This paper studies the algebraic structure of linear, cyclic, and constacyclic codes over the finite semi-local non-chain Frobenius ring R = R_1 + vR_1 , where R_1 = 𝔽_2[u]/⟨ u^4 - 1 ⟩ . A Gray map from R to R_1^2 is constructed. Beyond basic linearity, we prove that cyclic and constacyclic codes over R have a unique structural decomposition into two R_1 codes, making their construction and analysis straightforward. Furthermore, the study establishes a connection between these algebraic structures and DNA computing by introducing a mapping based on the u^2 -adic system. Also, a specific generator set is defined, enabling the construction of DNA codes that satisfy reversibility and reversible-complement constraints, which are essential for reliable biological applications. The results provide a combined algebraic framework for designing efficient DNA codes with desirable combinatorial properties.
In this article, we propose a Quadratic Uniform L1 (QUL1) numerical scheme for approximating the Caputo fractional derivative of order (0 < α < 1) and for solving Caputo fractional differential equations. The proposed method modifies only the initial quadrature weight of the classical L1 discretization while retaining the remaining convolution weights unchanged, thereby preserving the simplicity and straightforward implementation of the classical L1 method. A rigorous theoretical analysis is developed for the revised scheme. In particular, we establish finite time stability result and prove that the QUL1 approximation converges with order (O(τ^1-α)) under suitable regularity assumptions on the exact solution. Numerical experiments for both linear and nonlinear fractional differential equations confirm the theoretical convergence behaviour and demonstrate the robustness and practical effectiveness of the proposed method.
In this paper, a Bazykin-type predator-prey model is proposed. The dual fear effects, endemic disease prevalence within the prey population, and cooperative hunting behavior of predators are systematically incorporated into the governing dynamical equations. The existence of equilibria for the deterministic model is rigorously examined. And a detailed bifurcation analysis unveils a rich spectrum of complex dynamics, encompassing period-doubling cascades that culminate in chaos. The model is then extended to a stochastic framework. By employing Lyapunov function methods, we first rigorously establish the existence, uniqueness, and stochastically uniform boundedness of the global positive solution to the model. Building on this foundation, we further derive sufficient conditions governing the existence of an ergodic stationary distribution and the eventual extinction of the population. Numerical simulations are carried out to provide an intuitive validation of the theoretical findings.
In practical systems, the coefficient matrix often contains measurement or estimation errors, and the observed data may also exhibit fuzziness. Focusing on fuzzy matrix equations where the coefficient matrix is real and the right-hand side consists of LR fuzzy numbers, this paper simultaneously accounts for perturbations in both the coefficient matrix and the right-hand side, and proposes a novel fuzzy total least squares method. Based on the arithmetic of LR fuzzy numbers and the theory of generalized inverses, the original fuzzy matrix equation is decomposed into a central equation and a spread equation. Using the total least squares criterion and orthogonal projection techniques, the existence conditions and explicit expressions for the minimum-norm LR fuzzy TLS solution are derived. Furthermore, a parametric form of the solution set is given, a comparison with the fuzzy LS solution is carried out, and error bounds are established. Numerical examples demonstrate the effectiveness of the proposed method.
