Two Analytic Issues Arising from Models in Probability and Materials Science

This dissertation explores two distinct topics centered on mathematical models in probability theory and materials science. The first part investigates a series of functions derived from an adaptive algorithm designed to address the score-based secretary problem, a classic challenge in probability theory. This problem involves making immediate decisions to select the best candidate from a sequence of interviews. The algorithm aims to maximize the probability of selecting the optimal candidate based on observed scores. We prove two fundamental analytic properties of this sequence of functions as a theoretic support of the algorithm: first, the functions in the sequence each possess a unique maximum point, ensuring a well-defined optimal decision at every step; and second, the sequence of maximum points increases monotonically to infinity as n tends to infinity, suggesting that the algorithm becomes increasingly effective as the process improves. Although only single-variable calculus is used in the proofs, the argument is non-trivial and the results provide fresh insights into this adaptive approach for solving the classical problem. The second part revisits the 2-dimensional Smoluchowski equation, a mathematical model widely used to describe the orientational distribution of nematic liquid crystalline polymers, which are characterized by their ability to transition between isotropic and nematic phases and play a vital role in advanced applications such as liquid crystal displays, sensors, and other technologies. Focusing on the steady-state solutions, we present a simplified proof to characterize these solutions based on the intensity parameter. Specifically, if the intensity constant is less than or equal to 4, there exists a unique, trivial solution corresponding to the isotropic state. If the intensity constant is greater than 4, there are exactly two solutions -trivial and nontrivial- representing two distinct physical states: the isotropic and nematic phases of liquid crystals. This new proof relies on elementary calculus, making it more intuitive and accessible.