iosr-Jm   Volume-21 ~ Issue-5 ~ Sep. – Oct. 2025

Abstract : This paper presents a theoretical framework for understanding geometric objects through multiple representations in mathematics education. We establish mathematical foundations for transforming between different representational forms while maintaining the same geometric object and propose the concept of "representational fluency" as a theoretical construct for mathematics education. The primary contribution is a Translation Principle, which demonstrates that geometric constraints can be reformulated as the domain of a function, along with........

Keywords: geometric representation, mathematical equivalence, representational fluency, coordinate
transformations, mathematics education, theoretical framework.

[1]. Ainsworth, S. (2006). Deft: A Conceptual Framework For Considering Learning With Multiple Representations. Learning And Instruction, 16(3), 183-198.
[2]. Battista, M. T. (2007). The Development Of Geometric And Spatial Thinking. In F. K. Lester (Ed.), Second Handbook Of Research On Mathematics Teaching And Learning (Pp. 843-908). Information Age Publishing.
[3]. Duval, R. (2006). A Cognitive Analysis Of Problems Of Comprehension In Learning Of Mathematics. Educational Studies In Mathematics, 61(1-2), 103-131.
[4]. Edwards, C. H., & Penney, D. E. (2008). Calculus: Early Transcendentals (7th Ed.). Pearson Prentice Hall.
[5]. Goldin, G., & Shteingold, N. (2001). Systems Of Representations And The Development Of Mathematical Concepts. In A. A. Cuoco & F. R. Curcio (Eds.), The Roles Of Representation In School Mathematics (Pp. 1-23). National Council Of Teachers Of Mathematics.

Abstract : The chief interest of this article is to discuss non-archimedean pseudo- differential operator connected to coupled fractional Fourier transform. In this article, we some classes of p-adic complete inner product spaces, Bφ, k(Qp Qp), 0 k < ∞, connected to negative......

Keywords: Non-archimedean analysis, Pseudo-differential operators, Frac- tional Fourier transform, M-dissipative operators.

[1] Alexandra V Antoniouk, Andrei Yu Khrennikov, And Anatoly N Kochubei. Multidimensional Nonlinear Pseudo-Differential Evolution Equation With P- Adic Spatial Variables. Journal Of Pseudo-Differential Operators And Appli- Cations, 11:311–343, 2020.
[2] Andrei Khrennikov, Klaudia Oleschko, And Maria De Jesus Correa Lopez. Modeling Fluid’s Dynamics With Master Equations In Ultrametric Spaces Rep- Resenting The Treelike Structure Of Capillary Networks. Entropy, 18(7):249, 2016.
[3] Klaudia Oleschko And A Yu Khrennikov. Applications Of P-Adics To Geo- Physics: Linear And Quasilinear Diffusion Of Water-In-Oil And Oil-In-Water Emulsions. Theoretical And Mathematical Physics, 190(1):154–163, 2017.
[4] Ehsan Pourhadi, Andrei Khrennikov, Reza Saadati, Klaudia Oleschko, And María De Jesús Correa Lopez. Solvability Of The P-Adic Analogue Of Navier– Stokes Equation Via The Wavelet Theory. Entropy, 21(11):1129, 2019.
[5] Wilson A Zúñiga-Galindo. Pseudodifferential Equations Over Non- Archimedean Spaces, Volume 2174. Springer, 2016.

Abstract : This study investigates the relationship between Pell numbers and Pell-Lucas numbers, which follow the same recurrence relation but differ in initial conditions. The goal of this study is to establish and prove general identities connecting the two sequences through the Principle of Mathematical Induction. Several key identities involving sums, products, squares, and linear combinations were derived and validated.

Keywords: Pell numbers, Pell-Lucas numbers, relationship identities, principle of mathematical induction

[1]. M. Narayan Murty And Binayak Padhy, A Study On Pell And Pell-Lucas Numbers, Iosr Journal Of Mathematics, Volume 19, Issue 2 Series 1, Pp. 28-36, 2023. Doi: 10.9790/5728-1902012836
[2]. S.F.Santana And J.L. Diaz-Barrero, Some Properties Of Sums Involving Pell Numbers, Missouri Journal Of Mathematical Sciences, Doi.10.35834/2006/1801033, Vol.18, No.1, 2006.
[3]. O’regan, Gerard. Mathematical Induction And Recursion. In Guide To Discrete Mathematics: An Accessible Introduction To The History, Theory, Logic And Applications, Pp. 79-88. Cham: Springer International Publishing, 2021..

