We study the convergence properties of the series Ψs(α) := ∑ n≥1 ||n2α|| ns+1||nα|| with respect to the values of the real numbers α and s, where ||x|| is the distance of x to Z. For example when s ∈ (0, 1], the convergence of Ψs(α) strongly depends on the diophantine nature of α, mainly its irrationality exponent. We also conjecture that Ψs(α) is minimal at √ 5 for s ∈ (0, 1] and we present ev...

let $g$ be a finite group and let $text{cd}(g)$ be the set of all‎ ‎complex irreducible character degrees of $g$‎. ‎b‎. ‎huppert conjectured‎ ‎that if $h$ is a finite nonabelian simple group such that‎ ‎$text{cd}(g) =text{cd}(h)$‎, ‎then $gcong h times a$‎, ‎where $a$ is‎ ‎an abelian group‎. ‎in this paper‎, ‎we verify the conjecture for‎ ${f_4(2)}.$‎

Erdős and Lovász conjectured in 1968 that for every graph G with χ(G) > ω(G) and any two integers s, t ≥ 2 with s+ t = χ(G)+1, there is a partition (S, T ) of the vertex set V (G) such that χ(G[S]) ≥ s and χ(G[T ]) ≥ t. Except for a few cases, this conjecture is still unsolved. In this note we prove the conjecture for line graphs of multigraphs.

We build upon ideas developed in [9], as well as results of Greither on a strong form of Brumer’s Conjecture ([2]–[4]), and prove Rubin’s integral version of Stark’s Conjecture for imaginary abelian extensions of Q of odd prime power conductor. INTRODUCTION In the present paper, we prove Rubin’s integral version (Conjecture B, [10], §2.1) of Stark’s general conjecture (see Conjecture 5.1 in [12...

Graham Higman has defined coset diagrams for PSL(2,ℤ). These diagrams are composed of fragments, and the fragments are further composed of two or more circuits. Q. Mushtaq has proved in 1983 that existence of a certain fragment γ of a coset diagram in a coset diagram is a polynomial f in ℤ[z]. Higman has conjectured that, the polynomials related to the fragments are monic and for a fixed degree...

We pose a conjecture for the expected number of generators of the ideal of the union C of s general rational irreducible curves in P r. By using the computer we prove the conjecture for C of low degree d (e.g. if s = 1 for d 80 and if s 10 for d 40).