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 Title
 ON THE RAUTOMORPHISMS OF R(X).
 Creator

DOWLEN, MARY MARGARET., Florida State University
 Abstract/Description

Throughout, R is a commutative ring with identity and X is an indeterminate over R. We consider R{X}, the polynomial ring in one indeterminate over R, and G(R), the group of Rautomorphisms of R{X}. In particular, we consider the subring of R{X} left fixed by the group G(R), denoted by R{X}('G(R)). Let B(R) be the subgroup of G(R) such that (sigma) (ELEM) B(R) if and only if (sigma)(X) = a + bX, b a unit of R. If R is reduced, then G(R) = B(R); otherwise, B(R) (LHOOK) G(R). We prove in...
Show moreThroughout, R is a commutative ring with identity and X is an indeterminate over R. We consider R{X}, the polynomial ring in one indeterminate over R, and G(R), the group of Rautomorphisms of R{X}. In particular, we consider the subring of R{X} left fixed by the group G(R), denoted by R{X}('G(R)). Let B(R) be the subgroup of G(R) such that (sigma) (ELEM) B(R) if and only if (sigma)(X) = a + bX, b a unit of R. If R is reduced, then G(R) = B(R); otherwise, B(R) (LHOOK) G(R). We prove in Chapter I that R{X}('G(R)) = R{X}('B(R))., In Chapter I we also prove that for R to be properly contained in R{X}('G(R)), it is necessary that R/M is a finite field for some maximal ideal M of R. Hence, if R is a quasilocal ring with maximal ideal M and R/M is infinite, then R{X}('G(R)) = R., Let R be a quasilocal ring with maximal ideal M such that R/M is isomorphic to the Galois field with p('s) elements, where p is a prime integer and s (ELEM) Z('+). In Chapter II, we show that, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), where, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), In particular, we determine Z(,n){X}('G(Zn)) for n (ELEM) Z('+). Moreover, we prove that R{X}('G(R)) contains a nonconstant monic polynomial if and only if R is a 0dimensional SFTring., In Chapter III, we investigate R{X}('G(R)) for a von Neumann regular ring R. We obtain equivalent conditions for R{X}('G(R)) to contain a nonconstant monic polynomial; one of these is that {card(R/M)} is bounded for all maximal ideals M of R. Moreover, we prove that R is properly contained in R{X}('G(R)) if and only if R has a direct summand S such that S{X}('G(S)) contains a nonconstant monic polynomial. Finally, in Chapter III we construct a von Neumann regular ring B such that B/M is finite for infinitely many maximal ideals M of B, but B{X}('G(B)) = B., In Chapter IV, we show that for any commutative ring R with identity, R{X}('G(R)) contains a nonconstant monic polynomial if and only if R is 0dimensional, card(R/M) < N for some N (ELEM) Z('+) and for all maximal ideals M of R, and nilpotent elements have bounded order of nilpotency.
Show less  Date Issued
 1982, 1982
 Identifier
 AAI8218637, 3085307, FSDT3085307, fsu:74802
 Format
 Document (PDF)
 Title
 A Spectral Element Method to Price Single and MultiAsset European Options.
 Creator

Zhu, Wuming, Kopriva, David A., Huﬀer, Fred, Case, Bettye Anne, Kercheval, Alec N., Okten, Giray, Wang, Xiaoming, Department of Mathematics, Florida State University
 Abstract/Description

We develop a spectral element method to price European options under the BlackScholes model, Merton's jump diffusion model, and Heston's stochastic volatility model with one or two assets. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a GaussLobatto mesh within each element. This piecewise polynomial approximation allows an exact representation of the nonsmooth initial condition. For options with one asset under...
Show moreWe develop a spectral element method to price European options under the BlackScholes model, Merton's jump diffusion model, and Heston's stochastic volatility model with one or two assets. The method uses piecewise high order Legendre polynomial expansions to approximate the option price represented pointwise on a GaussLobatto mesh within each element. This piecewise polynomial approximation allows an exact representation of the nonsmooth initial condition. For options with one asset under the jump diffusion model, the convolution integral is approximated by high order GaussLobatto quadratures. A second order implicit/explicit (IMEX) approximation is used to integrate in time, with the convolution integral integrated explicitly. The use of the IMEX approximation in time means that only a block diagonal, rather than full, system of equations needs to be solved at each time step. For options with two variables, i.e., two assets under the BlackScholes model or one asset under the stochastic volatility model, the domain is subdivided into quadrilateral elements. Within each element, the expansion basis functions are chosen to be tensor products of the Legendre polynomials. Three iterative methods are investigated to solve the system of equations at each time step with the corresponding second order time integration schemes, i.e., IMEX and CrankNicholson. Also, the boundary conditions are carefully studied for the stochastic volatility model. The method is spectrally accurate (exponentially convergent) in space and second order accurate in time for European options under all the three models. Spectral accuracy is observed in not only the solution, but also in the Greeks.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd0513
 Format
 Thesis
 Title
 Modeling the Folding Pattern of the Cerebral Cortex.
 Creator

