# News

## Janet Anderson Prospectus

A Study of Arc-Strong Connectivity of Digraphs

We study the extremal and structural properties of graphs and digraphs closely related to subgraph and subdigraphs connectivity. We obtained a best possible bound for the size of strict digraphs that do not have a subdigraph with high arc strong-connectivity, but adding any additional arc to it will result in a subdigraph with high arc strong-connectivity. We also obtain several minimax theorems concerning subdigraph strength measures. We propose to continue investigating the strength of digraph networks of other typesin this direction.

Date: 11/02/2016

Time: 3:30PM-5:00PM

Place: 315 Armstrong Hall

All are welcome.

## Katie Horacek Prospectus

We study properties of Class 2 and edge-chromatic critical graphs.

Research Prospectus: View

Date: 10/26/2016

Time: 3:30PM-5:00PM

Place: 315 Armstrong Hall

All are welcome.

## Alumnus Gift

Two-time West Virginia University graduate Mark Roth recently demonstrated how alumni of multiple colleges can blend their entire academic experience through their philanthropic goals.

Taking a “one WVU” approach to his philanthropy, Roth sought to give back to both of his academic homes, the Eberly College of Arts and Sciences and the College of Business and Economics.

## Gary Seldomridge Retires

He received his Ph.D in 1992 from WVU and spent 40+ years working tirelessly to educate students.

## Deshler Named Big 12 Fellow

Jessica Deshler, associate professor of mathematics, has been awarded a research grant from the Office of the Provost as part of the 2016-2017 Big 12 Faculty Fellowship Program.

The program offers faculty members at institutions in the Big 12 Conference the opportunity to travel to other member institutions to pursue collaborative research in a wide variety of ways. Our university application specifies that award recipients may each use up to $2500 to "work on collaborative research, consult with faculty and students, offer a series of lectures or symposia, acquire new skills, or take advantage of a unique archive or laboratory facility."

## Renee Larue Defense

An Analysis of Student Approaches to Solving Optimization Problems in First Semester Calculus

The purpose of this dissertation is to investigate how students think about and understand optimization problems in first semester calculus. To solve an optimization problem, one must identify the quantity to be maximized or minimized and then construct an optimizing function modeling the scenario described in the problem, particularly paying attention to how the desired quantity varies under the given constraint. Once this optimizing function has been constructed, the problem solver uses calculus and algebra skills to analytically find the absolute maximum or minimum of the function in the realistic domain of the problem. These problems are notoriously difficult for students, but have been largely ignored by the research community.

To examine how students think about and understand optimization problems, I interviewed seven first semester calculus students as they solved two optimization problems and answered questions related to the optimization problem-solving process. Analysis of this interview data revealed six mathematical concepts that play a key role in students’ concept images of the optimizing function. In Paper 1, I describe how these mathematical concepts influence students’ problem-solving activities and their construction of the optimizing function.

In this dissertation I also have created an Optimization Problem-Solving Framework that describes the desired conceptual and analytical thought processes students should engage in while solving an optimization problem. In Paper 2, I describe this framework and present the results of analyzing students’ thought processes while solving a classic optimization problem. The students demonstrated evidence of engaging in pseudo-conceptual and pseudo-analytical thought processes as they solved the optimization problem, particularly in the orienting and planning phases.

Finally, in Paper 3, I describe students’ responses while doing an activity designed to assess their ability to connect the optimizing function they constructed to the graphical representation of the optimizing function. This analysis is used to frame a discussion of suggested teaching interventions to help students develop a conceptual understanding of the optimization problem-solving process.

Date: 5/2/2016

Time: 10:00AM-1:00PM

Place: 415 Armstrong Hall

All are welcome.

## Kristin Duling Qualifier

Ms. Kristin Duling will present her dissertation prospectus. All are welcome to her presentation and a question/answer period. Following the presentation, there will be an oral examination open only to the candidate and the PhD committee of the candidate.

Date: 5/2/2016

Time: 4:30PM-5:30PM

Place: 313 Armstrong Hall

## Fuller & Deshler Receive NSF Grant

In 2012, the President’s Council of Advisors on Science and Technology, an advisory group of the nation’s leading scientists and engineers appointed by President Obama, identified the need for more college graduates with degrees in science, technology, engineering and mathematics. Faculty at West Virginia University are addressing that need by making it a priority to see students through their education and onto rewarding STEM careers.

Please go here for the WVUToday story

## Jiaao Li Qualifier

Group Connectivity and Modulo Orientations of Graphs

All are welcome.

## Mohamed Amsaad Defense

Well-defined Lagrangian flows for absolutely continuous curves of probabilities on the real line.

It is known from Fluid Mechanics that the time evolution of a probability measure describing some physical quantity (such as the density of a fluid) is related to the velocity of the fluid by the continuity equation. This is known as the Eulerian description of fluid flow. Dually, the Lagrangian description uses the flow of the velocity field to look at the individual trajectories of particles. In the case of flows on the real line, only recently has it been discovered that "some sort'' of dual Lagrangian flow consisting of monotone maps is always available to match the Eulerian flow. The uniqueness of this "monotone flow'' among all possible "flows'' (quotation marks used precisely because traditionally it cannot be called a "flow'' unless it is unique) of the fluid velocity is the centerpiece of this dissertation.

The Lagrangian description of absolutely continuous curves of probability measures on the real line is analyzed in this thesis. Whereas each such curve admits a Lagrangian description as a well-defined flow of its velocity field, further conditions on the curve and/or its velocity are necessary for uniqueness. We identify two seemingly unrelated such conditions that ensure that the only flow map associated to the curve consists of a time-independent rearrangement of the generalized inverses of the cumulative distribution functions of the measures on the curve. At the same time, our methods of proof yield uniqueness within a certain class for the curve associated with a given velocity; that is, they provide uniqueness for the solution of the continuity equation within a certain class of curves. Our proposed approach is based on the connection between the flow equation and one-dimensional Optimal Transport.

This is based on joint work of the author with A. Tudorascu, in which some results on well-posedness (in the one-dimensional case) have been achieved. The results are presented in major conferences and published in a highly ranked mathematical journal.

All are welcome.

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