compound n2, n3, n7, n8, 1, 4, 5, 6, 9, 10-decaphenyltetradecahydrodipyrazino [2, 3-b: 2, 3-e] pyrazine-2, 3, 7, 8-tetraamine (4) was prepared by condensation of 1, 1′, 2, 2′-tetrakis (phenylamino) ethane (1d) and glyoxal in etoh or i-proh. also, reaction of 1d with glyoxal in equimolare in etoh resulted 5,6-bis (ethoxy)-n2, n3, 1, 4-tetraphenylhexahydro-2, 3-pyrazinediamine (5). the natures of...

We examine the behavior of the first-order rainbow for a coated sphere by using both ray theory and Aden-Kerker wave theory as the radius of the core a(12) and the thickness of the coating δ are varied. As the ratio δ/a(12) increases from 10(-4) to 0.33, we find three classes of rainbow phenomena that cannot occur for a homogeneous-sphere rainbow. For δ/a(12) ≲ 10(-3), the rainbow intensity is ...

Let $k>1$, and let $\mathcal{F}$ be a family of $2n+k-3$ non-empty sets edges in bipartite graph. If the union every $k$ members contains matching size $n$, then there exists an $\mathcal{F}$-rainbow $n$. Replacing by $2n+k-2$, result is true also for $k=1$, it can proved (for all $k$) both topologically relatively simple combinatorial argument. The main effort gaining last $1$, which makes ...

A path in an edge-colored graph is called a rainbow path if the edges on it have distinct colors. For k ≥ 1, the rainbow-k-connectivity of a graph G, denoted rck(G), is the minimum number of colors required to color the edges of G in such a way that every two distinct vertices are connected by at least k internally vertex-disjoint rainbow paths. In this paper, we study rainbow-k-connectivity in...

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by γc(G) + 2, where γc(G) is th...