Mathematics Commons^™

Quantum And Post-Newtonian Effects In The Anomalistic Period And The Mean Motion Of Celestial Bodies, Ioannis Haranas, Omiros Ragos, Ioannis Gkigkitzis, Ilias S. Kotsireas

Physics and Computer Science Faculty Publications

We study the motion of a secondary celestial body under the influence of the corrected gravitational force of a primary. We study the effect of quantum and relativistic corrections to the gravitational potential of a primary body acting on the orbiting body. More specifically, two equations are derived to approximate the perigee/perihelion/periastron time rate of change and its total variation over one revolution (i.e., the difference between the anomalistic period and the Keplerian period) under the influence of the quantum as well as post-Newtonian accelerations. Numerical results have been obtained for the artificial Earth satellite Molnya, Mercury, and, finally, the …

Unit Origami: Star-Building On Deltahedra, Heidi Burgiel

Mathematics Faculty Publications

This workshop provides instructions for folding the star-building unit – a modification of the Sonobe module for unit origami. Geometric questions naturally arise during this process, ranging in difficulty from middle school to graduate levels. Participants will learn to fold and assemble star-building units, then explore the structure of the eight strictly convex deltahedra.

Letters To Joel, David J. Stucki, Joel M. Stucki

Mathematics Faculty Scholarship

A professor and his students answer his brother’s question: Is mathematics art?

Being Smart About Parts, Nathaniel Stevens, S H. Steiner, R J. Mackay

Mathematics

When conducting a measurement system assessment study, practitioners want as precise an estimate of the true repeatability and reproducibility (R&R) as possible so they can correctly decide whether the measurement system is acceptable. They are, however, faced with cost and time constraints that restrict the number of parts and repeated measurements that can be used in the study. By incorporating freely available production measurements (baseline data), they can reduce the number of parts in the study to two or three and still obtain a better estimate of the R&R than they would have otherwise. To avoid bias, they must ensure …

Assessing Agreement Between Two Measurement Systems: An Alternative To The Limits Of Agreement Approach, Nathaniel Stevens, S H. Steiner, R J. Mackay

Mathematics

The comparison of two measurement systems is important in medical and other contexts. A common goal is to decide if a new measurement system agrees suitably with an existing one, and hence whether the two can be used interchangeably. Various methods for assessing interchangeability are available, the most popular being the limits of agreement approach due to Bland and Altman. In this article, we review the challenges of this technique and propose a model-based framework for comparing measurement systems that overcomes those challenges. The proposal is based on a simple metric, the probability of agreement, and a corresponding plot which …

Flipped Calculus: A Study Of Student Performance And Perceptions, Lori B. Ziegelmeir, Chad M. Topaz

Faculty Publications

No abstract provided.

Dynamics Of Planar Systems That Model Stage-Structured Populations, N. Lazaryan, Hassan Sedaghat

Mathematics and Applied Mathematics Publications

We study a general discrete planar system for modeling stage-structured populations. Our results include conditions for the global convergence of orbits to zero (extinction) when the parameters (vital rates) are time and density dependent. When the parameters are periodic we obtain weaker conditions for extinction. We also study a rational special case of the system for Beverton-Holt type interactions and show that the persistence equilibrium (in the positive quadrant) may be globally attracting even in the presence of interstage competition. However, we determine that with a sufficiently high level of competition, the persistence equilibrium becomes unstable (a saddle point) and …

Global Attractor Of Solutions Of A Rational System In The Plane, Miron B. Bekker, Martin Bohner, Hristo D. Voulov

Mathematics and Statistics Faculty Research & Creative Works

We consider a two-dimensional autonomous system of rational difference equations with three positive parameters. It was conjectured that every positive solution of this system converges to a finite limit. Here we confirm this conjecture, subject to an additional assumption.

2015 Oklahoma Research Day Full Program, Northeastern State University

Oklahoma Research Day Abstracts

This document contains all abstracts from the 2015 Oklahoma Research Day held at Northeastern State University.

