Mathematics Commons^™

Large And Small Data Blow-Up Solutions In The Trojan Y Chromosome Model, Rana D. Parshad, Matthew Beauregard, Eric M. Takyi, Thomas Griffin, Landrey Bobo

Faculty Publications

The Trojan Y Chromosome Strategy (TYC) is an extremely well investigated biological control method for controlling invasive populations with an XX-XY sex determinism. In [35, 36] various dynamical properties of the system are analyzed, including well posedness, boundedness of solutions, and conditions for extinction or recovery. These results are derived under the assumption of positive solutions. In the current manuscript, we show that if the introduction rate of trojan fish is zero, under certain large data assumptions, negative solutions are possible for the male population, which in turn can lead to finite time blow-up in the female and male populations. …

Quenching Estimates For A Non-Newtonian Filtration Equation With Singular Boundary Conditions, Matthew Beauregard, Burhan Selcuk

Faculty Publications

In this paper, the quenching behavior of the non-Newtonian filtration equation (φ(u))t = (|ux| r−2 ux)x with singular boundary conditions, ux (0, t) = u −p (0, t), ux (a, t) = (1 − u(a, t))−q is investigated. Various conditions on the initial condition are shown to guarantee quenching at either the left or right boundary. Theoretical quenching rates and lower bounds to the quenching time are determined when φ(u) = u and r = 2. Numerical experiments are provided to illustrate and provide additional validation of the theoretical estimates to the quenching rates and times.

Graph Pebbling Algorithms And Lemke Graphs, Charles A. Cusack, Aaron Green, Airat Bekmetjev, Mark Powers

Faculty Publications

Given a simple, connected graph, a pebbling configuration (or just configuration) is a function from its vertex set to the nonnegative integers. A pebbling move between adjacent vertices removes two pebbles from one vertex and adds one pebble to the other. A vertex r is said to be reachable from a configuration if there exists a sequence of pebbling moves that places at least one pebble on r. A configuration is solvable if every vertex is reachable. The pebbling number π(G) of a graph G is the minimum integer such that every configuration of size π(G) on G …

Finding The Symmetry Group Of An Lp With Equality Constraints And Its Application To Classifying Orthogonal Arrays, Andrew J. Geyer, Dursun A. Bulutoglu, Kenneth J. Ryan

Faculty Publications

Excerpt: For a given linear program (LP) a permutation of its variables that sends feasible points to feasible points and preserves the objective function value of each of its feasible points is a symmetry of the LP. The set of all symmetries of an LP, denoted by G^LP, is the symmetry group of the LP. Margot (2010) described a method for computing a subgroup of the symmetry group G^LP of an LP. This method computes G^LP when the LP has only non-redundant inequalities and its feasible set satisfies no equality constraints.

Deciphering The Transport Of Elastic Filaments By Antagonistic Motor Proteins, Stephanie Portet, Cecil Leduc, Sandrine Etienne-Manneville, J. C. Dallon

Faculty Publications

Intermediate filaments are long elastic fibres that are transported by microtubule-associated motor proteins kinesin and dynein inside the cell. How elastic filaments are efficiently transported by antagonistic motors is not well understood and difficult to measure with current experimental techniques. Adapting the tug-of-war paradigm for vesicle-like cargos, we develop a mathematical model to describe the motion of an elastic filament punctually bound to antagonistic motors. As observed in cells, up to 3 modes of transport are obtained; dynein-driven retrograde, kinesin-driven anterograde fast motions and a slow motion. Motor properties and initial conditions that depend on intracellular context, regulate the transport …

Atmospheric Propagation Of High Energy Lasers: Thermal Blooming Simulation, Jonathan Gustafsson, Benjamin F. Akers, Jonah A. Reeger, Sivaguru S. Sritharan

Faculty Publications

High Energy Laser (HEL) propagation through turbulent atmosphere is examined via numerical simulation. The beam propagation is modeled with the paraxial equation, which in turn is written as a system of equations for a quantum fluid, via the Madelung transform. A finite volume solver is applied to the quantum fluid equations, which supports sharp gradients in beam intensity. The atmosphere is modeled via a coupled advection-diffusion equation whose initial data have Kolmogorov spectrum. In this model the combined effects of thermal blooming, beam slewing, and deep turbulence are simulated.

