Volume-5 Issue-1, April 2025 – Indian Journal of Advanced Mathematics (IJAM)

B120005021025

IJAM Journal — Thu, 24 Apr 2025 10:08:24 +0000

The Indian Journal of Advanced Mathematics (IJAM) has ISSN 2582-8932 (online), an open-access, peer-reviewed, periodical half-yearly international journal, which is published by Lattice Science Publication (LSP) in April and October. The journal aims to publish high-quality peer–reviewed original articles in the area of Mathematics that covers Applied Mathematics, Arithmetic, Algebra, Calculus and Analysis, Geometry and Topology, Trigonometry, Combinatorics, Logic, Number Theory, Numerical Analysis, Dynamical Systems, Differential Equations, Mathematical Physics, Information Theory and Signal Processing, Probability and Statistics, Game Theory, Operations Research, Discrete Mathematics, Business Mathematics and Financial Mathematics. #Applied Mathematics #Arithmetic #Algebra #Calculus and Analysis #Geometry and Topology #Trigonometry #Combinatorics #Logic #Number Theory #Numerical Analysis #Dynamical Systems #Differential Equations #Mathematical Physics #Information Theory and Signal Processing #Probability and Statistics #Game Theory #Operations Research #Discrete Mathematics #Business Mathematics and Financial Mathematics #PhD ademic #Scopus #SCI #LatticeScience #Springer, #ScienceDirect #IEEE #Mendeley #Research #Scholarship #UGC #SSRN #LatticeScience #ESCI #Science #Journal #Conference #SSRN #PubLons

Study of Solution as Locally Asymptotic Attractivity for Nonlinear Functional Integral Equation
Shantaram N Salunkhe

Dr. Shantaram N Salunkhe, Associate Professor and Head, Department of Mathematics, Rani Laxmibai Mahavidyalaya Parola Dist. Jalgaon, (Maharashtra), India.

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© The Authors. Published by Lattice Science Publication (LSP). This is an open-access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Abstract: In this paper, we prove an existence result for local asymptotic attractivity of the solution to a nonlinear functional integral equation under certain conditions, which guarantee both the existence and asymptotic stability of solutions. An example is providing for the indication of natural realisations of abstract theories presented in the paper.

Keywords: Solution of Functional Integral Equation, Fixed Point Theorem, Locally Asymptotic Attractive Solution. Subject Classifications: 47H10, 34A60.
Scope of the Article: Applied Mathematics

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B120105021025

IJAM Journal — Mon, 21 Apr 2025 09:41:33 +0000

Subscription Inventive Over the S-Convex Set in Linear Space
Neelam Dubey

Neelam Dubey, Scholar, Department of Mathematics, Veer Kunwar Singh University, Ara (Bihar), India.

Abstract: In the context of the present paper, our focus revolves around the exploration and analysis of S-convex sets, which represent a fascinating and novel extension of the classical concept of convex sets. We define S-convex sets and discus their main properties as well as the corresponding S-convex hull. Moreover, we clarify the idea of S-linear combinations, which are essential to our research. This first work lays the foundation for a thorough study of S-convex sets and their characteristics. Several examples are provided to illustrate the necessity of the hypotheses underlying various theorems.

Keywords: Convexity, S-Convex Set, Convex Function, Linear Combination, S-linear Combinations, Linear Space, Contemporary Mathematics.
Scope of the Article: Applied Mathematics

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A119405010425

IJAM Journal — Fri, 18 Apr 2025 12:58:01 +0000

Rise of the Gole Number System
Ravi Revelly

Ravi Revelly, The Centre of Quantum Science and Technology, International Institute of Information Technology, Hyderabad, (Telangana), India.

Abstract: For centuries, the decimal number system served various applications, from basic counting to measuring astronomical distances. However, the efficient and humanfriendly representation of huge numbers remains a challenge. For instance, the distance between the Earth and the Moon is 384,400,000 meters, demanding nine digits in decimal representation. To address these challenges, this paper introduces a new number system called the Gole Number System. This new number system is based on an extended radix system, allowing for a compact and efficient representation of large numbers. Specifically, the Gole Number System, derived from the RNumber system with base 100, reduces the number of digits needed for large numbers, achieving a 50% reduction in representation length. By leveraging unique symbols, Gole number system provides compact numbers that can optimize digital displays, memory usage, and computational efficiency. It also offers a unique alignment with the decimal number system, thus making it more familiar to human cognitive abilities and allowing for easy comprehension of the Gole number’s value. This compactness can translate to greater efficiency in storing and transmitting data. Potential applications of this number system are data compression, compact displays, efficient indexing, and secure identification systems. This paper also outlines formal conversion steps and arithmetic operations within the Gole number system, establishing a rigorous mathematical framework for computational applications.

