Indian Journal of Advanced Mathematics (IJAM)
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.
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An Elementary Procedure in the Proof of Fermat's Last Theorem
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.
Copyright (c) 2025 P.N. Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
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
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 the universe. Furthermore, the existence of an inverse pie—always in conjugation and equilibrium with pie—is posited as essential for maintaining the balance of forces in the universe. Accordingly, this article presents a novel derivation of pie by simultaneously considering both the cosmos' direct space and the reciprocal (or inverse) space.
Copyright (c) 2025 Nishant Sahdev, Chinmoy Bhattacharya (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Hybrid Mean Value of General Quartic Gauss Sums and Three-Term Exponential Sums
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.
Copyright (c) 2025 Dr. Shikha Singh, Prof. Jagmohan Tanti (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM
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
Copyright (c) 2025 K. Srinivasa Raghava, Dr. R. Sivaraman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Numerical and Analytical Method based Mathematical Approach to Estimate Water Pollution over 3-D Region
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.
Adomian Decomposition Method Mathematics Subject Classification
Copyright (c) 2025 Tarjani Naik, Dr. Mukesh Patel, Dr. Rachna Patel (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Complex Dynamics of an Aquatic Tri-Trophic Food Chain System with Holling Type IV Functional Response
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.
Copyright (c) 2025 Dr. Partha Ghosh (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
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
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.
Copyright (c) 2025 I.A. Stepanov (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
A Comparative Study on the Effects of Vertical and Horizontal Magnetic Fields on Double Diffusive Convection with Soret Effect in Couple-Stressed Nanofluids
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.
Copyright (c) 2025 Suman Sindhwani (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Completed Functor S-1() of the Localization Functor S-1(), Isomorphism and Adjunction
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.
Copyright (c) 2025 Abdoulaye Mane, Mohamed Ben Maaouia, Mamadou Sanghare (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Study of Some Degree Five Identities of Type (4,1)
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.
Copyright (c) 2025 Dr. Abdoulaye DEMBEGA (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Existence and Uniqueness of Weak Solutions to the Fractional Navier-Stokes Equations in Turbulent Flows: A Study of Boundedness and Stability
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.
Copyright (c) 2025 Dr. Raghad Sahib Shamsah (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
An Innovative Approach to Find Remainder
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.
Copyright (c) 2025 Prof. Anil Kumar Sharma (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Some Results on Specificity of Possibility Distributions
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.
Copyright (c) 2025 Dr. Jayesh V. Karanjgaonkar (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Study of Solution as Locally Asymptotic Attractivity for Nonlinear Functional Integral Equation
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.
Solution of Functional Integral Equation
Copyright (c) 2025 Dr. Shantaram N Salunkhe (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Subscription Inventive Over the S-Convex Set in Linear Space
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.
Copyright (c) 2025 Neelam Dubey (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
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.
Copyright (c) 2025 Ravi Revelly (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
An Evaluation of the Dijkstra's Algorithm, Floyd's Algorithm and Ant Colony Optimization
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.
Copyright (c) 2025 Dr. A. J. Khan, Nikita Dewangan (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Analysis of Classical Special Beta & Gamma Functions in Engineering Mathematics and Physics
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,
A Pochhammer symbol in Applied Mathematics
Copyright (c) 2025 Dr. Pranesh Kulkarni (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
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.
Copyright (c) 2025 Dr. R. Sivaraman, J. Suganthi, P.N. Vijayakumar (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
An Elementary Proof for Fermat's Last Theorem using a Transformation Equation to Fermat's Equation
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 .
Mathematics Subject Classification 2010: 11A–XX
Copyright (c) 2025 P.N Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
An Elementary Proof for Fermat's Last Theorem using Three Distinct Odd Primes F, E and R
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.
Transformation Equations Two Fermat’s Equations
2010 Mathematics Subject Classification 2010: 11A–XX
New Concepts of Congruences Modulo 1/n and Positive Integers Modulo 1/n
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]
Copyright (c) 2025 Habib Lebsir (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
On Algebraic and Topological Properties of Some Fundamental Groups
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.
Copyright (c) 2025 Djerassem Laurent, Prof. Daniel Tieudjo (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
Impacts of Some Definitions on Algebra of Differential Operators for Noncommutative Algebras
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.
Copyright (c) 2025 Dr. SAMA Anzoumana, Sir Bénédic Kouamé Bah, Dr. Konan Mathias Kouakou (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
An Elementary Chapter in Number Theory: Proof of Fermat's Last Theorem
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.
Copyright (c) 2025 P. N Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
The basic properties of the th n Laplace-Riemann derivative of a function f at a point x is studied.
Mathematical Subject Classification 2020
Copyright (c) 2025 S. Deb (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
On the Model of Transport of Salt Solution in a Xylem
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.
Copyright (c) 2025 Evgeny Leonidovich Pankratov (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
A Proof for Fermat's Last Theorem using an Auxiliary Fermat's Equation
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.
Copyright (c) 2025 N Seetharaman (Author) https://creativecommons.org/licenses/by-nc-nd/4.0
An Elementary Proof for Fermat’s Last Theorem using Ramanujan-Nagell Equation
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.
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.
Copyright (c) 2024 https://creativecommons.org/licenses/by-nc-nd/4.0