Periodic And Chaotic Orbits Of A Discrete Rational System, N. Lazaryan, Hassan Sedaghat

Mathematics and Applied Mathematics Publications

We study a rational planar system consisting of one linear-affine and one linear-fractional difference equation. If all of the system’s parameters are positive (so that the positive quadrant is invariant and the system is continuous), then we show that the unique fixed point of the system in the positive quadrant cannot be repelling and the system does not have a snap-back repeller. By folding the system into a second-order equation, we find special cases of the system with some negative parameter values that do exhibit chaos in the sense of Li and Yorke within the positive quadrant of the plane.

Multiplication Rules For Schur And Quasisymmetric Schur Functions, Jennifer Anderson

Theses, Dissertations and Capstones

An important problem in algebraic combinatorics is finding expansions of products of symmetric functions as sums of symmetric functions. Schur functions form a well-known basis for the ring of symmetric functions. The Littlewood-Richardson rule was introduced to expand the product of two Schur functions as a positive sum of Schur functions. Remmel and Whitney introduced an algorithmic way to find the coefficients of Schur functions appearing in the expansion. Haglund et al. introduced quasisymmetric Schur functions as a refinement of Schur functions. For quasisymmetric Schur functions, the Littlewood-Richardson rule was introduced to expand the product of a Schur and quasisymmetric …

Qualitative Theory Of Functional Differential And Integral Equations, Muhammad Islam, Cemil Tunc, Mouffak Benchohra, Bingwen Lui, Samir H. Saker

Mathematics Faculty Publications

Functional differential equations arise in many areas of science and technology: whenever a deterministic relationship involving some varying quantities and their rates of change in space and/or time (expressed as derivatives or differences) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. In some cases, this differential equation (called an equation of motion) may be solved explicitly. In fact, differential equations play an important role in modelling virtually every physical, technical, biological, ecological, and epidemiological process, from celestial motion, to bridge design, …

Uniform Convergence On A Bakhvalov-Type Mesh Using The Preconditioningapproach. Technical Report, Thái Ahn Nhan, Relja Vulanović

Mathematics and Computer Science

The linear singularly perturbed convection-diffusion problem in one dimension is considered and its discretization on a Bakhvalov-type mesh is analyzed. The preconditioning technique is used to obtain the pointwise convergence uniform in the perturbation parameter.

Division Polynomials With Galois Group Su3(3).2 = G2(2), David P. Roberts

Mathematics Publications

We use a rigidity argument to prove the existence of two related degree twenty-eight covers of the projective plane with Galois group SU₃(3).2 ∼= G₂(2). Constructing corresponding two-parameter polynomials directly from the defining group-theoretic data seems beyond feasibility. Instead we provide two independent constructions of these polynomials, one from 3-division points on covers of the projective line studied by Deligne and Mostow, and one from 2-division points of genus three curves studied by Shioda. We explain how one of the covers also arises as a 2-division polynomial for a family of G₂ motives in the …

Analytic Structure Of The S-Matrix For Singular Quantum Mechanics, Horacio E. Camblong, Luis N. Epele, Huner Fanchiotti, Carlos A. García Canal

Physics and Astronomy

The analytic structure of the S-matrix of singular quantum mechanics is examined within a multichannel framework, with primary focus on its dependence with respect to a parameter (Ω) that determines the boundary conditions. Specifically, a characterization is given in terms of salient mathematical and physical properties governing its behavior. These properties involve unitarity and associated current-conserving Wronskian relations, time-reversal invariance, and Blaschke factorization. The approach leads to an interpretation of effective nonunitary solutions in singular quantum mechanics and their determination from the unitary family.

