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 Title
 The 1Type of Algebraic KTheory as a Multifunctor.
 Creator

Valdes, Yaineli, Aldrovandi, Ettore, Rawling, John Piers, Agashe, Amod S., Aluffi, Paolo, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and...
Show moreValdes, Yaineli, Aldrovandi, Ettore, Rawling, John Piers, Agashe, Amod S., Aluffi, Paolo, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

It is known that the category of Waldhausen categories is a closed symmetric multicategory and algebraic Ktheory is a multifunctor from the category of Waldhuasen categories to the category of spectra. By assigning to any Waldhausen category the fundamental groupoid of the 1type of its Ktheory spectrum, we get a functor from the category of Waldhausen categories to the category of Picard groupoids, since stable 1types are classified by Picard groupoids. We prove that this functor is a...
Show moreIt is known that the category of Waldhausen categories is a closed symmetric multicategory and algebraic Ktheory is a multifunctor from the category of Waldhuasen categories to the category of spectra. By assigning to any Waldhausen category the fundamental groupoid of the 1type of its Ktheory spectrum, we get a functor from the category of Waldhausen categories to the category of Picard groupoids, since stable 1types are classified by Picard groupoids. We prove that this functor is a multifunctor to a corresponding multicategory of Picard groupoids.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Valdes_fsu_0071E_14374
 Format
 Thesis
 Title
 Algorithms for Computing Congruences Between Modular Forms.
 Creator

Heaton, Randy, Agashe, Amod, Van Hoeij, Mark, Capstick, Simon, Aldrovandi, Ettore, Department of Mathematics, Florida State University
 Abstract/Description

Let $N$ be a positive integer. We first discuss a method for computing intersection numbers between subspaces of $S_{2}(Gamma_{0}(N),C)$. Then we present a new method for computing a basis of qexpansions for $S_{2}(Gamma_{0}(N),Q)$, describe an algorithm for saturating such a basis in $S_{2}(Gamma_{0}(N),Z)$, and show how these results have applications to computing congruence primes and studying cancellations in the conjectural Birch and SwinnertonDyer formula.
 Date Issued
 2012
 Identifier
 FSU_migr_etd4904
 Format
 Thesis
 Title
 Algorithms for Solving Linear Differential Equations with Rational Function Coefficients.
 Creator

Imamoglu, Erdal, van Hoeij, Mark, van Engelen, Robert, Agashe, Amod S. (Amod Sadanand), Aldrovandi, Ettore, Aluffi, Paolo, Florida State University, College of Arts and Sciences...
Show moreImamoglu, Erdal, van Hoeij, Mark, van Engelen, Robert, Agashe, Amod S. (Amod Sadanand), Aldrovandi, Ettore, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This thesis introduces two new algorithms to find hypergeometric solutions of second order regular singular differential operators with rational function or polynomial coefficients. Algorithm 3.2.1 searches for solutions of type: exp(∫ r dx) ⋅ ₂F₁ (a₁,a₂;b₁;f) and Algorithm 5.2.1 searches for solutions of type exp(∫ r dx) (r₀ ⋅ ₂F₁(a₁,a₂;b₁;f) + r₁ ⋅ ₂F´₁ (a₁,a₂;b₁;f)) where f, r, r₀, r₁ ∈ ℚ̅(̅x̅)̅ and a₁,a₂,b₁ ∈ ℚ and denotes the Gauss hypergeometric function. The algorithms use modular...
Show moreThis thesis introduces two new algorithms to find hypergeometric solutions of second order regular singular differential operators with rational function or polynomial coefficients. Algorithm 3.2.1 searches for solutions of type: exp(∫ r dx) ⋅ ₂F₁ (a₁,a₂;b₁;f) and Algorithm 5.2.1 searches for solutions of type exp(∫ r dx) (r₀ ⋅ ₂F₁(a₁,a₂;b₁;f) + r₁ ⋅ ₂F´₁ (a₁,a₂;b₁;f)) where f, r, r₀, r₁ ∈ ℚ̅(̅x̅)̅ and a₁,a₂,b₁ ∈ ℚ and denotes the Gauss hypergeometric function. The algorithms use modular reduction, Hensel lifting, rational function reconstruction, and rational number reconstruction to do so. Numerous examples from different branches of science (mostly from combinatorics and physics) showed that the algorithms presented in this thesis are very effective. Presently, Algorithm 5.2.1 is the most general algorithm in the literature to find hypergeometric solutions of such operators. This thesis also introduces a fast algorithm (Algorithm 4.2.3) to find integral bases for arbitrary order regular singular differential operators with rational function or polynomial coefficients. A normalized (Algorithm 4.3.1) integral basis for a differential operator provides us transformations that convert the differential operator to its standard forms (Algorithm 5.1.1) which are easier to solve.
Show less  Date Issued
 2017
 Identifier
 FSU_SUMMER2017_Imamoglu_fsu_0071E_13942
 Format
 Thesis
 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
 An Analytic Approach to Estimating the Required Surplus, Benchmark Profit, and Optimal Reinsurance Retention for an Insurance Enterprise.
 Creator

Boor, Joseph A. (Joseph Allen), Born, Patricia, Case, Bettye Anne, Tang, Qihe, Rogachev, Grigory, Okten, Giray, Aldrovandi, Ettore, Paris, Steve, Department of Mathematics,...
Show moreBoor, Joseph A. (Joseph Allen), Born, Patricia, Case, Bettye Anne, Tang, Qihe, Rogachev, Grigory, Okten, Giray, Aldrovandi, Ettore, Paris, Steve, Department of Mathematics, Florida State University
Show less  Abstract/Description

