Indian Journal of Advanced Mathematics (IJAM)
type: channel
description: 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. Authors can electronically submit articles throughout the year using the Article Submission System. Authors can format their article either in (i) a single-column format or (ii) as per the journal template. The submitted articles should not have been previously published or are currently under consideration for publication elsewhere. The journal does not accept brief or short notes for publication. The editors retain the right to reject any articles that lack quality or originality without sending them for review. All articles must fall within the journal’s scope and will undergo a double-anonymized peer-review process. Authors must confirm that they have read and understood the content of their submitted article and ensure that it meets acceptable English grammar and usage standards. To help with the proofreading process, authors can use tools like Grammarly or similar applications. As an open-access journal, authors must pay an Article Processing Charge (APC) to publish their articles and retain copyright. Additionally, authors should familiarise themselves with the editorial and publishing policies of the journal.
publisher: Lattice Science Publication (LSP)
language: en
publication name: Indian Journal of Advanced Mathematics (IJAM)
issn: 2582-8932
copyright: CC-BY-NC-ND 4.0
items: An Elementary Procedure in the Proof of Fermat's Last Theorem
items: Re-Evaluating the Value of π and Emerging New Concepts for Measuring Areas and Volumes Based on Translations and Rotations of Straight Lines, Planes, and Circles
items: Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums
items: An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM
items: Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region
items: Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response
items: Comment on ‘The Triple Product Rule and the Subtleties of the Mathematical Tools Used in Thermodynamics’ by Felipe Américo Reyes Navarro and Rafael Edgardo Carlos Reye
items: A Comparative Study on the Effects of Vertical and Horizontal Magnetic Fields on Double Diffusive Convection with Soret Effect in Couple-Stressed Nanofluids
items: Completed Functor S-1() of the Localization Functor S-1(), Isomorphism and Adjunction
items: Study of Some Degree Five Identities of Type (4,1)
items: Existence and Uniqueness of Weak Solutions to the Fractional Navier-Stokes Equations in Turbulent Flows: A Study of Boundedness and Stability
items: An Innovative Approach to Find Remainder
items: Some Results on Specificity of Possibility Distributions
items: Study of Solution as Locally Asymptotic Attractivity for Nonlinear Functional Integral Equation
items: Subscription Inventive Over the S-Convex Set in Linear Space
items: Rise of the Gole Number System
items: An Evaluation of the Dijkstra's Algorithm, Floyd's Algorithm and Ant Colony Optimization
items: Analysis of Classical Special Beta & Gamma Functions in Engineering Mathematics and Physics
items: A New Proof for Irrationality of π
items: An Elementary Proof for Fermat's Last Theorem using a Transformation Equation to Fermat's Equation
items: An Elementary Proof for Fermat's Last Theorem using Three Distinct Odd Primes F, E and R
items: New Concepts of Congruences Modulo 1/n and Positive Integers Modulo 1/n
items: On Algebraic and Topological Properties of Some Fundamental Groups
items: Impacts of Some Definitions on Algebra of Differential Operators for Noncommutative Algebras
items: An Elementary Chapter in Number Theory: Proof of Fermat's Last Theorem
items: On Laplace-Riemann Derivatives
items: On the Model of Transport of Salt Solution in a Xylem
items: A Proof for Fermat's Last Theorem using an Auxiliary Fermat's Equation
items: An Elementary Proof for Fermat’s Last Theorem using Ramanujan-Nagell Equation
items: On the Nörlund-Rice Integral Formula

An Elementary Procedure in the Proof of Fermat's Last Theorem
type: item
description: Pierre de Fermat first stated, around 1637, that for any integer n > 2, the equation an + bn = cn has no positive integer solutions, and he wrote the theorem in the margin of a copy of Arithmetica. His proof is available only for the equation a 4 + b 4 = c 4 for the exponent n = 4. Subsequently, Euler proved the theorem in the equation a 3 + b 3 = c 3 for the exponent n = 3. Taking the above two proofs of Fermat and Euler, it would suffice to prove the theorem for n = p, where p is any prime > 3. In this proof, we hypothesize all r, s and t as positive integerssatisfying the equation rp + sp = tp and establish a contradiction. We use another auxiliary equation, x 3 + y 3 = z 3 , and combine the two equations using transformation equations. Solving the transformation equations, we establish a contradiction, thereby proving the theorem.
subject: Articles
  topic: section
subject: Transformation Equations
  topic: keywords
creator: P.N. Seetharaman
rights: Copyright (c) 2025 P.N. Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 95
ending page: 100
doi: 10.54105/ijam.A1227.05021025

