Journal of Prime Research in Mathematics
Journal of Prime Research in Mathematics (JPRM) ISSN: 1817-3462 (Online) 1818-5495 (Print) is an HEC recognized, Scopus indexed, open access journal which provides a plate forum to the international community all over the world to publish their work in mathematical sciences. JPRM is very much focused on timely processed publications keeping in view the high frequency of upcoming new ideas and make those new ideas readily available to our readers from all over the world for free of cost. Starting from 2020, we publish one Volume each year containing two issues in June and December. The accepted papers will be published online immediate in the running issue. All issues will be gathered in one volume which will be published in December of every year.
bdus Salam School of Mathematical Sciences (ASSMS), GC University, Lahore
Lie’s classification of finite dimensional algebras of Vector Fields in CN
Brief proofs of classical results of Lie on finite dimensional subalgebras of vector fields in two and three variables are outlined. The results for algebras of maximal rank for vector fields in CN — N arbitrary — are also given.
Copyright (c) 2026 Hassan Azad, Indranil Biswas, Said Waqas Shah https://creativecommons.org/licenses/by-nc-nd/4.0
Insights on the Homogeneous 3-local representations of the Twin Groups
We provide a complete classification of the homogeneous 3-local representations of the twin group T_n, the virtual twin group VT_n, and the welded twin group WT_n, for all n\geq 4. Beyond this classification, we examine the main characteristics of these representations, particularly their irreducibility and faithfulness. More deeply, we show that all such representations are reducible, and most of them are unfaithful. Also, we find necessary and sufficient conditions of the first two types of the classified representations of T_n to be irreducible in the case n=4. The obtained results provide insights into the algebraic structure of these three groups.
Copyright (c) 2026 Mohamad N. Nasser https://creativecommons.org/licenses/by-nc-nd/4.0
New Approach to fractional Milne-type Inequalities
In the literature, there are several Milne-type inequalities by using the convex functions. In this study, we establish the upper and below bounds for fractional Milne-type inequalities by using functions whose second derivatives are bounded instead of convex functions. Moreover, we present new inequalities for Riemann integrals as special cases. We also present an example and a graph to illustrate the main results.
Copyright (c) 2026 Hüseyin Budak https://creativecommons.org/licenses/by-nc-nd/4.0
SA-Extending and WS-Rationally Extending Modules: Structure and Duality
We present a comprehensive study of several generalized module classes, including SA-extending, $\oplus$-s-extending, SA-supplement, and WS-rationally extending modules. Detailed definitions, fundamental properties, illustrative examples, and structural results are provided to clarify the relationships among these classes. Dual notions, such as SA-lifting, $\oplus$-s-lifting, and SA-co-supplement modules, are introduced to emphasize the interplay between extending and lifting properties. Hybrid classes, including semi-rationally extending modules, are also considered, with an analysis of their closure properties and behavior over artinian, semiperfect, and group rings. Several open problems and conjectures are formulated, highlighting directions for future research in module theory and its applications.
SA-extending modules; $\oplus$-s-extending modules; WS- rationally extending modules; SA-supplement modules; Lifting modules.
Copyright (c) 2026 Abdelkader Hamdouni https://creativecommons.org/licenses/by-nc-nd/4.0
Nontrivial Deformation of Vec(R)-Modules of Symbols
The action of Lie algebra of vector fields on the space of symbols is given by the Lie derivative. We compute the nonrelative cohomology space with respact the Lie subalgebra affine. We study nontrivial deformations of this action.