This paper presents an analysis of the capital needs, needed return on capital, and optimum reinsurance retention for insurance companies, all in the context where claims are either paid out or known with certainty within or soon after the policy period. Rather than focusing on how to estimate such values using Monte Carlo simulation, it focuses on closed form expressions and approximations for key quantities that are needed for such an analysis. Most of the analysis is also done using a...
Show moreThis paper presents an analysis of the capital needs, needed return on capital, and optimum reinsurance retention for insurance companies, all in the context where claims are either paid out or known with certainty within or soon after the policy period. Rather than focusing on how to estimate such values using Monte Carlo simulation, it focuses on closed form expressions and approximations for key quantities that are needed for such an analysis. Most of the analysis is also done using a distributionfree approach with respect to the loss severity distribution, so minimal or no assumptions surrounding the specific distribution are needed when analyzing the results. However, one key parameter, that is treated via an exhaustion of cases, involves the degree of parameter uncertainty, the number of separate lines of business involved. This is done for the no parameter uncertainty monoline compound Poisson distribution as well as situations involving (lognormal) severity parameter uncertainty, (gamma/negative binomial) count parameter uncertainty, the multiline compound Poisson case, and the compound Poisson scenario with parameter uncertainty, and especially parameter uncertainty correlated across the lines of business. It shows how the risk of extreme aggregate losses that is inherent in insurance operations may be understood (and, implicitly, managed) by performing various calculations using the loss severity distribution, and, where appropriate, key parameters driving the parameter uncertainty distributions. Formulas are developed that estimate the capital and surplus needs of a company(using the VaR approach), and therefore the profit needs of a company that involve tractable calculations. As part of that the process the benchmark loading for profit, reflecting both the needed financial support for the amount of capital to adequately secure to a given one year survival probability, and the amount needed to recompense investors for diversifiable risk is discussed. An analysis of whether or not the loading for diversifiable risk is needed is performed. Approximations to the needed values are performed using the moments of the capped severity distribution and analytic formulas from the frequency distribution as inputs into method of moments normal and lognormal approximations to the percentiles of the aggregate loss distribution. An analysis of the optimum reinsurance retention/policy limit is performed as well, with capped loss distribution/frequency distribution equations resulting from the relationship that the marginal profit (with respect to the loss cap) should be equal to the marginal expense and profit dollar loading with respect to the loss cap. Analytical expressions are developed for the optimum reinsurance retention. Approximations to the optimum retention based on the normal distribution were developed and their error analyzed in great detail. The results indicate that in the vast majority of practical scenarios, the normal distribution approximation to the optimum retention is acceptable. Also included in the paper is a brief comparison of the VaR (survival probability) and expected policyholder deficit (EPD) and TVaR approaches to surplus adequacy (which conclude that the VaR approach is superior for most property/casualty companies); a mathematical analysis of the propriety of insuring the upper limits of the loss distribution, which concludes that, even if unlimited funds were available to secure losses in capital and reinsurance, it would not be in the insured's best interest to do so. Further inclusions to date include a illustrative derivation of the generalized collective risk equation and a method for interpolating ``along'' a mathematical curve rather than directly using the values on the curve. As a prelude to a portion of the analysis, a theorem was proven indicating that in most practical situations, the n1st order derivatives of a suitable probability mass function at values L, when divided by the product of L and the nth order derivative, generate a quotient with a limit at infinity that is less than 1/n.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd4726
 Format
 Thesis
 Title
 Characteristic Classes and Local Invariants of Determinantal Varieties and a Formula for Equivariant ChernSchwartzMacPherson Classes of Hypersurfaces.
 Creator

Zhang, Xiping, Aluffi, Paolo, Piekarewicz, Jorge, Aldrovandi, Ettore, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and Sciences, Department...
Show moreZhang, Xiping, Aluffi, Paolo, Piekarewicz, Jorge, Aldrovandi, Ettore, Petersen, Kathleen L., Hoeij, Mark van, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

Determinantal varieties parametrize spaces of matrices of given ranks. The main results of this dissertation are computations of intersectiontheoretic invariants of determinantal varieties. We focus on the ChernMather and ChernSchwartzMacPherson classes, on the characteristic cycles, and on topologically motivated invariants such as the local Euler obstruction. We obtain explicit formulas in both the ordinary and the torusequivariant setting, and formulate a conjecture concerning the...
Show moreDeterminantal varieties parametrize spaces of matrices of given ranks. The main results of this dissertation are computations of intersectiontheoretic invariants of determinantal varieties. We focus on the ChernMather and ChernSchwartzMacPherson classes, on the characteristic cycles, and on topologically motivated invariants such as the local Euler obstruction. We obtain explicit formulas in both the ordinary and the torusequivariant setting, and formulate a conjecture concerning the effectiveness of the ChernSchwartzMacPherson classes of determinantal varieties. We also prove a vanishing property for the ChernSchwartzMacPherson classes of general group orbits. As applications we obtain formulas for the sectional Euler characteristic of determinantal varieties and the microlocal indices of their intersection cohomology sheaf complexes. Moreover, for a close embedding we define the equivariant version of the Segre class and prove an equivariant formula for the ChernSchwartzMacPherson classes of hypersurfaces of projective varieties.
Show less  Date Issued
 2018
 Identifier
 2018_Sp_Zhang_fsu_0071N_14521
 Format
 Thesis
 Title
 Chern Classes of Sheaves of Logarithmic Vector Fields for Free Divisors.
 Creator

