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...

In the study of convexity spaces, the most common convex invariants are based on notions of independence with respect to taking convex hulls. In [D.B. Parker, R.F. Westhoff and M.J. Wolf, Discuss. Math. Graph Theory 29 (2009), 51–69], H-independence, R-independence and convex independence were studied to prove results about the Helly number, Radon number and rank of a clone-free multipartite to...

We prove that if G = (VG, EG) is a finite, simple, and undirected graph with κ components and independence number α(G), then there exist a positive integer k ∈ N and a function f : VG → N0 with non-negative integer values such that f(u) ≤ dG(u) for u ∈ VG, α(G) ≥ k ≥ ∑ u∈VG 1 dG(u)+1−f(u) , and ∑ u∈VG f(u) ≥ 2(k − κ). This result is a best-possible improvement of a result due to Harant and Schi...

Let G be a graph. A function f : V (G) → {−1, 1} is a signed kindependence function if the sum of its function values over any closed neighborhood is at most k − 1, where k ≥ 2. The signed k-independence number of G is the maximum weight of a signed k-independence function of G. Similarly, the signed total k-independence number of G is the maximum weight of a signed total k-independence functio...

‎‎Let $G=(V‎, ‎E)$ be a simple graph with vertex set $V$ and edge set $E$‎. ‎A {em mixed Roman dominating function} (MRDF) of $G$ is a function $f:Vcup Erightarrow {0,1,2}$ satisfying the condition that every element $xin Vcup E$ for which $f(x)=0$ is adjacent‎‎or incident to at least one element $yin Vcup E$ for which $f(y)=2$‎. ‎The weight of an‎‎MRDF $f$ is $sum _{xin Vcup E} f(x)$‎. ‎The mi...