Student Research Colloquium

Rice Tungro Virus Disease is one of the most economically damaging vector-borne plant diseases in tropical countries like the Philippines. To manage its spread, farmers typically use cultural and chemical controls. In this study, the researchers used ordinary and impulsive differential equations to formulate two compartmental models with roguing and insecticide spraying as control strategies. The well-posedness and local stability of the models were analyzed. Moreover, Python was used to perform parameter sensitivity analysis and numerical simulations. The findings suggest that the impulsive model is a more accurate and viable approach than the continuous model. Specifically, roguing every 15 days and insecticide spraying every 30 days is the most optimal time interval variation for applying the controls. Furthermore, the researchers were also able to determine the optimal effectiveness rates that are not only mathematically valid but also practically effective in mitigating the disease.

Eating disorders have become much more prevalent and impactful on a global scale in recent years. Anorexia nervosa, bulimia nervosa, and binge eating disorder are the most common and the deadliest among all types of eating disorders. Hence, this paper aims to model the dynamics of developing these eating disorders. In this study, anorexia nervosa, bulimia nervosa, and binge eating disorder were considered as epidemic diseases. The researchers utilized ordinary differential equations to develop a compartmental model incorporating factors influencing the spread of the disease and intervention strategies affecting recovery. Control variables including awareness campaigns to address media influence and peer pressure, the availability of treatment resources such as therapy, medication, and individual willingness, and the building of support systems for recovered patients were incorporated to achieve optimal results. Methodologically, qualitative analyses and local stability of the model were examined. Furthermore, sensitivity analysis and numerical simulations were performed through Python programming software. 

A k-orbit polyhedron is a special type of abstract polytope defined by a group action acting on its flags. In this study, we establish the existence of k-orbit polyhedron for every even integer k ≥ 4, under the condition that the polyhedra exhibits non-degeneracy. Utilizing the concepts of combinatorial techniques and theorems of group theory as our method, we derive the k flag orbit equation. Next, we demonstrate the existence of even k-orbit polyhedra through explicit constructions, categorized into cases where k = 4, k = 6, k ≡ 0 mod 4, and k ≡ 2 mod 4, which encompass all even integer k ≥ 4. These findings have significant potential applications in diverse fields such as crystallography and combinatorial optimization. 

This paper undertakes an investigation into the Approximation of the Core Inverse of Square Real Matrices. The Core inverse, denoted as A is a unique matrix that satisfies specific conditions, crucial in mathematical applications when classical inversion is impossible. The study explores the Moore-Penrose inverse as a foundational concept. Methodologically, we initiate our inquiry by formulating an optimization problem akin to the methodology employed in approximating the Moore-Penrose inverse, and use Tikhonov Regularization technique to enhance the stability of solutions for ill-posed matrices. Our approach employs Gradient method, Symmetric Operators, and Proximity Operators. Ultimately, we provide a formula for approximating the Core inverse, contributing to Linear Algebra, Analysis, Mathematical Modeling and Optimization. 

Algebraic number theory delves into the arithmetic properties of algebraic number fields, particularly the ring of integers, ideals, units, and the extent of unique factorization. Within this field lies the modular arithmetic’s foundational principles that enable the exploration of k-potent elements in rings, where an element x satisfies the equation x^k=x for a fixed positive integer k. In this paper, we considered the k-potent elements in the ring of Quadratic Integers Modulo n, Z_n[sqrt. d] and we aim to count them. To achieve this, we need a strong understanding of modular arithmetic, as it simplifies complex problems by focusing on remainders when dividing integers by a specific modulus. This simplification helps identify patterns within integers and facilitates in examining complex arithmetic structures. Understanding these elements clarifies algebraic behavior in rings and contributes to the broader study of algebraic structures, aligning with the goals of algebraic number theory.