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Title: Higher order commutators in the method of orbits.
Creator: Kasprikova, Eva., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: Benson spaces of higher order are introduced extending the idea of N. Krugljak and M. Milman, A distance between orbits that controls commutator estimates and invertibilty of operators, Advances in Mathematics 182 (2004), 78-123. The concept of Benson shift operators is introduced and a class of spaces equipped with these operators is considered. Commutator theorems of higher order on orbit spaces generated by a single element are proved for this class. It is shown that these results apply to...
Show moreBenson spaces of higher order are introduced extending the idea of N. Krugljak and M. Milman, A distance between orbits that controls commutator estimates and invertibilty of operators, Advances in Mathematics 182 (2004), 78-123. The concept of Benson shift operators is introduced and a class of spaces equipped with these operators is considered. Commutator theorems of higher order on orbit spaces generated by a single element are proved for this class. It is shown that these results apply to the complex method of interpolation and to the real method of interpolation for the case q=1. Two new characterizations are presented of the domain space of the "derivation" operator in the context of orbital methods. Comparisons to the work of others are made, especially the unifying paper of M. Cwikel, N. Kalton, M. Milman and R. Rochberg, A United Theory of Commutator Estimates for a Class of Interpolation Methods, Advances in Mathematics 169 2002, 241-312.
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Date Issued: 2009
PURL: http://purl.flvc.org/FAU/2684304
Subject Headings: Operator theory, Interpolation spaces, Finite groups, Sporadic groups (Mathematics)
Format: Document (PDF)

Title: Message authentication in an identity-based encryption scheme: 1-Key-Encrypt-Then-MAC.
Creator: Amento, Brittanney Jaclyn, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: We present an Identity-Based Encryption scheme, 1-Key-Encrypt-Then-MAC, in which we are able to verify the authenticity of messages using a MAC. We accomplish this authentication by combining an Identity-Based Encryption scheme given by Boneh and Franklin, with an Identity-Based Non-Interactive Key Distribution given by Paterson and Srinivasan, and attaching a MAC. We prove the scheme is chosen plaintext secure and chosen ciphertext secure, and the MAC is existentially unforgeable.
Date Issued: 2010
PURL: http://purl.flvc.org/FAU/2796050
Subject Headings: Data encryption (Computer science), Public key cryptopgraphy, Public key infrastructure (Computer security)
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 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.
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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: 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 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.
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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: Computing automorphism groups of projective planes.
Creator: Adamski, Jesse Victor, Magliveras, Spyros S., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: The main objective of this thesis was to find the full automorphism groups of finite Desarguesian planes. A set of homologies were used to generate the automorphism group when the order of the plane was prime. When the order was a prime power Pa,a ≠ 1 the Frobenius automorphism was added to the set of homologies, and then the full automorphism group was generated. The Frobenius automorphism was found by using the planar ternary ring derived from a coordinatization of the plane.
Date Issued: 2013
PURL: http://purl.flvc.org/fau/fd/FA0004000
Subject Headings: Combinatorial group theory, Finite geometrics, Geometry, Projective
Format: Document (PDF)

Title: Detecting essential genes in microarray dataset with unequal number of gene probes.
Creator: Zhang, Wei, Qian, Lianfen, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: Microarray technology is a powerful approach for genomic research, which allows the monitoring of expressing profiles for tens of thousands genes in parallel and is already producing huge amounts of data. This thesis is motivated by a special microarray dataset for the bacteria Yersinia Pestis. It contains more than four thousands genes and each gene has different number of observations. The main purpose of this thesis is to detect essentially functional genes. Gene level adjusted multiple t...
Show moreMicroarray technology is a powerful approach for genomic research, which allows the monitoring of expressing profiles for tens of thousands genes in parallel and is already producing huge amounts of data. This thesis is motivated by a special microarray dataset for the bacteria Yersinia Pestis. It contains more than four thousands genes and each gene has different number of observations. The main purpose of this thesis is to detect essentially functional genes. Gene level adjusted multiple t‐test is proposed to handle the problem of unequal number of observations. Furthermore, a comparation study of our method with two other existing methods (Behrens‐Fisher method and Hotelling t‐square method) are presented. The comparison results show that our proposed methods is the best for identifying essential genes.
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Date Issued: 2012
PURL: http://purl.flvc.org/fau/fd/FA00004261
Format: Document (PDF)