Re-Evaluating the Value of π and Emerging New Concepts for Measuring Areas and Volumes Based on Translations and Rotations of Straight Lines, Planes, and Circles
type: item
description: The universal constant pie has been assigned a nonconverging value of approximately 3.145..., and this non-repeating, non-terminating decimal extends to millions—even trillions—of digits beyond the decimal point, as calculated by the most advanced computational methods. However, for thousands of years, mathematicians across different civilizations have attempted to determine the value of this fundamental constant in various ways. In geometry, angles are commonly expressed in terms of pie radians, and pie appears prominently in the formulas used to calculate the areas and volumes of curved geometrical figures such as circles, ellipses, spheres, cones, pyramids, and cylinders. Thus, pie has become an integral part of how we measure areas and volumes. Ancient scientists, in their efforts to understand and quantify the dimensions of curved or non-linear topological objects, sensed a recurring spatial constant underlying these forms. Over time, this invariant quantity came to be known as “pi.” The pursuit of its exact value has become a continuous endeavour spanning millennia, with each generation refining its estimation using increasingly sophisticated tools and methods. In this article, it is first emphasised that, although the mathematical parameters of area and volume have traditionally been expressed through formulas involving length, breadth, and height, their more profound physical significance has not been adequately explored. This work attempts to redefine area and volume fundamentally in terms of distance or length alone, offering new mathematical formulations for the areas and volumes of basic geometric shapes such as squares, cubes, circles, spheres, and others. Another critical shortcoming of conventional mathematics lies in its neglect of the reciprocal or inverse space of the universe when evaluating the value of pie. In recent research, it has been proposed that pie itself represents a circle whose radius corresponds to the smallest conceivable length in …
subject: Articles
  topic: section
subject: Pi (π)
  topic: keywords
subject: Reciprocal Geometry
  topic: keywords
subject: Area and Volume Redefinition
  topic: keywords
subject: Geometric Constants
  topic: keywords
subject: Cosmological Geometry
  topic: keywords
subject: non-Converging
  topic: keywords
subject: Planck Length
  topic: keywords
creator: Nishant Sahdev
creator: Chinmoy Bhattacharya
rights: Copyright (c) 2025 Nishant Sahdev, Chinmoy Bhattacharya (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 81
ending page: 94
doi: 10.54105/ijam.A1225.05021025

Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums
type: item
description: This paper explores the hybrid power mean of three-term exponential sums, incorporating weights derived from general quartic Gauss sums. By employing the theory of Dirichlet characters in conjunction with fundamental properties of classical Gauss sums, we establish several significant results. Furthermore, we determine the corresponding weight function for these three-term exponential sums, with particular applications in coding theory.
subject: Articles
  topic: section
subject: Dirichlet Character
  topic: keywords
subject: Quartic Gauss Sums
  topic: keywords
subject: Three-Term Exponential Sum
  topic: keywords
creator: Dr. Shikha Singh
creator: Prof. Jagmohan Tanti
rights: Copyright (c) 2025 Dr. Shikha Singh, Prof. Jagmohan Tanti (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 74
ending page: 80
doi: 10.54105/ijam.B1219.05021025

An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM
type: item
description: We describe a general method that proves irrationality statements from second-order equations and from the arithmetic geometric mean (AGM). Let L Greater than sign 0 and define the bump Bn,L (x)=Qnn[x(L-x)]n (0 less than or equal to x less than or equal to L), where L=P/Q element of Q is in lowest terms. If u solves a second-order equation on [0,L] and its Prufer phase turns by an integer multiple of pie, then repeated integration by parts shows that In:=integral L0 Bn,L(x)dx is an integer combination of endpoint jets and of a fixed index term. Hence In element of Z (or Dn In element of Z for a controlled denominator Dn that depends only on finitely many endpoint Taylor coefficients of the coefficients of the equation). A Beta-function estimate gives 0 Less-than sign In less than or equal to L2n+1 Qnn/(2n+1) Long Rightwards Arrow n Long Rightwards Arrow the lemniscate 0 so for large n we have 0 Less-than sign In Less-than sign 1, which contradicts integrality. This scheme yields: (i) the irrationality of pie
subject: Articles
  topic: section
subject: Irrationality Statements
  topic: keywords
subject: Coefficients
  topic: keywords
subject: Function Estimate
  topic: keywords
creator: K. Srinivasa Raghava
creator: Dr. R. Sivaraman
rights: Copyright (c) 2025 K. Srinivasa Raghava, Dr. R. Sivaraman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 70
ending page: 73
doi: 10.54105/ijam.B1224.05021025

Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region
type: item
description: One of the major environmental issues is water pollution, which harms aquatic ecosystems, human health, and the sustainability of natural resources. To solve this problem, reliable mathematical models are required that can predict the behaviour of pollutants in three-dimensional (3-D) water systems. In this study, the temporal and spatial fluctuations of pollutant concentration in water are estimated using a 3-D diffusion model. The work's goals are to investigate the gradual rise in pollution levels in a 3-D region and assess the accuracy of analytical and numerical methods for addressing diffusion-based pollution issues. Two distinct mathematical techniques are employed to determine the level of pollution in water at a 3-D location: the Crank-Nicolson (C-N) method, a numerical method, and the Adomian Decomposition Method (ADM), an analytical method. The data collected from an experiment conducted in a 3-D cuboid tank, where water serves as the medium and iodised saltwater solution as the pollutant, are used to derive the initial and boundary conditions for the presented mathematical approaches. The dispersion of pollutants over time and space can be directly observed in this experiment, yielding crucial data for validating the mathematical model. The results of the experiments show that during all the time intervals, there is a rise in water pollution at all 3-D locations. Furthermore, the insignificant error in the form of parts per million (PPM) difference obtained when comparing the outcomes of the C-N with Experimental data (Exp. data) and ADM methodologies demonstrates the effectiveness of the proposed mathematical model. The benefit of this research lies in the use of mathematical approaches and experimental data to investigate water pollution in a 3-D region. The study demonstrates the validity of the proposed model and its ability to forecast the spread of pollution in water accurately. This method is also applicable to larger water systems than the experi…
subject: Articles
  topic: section
subject: Water Pollution
  topic: keywords
subject: 3-D Diffusion Equation
  topic: keywords
subject: Crank-Nicolson Method
  topic: keywords
subject: Adomian Decomposition Method Mathematics Subject Classification
  topic: keywords
subject: 35K57
  topic: keywords
subject: 65N06
  topic: keywords
creator: Tarjani Naik
creator: Dr. Mukesh Patel
creator: Dr. Rachna Patel
rights: Copyright (c) 2025 Tarjani Naik, Dr. Mukesh Patel, Dr. Rachna Patel (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 50
ending page: 62
doi: 10.54105/ijam.B1222.05021025

Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response
type: item
description: Complex dynamics of a modified Hastings-Powell (HP) model (phytoplankton-zooplankton-fish) with Holling type IV functional response are investigated in this article. Boundedness of the system has been established. A detailed study of the boundary equilibrium points and their local stability has been carried out. The condition of uniform persistence of the system has also been derived. Dynamical complexities and subsequent changes in the states of the system have been portrayed using numerical simulation. Modified HP model with Holling type IV functional response gives rise to a similar type of chaotic dynamics (inverted 'teacup attractor’) as observed in the original HP model with Holling type II functional response. Chaotic or stable dynamics are also numerically verified using Sil'nikov eigenvalue analysis.
subject: Articles
  topic: section
subject: Holling type IV
  topic: keywords
subject: Boundedness
  topic: keywords
subject: Uniform persistence
  topic: keywords
subject: Chaos
  topic: keywords
creator: Dr. Partha Ghosh
rights: Copyright (c) 2025 Dr. Partha Ghosh (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 43
ending page: 49
doi: 10.54105/ijam.B1218.05021025

Comment on ‘The Triple Product Rule and the Subtleties of the Mathematical Tools Used in Thermodynamics’ by Felipe Américo Reyes Navarro and Rafael Edgardo Carlos Reye
type: item
description: It is shown that much of the criticism of my paper ‘The Triple Product Rule is Incorrect’ Indian Journal of Advanced Mathematics (IJAM), Vol. 1, No. 2, 2021. 1 − 3, https://doi.org/10.35940/ijam.B1102.101221 presented in [1] is incorrect.
subject: Articles
  topic: section
subject: Triple Product Rule
  topic: keywords
subject: Natural Variables
  topic: keywords
subject: Work
  topic: keywords
subject: Compression
  topic: keywords
subject: Function of Stat
  topic: keywords
creator: I.A. Stepanov
rights: Copyright (c) 2025 I.A. Stepanov (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 41
ending page: 42
doi: 10.54105/ijam.B1223.05021025

A Comparative Study on the Effects of Vertical and Horizontal Magnetic Fields on Double Diffusive Convection with Soret Effect in Couple-Stressed Nanofluids
type: item
description: This study aims to conduct a comparative analysis of double diffusive convection in a Soret-induced couple-stressed nanofluid layer under the influence of uniform vertical and horizontal magnetic fields. The linear stability analysis is based on the standard mode technique. The Galerkin method has been applied to find the critical Rayleigh number and the corresponding wave number in terms of various parameters. The effects of the Soret parameter, magnetic field, Lewis number, modified diffusivity ratio, Concentration Rayleigh-Darcy number, and Solutal Rayleigh number on the system's stability have been investigated. It has been observed that Stationary convection remains unaffected by the relaxation parameter, and the critical wave number is a function of both the couple stress parameter and the magnetic field; however, stationary convection is stabilised by the couple stress. In double diffusive convection under a magnetic field, the Darcy number also comes into play and has been observed to provide a stabilising effect on stationary convection. In comparison to ordinary fluids, convection sets up earlier in nanofluids. A comparative graphical analysis has been conducted to depict the effects of couple stress and Soret parameter on the stability of a nanofluid in the presence of both vertical and horizontal magnetic fields. It has been observed that the magnetic field still has a stabilising effect in both cases. However, in the case of a vertical magnetic field, the critical value of the stationary Rayleigh number is obtained at a significantly larger wave number compared to that in the case of a horizontal magnetic field, with all parameters remaining the same. The literature survey indicates that no study has investigated the comparison of the effect of vertical and horizontal magnetic fields on double diffusive convection in a couple stress nanofluid layer with a Soret factor. The present study examines the comparative analysis of the effect of vertical and hor…
subject: Articles
  topic: section
subject: Magnetic Field
  topic: keywords
subject: Rayleigh Number
  topic: keywords
subject: Couple-Stressed Nanofluid
  topic: keywords
subject: Soret Factor
  topic: keywords
creator: Suman Sindhwani
rights: Copyright (c) 2025 Suman Sindhwani (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 36
ending page: 40
doi: 10.54105/ijam.B1215.05021025