Striegel, Deborah A., Hurdal, Monica K., Steinbock, Oliver, Quine, Jack, Sumners, DeWitt, Bertram, Richard, Department of Mathematics, Florida State University
 Abstract/Description

The mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemicallydriven scenario, where an increase in intermediate progenitor cells in the subventricular zone (an area surrounding the lateral ventricles) correlates to gyral formation. This dissertation presents...
Show moreThe mechanism for cortical folding pattern formation is not fully understood. Current models represent scenarios that describe pattern formation through local interactions and one recent model is the intermediate progenitor model. The intermediate progenitor (IP) model describes a local chemicallydriven scenario, where an increase in intermediate progenitor cells in the subventricular zone (an area surrounding the lateral ventricles) correlates to gyral formation. This dissertation presents the Global Intermediate Progenitor (GIP) model, a theoretical biological model that uses features of the IP model and further captures global characteristics of cortical pattern formation. To illustrate how global features can effect the development of certain patterns, a mathematical model that incorporates a Turing system is used to examine pattern formation on a prolate spheroidal surface. Pattern formation in a biological system can be studied with a Turing reactiondiffusion system which utilizes characteristics of domain size and shape to predict which pattern will form. The GIP model approximates the shape of the lateral ventricle with a prolate spheroid. This representation allows the capture of a key shape feature, lateral ventricular eccentricity, in terms of the focal distance of the prolate spheroid. A formula relating domain scale and focal distance of a prolate spheroidal surface to specific prolate spheroidal harmonics is developed. This formula allows the prediction of pattern formation with solutions in the form of prolate spheroidal harmonics based on the size and shape of the prolate spheroidal surface. By utilizing this formula a direct correlation between the size and shape of the lateral ventricle, which drives the shape of the ventricular zone, and cerebral cortical folding pattern formation is found. This correlation is illustrated in two different applications: (i) how the location and directionality of the initial cortical folds change with respect to evolutionary development and (ii) how the initial folds change with respect to certain diseases, such as Microcephalia Vera and Megalencephaly Polymicrogyria Polydactyly with Hydrocephalus. The significance of the model, presented in this dissertation, is that it elucidates the consistency of cortical patterns among healthy individuals within a species and addresses interspecies variability based on global characteristics. This model provides a critical piece to the puzzle of cortical pattern formation.
Show less  Date Issued
 2009
 Identifier
 FSU_migr_etd0394
 Format
 Thesis
 Title
 QuasiMonte Carlo and Genetic Algorithms with Applications to Endogenous Mortgage Rate Computation.
 Creator

Shah, Manan, Okten, Giray, Goncharov, Yevgeny, Srinivasan, Ashok, Bellenot, Steve, Case, Bettye Anne, Kercheval, Alec, Kopriva, David, Nichols, Warren, Department of Mathematics...
Show moreShah, Manan, Okten, Giray, Goncharov, Yevgeny, Srinivasan, Ashok, Bellenot, Steve, Case, Bettye Anne, Kercheval, Alec, Kopriva, David, Nichols, Warren, Department of Mathematics, Florida State University
Show less  Abstract/Description

In this dissertation, we introduce a genetic algorithm approach to estimate the star discrepancy of a point set. This algorithm allows for the estimation of the star discrepancy in dimensions larger than seven, something that could not be done adequately by other existing methods. Then, we introduce a class of random digitpermutations for the Halton sequence and show that these permutations yield comparable or better results than their deterministic counterparts in any number of dimensions...
Show moreIn this dissertation, we introduce a genetic algorithm approach to estimate the star discrepancy of a point set. This algorithm allows for the estimation of the star discrepancy in dimensions larger than seven, something that could not be done adequately by other existing methods. Then, we introduce a class of random digitpermutations for the Halton sequence and show that these permutations yield comparable or better results than their deterministic counterparts in any number of dimensions for the test problems considered. Next, we use randomized quasiMonte Carlo methods to numerically solve a onefactor mortgage model expressed as a stochastic fixedpoint problem. Finally, we show that this mortgage model coincides with and is computationally faster than Citigroup's MOATS model, which is based on a binomial tree approach.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd0297
 Format
 Thesis
 Title
 Doublynullcobordant links.
 Creator

