Volume-5 Issue-2, October 2025 – Indian Journal of Advanced Mathematics (IJAM)

A122706010426

IJAM Journal — Tue, 28 Oct 2025 10:30:34 +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

An Elementary Procedure in the Proof of Fermat’s Last Theorem
P. N. Seetharaman

P.N. Seetharaman, Retired Executive Engineer, Energy Conservation Cell, Tamil Nadu State Electricity Board, Anna Salai, Chennai (Tamil Nadu), 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: 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 a4 + b4 = c 4 for the exponent n = 4. Subsequently, Euler proved the theorem in the equation a3 + b3 = c3 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 integers satisfying the equation rp + sp = tp and establish a contradiction. We use another auxiliary equation, x 3 + y3 = z3 , and combine the two equations using transformation equations. Solving the transformation equations, we establish a contradiction, thereby proving the theorem.

Keywords: Transformation Equations.
Scope of the Article: Applied Mathematics

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B122405021025

IJAM Journal — Thu, 23 Oct 2025 09:20:02 +0000

An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM
K. Srinivasa Raghava¹, R. Sivaraman²

¹K. Srinivasa Raghava, Research Associate, Department of Mathematics, Choolaimedu, Chennai (Tamil Nadu), India.

²Dr. R. Sivaraman, Associate Professor, Department of Mathematics, Dwaraka Doss Goverdhan Doss Vaishnav College, Arumbakkam, Chennai (Tamil Nadu), India.

Abstract: We describe a general method that proves irrationality statements from second-order equations and from the arithmetic geometric mean (AGM). Let 𝑳 > 𝟎 and define the bump 𝑩_𝒏,𝑳 (𝒙) = 𝑸^𝒏 𝒏! [𝒙(𝑳 − 𝒙)]^𝒏 (𝟎 ≤ 𝒙 ≤ 𝑳), where 𝑳 = 𝑷/𝑸 ∈ ℚ is in lowest terms. If 𝒖 solves a second-order equation on [𝟎,𝑳] and its Prüfer phase turns by an integer multiple of 𝝅, then repeated integration by parts shows that 𝑰𝒏: = ∫^𝑳_𝟎 𝑩_𝒏,𝑳 (𝒙)𝒖(𝒙)𝒅𝒙 is an integer combination of endpoint jets and of a fixed index term. Hence 𝑰_𝒏 ∈ ℤ (or 𝑫_𝒏𝑰_𝒏 ∈ ℤ for a controlled denominator 𝑫_𝒏 that depends only on finitely many endpoint Taylor coefficients of the coefficients of the equation). A Beta-function estimate gives 𝟎 < 𝑰_𝒏 ≤ 𝑳^{𝟐𝒏+𝟏} 𝑸^𝒏_𝒏!/ (𝟐_𝒏+𝟏)! ⟶𝒏→∞ 𝟎 so for large 𝒏 we have 𝟎 < 𝑰_𝒏 < 𝟏, which contradicts integrality. This scheme yields: (i) the irrationality of 𝝅 from 𝒖(𝒙) = 𝒔𝒊𝒏 𝒙 on [𝟎,𝝅]; (ii) a Sturm-Liouville version for analytic potentials with rational endpoint Taylor data and a halfturn of the Prüfer phase; and (iii) consequences for complete elliptic integrals, where the role of the index is played by the Legendre monodromy identity. In particular, for each 𝒌 ∈ (𝟎,𝟏) not all of 𝑲(𝒌),𝑲(𝒌 ′),𝑬(𝒌),𝑬(𝒌 ′) can be rational, and at 𝒌 = 𝟏/√𝟐 at least one of 𝑲(𝒌) or 𝑬(𝒌) is irrational.

Keywords: Irrationality Statements, Coefficients, Function Estimate.
Scope of the Article: Applied Mathematics

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B122305021025

IJAM Journal — Thu, 23 Oct 2025 09:06:40 +0000

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
I.A. Stepanov

I.A. Stepanov, Institute of Science and Innovative Technologies, Liepaja University, Liela 14, Liepaja, LV−3401, Latvia.

Abstract: 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.