Liao, Xia, Aluﬃ, Paolo, Reina, Laura, Klassen, Eric P., Aldrovandi, Ettore, Petersen, Kathleen, Department of Mathematics, Florida State University
 Abstract/Description

The thesis work we present here focuses on solving a conjecture raised by Aluffi about ChernSchwartzMacPherson classes. Let $X$ be a nonsingular variety defined over an algebraically closed field $k$ of characteristic $0$, $D$ a reduced effective divisor on $X$, and $U = X smallsetminus D$ the open complement of $D$ in $X$. The conjecture states that $c_{textup{SM}}(1_U) = c(textup{Der}_X(log D)) cap [X]$ in $A_{*}(X)$ for any locally quasihomogeneous free divisor $D$. We prove a stronger...
Show moreThe thesis work we present here focuses on solving a conjecture raised by Aluffi about ChernSchwartzMacPherson classes. Let $X$ be a nonsingular variety defined over an algebraically closed field $k$ of characteristic $0$, $D$ a reduced effective divisor on $X$, and $U = X smallsetminus D$ the open complement of $D$ in $X$. The conjecture states that $c_{textup{SM}}(1_U) = c(textup{Der}_X(log D)) cap [X]$ in $A_{*}(X)$ for any locally quasihomogeneous free divisor $D$. We prove a stronger version of this conjecture. We also report on work aimed at studying the Grothedieck class of hypersurfaces of low degree. In this work, we verified the Geometric ChevalleyWarning conjecture in several low dimensional cases.
Show less  Date Issued
 2013
 Identifier
 FSU_migr_etd7467
 Format
 Thesis
 Title
 ChernSchwartzMacpherson Classes of Graph Hypersurfaces and Schubert Varieties.
 Creator

Stryker, Judson P., Aluﬃ, Paolo, Van Engelen, Robert, Aldrovandi, Ettore, Hironaka, Eriko, Van Hoeij, Mark, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation finds some partial results in support of two positivity conjectures regarding the ChernSchwartzMacPherson (CSM) classes of graph hypersurfaces (conjectured by Aluffi and Marcolli) and Schubert varieties (conjectured by Aluffi and Mihalcea). Direct calculations of some of these CSM classes are performed. Formulas for CSM classes of families of both graph hypersurfaces and coefficients of Schubert varieties are developed. Additionally, the positivity of the CSM class of...
Show moreThis dissertation finds some partial results in support of two positivity conjectures regarding the ChernSchwartzMacPherson (CSM) classes of graph hypersurfaces (conjectured by Aluffi and Marcolli) and Schubert varieties (conjectured by Aluffi and Mihalcea). Direct calculations of some of these CSM classes are performed. Formulas for CSM classes of families of both graph hypersurfaces and coefficients of Schubert varieties are developed. Additionally, the positivity of the CSM class of certain families of these varieties is proven. The first chapter starts with an overview and introduction to the material along with some of the background material needed to understand this dissertation. In the second chapter, a series of equivalences of graph hypersurfaces that are useful for reducing the number of cases that must be calculated are developed. A table of CSM classes of all but one graph with 6 or fewer edges are explicitly computed. This table also contains Fulton Chern classes and Milnor classes for the graph hypersurfaces. Using the equivalences and a series of formulas from a paper by Aluffi and Mihalcea, a new series of formulas for the CSM classes of certain families of graph hypersurfaces are deduced. I prove positivity for all graph hypersurfaces corresponding to graphs with first Betti number of 3 or less. Formulas for graphs equivalent to graphs with 6 or fewer edges are developed (as well as cones over graphs with 6 or fewer edges). In the third chapter, CSM classes of Schubert varieties are discussed. It is conjectured by Aluffi and Mihalcea that all Chern classes of Schubert varieties are represented by effective cycles. This is proven in special cases by B. Jones. I examine some positivity results by analyzing and applying combinatorial methods to a formula by Aluffi and Mihalcea. Positivity of what could be considered the ``typical' case for low codimensional coefficients is found. Some other general results for positivity of certain coefficients of Schubert varieties are found. This technique establishes positivity for some known cases very quickly, such as the codimension 1 case as described by Jones, as well as establishing positivity for codimension 2 and families of cases that were previously unknown. An unexpected connection between one family of cases and a second order PDE is also found. Positivity is shown for all cases of codimensions 14 and some higher codimensions are discussed. In both the graph hypersurfaces and Schubert varieties, all calculated ChernSchwartzMacPherson classes were found to be positive.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd1531
 Format
 Thesis
 Title
 Closed Form Solutions of Linear Difference Equations.
 Creator

Cha, Yongjae, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

In this thesis we present an algorithm that finds closed form solutions for homogeneous linear recurrence equations. The key idea is transforming an input operator Linp to an operator Lg with known solutions. The main problem of this idea is how to find a solved equation Lg to which Linp can be reduced. To solve this problem, we use local data of a difference operator, that is invariant under the transformation.
 Date Issued
 2011
 Identifier
 FSU_migr_etd3960
 Format
 Thesis
 Title
 Constructing NonTrivial Elements of the ShafarevichTate Group of an Abelian Variety.
 Creator