affine Lie subalgebra, nontrivial Deformation,
Copyright (c) 2026 Khaled Basdouri https://creativecommons.org/licenses/by-nc-nd/4.0
AN ALGORITHM FOR MINIMAL GENERATING SET OF PARAMETRIC HOMOGENEOUS POLYNOMIAL IDEALS
We introduce comprehensive minimal generating systems (CMGS), a framework for computing minimal generating sets of homogeneous polynomial ideals with parametric coefficients, which is a non-trivial problem. While Schreyer’s syzygy theorem (1980) solved this problem for constant coefficients, the parametric case was recently addressed by the author’s MGSystem algorithm [9], which computes minimal generating sets for homogeneous parametric ideals forming Gr¨obner bases in the variables. In this paper, we extend these results to arbitrary homogeneous parametric polynomial ideals. Our approach combines completed Gr¨obner systems [6] with an extension of Schreyer’s theorem to partition the parameter space into regions where minimal generators are uniformly computed. The CMGS algorithm utilizes syzygy computations to systematically eliminate redundant generators, resulting in significant reductions in generating set sizes under parameter constraints. A Maple implementation demonstrates practical efficiency, with examples showing how parameter specializations simplify ideals. This work bridges a 40-year gap in computational algebra, offering theoretical advances (e.g., structural invariants of parametric ideals) and practical algorithms for applications in algebraic geometry, homological algebra, and symbolic computation.
Copyright (c) 2026 Dr. Mahdi Dehghani Darmian https://creativecommons.org/licenses/by-nc-nd/4.0
Topological Neighborhood Induced Ideal Convergence of Sequences of Sets
This paper investigates the concept of ideal convergence of sequences of sets through neighborhoods in topological context. The operators ∆−x and ∆+x are employed as fundamental tools for the establishment of nI-limit inferior and nI-limit superior. For sequence of sets, ‘sandwich’ theorem like description is presented under nI-convergence. Several inclusion properties of nI − lim inf, nI − lim sup and related notions are derived. It has been shown that in an I compact space, sequence of closed sets having I-intersection property always possess nI − lim inf. The scope and limitations of the theory are further tracked down through counterexamples.
Copyright (c) 2026 Prasenjit Bal https://creativecommons.org/licenses/by-nc-nd/4.0
Kernel Integral Legendre Polynomials and Approximations
This paper is concerned with deriving a new system of orthogonal polynomials whose inflection points coincide with their interior roots, primitives of Legendre polynomials,they appear as solutions of linear differential equation.We study orthogonality,and extremal properties and minimization and Fourier development involving of integral Legendre polynomials. There are some important properties and certain identities and extremal properties involving both associated integral Legendre polynomials. We have used mathematical induction to establish the relation between them.We also present some results for these orthogonal polynomials by using some properties of Jacobi polynomials.General expressions are found for the kernels polynomials associated to integral Legendre polynomials.These kernel polynomials can be used to describe the approximation of continuous functions by integral Legendre polynomials.They can be used for the representation of the n-th partial sum of the Fourier series expansion of integral Legendre polynomials in the form of an integral.We conclude the paper with some results on finite Fourier series expansion by using polynomials integrals of the kernels polynomials
Legendre polynomials,Integral Legendre polynoChristoffel-Darboux formula,Fourier series expansion,Best approximation problems.
Copyright (c) 2026 Adelhamid Rehouma, Hossein Jafari https://creativecommons.org/licenses/by-nc-nd/4.0
This paper introduces the concept of wp-semiring and investigates its algebraic properties. Closed weakly Stone and dense elements according to wp-complementation are defined, and their properties are proved. Connections between wp-semiring and other ringlike structures are verified. Structures of closed and dense elements are shown as orthopseudoring and presemiring, respectively.
Copyright (c) 2026 Eman Ghareeb Rezk https://creativecommons.org/licenses/by-nc-nd/4.0
Stochastic Differential Equations based on Perona-Malik Functions for Image Denoising
The aim of this work is to propose models based on stochastic differential equations with a good choice for both diffusion and drift terms to solve an image restoration problem. These proposed models are based on Barbu and Borkowski stochastic equations, where we exploit the Perona-Malik functions for the drift and diffusion terms. First, we show the derived model’s well posedness; then we present the related numerical results. These models give satisfactory experimental results in removing noise, improving and preserving the image structure compare to other well known approaches, such as [1], [2] and [4].
Stochastic Differential Equations, Analysis and Well-Posedness, Numerical Methods, Image Processing.