Sun, Biansheng., Florida State University
 Abstract/Description

Throughout, we work in the smooth category. We consider a special class of links and knots in $S\sp3$ which are transverse crosssections of trivial 2spheres in $S\sp4.$ They are called DoublyNullCobordant (DNC) links and DNC knots respectively. Closely related concepts are those of NullCobordant (NC) links and NC knots., We are interested in obtaining necessary conditions satisfied by DNC (NC) links and knots, and in constructing nontrivial links and knots which satisfy these conditions....
Show moreThroughout, we work in the smooth category. We consider a special class of links and knots in $S\sp3$ which are transverse crosssections of trivial 2spheres in $S\sp4.$ They are called DoublyNullCobordant (DNC) links and DNC knots respectively. Closely related concepts are those of NullCobordant (NC) links and NC knots., We are interested in obtaining necessary conditions satisfied by DNC (NC) links and knots, and in constructing nontrivial links and knots which satisfy these conditions., Based on analysis of various linking patterns of NC links, we are able to prove that any Hopf link of $\mu$ components is NC if and only if $\mu$ is odd. Another result obtained in this work is that there exists at least one pair of components of an NC link of an even number of components such that the linking number between these two components is zero. Various methods are employed in the geometric realizations of DNC links of a given number of components; we construct DNC links for any given number of components., The 2fold branched cyclic cover of $S\sp3$ branched along any link plays a fundamental role in detecting whether a given link is DNC or not since the cyclic branched cover embeds in $S\sp4$ if the given link is DNC. Considering the embedding problem of the cyclic branched cover leads to the hyperbolicity problem of the associated linking pairing on the homology of the cyclic branched cover. By investigating the torsion part of the first homology of the rfold cyclic branched cover of the 3sphere branched along a link, we are able to produce infinitely many NC links, none of which are DNC links. When specialized to knots, we also discover infinitely many NC knots, none of which are DNC., We also consider higher dimensional DNC links. We obtain a necessary condition for a higher dimensional link being DNC. Specifically, we prove that if L is a DNC (2m $$ 1)link of $\mu$ components with $m >$ 1, then L has a DNC Seifert matrix for any connected Seifert manifold of L.
Show less  Date Issued
 1995, 1995
 Identifier
 AAI9527942, 3088652, FSDT3088652, fsu:77454
 Format
 Document (PDF)
 Title
 A COMPARATIVE STUDY OF THE ABILITY OF FOURTH YEAR HIGH SCHOOL MATHEMATICSSTUDENTS TO USE THE PRINCIPLE OF MATHEMATICAL INDUCTION AND THE WELL ORDERING PRINCIPLE TO PROVE CONJECTURES.
 Creator

WARD, RONALD ALLISON., The Florida State University
 Date Issued
 1971, 1971
 Identifier
 AAI7210053, 2986635, FSDT2986635, fsu:71144
 Format
 Document (PDF)
 Title
 Variance Gamma Pricing of American Futures Options.
 Creator

Yoo, Eunjoo, Nolder, Craig A., Huﬀer, Fred, Case, Bettye Anne, Kercheval, Alec N., Quine, Jack, Department of Mathematics, Florida State University
 Abstract/Description

In financial markets under uncertainty, the classical BlackScholes model cannot explain the empirical facts such as fat tails observed in the probability density. To overcome this drawback, during the last decade, Lévy process and stochastic volatility models were introduced to financial modeling. Today crude oil futures markets are highly volatile. It is the purpose of this dissertation to develop a mathematical framework in which American options on crude oil futures contracts are priced...
Show moreIn financial markets under uncertainty, the classical BlackScholes model cannot explain the empirical facts such as fat tails observed in the probability density. To overcome this drawback, during the last decade, Lévy process and stochastic volatility models were introduced to financial modeling. Today crude oil futures markets are highly volatile. It is the purpose of this dissertation to develop a mathematical framework in which American options on crude oil futures contracts are priced more effectively than by current methods. In this work, we use the Variance Gamma process to model the futures price process. To generate the underlying process, we use a random tress method so that we evaluate the option prices at each tree node. Through fifty replications of a random tree, the averaged value is taken as a true option price. Pricing performance using this method is accessed using American options on crude oil commodity contracts from December 2003 to November 2004. In comparison with the Variance Gamma model, we price using the BlackScholes model as well. Over the entire sample period, a positive skewness and high kurtosis, especially in the shortterm options, are observed. In terms of pricing errors, the Variance Gamma process performs better than the BlackScholes model for the American options on crude oil commodities.
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd0691
 Format
 Thesis
 Title
 A Comparison Study of Principal Component Analysis and Nonlinear Principal Component Analysis.
 Creator

Wu, Rui, Magnan, Jerry F., Bellenot, Steven, Sussman, Mark, Department of Mathematics, Florida State University
 Abstract/Description

In the field of data analysis, it is important to reduce the dimensionality of data, because it will help to understand the data, extract new knowledge from the data, and decrease the computational cost. Principal Component Analysis (PCA) [1, 7, 19] has been applied in various areas as a method of dimensionality reduction. Nonlinear Principal Component Analysis (NLPCA) [1, 7, 19] was originally introduced as a nonlinear generalization of PCA. Both of the methods were tested on various...
Show moreIn the field of data analysis, it is important to reduce the dimensionality of data, because it will help to understand the data, extract new knowledge from the data, and decrease the computational cost. Principal Component Analysis (PCA) [1, 7, 19] has been applied in various areas as a method of dimensionality reduction. Nonlinear Principal Component Analysis (NLPCA) [1, 7, 19] was originally introduced as a nonlinear generalization of PCA. Both of the methods were tested on various artificial and natural datasets sampled from: "F(x) = sin(x) + x", the Lorenz Attractor, and sunspot data. The results from the experiments have been analyzed and compared. Generally speaking, NLPCA can explain more variance than a neural network PCA (NN PCA) in lower dimensions. However, as a result of increasing the dimension, the NLPCA approximation will eventually loss its advantage. Finally, we introduce a new combination of NN PCA and NLPCA, and analyze and compare its performance.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd0704
 Format
 Thesis
 Title
 Numerical Methods for Portfolio Risk Estimation.
 Creator

