Faculty/Graduate Speaker Session

We present a theorem on the approximation of the solutions of delay differential equations with continuously distributed delay with solutions of delay differential equations with discrete delays. We present numerical simulations of the trajectories of discrete delay differential equations and the dependence of their behavior for various delay amounts. We further simulate continuously distributed delays by considering discrete approximation of the continuous distribution.
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This paper presents a novel way of introducing undergraduate mathematics students to partial differential equations (PDEs) as part of a differential equations (DE) course. This method modifies the method of basic indefinite integration used to solve the ordinary differential equation (DE) y’ = f(x) in a way that works well for similarly simple PDEs.
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We discuss mayfly species with multiple generations each year and a mathematical framework for analyzing the stability of synchrony within each generation. We construct a circle map that sends the start time of a generation to its end time and analyze the conditions under which this map and its iterates have stable fixed points. Additionally, we use a model for fecundity to rate the evolutionary success of these stable fixed points, which we use to suggest why the linear growth rate we observe is in some sense optimal.
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For any integer $n$, let $U_n$ be the set of positive integers that are less than $n$ and relatively prime to $n$. If $U_n$ can be partitioned into $k$ subsets with equal sums, then we say that $n$ is a $k$-fold super totient number. We completely characterize all $k$-fold super totient numbers when $k\in\{2,3,5\}$, and we discuss on $k$-fold super totient numbers for a general $k$.
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In his delightful treatise entitled Mathematics For Human Flourishing, Dr. Francis Su of Harvey Mudd College and a past President of The Mathematical Association of America extrapolates why mathematics is truly a human endeavor and furnishes excerpts dealing with its power, beauty, exploration, permanence and struggle among its numerous attributes. In his book, puzzles and problems to ponder are interspersed. One of the problems is fascinating and is relevant for a course devoted to pre-service elementary teachers. It is entitled TOGGLING LIGHT SWITCHES. The text of the problem is as follows: Imagine 100 light bulbs, each numbered 1 through 100, all in a row, and all the lights off. Suppose you do the following: toggle all switches that are multiples of 1, then toggle all switches that are multiples of 2, then toggle all switches that are multiples of 3, etc., all the way to multiples of 100. (To toggle a switch means to flip it on if it’s off and off it it’s on.) When you are done, which lightbulbs are on and which are off? Do you see a pattern? Can you explain it? The goal of this talk is to seek resolution to the problem which is isomorphic to a popular problem known as The Locker Problem.
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For $p$ prime, the $p$-adic valuation of an integer $x$ is the highest power of $p$ which divides $x$. Consider a sequence of $p$-adic valuations, generated by a polynomial $f(n)$ with integer coefficients. We can represent this sequence using a tree diagram that could be infinite, finite, or even a single dot. For example, for $p=3$ the polynomial $f(n)=2n^2+81$ has a 3-adic valuation tree with two infinite branches, while the tree of $g(n)=2n^2+3n+18$ is a finite two-level tree. Moreover, a calculation can show that if $f(n)=an^2+bn+c$ where both $a$ and $b$ are congruent to $1 \pmod 3$ and c is congruent to $2 \pmod 3$, then its 3-adic valuation tree has only one node. In this talk, we will determine several properties of the $p$-adic valuation trees directly from the coefficients of the polynomial. The ``finiteness'' of the tree to start. But there are many more questions to ask: If the tree is infinite, how many infinite branches does it have? Can we determine the valuations at the terminating nodes? This is a joint work with REU students at Ursinus College, Summer 2021, Dr. Justin Trulen at Kentucky Wesleyan College, and his undergraduate student, Will Boultinghouse.
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We explore the notion that the principle of mathematical induction and the well-ordering principle are equivalent as noted in many textbooks introducing proof by induction. In particular, we first note the problems that occur with a naive substitution within Peano's axioms. We then formulate a set of axioms that "naturally" include the well-ordering principle and show that this axiom system is equivalent to the system given by Peano's axioms. Finally, we discuss some questions and observations for further study.
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Levels of sociality in nature vary widely from solitary species to complex multi-family societies. Increased levels of social interaction can allow for the spread of useful innovations and beneficial information, but can also facilitate the spread of harmful contagions, such as infectious diseases. In this talk, we will explore how coupled contagion processes can help shape the rules for interaction in complex social systems. We consider a model for the evolution of sociality strategies in the presence of both a beneficial and costly contagion, and study dynamics of this model at multiple timescales. We use a susceptible-infectious-susceptible (SIS) model to describe contagion spread for given sociality strategies, and then employ the adaptive dynamics framework to study the long-time evolution of the levels of sociality in the population. For a wide range of assumptions about the benefits and costs of infection, we identify a social dilemma: the evolutionarily-stable sociality strategy (ESS) produced by adaptive dynamics is distinct from the collective optimum -- the level of sociality that would be best for all individuals. In particular, the ESS level of social interaction is greater (respectively less) than the social optimum when the good contagion spreads more (respectively less) readily than the bad contagion. Our results shed light on how contagion shapes the evolution of social interaction, but reveals that evolution may not necessarily lead populations to social structures that are good for any or all. This project is joint work with Dylan H. Morris, Simon A. Levin, Daniel I. Rubenstein, and Pawel Romanczuk.
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The National Alliance for Doctoral Studies in the Mathematical Sciences ("the Math Alliance") has an established record of building a mathematical community dedicated to helping a diverse population successfully complete graduate programs in the mathematical sciences. In the past few years, new mathematical communities at a more regional level have taken root, inspired by the work and success of the Math Alliance. In this talk, I'll introduce you to the Pennsylvania Math Alliance, describe the kinds of things we've been doing, what we plan to do and invite you to join us if you're interested. For more information about the PA Math Alliance: https://pamathalliance.org/
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