Copyright (c) 2025 Radhia Halilou, Fatma Zohra Nouri https://creativecommons.org/licenses/by-nc-nd/4.0
Kummer Surface Arithmetic for Primality Testing
We use the arithmetic of the Kummer surface associated to the Jacobian of a hyperelliptic curve to study the primality of integers of the form λm,n := 4m^2*5^n − 1. We provide an algorithm capable of proving the primality or compositeness of most of the integers in these families and discuss in detail the necessary steps to implement this algorithm in a computer. The algorithm depends on an initial choice of a pair of integers. If λm,n is prime there is always a pair that works. We prove that the proportion of pairs that do not work to show compositeness for this algorithm is very small and decreases exponentially with n.
Copyright (c) 2026 Eduardo Ruiz Duarte, Marc Paul https://creativecommons.org/licenses/by-nc-nd/4.0
Common Fixed Point Results in Fuzzy b-Metric Space Using the Integrals as Application
In recent years, the notion of fuzzy mathematics has become one of the interesting field of research. In this article, using the idea of fuzzy b-metric space and contraction mappings, we prove Banach contraction theorem. Also applying the idea of different compatible mappings, unique common fixed point results in fuzzy b-metric space are introduced and these results are connected in terms of the integral as the application. Some interesting instances to highlight the significance of our research are presented. Previous findings from the pertinent study are presented as lemmas and corollaries. Our findings complement and add to a large number of existing findings
Fuzzy b-Metric Space ; Common Fixed Point; Convergent Sequence; Compatible Mapping; Integral Equation
Copyright (c) 2025 Mr. Thaneshor Bhandari, Dr. KB Manandhar, Prof. Dr. Kanhaiya Jha https://creativecommons.org/licenses/by-nc-nd/4.0
Numerical Computations of Fractional Differential Equations in Engineering using the Polynomial Least Squares Method
Fractional differential equations appear in the modeling of science and engineering phenomena including classical mechanics, quantum mechanics, thermodynamics, fluid mechanics, relativity theory and chemical engineering. In this paper, the least squares method is used to find the numerical solution of fractional differential equations appearing in mechanics and engineering. The increasing interest in applications of fractional calculus has motivated the development and study of numerical methods specifically created to solve fractional differential equations. A unique feature is that engineers, physicists and scientists come across processes which lead to involve fractional differential equations. When dealing with more complicated systems with no precise solutions other than approximations, numerical methods are relied on to obtain solutions. The suggested method is used to solve various linear and non-linear problems of constant-order. The proposed scheme is found to be computationally effective with fast convergence.
Caputo Fractional Derivative; Fractional Calculus; Numerical Analysis; Fractional Differential Equations; Polynomial Least Squares Method.
Copyright (c) 2025 Shakeel Ahmed, Saif Ullah, Faiza tul Rasool https://creativecommons.org/licenses/by-nc-nd/4.0
New Results on Singularly Perturbed Fractional KdV and KdV-Burgers Equations
This paper investigates traveling wave solutions for singularly perturbed fractional KdV equation and KdV– Burgers equation with a linear confinement term using Caputo fractional derivatives of order α ∈ (3, 4). Using the tanh method, we first derive exact solutions for the classical cases, recovering the well-known hyperbolic secant soliton for the KdV equation and a tanh-profile solution for the KdV–Burgers equation with confinement term. For the singularly perturbed cases, we employ matched asymptotic expansions to construct solutions that capture both the localized wave core and the algebraic decay in the outer region induced by the fractional derivatives. The inner case retains the classical soliton structure, and the outer solution exhibits a power-law tail of order ξ ^(3−α), reflecting the nonlocal effects of fractional derivatives. Our main results reveal that the fractional perturbation modifies the wave’s decay behavior, giving rise to nonlocal effects absent from the classical formulation.
