Mathematics Commons^™

Editorial: “Glocal”, “Glocavores”: Good Gadgetry?, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Mathematical Modeling And Dynamical Analysis Of The Operation Of The Hypothalamus - Pituitary - Thyroid (Hpt) Axis In Autoimmune (Hashimoto's) Thyroiditis, Balamurugan Pandiyan

Dissertations (1934 -)

This thesis is a mathematical modeling study of the operation of the negative feedback control through the hypothalamus-pituitary- thyroid (HPT) axis in autoimmune (Hashimoto's) thyroiditis. Negative feedback control through the HPT axis is a mechanism in which the high levels of thyroid hormone; free thyroxine (FT4) in the blood inhibits the secretion of the pituitary hormone, thyroid stimulating hormone (TSH) into the blood. Similarly, the low levels of free thyroxine (FT4) sensed by the pituitary gland and then it secretes thyroid stimulating hormone (TSH) into the blood. Autoimmune (Hashimoto's) thyroiditis is a disease in which the immune system turns against …

Springer Representations On The Khovanov Springer Varieties, Heather M. Russell, Julianna S. Tymoczko

Mathematics Sciences: Faculty Publications

Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S2)n. We show that if Xn is embedded antipodally in (S2)n then the natural Sn-action on (S2)n induces an Sn-representation on the image of H*(Xn). This representation is the Springer representation. Our construction admits an elementary (and geometrically natural) combinatorial description, which we use to prove that the Springer representation on H*(Xn) is irreducible in each degree. …

Smooth And Peaked Solitons Of The Camassa-Holm Equation And Applications, Darryl Holm, Rossen Ivanov

Articles

The relations between smooth and peaked soliton solutions are reviewed for the Camassa- Holm (CH) shallow water wave equation in one spatial dimension. The canonical Hamiltonian formulation of the CH equation in action-angle variables is expressed for solitons by using the scattering data for its associated isospectral eigenvalue problem, rephrased as a Riemann- Hilbert problem. The momentum map from the action-angle scattering variables T^∗(T^N) to the flow momentum (X^∗) provides the Eulerian representation of the N-soliton solution of CH in terms of the scattering data and squared eigenfunctions of its isospectral eigenvalue problem. The …

Using Performance Assessments To Connect Fractions And Rational Expressions: Noyce Scholars As Mentors To Pre-Service Elementary Teachers, Joy W. Darley, Michelle Cawthorn, Marlynn Griffin, Brian P. Koehler, James Lobue

Mathematical Sciences: Faculty Presentations (1991-2022)

At Georgia Southern University, Noyce scholars are not only being mentored, but they are also serving as mentors to preservice elementary teachers. One topic that proves to be problematic for many students is the conceptual understanding of fractions and rational expressions. Since our Noyce scholars with mathematics degrees will be teaching algebra, it is important that they are fluent with the arithmetic to algebra connection. In addition, it is crucial that these mathematics majors become stakeholders in mathematics education at the elementary school level.

Performance assessments can provide the structure necessary for assisting pre-service elementary teachers in firmly establishing the …

The Time At Which A Lévy Process Creeps, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We show that if a Levy process creeps then, as a function of u, the renewal function V (t, u) of the bivariate ascending ladder process (L⁻¹,H) is absolutely continuous on [0,∞) and left differentiable on (0,∞), and the left derivative at u is proportional to the (improper) distribution function of the time at which the process creeps over level u, where the constant of proportionality is d⁻¹_H, the reciprocal of the (positive) drift of H. This yields the (missing) term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As …

Path Decomposition Of Ruinous Behaviour For A General Lévy Insurance Risk Process, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We analyse the general Levy insurance risk process for Levy measures in the convolution equivalence class S^(alpha), alpha > 0, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level u → ∞, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as u -> ∞, conditional on ruin occurring, under our assumptions. Existing asymptotic results under the S^(alpha) assumption are synthesised and extended, and proofs …

Path Decomposition Of Ruinous Behaviour For A General Lévy Insurance Risk Process, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We analyse the general Levy insurance risk process for Levy measures in the convolution equivalence class

S^(alpha), alpha> 0, via a new kind of path decomposition. This yields a very general functional limit theorem as the initial reserve level u → ∞, and a host of new results for functionals of interest in insurance risk. Particular emphasis is placed on the time to ruin, which is shown to have a proper limiting distribution, as u → ∞, conditional on ruin occurring, under our assumptions. Existing asymptotic results under the S^(alpha)

assumption are synthesised and extended, and …

A Positive Monk Formula In The S1-Equivariant Cohomology Of Type A Peterson Varieties, Megumi Harada, Julianna Tymoczko

Mathematics Sciences: Faculty Publications

Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in type A_n−1, with respect to a natural S¹ -action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of …

The Time At Which A Lévy Process Creeps, Philip S. Griffin, Ross A. Maller

Mathematics - All Scholarship

We show that if a Levy process creeps then, as a function of u, the renewal function V (t, u) of the bivariate ascending ladder process (L⁻¹,H) is absolutely continuous on [0,∞) and left differentiable on (0,∞), and the left derivative at u is proportional to the (improper) distribution function of the time at which the process creeps over level u, where the constant of proportionality is d⁻¹_H , the reciprocal of the (positive) drift of H. This yields the (missing) term due to creeping in the recent quintuple law of Doney and Kyprianou (2006). As …

Income Inequality And Spatial Distribution : Firm Location, Product Quality And Welfare Of Poor., Namrata Gulati Dr.