Measuring Longitudinal Myelin Water Fraction In New Multiple Sclerosis Lesions, Wendy S. Vargas, Elizabeth Monohan, Sneha Pandya, Ashish Raj, Timothy Vartanian, Thanh D. Nguyen, Sandra M. Hurtado Rua, Susan A. Gauthier

Mathematics and Statistics Faculty Publications

Objectives: Investigating the potential of myelin repair strategies in multiple sclerosis (MS) requires an understanding of myelin dynamics during lesion evolution. The objective of this study is to longitudinally measure myelin water fraction (MWF), an MRI biomarker of myelin, in new MS lesions and to identify factors that influence their subsequent myelin content. Methods:Twenty-three MS patients were scanned with whole-brain Fast Acquisition with Spiral Trajectory and T2prep (FAST-T2) MWF mapping at baseline and median follow-up of 6 months. Eleven healthy controls (HC) confirmed the reproducibility of FAST-T2 in white matter regions of interests (ROIs) similar to a lesion size. A …

Topological Complexity In Protein Structures, Erica Flapan, Gabriella Heller '14

Pomona Faculty Publications and Research

For DNA molecules, topological complexity occurs exclusively as the result of knotting or linking of the polynucleotide backbone. By contrast, while a few knots and links have been found within the polypeptide backbones of some protein structures, non-planarity can also result from the connectivity between a polypeptide chain and inter- and intra-chain linking via cofactors and disulfide bonds. In this article, we survey the known types of knots, links, and non-planar graphs in protein structures with and without including such bonds and cofactors. Then we present new examples of protein structures containing Möbius ladders and other non-planar graphs as a …

Height Bounds On Zeros Of Quadratic Forms Over Q-Bar, Lenny Fukshansky

CMC Faculty Publications and Research

In this paper we establish three results on small-height zeros of quadratic polynomials over Q. For a single quadratic form in N ≥ 2 variables on a subspace of Q N , we prove an upper bound on the height of a smallest nontrivial zero outside of an algebraic set under the assumption that such a zero exists. For a system of k quadratic forms on an L-dimensional subspace of Q N , N ≥ L ≥ k(k+1) 2 + 1, we prove existence of a nontrivial simultaneous small-height zero. For a system of one or two inhomogeneous quadratic and …

Spherical 2-Designs And Lattices From Abelian Groups, Albrecht Böttcher, Lenny Fukshansky, Stephan Ramon Garcia, Hiren Maharaj

CMC Faculty Publications and Research

We consider lattices generated by finite Abelian groups. The main result says that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame.

Simultaneous Modelling Of Initial Conditions And Time Heterogeneity In Dynamic Networks: An Application To Foreign Direct Investments, Johan Koskinen, Alberto Caimo, Alessandro Lomi

Articles

In dynamic networks, the presence of ties are subject both to endogenous network dependencies and spatial dependencies. Current statistical models for change over time are typically defined relative to some initial condition, thus skirting the issue of where the first network came from. Additionally, while these longitudinal network models may explain the dynamics of change in the network over time, they do not explain the change in those dynamics. We propose an extension to the longitudinal exponential random graph model that allows for simultaneous inference of the changes over time and the initial conditions, as well as relaxing assumptions of …

Investigating The Engagement Of Mature Students With Mathematics Learning Support, Cormac Breen, Michael Carr, Mark Prendergast

Articles

The Maths Learning Support Centre (MLSC) in the Dublin Institute of Technology (DIT) provides free mathematical support to all DIT students. This support is primarily delivered through a drop-in service, where students can receive one-to-one tuition, without an appointment, in any area of mathematics. In the first semester of the 2013/14 academic year, a significant proportion of students that availed of this drop-in service were mature students enrolled in Engineering programmes. This is of particular interest as mature students constitute a relatively small proportion of the total student body, motivating a deeper study of the reasons for the high levels …

Vacuum Polarization On The Brane, Cormac Breen, Matthew Hewitt,, Elizabeth Winstanley, Adrian Ottewill

Articles

We compute the renormalized expectation value of the square of a massless, conformally coupled, quantum scalar field on the brane of a higher-dimensional black hole. Working in the AADD braneworld scenario, the extra dimensions are flat and we assume that the compactification radius is large compared with the size of the black hole. The four-dimensional on-brane metric corresponds to a slice through a higher-dimensional Schwarzschild-Tangherlini black hole geometry and depends on the number of bulk space-time dimensions. The quantum scalar field is in a thermal state at the Hawking temperature. An exact, closed-form expression is derived for the renormalized expectation …

Bounded, Asymptotically Stable, And L^1 Solutions Of Caputo Fractional Differential Equations, Muhammad Islam

