Asian Journal of Mathematical Sciences(AJMS)

A New Proof of the Pythagorean Theorem Using a Rhombus

Firdous Ahmad Khan — 2025-07-10

This short note presents a new proof of the Pythagorean Theorem using the concept of a rhombus. MSC Code: 97-01, 97G10 , 97G40.

ON THE USE OF BAYESIAN PROBABILITY NETWORKS WITH A REVIEW OF MALARIA CASE

Emmanuel Segun Oguntade — 2025-06-10

In spite of the versatile and general acceptability of estimation of disease cases from the various available methods in literature, incorporating model uncertainty remains an open issue. In this article, we derived a probability based graphical model using expert opinions in related studies on malaria and its hypothesized predictors with a Bayesian belief network (BBN). This approach is well applied in ecological studies and other environmental sciences in recent times for various estimations and predictions based on Bayesian reasoning. We gave a brief description of a BBN framework, its pros and cons, examine the principle of conditional independence. Also, we explore Markov Chain principles as it relates to a BBN formulation and useful guidelines for developing the preliminary structure of the network. We finally derived the topology of a BBN as a directed acyclic graph with malaria predictors as network nodes. We also illustrated the use of the network with an illustrative example.

Numerical Range Distortion Under Unitary Equivalence

Faith Mwangi Muthoni — 2025-06-10

This paper investigates the preservation properties of numerical ranges under unitary equivalencen transformations in operator theory. We establish that if operators T and S are unitarily equivalent via S = U ∗ TU , then their numerical ranges are identical: W (S) = W (T ). Beyond this fundamental equality, we prove that unitary equivalence preserves critical geometric properties of numerical ranges, including extreme points, exposed points, supporting lines, contact points, and the geometric multiplicity of boundary points. For normal operators, we demonstrate that numerical range equality characterizes unitary equivalence, providing a geometric criterion for this algebraic relation. Additionally, we show that the curvature of the boundary of numerical ranges remains invariant under unitary transformations. These results highlight the deep connection between the algebraic structure of operators and the geometric properties of their numerical ranges, contributing to our understanding of operator behavior under unitary equivalence.

BANACH'S FIXED POINT THEOREM ON THE EXISTENCE AND UNIQUENESS OF VOLTERRA INTEGRAL EQUATION SOLUTION

GANI GUNAWAN — 2025-06-10

An integral equation is an inversion of a differential equation. Therefore, an integral equation can be seen as the solution of a differential equation with a given condition at an interval. So that through this integral equation, it can be ensured that the existence of a solution to the differential equation can be ascertained.
In this article, it is shown in a functional analysis how Banach's fixed-point theorem can show the existence and uniqueness of the solution of integral equations, so that it can guarantee the existence of the solution of a differential equation.

COMPARATIVE STUDY OF TUMOR STAGE PREDICTION IN BREAST CANCER THROUGH FACTOR ANALYSIS AND MULTINOMIAL LOGISTIC REGRESSION

AWOGBEMI CLEMENT ADEYEYE1 — 2025-06-10

Breast cancer remains one of the leading causes of cancer-related morbidity and mortality among women globally. Early-stage detection of tumor of breast cancer assists in providing preventive measures against its spread to other parts of the body. In this study, factor analysis was employed in reducing the number of genes to fewer principal components. The Scree and Eigenvalue methods of selecting the number of principal components were employed for comparison purpose. Multinomial logistic regression model was employed to fit the stage of tumor of the breast cancer on the scores of the principal component variables along with the patient’s age and tumor size. The findings of the study showed that the eigenvalue approach outperformed the Scree approach ranging from the percentage of variance explained to the accuracy level. Hence, the eigenvalue method of selecting number of components to include in factor analysis is recommended.