Asian Research Journal of Mathematics

Roots of General Equation of Degree Five

Rajesh Kumar Maurya — Mon, 25 May 2026 00:00:00 +0000

We know formula for roots of general polynomial equations of degree two, three and four, also it has been proven that formula for general quintic could not written in the form of radicals, as on today there does not exist formula for the root of general quintic, In this article we shall find formula for roots of general equation of degree 5 which is not based on radicals.

Algebraic Structure of ς-Dual Constacyclic Codes of Length \(p^s\) over Non-Chain Ring

Somaiyah A. A. Abdulsattar, Arunkumar Patil — Tue, 26 May 2026 00:00:00 +0000

Let p be a prime and let R = \(\mathbb{F}_p\)m [u, v]/⟨u², v², uv−vu⟩ be a finite commutative non-chain ring with u² = v² = 0. We study ς-dual constacyclic codes of length ps over R, where ς is an automorphism of R. After explicitly characterizing the unit group of R and the full automorphism group Aut(R), we determine all admissible constacyclic shift constants λ. Exploiting the idea structure of R[x]/⟨xp^s− λ⟩, we derive generator polynomials for C^⊥ς in both the principal and non-principal cases—a distinction that arises precisely because R is non-chain. Because R is a non-chain ring, the ideal structure of the associated quotient ring gives rise to both principal and non-principal ideals, and we treat both cases separately, deriving explicit generator polynomials for each. We establish necessary and sufficient conditions on λ and ς for ς-self-dual codes to exist. These results generalize and unify prior work on constacyclic codes over chain rings such as \(\mathbb{F}_p\)m [u]/⟨u^k⟩, and yield new families of constacyclic codes with prescribed duality properties.

The Use of the Polya Method in Problem-Solving Skills of Students in General Education Mathematics Core Course

Mary Joyce C. Dueñas — Thu, 28 May 2026 00:00:00 +0000

The George Pólya method strengthens mathematical problem-solving by guiding students through structured thinking, encouraging independence, and improving overall learning outcomes. This study evaluated the extent to which the Polya method improved students’ mathematical problem-solving skills. It specifically assessed the respondents’ competency, performance, and challenges encountered in the four phases of Polya’s approach: understanding the problem, devising a plan, carrying out the plan, and looking back. It determines whether the Polya method is an effective teaching and learning approach that enhances students' problem-solving skills. The research employed a one-group pre-posttest mixed-methods design. The research uses a one-block comprising 34 students. This study administered a pre-test and post-test questionnaire for the quantitative part and later identified the challenges respondents encountered using Polya's stages through qualitative interviews via audio call. This study was conducted at Western Philippines University over three months (September 2025 to November 2025) during the first semester of the 2025-2026 academic year. The research instrument used in this study comprised two components: the first assessed respondents’ problem-solving ability across the Polya stages; it consisted of 10 arithmetic problem-solving questions, 10 for each pre-test and post-test, and underwent face and content validity testing. Using SPSS software, the acquired data were examined utilizing the Wilcoxon Signed Rank test, mean, standard deviation, and percentage. For the qualitative part, the research interviewed ten (10) students who scored below the third quartile regarding the challenges they encountered at each stage of the Polya method, and the data were interpreted using thematic coding. The majority of respondents were categorized as Level 1 (Needs Improvement) in the pre-test findings, indicating low skill across all stages of Polya's problem-solving process. The respondents' performance improved significantly across the following stages of the Polya method: understanding the problem, devising a plan, and carrying out the plan after the intervention (i.e., the Polya method). Even though there was progress, many respondents still struggled with looking backstage. The mean score percentage increased from 33.62% in the pre-test to 75.41% in the post-test, indicating a significant improvement in the respondents’ mathematical problem-solving skills. The results indicated a significant difference between pre-test and post-test scores across all four stages of Polya’s technique, yielding a p-value = 0.00, which is less than the 0.05 level of significance (p < 0.05). Polya’s approach to problem-solving is a useful way to teach and raise students’ proficiency in solving real-world mathematics problems. Nonetheless, ongoing practice and instructional support are advised, especially to enhance students' reflective thinking and assessment skills during the looking-back stage of Polya’s problem-solving process. Strengthening students' creative problem-solving skills develops more competent and independent problem-solvers and, on a larger scale, potentially improves students’ performance in mathematics on regional, national, and international mathematics proficiency assessments.

Assessing Assessment: Analyzing Classroom Assessment Practices of Mathematics Teacher Educators

Evans Atteh, Christipher Adjei Okpoti, Charles Kojo Assuah — Sat, 30 May 2026 00:00:00 +0000

Assessment in mathematics education is the systematic process of collecting and interpreting evidence of student learning to guide instruction, improve teaching practices, and evaluate student achievement and educational programs. The study investigated the types of assessment methods employed, the underlying rationales for these practices, and the cognitive levels of assessment questions used. The study adopted qualitative case study design to examined the classroom assessment practices of 14 mathematics teacher educators across three colleges of education in Ghana. Data were collected through semi-structured interviews and analysis of assessment portfolios, with findings analyzed using thematic analysis, frequencies and percentages. Mathematics teacher educators utilized diverse assessment methods, including assignments, quizzes, group work, presentations, and project work. However, while these practices effectively serve the purposes of grading and identifying student learning difficulties to inform instructional adjustments, the assessment questions predominantly assess lower-order cognitive skills such as recall and basic comprehension. The findings have significant implications for college administration and teacher educators in supporting the development of effective assessment practices among preservice teachers. To enhance assessment quality and prepare preservice teachers for contemporary classroom practice, the study recommends that mathematics teacher educators deliberately balance lower-order questions with higher-order questions that assess application, analysis, evaluation, and creation. This balanced approach will ensure that preservice teachers develop competency in assessing not only factual knowledge but also critical thinking and problem-solving abilities that are essential competencies for effective mathematics instruction and student learning in the 21st century.