Stochastic Modeling Reveals How Motor Protein And Filament Properties Affect Intermediate Filament Transport, J. C. Dallon, Cecil Leduc, Sandrine Etienne-Manneville, Stephanie Portet

Faculty Publications

Intermediate filaments are a key component of the cytoskeleton. Their trans- port along microtubules plays an essential role in the control of the shape and structural organization of cells. To identify the key parameters responsible for the control of intermediate filament transport, we generated a model of elastic filament transport by microtubule-associated dynein and kinesin. The model is also applicable to the transport of any elastically-coupled cargoes. We inves- tigate the effect of filament properties such as number of motor binding sites, length, and elasticity on motion of filaments. Additionally, we consider the ef- fect of motor properties, i.e. off …

The Knill Graph Dimension From The Minimum Clique Cover, Kassahun Betre, Evatt Salinger

Faculty Publications

In this paper we prove that the recursive (Knill) dimension of the join of two graphs has a simple formula in terms of the dimensions of the component graphs: dim(G1+G2)=1+dimG1+dimG2. We use this formula to derive an expression for the Knill dimension of a graph from its minimum clique cover. A corollary of the formula is that a graph made of the arbitrary union of complete graphs KN of the same order N will have dimension N−1. We finish by finding lower and upper bounds on the Knill dimension of a graph in terms of its clique number.

Two-Pebbling And Odd-Two-Pebbling Are Not Equivalent, Charles A. Cusack, Airat Bekmetjev, Mark Powers

Faculty Publications

Let G be a connected graph. A configuration of pebbles assigns a nonnegative integer number of pebbles to each vertex of G. A move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. A configuration is solvable if any vertex can get at least one pebble through a sequence of moves. The pebbling number of G, denoted π(G), is the smallest integer such that any configuration of π(G) pebbles on G is solvable. A graph has the two-pebbling property if after placing more than 2π(G) -- q pebbles on G …

Diagonalization Of 1-D Differential Operators With Piecewise Constant Coefficients Using The Uncertainty Principle, Sarah D. Long, Somayyeh Sheikholeslami, James V. Lambers, Carley Walker

Faculty Publications

A highly accurate and efficient numerical method is presented for computing the solution of a 1-D time-dependent partial differential equation in which the spatial differential operator features a piecewise constant coefficient defined on n" role="presentation"> pieces, in either self-adjoint and non-self-adjoint form, on a finite interval with periodic boundary conditions. The Uncertainty Principle is used to estimate the eigenvalues of the operator. Then, these estimates are used to construct a basis of eigenfunctions for use with a spectral method. The solution is presented as a truncated eigenfunction expansion, where each eigenfunction is a wave function that changes frequencies at the …

A Variable Nonlinear Splitting Algorithm For Reaction Diffusion Systems With Self- And Cross-Diffusion, Matthew Beauregard, Joshua L. Padgett

Faculty Publications

Self- and cross-diffusion are important nonlinear spatial derivative terms that are included into biological models of predator-prey interactions. Self-diffusion models overcrowding effects, while cross-diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting method is presented that directly incorporates both self- and cross-diffusion into a computational efficient design. The numerical analysis guarantees the accuracy and demonstrates appropriate criteria for stability. Numerical experiments display its efficiency and accuracy

Case–Control Research Study Of Auto-Brewery Syndrome, Barbara Jean Cordell, Anup Kanodia, Gregory K. Miller

Faculty Publications

Background: Auto-brewery syndrome (ABS), also known as Gut Fermentation Syndrome and Endogenous Ethanol Fermentation, is afflicting people worldwide, but little is known about ABS patients’ demographics, health history, lifestyle factors, and diet.

Method: We conducted a broad-based case–control survey study on 52 patients known to have a diagnosis of ABS and their household members. The research compares the symptomatic group (N ¼ 28) to the asymptomatic group (N ¼ 18) regarding lifestyle and health, diet, and medical history.

Results: With a response rate of 88% and using rank-sum tests, the data demonstrate that patients with ABS have …

A Limiting Process To Invert The Gauss-Radon Transform, Jeremy Becnel

Faculty Publications

In this work we extend the finite dimensional Radon transform [23] to the Gaussian measure. We develop an inversion formula for this GaussRadon transform by way of Fourier inversion formula. We then proceed to extend these results to the infinite dimensional setting.

Recovering A Random Variable From Conditional Expectations Using Reconstruction Algorithms For The Gauss Radon Transform, Jeremy Becnel, Daniel Riser-Espinoza

Faculty Publications

The Radon transform maps a function on n-dimensional Euclidean space onto its integral over a hyperplane. The fields of modern computerized tomography and medical imaging are fundamentally based on the Radon transform and the computer implementation of the inversion, or reconstruction, techniques of the Radon transform. In this work we use the Radon transform with a Gaussian measure to recover random variables from their conditional expectations. We derive reconstruction algorithms for random variables of unbounded support from samples of conditional expectations and discuss the error inherent in each algorithm.