Zhang, Jianke, Kercheval, Alec, Huﬀer, Fred, Gallivan, Kyle, Beaumont, Paul, Nichols, Warren, Department of Mathematics, Florida State University
 Abstract/Description

In portfolio risk management, a global covariance matrix forecast often needs to be adjusted by changing diagonal blocks corresponding to specific submarkets. Unless certain constraints are obeyed, this can result in the loss of positive definiteness of the global matrix. Imposing the proper constraints while minimizing the disturbance of offdiagonal blocks leads to a nonconvex optimization problem in numerical linear algebra called the Weighted Orthogonal Procrustes Problem. We analyze...
Show moreIn portfolio risk management, a global covariance matrix forecast often needs to be adjusted by changing diagonal blocks corresponding to specific submarkets. Unless certain constraints are obeyed, this can result in the loss of positive definiteness of the global matrix. Imposing the proper constraints while minimizing the disturbance of offdiagonal blocks leads to a nonconvex optimization problem in numerical linear algebra called the Weighted Orthogonal Procrustes Problem. We analyze and compare two local minimizing algorithms and offer an algorithm for global minimization. Our methods are faster and more effective than current numerical methods for covariance matrix revision.
Show less  Date Issued
 2007
 Identifier
 FSU_migr_etd0542
 Format
 Thesis
 Title
 MANIFOLD FACTORS THAT ARE THE CELLLIKE IMAGE OF A MANIFOLD.
 Creator

KUTTER, MARY YEILDING., Florida State University
 Abstract/Description

F. Waldhausen defines a kfold end structure on a space X to be an ordered ktuple of continuous maps xj :X(>)R('+), 1 (LESSTHEQ) j (LESSTHEQ) k (where R('+) is the euclidean half line) yielding a map x:X(>)(R)('k). The pairs (X,x) are made into the category E('k) of spaces with kfold end structure. Attachments and expansions in E('k) are defined by induction on k, where elementary attachments and expansions in E('0) have their usual meaning. For Z (epsilon) E('k), the category E('k)/Z...
Show moreF. Waldhausen defines a kfold end structure on a space X to be an ordered ktuple of continuous maps xj :X(>)R('+), 1 (LESSTHEQ) j (LESSTHEQ) k (where R('+) is the euclidean half line) yielding a map x:X(>)(R)('k). The pairs (X,x) are made into the category E('k) of spaces with kfold end structure. Attachments and expansions in E('k) are defined by induction on k, where elementary attachments and expansions in E('0) have their usual meaning. For Z (epsilon) E('k), the category E('k)/Z consists of pairs (X,i) where i:Z(>)X is an inclusion in E('k) such that there exists an attachment from i(z) to X. And E('k)//Z is the category whose objects are triples (X,i,r) with (X,i) (epsilon) E('k)/z and r:X(>)Z a retraction. An infinite complex over Z is a sequence of inclusions in E('k)//Z, X = {X(,1))(Y,y) in E('k) can be madebounded with respect to equivalent kfold end structures x',y' onX,Y respectively. When X (epsilon) S(,1)(R('k)), that fact can be used to extendthe guaranteed deformation X(SQUIGARR)R('k) in E('k) to a proper deformation(')X(SQUIGARR)D('k) where, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI), is the associated compactification of X. It is shown that after embedding (')X in R('n) for n large enough, and choosing a regular neighborhood (')N of (')X, that ((')N,D('k)) is a proper unknotted ball pair. The result proves, when R('k) is given the natural product kfold end structure, Waldhausen's group S(,1)(R('k)) = 0. An exact sequence established by M. Petty is applied to show S(,0)(R('k)) is also trivial. As a consequence, we show that when X is a generalized qmanifold (q (GREATERTHEQ) 5) with singular set S(X) a polyhedron, XxR a piecewiselinear (q+1)manifold, then X is the celllike image of a manifold.
Show less  Date Issued
 1982, 1982
 Identifier
 AAI8306166, 3085519, FSDT3085519, fsu:75011
 Format
 Document (PDF)
 Title
 A NUMERICAL AND ANALYTICAL STUDY OF DRAG ON A SPHERE IN OSEEN'S APPROXIMATION.
 Creator

