Physical Sciences and Mathematics Commons^™

The Regular Excluded Minors For Signed-Graphic Matroids, Hongxun Qin, Dan Slilaty, Xiangqian Zhou

Mathematics and Statistics Faculty Publications

We show that the complete list of regular excluded minors for the class of signed-graphic matroids is M*(G₁),...,M*(G₂₉),R₁₅,R₁₆. Here G₁,...,G₂₉ are the vertically 2-connected excluded minors for the class of projective-planar graphs and R₁₅ and R₁₆ are two regular matroids that we will define in the article.

Leading Students To Investigate Diffusion As A Model Of Brine Shrimp Movement, Brynja R. Kohler, Rebecca J. Swank, James W. Haefner, James A. Powell

Mathematics and Statistics Faculty Publications

Integrating experimental biology laboratory exercises with mathematical modeling can be an effective tool to enhance mathematical relevance for biologists and to emphasize biological realism for mathematicians. This paper describes a lab project de-signed for and tested in an undergraduate biomathematics course. In the lab, students follow and track the paths of individual brine shrimp confined in shallow salt water in a Petri dish. Students investigate the question, “Is the movement well characterized as a 2-dimensional random walk?” Through open, but directed discussions, students derive the corresponding partial differential equation, gain an understanding of the solution behavior, and model brine shrimp …

A Sensitivity Matrix Methodology For Inverse Problem Formulation, Ariel Cintron-Arias, H. Banks, Alex Capaldi, Alun Lloyd

Mathematics and Statistics Faculty Publications

We propose an algorithm to select parameter subset combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal values of parameters are used to construct synthetic data sets, and explore the effects of removing certain parameters from those to be estimated using OLS procedures. We quantify these effects in a score for a vector parameter defined using the norm of the …

Generalized Shifts On Cartesian Products, M. Rajagopalan, K. Sundaresan

Mathematics and Statistics Faculty Publications

It is proved that if E, F are infinite dimensional strictly convex Banach spaces totally incomparable in a restricted sense, then the Cartesian product E×F with the sum or sup norm does not admit a forward shift. As a corollary it is deduced that there are no backward or forward shifts on the Cartesian product`p1×`p2,1< p16=p2<∞, with the supremum norm thus settling a problem left open in Rajagopalan and Sundaresan in J. Analysis 7 (1999(, 75-81 and also a problem stated as unsolved in Rassias and Sundaresan.

Development And Testing Of The Gait Assessment And Intervention Tool (G.A.I.T.): A Measure Of Coordinated Gait Components, J. J. Daly, J. Nethery, J. P. Mccabe, I. Brenner, J. Rogers, J. Gansen, K. Butler, R. Burdsall, K. Roenigk, John Holcomb

Mathematics and Statistics Faculty Publications

Recent neuroscience methods have provided the basis upon which to develop effective gait training methods for recovery of the coordinated components of gait after neural injury. We determined that there was not an existing observational measure that was, at once, adequately comprehensive, scored in an objectively-based manner, and capable of assessing incremental improvements in the coordinated components of gait. Therefore, the purpose of this work was to use content valid procedures in order to develop a relatively inexpensive, more comprehensive measure, scored with an objectively-based system, capable of incrementally scoring improvements in given items, and that was both reliable and …

A Berry-Esseen Theorem For Sample Quantiles Under Weak Dependence, S. N. Lahiri, Shuxia Sun

Mathematics and Statistics Faculty Publications

This paper proves a Berry-Esseen theorem for sample quantiles of strongly-mixing random variables under a polynomial mixing rate. The rate of normal approximation is shown to be O(n^-1/2) as n -> infinity, where n denotes the sample size. This result is in sharp contrast to the case of the sample mean of strongly-mixing random variables where the rate O(n^-1/2) is not known even under an exponential strong mixing rate. The main result of the paper has applications in finance and econometrics as financial time series important data often are heavy-tailed and quantile …

Properties Of Matrix Variate Beta Type 3 Distribution, Arjun K. Gupta, Daya K. Nagar

Mathematics and Statistics Faculty Publications

We study several properties of matrix variate beta type 3 distribution. We also derive probability density functions of the product of two independent random matrices when one of them is beta type 3. These densities are expressed in terms of Appell’s first hypergeometric function F1 and Humbert’s confluent hypergeometric function Φ1 of matrix arguments. Further, a bimatrix variate generalization of the beta type 3 distribution is also defined and studied.

Periodic Traveling Waves In Sirs Endemic Models, Tong Li, Yi Li, Herbert W. Hethcote

Mathematics and Statistics Faculty Publications

Mathematical models are used to determine if infection wave fronts could occur by traveling geographically in a loop around a region or continent. These infection wave fronts arise by Hopf bifurcation for some spatial models for infectious disease transmission with distributed-contacts. Periodic traveling waves are shown to exist for the spatial analog of the SIRS endemic model, in which the temporary immunity is described by a delay, but they do not exist in a similar spatial SIRS endemic model without a delay. Specifically, we found that the ratio of the delay ω in the recovered class and the average infectious …

A Direct Solution Of The Robin Inverse Problem, Weifu Fang, Suxing Zeng

Mathematics and Statistics Faculty Publications

We present a direct, linear boundary integral equation method for the inverse problem of recovering the Robin coefficient from a single partial boundary measurement of the solution to the Laplace equation.

A Note On The Positive Solutions Of An Inhomogeneous Elliptic Equation On Rn, Yinbin Deng, Yi Li, Fen Yang

Mathematics and Statistics Faculty Publications

This paper is contributed to the elliptic equation

(0.1) Δu+K(|x|)up+μf(|x|)=0,

where p>1, x∈Rn, n⩾3, and μ⩾0 is a constant. We study the structure of positive radial solutions of (0.1) and obtain the uniqueness of solution decaying faster than r−m at ∞ if μ is small enough under some assumptions on K and f, where m is the slow decay rate.

Superposition Formulas For Darboux Integrable Exterior Differential Sys-Tems, Ian M. Anderson, Mark E. Fels, Peter J. Vassiliou

Mathematics and Statistics Faculty Publications

In this article we solve an inverse problem in the theory of quotients for differential equations. We characterize a family of exterior differential systems that can be written as a quotient of a direct sum of two associated systems that are constructed from the original. The fact that a system can be written as a quotient can be used to find the general solution to these equations. Some examples are given to demonstrate the theory.

Zpc Matrices And Zero Cycles, Marina Arav, Frank Hall, Zhongshan Li, Bhaskara Rao

Mathematics and Statistics Faculty Publications

Let H be an m x n real matrix and let Z_i be the set of column indices of the zero entries of row i of H. Then the conditions |Z_k ∩(U^k-1 _i=1 Z_i)| ≤ 1 for all k (2 ≤ k ≤ m) are called the (row) Zero Position Conditions (ZPCs). If H satisfies the ZPC, then H is said to be a (row) ZPC matrix. If H^T satisfies the ZPC, then H is said to be a column ZPC matrix. The real matrix H is said to …

Toric Surface Codes And Minkowski Length Of Polygons, Ivan Soprunov, Jenya Soprunova

Mathematics and Statistics Faculty Publications

In this paper we prove new lower bounds for the minimum distance of a toric surface code CP defined by a convex lattice polygon P⊂R2. The bounds involve a geometric invariant L(P), called the full Minkowski length of P. We also show how to compute L(P) in polynomial time in the number of lattice points in P.