Equiangular Tight Frames From Group Divisible Designs, Matthew C. Fickus, John Jasper

Faculty Publications

An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem in many cases. In this paper, we observe that many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known …

A Maximum Entropy Method For Solving The Boundary Value Problem Of Second Order Ordinary Differential Equations, Congming Jin, Jiu Ding

Faculty Publications

We propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations, which has a non-negative solution. The method applies the maximum entropy principle to approximating the solution numerically. The theoretical analysis and numerical examples show that our method is convergent.

A Deterministic-Stochastic Method For Computing The Boltzmann Collision Integral In O(Mn) Operations, Alexander Alekseenko, Truong Nguyen, Aihua W. Wood

Faculty Publications

We developed and implemented a numerical algorithm for evaluating the Boltzmann collision integral with O(MN) operations, where N is the number of the discrete velocity points and M < N. At the base of the algorithm are nodal-discontinuous Galerkin discretizations of the collision operator on uniform grids and a bilinear convolution form of the Galerkin projection of the collision operator. Efficiency of the algorithm is achieved by applying singular value decomposition compression of the discrete collision kernel and by approximating the kinetic solution by a sum of Maxwellian streams using a stochastic likelihood maximization algorithm. Accuracy of the method is …

Evaluating Annual Fixed Wing Maintenance Costs, Kirsten Bunecke, Edward D. White, Jonathan D. Ritschel, Brett A. Bush

Faculty Publications

This article serves two purposes: first, to empirically model the annual percentage increase of operations and support (O&S) costs for fixed wing aircraft; and second, to place into the archival realm another reference for other researchers to consider when investigating other O&S topics. For this study of 21 different airframes grouped together at the Mission- Design Series (MDS), maintenance subcategories 3.1 Consumable Materials and Repair Parts, 3.2 Depot Level Repairables (DLR), and 3.4 Depot Maintenance accounted for an average 91.5% of annual organic maintenance costs. For contractor-maintained MDS, subcategory 3.7, Contractor Logistics Support (CLS) accounted for 96.6% of annual maintenance …

An Extension Of The Method Of Brackets: Part 1, Ivan Gonzalez, Karen T. Kohl, Lin Jiu, Victor H. Moll

Faculty Publications

The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients an have meromorphic representations for n 2 C, but might vanish or blow up when n 2 N. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.

On The Strong Chromatic Index Of Sparse Graphs, Philip Deorsey, Jennifer Diemunsch, Michael Ferrara, Nathan Graber, Stephen Hartke, Sogol Jahanbekam, Bernard Lidicky, Luke Nelsen, Derrick Stolee, Eric Sullivan

Faculty Publications

The strong chromatic index of a graph G, denoted χ′s(G), is the least number of colors needed to edge-color G so that edges at distance at most two receive distinct colors. The strong list chromatic index, denoted χ′s,ℓ(G), is the least integer k such that if arbitrary lists of size k are assigned to each edge then G can be edge-colored from those lists where edges at distance at most two receive distinct colors.We use the discharging method, the Combinatorial Nullstellensatz, and computation to show that if G is a subcubic planar graph with girth(G)≥41 then χ′s,ℓ(G)≤5, answering a question …

On The Adjacency Spectra Of Hypertrees, Gregory J. Clark, Joshua N. Cooper

Faculty Publications

We show that λ is an eigenvalue of a k-uniform hypertree (k ≥ 3) if and only if it is a root of a particular matching polynomial for a connected induced subtree. We then use this to provide a spectral characterization for power hypertrees. Notably, the situation is quite different from that of ordinary trees, i.e., 2-uniform trees. We conclude by presenting an example (an 11 vertex, 3-uniform non-power hypertree) illustrating these phenomena.

The Method Of Particular Solutions Using Trigonometric Basis Functions, Zhaolu Tian, Xinxiang Li, C.M. Fan, Ching-Shyang Chen

Faculty Publications

In this paper, the method of particular solutions (MPS) using trigonometric functions as the basis functions is proposed to solve two-dimensional elliptic partial differential equations. The inhomogeneous term of the governing equation is approximated by Fourier series and the closed-form particular solutions of trigonometric functions are derived using the method of undetermined coefficients. Once the particular solutions for the trigonometric basis functions are derived, the standard MPS can be applied for solving partial differential equations. In comparing with the use of radial basis functions and polynomials in the MPS, our proposed approach provides another simple approach to effectively solving two-dimensional …

Identifying Prevalent Mathematical Pathways To Engineering In South Carolina, Eliza Gallagher, Christy Brown, D. Andrew Brown, Kristin Kelly Frady, Patrick Bass, Michael A. Matthews, Thomas T. Peters, Robert J. Rabb, Ikhalfani Solan, Ronald W. Welch, Anand K. Gramopadhye

