Resonance free domain for a system of Schrödinger operators with energy-level crossings

We consider a [Formula: see text] system of 1D semiclassical differential operators with two Schrödinger operators in the diagonal part and small interactions of order [Formula: see text] in the off-diagonal part, where [Formula: see text] is a semiclassical parameter and [Formula: see text] is a constant larger than [Formula: see text]. We study the absence of resonance near a non-trapping energy for both Schrödinger operators in the presence of crossings of their potentials. The width of resonances is estimated from below by [Formula: see text] and the coefficient [Formula: see text] is given in terms of the directed cycles of the generalized bicharacteristics induced by two Hamiltonians.

The Italian Domination Numbers of Some Products of Directed Cycles

An Italian dominating function on a digraph D with vertex set V(D) is defined as a function f:V(D)→{0,1,2} such that every vertex v∈V(D) with f(v)=0 has at least two in-neighbors assigned 1 under f or one in-neighbor w with f(w)=2. In this article, we determine the exact values of the Italian domination numbers of some products of directed cycles.

Feedback Arc Number and Feedback Vertex Number of Cartesian Product of Directed Cycles

For a digraph D, the feedback vertex number τD, (resp. the feedback arc number τ′D) is the minimum number of vertices, (resp. arcs) whose removal leaves the resultant digraph free of directed cycles. In this note, we determine τD and τ′D for the Cartesian product of directed cycles D=Cn1→□Cn2→□…Cnk→. Actually, it is shown that τ′D=n1n2…nk∑i=1k1/ni, and if nk≥…≥n1≥3 then τD=n2…nk.