This paper investigates spacetime manifolds admitting a pseudo‐ W 2 curvature tensor. We show that a pseudo‐ W 2 flat spacetime is an Einstein manifold and therefore has constant curvature. Moreover, when the manifold satisfies the Einstein field equations (EFE), with a cosmological constant, the associated energy–momentum tensor is covariantly constant. We further determine the length of the Ricci operator in a pseudo‐ W 2 flat perfect fluid spacetime and prove that, under the EFE without a cosmological constant, such a spacetime possesses constant energy density and isotropic pressure. Finally, an explicit example of a spacetime admitting a pseudo‐W2 curvature tensor is constructed to validate the results using differential equations.

Topological invariants are key tools for studying the physicochemical and thermodynamic properties of chemical compounds. Recently, a new bond‐additive distance‐based graph invariant called the Mostar index has been developed. It measures the importance of individual edges and the graph as a whole. It is denoted and defined as . This invariant is helpful to characterize the structure of a given connected graph G . In this invariant, the quantity n x ( x y ) means the number of vertices closer to x than y . In the present study, first, we have considered the Mostar index of extremal unicyclic graphs of n vertices with given diameter. Second, we have determined all unicyclic graphs that contain the second maximum Mostar index.

The use of graph‐theoretic parameters is essential in interpreting the structural robustness, efficiency, and functionality of complex networks. Specifically, edge‐based metrics are more useful in interpreting the susceptibility of links and the concept of upstream impact, as well as structural robustness. Such metrics are essential in a variety of fields such as network security, communication, and optimization theory. In the same way, a new edge‐based invariant is introduced in this research paper, which is known as the b -edge differential . The b‐edge differential is essentially linked to b ‐differential of a graph. Let G = ( V , E ) be a simple connected graph. For any subset of edges X ⊆ E , we associate the set B ( X ) = { e ∈ E \ X : e shares a common vertex with at least one edge in X }, that is, B ( X ) captures all edges outside X which are adjacent to X . The quantity ∂ B E ( X ) = | B ( X )| is called the b ‐edge differential of X . We then define the b ‐edge differential of a graph G by ∂ B E ( G ) = max X ⊆ E ∂ B E ( X ). This parameter calculates the maximum value of the indirect dominances, or coverage, which can be achieved by a given set of edges. It serves as a useful tool in the identification of critical edges as well as the maximization of influence in networks. The problem can also be used in the applications involving the maximization of profit, traffic routing, and vulnerability evaluation for communication networks. We carry out the explicit computation of the b ‐edge differential for some classical types of graphs, such as paths, cyclic graphs, wheel graphs, complete graphs, star graphs, double‐star graphs, comb graphs, complete bipartite graphs, ladder graphs, and triangular ladder graphs. The work described in this article expands the relevance of differential parameters in the theory of graphs, with promising research directions for theoretical and application aspects of network science.

This paper defines a strong convertible nonconvex (SCN) function for solving the unconstrained optimization problems with the nonconvex or nonsmooth (nondifferentiable) function. First, the concept of SCN function is defined, where the SCN functions are nonconvex or nonsmooth. Second, the operational properties of SCN functions are proved, including addition, multiplication, and compound operations. Third, an SCN optimization with the SCN function and its penalty function is defined. The optimization condition, exactness, and stability of the penalty function for SCN optimization are proved. This paper provides a way for solving unconstrained nonconvex or nonsmooth (nondifferentiable) optimization problems to avoid using subdifferentiation or smoothing techniques.

The Hirota–Maccari (HM) system is a fundamental model in wave propagation that has been widely utilized to investigate complex nonlinear phenomena in nonlinear optics, optical communications, and mathematical physics. The HM system sheds light on critical insights into soliton dynamics, wave interactions, and other nonlinear effects. Therefore, the main goal of this work is to compile new soliton structures to the HM system in hyperbolic, trigonometric, exponential, and rational forms, both single and combined version, by using three robust analytical approaches: the new extended rational sinh‐Gordon equation expansion method (ShGEEM), the ( G ′ / G , 1/ G )‐expansion method, and the new extended hyperbolic function method (EHFM). The solutions thus obtained are also tested for their validity and accuracy using Mathematica. In order to exhibit the physical properties of the soliton pulses, we depict 2D, 3D, and contour graphs by using suitable values of the parameters. These methods play a major role compared with other methods in the literature because new and more general solutions are obtained with additional free parameters. Remarkably, all the known solutions are special cases of our generalized solutions. An important feature of the proposed methods is their simplicity, robustness, and computational efficiency that can be widely extended to nonlinear partial differential equations in all scientific disciplines. The efficiency of this framework thus indicates at its application in addressing complicated problems in the future.