Title: The Finite Abelian Hidden Subgroup Problem.
Creator: Losert, Bernd, Magliveras, Spyros S., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: The hidden subgroup problem has been an active topic of research in quantum computing for over the past 10 years. Out of all the literature that is out there on this topic, there are very few survey articles which discuss most or all of the details concerning the Abelian hidden subgroup problem. As a matter of fact , many articles [17, 26 , 22, 45] claim that an eff-icient quantum algorithm for the Abelian hidden subgroup problem is folklore. To quote Jozsa [21]: " ... the detailed...
Show moreThe hidden subgroup problem has been an active topic of research in quantum computing for over the past 10 years. Out of all the literature that is out there on this topic, there are very few survey articles which discuss most or all of the details concerning the Abelian hidden subgroup problem. As a matter of fact , many articles [17, 26 , 22, 45] claim that an eff-icient quantum algorithm for the Abelian hidden subgroup problem is folklore. To quote Jozsa [21]: " ... the detailed description of an efficient quantum algorithm for the general abelian hidden subgroup problem seems not to have been described in the literature." Apart from banishing this folklore, the aim of this work is to serve as a monograph about the Abelian hidden subgroup, discussing many of the finer points that are ignored in the literature so as to make it accessible and comprehensible to the mathematically mature reader.
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Date Issued: 2010
PURL: http://purl.flvc.org/fau/fd/FA00000790
Format: Document (PDF)

Title: Simulation study on option pricing under jump diffusion models.
Creator: Rodrigues, Justin, Long, Hongwei, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: The main objective of this thesis is to simulate, evaluate and discuss several methods for pricing European-style options. The Black-Scholes model has long been considered the standard method for pricing options. One of the downfalls of the Black-Scholes model is that it is strictly continuous and does not incorporate discrete jumps. This thesis will consider two alternate Levy models that include discretized jumps; The Merton Jump Diffusion and Kou's Double Exponential Jump Diffusion. We...
Show moreThe main objective of this thesis is to simulate, evaluate and discuss several methods for pricing European-style options. The Black-Scholes model has long been considered the standard method for pricing options. One of the downfalls of the Black-Scholes model is that it is strictly continuous and does not incorporate discrete jumps. This thesis will consider two alternate Levy models that include discretized jumps; The Merton Jump Diffusion and Kou's Double Exponential Jump Diffusion. We will use each of the three models to price real world stock data through software simulations and explore the results.Keywords: Levy Processes, Brownian motion, Option pricing, Simulation, Black-Scholes, Merton Jump Diffusion, Kou, Kou's Double Exponential Jump Diffusion.
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Date Issued: 2013
PURL: http://purl.flvc.org/fau/fd/FA0004051
Subject Headings: Finance -- Mathematical models, Levy processes, Prices -- Econometric models, Statistical physics, Stochastic processes, Valuation -- Econometric models
Format: Document (PDF)

