Conceptually graph vulnerability relates to the study of graphintactness when some of its elements are removed. The motivation forstudying vulnerability measures is derived from design and analysisof networks under hostile environment. Graph tenacity has been anactive area of research since the the concept was introduced in1992. The tenacity T(G) of a graph G is defined asbegin{center} $T(G)=di...

Ever since the work of Richardson in the mid-60s1, it has been recognized that the Pairing Model (PM) is exactly solvable, even in the presence of non-degenerate single-particle levels. In recent years, there has been a resurgence of interest in the PM, with several applications reported that build on this exact solvability 2. This talk reviews two recent applications of the PM in nuclear physi...

Let (Xn, Yn)n 1 denote the positive integer solutions of Pell’s equation X 2 DY 2 = 1 or X2 DY 2 = 1. We introduce the Dirichlet series ⇣X(s) = P1 n=1 1/X s n and ⇣Y (s) = P1 n=1 1/Y s n for <(s) > 0 and prove that both functions do not satisfy any nontrivial algebraic di↵erential equation. For any positive integers s1 and s2 the two numbers ⇣X(2s1) and ⇣Y (2s2) are algebraically independent ov...

We obtain sharp asymptotic estimates on the number of n × $n \times n$ contingency tables with two linear margins C $Cn$ and B $BCn$ . The results imply a second-order phase transition such tables, critical value at c : = 1 + / $B_{c}:=1 \sqrt {1+1/C}$ As consequence, for > $B>B_{c}$ , we prove that classical independence heuristic leads to large undercounting.