a set $wsubseteq v(g)$ is called a resolving set for $g$, if for each two distinct vertices $u,vin v(g)$ there exists $win w$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. the minimum cardinality of a resolving set for $g$ is called the metric dimension of $g$, and denoted by $dim(g)$. in this paper, it is proved that in a connected graph $...

| In order to improve the eeciency of the Secure Electronic Transaction (SET) protocol, Hanaoka et al. proposed a light weight version of the SET protocol, called \LITESET"at ACISP '98. They claimed that LITESET has about 50% reduction in computational time and about 80% reduction in communication overhead by using a signcryp-tion scheme. We propose another way to construct an eecient SET proto...

In this paper we present some properties of set-norm exhaustive set multifunctions and also of atoms and pseudo-atoms of set multifunctions taking values in the family of non-empty subsets of a commutative semigroup with unity.

Prove that any strictly monotone sequence (Uα)α<γ of open subsets of X has countable length, i.e. γ is countable. Hint: Use the same idea as in the proof of (a). (c) Show that every monotone sequence (Uα)α<ω1 open subsets of X eventually stabilizes, i.e. there is γ < ω1 such that for all α < ω1 with α ≥ γ, we have Uα = Uγ. Hint: Use the regularity of ω1. (d) Conclude that parts (a), (b) and (c)...

Let A be a subset of a ring with cardinality |A| = N . The sum set and the product set are 2A = A+A = {a1 +a2 | ai ∈ A} and A = AA = {a1a2 | ai ∈ A}. The study of sum-product sets is to estimate the sizes of 2A and A asymptotically when N is very large. Since Erdős and Szemerédi conjectured that if A ⊂ Z, at least one of |2A| and |A2| should be bigger than cN for any constant c, a lot of work h...

In this paper a new class of sets called  regular generalized  $delta$-closed set (briefly rg$delta$-closed set)is introduced and its properties are studied. Several examples are provided to illustrate the behaviour of these new class of sets.