LEE, SANG MYUNG., Florida State University
 Abstract/Description

We have investigated the properties of the drag coefficient C(,D) of a sphere according to Oseen's linearization of the equations of viscous incompressible flow. We have treated C(,D) as a complex function of complex Reynolds number with an aim of determining its asymptotic behavior as R(>)(INFIN). C(,D) has a doubly infinite array of simple poles in the left half complex Rplane, each of which lies close to one of the zeros of the spherical Bessel Function K(,m+1/2)(R), for some positive...
Show moreWe have investigated the properties of the drag coefficient C(,D) of a sphere according to Oseen's linearization of the equations of viscous incompressible flow. We have treated C(,D) as a complex function of complex Reynolds number with an aim of determining its asymptotic behavior as R(>)(INFIN). C(,D) has a doubly infinite array of simple poles in the left half complex Rplane, each of which lies close to one of the zeros of the spherical Bessel Function K(,m+1/2)(R), for some positive integer m. These zeros of K(,m+1/2)(R) are the poles of the heat transfer coefficient C(,H)(R) that arises from a simple problem studied by Illingworth (1963). Wu's (1956) analysis of a shortwave scattering problem shows that C(,H)(R) has, for large R, an asymptotic expansion in powers of R('2/3). Numerical computations showed that the same form of expansion works well for C(,D)(R). However, the asymptotic behavior of C(,D)(R) is represented better still by including, in the expansion, an additional term that decays more slowly than R('2/3). The coefficients of this(' )presumed expansion have been estimated by fitting values of C(,D)(R) in the interval 5 < R < 21., The(' )smallReynoldsnumber series for C(,D)(R) has also been extended to 66 terms in double precision. The validity and effectiveness of the techniques used by Van Dyke in extending and improving this series, which is known to be valid only within (VBAR)R(VBAR) = 1.04543, have been examined.
Show less  Date Issued
 1984, 1984
 Identifier
 AAI8501833, 3085973, FSDT3085973, fsu:75459
 Format
 Document (PDF)
 Title
 A STUDY OF STRONG SRINGS AND PRUEFER VMULTIPLICATION DOMAINS.
 Creator

MALIK, SAROJ BALA., The Florida State University
 Abstract/Description

In this work two types of rings have been studied, strong Srings and Prufer vmultiplication domains. Let R be a Prufer domain then R{X} is a strong Sring. For an integrally closed domain R, each tideal is a finite type videal if and only if each prime tideal is a finite type videal. The semigroup ring R{X;S} is a Prufer vmultiplication domain if and only if R and K{X;S} are. A PVMD is an Sdomain.
 Date Issued
 1979, 1979
 Identifier
 AAI8017668, 2989587, FSDT2989587, fsu:74094
 Format
 Document (PDF)
 Title
 THE EIGENVALUES OF THE SPHEROIDAL WAVE EQUATION AND THEIR BRANCH POINTS.
 Creator

GUERRIERI, BRUNO., Florida State University
 Abstract/Description

A comprehensive account is given of the behavior of the eigenvalues of the spheroidal wave equation as functions of the complex variable c('2). The convergence of their smallc('2) expansions is limited by an infinite sequence of rings of branch points of square root type at which adjacent eigenvalues of the same type become equal. Known asymptotic formulas are shown to account for how and where the eigenvalues become equal. These asymptotic series for the eigenvalues apply beyond the rings...
Show moreA comprehensive account is given of the behavior of the eigenvalues of the spheroidal wave equation as functions of the complex variable c('2). The convergence of their smallc('2) expansions is limited by an infinite sequence of rings of branch points of square root type at which adjacent eigenvalues of the same type become equal. Known asymptotic formulas are shown to account for how and where the eigenvalues become equal. These asymptotic series for the eigenvalues apply beyond the rings of branch points; we show how they can now be identified with specific eigenvalues.
Show less  Date Issued
 1982, 1982
 Identifier
 AAI8208739, 3085212, FSDT3085212, fsu:74707
 Format
 Document (PDF)
 Title
 An Analysis of Conjugate Harmonic Components of Monogenic Functions and Lambda Harmonic Functions.
 Creator

BallengerFazzone, Brendon Kerr, Nolder, Craig, Harper, Kristine, Aldrovandi, Ettore, Case, Bettye Anne, Quine, J. R. (John R.), Ryan, John Barry, Florida State University,...
Show moreBallengerFazzone, Brendon Kerr, Nolder, Craig, Harper, Kristine, Aldrovandi, Ettore, Case, Bettye Anne, Quine, J. R. (John R.), Ryan, John Barry, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Clifford analysis is seen as the higher dimensional analogue of complex analysis. This includes a rich study of Clifford algebras and, in particular, monogenic functions, or Cliffordvalued functions that lie in the kernel of the CauchyRiemann operator. In this dissertation, we explore the relationships between the harmonic components of monogenic functions and expand upon the notion of conjugate harmonic functions. We show that properties of the even part of a Cliffordvalued function...
Show moreClifford analysis is seen as the higher dimensional analogue of complex analysis. This includes a rich study of Clifford algebras and, in particular, monogenic functions, or Cliffordvalued functions that lie in the kernel of the CauchyRiemann operator. In this dissertation, we explore the relationships between the harmonic components of monogenic functions and expand upon the notion of conjugate harmonic functions. We show that properties of the even part of a Cliffordvalued function determine properties of the odd part and vice versa. We also explore the theory of functions lying in the kernel of a generalized Laplace operator, the λLaplacian. We explore the properties of these socalled λharmonic functions and give the solution to the Dirichlet problem for the λharmonic functions on annular domains in Rⁿ.
Show less  Date Issued
 2016
 Identifier
 FSU_2016SP_BallengerFazzone_fsu_0071E_13136
 Format
 Thesis
 Title
 Boundaries of groups.
 Creator