Copyright (c) 2025 Zoubir Dahmani, Iqbal Jebril https://creativecommons.org/licenses/by-nc-nd/4.0
Mathematical Modeling of Anticancer Drugs Using Distance-based Topological Indices
We examine the suitability of distance-based topological indices in modeling and forecasting the physicochemical properties of a selection of key drugs like Trastuzumab, Tamoxifen, paclitaxel, Ibritumomab, Gemcitabine, Fluorouracil (5-FU), and Doxorubicin. Four different indices—Mostar, Szeged, Trinajstić, and Padmakar–Ivan (PI)—were derived from molecular graphs for each drug. Quadratic models showed appreciable correlations between these indices and principal physicochemical properties like molecular weight and melting points. Topological indices are seen to forecast physicochemical properties effectively, making them useful aids in drug design, optimization, and discovery. The mathematical models shed light on structural complications affecting drug efficacy and pharmacokinetics, proving graph-theoretical methods effective in predicting.
Copyright (c) 2025 Muhammad Faisal Nadeem, Abdulrahman Ali Alsolami https://creativecommons.org/licenses/by-nc-nd/4.0
CRITIC-MABAC approach with Probabilistic Uncertain Linguistic q-Rung Orthopair Fuzzy Sets for Selecting the best Cloud Storage Service Alternative
Decision-making (DM) often experience challenges because of ambiguity and uncertainty in practical life. It is normally not easy to give clear values to data in such conditions. Probabilistic uncertain linguistic $q$-rung orthopair fuzzy sets (PUL$q$-ROFSs) constitute a flexible and applicable instrument that allows coping with the inherent ambiguity and fuzziness. These sets provide a wider methodology for addressing complex DM issues. A powerful solution for addressing such issues is the decomposition of the advantages of the multi-attributive border approximation area comparison (MABAC) method and the criteria importance through inter-criteria correlation (CRITIC) method. CRITIC is efficient when distributing weights of criteria based on interrelations, meanwhile MABAC is famous for its high rank towards options for their distance from the approximation area. A combination of these techniques leads to a thorough and systematic framework for the solution of uncertain DM problems. Based on this background, we extend the CRITIC-MABAC methodology to PUL$q$-ROFSs. This methodology can tackle a real-world use case: choosing the best cloud storage service alternative. We use the MABAC technique to rank alternatives and the CRITIC method to weight criteria with the aim of making our results valid and meaningful. We confirm the practical aspect of the methodology and evaluate its performance on a real-world DM situation, comparing the results with those yielded by previous techniques. Our findings highlight the impressive performance and the applicability of the presented CRITIC-MABAC approach.
Decision-making; CRITIC-MABAC; Probablistic linquistic fuzzy sets; $q$-rung orthopair fuzzy sets; Cloud storage service
Copyright (c) 2025 Uzma Ahmad, Saira Hameed, Muhammad Faisal Shabir, Ayesha Khan https://creativecommons.org/licenses/by-nc-nd/4.0
An Optimized Decomposition Method for Solving the Fractional-Order Burger’s Equation
In this study, the optimized decomposition method (ODM) is employed to investigate nonlinear fractional partial differential equations. Specifically, the method is applied to construct approximate analytical solutions of the fractional nonlinear Burgers’ equation (FBE), with the fractional derivatives defined in the Caputo sense. A comparative analysis is carried out with the Sumudu variational iteration method (SVIM). The numerical simulations demonstrate the robustness and efficiency of the proposed method, confirming its ease of implementation, reliability, and capability in capturing the essential dynamical behavior of the system.
Burger’s equation, fractional calculus, Caputo derivative, optimized decomposition method, nonlinear partial differential equations.
