We show that ${{L}^{\infty }}\left( \mu \right)$, in its capacity as multiplication operators on ${{L}^{p}}\left( \mu \right)$, is minimal as a $p$-operator space for a decomposable measure $\mu $. We conclude that ${{L}^{1}}\left( \mu \right)$ has a certain maximal type $p$-operator space structure that facilitates computations with ${{L}^{1}}\left( \mu \right)$ and the projective tensor product.

We prove that if $f\,:\,{{\mathbb{R}}^{N}}\,\to \,\overline{\mathbb{R}}$ is quasiconvex and $U\,\subset \,{{\mathbb{R}}^{N}}$ is open in the density topology, then $\underset{U}{\mathop{\sup }}\,f=\text{ess}\,\underset{U}{\mathop{\sup }}\,f$, while ${{\inf }_{U}}\,f\,=\,\text{ess}\,{{\inf }_{U}}\,f$ if and only if the equality holds when $U\,\subset \,{{\mathbb{R}}^{N}}$. The first (second) property is typical of $\text{lsc}\,\text{(usc)}$ functions, and, even when $U$ is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions.

This property ensures that the pointwise extrema of $f$ on any nonempty density open subset can be arbitrarily closely approximated by values of $f$ achieved on “large” subsets, which may be of relevance in a variety of situations. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.

In this paper we obtain a characterization for all bipartite zero-divisor graphs of commutative rings $R$ with 1 such that $R$ is finite or $\left| \text{Nil}\left( R \right) \right|\,\ne \,2$.

In this paper, we obtain a lower bound for the length of closed geodesics on an arbitrary closed Finsler manifold.

We correct two clerical errors made in the paper ”A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold”.

We provide an explicit formula for the generalized cyclic shuffle map for cylindrical modules. Using this formula we give a combinatorial proof of the generalized cyclic Eilenberg–Zilber theorem.