Biswas, Saikat, Agashe, Amod, Aggarwal, Sudhir, Hironaka, Eriko, Van Hoeij, Mark, Aldrovandi, Ettore, Department of Mathematics, Florida State University
 Abstract/Description

The ShafarevichTate group of an elliptic curve is an important invariant of the curve whose conjectural finiteness can sometimes be used to determine the rank of the curve. The second part of the Birch and SwinnertonDyer (BSD) conjecture gives a conjectural formula for the order of the ShafarevichTate group of a elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing non...
Show moreThe ShafarevichTate group of an elliptic curve is an important invariant of the curve whose conjectural finiteness can sometimes be used to determine the rank of the curve. The second part of the Birch and SwinnertonDyer (BSD) conjecture gives a conjectural formula for the order of the ShafarevichTate group of a elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing nontrivial elements of the ShafarevichTate group of an elliptic curve by means of the MordellWeil group of an ambient curve. In this thesis, we extract a general theorem out of Cremona and Mazur's work and give precise conditions under which such a construction can be made. We then give an extension of our result which provides new theoretical evidence for the BSD conjecture. Finally, we prove a theorem that gives an alternative method to potentially construct nontrivial elements of the ShafarevichTate group of an elliptic curve by using the component groups of a second curve.
Show less  Date Issued
 2011
 Identifier
 FSU_migr_etd3717
 Format
 Thesis
 Title
 Effective Methods in Intersection Theory and Combinatorial Algebraic Geometry.
 Creator

Harris, Corey S. (Corey Scott), Chicken, Eric, Aldrovandi, Ettore, Kim, Kyounghee, Petersen, Kathleen L., Florida State University, College of Arts and Sciences, Department of...
Show moreHarris, Corey S. (Corey Scott), Chicken, Eric, Aldrovandi, Ettore, Kim, Kyounghee, Petersen, Kathleen L., Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This dissertation presents studies of effective methods in two main areas of algebraic geometry: intersection theory and characteristic classes, and combinatorial algebraic geometry. We begin in chapter 2 by giving an effective algorithm for computing Segre classes of subschemes of arbitrary projective varieties. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space. To our knowledge, this...
Show moreThis dissertation presents studies of effective methods in two main areas of algebraic geometry: intersection theory and characteristic classes, and combinatorial algebraic geometry. We begin in chapter 2 by giving an effective algorithm for computing Segre classes of subschemes of arbitrary projective varieties. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space. To our knowledge, this is the first algorithm to be able to compute Segre classes in projective varieties with arbitrary singularities. In chapter 3, we generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field, proving that the more general class of r.c. monomial subschemes in arbitrarily singular varieties can be principalized by a sequence of blowups at codimension 2 r.c. monomial centers. The main result of chapter 4 is a classification of the monomial Cremona transformations of the plane up to conjugation by certain linear transformations. In particular, an algorithm for enumerating all such maps is derived. In chapter 5, we study the multiview varieties and compute their ChernMather classes. As a corollary we derive a polynomial formula for their Euclidean distance degree, partially addressing a conjecture of Draisma et al. [35]. In chapter 6, we discuss the classical problem of counting planes tangent to general canonical sextic curves at three points. We investigate the situation for real and tropical sextics. In chapter 6, we explicitly compute equations of an Enriques surface via the involution on a K3 surface.
Show less  Date Issued
 2017
 Identifier
 FSU_2017SP_Harris_fsu_0071E_13829
 Format
 Thesis
 Title
 Factoring Univariate Polynomials over the Rationals.
 Creator

Novocin, Andrew, Van Hoeij, Mark, Van Engelen, Robert, Agashe, Amod, Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

This thesis presents an algorithm for factoring polynomials over the rationals which follows the approach of the van Hoeij algorithm. The key theoretical novelty in our approach is that it is et up in a way that will make it possible to prove a new complexity result for this algorithm which was actually observed on prior algorithms. One difference of this algorithm from prior algorithms is the practical improvement which we call early termination. Our algorithm should outperform prior...
Show moreThis thesis presents an algorithm for factoring polynomials over the rationals which follows the approach of the van Hoeij algorithm. The key theoretical novelty in our approach is that it is et up in a way that will make it possible to prove a new complexity result for this algorithm which was actually observed on prior algorithms. One difference of this algorithm from prior algorithms is the practical improvement which we call early termination. Our algorithm should outperform prior algorithms in many common classes of polynomials (including irreducibles).
Show less  Date Issued
 2008
 Identifier
 FSU_migr_etd2515
 Format
 Thesis
 Title
 Finding All Bessel Type Solutions for Linear Differential Equations with Rational Function Coefficients.
 Creator

Yuan, Quan, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

A linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bessel functions, change of variables, algebraic operations and exponential integrals. For second order equations with rational function coefficients, the function f of change of variables must be a rational function or the square root of a rational function. An algorithm was given by Debeerst, van Hoeij, and Koepf, that can compute Bessel type solutions if and only if...
Show moreA linear differential equation with rational function coefficients has a Bessel type solution when it is solvable in terms of Bessel functions, change of variables, algebraic operations and exponential integrals. For second order equations with rational function coefficients, the function f of change of variables must be a rational function or the square root of a rational function. An algorithm was given by Debeerst, van Hoeij, and Koepf, that can compute Bessel type solutions if and only if change of variables is a rational function. In this thesis we extend this work to the square root case, resulting in a complete algorithm to find all Bessel type solutions. This algorithm can be easily extended to a Whittaker/Kummer solver. Combine the two algorithms, we can get a complete algorithm for all 0F1 and 1F1 type solutions. We also use our algorithm to analyze the relation between Bessel functions and Heun functions.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5296
 Format
 Thesis
 Title
 Hypergeometric Solutions of Linear Differential Equations with Rational Function Coefficients.
 Creator