Copyright (c) 2025 abubker ahmed, Khalid Suliman Aboodh https://creativecommons.org/licenses/by-nc-nd/4.0
Image Encryption Using Chaotic Logistic Map and Finite Field Inversion for Dynamic S-Box Design
We present a new method for constructing dynamic S-boxes that combines the chaotic behavior with Galois Field arithmetic over GF(28). The approach employs the logistic map Xn+1 = μXn(1 − Xn), where X0 = 0.5 and 3.57 ≤ μ ≤ 3.99 to generate values, which are then reduced modulo 256 and interpreted as field elements. Using finite field inversion with an irreducible polynomial, bijective S-boxes are obtained. The proposed S-boxes are applied in an image encryption scheme to enhance security. Experimental evaluation shows that our designs are highly resistant against cryptanalytic attacks and improves the quality of encrypted images, offering a reliable way to protect sensitive image data during transmission.
S-box, chaotic maps, logistic map, Galois Field GF(28), cryptography, nonlinearity, differential probability, linear probability, image encryption, dynamic S-box, lightweight cryptosystem
Copyright (c) 2025 Anam Rani, Muhammad Sham Ul Haq, Syed Ahtsham Ul Haq Bokhary, Usman Ali https://creativecommons.org/licenses/by-nc-nd/4.0
Leap Edge Eccentricity Connectivity Index of PAMAM and Porphyrin-Cored Dendrimers
With the rapid advancement of technology, computer-aided methods are increasingly employed to study the structural properties of chemical compounds. One such approach involves predicting chemical behavior using topological indices—numerical descriptors derived from graph-theoretic representations of molecular structures. In this paper, the leap edge eccentricity connectivity index is introduced distance-based topological index that can be regarded as both the edge version of the leap eccentric connectivity index and the leap version of the edge eccentric connectivity index and investigates the leap edge eccentricity connectivity index (LEECI) for Polyamidoamine (PAMAM) dendrimers and porphyrin-cored dendrimers through exact analytical computations. By modeling these nanostructures as molecular graphs, LEECI values are derived for multiple dendrimer generations and architectures. The results reveal strong correlations between branching complexity and index growth, highlighting LEECI as a promising descriptor in computational nanomaterial characterization and drug delivery design. These findings provide a foundation for integrating LEECI into predictive models linking molecular topology with experimental bioactivity.
Copyright (c) 2025 ÖZGE ÇOLAKOĞLU, Yusuf Ayday https://creativecommons.org/licenses/by-nc-nd/4.0
Limit Cycles for a Class of Planar Differential Systems with a Degenerate Singular Point
This paper investigates a class of planar differential systems characterized by a degenerate singular point. We demonstrate that this class is Liouville integrable and explicitly derive a first integral using an Abel equation of the second kind. Moreover, through an analysis based on the Poincaré return map, we establish the existence of two non-algebraic limit cycles or a single algebraic limit cycle near the degenerate point. The occurrence of these limit cycles is shown to depend sensitively on the system's parameters.
Copyright (c) 2025 Abdelkrim Kna, Ahmed bendjeddou, Omar Chaalal, Akca Haydar https://creativecommons.org/licenses/by-nc-nd/4.0
An Efficient ElGamal Cryptosystem Based on Prime Power Moduli and Modular Key Exchange
In this paper, we give a security enhancement of ElGamal cryptosystem based on the use of the group of units U(n), where n = pt or 2pt, where p is an odd prime number and t is a positive integer dynamically determined by users using a modification of the Diffie-Hellman key exchange protocol. The paper presents in details the procedures of the key generation, encryption, decryption and digital signature for the proposed cryptosystem. As applications, we apply procedures of this cryptosystem over plaintexts and implemented its algorithms in the SageMath Software. Finally, we provide results concerning its complexity evaluation, its expansion rate, and its performance timing. The results show that this system is suitable for scalable real-time encryption applications as it is indicating the successful optimization performance and robust integration of security features.
Cryptography, public key cryptosystem, ElGamal cryptosystem, signature scheme, prime numbers, Hash function, group of units, Diffie--Hellman key exchange.
Copyright (c) 2025 Hayder R. Hashim https://creativecommons.org/licenses/by-nc-nd/4.0
Refinements of the Fractional Hermite-Hadamard Inequality via Arbitrary Means
This paper aims to refine the fractional Hermite-Hadamard inequality by employing arbitrary means defined on a given interval. Using weighted integral techniques and properties of convex functions, new bounds are established that improve upon existing results.