Keywords: Triple Product Rule, Natural Variables, Work, Compression, Function of Stat.
Scope of the Article: Calculus

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B122205021025

IJAM Journal — Thu, 23 Oct 2025 09:00:04 +0000

Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region
Tarjani Naik¹, Mukesh Patel², Rachna Patel³

¹Tarjani Naik, Scholar, Department of Mathematics, Uka Tarsadia University, Bardoli (Gujarat), India.

²Dr. Mukesh Patel, Assistant Professor, Department of Mathematics, Uka Tarsadia University, Bardoli (Gujarat), India.

³Dr. Rachna Patel, Assistant Professor, Department of Computer Engineering, CGPIT, Uka Tarsadia University, Bardoli (Gujarat), India.

Abstract: 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 experimental tank. The research enhances mathematical solutions to diffusion equations and provides valuable insights for developing pollution control measures, assessing water quality, and promoting sustainable water management.

Keywords: Water Pollution, 3-D Diffusion Equation, CrankNicolson Method, Adomian Decomposition Method Mathematics Subject Classification: 35K57, 65N06
Scope of the Article: Applied Mathematics

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B121905021025

IJAM Journal — Thu, 23 Oct 2025 08:52:51 +0000

Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums
Shikha Singh¹, Jagmohan Tanti²

¹Dr. Shikha Singh, Assistant Professor, Department of Mathematics, Central University of Jharkhand, Ranchi (Jharkhand), India.

²Prof. Jagmohan Tanti, Department of Mathematics, Central University of Jharkhand, Ranchi (Jharkhand), India.

Abstract: 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.

Keywords: Dirichlet Character, Quartic Gauss Sums, Three Term Exponential Sum. MSC: 11L05.
Scope of the Article: Number Theory

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B121805021025

IJAM Journal — Thu, 23 Oct 2025 06:40:44 +0000

Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response
Partha Ghosh

Dr. Partha Ghosh, Department of Mathematics, Abhedananda, Mahavidyalaya, Sainthia, Birbhum, West Bengal, India.

Abstract: 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.

Keywords: Holling type IV, Boundedness, Uniform persistence, Chaos. Mathematics Subject Classification 2010: 92 XX
Scope of the Article: Dynamical Systems

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B121705021025

IJAM Journal — Thu, 23 Oct 2025 06:04:35 +0000

An Elementary Chapter in Number Theory: Proof of Fermat’s Last Theorem
P. N. Seetharaman

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

Abstract: 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 a⁴ + b⁴ = c⁴ for the exponent n = 4. Subsequently, Euler proved the theorem in the equation a³ + b³ = c³ 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, x³ + y³ = z³, and combine the two equations using transformation equations. Solving the transformation equations, we establish a contradiction, thereby proving the theorem.

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

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B121505021025

IJAM Journal — Thu, 23 Oct 2025 05:57:09 +0000

A Comparative Study on the Effects of Vertical and Horizontal Magnetic Fields on Double Diffusive Convection with Soret Effect in Couple-Stressed Nanofluids
Suman Sindhwani

Suman Sindhwani, Associate Professor, Department of Mathematics, Hindu Girls College, Sonipat (Haryana), India.

Abstract: 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 horizontal magnetic fields on Soret-induced double diffusive convection in a couple-stress nanofluid horizontal layer.

Keywords: Magnetic Field, Rayleigh Number, Couple-Stressed Nanofluid, Soret Factor.
Scope of the Article: Mathematical Physics

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B121405021025

IJAM Journal — Thu, 23 Oct 2025 05:38:43 +0000

Completed Functor 𝑆⁻¹ ̂() of the Localization Functor 𝑆⁻¹(), Isomorphism and Adjunction
Abdoulaye Mane¹, Mohamed Ben Maaouia², Mamadou Sanghare³

¹Abdoulaye Mane, Department of Mathématiques, Université Gaston Berger, Saint-Louis, Senegal.

²Mohamed Ben Maaouia, Laboratory of Algebra, Codes And Cryptography Applications (LACCA), UFR-SAT, University Gaston Berger (UGB), Saint-Louis, Senegal.

³Mamadou Sanghare, Doctoral School of Mathematics-Computer – UCAD-Sénégal, University Cheikh Anta Diop of Dakar, Dakar, Senegal.