Kunwar, Vijay Jung, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Hironaka, Eriko, Petersen, Kathleen, Department of Mathematics, Florida State...
Show moreKunwar, Vijay Jung, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Hironaka, Eriko, Petersen, Kathleen, Department of Mathematics, Florida State University
Show less  Abstract/Description

Let L be a second order linear differential equation with rational function coefficients. We want to find a solution (if that exists) of L in terms of 2F1hypergeometric function. This thesis presents two algorithms to find such solution in the following cases: 1. L has five regular singularities where at least one of them is logarithmic. 2. L has hypergeometric solution of degree three, i.e, L is solvable in terms of 2F1(a,b;c  f) where f is a rational function of degree three.
 Date Issued
 2014
 Identifier
 FSU_migr_etd9021
 Format
 Thesis
 Title
 Intersection Numbers of Divisors in Graph Varieties.
 Creator

Jones, Deborah, Aluffi, Paolo, Aldrovandi, Ettore, Hironaka., Eriko, Klassen, Eric, Reina, Laura, Department of Mathematics, Florida State University
 Abstract/Description

This dissertation studies certain intersection numbers of exceptional divisions arising from blowing up subspaces of lattices associated to graphs. These permit the computation of the Segre class of a scheme associated to the graph/lattice. Explicit formulas are provided for lattices associated to trees and several patterns among these numbers are explored. The problem can be related to the study of socalled Cremona transformations. It is shown that the geometry of such transformations...
Show moreThis dissertation studies certain intersection numbers of exceptional divisions arising from blowing up subspaces of lattices associated to graphs. These permit the computation of the Segre class of a scheme associated to the graph/lattice. Explicit formulas are provided for lattices associated to trees and several patterns among these numbers are explored. The problem can be related to the study of socalled Cremona transformations. It is shown that the geometry of such transformations explain a certain symmetry pattern we discovered.
Show less  Date Issued
 2003
 Identifier
 FSU_migr_etd3426
 Format
 Thesis
 Title
 Lagrangian Specialization via Log Resolutions and SchwartzMacPherson Chern Classes.
 Creator

Adams, William J. L. (William James Louis), Aluffi, Paolo, Rawling, J. Piers, Aldrovandi, Ettore, Kim, Kyounghee, Agashe, Amod Sadanand, Florida State University, College of...
Show moreAdams, William J. L. (William James Louis), Aluffi, Paolo, Rawling, J. Piers, Aldrovandi, Ettore, Kim, Kyounghee, Agashe, Amod Sadanand, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

This dissertation covers several topics around the idea of the SchwartzMacPherson Chern classes, which were independently constructed by M.H. Schwartz around 1965 and R. MacPherson in the early 1970's. First we review a more recent construction of SchwartzMacPherson Chern class due to G. Kennedy using projective Lagrangian conormals and make explicit some details not found in that work. Around 1980, J.L. Verdier obtained a specialization formula for SchwartzMacPherson Chern classes, which...
Show moreThis dissertation covers several topics around the idea of the SchwartzMacPherson Chern classes, which were independently constructed by M.H. Schwartz around 1965 and R. MacPherson in the early 1970's. First we review a more recent construction of SchwartzMacPherson Chern class due to G. Kennedy using projective Lagrangian conormals and make explicit some details not found in that work. Around 1980, J.L. Verdier obtained a specialization formula for SchwartzMacPherson Chern classes, which was recovered by Kennedy again using the Lagrangian setting and FultonMacPherson intersection theory, elaborating on work of C. Sabbah. A verbatim reading of this approach reveals several subtleties. We introduce an alternate blowup construction to address some of these subtleties, as well as an alternate definition based upon the work of C. Sabbah. We then prove that either of these constructions indeed provides a complete alternative proof of Verdier's specialization formula. In the early 2010's P. Aluffi revisited Verdier's work and developed a more calculable approach to specialization utilizing constructible functions and the Weak Factorization Theorem of Abramovich, Karu, Matsuki, and Wlodarczyk. Aluffi's construction gives an explicit formula when in the case of a divisor with normal crossings and nonsingular components. We take those ideas to the Lagrangian setting to define a new Langrangian cycle, the Asp cycle. All of the proofs in that section utilize Langrangians. We prove that the Asp cycle agrees with the specialization cycle introduced by Sabbah and Kennedy in the special case in which the subvariety is the central fiber of a family over a smooth curve. Thus the Asp cycle may be viewed as a generalization of the KennedySabbah cycle. We give a corresponding generalization of Verdier's specialization formula.
Show less  Date Issued
 2015
 Identifier
 FSU_migr_etd9535
 Format
 Thesis
 Title
 NButterflies: Modeling Weak Morphisms of Strict NGroups.
 Creator

Dungan, Gregory John, Aldrovandi, Ettore, Piekarewicz, Jorge, Agashe, Amod, Aluffi, Paolo, Petersen, Kathleen, Department of Mathematics, Florida State University
 Abstract/Description