Completed Functor S-1() of the Localization Functor S-1(), Isomorphism and Adjunction
type: item
description: This article serves as a continuation of our previous work 1, which remains our primary reference for investigating specific homological properties with completion. Let the rings not be necessarily commutative and the modules be the unitary left (resp. right) modules. Let (G, (Gn) N element of N) be a filtered normal group equipped with the group topology associated with the filtration (Gn)nEN formed of normal subgroups and C(G) the set of Cauchy sequences with values in G. We define an equivalence relation R on C(G) by: (xn)R(yn) (xn)-(yn)= (xn-yn) converges to 0, noted by (xn-yn) 0. The quotient set C(G)/R:+{(xn)| (xn)element of C(G)} denoted G is equipped with a group structure and is called the completed groupe of G. For any filtered ring (resp. left A-module) (A, (In)nelement ofN (resp.(M,(Mn)nelement of N), the completed group A (resp. M) is equipped with a ring structure (resp. A-module) by (an ) x (bn ) = (anbn ) (resp.(an).(mn)=(an.mn)) where (an), (bn) element of A(resp. (mn ) element of M) called completed ring (resp. module) of A (resp. M). And for all saturated multiplicative subset S of A that satisfies the left Ore conditions, S = {(xn) element of A ∣ (xn ) not-equal 0 and ∃ n0 element of N, n greater than or equal to n0,xn element of S} is a saturated multiplicative subset of A that satisfies the left Ore conditions 1. Among the main results of this article, we have : – the functors S-1 () is isomorphic toS-1(A) tensor product A−. and S-1 () is isomorphic to S-1(A) tensor productA−. – the functors Home A(S-1A tensor productAM,−) and HOM A(S-1Atensor productA M,−) are isomorphic. – the functors S-1Atensor productA – and HomA(S-1A,−) are adjoints. This Study Allows How Establish a Relationship Between Completion [2] and Localization [4] Under the Assumptions of a Topological Structure.
subject: Articles
  topic: section
subject: Ring
  topic: keywords
subject: Modules
  topic: keywords
subject: Filtration
  topic: keywords
subject: Completion
  topic: keywords
subject: Ore Condition
  topic: keywords
subject: Localization
  topic: keywords
subject: Isomorphisms
  topic: keywords
subject: Categories
  topic: keywords
subject: Functors
  topic: keywords
subject: Completed Functor
  topic: keywords
subject: Adjunction
  topic: keywords
creator: Abdoulaye Mane
creator: Mohamed Ben Maaouia
creator: Mamadou Sanghare
rights: Copyright (c) 2025 Abdoulaye Mane, Mohamed Ben Maaouia, Mamadou Sanghare (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 27
ending page: 35
doi: 10.54105/ijam.B1214.05021025

Study of Some Degree Five Identities of Type (4,1)
type: item
description: In non-associative algebra, irreducible identities of degree five are the least studied. Following Osborn's studies, the only identity of type (5) has generated very little literature, as seen in ([1]) and ([2]). Hence, our interest in identities of the following type. The purpose of this study is to enable us to consider a baric case study at a later stage, such as in ([3]) and ([4]). This paper is devoted to the study of three of type (4,1), taken from the families of irreducible degree five identities of Osborn. We conduct this study in the presence of an idempotent, through a Peirce decomposition, depending on whether the Peirce polynomial is reducible or not over the base field of the algebra. In each studied case, we find two orthogonal subalgebras. Therefore, in the first two studied identities, we manage to show that there is a homomorphism over one of the subspaces of A, whose kernel is an ideal.
subject: Articles
  topic: section
subject: Idempotent
  topic: keywords
subject: Linearization
  topic: keywords
subject: Peirce Decomposition
  topic: keywords
creator: Dr. Abdoulaye DEMBEGA
rights: Copyright (c) 2025 Dr. Abdoulaye DEMBEGA (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 18
ending page: 26
doi: 10.54105/ijam.B1212.05021025

