We define and examine in-depth the notion of a weakly reversible ring showing that all weakly reversible rings are abelian McCoy rings and so, in particular, they are abelian 2-primal rings. Moreover, we construct a weakly reversible ring which is not reversible. We also prove that, if R is a weakly reversible ring, then the polynomial ring R[x] is strongly AB. Thus, in particular, the weakly reversible ring R is zip if, and only if, R[x] is zip. We, moreover, establish that if R is a weakly reversible ring and every prime ideal of R is maximal, then both R and R[x] are AB rings.
The main focus of this study is the investigation of a time fractional higher-order Broer-Kaup (BK) system which, to the best of our knowledge, has not been reported previously in terms of Riemann-Liouville (RL) and Caputo fractional derivatives. Performing the similarity reduction formalism, the time fractional higher-order BK system is reduced into a system of fractional nonlinear ODEs. Moreover, the power series solutions are derived, including their convergence analysis. By considering the obtained symmetries, conserved quantities of the system under consideration are formulated based on the nonlinear self-adjointness property for fractional differential equations, and the nontriviality property is also checked. In view of the Sumudu-Adomian decomposition method, we obtain a set of analytical solutions. To show the effectiveness of this employed method, we analyze the influence of the fractional order and explore the dynamic behavior of the constructed solutions through graphical representations.
Let (X,d,μ ) be an RD-space satisfying the doubling condition in the sense of Coifman and Weiss and a certain reverse doubling condition. In this setting, the authors first investigate a weighted John–Nirenberg inequality for the space BMO_θ ,φ(X) introduced in Lu, G.; Wang, M.: Characterizations of spaces BMO_θ ,φ(X) on RD-spaces., which extends some known results on BMO -type spaces on Euclidean spaces and RD-spaces. Furthermore, some useful characterizations of spaces BMO_θ ,φ(X) are established.
In this research paper, we study various central differential identities on prime rings with anti-automorphism involving generalized derivations. The forms of these maps satisfying some special identities are examined utilizing the theory of functional identities.
Systems of Lane–Emden-type equations describe the modelling of catalytic diffusion reactions, the concentration of carbon substrate and oxygen, the steady state concentration of carbon dioxide, dusty fluid models, and pattern formation. In this article, we explore a collocation method that is based on mth-degree shifted Jacobi polynomials _L,m^(α ,β )(t):= P_m^(α ,β )( 2t/L-1) (t∈ [0,L],L>0;α ,β∈ (-1,∞ )) to obtain numerical solutions of systems of Lane–Emden-type equations. The proposed method assumes the solution in the form of shifted Jacobi polynomial series; thus, the differentiation formula for the shifted Jacobi polynomial is applied upon substituting the assumed series solution into the proposed problem. Collocating at the nodes of the shifted Jacobi–Gauss interpolation in the interval [0, L], we obtain a set of nonlinear algebraic equations, which are subsequently solved for the expansion coefficients using Newton’s iteration method. The convergence and error analysis of the proposed collocation scheme is presented. Two special systems of nonlinear singular initial value problems of Lane–Emden-type (with different values of parameters α ,β ) are presented to demonstrate the proposed method’s reliability, effectiveness, and accuracy. The obtained numerical solutions are compared with the exact solutions and other published results. Our results are in excellent agreement with the solutions under comparison, which is a clear indication that the proposed method is highly reliable, effective, and accurate for nonlinear singular initial value problems.
This paper investigates the fractal dimension of continuous and differentiable curves through the construction of function sequences defined on the unit closed interval [0,1] . Sequences composed of piecewise linear functions and their Box dimensions are analyzed. Under appropriate conditions, it is shown that the Box dimension of the graphs in ℝ^2 exceeds the Box dimension of the corresponding intersection set in ℝ by one, when infinitely many lines intersect the entire interval [0,1] . Additionally, in cases where the lines intersect only a portion of [0,1] , an upper bound for the Box dimension has been established using analogous methods.
This paper presents a general class of fast shift-splitting preconditioners for solving nonsymmetric saddle point problems. By selecting some suitable matrix splitting schemes, we propose general fast shift-splitting preconditioners. The convergence of the corresponding matrix splitting iteration method is also analyzed. Additionally, we discuss the spectral properties of the preconditioned matrix. With suitable choices of the splitting schemes, numerical experiments are given to demonstrate the efficiency and flexibility.
In this paper, we introduce the notions of packing polynomial entropy for subsets and local measure-theoretic polynomial entropy for Borel probability measures. We investigate the relationship between Bowen polynomial entropy and packing polynomial entropy, and establish both variational and inverse variational principles for the packing polynomial entropy of subsets.
We study the composition of F. R. Cohen map B_n→ B_n k with the Tong-Yang-Ma representation B_nk→ GL_nk(ℤ[t^± 1]) in the case k=2 , where B_n is the braid group on n strings. We prove that the transpose of the representation thus obtained of degree 2n is the direct sum of two copies of Tong-Yang-Ma representations, each of degree n, with t replaced by t^2 .
In this paper, we introduce and study conformal algebras with brackets (CAWB), which generalize algebras with brackets to the conformal setting. After establishing their definition and basic properties, we provide an equivalent characterization of quadratic CAWB. We construct dendriform-CAWB structures via 𝒪 -operators. These results extend the operator approach of algebras with brackets to the conformal framework, revealing new structural connections with conformal algebra theory and related algebraic systems.