Doctoral Theses

The key idea explored in this thesis is the following: though being poor in itself is a huge disadvantage, the situation might be influenced considerably by the type of neighbourhood the poor lives in as private establishments like educational institutions, health care facilities or credit institutions take both the location and income mix of people into account while making strategic decisions like whether to enter into the neighbourhood at all, and, upon entry, what price and quality to choose for their products and services. Is staying with the rich a virtue for the poor or a source of resentment? Are …

Polynomial Generalizations Of Two-Variable Ramanujan Type Identities, James Mclaughlin, Andrew V. Sills

Mathematics Faculty Publications

No abstract provided.

Hamilton-Waterloo Problem With Triangle And C9 Factors, David C. Kamin

Dissertations, Master's Theses and Master's Reports - Open

The Hamilton-Waterloo problem and its spouse-avoiding variant for uniform cycle sizes asks if Kv, where v is odd (or Kv - F, if v is even), can be decomposed into 2-factors in which each factor is made either entirely of m-cycles or entirely of n-cycles. This thesis examines the case in which r of the factors are made up of cycles of length 3 and s of the factors are made up of cycles of length 9, for any r and s. We also discuss a constructive solution to the general (m,n) case which fixes r and s.

Nonlocal Conditions For Differential Inclusions In The Space Of Functions Of Bounded Variations, Ravi P. Agarwal, Abdelkader Boucherif

Mathematics and System Engineering Faculty Publications

We discuss the existence of solutions of an abstract differential inclusion, with a right-hand side of bounded variation and subject to a nonlocal initial condition of integral type.

Distinguishability Of Infinite Groups And Graphs, Simon M. Smith, Thomas W. Tucker, Mark E. Watkins

Mathematics - All Scholarship

The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its full automorphism group acting on its vertex set. A connected graph T is said to have connectivity 1 if there exists a vertex alpha in VT such that T \ {alpha} is not connected. For alpha in V , an orbit of the point stabilizer G_alpha is called a suborbit …

A Multi-Frequency Inverse Source Problem For The Helmholtz Equation, Sebastian Ignacio Acosta

Theses and Dissertations

The inverse source problem for the Helmholtz equation is studied. An unknown source is to be identified from the knowledge of its radiated wave. The focus is placed on the effect that multi-frequency data has on establishing uniqueness. In particular, we prove that data obtained from finitely many frequencies is not sufficient. On the other hand, if the frequency varies within an open interval of the positive real line, then the source is determined uniquely. An algorithm is based on an incomplete Fourier transform of the measured data and we establish an error estimate under certain regularity assumptions on the …

Solvability Of A Quadratic Hammerstein Integral Equation In The Class Of Functions Having Limits At Infinity, Ravi P. Agarwal, Józef Banas̈, Kamil Banas̈, Donal O'Regan

Mathematics and System Engineering Faculty Publications

The goal of the paper is to prove that a quadratic Hammerstein integral equation has solutions in the class of real functions denned, bounded, continuous on the real half-axis and having limits at infinity. The main tools used in our investigations are the technique of measures of noncompactness and the Darbo fixed point theorem. We provide an example illustrating our theory. © 2011 Rocky Mountain Mathematics Consortium.

Three-Dimensional Galois Representations And A Conjecture Of Ash, Doud, And Pollack, Vinh Xuan Dang

Theses and Dissertations

In the 1970s and 1980s, Jean-Pierre Serre formulated a conjecture connecting two-dimensional Galois representations and modular forms. The conjecture came to be known as Serre's modularity conjecture. It was recently proved by Khare and Wintenberger in 2008. Serre's conjecture has various important consequences in number theory. Most notably, it played a key role in the proof of Fermat's last theorem. A natural question is, what is the analogue of Serre's conjecture for higher dimensional Galois representations? In 2002, Ash, Doud and Pollack formulated a precise statement for a higher dimensional analogue of Serre's conjecture. They also provided numerous computational examples …

Asymptotic Distributions Of The Overshoot And Undershoots For The Lévy Insurance Risk Process In The Cramér And Convolution Equivalent Cases, Philip S. Griffin, Ross A. Maller, Kees Van Schaik

Mathematics - All Scholarship

Recent models of the insurance risk process use a Levy process to generalise the traditional Cramer-Lundberg compound Poisson model. This paper is concerned with the behaviour of the distributions of the overshoot and undershoots of a high level, for a Levy process which drifts to -infinity and satis es a Cramer or a convolution equivalent condition. We derive these asymptotics under minimal conditions in the Cramer case, and compare them with known results for the convolution equivalent case, drawing attention to the striking and unexpected fact that they become identical when certain parameters tend to equality. Thus, at least regarding …

Non-Commutative Desingularization Of Determinantal Varieties, Ii, Ragar-Olaf Buchweitz, Graham J. Leuschke, Michel Van Den Bergh

Mathematics - All Scholarship

In our paper "Non-commutative desingularization of determinantal varieties, I" we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction we asserted that the results could be generalized to determinantal varieties defined by non-maximal minors, at least in characteristic zero. In this paper we prove the existence of non-commutative resolutions in the general case in a manner which is still characteristic free. The explicit description of the resolution by generators and relations is deferred to a later paper. As an application of our results we prove that there is a fully faithful …