Abstract : This study explores the impact of solving the Rubik’s Cube on spatial reasoning abilities using inferential statistical methods. Within a sample of 30 participants there existed two cohorts, cubers (n = 15) and non-cubers (n = 15). A two-sample t-test was employed to analyse the outcomes after both groups completed a standardized spatial reasoning test. The findings suggested that cubers scored far higher in spatial reasoning (M = 72.74%) compared to non-cubers (M = 56.63%) because the t-statistic equaled 2.962 as p = 0.006, also this indicates a statistically important.....

[1]. Https://Www.Centraltest.Com/Blog/Spatial-Reasoning-Often-Overlooked-Key-Asset
[2]. Https://Www.Graphpad.Com/Quickcalcs/Pvalue1/
[3]. Https://Www.Assessmentday.Co.Uk/Free/Spatial/1/Index.Php?_Gl=1*1a8i5at*_Gcl_Au*Mtmxotkwntmxmc4xnzq3mza0mzcy*_Ga*Ndu2nti2mdu0lje3ndczmdqznzi.*_Ga_S04nddmhwq*Cze3ndg3otu5mjgkbzikzzekdde3ndg3otu5mzykajuyjgwwjggw
[4]. Https://Psycnet.Apa.Org/Record/2010-16524-002
[5]. Https://Www.Scirp.Org/Reference/Referencespapers?Referenceid=1359229.

Abstract : This paper represents an approximate solution of the fingero phenomenon with inclination. The fingering phenomenon is a well-known instability phenomenon that occurs when a fluid contained in a porous medium is displaced by another of lower viscosity, leading to the formation of perturbations or "fingers" that advance rapidly through the medium. This phenomenon is particularly significant in petroleum engineering, where water injection is commonly used for.....

Key Words: Fingero phenomenon; Homogeneous porous medium; Inclined porous medium; Double phase; Schmidt method; Immiscible fluids; Capillary pressure; Phase saturation

[1]. Scheidegger, The Physics Of Flow Through Porous Media (3rd Edition), Toronto: University Of Toronto Press, 1960, Pp. 229-230.
[2]. E. Scheidegger And E. F. Johnson, "The Statistical Behaviour Of Instabiliies In Displacement Processes In Porous Media," Canadian Journal Of Physics , Vol. 39, No. 2, Pp. 326-334, 1961.
[3]. Verma, "Motion Of Immiscible Liquids In A Cracked Heterogeneous Porous Medium With Capillary Pressure," Rev. Roun. Science And Techlogy Ser. Mecan. Appl., Vol. 13, No. 2, Pp. 277-292, 1968.
[4]. E. Scheidegger, "Growth Of Instabilities On Displacement Fronts In Porous Media," Physics Of Fluids, Vol. 3, No. 1, Pp. 94-104, 1960.
[5]. R. Chuoke, P. V. Meurs And C. V. D. Poel, "The Instability Of Slow, Immiscible, Viscous Liquid-Liquid Displacements In Permeable Media," Petroleum Transactions, Vol. 216, No. 1, Pp. 188-194, 1959

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Abstract : This paper examines the solvability of two non-linear Diophantine equations, 13x + 8y = z² and 5x + 19y = z², within the domain of non-negative integers. Building on the foundational principles of number theory, including the theory of quadratic residues and modular arithmetic, we apply rigorous analytical methods to demonstrate the lack of solutions in ℕ. The results support earlier findings on similar exponential Diophantine forms and further reinforce the use of parity, modularity, and known lemmas for exploring integer constraints.

Key Words: Non-linear Diophantine Equation, Quadratic Forms, Modular Arithmetic, Number Theory, Integer Solutions

[1]. Baker, A., & Wüstholz, G. (2007). Logarithmic Forms And Diophantine Geometry. Cambridge University Press.
[2]. Burton, D. M. (2011). Elementary Number Theory (7th Ed.). Mcgraw-Hill.
[3]. Dickson, L. E. (2005). History Of The Theory Of Numbers, Vol. II. Dover Publications.
[4]. Fergy, J., & Rabago, T. (2016). On The Diophantine Equation 2x + 17y = Z². Int. J. Pure And Appl. Math, 2(2), 65–69.
[5]. Ireland, K., & Rosen, M. (1990). A Classical Introduction To Modern Number Theory. Springer.