Keywords: Number Theory, Data Compression, Compact Number Representation, Optimal Numeric Encoding, Numerical Optimization, Alternative Number Systems, RNumbers, Gole Number System.
Scope of the Article: Number Theory

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A119805010425

IJAM Journal — Mon, 14 Apr 2025 04:31:36 +0000

An Evaluation of the Dijkstra’s Algorithm, Floyd’s Algorithm and Ant Colony Optimization
A. J. Khan¹, Nikita Dewangan²

¹Dr. A. J. Khan, Associate Professor, Department of Mathematics, MATS University, Raipur (Chhattisgarh), India.

²Nikita Dewangan, Scholar, Department of Mathematics, MATS University, Raipur (Chhattisgarh), India.

Abstract: Travel is the movement of people between distant geographical locations. It can help enhance our well-being, increase our understanding, facilitate socialising with new individuals, promote relaxation and unwinding, encourage adventure, create memories, improve both mental and physical health, and immerse ourselves in different cultures. In this paper, we briefly explain the concept of the shortest path and its various types, which are applicable in practical life. The shortest path problem is a flexible and crucial tool in multiple areas, facilitating effective route planning, network optimisation, and resource allocation. There are many popular algorithms for solving the shortest distance path problem and its variations. We discuss how to create the most efficient route and introduce Dijkstra’s algorithm, Floyd’s algorithm, and Ant Colony Optimisation to minimise the overall path expense, which could be measured by distance, time, or another factor in the given scenario. We use two scenarios to evaluate Dijkstra’s algorithm, Floyd’s algorithm, and Ant Colony Optimisation, determining the shortest route in practical situations to enhance efficiency in solving the same issue.

Keywords: Ant Colony Optimization, Dijkstra’s algorithm, Floyd’s algorithm, Shortest Path, Travelling.
Scope of the Article: Operation Research

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A119605010425

IJAM Journal — Mon, 14 Apr 2025 04:23:53 +0000

A New Proof for Irrationality of π
R. Sivaraman¹, J. Suganthi², P.N. Vijayakumar³

¹Dr. R. Sivaraman, Associate Professor, Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Arumbakkam, Chennai (Tamil Nadu), India.

²J. Suganthi, Assistant Professor and Head, Department of Mathematics, S.S.K.V. College of Arts and Science for Women, Kilambi, Kanchipuram (Tamil Nadu), India.

³P.N. Vijayakumar, B.T. Assistant in Mathematics, Gopalapuram Boys Higher Secondary School, Chennai (Tamil Nadu), India.

Abstract: Ever since Lambert proved that π is irrational in 18th century, various mathematicians have provided lots of wonderful proofs. To this day, π remains one of the most significant real numbers among all real numbers. In this paper, we try to prove that π is irrational in a new and elementary way. In doing so, we have obtained new rational approximations for π.

Keywords: Irrational, Mediant, Closed Intervals, Rational Approximations.
Scope of the Article: Arithmetic

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A119505010425

IJAM Journal — Mon, 14 Apr 2025 04:17:22 +0000

Analysis of Classical Special Beta & Gamma Functions in Engineering Mathematics and Physics
Pranesh Kulkarni

Dr. Pranesh Kulkarni, Assistant Professor, Department of Mathematics, T. John Institute of Technology, Bannerghatta Road Gottigere Bangalore (Karnataka), India.

Abstract: In many areas of applied mathematics, various types of Special functions have become essential tools for Scientists and engineers. Both Beta and Gamma functions are crucial to calculus, as complex integrals can be reduced to a simpler form. In physics and engineering, problems require a detailed knowledge of applied mathematics and an understanding of special functions, such as the gamma and beta functions. The topic of special functions is critical and constantly expanding with the emergence of new problems in the applied sciences. In this article, we describe the basic theory of gamma and beta functions, their interconnections, and their applications to engineering problems.to compute and depict scattering amplitude in Reggae trajectories. We aim to demonstrate that the extension of the classical beta function has numerous applications. It helps in providing new extensions of the beta distribution, as well as new extensions of the Gauss hypergeometric functions and the confluent hypergeometric function, generating relations, and extending Riemann-Liouville derivatives. In this Article, we develop some elementary properties of the beta and gamma functions. We provide multiple proofs for some results. Often, one proof generalizes and others do not. We briefly discuss the finite field analogy of the gamma and beta functions. These are called Gauss and Jacobi sums and are important in number theory. We demonstrate how they can be used to prove Fermat’s theorem, which states that a prime of the form 4n + 1 is expressible as a sum of two squares. We also treat a simple multidimensional extension of a beta integral,

Keywords: Beta Function, Gamma Function, A Pochhammer symbol in Applied Mathematics, Engineering.
Scope of the Article: Applied Mathematics

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B117804021024

IJAM Journal — Sat, 29 Mar 2025 10:40:59 +0000

Impacts of Some Definitions on Algebra of Differential Operators for Noncommutative Algebras
SAMA Anzoumana¹, Bah S.B. Kouame², Kouakou Konan Mathias³

¹Dr. SAMA Anzoumana, Département de Mathématiques, Classes Universitaires Préparatoires aux Grandes Ecoles (CUPGE) Université Polytechnique de San-Pédro, Côte d’Ivoire. 01 BP 1800 San-Pédro 01.

