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 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. / A Thesis submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Master of Science. / Degree Awarded: Spring semester, 2007. / Date of Defense: December 1, 2006. / FUV, Singular Value Decomposition, Variance, Principal Component Analysis, PCA, Neural Network, NN, Nonlinear Principal Component Analysis, NLPCA, Dimension Reduction, SVD / Includes bibliographical references. / Jerry F. Magnan, Professor Directing Thesis; Steven Bellenot, Committee Member; Mark Sussman, Committee Member.

In portfolio risk management, a global covariance matrix forecast often needs to be adjusted by changing diagonal blocks corresponding to specific sub-markets. 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 off-diagonal blocks leads to a non-convex 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. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Degree Awarded: Spring Semester, 2007. / Date of Defense: March 30, 2007. / Weighted Orthogonal Procrustes Problem, Portfolio Risk, Total Risk, Optimization, Positive Definite / Includes bibliographical references. / Alec Kercheval, Professor Directing Dissertation; Fred Huffer, Outside Committee Member; Kyle Gallivan, Committee Member; Paul Beaumont, Committee Member; Warren Nichols, Committee Member.

F. Waldhausen defines a k-fold end structure on a space X to be an ordered k-tuple 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 k-fold 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 k-fold 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 k-fold 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 q-manifold (q (GREATERTHEQ) 5) with singular set S(X) a polyhedron, XxR a piecewise-linear (q+1)-manifold, then X is the cell-like image of a manifold. / Source: Dissertation Abstracts International, Volume: 43-12, Section: B, page: 4012. / Thesis (Ph.D.)--The Florida State University, 1982.

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 R-plane, 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 short-wave 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(' )small-Reynolds-number 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. / Source: Dissertation Abstracts International, Volume: 45-11, Section: B, page: 3528. / Thesis (Ph.D.)--The Florida State University, 1984.

Source: Dissertation Abstracts International, Volume: 41-01, Section: B, page: 0227. / Thesis (Ph.D.)--The Florida State University, 1979.

In this work two types of rings have been studied, strong S-rings and Prufer v-multiplication domains. Let R be a Prufer domain then R{X} is a strong S-ring. For an integrally closed domain R, each t-ideal is a finite type v-ideal if and only if each prime t-ideal is a finite type v-ideal. The semigroup ring R{X;S} is a Prufer v-multiplication domain if and only if R and K{X;S} are. A PVMD is an S-domain. / Source: Dissertation Abstracts International, Volume: 41-02, Section: B, page: 0590. / Thesis (Ph.D.)--The Florida State University, 1979.

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 small-c('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. / Source: Dissertation Abstracts International, Volume: 42-11, Section: B, page: 4449. / Thesis (Ph.D.)--The Florida State University, 1982.

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 Clifford-valued functions that lie in the kernel of the Cauchy-Riemann 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 Clifford-valued 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 λ-Laplacian. We explore the properties of these so-called λ-harmonic functions and give the solution to the Dirichlet problem for the λ-harmonic functions on annular domains in Rⁿ. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2016. / April 11, 2016. / Clifford analysis, Harmonic analysis, Monogenic functions / Includes bibliographical references. / Craig A. Nolder, Professor Directing Thesis; Kristine C. Harper, University Representative; Ettore Aldrovandi, Committee Member; Bettye Anne Case, Committee Member; John R. Quine, Committee Member; John Ryan, Committee Member.

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 geodesic rays in the space. A technique used for studying this behavior is to compactify the space by adding the endpoints of geodesic rays--i.e. the boundary of the space. / Several new theorems in group theory were proven only after the introduction of these geometric methods--for instance, the Scott conjecture--and 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 non-positively curved groups. / The first task is to define a notion of non-positive 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 non-positively curved groups can be defined as those that act geometrically on one of these non-positively curved spaces. / My research has focused on studying the boundary of the non-positively 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? / Source: Dissertation Abstracts International, Volume: 57-04, Section: B, page: 2612. / Major Professor: Philip Bowers. / Thesis (Ph.D.)--The Florida State University, 1996.

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 4-plat 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 4-plats are counted. Many of these results have appeared in $\lbrack$ES1$\rbrack$. / A knot or (2-component) link L can be factored (non-uniqely) into the sum of two 2-string 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 4-plats, 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 two-fold branched covering spaces, Dehn surgery, and the classification of certain 3-manifolds (Lens spaces and Seifert fiber spaces). This tangle calculus is applied to a model for site-specific 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. / Source: Dissertation Abstracts International, Volume: 49-10, Section: B, page: 4348. / Major Professor: De Witt Sumners. / Thesis (Ph.D.)--The Florida State University, 1988.