A Theoretical Exploration of Paraletrix Calculus as an Extension of Rhotrix Mathematics
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94-103
Digital Object Identifier:
10.58578/amjsai.v3i1.9100
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Authors:
Isa Yahaya1, Salisu Lukunti2, Umar Mujahid Aliyu3, Imafidor Hassan Ibrahim4, Mohammed Abubakar Kolo5, Sulaiman Ahmad6, Nura Hashim7, Mohammed Yusuf Marafa8 
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Abstract
This paper, titled A Theoretical Exploration of Paraletrix Calculus as an Extension of Rhotrix Mathematics, builds upon earlier studies in generalized matrix theory by extending the structural and operational framework of non-standard matrix-like objects. Atanassov and Shannon [1] first introduced matrix-tertions and matrix-ngittrets as entities that interpolate between 2-dimensional vectors and 2×2 matrices, thereby enriching the conceptual landscape of generalized matrices. Ajibade [2] subsequently advanced the field by proposing thotrices as intermediates between 2×2 and 3×3 matrices, while further developments in rhotrix theory have established various multiplication techniques, such as heart-oriented and row–column multiplications—and yielded several important results. Recognizing the diversity of both rectangular and square matrices, the paraletrix structure was formulated as a generalization of the thotrix, allowing unequal numbers of rows and columns and thus providing a more flexible algebraic setting. This study extends the mathematical framework by introducing differentiation and integration within paraletrix calculus, defining these operations for paraletrix-valued functions with respect to an independent variable. In doing so, it lays the groundwork for a coherent calculus on paraletrices as a theoretical extension of rhotrix mathematics and generalized matrix theory.
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References
[2]. Ajibade, A. O. (2003). The concept of rhotrix in mathematical enrichment. International Journal of Mathematical Education in Science and Technology, 34(2), 175–179. https://doi.org/10.1080/0020739021000053828
[3]. Sani, B. (2004). An alternative method for multiplication of rhotrices. International Journal of Mathematical Education in Science and Technology, 35(5), 777–781. https://doi.org/10.1080/00207390410001716577
[4]. Sani, B. (2007). The row–column multiplication of high dimensional rhotrices. International Journal of Mathematical Education in Science and Technology, 38(5), 657–662. https://doi.org/10.1080/00207390601035245
[5]. Aminu, A., & Michael, O. (2014). An introduction to the concept of paraletrix, a generalization of rhotrix. Afrika Matematika. Advance online publication. https://doi.org/10.1007/s13370-014-0251-1
[6]. Aminu, A., & Michael, O. (2015). An introduction to the concept of paraletrix, a generalization of rhotrix. Afrika Matematika, 26(5–6), 871–885. https://doi.org/10.1007/s13370-014-0251-1
[7]. Ndubuisi, R. U., Abubakar, R. B., & Udoaka, O. G. (2024). Some results on heart-oriented paraletrix ring. Journal of Applied Mathematics and Computation, 8(1), 1–6. https://doi.org/10.26855/jamc.2024.03.001