²Sir Bénédic Kouamé Bah, UFR-MI, Laboratoire de Mathématiques et Applications Université Félix Houphouët-Boigny d’Abidjan, Côte d’Ivoire 22 BP 582 Abidjan 22.

³Dr. Konan Mathias Kouakou, UFR-MI, Laboratoire de Mathématiques et Applications Université Félix Houphouët-Boigny d’Abidjan, Côte d’Ivoire 22 BP 582 Abidjan 22.

Abstract: Rings of differential operators are one of the most essential noncommutative (associative) algebras. They play a crucial role in the representation theory of Lie algebras and the algebraic analysis of systems of partial differential equations. However, If A is a commutative and unitary algebra on a field k, Grothendieck defined the ring of differential operators on the algebra A, denoted by D(A), as follows: D(A):=∪Dⁿ(A), where D⁻¹(A)=0 and for n∈ℕ, (1) Dⁿ(A):={u∈ Endk(A):[u, a]=ua-au ∈ Dⁿ⁻¹(A),∀ a∈ A}. In this paper, we show that with this definition, the algebra of differential operators is no longer rich when it is a noncommutative algebra.

Keywords: Differential Operators, Weyl Algebra, Noncommutative Algebra, Derivation on an Algebra.
Scope of the Article: Algebra

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B117704021024

IJAM Journal — Sat, 29 Mar 2025 10:35:10 +0000

On Algebraic and Topological Properties of Some Fundamental Groups
Laurent Djerassem¹, Daniel Tieudjo²

¹Djerassem Laurent, Department of Mathématiques, Université de Ndjamena Tchad, Faculté des Sciences Exactes et Appliquées Ndjamena Tchad, Ndjamena, Commune de Ndjamena, Chad.

²Prof. Daniel Tieudjo, Associate Professor, Department of Mathématiques, Université de Ngaoundere, Ngaoundere Cameroon.

Abstract: In this paper, we study the algebraic and topological properties of some topological spaces. We note that the fundamental group of a topological group is abelian, and we study some spaces of the same homotopy type as the unit circle S¹. The basic group of the unit circle S1 is isomorphic to the additive group of integers. We say that a topological space is simply connected if it is path-connected and has a trivial fundamental group. We show that the fundamental group of an n-punctured plane is free, and we characterise some surfaces as topologically distinct.

Keywords: Homeomorphism, Path, Loop, Topological Group, Fundamental Group, Free Product.
Scope of the Article: Topology

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A119205010425

IJAM Journal — Sat, 29 Mar 2025 10:03:13 +0000

An Elementary Proof for Fermat’s Last Theorem using a Transformation Equation to Fermat’s Equation
P. N. Seetharaman

P.N Seetharaman, Retired Executive Engineer, Energy Conservation Cell), Tamil Nadu State Electricity Board, Anna Salai, Chennai (Tamil Nadu), India.

Abstract: Fermat’s Last Theorem states that there are no positive integers x, y and z satisfying the equation x n + y n = z n , where n is any integer > 2. Around 1637, Fermat proved that there are non-zero solutions to the above equation with n = 4. In the 18th century Euler treated the case n = 3, thereby reducing the proof for the case of a prime exponent ≥ 5 in this proof we hypothesize that r, s and t are positive integers satisfying the equation rp + sp = tp , where p is any prime >3 and establish a contradiction. We use an auxiliary equation, x³ + y³ = z³, and create transformation equations. Solving the transformation equations, we prove that only a trivial solution exists in the primary equation r p + sp = tp .

Keywords: Transformation Equations. Mathematics Subject Classification 2010: 11A–XX.
Scope of the Article: Applied Mathematics

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A119105010425

IJAM Journal — Sat, 29 Mar 2025 09:57:02 +0000

An Elementary Proof for Fermat’s Last Theorem using Three Distinct Odd Primes F, E and R
P. N. Seetharaman

P.N Seetharaman, Retired Executive Engineer, Energy Conservation Cell), Tamil Nadu State Electricity Board, Anna Salai, Chennai (Tamil Nadu), India.

Abstract: In number theory, Fermat’s Last Theorem states that no three positive integers a, b and c satisfy the equation a n + b n = c n where n is any integer > 2. Fermat and Euler had already proved that there are no integral solutions to the equations x 3 + y3 = z3 and x4 + y4 = z4 . Hence, it would suffice to prove the theorem for the index n = p, where p is any prime greater than 3. In this proof, we have hypothesised that r, s, and t are positive integers in the equation rp + sp = tp, where p is any prime greater than 3, and we prove the theorem using the method of contradiction. We have used auxiliary equations x³ + y³ = z³, along with the primary equation rp + sp = tp, which are connected using a transformation equation through the parameters. Solving the transformation equations, we get the result rst = 0, showing that only a trivial solution exists in the central equation.

Keywords: Transformation Equations Two Fermat’s Equations. 2010 Mathematics Subject Classification 2010: 11A–XX.
Scope of the Article: Applied Mathematics

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Volume-5 Issue-1, April 2025 – Indian Journal of Advanced Mathematics (IJAM)

B120005021025

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