A Math Research Project Inspired by Twin Motherhood

The phenomenon of twins, triplets, quadruplets, and other higher order multiples has fascinated humans for centuries and has even captured the attention of mathematicians who have sought to model the probabilities of multiple births. However, there has not been extensive research into the phenomenon of polyovulation, which is one of the biological mechanisms that produces multiple births. In this paper, I describe how my own experience becoming a mother to twins led me on a quest to better understand the scientific processes going on inside my own body and motivated me to conduct research on polyovulation frequencies. An overview of the previous mathematical research on multiple births, as well as my own contributions involving polyovulation, is presented. Furthermore, I discuss more generally how motherhood can influence and enrich the research agendas of mathematicians

Carcassonne in the Classroom

“What is the probability of choosing a green ball from an urn with three blue balls,five green balls, and seven yellow balls?” Many students not only struggle to engage with this sort of question but are left wondering why the world of mathematics is obsessed with balls and urns. The variety of approaches to choose from makes probability a difficult subject for many students, yet probability is an important part of quantitative literacy since it is prevalent in everyday life. Finding ways to clarify probability for undergraduates is key to a successful mathematics experience. One strategy for increasing student engagement with probability concepts is to teach probability through its application to games. Many previous works have investigated the use of Markov chains to model board games, such as Chutes and Ladders [2, 4, 5], Monopoly [1, 3], and Risk [6, 7, 8]. While these works have primarily focused on understanding the various games for their own sake, in this article we focus on using the board game Carcassonne in the classroom as a path for students to learn about probability through a more interesting context. In particular, we give a sequence of increasingly difficult probability problems derived from Carcassonne that can be used in a wide range of undergraduate mathematics courses

Minimal Noise-Induced Stabilization of One-Dimensional Diffusions

The phenomenon of noise-induced stabilization occurs when an unstable deterministic system of ordinary differential equations is stabilized by the addition of randomness into the system. In this paper, we investigate under what conditions one-dimensional, autonomous stochastic differential equations are stable, where we take the notion of stability to be that of global stochastic boundedness. Specifically, we find the minimum amount of noise necessary for noise-induced stabilization to occur when the drift and noise coefficients are power, polynomial, exponential, or logarithmic functions

Probabilistic Analysis of Polyovulation Frequencies

Polyovulation is the production of more than one ovum, or egg, during a single menstrual cycle. This paper examines the probability of the human ovarian system ovulating $k$ eggs during a single cycle, for $k\geq0$. In order to obtain precise estimates for the probability of polyovulation, we use U.S. birth data from the 1950\u27s (before the introduction of artificial reproductive technologies). However, to utilize birth data, we model the various processes that eggs undergo in order to result in a live birth, including fertilization, possible division, implantation, and potential miscarriage. Our model produces novel estimates for the probability that a fertilized egg divides, as well as for the zygosity type frequencies of twins, triplets, and quadruplets

Propagating Lyapunov Functions to Prove Noise--induced Stabilization

We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to a unique invariant probability measure to which the system converges at a uniform, exponential rate. These facts are established primarily through the construction of a Lyapunov function which we generate as the solution to a sequence of Poisson equations. Unlike a number of other works, however, our Lyapunov function is constructed in a systematic way, and we present a meta-algorithm we hope will be applicable to other problems. We conclude by proving positivity properties of the transition density by using Malliavin calculus via some unusually explicit calculations.Comment: 41 pages, 3 figures Added picture to this version and simplified the control theory discussion significantl

Minimal Noise-Induced Stabilization of One-Dimensional Diffusions

The phenomenon of noise-induced stabilization occurs when an unstable deterministic system of ordinary differential equations is stabilized by the addition of randomness into the system. In this paper, we investigate under what conditions one-dimensional, autonomous stochastic differential equations are stable, where we take the notion of stability to be that of global stochastic boundedness. Specifically, we find the minimum amount of noise necessary for noise-induced stabilization to occur when the drift and noise coefficients are power, polynomial, exponential, or logarithmic functions

Effects of Environmental Factors on Candida albicans Morphology: A Focus On Estrogen and Microgravity

C. albicans is one of the commensal fungi living in the human intestinal tract in a harmless spore form. In its filamentous form, C. albicans becomes invasive and penetrates the human body, which can cause serious health issues. In vitro factors such as change in temperature or pH are known to induce morphology shift in C. albicans. Interestingly, microgravity has been reported to decrease the human immunity and increase gene virulence expression in C. albicans. During sepsis, high levels of estrogen are reported and the risk of candidiasis also increases. Within present work, we tested the effect of microgravity and estrogen on the shift of morphology (spore to filamentous). C. albicans were grown in minimum media for 3 days in presence or absence of 0.1 nM estrogen. In addition, two other groups of C. albicans were subjected to microgravity for 3 days, using a clinostat, in presence or in absence of estrogen. For each condition, 5 random pictures were taken and scored 1 for the presence and 0 for absence of filament. Experiments were conducted in duplicate. Our results show that subjecting C. albicans to microgravity significantly increase the number of filaments compared to control (9.59±2.77 versus 1.68±1.93, P\u3c0.001, unpaired t-test), whereas estrogen did not significantly affect the number of filaments compared to control (2.66±1.61 versus 1.68±1.93, p=0.6, unpaired t-test). Finally, there was no significant effect of estrogen found on the number of filament when C. albicans was exposed to microgravity plus estrogen versus microgravity alone (8.0±2.76 versus 9.59±2.77). In conclusion, we have found that simulated microgravity dramatically increases the number of filaments, and estrogen at 0.1 nM has no effect on the number of filaments in our experimental conditions

A Generalized Lyapunov Construction for Proving Stabilization by Noise

<p>Noise-induced stabilization occurs when an unstable deterministic system is stabilized by the addition of white noise. Proving that this phenomenon occurs for a particular system is often manifested through the construction of a global Lyapunov function. However, the procedure for constructing a Lyapunov function is often quite ad hoc, involving much time and tedium. In this thesis, a systematic algorithm for the construction of a global Lyapunov function for planar systems is presented. The general methodology is to construct a sequence of local Lyapunov functions in different regions of the plane, where the regions are delineated by different behaviors of the deterministic dynamics. A priming region, where the deterministic drift is directed inward, is first identified where there is an obvious choice for a local Lyapunov function. This priming Lyapunov function is then propagated to the other regions through a series of Poisson equations. The local Lyapunov functions are lastly patched together to form one smooth global Lyapunov function.</p><p>The algorithm is applied to a model problem which displays finite time blow up in the deterministic setting in order to prove that the system exhibits noise-induced stabilization. Moreover, the Lyapunov function constructed is in fact what we define to be a super Lyapunov function. We prove that the existence of a super Lyapunov function, along with a minorization condition, implies that the corresponding system converges to a unique invariant probability measure at an exponential rate that is independent of the initial condition.</p>Dissertatio