University-wide, Talks & Lectures, Open to the Public
21 Sep 2020
Zoom Webcast

2020 Student Seminar Research Presentations

Department of Mathematical Sciences

The recipients of summer research funds, Jeffrey Slepoi, Michael Tamarov, Alicia Muth, and Volodymyr Volchenko will report their research findings and give 25 minute presentations with short discussions immediately following.

Generalized Fractional Bessel Equation

Research project proposal

Jeffrey Sleopi


Bessel equation has been around since 1817 when F.W. Bessel derived it during an investigation of planetary motion. Since then it proved to be useful in various sciences. Fractional Bessel equation was suggested as a more precise description of some processes in physics. This presentation covers generalization of Bessel equation with fractional derivatives of Caputo and Riemann-Liouville types.  We discover solutions in a form of series, uniqueness of solution, necessary conditions for the existence of solution as a series. We essentially develop further the results obtained by Rodriguez et. al. who investigated the standard fractional Bessel equation and Okrasinski with Plociniczak who expressed solution in more general terms and performed asymptotic analysis of the solution. Some other theoretical findings are presented on series’ uniqueness and multi-series solutions in unusual cases. Examples illustrate the findings.

Existence of an Infinite Open Cluster in a Decaying Bernoulli Percolation Model

Research project proposal

Michael Tamarov


In a standard model of Bernoulli percolation, the edges of a 2-dimensional square integer lattice are taken to be open with constant probability p independently at random. It is known that there exists an infinite open cluster of edges if and only if p is greater than 1/2. For this research paper, we consider models where the probabilities of the edges being open is not constant, but decays with distance from the origin or an axis. We examine several such models and find conditions on the decay of the probabilities to guarantee the existence of an infinite open cluster (percolation). The principle result of the paper concerns the case where the probabilities decay to 1/2, where we establish a lower bound on the decay rate to guarantee percolation.

Distance-l, k-Component Order Neighbor Connectivity
A Generalization of Distance Domination

Research project proposal

Alicia Muth


The component order neighbor connectivity of a graph, a measure of the vulnerability of a network, is the number of nodes that need to fail and consequently subvert their adjacent neighbors in order to produce a component of order less than some given threshold. The instance where the threshold value is one coincides with domination, and has been widely studied. One variation of domination is Distance Domination. This parameter allows a failed node to not only subvert their adjacent neighbors, but neighbors at most a fixed distance away. The goal of this paper is to explore distance domination and determine which properties carry over to higher threshold values and which do not. A particularly important result is the algorithm for the distance-l, k-component order neighbor connectivity number of arbitrary trees.

Financial market efficiency. Equilibrium and investors’ behaviour

Research project proposal

Volodymyr Volchenko


The efficient-market hypothesis (EMH) stands on a crucial assumption that stocks are traded at its fair value. However, it is argued both empirically and theoretically. The key idea of EMH is that price of a particular asset reflects all information about the market. But if the price is different from the one that furnishes equilibrium in the market, then a disbalance in demand and supply creates an opportunity for an arbitrage. Is the existence of an arbitrage a contradiction to market’s efficiency and, moreover, rationality? The research project aims to show a mechanism of investors’ interactions and choices on the financial market in time investigating how the market equilibrium could be furnished in dynamics and to poses the insights of market’s efficiency.


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