Kayode Oluwasegun

This report addresses the challenge of determining optimal mentor-mentee pairings for the University of Delaware’s GradLEAP program. Two mathematical models are developed—a linear programming approach solved using the Hungarian Method, and a stable marriage formulation solved via the Gale-Shapley algorithm. Compatibility scores, calculated using responses to a structured survey, incorporate both weighted and uniform criteria such as academic discipline, career interests, and age preferences. A revised survey introduces a Likert scale for ranking preferences, allowing for greater flexibility in determining weights and improving match quality. The performance of both models is evaluated using real and synthetically generated data, with results indicating that the Hungarian Method yields more consistent matches while the Gale-Shapley method is more likely to produce highly compatible pairings. The study also explores classification models using decision trees and random forests, with an eye toward future development of regression-based scoring systems. The final deliverables include a restructured survey, a user-friendly executable program, and documentation for use in future iterations of the GradLEAP program.

This paper explores the oceanic wave characteristics exhibited by a modified integrable generalized [ ]-dimensional nonlinear Schrödinger system of equations through variable coefficients. Two newly modified methods, specifically the improved nonlinear Ricatti equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary waves solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This paper illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of soliton, rational, trigonometric, hyperbolic and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering.

This paper explores the oceanic wave characteristics exhibited by a modified integrable generalized (2+1)-dimensional nonlinear Schrodinger system of equations through variable coefficients. Two newly modified methods, specifically the improved nonlinear Ricatti equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary waves solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This paper illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of soliton, rational, trigonometric, hyperbolic and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering.

With the introduction of mathematics into biological research, patterns arising from biological problems can be effectively studied. The complex interactions involving biological systems and invading pathogens can be described using mathematical frameworks of partial differential equations (PDEs). More specifically, the spread of new or invasive species can be set up as a population model with a freely moving boundary. Proving the existence of solutions to this model helps us scientists to have a solid understanding of how the population works and predict how it changes in the future. Generally speaking, the mathematical modeling of ecological problems is difficult. In this work, we propose a different approach to the understanding of the spreading of species. We have introduced a special case of the well-known Stefan condition as the equation governing the free boundary. Our study therefore seeks to develop rigorous mathematical frameworks underpinning existence theorems and see the role of free boundaries in shaping the behavior of populations. We have employed energy methods with the goal of building a general framework that will be highly portable for use in other, related problems in the modeling of infectious diseases. Invading fungal pathogens like Candida albicans cause over 150 million mucosal infections and nearly 200,000 deaths per year due to invasive and disseminated disease in susceptible populations. This work represents significant advancements in resolving a long-standing mathematical problem and addressing one of the most important health-related issues nations are grappling with.