Existence and Uniqueness of Weak Solutions to the Fractional Navier-Stokes Equations in Turbulent Flows: A Study of Boundedness and Stability
type: item
description: This study employs fractional analysis to investigate the existence and uniqueness of weak solutions to the fractional NavierStokes equations in three-dimensional turbulent flows, thereby addressing the complexity inherent in turbulent regimes. Turbulence presents significant challenges in fluid dynamics, characterised by chaotic and erratic motion that confounds conventional modelling techniques. By reformulating the NavierStokes equations using fractional derivatives, we capture non-local effects and memory phenomena, thereby enhancing the mathematical representation of fluid behaviour. First, we transform the Navier-Stokes equations to demonstrate the utility of fractional analysis. The second thing we do is show that there are weak answers in some situations. This means that we can use our models with starting data that isn't stable. This is the third thing we do. We show that these answers are unique. This proves that our models are right. Ultimately, we demonstrate that weak solutions remain bounded over time under specific conditions regarding the initial data and external forces. When the Reynolds number approaches a critical level, we investigate its stability. This helps us understand how smooth flow can become rough flow. The findings of this research not only advance the theoretical understanding of weak solutions to the fractional Navier-Stokes equations but also have practical implications for modeling complex fluid systems. By linking fractional derivatives with turbulent flows, this work contributes to the broader field of fluid dynamics, paving the way for future investigations in applied mathematics and engineering. Ultimately, this exploration enhances our understanding of turbulence and its mathematical foundations, emphasising the importance of fractional calculus in accurately modelling fluid dynamics.
subject: Articles
  topic: section
subject: Navier-Stokes Equations
  topic: keywords
subject: Weak Solutions
  topic: keywords
subject: Turbulent Flows
  topic: keywords
subject: Existence and Uniqueness
  topic: keywords
subject: Boundedness
  topic: keywords
subject: Stability Analysis
  topic: keywords
creator: Dr. Raghad Sahib Shamsah
rights: Copyright (c) 2025 Dr. Raghad Sahib Shamsah (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 8
ending page: 17
doi: 10.54105/ijam.B1211.05021025

An Innovative Approach to Find Remainder
type: item
description: Let r be the remainder when x! is divided by p, (x, p) ∈ N, x < p. Then the value of r is given by the minimum value of k for which (−1) (m-1) (m-1)! k + 1 = 0(mod p), where m = p − x, k ∈ N.
subject: Articles
  topic: section
subject: Remainder
  topic: keywords
subject: Number Theorem
  topic: keywords
subject: Factorial Remainder
  topic: keywords
subject: Remainder using Wilson’s Theorem
  topic: keywords
creator: Prof. Anil Kumar Sharma
rights: Copyright (c) 2025 Prof. Anil Kumar Sharma (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 6
ending page: 7
doi: 10.54105/ijam.B1205.05021025

Some Results on Specificity of Possibility Distributions
type: item
description: Specificity of a possibility distribution is akin to the entropy of a probability distribution. It serves an essential purpose to zero in on the maximum probability observation. However, when we discuss the existing definition of possibility distribution, it lacks applicability in real-world problems; hence, specificity also becomes an underrated measure for gauging the degree of uncertainty in a possibility distribution. In this paper, we present new findings on the specificity of a possibility distribution, resulting from our research on data-based semantic information analysis in hybrid human-machine systems. In this research, we propose a new frequency-based possibility and probability measure and formalise a new method for fitting restrictions on data or information available in the system. We will demonstrate that the proposed formula is superior to existing specificity measures and discuss various applications of specificity measures in solving problems related to hybrid systems. We shall summarise this paper by providing a real-world application of the proposed measure.
subject: Articles
  topic: section
subject: Possibility Distribution
  topic: keywords
subject: Restriction
  topic: keywords
subject: Specificity
  topic: keywords
subject: Hybrid System
  topic: keywords
subject: Semantic Information
  topic: keywords
creator: Dr. Jayesh V. Karanjgaonkar
rights: Copyright (c) 2025 Dr. Jayesh V. Karanjgaonkar (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 1
ending page: 5
doi: 10.54105/ijam.B1206.05021025

Study of Solution as Locally Asymptotic Attractivity for Nonlinear Functional Integral Equation
type: item
description: In this paper we proved an existence result for local asymptotic attractivity of the solution for a nonlinear functional integral equation under certain condition which gives the existence as well as existence of the asymptotic stability of solutions. An example is providing for indicating the natural realizations of abstract theory presented in the paper.
subject: Articles
  topic: section
subject: Solution of Functional Integral Equation
  topic: keywords
subject: Fixed Point Theorem
  topic: keywords
subject: Locally Asymptotic Attractive Solution
  topic: keywords
subject: Subject Classifications: 47H10
  topic: keywords
subject: 34A60
  topic: keywords
creator: Dr. Shantaram N Salunkhe
rights: Copyright (c) 2025 Dr. Shantaram N Salunkhe (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 50
ending page: 54
doi: 10.54105/ijam.B1200.05010425

Subscription Inventive Over the S-Convex Set in Linear Space
type: item
description: 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 concept 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. A number of examples are included to show the necessity of the hypotheses of various theorems.
subject: Articles
  topic: section
subject: Convexity
  topic: keywords
subject: S-Convex Set
  topic: keywords
subject: Convex Function
  topic: keywords
subject: Linear Combination
  topic: keywords
subject: S-linear Combinations
  topic: keywords
subject: Linear Space
  topic: keywords
subject: Contemporary Mathematics
  topic: keywords
creator: Neelam Dubey
rights: Copyright (c) 2025 Neelam Dubey (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 47
ending page: 49
doi: 10.54105/ijam.B1201.05010425