Abstract : Let 𝑛 ≥ 1 be an integer and let 𝑆 be the set of unitary divisors of 𝑛, other than 𝑛. Then the set.....

Keywords: Unitary divisor, Cayley graph, unitary divisor Cayley graph, bipartite graph, Hamilton cycle, Eulerian graph

[1] Apostol, Tom M., Introduction To Analytic Number Theory, Springer International Student Edition, (1989).
[2] Berrizbeitia, P. And Giudici, R.E., Counting Pure K-Cycles In Sequences Of Cayley Graphs, Discrete Math., 149(1996), 11-18.
[3] Berrizbeitia, P. And Giudici, R.E., On Cycles In Sequences Of Unitary Cayley Graphs, To Appear. (Reporte Technico No. 01-95, Universidad Simon Bolivar, Dpto. De Mathematicas) Caracas, Venezula, 1995.
[4] Bondy, J.A. And Murty, U.S.R., Graph Theory With Applications, Macmillan, London (1976).
[5] Dejter, I. And Giudici, R.E., On Unitary Cayley Graphs. JCMCC, 18, 1995, 121-124..

Abstract : In this paper, we prove common fixed point theorem for three self mappings using compatible condition in G-metric spaces.....

Keywords: Common fixed point, compatible mappings, G-metric spaces

[1]. Badshah V.H., Gupta Nirmala, Pariya Akhilesh, Sindersiya Antima, “Common Fixed Point Theorem In G-Metric Spaces”, International Journal Of Mathematics Trends And Technology, Vol. 47 (2017), Pp 101-104.
[2]. Jungck G., “Commuting Mappings And Fixed Point”, Amer. Math. Monthly 83 (1976), Pp 261-263.
[3]. Manaro S., Kumar S. And Bhatiya S.S., “Weakly Compatible Maps Of Type (A) In G-Metric Spaces”, Demonstr. Mathe 45(4) (2012), Pp 901-908.
[4]. Mustafa Z. And Sims B., “Some Remarks Concerning D-Metric Spaces”, Proc. Int. Conf. On Fixed Point Theory, Appl. Valencia (Spain), July (2004), Pp 189-198.
[5]. Mustafa Z. And Sims B., “A New Approach To Generalized Metric Spaces”, Journal Of Nonlinear And Convex Analysis Vol. 7 (2), (2006), Pp289–297...

Abstract : In this paper we prove a common fixed point theorem for compatible mappings of type (K) that satisfy an integral-type inequality within the framework of intuitionistic fuzzy metric spaces. Additionally, a common fixed point result is derived for self-mappings in such spaces under the same integral-type inequality conditions. In this paper we will generalize the result of Tenguria A., Rajput A. and Mandwariya V. [17]......

Keywords: Intuitionistic Fuzzy Metric Space, Fixed Point, Compatible Maps of Type (K).

[1]. Atanassov K., Intuitionistic Fuzzy Sets, Fuzzy Sets And System, 20, (1986), 87-96.
[2]. Coker D., An Introduction To Intuitionistic Fuzzy Topological Spaces, Fuzzy Sets And System, 88 (1997), 81-89.
[3]. Deng Z. K., Fuzzy Pseudo-Metric Spaces, J. Math. Anal. Appl., 86 (1982), 74 - 95.
[4]. Erceg M. A., Metric Spaces In Fuzzy Set Theory, J. Math. Anal. Appl., 69 (1979),338- 353.
[5]. Fang J.X., On Some Result Theorems In Fuzzy Metric Spaces, Fuzzy Sets And Systems, (1992), 107-113.
[6]. George A. & Veeramani P., On Some Results In Fuzzy Metric Spaces, Fuzzy Sets And Systems, 64(1994), 395-399

Abstract : In Kenya, over 60% of SMEs collapse annually despite government support, with the COVID-19 pandemic worsening the situation as 51.28% went bankrupt, 15.38% refinanced, and 12.82% exited. Handicraft enterprises in Nairobi City County (NCC) face similar challenges but remain under-researched despite their economic potential. Their growth is constrained by limited funding, restrictive policies, inadequate knowledge, and poor adoption of technology. Statistics indicate that three out of every five handicraft businesses fail within six months, and 80% collapse......