Abstract: 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, (𝑮_𝒏)_𝒏∈ℕ) be a filtered normal group equipped with the group topology associated with the filtration (𝑮_𝒏)_𝒏∈ℕ 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: (𝒙_𝒏)𝓡(𝒚_𝒏) ⇔ (𝒙_𝒏) − (𝒚_𝒏) = (𝒙_𝒏 − 𝒚_𝒏) converges to 0, noted by (𝒙_𝒏 − 𝒚_𝒏 ) → 𝟎. The quotient set 𝓒(𝑮)/𝓡: = {(𝒙_𝒏 ̂) ∣ (𝒙_𝒏) ∈ 𝓒(𝑮)} denoted 𝑮̂ is equipped with a group structure and is called the completed groupe of 𝑮. For any filtered ring (resp. left 𝑨-module) (𝑨, (𝑰_𝒏 )_𝒏∈ℕ) (resp. (𝑴, (𝑴_𝒏 )_𝒏∈ℕ) ), the completed group 𝑨̂ (resp. 𝑴̂ ) is equipped with a ring structure (resp. 𝑨̂-module) by (𝒂_𝒏 ̂) ×̂ (𝒃_𝒏 ̂) = (𝒂_𝒏𝒃_𝒏 ̂) (𝒓𝒆𝒔𝒑. (𝒂_𝒏) ⋅ (𝒎_𝒏 ̂) = (𝒂_𝒏 ⋅ 𝒎_𝒏 ̂ )) where (𝒂_𝒏 ̂), (𝒃_𝒏 ̂) ∈ 𝑨̂ (resp. (𝒎_𝒏 ̂) ∈ 𝑴̂) called completed ring (resp. module) of 𝑨 (resp. 𝑴). And for all saturated multiplicative subset 𝑺 of 𝑨 that satisfies the left Ore conditions, 𝑺̂ = {(𝒙_𝒏 ̂) ∈ 𝑨̂ ∣ (𝒙_𝒏 ̂) ≠ 𝟎̂ and ∃𝒏₀ ∈ ℕ, 𝒏 ≥ 𝒏₀, 𝒙_𝒏 ∈ 𝑺} is a saturated multiplicative subset of 𝑨̂ that satisfies the left Ore conditions 1. Among the main results of this article, we have : – the functors 𝑺̂^−𝟏 ̂() is isomorphic to 𝑺̂^−𝟏(𝑨̂)⊗_𝑨̂−. and 𝑺̂^−𝟏() is isomorphic to 𝑺̂^−𝟏(𝑨) ⊗_𝑨̂−. – the functors 𝑯𝒐𝒎_𝑨̂(𝑺̂^−𝟏𝑨 ⊗_𝑨̂ 𝑴̂,−) and 𝑯𝒐𝒎_𝑨̂(𝑺̂^−𝟏𝑨̂⊗_𝑨̂ 𝑴,−) are isomorphic. – the functors 𝑺̂^−𝟏𝑨⊗_𝑨̂ – and 𝑯𝒐𝒎_𝑨̂(𝑺̂^−𝟏𝑨̂,−) are adjoints. This Study Allows How Establish a Relationship Between Completion [2] and Localization [4] Under the Assumptions of a Topological Structure.

Keywords: Ring, Modules, Filtration, Completion, Ore Condition, Localization, Isomorphisms, Categories, Functors, Completed Functor, Adjunction.
Scope of the Article: Algebra

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B121205021025

IJAM Journal — Thu, 23 Oct 2025 05:17:32 +0000

Study of Some Degree Five Identities of Type (4,1)
Abdoulaye DEMBEGA

Dr. Abdoulaye DEMBEGA, Researcher, Department of Mathématiques, Université Norbert ZONGO, Koudougou, Burkina Faso.

Abstract: 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.

Keywords: Idempotent, Linearization, Peirce Decomposition.
Scope of the Article: Algebra

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Volume-5 Issue-2, October 2025 – Indian Journal of Advanced Mathematics (IJAM)

A122706010426

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B122405021025

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B122305021025

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B122205021025

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B121905021025

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B121705021025

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B121505021025

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B121405021025

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B121205021025

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