Butterflies are an algebraic model of the morphisms of the homotopy category of crossed modules and were originally introduced by Behrang Noohi. Crossed complexes are algebraic structures which generalize crossed modules. The following dissertation is concerned with adapting butterflies to the full subcategory of crossed complexes called reduced ncrossed complexes.
 Date Issued
 2014
 Identifier
 FSU_migr_etd8975
 Format
 Thesis
 Title
 On Elliptic Fibrations and FTheory Compactifications of String Vacua.
 Creator

Fullwood, James, Aluﬃ, Paolo, Reina, Laura, Van Hoeij, Mark, Aldrovandi, Ettore, Hironaka, Eriko, Department of Mathematics, Florida State University
 Abstract/Description

We investigate some algebrogeometric aspects of several families of elliptic fibrations relevant for Ftheory model building along with some physical applications. In particular, we compute topological invariants of elliptic fibrations via `SethiVafaWitten formulas', which relate the given invariant of the total space of the fibration to invariants of the base. We find that these invariants can often be computed in a baseindependent manner, and moreover, can be computed for all possible...
Show moreWe investigate some algebrogeometric aspects of several families of elliptic fibrations relevant for Ftheory model building along with some physical applications. In particular, we compute topological invariants of elliptic fibrations via `SethiVafaWitten formulas', which relate the given invariant of the total space of the fibration to invariants of the base. We find that these invariants can often be computed in a baseindependent manner, and moreover, can be computed for all possible dimensions of a base at once. As such, we construct generating series $f(t)$ corresponding to each invariant such that the coefficient of $t^k$ encodes the invariant of the elliptic fibration over a base of dimension $k$, solely in terms of invariants of the base. From the Ftheory perspective, we highlight aspects of elliptic fibrations other than Weierstrass models, and construct a new orientifold limit of Ftheory associated with $D_5$ fibrations, i.e., elliptic fibrations whose elliptic fiber is realized via a complete intersection of two quadrics in $\mathbb{P}^3$. We verify tadpole relations as predicted by the (conjectural) equivalence between Ftheory and typeIIB, as well as `universal tadpole relations', which are mathematical generalizations of the tadpole relations predicted by the physics of Ftheory. We also simplify formulas for invariants of CalabiYau fourfolds, and suggest that all Hodge numbers of CalabiYau fourfolds depend linearly on $c_1(B)^3$, where $B$ is the base of the fibration.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd4848
 Format
 Thesis
 Title
 On Picard 2Stacks and Length 3 Complexes of Abelian Sheaves.
 Creator

Tatar, Ahmet Emin, Aldrovandi, Ettore, Capstick, Simon, Agashe, Amod, Aluﬃ, Paolo, Klassen, Eric, Department of Mathematics, Florida State University
 Abstract/Description

In Seminaire de Geometrie Algebrique 4 (SGA4), Expose XVIII, Pierre Deligne proves that to any Picard stack one can associate a complex of abelian sheaves of length 2. He also studies the morphisms between such stacks and shows that such a morphism defines a class of fractions in the derived category of complexes of abelian sheaves of length 2. From these two preliminary results, he finally deduces that the derived category of complexes of abelian sheaves of length 2 is equivalent to the...
Show moreIn Seminaire de Geometrie Algebrique 4 (SGA4), Expose XVIII, Pierre Deligne proves that to any Picard stack one can associate a complex of abelian sheaves of length 2. He also studies the morphisms between such stacks and shows that such a morphism defines a class of fractions in the derived category of complexes of abelian sheaves of length 2. From these two preliminary results, he finally deduces that the derived category of complexes of abelian sheaves of length 2 is equivalent to the category of Picard stacks with morphisms being the isomorphism classes. In this dissertation, we generalize his work, following closely his steps in SGA4, to the case of Picard 2stacks. But this generalization requires first a clear description of a Picard 2category as well as of a 2functor between such 2categories that respects Picard structure. Once this has been done, we can talk about category of Picard 2stacks and prove that the derived category of complexes of abelian sheaves of length 3 is equivalent to the category of Picard 2stacks.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd1674
 Format
 Thesis
 Title
 Periods and Motives: Applications in Mathematical Physics.
 Creator

Li, Dan, Marcolli, Matilde, Reina, Laura, Aluﬃ, Paolo, Agashe, Amod, Aldrovandi, Ettore, Department of Mathematics, Florida State University
 Abstract/Description

The study of periods arose in number theory and algebraic geometry, periods are interesting transcendental numbers like multiple zeta values, on the other hand periods are integrals of algebraic differential forms over domains described by algebraic relations. Viewed as abstract periods, we also consider their relations with motives. In this work, we consider two problems in mathematical physics as applications of the ideas and tools from periods and motives. We first consider the algebro...
Show moreThe study of periods arose in number theory and algebraic geometry, periods are interesting transcendental numbers like multiple zeta values, on the other hand periods are integrals of algebraic differential forms over domains described by algebraic relations. Viewed as abstract periods, we also consider their relations with motives. In this work, we consider two problems in mathematical physics as applications of the ideas and tools from periods and motives. We first consider the algebrogeometric approach to the spectral theory of Harper operators in solid state physics. When the parameters are irrational, the compactification of its Bloch variety is an indprovariety, which is a Cantorlike geometric space and it is compatible with the picture of Hofstadter butterfly. On each approximating component the density of states of the electronic model can be expressed in terms of period integrals over Fermi curves, which can be explicitly computed as elliptic integrals or periods of elliptic curves. The above density of states satisfies a PicardFuchs equation, whose solutions are generally given by hypergeometric functions. We use the idea of mirror maps as in mirror symmetry of elliptic curves to derive a qexpansion for the energy level based on the PicardFuchs equation. In addition, formal spectral functions such as the partition function are derived as new period integrals. Secondly, we consider generalized Feynman diagram evaluations of an effective noncommutative field theory of the PonzanoRegge model coupled with matter in loop quantum gravity. We present a parametric representation in a linear kapproximation of the effective field theory derived from a kdeformation of the PonzanoRegge model and define a generalized Kirchhoff polynomial with kcorrection terms. Setting k equal to 1, we verify that the number of points of the corresponding hypersurface of the tetrahedron over finite fields does not fit polynomials with integer coefficients by computer calculations. We then conclude that the hypersurface of the tetrahedron is not polynomially countable, which possibly implies that the hypersurface of the tetrahedron as a motive is not mixed Tate.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5390
 Format
 Thesis
 Title
 Predegree Polynomials of Plane Configurations in Projective Space.
 Creator