Keywords: Financial literacy, Investment decisions, Handicraft enterprises, financial performance, Strategic investment decisions, Budgetary investment decisions, Expansion investments decisions

[1]. Amaya (2021). Five-Stage Model For Firms’ Expansion Decisions. The Wolfson School Of Mechanical And Manufacturing Engineering. Loughborough University, Leicestershire LE11 3TU
[2]. Ackermann, N. & Loy, B. (2022). How Do Short-Term Investments Work? Retrieved From:
Https://Study.Com/Learn/Lesson/Short-Term-Investments-Treasuries-Examples.Html
[3]. Ahinful, G. S., Boakye, J. D., & Osei Bempah, N. D. (2021). Determinants Of Smes’ Financial Performance: Evidence From An Emerging Economy. Journal Of Small Business & Entrepreneurship, 1-24.
[4]. Alkaraan, F. (2020). Strategic Investment Decisions: Organizational Culture And Financial Considerations. Journal Of Business Research, 112, 456-467. Https://Doi.Org/10.1016/J.Jbusres.2019.11.041
[5]. Amuko, D. A. (2015). Smess’ Investment Decisions: An Assessme

Abstract : Mathematical modelling provides a powerful framework for analysing and predicting dynamics of addiction to a drug. This study presents a mathematical model to evaluate the impact of education on drug addiction. The analysis highlights how increased awareness and preventive education can reduce addiction rates and promote rehabilitation. The model employs a compartmental approach, consisting of six non-linear differential equations, to analyze these dynamics. The basic reproduction number is derived to determine the threshold for the persistence or elimination......

Keywords: Mathematical Model, Reproduction Number, Next Generation Matrix, Global Stability, Lyapunov Function

[1] Andrawus, J., Ibrahim, K. G., Abdullahi, I., Abubakar, A., & Maiwa, S. I. (2024). Mathematical Modelling On Drug Addiction With Awareness Control. Journal Of Advanced Science And Optimization Research.
[2] Andrawus, J., Iliyasu Muhammad, A., Akawu Denue, B., Abdul, H., Yusuf, A., & Salahshour, S. (2024). Unraveling The Importance Of Early Awareness Strategy On The Dynamics Of Drug Addiction Using Mathematical Modeling Approach. Chaos: An Interdisciplinary Journal Of Nonlinear Science, 34(8).
[3] Babaei, A., Jafari, H., & Liya, A. (2020). Mathematical Models Of HIV/AIDS And Drug Addiction In Prisons. The European Physical Journal Plus, 135(5), 1-12.
[4] Binuyo, A. O. (2021). Mathematical Modelling Of The Addiction Of Drug Substances Among Students In Tertiary Institutions In Nigeria.
[5] Bansal, K., Mathur, T., Singh, N. S. S., & Agarwal, S. (2022). Analysis Of Illegal Drug Transmission Model Using Fractional Delay Differential Equations. AIMS Mathematics, 7(10), 18173-18193.

Abstract : For each positive integer n, let Zn be the additive group of integers modulo n and let S be the set of all numbers less than n and relatively prime to n. The Euler totient Cayley graph G(Zn, ) is defined as the graph whose vertex set V is the set Zn = { 0, 1, 2, …, n–1} and the edge set is given by E = { ( x, y) / ( x –y  S, or, y – x  S }. In a graph G, a vertex v and an edge e in G are said to cover each other if they are incident. An edge cover of a Graph G is a set of edges covering all the vertices of G. A minimum edge cover is the one with minimum cardinality. The......

Keywords:Euler totient Function, Cayley graph, Edge cover, Edge covering number, edge dominating set, edge domination number Matching number.

[1]. Allan, R. B., And Laskar, R., On Domination And Independent Domination Number Of A Graph , Discrete Math , 23(1978), 73-76.
[2]. Apostol, Tom M., Introduction To Analytic Number Theory, Springer International Student Edition , (1989).
[3]. Arumugam, S., Uniform Domination In Graphs, National Seminar On Graph Theory And Its Applications , January(1993).
[4]. Berege, C., Theory Of Graphs And Its Applications, Methuen ; London (1962).
[5]. Bondy, J.A., And Murty , U.S.R., Graph Theory With Applications, Macmillan , London (1976).