Mathematics Faculty Publications

The existence of bounded solutions, asymptotically stable solutions, and L¹ solutions of a Caputo fractional differential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular and hence the standard techniques that are normally applied on Volterra integral equations do not apply here. This hurdle is overcomed using a resolvent equation and then applying some known properties of the resolvent. In the analysis Schauder's fixed point theorem and Liapunov's method have been employed. …

Maximizing The Impact Of Digital Supports In Mathematics Learning Support In Higher Education – An Overview Of The 9th Annual Imlsn Workshop, Cormac Breen, Anthony Cronin

Articles

In this article we give a short description of the 9th Annual Workshop of the Irish Mathematics Learning Support Network (IMLSN). The workshop theme was ‘Maximizing the impact of digital supports in Mathematics Learning Support in Higher Education’. We briefly describe the Irish Mathematics Learning Support Network (IMLSN) and outline the factors that motivated this workshop theme. We will also discuss the presentations, some of the issues that were raised during the workshop and we close with some brief conclusions on this very successful event.

Efficient Computational Strategies For Doubly Intractable Problems With Applications To Bayesian Social Networks, Alberto Caimo, Antonietta Mira

Articles

Powerful ideas recently appeared in the literature are adjusted and combined to design improved samplers for doubly intractable target distributions with a focus on Bayesian exponential random graph models. Different forms of adaptive Metropolis–Hastings proposals (vertical, horizontal and rectangular) are tested and merged with the delayed rejection (DR) strategy with the aim of reducing the variance of the resulting Markov chain Monte Carlo estimators for a given computational time. The DR is modified in order to integrate it within the approximate exchange algorithm (AEA) to avoid the computation of intractable normalising constant that appears in exponential random graph models. This …

Permutation Invariant Lattices, Lenny Fukshansky, Stephan Ramon Garcia, Xun Sun

Pomona Faculty Publications and Research

We say that a Euclidean lattice in Rⁿ is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group S_n, i.e., if the lattice is closed under the action of some non-identity elements of S_n. Given a fixed element T E S_n, we study properties of the set of all lattices closed under the action of T: we call such lattices T-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio in [7,8], which we previously studied in [1]. Continuing our investigation, we discuss some basic properties of …

Derivatives Pricing With Accelerated Trinomial Trees, Conall O'Sullivan, Stephen O'Sullivan

Articles

Accelerated Trinomial Trees (ATTs) are a derivatives pricing lattice method that circumvent the restrictive time step condition inherent in standard trinomial trees and explicit finite difference methods (FDMs) in which the time step must scale with the square of the spatial step. ATTs consist of L uniform supersteps each of which contains an inner lattice/trinomial tree with N non-uniform subtime steps. Similarly to implicit FDMs, the size of the superstep in ATTs, a function of N, are constrained primarily by accuracy demands. ATTs can price options up to N times faster than standard trinomial trees (explicit FDMs). ATTs can be …

Polynomials Occuring In Generating Function Identities For B-Ary Partitions, David Dakota Blair

Graduate Student Publications and Research

Let p_b(n) be the number of integer partitions of n whose parts are powers of b. For each m there is a generating function identity:

f_m(b,q)\sum_{n} p_b(n) q^n = (1-q)^m \sum_{n} p_b(b^m n q)q^n

where n ranges over all integer values. The proof of this identity appears in the doctoral thesis of the author. For more information see http://dakota.tensen.net/2015/rp/.

This dataset is a JSON object with keys m from 1 to 23 whose values are f_m(b,q).

Why Right-Brain Cultures Are More Flexible: A Possible Explanation Of Yu. Manin's Observation, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Yuri Manin, a renowned mathematician, observed that it is much easier for a person raised in a right-brain culture to adjust to the left-brain environment than vice versa. In this paper, we provide a possible explanation for this phenomenon.

When An Idea Comes, Write It Down Right Away: Mathematical Justification Of Vladimir Smirnov's Advice, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Among several advices to students, Vladimir Smirnov, a renowned Russian mathematician, suggested that when an idea comes, it is better to write it down right away. In this paper, we provide a quantitative justification for this advice.