https://thejournalshouse.com/index.php/Journal-Maths-Stats Advanced Research Publications en-US Journal of Advanced Research in Applied Mathematics and Statistics 2455-7021 https://thejournalshouse.com/index.php/Journal-Maths-Stats/article/view/2284 <p>This research introduces Algebras, a novel algebraic framework defined by a set and a binary operation that adheres to the principles of Asymmetry, Transitive Difference, and Left Identity/Dominance. The study systematically develops Subalgebras as closed subsets, showing that they uphold all the axioms of Algebras and that their non-empty intersections also constitute subalgebras. This concept leads to the notion of a uniquely smallest generated subalgebra. Moreover, the study introduces Normal Sub algebras, a specific type of sub algebra that exhibits a "Left Absorption Property," meaning that for any element x from the main algebra and n from the normal sub algebra, indicating strong left-invariance. Additionally, Isomorphisms are defined as bijective mappings that preserve operations between Algebras. Theorems demonstrate that these isomorphisms maintain all fundamental axioms and the subalgebra structure. It is also established that the inverse of an Isomorphism is itself an Isomorphism, confirming algebraic equivalence. This foundational research aims to classify and understand the structural relationships within Algebras.</p> <p><strong>How to cite this article:</strong></p> <p>Perarasan, M. Vasuki, Mishra A k , B. Mohamed Harif, A. Dinesh Kumar, Celestin M, An Introduction to HP-Algebras: Axiomatic Foundations and Basic Theory, <em>J Adv Res Appl Math Stat, Vol 11, Issue 1&amp;2 2026: Pg. No. 14-25.</em></p> <p><strong>DOI : https://doi.org/10.24321/2455.7021.202606</strong></p> Dr Anjay Kumar Mishra K. Perarasan M. Vasuki B. Mohamed Harif A. Dinesh Kumar Mbonigaba Celestin Copyright (c) 2026 Journal of Advanced Research in Applied Mathematics and Statistics 2026-07-17 2026-07-17 11 3&4 14 25 https://thejournalshouse.com/index.php/Journal-Maths-Stats/article/view/2296 <p>Among the great mathematicians of India, there is a renowned name “Brahmagupta”. Brahmagupta’s formula is the well-known <br>result in geometry. This paper is based on the ancient Indian mathematics, and their work, which mainly focuses on Brahmagupta formula. The formulae enunciated by Brahmagupta for the area of a cyclic quadrilateral is proved with an easier method. In addition, various formulas for the area of different types of quadrilaterals are derived using the Brahmagupta’s formula. The applications of Brahmagupta formula, and its relevance in modern era, have also been discussed in the paper. In this entire review, we give a detailed overview of Brahmgupta formulas, particularly, we describe how we can use this formula to <br>obtain area of different geometrical shapes that have been widely used in the school mathematics.</p> <p><strong>How to cite this article:</strong> <br>Kaushik R, Purwar K, Singh R B, Review and <br>Applications of Brahmagupta’s Formula: A <br>Priceless Morsel in Ancient Indian Mathematics, <br>Vol 11, Issue 3&amp;4, 2026: Pg. No. 1-13.</p> <p><strong> DOI:</strong> https://doi.org/ 10.24321/2455.7021.202605</p> Rajat Kaushik Kavyansh Purwar Ram Bharat Singh Copyright (c) 2026 Journal of Advanced Research in Applied Mathematics and Statistics 2026-07-15 2026-07-15 11 3&4 1 13