The commuting probability of a finite group G $G$ is the that two randomly chosen elements commute. Let S ⊆ ( 0 , 1 ] $S\subseteq (0,1]$ denote set all possible probabilities groups. We prove ∪ { } $S\cup \lbrace 0\rbrace$ closed, which was conjectured by Joseph in 1977.

Abstract In this work the $\ell_q$ -norms of points chosen uniformly at random in a centered regular simplex high dimensions are studied. Berry–Esseen bounds regime $1\leq q < \infty$ derived and complemented by non-central limit theorem together with moderate large deviations case where $q=\infty$ . An application to intersection volume an $\ell_p^n$ -ball is also carried out.

Many numerical simulations in quantum (bilinear) control use the monotonically convergent algorithms of Krotov (introduced by Tannor in [12]), Zhu & Rabitz ([11]) or the general form of Maday & Turinici ([13]). This paper presents an analysis of the limit set of controls provided by these algorithms and a proof of convergence in a particular case.

In [1] Agnew presented the following theorem: if x is a bounded sequence and A is a regular summability matrix, then there exists a subsequence y of x such that each limit point of x is a limit point of Ay. Fridy [2] extended this result by replacing subsequence with rearrangement. Keagy [3] presented two theorems that strengthened the results of both Agnew and Fridy. This was accomplished by w...