Copyright (c) 2025 Journal of Prime Research in Mathematics https://creativecommons.org/licenses/by-nc-nd/4.0
Optimal Control and Bifurcation Analysis of Cholera Model
Cholera presents a significant public health challenge, especially in regions lacking access to clean water and proper sanitation. This study aims to develop an optimal control framework to effectively mitigate the dynamic spread of cholera and inform more efficient intervention strategies. Initially, we establish the well-posedness of the cholera transmission model, demonstrating the positivity and boundedness of solutions within a specified domain. Utilizing the next-generation matrix approach, we compute the basic reproduction number, a critical epidemiological threshold for disease dynamics. Upon obtaining a value below one, we employ the Jacobian matrix, Metzler matrix, and Lyapunov function analysis to verify the local and global stability of the cholera-free equilibrium. Sensitivity analysis highlights the significant impact of certain model parameters, such as the transmission rate and treatment efficacy, on cholera control. Leveraging the Pontryagin minimum principle, we formulate an optimal control problem to derive the most effective combination of prevention and treatment strategies. Numerical simulations illustrate that the optimal control approach, involving the simultaneous implementation of both interventions, surpasses standalone prevention or treatment measures in reducing the disease burden. This study underscores the importance of integrating prevention and treatment strategies for the effective mitigation of cholera outbreaks. The proposed optimal control framework provides a systematic approach for evaluating the impact of various interventions and informing public health policies. Future work will involve model extensions to incorporate spatial heterogeneity, fractional derivatives, and real-world data integration, enhancing the applicability and robustness of the control strategies.
Copyright (c) 2026 Tulu Leta Tirfe, Legesse Lemecha Obsu, Eshetu Dadi Gurmu, Agbata Benedict Celestine, Shyamsunder https://creativecommons.org/licenses/by-nc-nd/4.0
Modified-Laplace Transform Based Variational Iteration Method for Solving Nonlinear Fractional Order Differential Equations of Caputo-Fabrizio Type
Fractional-order differential equations have become vital tools for accurately modelling processes with memory and hereditary properties in various fields such as engineering, physics and applied mathematics. However, solving such equations—particularly those involving the Caputo-Fabrizio derivative—poses analytical and numerical challenges due to their nonlocal behaviour. This study introduces a novel hybrid computational technique: the modified Laplace transform-based variational iteration method (MLTVIM), designed specifically to tackle these challenges efficiently. The method synergistically combines the modified Laplace transform, which simplifies fractional-order equations into algebraic form, with the variational iteration method, which effectively handles the nonlinear components. A strong theoretical foundation underpins the development of the method, including convergence analysis and supporting theorems that confirm its mathematical soundness. To validate the effectiveness of MLTVIM, the approach is applied to several benchmark fractional differential equations. The results of numerical experiments demonstrate that the proposed method outperforms traditional techniques in terms of accuracy, speed of convergence, and computational efficiency. The error analysis confirms that MLTVIM achieves lower approximation errors, making it a robust and precise tool for modelling complex dynamic systems. This research contributes a reliable and powerful approach to solving complex fractional models, offering significant potential for applications in science and engineering where memory-dependent behavior is prevalent.
Fractional Modified Laplace Transform Scheme
Copyright (c) 2025 Razaq A. ODERINU, Waliu A. TIJANI, John AKANNI, Kafilat A. SALAUDEEN, Musilimu TAIWO, Fatmawati Fatmawati https://creativecommons.org/licenses/by-nc-nd/4.0
New perspectives on Hepatitis C model with Piecewise Derivative: Deterministic and Stochastic Approaches
In this paper, we have extensively explored deterministic-stochastic approaches of the Hepatitis C model using piecewise fractional operators techniques. This new concept leads us to capture many behaviors from cross-over to random processes. The piecewise differential operators that can be constructed from deterministic to stochastic patterns showed us their innovative aspects. To illustrate the effectiveness of the proposed approach, graphical representations for such a model are provided. As a result, it is concluded that these operators allow us to adapt more effectively to the realistic tendencies of real-world problems. Due to the property of capturing different behaviours at different time intervals, these operators have opened new doors for research in different disciplines.
