Journal of Advances in Mathematics and Computer Science

A Mathematical Model for Teacher–Student Interaction and Student Learning Dynamics

Rajat Kaushik — 2026-08-19

The interaction between teachers and students is an important determinant of learning outcomes and the overall effectiveness of an education system. In this paper, we formulate and analyse a nonlinear ordinary differential equation model to study the dynamics of teacher–student interaction and student learning. The student population is divided into two categories: students needing additional academic support and students with relatively higher levels of learning achievement, while teacher effectiveness is treated as a dynamic variable. The model considers student progression due to effective teacher intervention, student disengagement, teacher performance, and the negative impact of heavy workload on teaching effectiveness. Positivity and boundedness of the model solutions are established, and possible equilibrium states are identified. We analyse the local stability of the equilibria using standard dynamical-systems techniques and obtain a threshold quantity that characterises the conditions for the persistence of effective learning. Sensitivity analysis is conducted to determine which parameters have the greatest influence on student learning outcomes. Numerical simulations are presented to demonstrate the effects of teaching effectiveness, student workload, and learning transition rates on the long-term behaviour of the system. The findings suggest that improved teacher effectiveness and an appropriate teacher–student ratio can make a substantial difference to learning gains. The proposed model provides a mathematical framework for understanding educational dynamics and may help to evaluate strategies aimed at improving teaching effectiveness and student achievement.

Eigen Spectrum of k− Uniform Loose Cyclic Hypergraphs

Santhosh Kumar N. — 2026-08-19

Hypergraphs extend ordinary graphs by allowing a hyperedge to connect more than two vertices. A hypergraph is k -uniform when each hyperedge contains exactly k vertices, and it is loose cyclic when the hyperedges are arranged cyclically so that consecutive hyperedges share exactly one vertex while non-consecutive hyperedges are disjoint. This study examines the possible k-uniform loose cyclic hypergraphs in relation to the number of vertices and develops a computational procedure for determining their spectral properties. For a loose cyclic hypergraph H = (V, E) with n vertices and m hyperedges, the relation n = m (k-1) is used to describe admissible configurations. An adjacency matrix is formed by assigning each off-diagonal entry according to the number of hyperedges containing the corresponding pair of vertices. A Python-based procedure is then used to construct the adjacency matrix for admissible parameter choices and to compute its eigenvalues and eigenvectors. The method is illustrated using a 4-uniform loose cyclic hypergraph on 15 vertices with five hyperedges. The resulting 15 × 15 adjacency matrix and its eigenvalues demonstrate the computational implementation of the procedure. The study provides a systematic matrix-based approach for obtaining the eigen spectrum of uniform loose cyclic hypergraphs when closed-form expressions are difficult to derive, while retaining the structural conditions that define the loose cyclic arrangement.

Cycle Index for Symmetric Group Acting on Cartesian Product of Two, Three and Four Sets

Grace Wakesho Kivunga — 2026-08-21

This study derives cycle-index expressions for the symmetric group acting on the Cartesian products of two, three, and four copies of a finite set. The analysis extends established results for ordered pairs, triples, and higher-order ordered subsets by incorporating contributions that arise when entries are repeated or are drawn from different cycles of a permutation. For the Cartesian product of two sets, the derivation separates ordered pairs into three cases, including the additional contribution from pairs of the form (a, a). For the Cartesian product of three sets, six contributions are considered, with the first three accounting for triples containing repeated entries and the remaining cases corresponding to previously established ordered-triple configurations. For the Cartesian product of four sets, nine cases are examined. Four of these cases provide additional contributions associated with repeated entries, while the remaining cases correspond to established configurations for ordered four-element subsets. In each setting, the relevant monomial contributions are combined over the applicable cycle structures to obtain the corresponding cycle-index formula. Illustrative examples for S6 and S7 are provided for the Cartesian products of two, three, and four sets. The results organise the contributions required for these induced actions and show how repeated entries alter the cycle-index calculations within the framework developed in the manuscript.