Rise of the Gole Number System
type: item
description: For centuries, the decimal number system served various applications, from basic counting to measuring astronomical distances. However, the efficient and humanfriendly representation of extremely large 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 ability to easily comprehend the value of the Gole number. 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.
subject: Articles
  topic: section
subject: Number Theory
  topic: keywords
subject: Data Compression
  topic: keywords
subject: Compact Number Representation
  topic: keywords
subject: Optimal Numeric Encoding
  topic: keywords
subject: Numerical Optimization
  topic: keywords
subject: Alternative Number Systems
  topic: keywords
subject: RNumbers
  topic: keywords
subject: Gole Number System
  topic: keywords
creator: Ravi Revelly
rights: Copyright (c) 2025 Ravi Revelly (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 43
ending page: 46
doi: 10.54105/ijam.A1194.05010425

An Evaluation of the Dijkstra's Algorithm, Floyd's Algorithm and Ant Colony Optimization
type: item
description: Travel is the movement of people between distant geographical locations. It can assist in enhancing our well-being, increasing our understanding, socializing with new individuals, relaxing and unwinding, seeking adventure, creating memories, improving mental and physical health, and immersing ourselves in different cultures. In this paper, we briefly explain the shortest path and their types applicable in practical life. The shortest path problem is a flexible and crucial instrument in different areas, facilitating effective route planning, network optimization, and resource distribution. 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 Optimization to decrease the overall path expense, which could be distance, time, or another factor for the given scenario. We use two scenarios and evaluate Dijkstra’s algorithm, Floyd’s algorithm, and Ant Colony Optimization to determine the shortest route in practical situations to enhance efficiency in solving the identical issue.
subject: Articles
  topic: section
subject: Ant Colony Optimization
  topic: keywords
subject: Dijkstra’s algorithm
  topic: keywords
subject: Floyd’s algorithm
  topic: keywords
subject: Shortest Path
  topic: keywords
subject: Travelling
  topic: keywords
creator: Dr. A. J. Khan
creator: Nikita Dewangan
rights: Copyright (c) 2025 Dr. A. J. Khan, Nikita Dewangan (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 38
ending page: 42
doi: 10.54105/ijam.A1198.05010425

Analysis of Classical Special Beta & Gamma Functions in Engineering Mathematics and Physics
type: item
description: 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 very important in calculus as complex integrals can be moderated into simpler form. In physics and engineering problems require a detailed knowledge of applied mathematics and an understanding of special functions such as gamma and beta functions. The topic of special functions is very important and it is constantly expanding with the existence of new problems in the applied Sciences in this article, we describe the basic theory of gamma and beta functions, their connections with each other and their applicability to engineering problems.to compute and depict scattering amplitude in Reggae trajectories. Our aim is to illustrate the extension of the classical beta function has many uses. It helps in providing new extensions of the beta distribution, providing new extensions of the Gauss hyper geometric functions and confluent hyper geometric function and generating relations, and extension of Riemann-Lowville derivatives. In this Article, we develop some elementary properties of the beta and gamma functions. We give more than one proof 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 show how they can be used to prove Fermat's theorem 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,
subject: Articles
  topic: section
subject: Beta Function
  topic: keywords
subject: Gamma Function
  topic: keywords
subject: A Pochhammer symbol in Applied Mathematics
  topic: keywords
subject: Engineering
  topic: keywords
creator: Dr. Pranesh Kulkarni
rights: Copyright (c) 2025 Dr. Pranesh Kulkarni (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 35
ending page: 37
doi: 10.54105/ijam.A1195.05010425

A New Proof for Irrationality of π
type: item
description: Ever since Lambert proved that π is irrational in 18th century, lots of wonderful proofs have been provided by various mathematicians. To this day, π remains as one of the most significant and important real number 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.
subject: Articles
  topic: section
subject: Irrational
  topic: keywords
subject: Mediant
  topic: keywords
subject: Closed Intervals
  topic: keywords
subject: Rational Approximations
  topic: keywords
creator: Dr. R. Sivaraman
creator: J. Suganthi
creator: P.N. Vijayakumar
rights: Copyright (c) 2025 Dr. R. Sivaraman, J. Suganthi, P.N. Vijayakumar (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 32
ending page: 34
doi: 10.54105/ijam.A1196.05010425

An Elementary Proof for Fermat's Last Theorem using a Transformation Equation to Fermat's Equation
type: item
description: 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 x 3 + y3 = z3 and create transformation equations. Solving the transformation equations we prove that only a trivial solution exists in the main equation r p + sp = tp .
subject: Articles
  topic: section
subject: Transformation Equations
  topic: keywords
subject: Mathematics Subject Classification 2010: 11A–XX
  topic: keywords
creator: P.N Seetharaman
rights: Copyright (c) 2025 P.N Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 27
ending page: 31
doi: 10.54105/ijam.A1192.05010425

