Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambd...

Let R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I ⊕R/J is a cyclic R-module, then I + J = R. The rings R such that R/I ⊕R/J is a cyclic R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings: fields, rings isomorphic to K ×L for the fields K and L, and special principal ideal rings (R,M) such thatM...

In this paper we introduce remainder cordial labeling of graphs. Let $G$ be a $(p,q)$ graph. Let $f:V(G)rightarrow {1,2,...,p}$ be a $1-1$ map. For each edge $uv$ assign the label $r$ where $r$ is the remainder when $f(u)$ is divided by $f(v)$ or $f(v)$ is divided by $f(u)$ according as $f(u)geq f(v)$ or $f(v)geq f(u)$. The function$f$ is called a remainder cordial labeling of $G$ if $left| e_{...

In this paper we primarily introduce the local version of star-Menger property, namely locally property (a space with is called star-Menger) and present some important topological observations. Certain interactions between new notion are also observed. Some observations on effectively Pixley-Roy hyperspaces (introduced here) obtained. Preservation like properties under several operations interp...

In this paper we generalize the remainder cordial labeling, called $k$-remainder cordial labeling and investigate the $4$-remainder cordial labeling behavior of certain graphs.

In this paper, we defined the congruence relation and proved its fundamental properties on the base of some useful theorems. Then we proved the existence of solution and the number of incongruent solution to a linear congruence and the linear congruent equation class, in particular, we proved the Chinese Remainder Theorem. Finally, we defined the complete residue system and proved its fundament...

A secret sharing scheme derives from a given secret certain shares (or shadows) which are distributed to users. The secret can be recovered only by certain predetermined groups. In the first secret sharing schemes only the number of the participants in the reconstruction phase was important for recovering the secret. Such schemes have been referred to as threshold secret sharing schemes. In the...