Mathematics Commons^™

A Numerical Method For Studying Thermal Deformation In 3d Double-Layered Thin Films With Imperfect Interfacial Thermal Contact Exposed To Ultrashort-Pulsed Lasers, Runzhou Liu

Doctoral Dissertations

Micro heat transfer induced by Ultrashort-pulsed lasers is an important research topic in mechanical engineering and material science. In order to apply ultrashort-pulsed lasers successfully, studying the thermal deformation in double-layered thin films with imperfect thermal interfacial contact induced by ultrashort-pulsed lasers is important for preventing thermal damage. For the ultrashort-pulsed laser, the thermal damage is different from that caused by the long-pulsed lasers, and ultrafast cracks occur after heating.

This dissertation presents a new finite difference method for investigating the thermal deformation in a 3D gold-chromium thin film with imperfect interfacial thermal contact exposed to ultrashort-pulsed lasers. The method …

A Characterization Of Ramsey Graphs For R(3,4), Nicholas M. Richardson

Doctoral Dissertations

The Ramsey number R(ω, α) is the minimum number n such that every graph G with |V(G)| ≥ n has an induced subgraph that is isomorphic to a complete graph on ω vertices, Kω, or has an independent set of size α, Nα. Graphs having fewer than n vertices that have no induced subgraph isomorphic to K ω or Nα form a class of Ramsey graphs, denoted ℜ(ω, α). This dissertation establishes common structure among several classes of Ramsey graphs and establishes the complete list of ℜ(3, 4).

The process used to …

Applications Of Variational Analysis To A Generalized Heron Problem, Boris S. Mordukhovich, Nguyen Mau Nam, Juan Salinas Jr

Mathematics Research Reports

This paper is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here we propose and study, from both qualitative and numerical viewpoints, the following optimal location problem as well as its further extensions: on a given nonempty subset of a Banach space, find a point such that the sum of the distances from it to n given nonempty subsets of this space is minimal. This is a generalized version of the classical Heron problem: on a given straight line, find a point …

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 …

Practical Rationality, The Disciplinary Obligation, And Authentic Mathematical Work: A Look At Geometry, Deborah Moore-Russo, Michael Weiss

The Mathematics Enthusiast

Grossman and McDonald (2008) recently argued that the research community needs to move its “attention beyond the cognitive demands of teaching … to an expanded view of teaching that focuses on teaching as a practice (p. 185).” Building on the work of Bourdieu (Bourdieu and Wacquent, 1992; Bourdieu, 1985, 1998), Herbst and Chazan (2003, 2006) have written about mathematics teaching as a practice, just as law and medicine are considered practices, in an attempt to better understand the rationality that produces, regulates, and sustains mathematics instruction. This practical rationality is the commonly held system of dispositions or the “feel for …

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

The Mathematics Enthusiast

No abstract provided.

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 …

Base-Free Formulas In The Lattice-Theoretic Study Of Compacta, Paul Bankston

Mathematics, Statistics and Computer Science Faculty Research and Publications

The languages of finitary and infinitary logic over the alphabet of bounded lattices have proven to be of considerable use in the study of compacta. Significant among the sentences of these languages are the ones that are base free, those whose truth is unchanged when we move among the lattice bases of a compactum. In this paper we define syntactically the expansive sentences, and show each of them to be base free. We also show that many well-known properties of compacta may be expressed using expansive sentences; and that any property so expressible is closed under inverse limits and …

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 …

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 …