, . . . , Sn is dened as the intersection of T1, . . . , Tn. Our main contribution is a hardware...
Show moreThe aim of this work is to investigate an algebraic attack on block ciphers called Multiple Right Hand Sides (MRHS). MRHS models a block cipher as a system of n matrix equations Si := Aix = [Li], where each Li can be expressed as a set of its columns bi1, . . . , bisi . The set of solutions Ti of Si is dened as the union of the solutions of Aix = bij , and the set of solutions of the system S1, . . . , Sn is dened as the intersection of T1, . . . , Tn. Our main contribution is a hardware platform which implements a particular algorithm that solves MRHS systems (and hence block ciphers). The case is made that the platform performs several thousand orders of magnitude faster than software, it costs less than US$1,000,000, and that actual times of block cipher breakage can be calculated once it is known how the corresponding software behaves. Options in MRHS are also explored with a view to increase its efficiency.
Show less  Date Issued
 2010
 PURL
 http://purl.flvc.org/FAU/2976444
 Subject Headings
 Ciphers, Cryptography, Data encryption (Computer science), Computer security, Coding theory, Integrated circuits, Design and construction
 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
 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
 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
 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
 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
 Asymmetric information in fads models in Lâevy markets.
 Creator
 Buckley, Winston S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Fads models for stocks under asymmetric information in a purely continuous(GBM) market were first studied by P. Guasoni (2006), where optimal portfolios and maximum expected logarithmic utilities, including asymptotic utilities for the informed and uninformed investors, were presented. We generalized this theory to Lâevy markets, where stock prices and the process modeling the fads are allowed to include a jump component, in addition to the usual continuous component. We employ the methods of...
Show moreFads models for stocks under asymmetric information in a purely continuous(GBM) market were first studied by P. Guasoni (2006), where optimal portfolios and maximum expected logarithmic utilities, including asymptotic utilities for the informed and uninformed investors, were presented. We generalized this theory to Lâevy markets, where stock prices and the process modeling the fads are allowed to include a jump component, in addition to the usual continuous component. We employ the methods of stochastic calculus and optimization to obtain analogous results to those obtained in the purely continuous market. We approximate optimal portfolios and utilities using the instantaneous centralized and quasicentralized moments of the stocks percentage returns. We also link the random portfolios of the investors, under asymmetric information to the purely deterministic optimal portfolio, under symmetric information.
Show less  Date Issued
 2009
 PURL
 http://purl.flvc.org/FAU/3337187
 Subject Headings
 Investments, Mathematical models, Capital market, Mathematical models, Finance, Mathematical models, Information theory in economics, Capital asset pricing model, Lâevy processes
 Format
 Document (PDF)
 Title
 AUC estimation under various survival models.
 Creator
 Chang, Fazhe., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

In the medical science, the receiving operationg characteristic (ROC) curve is a graphical representation to evaluate the accuracy of a medical diagnostic test for any cutoff point. The area under the ROC curve (AUC) is an overall performance measure for a diagnostic test. There are two parts in this dissertation. In the first part, we study the properties of biExponentiated Weibull models. FIrst, we derive a general moment formula for single Exponentiated Weibull models. Then we move on to...
Show moreIn the medical science, the receiving operationg characteristic (ROC) curve is a graphical representation to evaluate the accuracy of a medical diagnostic test for any cutoff point. The area under the ROC curve (AUC) is an overall performance measure for a diagnostic test. There are two parts in this dissertation. In the first part, we study the properties of biExponentiated Weibull models. FIrst, we derive a general moment formula for single Exponentiated Weibull models. Then we move on to derive the precise formula of AUC and study the maximus likelihood estimation (MLE) of the AUC. Finally, we obtain the asymptotoc distribution of the estimated AUC. Simulation studies are used to check the performance of MLE of AUC under the moderate sample sizes. The second part fo the dissertation is to study the estimation of AUC under the crossing model, which extends the AUC formula in Gonen and Heller (2007).
Show less  Date Issued
 2012
 PURL
 http://purl.flvc.org/FAU/3359287
 Subject Headings
 Receiver operating characteristic curves, Medical screening, Statistical methods, Diagnosis, Statistical methods, Smoothing (Statistics)
 Format
 Document (PDF)
 Title
 AuslanderReiten theory for systems of submodule embeddings.
 Creator
 Moore, Audrey., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

In this dissertation, we will investigate aspects of AuslanderReiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute AuslanderReiten quivers of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the dissertation is an adaptation to this situation of the Auslander and RingelTachikawa Theorem which states that for an artinian ring R of finite...
Show moreIn this dissertation, we will investigate aspects of AuslanderReiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute AuslanderReiten quivers of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the dissertation is an adaptation to this situation of the Auslander and RingelTachikawa Theorem which states that for an artinian ring R of finite representation type, each Rmodule is a direct sum of finitelength indecomposable Rmodules. In cases where this applies, the indecomposable objects obtained in the AuslanderReiten quiver give the building blocks for the objects in the category. We also briefly discuss in which cases systems of submodule embeddings form a Frobenius category, and for a few examples explore pointwise CalabiYau dimension of such a category.
Show less  Date Issued
 2009
 PURL
 http://purl.flvc.org/fcla/dt/210496
 Subject Headings
 Artin algebras, Rings (Algebra), Representation of algebras, Embeddings (Mathematics), Linear algebraic groups
 Format
 Document (PDF)