Ruane, Kim E., Florida State University
 Abstract/Description

In recent years, the theory of infinite groups has been revolutionized by the introduction of geometric methods. In his foundational paper, "Hyperbolic Groups", Gromov outlines a geometric group theory which provides tools for studying a wide class of groups meant to generalize the classical groups coming from Riemannian geometry. In this setting, the metric geometry of the space is used to study the algebraic properties of the group. One aspect of the metric geometry is the behavior of...
Show moreIn recent years, the theory of infinite groups has been revolutionized by the introduction of geometric methods. In his foundational paper, "Hyperbolic Groups", Gromov outlines a geometric group theory which provides tools for studying a wide class of groups meant to generalize the classical groups coming from Riemannian geometry. In this setting, the metric geometry of the space is used to study the algebraic properties of the group. One aspect of the metric geometry is the behavior of geodesic rays in the space. A technique used for studying this behavior is to compactify the space by adding the endpoints of geodesic raysi.e. the boundary of the space., Several new theorems in group theory were proven only after the introduction of these geometric methodsfor instance, the Scott conjectureand many known theorems can be given new, elegant geometric proofs. With the success of this approach, Gromov wrote a second paper which gives certain minimum requirements for a theory including certain nonpositively curved groups., The first task is to define a notion of nonpositive curvature that will generalize the classical Riemannian notion. One proposed notion goes back to the work of Alexandroff and Topogonov wherein they compare the triangles in a given geometry to the triangles in Euclidean geometry and ask that those in the former be as least as thin as those in the latter. Then a class of nonpositively curved groups can be defined as those that act geometrically on one of these nonpositively curved spaces., My research has focused on studying the boundary of the nonpositively curved spaces which admit geometric actions by a group. The overriding question is a question in Gromov's second paper: If a group acts geometrically on two such spaces, then do they have homeomorphic boundaries?
Show less  Date Issued
 1996, 1996
 Identifier
 AAI9627212, 3088922, FSDT3088922, fsu:77721
 Format
 Document (PDF)
 Title
 On knots and tangles.
 Creator

Ernst, Claus., Florida State University
 Abstract/Description

Recent developments in knot theory provide a method for computing the crossover number for special types of knots and links ($\lbrack$K1$\rbrack$,$\lbrack$LT$\rbrack$,$\lbrack$MuT2$\rbrack$). With this information, questions involving the asymptotic behavior of knots with a fixed crossover number (as the crossover number goes to infinity) can be addressed. An exact count of 4plat knots and links is obtained, thus proving that the number of prime knots grows at least exponentially. Further, a...
Show moreRecent developments in knot theory provide a method for computing the crossover number for special types of knots and links ($\lbrack$K1$\rbrack$,$\lbrack$LT$\rbrack$,$\lbrack$MuT2$\rbrack$). With this information, questions involving the asymptotic behavior of knots with a fixed crossover number (as the crossover number goes to infinity) can be addressed. An exact count of 4plat knots and links is obtained, thus proving that the number of prime knots grows at least exponentially. Further, a lower bound of the number of Montesinos knots is produced and some special classes of 4plats are counted. Many of these results have appeared in $\lbrack$ES1$\rbrack$., A knot or (2component) link L can be factored (nonuniqely) into the sum of two 2string tangles, say A and B. We write L = A + B. Given a system of equations of this kind, some of the knots and tangles involved are treated as known quantities, others as unknown quantities. We want to solve the system for the unknowns. If all tangles involved are rational and all knots and links are 4plats, we can always find all possible solutions. This is called rational tangle calculus. In more general equations, some partial answers are obtained. The main techniques are the theory of twofold branched covering spaces, Dehn surgery, and the classification of certain 3manifolds (Lens spaces and Seifert fiber spaces). This tangle calculus is applied to a model for sitespecific DNA recombination. Most of the results involving tangle calculus will appear in $\lbrack$ES2$\rbrack$., In the last chapter I compile a table of all arborescent tangles with less than 6 crossings in a minimal projection. The chirality of these tangles is determined.
Show less  Date Issued
 1988, 1988
 Identifier
 AAI8827881, 3161671, FSDT3161671, fsu:77870
 Format
 Document (PDF)
 Title
 Finite dimensional Hopf algebras.
 Creator