Faculty Publications

National data indicate that initial mathematics course placement in college is a strong predictor of persistence to degree in engineering, with students placed in calculus persisting at nearly twice the rate of those placed below calculus. Within the state of South Carolina, approximately 95% of engineering-intending students who initially place below calculus are from in-state. In order to make systemic change, we are first analyzing system-wide data to identify prevalent educational pathways within the state, and the mathematical milestones along those pathways taken by students in engineering and engineering-related fields. This paper reports preliminary analysis of that data to understand …

Additive List Coloring Of Planar Graphs With Given Girth, Axel Brandt, Sogol Jahanbekam, Jennifer Diemunsch

Faculty Publications

An additive coloring of a graph G is a labeling of the vertices of G from {1,2,...,k} such that two adjacent vertices have distinct sums of labels on their neighbors. The least integer k for which a graph G has an additive coloring is called the additive coloring number of G, denoted \luck(G). Additive coloring is also studied under the names lucky labeling and open distinguishing. In this paper, we improve the current bounds on the additive coloring number for particular classes of graphs by proving results for a list version of additive coloring. We apply the discharging method and …

Numerical Methods For A Two-Species Competition-Diffusion Model With Free Boundaries, Shuang Liu, Xinfeng Liu

Faculty Publications

The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population and with competition of two species. To solve these systems numerically, new numerical challenges arise from the competition of two species due to the interaction of their free boundaries. On the one hand, extremely small time steps are usually needed due to the stiffness of the system. On the other hand, it is always difficult to efficiently and accurately handle the moving boundaries especially with competition of two species. To overcome these numerical difficulties, we introduce …

Particular Solutions Of Products Of Helmholtz-Type Equations Using The Matern Function, Xiao-Yan Liu, Ching-Shyang Chen, Wen Li, Ming Li

Faculty Publications

We derive closed-form particular solutions for Helmholtz-type partial differential equations. These are derived explicitly using the Matern basis functions. The derivation of such particular solutions is further extended to the cases of products of Helmholtz-type operators in two and three dimensions. The main idea of the paper is to link the derivation of the particular solutions to the known fundamental solutions of certain differential operators. The newly derived particular solutions are used, in the context of the method of particular solutions, to solve boundary value problems governed by a certain class of products of Helmholtz-type equations. The leave-one-out cross validation …

Modeling Of First-Order Photobleaching Kinetics Using Krylov Subspace Spectral Methods, Somayyeh Sheikholeslami, James V. Lambers

Faculty Publications

We solve the first order 2-D reaction–diffusion equations which describe binding-diffusion kinetics using the photobleaching scanning profile of a confocal laser scanning microscope, approximated by a Gaussian laser profile. We show how to solve the first-order photobleaching kinetics partial differential equations (PDEs) using a time-stepping method known as a Krylov subspace spectral (KSS) method. KSS methods are explicit methods for solving time-dependent variable-coefficient partial differential equations. They approximate Fourier coefficients of the solution using Gaussian quadrature rules in the spectral domain. In this paper, we show how a KSS method can be used to obtain not only an approximate numerical …

Weak Dynamic Coloring Of Planar Graphs, Caroline Accurso, Vitaliy Chernyshov, Leaha Hand, Sogol Jahanbekam, Paul Wenger

Faculty Publications

The k-weak-dynamic number of a graph G is the smallest number of colors we need to color the vertices of G in such a way that each vertex v of degree d(v) sees at least min{k, d(v)} colors on its neighborhood. We use reducible configurations and list coloring of graphs to prove that all planar graphs have 3-weak-dynamic number at most 6.

Improved Kansa Rbf Method For The Solution Of Nonlinear Boundary Value Problems, Malgorzata A. Jankowska, Andreas Karageorghis, Ching-Shyang Chen

Faculty Publications

We apply the Kansa–radial basis function (RBF) collocation method to two-dimensional nonlinear boundary value problems. In it, the solution is approximated by a linear combination of RBFs and the governing equation and boundary conditions are satisfied in a collocation sense at interior and boundary points, respectively. The nonlinear system of equations resulting from the Kansa–RBF discretization for the unknown coefficients in the RBF approximation is solved by directly applying a standard nonlinear solver. In a natural way, the value of the shape parameter in the RBFs employed in the approximation may be included in the unknowns to be determined. The …

A Brief Introduction To Equi-Chordal And Equi-Isoclinic Tight Fusion Frames, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Cody E. Watson

Faculty Publications

Equi-chordal and equi-isoclinic tight fusion frames (ECTFFs and EITFFs) are both types of optimal packings of subspaces in Euclidean spaces. In the special case where these subspaces are one-dimensional, ECTFFs and EITFFs both correspond to types of optimal packings of lines known as equiangular tight frames. In this brief note, we review some of the fundamental ideas and results concerning ECTFFs and EITFFs.