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Articles 1 - 7 of 7
Full-Text Articles in Physical Sciences and Mathematics
Random Walks In The Quarter Plane: Solvable Models With An Analytical Approach, Harshita Bali, Enrico Au-Yeung
Random Walks In The Quarter Plane: Solvable Models With An Analytical Approach, Harshita Bali, Enrico Au-Yeung
DePaul Discoveries
Initially, an urn contains 3 blue balls and 1 red ball. A ball is randomly chosen from the urn. The ball is returned to the urn, together with one additional ball of the same type (red or blue). When the urn has twenty balls in it, what is the probability that exactly ten balls are blue? This is a model for a random process. This urn model has been extended in various ways and we consider some of these generalizations. Urn models can be formulated as random walks in the quarter plane. Our findings indicate that for a specific type …
Mapping Polynomial Dynamics, Devin Becker
Mapping Polynomial Dynamics, Devin Becker
DePaul Discoveries
We explore the complex dynamics of a family of polynomials defined on the complex plane by f(z) = azm(1+z/d)d where a is a complex number not equal to zero, and m and d are at least 2. These functions have three finite critical points, one of which has behavior that differs as we change our parameter values. We analyze the dynamical behavior at this critical point, with a particular interest in the structures that appear in the filled Julia set K(f) and the basin of infinity A_{\infty}(f). The behavior of the family is extremely sensitive to our …
Triangulation And Finite Element Method For A Variational Problem Inspired By Medical Imaging, Tim Komperda, Enrico Au-Yeung
Triangulation And Finite Element Method For A Variational Problem Inspired By Medical Imaging, Tim Komperda, Enrico Au-Yeung
DePaul Discoveries
We implement the finite element method to solve a variational problem that is inspired by medical imaging. In our application, the domain of the image does not need to be a rectangle and can contain a cavity in the middle. The standard approach to solve a variational problem involves formulating the problem as a partial differential equation. Instead, we solve the variational problem directly, using only techniques available to anyone familiar with vector calculus. As part of the computation, we also explore how triangulation is a useful tool in the process.
Spring 2021
Scientia
From the Dean: A Decade of Purpose and Progress; Lab Notes: Alumna Wins Gordon Bell Special Prize, New Scholarships, Vaccination Site Volunteers; Women in Science Lecture, National Institutes of Health Grants, "Unequal Cities" Research; All Hands on Deck: Inspired pandemic approaches showcase interdisciplinary acumen in action; Unlocking Potential: Christopher Beasley thinks psychology is key to academic transformation for the formerly incarcerated; Puzzle Master: Bridget Tenner goes to pieces solving problems in cutting-edge mathematics
The Mystery Of Frobenius Symmetry, Maciej Piwowarczyk
The Mystery Of Frobenius Symmetry, Maciej Piwowarczyk
DePaul Discoveries
In this project we studied the mathematical concept of the Frobenius number and some curious patterns that come with it. One common example of the Frobenius number is the Coin Problem: If handed two denominations of coins, say 4¢ and 5¢, and asked to create all possible values, we will eventually find ourselves in a position where we can make any value. With 4¢ and 5¢ coins, we can create any value above 11¢, but not 11¢ itself. So, that makes 11 the Frobenius number of 4 and 5. What we explore in this paper is a pattern we call …
United States Population Future Estimates And Long-Term Distribution, Sean P. Brogan
United States Population Future Estimates And Long-Term Distribution, Sean P. Brogan
DePaul Discoveries
The population of the United States has always increased year over year. Even now with decreasing birth rates, the overall population continues to grow when looking at conventional models. The present study specifically examines what would happen to the U.S. population if we were to maintain the current birth and survival rates into the future. By 2050, our research shows that the U.S. population will become much older and cease to grow at all.
The Search For The Cyclic Sieving Phenomenon In Plane Partitions, William J. Asztalos
The Search For The Cyclic Sieving Phenomenon In Plane Partitions, William J. Asztalos
DePaul Discoveries
The efforts of this research project are best understood in the context of the subfield of dynamical combinatorics, in which one enumerates a set of combinatorial objects by defining some action to guide the search for underlying structures. While there are many examples with varying degrees of complexity, the necklace problem, which concerns the possible unique configurations of beads in a ring up to rotational symmetry, is a well-known example. Though this sort of approach to enumeration has been around for a century or more, activity in this area has intensified in the last couple of decades. Perhaps the most …