Title: Password-authenticated two-party key exchange with long-term security.
Creator: Gao, WeiZheng., Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: In the design of two-party key exchange it is common to rely on a Die-Hellman type hardness assumption in connection with elliptic curves. Unlike the case of nite elds, breaking multiple instances of the underlying hardness assumption is here considered substantially more expensive than breaking a single instance. Prominent protocols such as SPEKE [12] or J-PAKE [8, 9, 10] do not exploit this, and here we propose a password-authenticated key establishment where the security builds on the...
Show moreIn the design of two-party key exchange it is common to rely on a Die-Hellman type hardness assumption in connection with elliptic curves. Unlike the case of nite elds, breaking multiple instances of the underlying hardness assumption is here considered substantially more expensive than breaking a single instance. Prominent protocols such as SPEKE [12] or J-PAKE [8, 9, 10] do not exploit this, and here we propose a password-authenticated key establishment where the security builds on the intractability of solving a specied number of instances v of the underlying computational problem. Such a design strategy seems particularly interesting when aiming at long-term security guarantees for a protocol, where expensive special purpose equipment might become available to an adversary. In this thesis, we give one protocol for the special case when v = 1 in the random oracle model, then we provide the generalized protocol in the random oracle model and a variant of the generalized protocol in the standard model for v being a polynomial of the security parameter `.
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Date Issued: 2012
PURL: http://purl.flvc.org/FAU/3342105
Subject Headings: Data encryption (Computer science), Computer networks (Security measures), Software protection, Computers, Access control, Passwords
Format: Document (PDF)

Title: Minimal zero-dimensional extensions.
Creator: Chiorescu, Marcela, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: The structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are...
Show moreThe structure of minimal zero-dimensional extension of rings with Noetherian spectrum in which zero is a primary ideal and with at most one prime ideal of height greater than one is determined. These rings include K[[X,T]] where K is a field and Dedenkind domains, but need not be Noetherian nor integrally closed. We show that for such a ring R there is a one-to-one correspondence between isomorphism classes of minimal zero-dimensional extensions of R and sets M, where the elements of M are ideals of R primary for distinct prime ideals of height greater than zero. A subsidiary result is the classification of minimal zero-dimensional extensions of general ZPI-rings.
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Date Issued: 2009
PURL: http://purl.flvc.org/FAU/210447
Subject Headings: Algebra, Abstract, Noetherian rings, Commutative rings, Modules (Algebra), Algebraic number theory
Format: Document (PDF)

Title: Detailed study of "polynomial structures in order statistics distributions".
Creator: Liu, Chih-Fan, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: This thesis is based on the paper "Polynomial Structures in Order Statistics Distributions" by M. Denuit, Cl. Lefevre and Ph. Picard. We study the exact order distributions for i.i.d. random variables with arbitrary common law. The left tail distributions can be written as Abel-Gontcharoff polynomials and the right tail distributions can be expressed by Appell polynomials. The polynomial structure makes it easier to obtain closed forms and recursive methods for evaluating the distribution of...
Show moreThis thesis is based on the paper "Polynomial Structures in Order Statistics Distributions" by M. Denuit, Cl. Lefevre and Ph. Picard. We study the exact order distributions for i.i.d. random variables with arbitrary common law. The left tail distributions can be written as Abel-Gontcharoff polynomials and the right tail distributions can be expressed by Appell polynomials. The polynomial structure makes it easier to obtain closed forms and recursive methods for evaluating the distribution of frequently occurring statistics related to empirical distribution functions.
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Date Issued: 2004
PURL: http://purl.flvc.org/fcla/dt/13116
Subject Headings: Mathematics
Format: Document (PDF)

Title: Decay for time-dependent Schroedinger equations.
Creator: Zhou, Zhen, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: We study the decay in time of solutions of Schrodinger equations of the type du/du=idelta u+iV(t)u, establishing that for small potentials and initial data in L1 the solution u satisfies sup[u(x,t)](x element of R)
Show moreWe study the decay in time of solutions of Schrodinger equations of the type du/du=idelta u+iV(t)u, establishing that for small potentials and initial data in L1 the solution u satisfies sup[u(x,t)](x element of R)Show less
Date Issued: 1996
PURL: http://purl.flvc.org/fcla/dt/12463
Subject Headings: Mathematics
Format: Document (PDF)