An Elementary Proof for Fermat's Last Theorem using Three Distinct Odd Primes F, E and R
type: item
description: 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 > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation r p + sp = tp where p is any prime >3 and prove the theorem using the method of contradiction. We have used an Auxiliary equations x 3 + y3 = z3 along with the main equation rp + sp = tp , which are connected by means of transformation equation through the parameters. Solving the through transformation equations we get the result rst = 0, showing that only a trivial solution exists in the main equation.
subject: Articles
  topic: section
subject: Transformation Equations Two Fermat’s Equations
  topic: keywords
subject: 2010 Mathematics Subject Classification 2010: 11A–XX
  topic: keywords
creator: P.N Seetharaman
rights: Copyright (c) 2025 P.N Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 22
ending page: 26
doi: 10.54105/ijam.A1191.05010425

New Concepts of Congruences Modulo 1/n and Positive Integers Modulo 1/n
type: item
description: The aim of this article is the creation of new Mathematical beings: The positive integers Modulo 1/n. I hope that they will be used by all the scientific community in general and the computer scientists (cryptographers) particularly. In the other hand, the paper contributes to enlarge the field of number theory and algebraic sets. In this paper, I present a new type of congruences on IN: the congruences modulo 1/N and The Sets of positive integers modulo 1/N denoted respectively = [1/N] AND IN 1/n, n € IN* The Work is divided into two Sections : Section A: It’s composed of: ▪ A fundamental Theorem (With Proof) ▪ A fundamental relation of equivalence on IN denoted = [1/n] ▪ The definition of the sets IN1/n , n € IN* (with examples) ▪ The definition of the binary operations*and ∆ on IN ▪ The definition of the binary operations *and ∆ on IN 1/n, n € IN* ▪ The presentation of the semi-groups (IN ; *) ; (IN ; ∆) ; ( IN1/n ;*); ( IN 1/n ; ∆ Section B: It’s Composed of an Appendice with: I) The definition and graph of the function: r: IN* → IN* x → n0 Where n0 is the rank of x . II) The definition and graph of the function: f: IN* → IN* n |→ [100, 1/n]
subject: Articles
  topic: section
subject: Congruences Modulo
  topic: keywords
subject: Positive integers Modulo
  topic: keywords
subject: Integer Part
  topic: keywords
subject: Partition
  topic: keywords
subject: Equivalence Classes
  topic: keywords
subject: Rank
  topic: keywords
subject: Amplitude
  topic: keywords
subject: Semi-group
  topic: keywords
creator: Habib Lebsir
rights: Copyright (c) 2025 Habib Lebsir (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 11
ending page: 21
doi: 10.54105/ijam.A1188.05010425

On Algebraic and Topological Properties of Some Fundamental Groups
type: item
description: 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 with 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 a n-punctured plane is free and we characterize some surfaces as topologically distinct.
subject: Articles
  topic: section
subject: Homeomorphism
  topic: keywords
subject: Path
  topic: keywords
subject: Loop
  topic: keywords
subject: Topological Group
  topic: keywords
subject: Fundamental Group
  topic: keywords
subject: Free Produc
  topic: keywords
creator: Djerassem Laurent
creator: Prof. Daniel Tieudjo
rights: Copyright (c) 2025 Djerassem Laurent, Prof. Daniel Tieudjo (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 5
ending page: 10
doi: 10.54105/ijam.B1177.05010425

Impacts of Some Definitions on Algebra of Differential Operators for Noncommutative Algebras
type: item
description: Rings of differential operators are one of the most important noncommutative (associative) algebras. They play an important 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.
subject: Articles
  topic: section
subject: Differential Operators
  topic: keywords
subject: Weyl Algebra
  topic: keywords
subject: Noncommutative Algebra
  topic: keywords
subject: Derivation on an Algebra
  topic: keywords
creator: Dr. SAMA Anzoumana
creator: Sir Bénédic Kouamé Bah
creator: Dr. Konan Mathias Kouakou
rights: Copyright (c) 2025 Dr. SAMA Anzoumana, Sir Bénédic Kouamé Bah, Dr. Konan Mathias Kouakou (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-30
publication date: 2025-04-30
starting page: 1
ending page: 4
doi: 10.54105/ijam.B1178.05010425

