This article study a coupled one-dimensional hyperbolic system with Lord-Shulman thermal heat law on the first equation, and a general weak inherent damping, along with time-dependent coefficient and time-varying delay term on the second equation. Using the multiplier method, through a suitable Lyapunov functional, we prove a general decay stability result that encompasses both exponential and polynomial decay estimates as special cases. This result extends and generalizes existing findings in the literature related to the system under study.

In recent years, complex fuzzy sets and $N$-fuzzy sets have gained substantial attention within the research community, leading to their extensive utilization in different algebraic systems including, groups, rings, and modules. This research work presents a pioneering mathematical framework, the complex $N$-fuzzy set, which combines the complex fuzziness of parameters with the negative fuzziness of data. Complex $N$-fuzzy sets represent a recent generalization of complex fuzzy sets and $N$-fuzzy sets in which the membership degree of each element is allowed to take values in a complex negative domain of the form $\left[-1,0\right]+[-1,0]i$. The real part represents a negative, counter-supportive or inhibitive degree of membership, while the imaginary part models an independent direction of orthogonal uncertainty, conflicting evidence, or dual hesitation. Based on this new structure, we introduce and examine the notions of complex $N$-fuzzy characteristic functions, level ( $\alpha ,\beta$)-cut, and the product of two complex $N$-fuzzy sets. Also, we present some fundamental operations of complex $N$-fuzzy sets and certain examples of them. In addition, this paper presents the novel mathematical framework of complex $N$-fuzzy semigroups, an algebraic structure that arises by superimposing complex $N$-fuzzy sets on crisp semigroup theory. Here, we present the notions of complex $N$-fuzzy sub-semigroups, complex $N$-fuzzy left (right) ideals, and complex $N$-fuzzy ideals of semigroups. We study certain theorems and their corresponding proofs to underpin these foundational concepts. Furthermore, we investigate some characterizations of these concepts using level ( $\alpha ,\beta$)-cut, and the product of two complex $N$-fuzzy sets. Lastly, we use complex $N$-fuzzy semigroup structure in real life industrial systems.

The present research is aim at investigating a thermoelastic Timoshenko system with an infinite memory term on the shear force while the bending moment is under the influence of a thermoelastic dissipation governed by Fourier's law. We prove that the system's stability holds for a broader class of relaxation functions. Under this class of relaxation functions $h$ at infinity, we establish a relation between the decay rate of the solution and the growth of $h$ at infinity. Moreover, we drop the boundedness assumptions on the history data. We employ Neumann-Dirichlet-Neumann boundary conditions for our result. In comparison to the bulk of results in the literature, which frequently enforce the "equal-wave-speed" constraint, the present result shows that the infinite memory of the beam and the thermal damping are strong enough to guarantee stability without any conditions on the parameters.