Williams, Roselyn Elaine., Florida State University
 Abstract/Description

Let k be an algebraically closed field of characteristic 0. This thesis develops techniques used to determine the structure of a finite dimensional Hopf algebra over k. The Hopf algebras of dimension $\leq$11 are classified., Let p be a prime number, r a positive integer, and n = p$\sp{\rm r}1$. Let GF(p$\sp{\rm r}$) be the Galois field of order p$\sp{\rm r}$. Let G = GF(p$\sp{\rm r}$) $\times$ $\sb\varphi$ GF(p$\sp{\rm r}$)$\sp\cdot$ be the semidirect product of GF(p$\sp{\rm r}$) and GF(p$...
Show moreLet k be an algebraically closed field of characteristic 0. This thesis develops techniques used to determine the structure of a finite dimensional Hopf algebra over k. The Hopf algebras of dimension $\leq$11 are classified., Let p be a prime number, r a positive integer, and n = p$\sp{\rm r}1$. Let GF(p$\sp{\rm r}$) be the Galois field of order p$\sp{\rm r}$. Let G = GF(p$\sp{\rm r}$) $\times$ $\sb\varphi$ GF(p$\sp{\rm r}$)$\sp\cdot$ be the semidirect product of GF(p$\sp{\rm r}$) and GF(p$\sp{\rm r}$)$\sp\cdot$ relative to the homomorphism $\varphi$:GF(p$\sp{\rm r}$)$\sp\cdot$ $\to$ AutGF(p$\sp{\rm r}$) defined by $\varphi$(x)(v) = xv for v$\in$ GF(p$\sp{\rm r}$) and x$\in$ GF(p$\sp{\rm r}$)$\sp\cdot$. A Hopf algebra H of dimension n$\sp2$(n + 1) is constructed which contains a Hopf subalgebra isomorphic to (kG)*. H is shown to be isomorphic to its linear dual.
Show less  Date Issued
 1988, 1988
 Identifier
 AAI8909946, 3161752, FSDT3161752, fsu:77951
 Format
 Document (PDF)
 Title
 On determinants of Laplacians and multiple gamma functions.
 Creator

Choi, Junesang., Florida State University
 Abstract/Description

In recent years the problem of evaluating the determinants of Laplacians on Riemannian manifolds has received considerable attention. The theory of multiple gamma functions play an important role in computations of determinants of Laplacians on manifolds of constant curvature. These functions were introduced by E. W. Barnes in about 1900., We are particularly interested in the functional determinant for the nsphere S$\sp{n}$ with the standard metric. For all n we give a factorization it into...
Show moreIn recent years the problem of evaluating the determinants of Laplacians on Riemannian manifolds has received considerable attention. The theory of multiple gamma functions play an important role in computations of determinants of Laplacians on manifolds of constant curvature. These functions were introduced by E. W. Barnes in about 1900., We are particularly interested in the functional determinant for the nsphere S$\sp{n}$ with the standard metric. For all n we give a factorization it into multiple gamma functions and use this factorization to compute nice closed form expressions for the determinant in cases n = 1, 2 and 3., In the course of this investigation we give a new proof of the multiplication formulas for the simple and double gamma functions.
Show less  Date Issued
 1991, 1991
 Identifier
 AAI9123529, 3162286, FSDT3162286, fsu:78446
 Format
 Document (PDF)
 Title
 Incompressible surfaces in punctured Klein bottle bundles.
 Creator

Raspopovic, Pedja., Florida State University
 Abstract/Description

All punctured Klein bottle bundles over S$\sp1$ are classified. For each of those, all their twosided incompressible surfaces are described, up to isotopy. This is used to obtain information on Dehn fillings of the bundles. For example, there is a manifold M with a nontrivial knot k in it, so that infinitely many Dehn surgeries on k yield M.
 Date Issued
 1990, 1990
 Identifier
 AAI9103112, 3162125, FSDT3162125, fsu:78323
 Format
 Document (PDF)
 Title
 Topics in quantum groups.
 Creator

Wen, John Fengping., Florida State University
 Abstract/Description

It has been shown that quasitriangular Hopf algebras (QTHAs) have been increasingly playing important roles in many areas of mathematics and physics. Some people believe that the theory of quantum groups will be the group theory of next century. The main goal of this thesis is to develop methods to determine the quasitriangular structures (Rmatrices) of a finite dimensional Hopf algebra over a field. The primary research that I have done in this thesis touched quantum groups from several...
Show moreIt has been shown that quasitriangular Hopf algebras (QTHAs) have been increasingly playing important roles in many areas of mathematics and physics. Some people believe that the theory of quantum groups will be the group theory of next century. The main goal of this thesis is to develop methods to determine the quasitriangular structures (Rmatrices) of a finite dimensional Hopf algebra over a field. The primary research that I have done in this thesis touched quantum groups from several directions. We prove the main results in this thesis that are stated as follows. Let H be a finite dimensional Hopf algebra over a field k. If H is unimodular, then the Rmatrices of H can be embedded in the center of the quantum double D(H), a QTHA associated to H that was discovered by Drinfel'd. If H is cosemisimple (equivalently, if the dual algebra of H is semisimple), then the Rmatrices of H correspond to central idempotents in D(H). Hence, for a finite dimensional cosemisimple Hopf algebra H (such as the group algebra of a finite group), one can possibly locate all the Rmatrices among the set of central idempotents of D(H), which is a finite set in many general contexts. We will see that there are many nontrivial Rmatrices arising from finite nonabelian groups. Nontrivial Rmatrices of nonabelian group algebras allow us to use groups to construct quantum groups.
Show less  Date Issued
 1996, 1996
 Identifier
 AAI9622873, 3088888, FSDT3088888, fsu:77687
 Format
 Document (PDF)
 Title
 Singular complex periodic solutions of van der Pol's equation and uniform approximations for the solution of Lagerstrom's model problem.
 Creator

