For any 2-distance set X in the n-dimensional binary Hamming spaceHn, let ΓX be the graph with X as the vertex set and with two vertices adjacent if and only if the distance between them is the smaller of the two nonzero distances in X. The binary spherical representation number of a graph Γ, or bsr Γ , is the least n such that Γ is isomorphic to ΓX , where X is a 2-distance set lying on a sphe...

For a given graph G, the square of G, denoted by G2, is a graph with the vertex set V(G) such that two vertices are adjacent if and only if the distance of these vertices in G is at most two. A graph G is called squared if there exists some graph H such that G= H2. A function f:V(G) {0,1,2…, k} is called a coloring of G if for every pair of vertices x,yV(G) with d(x,y)=1 we have |f(x)-f(y)|2 an...

In [3] a geometrical construction was defined from which a generalization of sectrix of Ceva comes. Let e be a line given by the origin O and angle α between axis x+ and e, as the angle of polar coordinates of e (Figure 3). Let the point A0 coincide with O. Let the point A1 be given on e such that the distance between the points O and A1 is 1. Let the point A2 be on axis x such that the distanc...

The extent of where magnetic reconnection (MR), the dominant process responsible for energy and plasma transport into the magnetosphere, operates across Earth's dayside magnetopause has previously been only indirectly shown by observations. We report the first direct evidence of X-line structure resulting from the operation of MR at each of two widely separated locations along the tilted, subso...

This paper proposes a simple, empirically derived, unifying structure for the inner regions of quasars. This structure is constructed to explain the broad absorption line (BAL) regions, the narrow ‘associated’ ultraviolet and X-ray ‘ionized’ absorbers (NALs); and is also found to explain the broad emission line regions (BELR), and several scattering features, including a substantial fraction of...

Abstract Smaller and more complex three-dimensional periodic nanostructures are part of the next generation integrated electronic circuits. Additionally, decreasing dimensions increases effect line-edge roughness on performance nanostructures. Efficient methods for characterizing required process control. Here, extreme-ultraviolet (EUV) scatterometry is exploited analysis from In line with prev...

It is well know that for globally contractive autonomous systems, there exists a unique equilibrium and the distance to the equilibrium evaluated along any trajectory decreases exponentially with time. We show that, additionally, the magnitude of the velocity evaluated along any trajectory decreases exponentially, thus giving an alternative choice of Lyapunov function. 1 Main Result Consider th...