A fundamental result, due to Sabidussi and Vizing, states that every connected graph has a unique prime factorization relative to the Cartesian product; but disconnected graphs are not uniquely prime factorable. This paper describes a system of modular arithmetic on graphs under which both connected and disconnected graphs have unique prime Cartesian factorizations.

In this paper, using Bottema's inequality for two triangles and other results, the generalization of an inequality involving the medians and angle-bisectors of the triangle is proved. This settles affirmatively a problem posed by J-Liu.

Hyperplanes in $\mathbb{R}^{n}$ are extended to affine subspaces of $\ell^{2}$, independently of the Axiom of Choice. These affine subspaces form a set of uncountably many mutually disjoint, dense and convex subsets of $\ell^{2}$. A homeomorphism maps $\ell^{2}$ to the sum of these sets. Further, any sphere $S$ in $\ell^{2}$ contains an uncountable collection of mutually disjoint and path connected subsets, each of which is dense in $S$.

In the present paper, we offer a new form of firm continuity, called firm contra-continuity, by which we characterize strongly S-closed spaces. Moreover, we investigate the basic properties of firmly contra-continuous functions. We also introduce and investigate the notion of locally contra-closed graphs.

A generalized martingale betting strategy is analyzed for which bets are increased by a factor of $m \geq 1$ after each loss, but return to the initial bet amount after each win. The average amount bet and the average final fortune are derived for sequences of $n$ bets, for the number of bets $T$ that results in the first win, and for $\min (T,\;n)$.

In this article, we introduce $s$-cluster sets of functions and multifunctions and investigate their various properties. Such investigations provide a new technique for studying $s$-closed spaces [7] and weakly Hausdorff spaces [9].

In this paper we obtain a characterization of the $i$th Wielandt subgroup of a finite group. We then use this to find relations between the Wielandt length of a finite group and that of some of its factor groups. This in turn is used to obtain an upper bound for the Wielandt lengths of finite $p$-groups.