In this paper, we investigate symmetry properties of lightlike hypersurfaces in indefinite Sasakian manifolds, tangent to the structure vector field. Theorems on locally symmetric, semi-symmetric and Ricci semi-symmetric lightlike hypersurfaces are obtained. We show that, under some conditions, these hypersurfaces are totally geodesic. The non-existence conditions of specific lightlike hypersurfaces are given. We prove, under a certain condition, that in lightlike hypersurfaces of an indefinite Sasakian space form, tangent to the structure vector field, the local symmetry and semi-symmetry notions are equivalent. This equivalence is extended to the Ricci semi-symmetry notion when the lightlike hypersurfaces are considered to be $\eta$-totally umbilical.

This paper addresses necessary and sufficient factorizability conditions for classes of second order linear ordinary differential equations (ODEs) characterized by the degrees of their corresponding polynomial functions coefficients. A pure algebraic method is used to solve a system of linear algebraic equations whose solutions satisfy a compatibility criterion and generate two first order differential operators factorizing the considered second order differential operator. Concrete examples are probed, including special cases of Böcher ODEs like Heun, extensions of Wangerin and Heine’s differential equations.

For a graph $G$, a subset $S$ of $V(G)$ is called a shredder if $G-S$ consists of three or more components. We show that if $G$ is a 9-connected graph of order at least 67, then the number of shredders of cardinality 9 of $G$ is less than or equal to $(2|V(G)|-9)/3$.

The average number of ciphertexts per one session ${\overline{c}}_T$ is an indicator of aptitude of a key-management tree $T$ for a given frequency distribution of communication. In this paper, we estimate the weighted mean of ${\overline{c}}_T$ over all $T$’s with respect to appropriate weights as a criterion of aptitude of a key-management tree. We also give the concrete value of that mean in the case of equal weights.

It is proved that $\{{\mathrm{\Delta }}^2+\kappa |x{|}^{-4};\kappa \in \sum _{}^{\mathrm{c}}\}$ in $L^2({\mathbf{ℝ}}^N)$ forms a holo-morphic family of type (A), where $\sum$ is a closed and convex subset of $ℂ$. In particular, the $m$-accretivity of ${\mathrm{\Delta }}^2+\kappa |x{|}^{-4}$ in $L^2({\mathbf{ℝ}}^N)$ is established as an application of the perturbation theorem for linear $m$-accretive operators. The key lies in two inequalities derived by positive semi-definiteness of Gram matrix.