Hepatit-C model, piecewise derivative, stochastic approaches, existence and uniqueness.
Copyright (c) 2025 İPEK KORKMAZ, Seda IGRET ARAZ https://creativecommons.org/licenses/by-nc-nd/4.0
Mathematical Properties of Inverse Sum Index Eccentric Coindices of Graphs
Essential and widely studied topological indices, including the well-known Zagreb indices (M1 and M2), and the newly proposed Inverse Sum Indeg Eccentricity Index (ξISI), To ensure the contributions of all edges within a graph are effectively considered. By emphasizing on the total eccentricity of non-adjacent vertices, Hua et al. introduced the eccentric connectivity coindex (ξc). Inspired by their contributions, we introduce the inverse sum indeg eccentric coindex (ξISI), which is defined as the ratio of the product of the eccentricities to the sum of the eccentricities for all isolated pair of vertex in a connected graph. This study primarily aims to establish various bounds for ξISI in finite simple graphs and derives the values of the proposed indices for two specific graph constructions. Additionally, we present a comprehensive set of relationships for ξISI using several graph products.
Copyright (c) 2024 Journal of Prime Research in Mathematics https://creativecommons.org/licenses/by-nc-nd/4.0
New Results Concerned with Fixed Point Theory in Controlled Metric Spaces
This paper introduces a new sufficient condition for Cauchyness on sequences in controlled metric spaces. This sufficient condition can improve many pre-existing fixed point results under weaker hypotheses. Applying this sufficient condition, I establish some fixed point theorems involving nonlinear Ćirić type contraction via control functions in controlled metric spaces and present numerical examples to show the validity
Copyright (c) 2025 Chol Jin Kil https://creativecommons.org/licenses/by-nc-nd/4.0
Continuous automorphisms and an equivalence relation In KX
Let K be an arbitrary commutative field and let R=KXR=KX be the ring of formal power series in one variable. Let GRGR be the set of all power series of the form u=Xvu=Xv, where vv is a unity in RR. Relative to the usual composition GRGR becomes a topological group with respect to the X−X−adic topology of RR. We also study an equivalence relation on RR. Let R=KXR=KX be the ring of formal power series in one variable over a fixed commutative field KK. We denote by ordf=min{i:ai≠0}ordf=min{i:ai≠0} for any f∈Rf∈R . It is well known that ordfordf is a valuation on RR and RR becomes a complete topological ring relative to the topology induced by this valuation. Let GG={u∈R:ordu=1}GG={u∈R:ordu=1} and for u,v∈RGu,v∈RG we denote (uov)(X)=v(u(X))(uov)(X)=v(u(X)), a new element of RGRG.
Copyright (c) 2005 Journal of Prime Research in Mathematics
In this paper, we introduce the concept of morphisms between two grey sets and defined a new category, namely, GSet, of grey sets and grey morphisms. We investigate some categorical notions such product, coproduct, pullback, and pushout. Additionally, we study equalizer and coequalizer for pairs of grey morphisms and show that any grey set has an injective hull.
Formulas of the solutions of a solvable system of nonlinear difference equations
where n ∈ N0, p, q, r ∈ N, f, g, h : D → R are continuous one-to-one functions on D ⊆ R, the coefficients (an)n∈N0 , (bn)n∈N0 ,(cn)n∈N0 ,(dn)n∈N0 ,(sn)n∈N0 ,(tn)n∈N0 are non-zero real numbers and the initial values x−i , y−i , z−i , i = 0, 1, 2, are real numbers. We will give explicit formulas for well-defined solutions of the aforementioned system in both variable and constant cases of the coefficients. As an application, we will deduce the formulas of the solutions of the particular system obtained from the general one by taking f(x) = g(x) = h(x) = x.