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- Title
- 2012-2013 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
- 2012-2013
- PURL
- http://purl.flvc.org/fau/fd/FA00007690
- Format
- Document (PDF)
- Title
- 2010-2011 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
- 2010-2011
- PURL
- http://purl.flvc.org/fau/fd/FA00007683
- Format
- Document (PDF)
- Title
- 2013-2014 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
- 2013-2014
- PURL
- http://purl.flvc.org/fau/fd/FA00007697
- Format
- Document (PDF)
- Title
- 2014-2015 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
- 2014-2015
- PURL
- http://purl.flvc.org/fau/fd/FA00007704
- Format
- Document (PDF)
- Title
- 2009-2010 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
- 2009-2010
- PURL
- http://purl.flvc.org/fau/fd/FA00007676
- Format
- Document (PDF)
- Title
- 2015-2016 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
- 2015-2016
- PURL
- http://purl.flvc.org/fau/fd/FA00007711
- Format
- Document (PDF)
- Title
- 2016-2017 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
- 2016-2017
- PURL
- http://purl.flvc.org/fau/fd/FA00007718
- 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
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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 so-called free mappings, a powerful tool for factorization of a wide class of abelian and non-abelian 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 so-called free mappings, a powerful tool for factorization of a wide class of abelian and non-abelian 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
- 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
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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 long-term 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
- Auslander-Reiten theory for systems of submodule embeddings.
- Creator
- Moore, Audrey., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
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In this dissertation, we will investigate aspects of Auslander-Reiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute Auslander-Reiten 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 Ringel-Tachikawa Theorem which states that for an artinian ring R of finite...
Show moreIn this dissertation, we will investigate aspects of Auslander-Reiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute Auslander-Reiten 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 Ringel-Tachikawa Theorem which states that for an artinian ring R of finite representation type, each R-module is a direct sum of finite-length indecomposable R-modules. In cases where this applies, the indecomposable objects obtained in the Auslander-Reiten 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 Calabi-Yau 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)
- 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
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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 quasi-centralized 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
- An Algorithmic Approach to Tran Van Trung's Basic Recursive Construction of t-Designs.
- Creator
- Lopez, Oscar A., Magliveras, Spyros S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
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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 t-designs 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 t-designs 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" t-designs or resolutions whose parameters satisfy a system of non-linear 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 2-resolutions, 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, t-designs
- 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 e-Hellman (ECDH) key exchange algorithm are widely used in practice today for their e ciency and small key sizes. More recently, the Supersingular Isogeny-based Di e-Hellman (SIDH) algorithm provides a method of exchanging keys which is conjectured to be secure in the post-quantum 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 e-Hellman (ECDH) key exchange algorithm are widely used in practice today for their e ciency and small key sizes. More recently, the Supersingular Isogeny-based Di e-Hellman (SIDH) algorithm provides a method of exchanging keys which is conjectured to be secure in the post-quantum 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 d-MUL, a multidimensional scalar multiplication algorithm which is uniform in its operations and generalizes the well known 1-dimensional Montgomery ladder addition chain and the 2-dimensional 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 memory-e 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 join-irreducible attractor-repeller pairs in computing a Lyapunov function, rather than having to use all possible attractor-repeller 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 gradient-like 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 non-invertible 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 networks--Security measures., Computer security., Computers--Access control--Code words., Cyberinfrastructure., Computer network architectures., Cryptography., Number theory--Data 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 QUANTUM-SAFE 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, large-scale communication networks with strong security guarantees, based on quantum-mechanical properties. Quantum cryptography builds on the idea of exploiting physical properties to establish secure cryptographic operations. A particular quantum-based 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, large-scale communication networks with strong security guarantees, based on quantum-mechanical properties. Quantum cryptography builds on the idea of exploiting physical properties to establish secure cryptographic operations. A particular quantum-based 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 error-correction for the optical channel and establishing a strong provable security guarantee. Integrating a state-of-the-art 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 so-called "Weyl ensemble" of...
Show moreThis thesis is composed of three main parts. Each chapter is concerned with characterizing some properties of a random ensemble or stochastic process. The properties of interest and the methods for investigating them di er between chapters. We begin by establishing some asymptotic results regarding zeros of random harmonic mappings, a topic of much interest to mathematicians and astrophysicists alike. We introduce a new model of harmonic polynomials based on the so-called "Weyl ensemble" of random analytic polynomials. Building on the work of Li and Wei [28] we obtain precise asymptotics for the average number of zeros of this model. The primary tools used in this section are the famous Kac-Rice formula as well as classical methods in the asymptotic analysis of integrals such as the Laplace method. Continuing, we characterize several topological properties of this model of harmonic polynomials. In chapter 3 we obtain experimental results concerning the number of connected components of the orientation-reversing region as well as the geometry of the distribution of zeros. The tools used in this section are primarily Monte Carlo estimation and topological data analysis (persistent homology). Simulations in this section are performed within MATLAB with the help of a computational homology software known as Perseus. While the results in this chapter are empirical rather than formal proofs, they lead to several enticing conjectures and open problems. Finally, in chapter 4 we address an industry problem in applied mathematics and machine learning. The analysis in this chapter implements similar techniques to those used in chapter 3. We analyze data obtained by observing CAN tra c. CAN (or Control Area Network) is a network for allowing micro-controllers inside of vehicles to communicate with each other. We propose and demonstrate the e ectiveness of an algorithm for detecting malicious tra c using an approach that discovers and exploits the natural geometry of the CAN surface and its relationship to random walk Markov chains.
Show less - Date Issued
- 2018
- PURL
- http://purl.flvc.org/fau/fd/FA00013118
- Subject Headings
- Stochastic processes, Harmonic functions, Random point cloud
- Format
- Document (PDF)
- Title
- CHARACTERIZATIONS OF LINEAR ISOMETRIES ON COMPLEX SEQUENCE SPACES.
- Creator
- Babun Codorniu, Omar, Zhang, Xiao-Dong, 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
- Computing topological dynamics from time series.
- Creator
- Wess, Mark., Charles E. Schmidt College of Science, Department of Mathematical Sciences
- Abstract/Description
-
The topological entropy of a continuous map quantifies the amount of chaos observed in the map. In this dissertation we present computational methods which enable us to compute topological entropy for given time series data generated from a continuous map with a transitive attractor. A triangulation is constructed in order to approximate the attractor and to construct a multivalued map that approximates the dynamics of the linear interpolant on the triangulation. The methods utilize...
Show moreThe topological entropy of a continuous map quantifies the amount of chaos observed in the map. In this dissertation we present computational methods which enable us to compute topological entropy for given time series data generated from a continuous map with a transitive attractor. A triangulation is constructed in order to approximate the attractor and to construct a multivalued map that approximates the dynamics of the linear interpolant on the triangulation. The methods utilize simplicial homology and in particular the Lefschetz Fixed Point Theorem to establish the existence of periodic orbits for the linear interpolant. A semiconjugacy is formed with a subshift of nite type for which the entropy can be calculated and provides a lower bound for the entropy of the linear interpolant. The dissertation concludes with a discussion of possible applications of this analysis to experimental time series.
Show less - Date Issued
- 2008
- PURL
- http://purl.flvc.org/FAU/186294
- Subject Headings
- Algebraic topology, Graph theory, Fixed point theory, Singularities (Mathematics)
- Format
- Document (PDF)