Suppose $X$ is a simple graph. The $X-$join $Gamma$ of a set ofcomplete or empty graphs ${X_x }_{x in V(X)}$ is a simple graph with the following vertex and edge sets:begin{eqnarray*}V(Gamma) &=& {(x,y) | x in V(X) & y inV(X_x) },\ E(Gamma) &=& {(x,y)(x^prime,y^prime) | xx^prime in E(X) or else x = x^prime & yy^prime in E(X_x)}.end{eqnarray*}The $X-$join graph $Gamma$ is said to be re...

(2)1 For every set X and for all functions f , g such that X ⊆ dom f and f ⊆ g holds f X = g X . (3) For every non empty set A and for every set b such that A 6= {b} there exists an element a of A such that a 6= b. (4) For all sets X , Y holds every non empty subset of X→̇Y is a non empty functional set. (5) Let B be a non empty functional set and f be a function. Suppose f = ⋃ B. Then dom f = ⋃...

We introduce X is without zero as a synonym of X has non empty elements. We introduce X has zero as an antonym of X has non empty elements. Let us mention that R has zero and N has zero. Let us observe that there exists a set which is non empty and without zero and there exists a set which is non empty and has zero. Let us observe that there exists a subset of R which is non empty and without z...

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328` points in the plane contains an empty pentagon or ` collinear points. This is optimal up to a constant factor since the (` − 1) × (` − 1) grid contains no empty pentagon and no ` collinear points. ...

This thesis concentrates on the vulnerabilities of the RSA Cryptographic Algorithm when it is not securely implemented. While it has been proven that a brute force attack on the algorithm is not practical there remain aspects of the algorithm that require proper use to prevent back-door attacks. The attacks performed in this thesis attempt to exploit both mathematical and inherent timing vulner...