In this paper, we introduce and study the notion of strongly $\phi$-$w$-flat modules. The $\phi$-$w$-weak global dimension $\phi$-$w$-w.gl.dim$(R)$ of an $\NP$-ring $R$ is also introduced and studied. We characterize $\phi$-\Prufer\ $v$-multiplication strong $\phi$-rings in terms of strongly $\phi$-$w$-flat modules.

In this paper, we introduce a method for constructing continuous sections of the core fundamental groupoid bundle via a special group action induced by a regular topological group action on topological space. In this case each orbit determines a continuous section of the bundle. We investigate the algebraic and topological properties of the resulting sections and show that every section obtained through this construction is left-invariant (or $G$-invariant) with respect to the regularly acting topological group. Furthermore, the collection of all such orbit-derived sections forms a group that is naturally related to the fundamental groups of the underlying space. We establish that the associated group of sections yields a topological invariant for the class of spaces considered. In addition, we study induced maps arising from this construction and examine their behaviour with respect to several standard maps, including covering maps, homeomorphisms, deformation retracts, topological group homomorphisms, and bundle maps.

This paper is concerned with a new coupled KdV equation in the framework of the prolongation structure theory. By means of embedding the prolongation algebra into $sl(n+1,\mathbb{C})$, we present the a set of commutation relations of generators for of the Lie algebra determined by the prolongation structure. Moreover, its Lax representation By the way, Lax representation for the coupled KdV equation has been investigated.

Let $S$ be a semigroup. The \textit{co-intersection graph} of an $S$-act $A$, denoted by $G(A)$, is an undirected simple graph whose vertices are the non-trivial subacts of $A$ and two distinct vertices are adjacent if and only if their union is not equal to $A$. In this paper our focus is on exploring the interplay between the act-theoretic properties of $A$ and the graph-theoretic properties of $G(A)$. We characterize that every (free, projective, cofree) $S$-act for which the co-intersection graph of subacts is not connected. Also we determine the diameter and the girth of $G(A)$. Moreover, we study the clique number and the chromatic number of $G(A)$.

In 2023, F. Abbas introduced a new type of generalized open sets in the point set topological space and it is shown that, the collection of these sets form a topology on the original set. This topology is finer than the original topology. Is it possible to obtain a topology which is more finer than Abbas's topology? To solve the question, some generalized method for obtaining the new topologies have been introduced and it is shown that interior operator is not the only operator for obtaining the new topology. Through this write up, the relations between locally closed sets with the locally closed sets of the obtaining topologies have also been considered.

For $ 0 \leq \alpha \leq 1$, $\beta <1$ and $\gamma \geq 0$, the class $\mathcal{W}^{\varphi} _{\beta } (\alpha ,\gamma )$ satisfies the condition $ \operatorname{Re} \left(e^{i \varphi} \left[(1-\alpha+2 \gamma) \frac{f(z)}{z}+(\alpha-2 \gamma) f^{\prime}(z)+\gamma z f^{\prime \prime}(z)-\beta \right]\right)>0, \, \varphi \in \mathbb{R}$ with $\lvert \varphi \lvert < \frac{\pi}{2}$, $z \in \mathbb{D} $, is taken into consideration. We find the bound for $n^{th}$ coefficient of $f \in \mathcal{W}^{\varphi} _{\beta } (\alpha ,\gamma )$ and sharp bounds of coefficient functional, namely Zalcman functional, second order Hankel determinant, third order Toeplitz and Hermitian Toeplitz determinants. Bounds on difference of successive coefficients are also determined.

In this paper, we search certain distribution functions which become identical to the formal power series which is transformed via extended Borel-Leroy transformation operator at definite non-zero real points. It is also proved that the inverse extended Borel-Leroy (IEBL) transformation of the formal power series has non-zero real zeroes. In this connections, some of the generating functions in terms of the Wright function and Voigt function are determined via EBL transformation of the formal power series along-with applications of Lagrange's expansion formula and other known techniques.

A star coloring of a graph $G$ is a proper vertex coloring in which every path on four vertices in $G$ is not bicolored. The star chromatic number $\chi_{s}\left(G\right)$ of $G$ is the least number of colors needed to star color $G$. In this paper, we find the exact values of the star chromatic number of neighbourhood corona product of path graph with path graph $P_m \star P_n$, path graph with star graph $P_m\star K_{1,n}$, path graph with cycle graph $P_m \star C_n$, path graph with complete graph $P_m\star K_n$, path graph with wheel graph $P_m\star W_n$ path graph with Sunlet graph $P_m\star S_n$. Also, we find the star chromatic number of neighbourhood corona product of star graph with star graph $K_{1,m}\star K_{1,n}$ and complete graph with complete graph $K_m \star K_n$.

We introduce and investigate the notion of weak projection invariant stable modules for left exact preradicals in the category of right modules. We deal with structural properties of this new class of modules. In this direction, we have indecomposable decompositions of the special class of the former class of modules via some module theoretical properties. As a consequence, we determine conditions under which the finite exchange property implies the full exchange property for the new class of modules.