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Pages
 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), 78123. 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), 78123. 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, 241312.
Show less  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 identitybased encryption scheme: 1KeyEncryptThenMAC.
 Creator
 Amento, Brittanney Jaclyn, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

We present an IdentityBased Encryption scheme, 1KeyEncryptThenMAC, in which we are able to verify the authenticity of messages using a MAC. We accomplish this authentication by combining an IdentityBased Encryption scheme given by Boneh and Franklin, with an IdentityBased NonInteractive 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 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
 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
 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.
Show less  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 efficient 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 efficient 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.
Show less  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 Europeanstyle options. The BlackScholes model has long been considered the standard method for pricing options. One of the downfalls of the BlackScholes 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 Europeanstyle options. The BlackScholes model has long been considered the standard method for pricing options. One of the downfalls of the BlackScholes 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, BlackScholes, Merton Jump Diffusion, Kou, Kou's Double Exponential Jump Diffusion.
Show less  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
 Passwordauthenticated twoparty key exchange with longterm security.
 Creator
 Gao, WeiZheng., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

In the design of twoparty key exchange it is common to rely on a DieHellman 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 JPAKE [8, 9, 10] do not exploit this, and here we propose a passwordauthenticated key establishment where the security builds on the...
Show moreIn the design of twoparty key exchange it is common to rely on a DieHellman 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 JPAKE [8, 9, 10] do not exploit this, and here we propose a passwordauthenticated 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 longterm 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 `.
Show less  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 zerodimensional extensions.
 Creator
 Chiorescu, Marcela, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

The structure of minimal zerodimensional 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 onetoone correspondence between isomorphism classes of minimal zerodimensional extensions of R and sets M, where the elements of M are...
Show moreThe structure of minimal zerodimensional 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 onetoone correspondence between isomorphism classes of minimal zerodimensional 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 zerodimensional extensions of general ZPIrings.
Show less  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, ChihFan, 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 AbelGontcharoff 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 AbelGontcharoff 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.
Show less  Date Issued
 2004
 PURL
 http://purl.flvc.org/fcla/dt/13116
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 Decay for timedependent 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 moreShow less  Date Issued
 1996
 PURL
 http://purl.flvc.org/fcla/dt/12463
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 DiffieHellman 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 DiffieHellman key exchange protocol. We survey the current state of security for the DiffieHellman key exchange protocol. We also motivate the reader to think about the DiffieHellman key exchange in terms of group automorphisms. In the second chapter we study two key exchange protocols similar to the DiffieHellman 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 DiffieHellman key exchange protocol. We survey the current state of security for the DiffieHellman key exchange protocol. We also motivate the reader to think about the DiffieHellman key exchange in terms of group automorphisms. In the second chapter we study two key exchange protocols similar to the DiffieHellman key exchange protocol using an abelian subgroup of the automorphism group of a nonabelian group. We also generalize group no. 92 of the HallSenior 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.
Show less  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 threeelement 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.
Show less  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
 WinkowskaNowak, 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 (sigmacomplete) homomorphisms and full (sigmacomplete) 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 (sigmacomplete) homomorphisms and full (sigmacomplete) 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.
Show less  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 outliers 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 outliers 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.
Show less  Date Issued
 2002
 PURL
 http://purl.flvc.org/fcla/dt/12883
 Subject Headings
 WindsSpeedFloridaOkeechobee, Lake, Okeechobee, Lake (Fla )Environmental conditions
 Format
 Document (PDF)
 Title
 Galois groups of prime degree and the O'NanScott 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'NanScott 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.
Show less  Date Issued
 1996
 PURL
 http://purl.flvc.org/fcla/dt/15289
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 Inputtostate stability properties for discretetime 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 inputtostate stability (scISS) property and related characterizations for discretetime 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 continuoustime nonlinear system can be extended to discretetime case.
 Date Issued
 1999
 PURL
 http://purl.flvc.org/fcla/dt/15688
 Subject Headings
 Discretetime 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 delaycoordinate 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 delaycoordinate 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.
Show less  Date Issued
 1995
 PURL
 http://purl.flvc.org/fcla/dt/15213
 Subject Headings
 Timeseries analysisMathematical 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 BasteroMilmanRuiz remain valid m the metric setting. We also show that the symmetrization by truncation method of MartfnMilmanPustylnik can be implemented in our generalized setting. Furthermore we also show that our methods can be adapted to deal with nondoubling measures.
Show less  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)