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.

Oscillation Criteria for First-Order Non-Linear Neutral Difference Equations with Several Delay Arguments

D. Palanisamy — 2026-08-24

This paper investigates the oscillatory behaviour of the following first-order non-linear neutral delay difference equation involving several delay arguments that are not necessarily monotone:

\[\Delta\left[\omega(v)-\sum_{i=1}^r \rho_i(v) \omega\left(\tau_i(v)\right)\right]+\sum_{i=1}^k q_i(v) g_i\left(\omega\left(\phi_i(v)\right)\right)=0 ; \quad v \geq v_0 .(*)\]

where {pi (v) } (1\(\le\) i \(\le\) r) and {qi (v) } (1 \(\le\) i \(\le\) k ) are sequences of real numbers, and {Ti (v) } (1 \(\le\) i \(\le\) r} and {\(\phi\)i (v) } (1 \(\le\) i \(\le\) k) are sequences of positive integers. Under appropriate assumptions on the neutral coefficients, delay sequences, and non-linear factors, new sufficient conditions are established to guarantee that every solution of the considered equation is oscillatory. The study uses the development of suitable auxiliary sequences, comparison methods, and exponential estimates for eventually positive solutions. Several auxiliary lemmas are established to obtain the main oscillation theorems formulated in terms of lim inf and lim sup conditions. The criteria obtained generalise and improve a number of existing oscillation results in the literature by allowing the simultaneous presence of several non-monotone delay arguments and non-linear neutral terms.

Deep Model Combinations Based on Structured Collaborative Learning for Pancreas Image Segmentation

Jinn-Yi Yeh — 2026-08-28

Pancreas segmentation from computed tomography (CT) images is challenging because of anatomical variability, irregular morphology, and the complexity of surrounding abdominal structures. Although deep neural networks can provide strong segmentation performance, their computational demands and prolonged training can limit use in resource-constrained settings. This study evaluates Structured Knowledge Distillation via Collaborative Learning (SKDCL), a teacher-free, single-stage framework that integrates collaborative learning with structured distillation for lightweight pancreas segmentation. Six architectures—UNet, PSPNet, DeepLabv3+, ERFNet, ESPNet, and ENet—were evaluated in paired combinations under consistent experimental settings. The NIH Pancreas-CT dataset comprised 6,882 abdominal CT images from 82 patients and was divided into training, validation, and test sets in proportions of 70%, 15%, and 15%, respectively. Structured distillation incorporated region-level and prediction-level information to strengthen spatial representation learning, while performance was assessed using the Dice similarity coefficient (DSC) and total training time. Collaborative learning improved segmentation performance relative to independent model training, and structured distillation produced further gains across the evaluated combinations. The highest DSC was 93.56% for ESPNet, while ENet achieved 92.83% in a structured collaborative pairing. These findings show that SKDCL can support competitive segmentation performance in lightweight two-dimensional models without requiring a pre-trained teacher network. The framework therefore offers a computationally efficient approach to pancreas CT segmentation, although validation on additional datasets and extension to three-dimensional architectures remain necessary.

Modeling the Adoption of Artificial Intelligence (AI) Learning Tools among University Students Using the SEIR Model

Benjamin Adu Obeng — 2026-09-02

Background: The increasing integration of artificial intelligence (AI) learning tools in higher education has created a need to understand how their adoption spreads and is sustained among university students.

Aims: This study examines how AI learning tools spread among university students and identifies the conditions that support or limit their continued adoption.

Methodology/Design/Approach: An SEIR-based mathematical model was developed to represent students' progression from being unaware of AI tools to becoming aware of them, using them, and becoming fully integrated users. The model was analysed using equilibrium and stability analysis, sensitivity analysis, and numerical simulations.

Findings: The results show that AI adoption can become widespread when R₀> 1. For example, when R₀= 3.463, the number of students using and becoming integrated with AI tools increases considerably over time. However, when R₀= 0.16 < 1, adoption gradually declines, showing that AI use may not be sustained without sufficient awareness and continued support. The sensitivity results also indicate that the rate of active AI use plays an important role in encouraging greater student integration.

Limitation: The model assumes that students interact in relatively similar ways and that the model parameters remain constant. It therefore does not fully account for differences among students, institutional settings, or changes in the learning environment.

Originality: The study provides a different perspective on AI adoption by using a mathematical model to examine how AI use can spread across a university student population rather than focusing only on individual students' willingness or intention to use AI. The findings can help universities understand the conditions needed to encourage sustained and responsible adoption of AI learning tools.