Tzigantchev, Dimitre G. (Dimitre Gueorguiev), Aluﬃ, Paolo, Reina, Laura, Aldrovandi, Ettore, Klassen, Eric, Seppälä, Mika, Department of Mathematics, Florida State University
 Abstract/Description

We work over an algebraically closed ground field of characteristic zero. The group of PGL(4) acts naturally on the projective space P^N parameterizing surfaces of a given degree d in P^3. The orbit of a surface under this action is the image of a rational map from P^15 to P^N. The closure of the orbit is a natural and interesting object to study. Its predegree is defined as the degree of the orbit closure multiplied by the degree of the above map restricted to a general P^j , j being the...
Show moreWe work over an algebraically closed ground field of characteristic zero. The group of PGL(4) acts naturally on the projective space P^N parameterizing surfaces of a given degree d in P^3. The orbit of a surface under this action is the image of a rational map from P^15 to P^N. The closure of the orbit is a natural and interesting object to study. Its predegree is defined as the degree of the orbit closure multiplied by the degree of the above map restricted to a general P^j , j being the dimension of the orbit. We find the predegrees and other invariants for all surfaces supported on unions of planes. The information is encoded in the socalled adjusted predegree polynomials, which possess nice multiplicative properties allowing us to easily compute the predegree (polynomials) of various special plane configurations. The predegree has both a combinatorial and geometric significance. The results obtained in this thesis would be a necessary step in the solution of the problem of computing predegrees for all surfaces.
Show less  Date Issued
 2006
 Identifier
 FSU_migr_etd1747
 Format
 Thesis
 Title
 Principal Elements of MixedSign Coxeter Systems.
 Creator

Armstrong, Johnathon Kyle, Hironaka, Eriko, Petersen, Kathleen, Chicken, Eric, Aldrovandi, Ettore, Bellenot, Steven, Van Hoeij, Mark, Department of Mathematics, Florida State...
Show moreArmstrong, Johnathon Kyle, Hironaka, Eriko, Petersen, Kathleen, Chicken, Eric, Aldrovandi, Ettore, Bellenot, Steven, Van Hoeij, Mark, Department of Mathematics, Florida State University
Show less  Abstract/Description

In this thesis we generalize results from classical Coxeter systems to mixedsign Coxeter systems which are denoted by a triple (W,S,B)consisting of a reflection group W, a distinguished set of generators Sfor the group for W, and a bilinear form Bon R n. A generator s i in the set S is defined to negate the ith basis vector of R n and fix the set of vectors v which are orthogonal relative to B. Classical Coxeter theory works in this fashion, here we generalize this notion to encompass both...
Show moreIn this thesis we generalize results from classical Coxeter systems to mixedsign Coxeter systems which are denoted by a triple (W,S,B)consisting of a reflection group W, a distinguished set of generators Sfor the group for W, and a bilinear form Bon R n. A generator s i in the set S is defined to negate the ith basis vector of R n and fix the set of vectors v which are orthogonal relative to B. Classical Coxeter theory works in this fashion, here we generalize this notion to encompass both Coxeter systems in addition to mixedsign Coxeter systems. As in classical Coxeter theory, we show that the bilinear form may be used to compute an element of the reflection group called a principal element. In classical Coxeter groups, the principal elements have been shown to have special properties. The socalled deletion condition is a property of classical Coxeter systems which allows Coxeter groups to have a presentation which only depends on pairwise relationships between generators. Here, we show that mixedsign Coxeter systems do not generally have the deletion condition. We give a correspondence between a graph $\Gamma$ and the reflection system (W,S,B). We refer to the reflection group associated to &Gamma by W (&Gamma). We show an isomorphism of mixedsign Coxeter groups; explicitly if &Gamma is a bipartite mixedsign Coxeter graph and &Gamma is the mixedsign Coxeter graph with all the nodes of &Gamma negated then (W,S,B(&Gamma)) and (W,S,B(&Gamma)) are conjugate reflection systems. Furthermore, we indicate the the bipartite condition is necessary. We show a class of examples; odd cycles with all negative nodes where negating all the nodes gives a reflection system which is not conjugate. Additionally, we show that the spectral radius of mixedsign Coxeter elements are not bounded below by the bipartite eigenvalue of the mixedsign Coxeter system, this is another distinguishing feature of mixedsign Coxeter systems from their classical counterparts and provides an interesting avenue of research to pursue in the future.
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd4697
 Format
 Thesis
 Title
 Solutions of Second Order Recurrence Relations.
 Creator