Transient Velocity Analysis of Three-Phase Immiscible Maxwell Fluids in Clear and Porous Cylindrical Media

Emeng Emeng Iyeme — Sat, 30 May 2026 00:00:00 +0000

The transient flow behaviour of three immiscible fractional Maxwell fluids in a cylindrical domain was investigated. The domain comprises a central clear region and two surrounding porous layers. An incompressible and one-dimensional flow is assumed. This flow is driven by a quick, unexpected motion of the cylinder. A viscoelastic memory effect was captured by using fractional relations to formulate the governing equations. These equations are then non-dimensionalised to introduce parameters such as the Reynolds and Darcy numbers, porosity coefficient, and relaxation time. An analytical solution of the resulting system of equations using Olayiwola’s Generalised Polynomial Approximation Method (OGPAM) is presented. Velocity expressions are obtained for each region and used in the analysis of the effects of inertia, permeability, porosity, and viscoelastic memory on the fluid's motion. Graphical summaries using Maple 2018 reveal that velocity increases in the clear region and decreases in the porous layers due to the effect of resistance. Flow velocity is enhanced with the increase in Reynolds and Darcy numbers, while fluid motion in the porous region is aided by higher porosity. The importance of memory effects in viscoelasticity is emphasised as relaxation time significantly influences transient behaviour. The model is physically consistent, as continuity of velocity across interfaces is maintained. The study demonstrates the effectiveness of OGPAM in providing semi-analytical solutions and reducing computational complexity. The study can be applied to biomedical systems, enhanced oil recovery processes, and porous media flow.

A Hierarchical Approach to Nonlinear Rational and \(\varphi\)–Contractive Mappings in Fuzzy Metric Spaces with Applications

Makhan Satpute, Manoj Ughade, Rashmi Tiwari, Kamal Vadhawa — Tue, 02 Jun 2026 00:00:00 +0000

In this paper, we develop a unified framework of nonlinear contractive mappings in fuzzy metric spaces (\(\mathfrak{X}\), ℳ, * ) by integrating classical contraction principles such as Banach-type, Kannan-type, and Reich-type contractions with newly introduced rational and \(\varphi\)–nonlinear contractive conditions. The proposed \(\varphi\)-nonlinear contraction, governed by a control function \(\varphi\) : [0, 1] → [0, 1] satisfying \(\varphi\)(s) > s for all s ∈ (0, 1), provides a significant generalization of standard linear contraction models. Under suitable conditions, we establish existence and uniqueness of fixed points for self-mappings in complete fuzzy metric spaces and develop a hierarchy of contractions that clarifies the structural relationships among the introduced classes. The theoretical results are supported by illustrative examples and applications to nonlinear integral equations and fractional differential equations, demonstrating the effectiveness of the proposed framework in addressing problems arising in nonlinear and applied analysis.

Mathematical Modeling of Mentor–Mentee Relationship via Non-linear Ordinary Differential Equations

Rajat Kaushik, Aji Thomas, A. K. Garg, Ram Keval — Wed, 03 Jun 2026 00:00:00 +0000

In the teaching profession, mentoring plays a vital role in promoting effective learning and professional development. The mentor–mentee relationship acts as an important mechanism for knowledge sharing, skill enhancement, and collaborative problem-solving. In this study, we propose a two-dimensional ordinary differential equation (ODE) model to investigate the dynamics of mentor–mentee interactions. The proposed model is rigorously analyzed for positivity, boundedness, uniform permanence, and sensitivity with respect to key parameters. Furthermore, the local and global stability of the positive equilibrium point is examined using mathematical techniques. Numerical simulations are performed to support the analytical findings and to illustrate the significant impact of mentor–mentee relationships on the overall system dynamics.

On an Infinite-server Queue with Consultation by a Consultant

T. Resmi — Wed, 03 Jun 2026 00:00:00 +0000

This paper analyzes an infinite-server queueing model with consultation by a consultant. The consultant does not directly serve customers, but instead provides consultation to the servers. The arrivals to the system and requirement of consultation follow independent Poisson processes. The service times at the servers are assumed to follow mutually independent and identically distributed exponential random variables. The duration of consultation is an exponentially distributed random variable. Using Neuts Rao’s truncation method, we derive explicit expression for the stability of the system with a sufficiently large level N and an appropriate truncation error ϵ. A special case of the model corresponding to zero consultation rate is considered, and it is shown that the system reduces to the classical M/M/∞ queueing model. Several important performance measures are analyzed numerically, followed by graphical illustrations and a sensitivity analysis of the system parameters.