In this paper, we investigate the existence of completions and Fréchet universal spaces within the framework of ‐metric spaces. We show that every ‐metric space has a completion and that, for any family of ‐metric spaces, there exists a Fréchet universal space.

We derive the left‐ and right‐sided fractional Hermite–Hadamard ( H – H )‐type inequalities for harmonic convex mappings from the left‐ and right‐sided fractional integral operators possessing exponential kernels. In addition, we introduce two variants of fractional equalities that are further deployed with the idea differentiable harmonic convex mappings to explore refine fractional bounds for trapezoid and midpoint‐type integral inequalities. We use Power‐mean, Hölder’s, and improved Hölder inequality to derive new and refined fractional inequalities. In addition, we make the important connections between the obtained results and those classical integrals very clear. Strong visual illustrations make it evident how accurate and superior the provided technique is. The study includes several applications of special means and numerical quadrature rules. Our results generalize and extend some existing ones in the literature.

In this paper, we have developed a discrete‐time predator–prey system to discuss the theoretical basis of guava‐pest management, incorporating the novel neem‐leaf treatment ( k ), the key bifurcation parameter, logistic prey growth, and the Holling Type‐II functional response. The continuous counterpart has been discretized by using the forward Euler scheme to align with pulsed, seasonal management operations. We concluded that the system observes the complex dynamics, including the flip and Neimark–Sacker bifurcation, which demonstrate the shift of a stable regime to periodic, oscillatory behavior and then to chaos. The detailed study of bifurcation and stability concluded that neem intensity ( k ) plays the key role of bifurcation parameter, and we have developed a parameter threshold at which the system either loses stability or can be recovered and finally delay the chaos. Our theoretical results have been justified by the numerical examples along with the bifurcation diagrams and maximum Lyapunov exponent that demonstrate how the interplay between treatment intensity ( k ) and application frequency ( δ ) interact to mitigate the long‐term behavior. In spite of this work being purely theoretical, we have concluded that a novel mathematical framework in the context of bifurcation theory and utilizing the biologically meaningful control parameters plays a significant role in the ecological modeling. The findings in this paper provide a structure for intervention methodologies and build a foundation for future empirical validation, parameter estimation, and use in different crop‐pest systems within integrated pest management (IPM).

Cyclotomic classes of finite abelian groups have been extensively investigated for many decades, largely because of their nice algebraic structure and the breadth of their theoretical and practical applications. They naturally arise in diverse areas of mathematics, ranging from number theory and polynomial factorization to the decomposition of group algebras, and they play a particularly important role in the study of algebraic coding theory, where they provide fundamental tools for analyzing cyclic and abelian codes as well as their generalizations. Motivated by these connections, the present work concentrates on certain special types of q 2 ‐cyclotomic classes arising in a direct product of finite abelian groups, with q denoting a prime power. The main contributions of this work include a detailed characterization and enumeration of these cyclotomic classes in terms of the corresponding classes in their component groups. Several concrete examples and explicit computations are provided to illustrate the general results and to demonstrate how the cyclotomic structure behaves under group product decompositions. As an application, the paper revisits the enumeration of Hermitian complementary dual and Hermitian self‐dual abelian codes defined over group algebras, presenting refined counting formulas derived from the structure of q 2 ‐cyclotomic classes. These results not only deepen our understanding of the interplay between group theory and coding theory but also contribute to the ongoing development of algebraic coding techniques grounded in group‐theoretic methods.

A connection between cryptography and polynomial functions is extremely significant. Mathematical performance of polynomials helps to enhance the cryptographic primitives, which are trustworthy as well as straightforward representation tools, in everyday use. In this research, explicit topological sequences Q f which correspond to degree‐based, distance‐based, and degree‐distance‐based indices f have been developed for ‐order relatively prime graphs Γ of finite commutative weak inverse property quasigroups, . Through the utilization of Lagrange interpolation and polynomial approximation for introducing quasigroup curves , it additionally gives polynomial formulations that encode the structural complexity of graphs interconnected with commutative weak inverse property quasigroups. Compared to previous investigations, the current research highlights a systematic polynomial‐sequence connection. Quasigroup curves, cryptographic primitives, and graph invariants are all integrated together in a single framework. For lightweight cryptographic parameterization, the utilization of commutative weak inverse property quasigroups, relatively prime graphs, and polynomial modeling is an entirely new mathematical and technical advancement not previously seen in the literature.