The matrix optics formalism is applied to show that a simple Gaussian equation enough relate object and image distances in an optical system consisting of arbitrary number thick lenses cascade immersed air. First, the case single lens was studied. Applying sequentially matrixes corresponding refraction displacement ray, imposing certain conditions on behavior bunch rays being refracted by lens,...

We study (approximate) null-controllability of parabolic equations in $L_p(\mathbb{R}^d)$ and provide explicit bounds on the control cost. In particular we consider systems form $\dot{x}(t) = -A_p x(t) + \mathbf{1}_E u(t)$, $x(0) x_0\in L_p (\mathbb{R}^d)$, with interior a so-called thick set $E \subset \mathbb{R}^d$, where $p\in [1,\infty)$, $A$ is an elliptic operator order $m \in \mathbb{N}$...

In 1985, Chor et al 2] conjectured that the only 1-resilient symmetric functions are the exclusive-or of all n variables and its negation. In this note the existence of symmetric resilient functions is shown to be equivalent to the existence of a solution to a simultaneous subset sum problem. Then, using arithmetic properties of certain binomial coeecients, an innnite class of counterexamples t...

Let a topological space. An intersection graph on a topological space , which denoted by ‎ , is an undirected graph which whose vertices are open subsets of and two vertices are adjacent if the intersection of them are nonempty. In this paper, the relation between topological properties of  and graph properties of ‎  are investigated. Also some classifications and representations for the graph ...