We present Saharon Shelah's Stability Spectrum and Homogeneity Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the first order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the point of view is contemporary and some...

We give a decomposition theorem for Platonic graphs over finite fields and use this to determine the spectrum of these graphs. We also derive estimates for the isoperimetric numbers of the graphs. THE SPECTRUM OF PLATONIC GRAPHS OVER FINITE FIELDS MICHELLE DEDEO, DOMINIC LANPHIER, AND MARVIN MINEI Abstract. We give a decomposition theorem for Platonic graphs over finite fields and use this to d...

Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exist unit elements $u,v$ of $R$ such that $x+uyv$ is a unit of $R$. In this paper, basic properties of $Gamma(R)$ are studied. We investigate connectivity and the girth of $Gamma(R)$, where $...

in this paper, a new design of optical channel drop filter based on two- dimensional photonic crystal ring resonators with triangular lattice is proposed. the rods of this structure is silicon with the refractive index nsi=3.46 and the surrounding environment is air with the refractive index of nair=1.the widest photonic band gap obtains for the filling ratio of r/a = 0.2. the filter’s transmis...