Tajdari, Mohammad Sina., Florida State University
 Abstract/Description

Two problems are studied. First, the analytic continuation of the real periodic solutions of van der Pol's equation to complex values of the damping parameter $\varepsilon$ are discussed. This continuation shows the existence of an infinite family of singular complex periodic solutions associated with values of $\varepsilon$ lying on two curves $\Gamma$ and $\bar\Gamma$ (see Figure 8) located symmetrically in the $\varepsilon$plane. These singular solutions are found to cause the existence...
Show moreTwo problems are studied. First, the analytic continuation of the real periodic solutions of van der Pol's equation to complex values of the damping parameter $\varepsilon$ are discussed. This continuation shows the existence of an infinite family of singular complex periodic solutions associated with values of $\varepsilon$ lying on two curves $\Gamma$ and $\bar\Gamma$ (see Figure 8) located symmetrically in the $\varepsilon$plane. These singular solutions are found to cause the existence of the moving singularities of the PoincareLindstedt series for the real limit cycle which were developed at great length by Andersen and Geer (7), and were analyzed, using Pade approximants, by Dadfar et al. (10). A numerical method for the computation of these singular solutions is described. In addition, an asymptotic description of them for large values of $\vert\varepsilon\vert$ is obtained using the method of matched asymptotic expansions. Our results suggest that the existence of the complex singular solutions may, in general, play an important role in the utility of computergenerated perturbation expansions at moderate or large values of the perturbation parameter., Our second study involves a model, due to Lagerstrom, of the steady flow of a viscous incompressible fluid past an object in (m + 1) dimensions. The model is a nonlinear boundaryvalue problem in the range $\varepsilon \leq x $ 0 and $m >$ 0. Our results suggest that a similar iteration may be an effective method of approximation of viscous flows at moderate Reynolds numbers.
Show less  Date Issued
 1990, 1990
 Identifier
 AAI9100068, 3162084, FSDT3162084, fsu:78282
 Format
 Document (PDF)
 Title
 On algebraic and analytic properties of Jacobian varieties of Riemann surfaces.
 Creator

Zhang, Liang., Florida State University
 Abstract/Description

The main purpose of this dissertation is to study some basic properties of Riemann surfaces. The Jacobian of a Riemann surface is one of the most important algebraic and analytic characteristics for the surface. Related to Jacobian of a Riemann surface are Riemann Period Matrix, Jacobian Lattice, and Jacobian Variety. There are algebraic and analytic aspects of study of Jacobians., In Chapter 2, we will establish a number of algebraic properties of complex lattices and tori that are...
Show moreThe main purpose of this dissertation is to study some basic properties of Riemann surfaces. The Jacobian of a Riemann surface is one of the most important algebraic and analytic characteristics for the surface. Related to Jacobian of a Riemann surface are Riemann Period Matrix, Jacobian Lattice, and Jacobian Variety. There are algebraic and analytic aspects of study of Jacobians., In Chapter 2, we will establish a number of algebraic properties of complex lattices and tori that are fundamental to the study of Jacobians of Riemann surfaces. In Chapter 3, we will consider an extremal problem of Riemann surface. The problem was first studied by Buser and Sarnak (BS), who introduced the concept of maximal minimal norms of the Jacobian lattices of Riemann surfaces, and obtained a number of properties for a Riemann surface of large genus. We will answer to their conjecture that Klein's surface would be an absolute extremal Riemann surface in the case of genus 3, and prove that their conjecture is not really true. To complete our proof, we will first give out a sufficient, possibly necessary, condition for a Riemann surface to have a maximal minimal norm of its Jacobian lattice, and then prove that, based on the results of Quine (Q2), there exists a local extremal Riemann surface in the case of genus 3 that has a bigger minimal norm than Klein's surface does. It seems to be extremely difficult to find out all the extremal Riemann surfaces no matter how one nontrivially defines the extremality., Among all the essential work to this paper are the results obtained by Rauch and Lewittes (RL), Quine (Q1) (Q2), and the classic discussions of perfect and eutactic forms introduced by Voronoi (VO) and studied extensively by Barnes (BA3) and many other mathematicians (CS2)., Buser and Sarnak in their paper (BS) have obtained a number of interesting characteristics for surfaces of large genus. Our discussion is very computational and probably not applicable to higher genus case. We will also provide some information on Riemann surfaces of genus 3 such as Klein's surface and Fermat's surface, to hopefully help further study in this area.
Show less  Date Issued
 1995, 1995
 Identifier
 AAI9544338, 3088763, FSDT3088763, fsu:77562
 Format
 Document (PDF)