Title: Diffie-Hellman key exchange protocol, its generalization and nilpotent groups.
Creator: Mahalanobis, Ayan, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: This dissertation has two chapters. In the first chapter we talk about the discrete logarithm problem, more specifically we concentrate on the Diffie-Hellman key exchange protocol. We survey the current state of security for the Diffie-Hellman key exchange protocol. We also motivate the reader to think about the Diffie-Hellman key exchange in terms of group automorphisms. In the second chapter we study two key exchange protocols similar to the Diffie-Hellman key exchange protocol using an...
Show moreThis dissertation has two chapters. In the first chapter we talk about the discrete logarithm problem, more specifically we concentrate on the Diffie-Hellman key exchange protocol. We survey the current state of security for the Diffie-Hellman key exchange protocol. We also motivate the reader to think about the Diffie-Hellman key exchange in terms of group automorphisms. In the second chapter we study two key exchange protocols similar to the Diffie-Hellman key exchange protocol using an abelian subgroup of the automorphism group of a non-abelian group. We also generalize group no. 92 of the Hall-Senior table, for arbitrary prime p and study the automorphism group of these generalized group. We show that for those groups, the group of central automorphisms is an abelian group. We use these central automorphisms for the key exchange we are studying. We also develop a signature scheme.
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Date Issued: 2005
PURL: http://purl.flvc.org/fcla/dt/12154
Subject Headings: Mathematics, Computer Science
Format: Document (PDF)

Title: Enumeration of lattice paths using finite operator calculus.
Creator: Humphreys, Katherine L. B., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: This dissertation discusses umbral calculus and lattice path enumeration and then continues by explicitly enumerating weighted directed lattice paths staying above a boundary using finite operator calculus. In Part I we discuss the history and representative results of the two topics. We separate umbral calculus into two fields, classical umbral calculus and finite operator calculus, and attempt to correct their intertwined histories. We discuss the beginnings of lattice path enumeration and...
Show moreThis dissertation discusses umbral calculus and lattice path enumeration and then continues by explicitly enumerating weighted directed lattice paths staying above a boundary using finite operator calculus. In Part I we discuss the history and representative results of the two topics. We separate umbral calculus into two fields, classical umbral calculus and finite operator calculus, and attempt to correct their intertwined histories. We discuss the beginnings of lattice path enumeration and survey the types of lattice path enumeration problems and solution methods found in the literature. In Part II, we give necessary conditions of a step set or of its equivalent operator equation such that the path count functions coincide with Sheffer polynomials where the path counts are nonzero. We derive the polynomials from an expansion theorem that includes a polynomial basis and initial conditions. The polynomial basis is derived from a known basic sequence with a transfer formula and a linear operator equation based on the step set. The initial conditions are functionals on the polynomials designed to vanish when evaluated along the boundary line for all but finitely many values. We solve lattice path enumeration problems with four types of boundary conditions and various step sets. We work out general solutions for paths that stay in the first quadrant, paths that stay in the first quadrant and above a line with an integer slope, and paths that can reach the boundary with an additional privileged access step set. We count the number of paths, and in one example we count the paths refined by the number of times they contact the boundary. We explore step sets including a general three-element step set, weighted finite step sets, weighted infinite step sets, and step sets that include paths as steps called pathlets. We research if our methods still give explicit solutions as we complicate and expand the step sets. The example sections include fourteen explicitly worked out problems. Part II of the dissertation includes and extends the three papers on the subject by Humphreys and Niederhausen written between 2000 and 2004.
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Date Issued: 2005
PURL: http://purl.flvc.org/fcla/dt/12165
Subject Headings: Mathematics
Format: Document (PDF)