Defining a graph that establishes a connection between ring theory and graph theory is an interesting area of research in mathematics. In this paper, we establish a relationship between graphs and seminearrings, which are generalizations of rings. We propose an equiprime graph of a seminearring S, which handles both the binary operations of S. It’s interesting to see that the classical zero-divisor graph is a special case of an equiprime (e-prime) graph. Further, other prime graphs are studied in detail, and related results are presented. Additionally, we examine the conditions under which the ideal becomes a vertex cover of the e-prime graph and the dominating set of its line graph.
A unit u in a ring R is called exceptional if 1-u is also a unit. Such units have been previously studied from a Number Theory perspective. In this paper, we investigate exceptional units from the standpoint of Ring Theory, with particular emphasis on matrix rings, where several characterizations are provided. We define and explore a special subclass of exceptional units, termed complementable units. An exceptional unit u is called complementable if u(1-u)=1 . We determine the residue class rings that contain complementable units and identify such units in certain matrix rings.
We present a general framework for time-fractional optimal control problems posed in non-normable state spaces, formulated as projective limits of Hilbert scales. The system dynamics are governed by a Caputo time-fractional evolution equation with a generator A satisfying level-wise monotonicity and resolvent estimates. Controls may act in the domain or on the boundary via operators that are only well-defined in the distributional sense. We establish well-posedness of the state equation using level-wise Galerkin approximations, prove a fractional integration-by-parts identity adapted to the projective-limit setting, and derive the associated adjoint equation. The optimal control is characterized by a projection formula that remains valid for both distributed and boundary control scenarios. A consistent discretization strategy is proposed, ensuring the discrete state and adjoint operators satisfy the same duality relations as in the continuous theory. Numerical experiments—including scalar fractional ODEs and PDEs with both distributed and boundary control—validate the theoretical results, demonstrating convergence and computational efficiency. Potential extensions include time-dependent operators, measure-valued controls, and free-final-time problems.
A modification of the Einstein–Hilbert Lagrangian by introducing a coupling between the Weyl tensor and the stress-energy tensor was proposed to explain flat galactic rotation curves without the exotic (non-baryonic) dark matter (DM) [25]. The proposed coupling constant was previously determined by fitting the rotational velocities of the Milky Way and M31 modeled with constant density, yielding the same coupling constant for both [5, 32]. In this work, we have modified the formalism for a variable density by modeling the galactic systems with realistic, spherically symmetric and radially varying density profiles for the baryonic matter and this analysis is applied to seven edge-on spiral galaxies of the local cluster [8, 10, 18, 23, 35, 45, 51] and the Milky Way.
Analysis of coupled Poisson–Schrödinger equations is of substantial research interest in nano-scale semiconductor device modeling as it effectively captures the interdependence of quantum mechanical effects of charge carriers and electrostatic potential. In this article, we present an iterative mixed numerical scheme with a unified approach of finite difference—finite element method on layer-adapted meshes to comply with the singularly perturbed nature of Schrödinger equation. The latter being a singularly perturbed reaction-diffusion equation with non-homogeneous Neumann boundary conditions, the coupled equation is subjected to an extensive numerical analysis employing the proposed iteratively coupled scheme using the singular perturbation approach. The framing of a novel scheme that befits the singularly perturbed nature of the Schrödinger equation and executing an unprecedented singular perturbation analysis of an iterative scheme that decouples a singularly perturbed reaction diffusion equation with an ordinary differential equation demonstrates the novelty of the study. The errors associated with the proposed scheme are estimated and validated by comparing the results of computational works on MoS_2 transistors of channel length of 20 nm. Also, the mixed numerical scheme is illustrated for two-dimensional coupled Poisson-Schrödinger equations subject to appropriate boundary conditions, and the computational results are presented.
In this article, we have designed a fifth-order conformable iterative method to solve nonlinear and transcendental equations. The proposed method exhibits superior stability, efficiency, computational performance, and convergence speed compared to the existing Newton and Traub methods. This is demonstrated through mathematical models such as the Shockley diode equation, biometric human arm motion, and signal processing models. Furthermore, we have generalized our proposed method to solve systems of nonlinear equations and conducted a detailed convergence analysis. Numerical experiments have been performed, and the Approximated computational order of convergence (ACOC) validates the theoretical findings. Additionally, in some numerical experiments, it is observed that conformable iterative methods outperform classical iterative methods. Moreover, the stability of the proposed method is illustrated using convergence plane analysis.
This article presents a generalization of the entropy and its complementary measure, called generalized fractional extropy and entropy. We compare the two measures, highlight their features, and establish several bounds. Moreover, the approximation of the generalized fractional extropy is discussed. In addition, the differential generalized fractional extropy of its approximation is given. Finally, we demonstrate the application of the proposed measures in a pattern recognition problem, where the softmax function is employed.
Time-fractional diffusion (TFD) equations have emerged as powerful tools for modeling various physical and engineering processes, particularly in signal smoothing applications. This paper presents an efficient numerical method for solving TFD equations based on the operational matrix approach using Bernstein polynomials. The operational matrix of fractional derivatives is systematically derived and represented as a product of structured matrices to enhance computational efficiency. With the help of this matrix and the collocation method, the TFD equation is transformed into a system of algebraic equations, significantly simplifying the numerical procedure. A perturbation-based technique is employed to analyze the stability of the proposed scheme. Numerical results confirm the accuracy, stability, and effectiveness of the method, highlighting its potential for solving a broad class of fractional diffusion problems.