An Elementary Chapter in Number Theory: Proof of Fermat's Last Theorem
type: item
description: Pierre de Fermat first stated around 1637 that for any integer n > 2, the equation an + bn = cn has no positive integer solutions, and he said the theorem in the margin of a copy of Arithmetica. His proof is available only for the equation a 4 + b 4 = c 4 for the exponent n = 4. Subsequently, Euler proved the theorem in the equation a 3 + b 3 = c 3 for the exponent n = 3. Taking the above two proofs of Fermat and Euler, it would suffice to prove the theorem for n = p, where p is any prime > 3. In this proof, we hypothesize all r, s and t as positive integerssatisfying the equation rp + sp = tp and establish a contradiction. We use another auxiliary equation, x 3 + y 3 = z 3 , and combine the two equations using transformation equations. Solving the transformation equations, we establish a contradiction, thereby proving the theorem.
subject: Articles
  topic: section
subject: Transformation Equations
  topic: keywords
subject: Mathematics Subject Classification
  topic: keywords
subject: 2010
  topic: keywords
subject: 11A–XX
  topic: keywords
creator: P. N Seetharaman
rights: Copyright (c) 2025 P. N Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-10-30
publication date: 2025-10-30
starting page: 63
ending page: 69
doi: 10.54105/ijam.B1217.05021025

On Laplace-Riemann Derivatives
type: item
description: The basic properties of the th n Laplace-Riemann derivative of a function f at a point x is studied.
subject: Articles
  topic: section
subject: Laplace-Riemann Derivatives
  topic: keywords
subject: Riemann - Derivatives
  topic: keywords
subject: Derivative
  topic: keywords
subject: Continuity
  topic: keywords
subject: Mathematical Subject Classification 2020
  topic: keywords
subject: 26E99
  topic: keywords
subject: 28E99
  topic: keywords
creator: S. Deb
rights: Copyright (c) 2025 S. Deb (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-25
publication date: 2025-04-25
starting page: 25
ending page: 30
doi: 10.54105/ijam.D1124.04021024

On the Model of Transport of Salt Solution in a Xylem
type: item
description: We consider a model for vertical lifting of an aqueous solution of salts in a xylem. We analyzed changes of movement of the considered solution depending on various parameters. We consider an analytical approach for analysis of the considered model. The approach gives a possibility to take into account changes of parameters in space and time, as well as the nonlinearity of the considered processes.
subject: Articles
  topic: section
subject: this
  topic: keywords
creator: Evgeny Leonidovich Pankratov
rights: Copyright (c) 2025 Evgeny Leonidovich Pankratov (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-25
publication date: 2025-04-25
starting page: 16
ending page: 18
doi: 10.54105/ijam.B1173.04021024

A Proof for Fermat's Last Theorem using an Auxiliary Fermat's Equation
type: item
description: Fermat’s Last Theorem states that there exists no three positive integers x, y and z satisfying the equation x n + y n = z n , where n is any integer > 2. Fermat and Euler had already proved the theorem for the exponents n = 4 and n = 3 in the equations x 4 + y 4 = z 4 and x 3 + y 3 = z 3 respectively. Hence taking into account of the same, it is enough to prove the theorem for the exponent n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation r p + s p = t p where p is any prime >3 and prove the theorem by the method of contradiction. To support the proof in the above equation we have used an Auxiliary equation x 3 + y 3 = z 3 . The two equations are linked by means of transformation equations. Solving the transformation equations we prove the theorem.
subject: Articles
  topic: section
subject: this
  topic: keywords
creator: N Seetharaman
rights: Copyright (c) 2025 N Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-25
publication date: 2025-04-25
starting page: 19
ending page: 24
doi: 10.54105/ijam.A1182.04021024

An Elementary Proof for Fermat’s Last Theorem using Ramanujan-Nagell Equation
type: item
description: Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesize that all r, s and t are non-zero integers in the equation rp + sp = tp and establish a contradiction in this proof. Just for supporting the proof in the above equation, we have used another equation x3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 a non-zero integer; z and z2 irrational. We create transformed equations to the above two equations through parameters, into which we have incorporated the Ramanujan – Nagell equation. Solving the transformed equations we prove the theorem.
subject: Articles
  topic: section
subject: Transformed Fermat’s
  topic: keywords
subject: Equations through Parameters
  topic: keywords
subject: Mathematics Subject
  topic: keywords
subject: Classification 2010
  topic: keywords
creator: N Seetharaman
rights: Copyright (c) 2025 N Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-25
publication date: 2025-04-25
starting page: 10 to 15
ending page: 10 to 15
doi: 10.54105/ijam.B1180.04021024

On the Nörlund-Rice Integral Formula
type: item
description: After introducing the famous Nörlund-Rice integral formula, we apply it to Laguerre polynomials, Melzak’s relation, and Stirling numbers of the second kind to obtain nice expressions.
subject: Articles
  topic: section
subject: Stirling numbers
  topic: keywords
subject: Nörlund-Rice integral
  topic: keywords
subject: Melzak’s identity
  topic: keywords
subject: Laguerre polynomials
  topic: keywords
creator: Dr. R. Sivaraman
creator: Muniru Iddrisu
creator: Prof. J. López-Bonilla
rights: Copyright (c) 2024 https://creativecommons.org/licenses/by-nc-nd/4.0
date: 2025-04-25
publication date: 2025-04-25
starting page: 7
ending page: 9
doi: 10.54105/ijam.B1175.04021024