Title: On Boolean algebras and their role in analysis.
Creator: Winkowska-Nowak, Katarzyna, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: The relations between complete and $\sigma$-complete covers of a Boolean algebra are examined. The Dedekind completion of a Boolean algebra is shown to be a quotient of any complete cover. Atoms of a Boolean algebra correspond to atoms of the Dedekind completion hence the Dedekind completion of an atomic Boolean algebra is isomorphic to the power set of the set of all atoms. There exists a correspondence between complete (sigma-complete) homomorphisms and full (sigma-complete) ideals. The...
Show moreThe relations between complete and $\sigma$-complete covers of a Boolean algebra are examined. The Dedekind completion of a Boolean algebra is shown to be a quotient of any complete cover. Atoms of a Boolean algebra correspond to atoms of the Dedekind completion hence the Dedekind completion of an atomic Boolean algebra is isomorphic to the power set of the set of all atoms. There exists a correspondence between complete (sigma-complete) homomorphisms and full (sigma-complete) ideals. The explicit form of the Dedekind completion is given for the Boolean algebra generated by all semiopen subintervals of [0,1) as the atomless, complete Boolean algebra of all regularly closed subsets of [0,1). A compatible topology for a Boolean algebra is a topology for which addition and multiplication are continuous. The properties concerning products, quotients, subspaces and uniform completions of topological Boolean algebras are examined. Compact algebras are isomorphic and homeomorphic with power sets, endowed with the product topology. Measure algebras endowed with the weak* topology are compatible if and only if the underlying measure is purely atomic. A new proof of Stone Representation Theorem for a field of sets is given, providing a tool for establishing relations between Stone representation spaces of algebras, covers, subalgebras and quotients.
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Date Issued: 1996
PURL: http://purl.flvc.org/fcla/dt/12454
Subject Headings: Mathematics
Format: Document (PDF)

Title: Wind speed analysis for Lake Okeechobee.
Creator: Hu, Mingyan, Florida Atlantic University, Qian, Lianfen, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: In this thesis, we analyze wind speeds collected by South Florida Water Management District at stations L001, L005, L006 and LZ40 in Lake Okeechobee from January 1995 to December 2000. There are many missing values and out-liers in this data. To impute the missing values, three different methods are used: Nearby window average imputation, Jones imputation using Kalman filter, and EM algorithm imputation. To detect outliers and remove impacts, we use ARIMA models of time series. Innovational...
Show moreIn this thesis, we analyze wind speeds collected by South Florida Water Management District at stations L001, L005, L006 and LZ40 in Lake Okeechobee from January 1995 to December 2000. There are many missing values and out-liers in this data. To impute the missing values, three different methods are used: Nearby window average imputation, Jones imputation using Kalman filter, and EM algorithm imputation. To detect outliers and remove impacts, we use ARIMA models of time series. Innovational and additive outliers are considered. It turns out that EM algorithm imputation is the best method for our wind speed data set. After imputing missing values, detecting outliers and removing the impacts, we obtain the best models for all four stations. They are all in the form of seasonal ARIMA(2, 0, 0) x (1, 0, 0)24 for the hourly wind speed data.
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Date Issued: 2002
PURL: http://purl.flvc.org/fcla/dt/12883
Subject Headings: Winds--Speed--Florida--Okeechobee, Lake, Okeechobee, Lake (Fla )--Environmental conditions
Format: Document (PDF)

Title: Galois groups of prime degree and the O'Nan-Scott theorem.
Creator: Kaufman, J., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: Discussion begins with a modular method for determining cycle types of permutations in the Galois group of a given separable irreducible polynomial over Q. As the Galois group is a transitive permutation group on the n roots of its irreducible polynomial, a list of all transitive groups of degree n, together with the cycle type distributions of each group, allows a probablistic determination of the group in a process of elimination. In the case of prime degree, transitive groups are primitive...
Show moreDiscussion begins with a modular method for determining cycle types of permutations in the Galois group of a given separable irreducible polynomial over Q. As the Galois group is a transitive permutation group on the n roots of its irreducible polynomial, a list of all transitive groups of degree n, together with the cycle type distributions of each group, allows a probablistic determination of the group in a process of elimination. In the case of prime degree, transitive groups are primitive and by the O'Nan-Scott theorem are of restricted form. Theory is presented in order to arrive at these results and others, so that in conjunction with the classification theorem on finite simple groups, it is possible (in principle) to list all primitive permutation groups of particular prime degree. The case of degree 17 is examined to obtain a list of the transitive permutation groups of degree seventeen, as well as the cycle type of distributions of the groups identified.
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Date Issued: 1996
PURL: http://purl.flvc.org/fcla/dt/15289
Subject Headings: Mathematics
Format: Document (PDF)