Levy, Giles, Van Hoeij, Mark, Van Engelen, Robert A., Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

This thesis presents three algorithms each of which returns a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gttransformations). The first and third algorithm are new and the second algorithm is an improvement over prior algorithms for the second order case. The first algorithm `Find 2F1' finds a gttransformation to a recurrence relation satisfied by a hypergeometric series u(n) = hypergeom([a+n, b],[c],z), if such...
Show moreThis thesis presents three algorithms each of which returns a transformation from a base equation to the input using transformations that preserve order and homogeneity (referred to as gttransformations). The first and third algorithm are new and the second algorithm is an improvement over prior algorithms for the second order case. The first algorithm `Find 2F1' finds a gttransformation to a recurrence relation satisfied by a hypergeometric series u(n) = hypergeom([a+n, b],[c],z), if such a transformation exists. The second algorithm `Find Liouvillian' finds a gttransformation to a recurrence relation of the form u(n+2) + b(n)u(n) = 0 for some b(n) in C(n), if such a transformation exists. The third algorithm `Database Solver' takes advantage of a large database of sequences, `The OnLine Encyclopedia of Integer Sequences' maintained by Neil A. J. Sloane at AT&T Labs Research. It employs this database by using the recurrence relations that they satisfy as base equations from which to return a gttransformation, if such a transformation exists.
Show less  Date Issued
 2010
 Identifier
 FSU_migr_etd3099
 Format
 Thesis
 Title
 Solving Linear Differential Equations in Terms of Hypergeometric Functions by ₂Descent.
 Creator

Fang, Tingting, Van Hoeij, Mark, Van Engelen, Robert A., Agashe, Amod, Aldrovandi, Ettore, Aluﬃ, Paolo, Department of Mathematics, Florida State University
 Abstract/Description

Let L be a linear ordinary differential equation with coefficients in C(x). This thesis presents algorithms to solve L in closed form. The key part of this thesis is 2descent method, which is used to reduce L to an equation that is easier to solve. The starting point is an irreducible L, and the goal of 2descent is to decide if L is projectively equivalent to another equation $\tilde{L}$ that is defined over a subfield C(f) of C(x). Although part of the mathematics for 2descent has already...
Show moreLet L be a linear ordinary differential equation with coefficients in C(x). This thesis presents algorithms to solve L in closed form. The key part of this thesis is 2descent method, which is used to reduce L to an equation that is easier to solve. The starting point is an irreducible L, and the goal of 2descent is to decide if L is projectively equivalent to another equation $\tilde{L}$ that is defined over a subfield C(f) of C(x). Although part of the mathematics for 2descent has already been treated before, a complete implementation could not be given because it involved a step for which we do not have a complete implementation. Our key novelty is to give an approach that is fully implementable. We describe and implement the algorithm for order 2, and show by examples that the same also work for higher order. By doing 2descent for L, the number of true singularities drops to at most n/2 + 2 (n is the number of true singularities of L). This provides us ways to solve L in closed form(e.g.in terms of hypergeometric funtions).
Show less  Date Issued
 2012
 Identifier
 FSU_migr_etd5350
 Format
 Thesis
 Title
 Third Order AHypergeometric Functions.
 Creator

Xu, Wen, Hoeij, Mark van, Reina, Laura, Agashe, Amod S. (Amod Sadanand), Aldrovandi, Ettore, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of...
Show moreXu, Wen, Hoeij, Mark van, Reina, Laura, Agashe, Amod S. (Amod Sadanand), Aldrovandi, Ettore, Aluffi, Paolo, Florida State University, College of Arts and Sciences, Department of Mathematics
Show less  Abstract/Description

To solve globally bounded order $3$ linear differential equations with rational function coefficients, this thesis introduces a partial $_3F_2$solver (Section~\ref{3F2 type solution}) and $F_1$solver (Chapter~\ref{F1 solver}), where $_3F_2$ is the hypergeometric function $_3F_2(a_1,a_2,a_3;b_1,b_2\,\,x)$ and $F_1$ is the Appell's $F_1(a,b_1,b_2,c\,\,x,y).$ To investigate the relations among order $3$ multivariate hypergeometric functions, this thesis presents two multivariate tools:...
Show moreTo solve globally bounded order $3$ linear differential equations with rational function coefficients, this thesis introduces a partial $_3F_2$solver (Section~\ref{3F2 type solution}) and $F_1$solver (Chapter~\ref{F1 solver}), where $_3F_2$ is the hypergeometric function $_3F_2(a_1,a_2,a_3;b_1,b_2\,\,x)$ and $F_1$ is the Appell's $F_1(a,b_1,b_2,c\,\,x,y).$ To investigate the relations among order $3$ multivariate hypergeometric functions, this thesis presents two multivariate tools: compute homomorphisms (Algorithm~\ref{hom}) of two $D$modules, where $D$ is a multivariate differential ring, and compute projective homomorphisms (Algorithm~\ref{algo ProjHom}) using the tensor product module and Algorithm~\ref{hom}. As an application, all irreducible order $2$ subsystems from reducible order $3$ systems turn out to come from Gauss hypergeometric function $_2F_1(a,b;c\,\,x)$ (Chapter~\ref{chapter applications}).
Show less  Date Issued
 2017
 Identifier
 FSU_FALL2017_XU_fsu_0071E_14234
 Format
 Thesis