# News

On the Matroid Intersection Conjecture

Abstract: In this dissertation, we investigate the Matroid Intersection Conjecture for pairs of matroids on the same ground set, proposed by Nash-Williams in 1990. Originally, the conjecture was stated for finitary matroids only, but we consider it for general matroids and introduce new approaches to attack the conjecture.

The first approach is to consider the situation when it is possible to make a finite modification to the matroids after which the pair satisfies the conjecture. In such a situation we say that the pair has the “Almost Intersection Property”. We prove that any pair of matroids with the Almost Intersection Property must satisfy the Matroid Intersection Conjecture. Using this result we prove that the Matroid Intersection Conjecture is true in the case when one of the matroids has finite rank and also in the case when one of the matroids is a patchwork matroid.

Our second new approach is inspired by the proof of the general version of König’s Theorem for bipartite graphs. That result implies that the Matroid Intersection Conjecture is true for pairs of partition matroids. We develop some new techniques that generalize the "critical set" approach used in the proof of the countable version of König’s Theorem. Our results enable us to prove that the Matroid Intersection Conjecture is true for a pair of singular matroids on a set that is infinitely countable. A matroid is singular when it is a direct sum of matroids such that each term of the sum is a uniform matroid either of rank one or of co-rank one.

Date: 6/26/2017
Time: 2:00PM-4:00PM
Place: 207 Armstrong Hall

All are welcome.

Date, Location:
2017-06-26

## Jaxon Lee Capstone

Analytic Approaches to Common Means and their Inequalities

Abstract: Means are measurements of centrality, commonly used in science, engineering, statistics, and mathematics. There is a plethora of means and they measure centrality differently as to obtain the most pertinent information to the question at hand. We will discuss a specific subclass of means called Pythagorean means, which include the arithmetic, geometric, and harmonic means. We will also discuss the root mean squared, as these four means are the most common means to run into in science, engineering, and statistics applications. Lastly, we will show how the value of each of the means is related to one another through the inequality relationship between them.

Date: 6/21/2017
Time: 4:00PM-5:00PM
Place: 123 Armstrong Hall

All are welcome.

Date, Location:
2017-06-21

Date, Location:
2017-06-01

## Ariel Sitler Masters Project

Abstract: Unmanned Aerial Vehicles (UAVs) are a valuable tool for image acquisition in the field of 3D mapping. UAVs are a low-cost alternative to manned aerial vehicles, and can be affixed with digital cameras capable of collecting high quality images of structures and terrain for the creation of 3D models. This research is an extension of previous work which used a genetic algorithm and space partitioning to optimize UAV flight paths over areas with structures. Here, the focus is on optimizing the structure mapping portion of the flight plan by imposing the grid technology used for the area portion in the previous work to a building face, then using the grid size, camera specifications, and desired resolution to determine the optimal offset distance from the building.

Date: 5/3/2017
Time: 1:00PM-2:00PM
Place: 313 Armstrong Hall

All are welcome.

Date, Location:
2017-05-03

Math undergraduate Tony Allen awarded NSF Summer Fellowship. Please congratulate him. Great work Tony!

Date, Location:
2017-04-26

## Fatma Mohamed Defense

On Some Parabolic Type Problems from Thin Film Theory and Chemical Reaction-Diffusion Networks

Abstract: In the first part of this dissertation we study the evolution of a thin film of fluid modeled by the lubrication approximation for thin viscous films. We prove existence of (dissipative) strong solutions for the Cauchy problem when the sub-diffusive exponent ranges between 3/8 and 2; then we show that these solutions tend to zero at rates matching the decay of the source-type self-similar solutions with zero contact angle. We introduce the weaker concept of dissipative mild solutions and we show that in this case the surface-tension energy dissipation is the mechanism responsible for the \$H^1\$--norm decay to zero of the thickness of the film at an explicit rate. Relaxed problems, with second-order nonlinear terms of porous media type are also successfully treated by the same means.

In the second part we are concerned with the convergence of a certain space-discretization scheme (the so-called method of lines) for mass-action reaction-diffusion systems. First, we start with a toy model, namely AB and prove convergence of the method of lines for this linear case (weak convergence in L^2 is enough in this case). Then we show that solutions of the chemical reaction-diffusion system associated to A+BC in one spatial dimension can be approximated in L^2 on any finite time interval by solutions of a space discretized ODE system which models the corresponding chemical reaction system replicated in the discretization subdomains where the concentrations are assumed spatially constant. Same-species reactions through the virtual boundaries of adjacent subdomains lead to diffusion in the vanishing limit. We show convergence of our numerical scheme by way of a consistency estimate, with features generalizable to reaction networks other than the one considered here, and to multiple space dimensions.

Date: 4/20/2017
Time: 12:30PM-2:30PM
Place: 315 Armstrong Hall

All are welcome.

Date, Location:
2017-04-20

## Mushtaq Abd Al-Rahem Defense

A Multidimensional Technique for Measuring Consensus Within Groups via Conditional Probability

Abstract: A recent increase in the use of the term “consensus” in various fields has led researchers to develop various ways to measure the consensus within and across groups depending on the areas. Numerous studies use the mean or the variance alone as a measure of consensus, or lack of consensus. Most of the time, high variance is viewed as more disagreement in a group. Using the variance as a measure of disagreement is meaningful in an exact comparison cases (same group, same mean). However, it could be meaningless when it is used to compare groups that have different sizes, or if the mean is different. In this thesis, we establish the fact that the range of the variance is a function of the mean, we present a new index of disagreement and measure of consensus that depend on both, the mean and the variance, by utilizing the conditional distribution of the variance for a given mean. Initially, this new index is developed for comparison of data collected using a Likert scale of size five. This new measure is compared with the results of two other known measures, to show that in some cases they agree, but in other cases the new measure provides additional information. Next, to facilitate generalization, a new algorithmic method to determine the index using a geometric approach is presented. The geometric approach makes it easier to compute the measure of consensus and provides the foundational ideas for generalizing the measure to Likert scales for any n. Finally, a multidimensional computational technique was developed to provided the final step of generalization to Likert scales of any n.

Date: 4/10/2017
Time: 3:00PM-5:00PM
Place: 313 Armstrong Hall

All are welcome.

Date, Location:
2017-04-10

The Eberly College of Arts and Sciences has named two recipients of the 2016-17 Outstanding Researcher Award: Christina Duncan and Casian Pantea.

Congratulate Professor Pantea the next time you see him!

Date, Location:
2017-04-10

## Junior Math Competition

This competition will be open to all students grades 7-12. The questions will be at AMC difficulty level.

Casian Pantea: cpantea@math.wvu.edu
Charis Tsikkou: tsikkou@math.wvu.edu
Megan Henry: mhott1@mix.wvu.edu

Date, Location:
2017-04-08

## Janet Anderson Defense

A Study of arc strong connectivity of digraphs

Abstract: This dissertation studies the extremal, structural and minimax properties related to digraph arc strong connectivity. Motivated by the former researches by Mader and Matula, we defined the digraph arc strength of a digraph \$D\$ to be the maximum arc strong connectivity of subdigraphs of \$D\$, and investigate extremal conditions for a strict digraph that is saturated with respect to the condition that it digraph arc strength is bounded by a positive integer \$k\$. Extremal size of such digraphs are obtained and structural characterization of the extremal digraphs are also obtained. In addition, a minimax duality theorem to determine the digraph arc strength of a digraph is found. Similar minimax results on related desity functions of digraphs are also obtained.

Date: 4/7/2017
Time: 3:30PM-5:00PM
Place: 315 Armstrong Hall

All are welcome.

Date, Location:
2017-04-07