Title: Input-to-state stability properties for discrete-time nonlinear systems.
Creator: Gao, Kehan, Florida Atlantic University, Wang, Yuan, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: In this thesis, we study the input-to-state stability (scISS) property and related characterizations for discrete-time nonlinear systems. Variations of scISS property were employed in solving particular control problems. The main contribution of this work is to provide a detailed analysis on the relations among various types of notations related to system stability and show that most scISS results for continuous-time nonlinear system can be extended to discrete-time case.
Date Issued: 1999
PURL: http://purl.flvc.org/fcla/dt/15688
Subject Headings: Discrete-time systems, Nonlinear systems, Stability, Control theory
Format: Document (PDF)

Title: Time series analysis and correlation dimension estimation: Mathematical foundation and applications.
Creator: Jiang, Wangye, Florida Atlantic University, Ding, Mingzhou, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: A time series is a data set of a single quantity sampled at intervals T time units apart. It is widely used to represent a chaotic dynamical system. The correlation dimension measures the complexity of a dynamical system. Using the delay-coordinate map and the extended GP algorithm one can estimate the correlation dimension of an experimental dynamical system from measured time series. This thesis discusses the mathematical foundation of the methods and the corresponding applications. The...
Show moreA time series is a data set of a single quantity sampled at intervals T time units apart. It is widely used to represent a chaotic dynamical system. The correlation dimension measures the complexity of a dynamical system. Using the delay-coordinate map and the extended GP algorithm one can estimate the correlation dimension of an experimental dynamical system from measured time series. This thesis discusses the mathematical foundation of the methods and the corresponding applications. The embedding theorems and their relationship with dimension preservation are reviewed in detail, but more attention is focussed on the concept development.
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Date Issued: 1995
PURL: http://purl.flvc.org/fcla/dt/15213
Subject Headings: Time-series analysis--Mathematical models, Chaotic behavior in systems
Format: Document (PDF)

Title: Sobolev Inequalities.
Creator: Kalis, Jan, Florida Atlantic University, Milman, Mario, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description: Recently a rich theory of Sobolev spaces on metric spaces has been developed. It. turas out that many relevant results from the classical theory have their counterparts in the mcnic setting ( cf. [P. Hajlasz and P. Koskela. Sobokv met Poincare), Mern. Arner. Math. Soc. 145 (2000), no. 6888, x+101pp]). In this thesis we prove sharp Sobolev inequalities in the context of metric spaces. Our approach is ba....,ed on two recent papers, [J. Baster·o and M. Milman and F. Ruiz, A note on L(oc, q)...
Show moreRecently a rich theory of Sobolev spaces on metric spaces has been developed. It. turas out that many relevant results from the classical theory have their counterparts in the mcnic setting ( cf. [P. Hajlasz and P. Koskela. Sobokv met Poincare), Mern. Arner. Math. Soc. 145 (2000), no. 6888, x+101pp]). In this thesis we prove sharp Sobolev inequalities in the context of metric spaces. Our approach is ba....,ed on two recent papers, [J. Baster·o and M. Milman and F. Ruiz, A note on L(oc, q) spaces and Sobolev embeddings, Indiana Univ. Math. J. 52 (2003), no. 5, 1215- 1230] and [J. Martfn and M. Milman and E. Pustylnik, Sobolev inequalities: symmetrization and self improvement via truncation, to appear in J. Funct. Anal.]. These authors establish sharp Sobolev embeddings in the Euclidean setting using symmetrization. Using suitable maximal operators and covering lemmas we show that these symmetrization inequalities of Bastero-Milman-Ruiz remain valid m the metric setting. We also show that the symmetrization by truncation method of Martfn-Milman-Pustylnik can be implemented in our generalized setting. Furthermore we also show that our methods can be adapted to deal with non-doubling measures.
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Date Issued: 2007
PURL: http://purl.flvc.org/fau/fd/FA00000863
Subject Headings: Sobolev spaces, Inequalities (Mathematics), Nonlinear theories, Potential theory (Mathematics)
Format: Document (PDF)

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