An exact upper bound for sums of element orders in non-cyclic finite groups

Denote the sum of element orders in a finite group G by $\psi (G)$ and let $C_n$ denote the cyclic group of order n. Suppose that G is a non-cyclic finite group of order n and q is the least prime divisor of n. We proved that $\psi (G)\le \frac{7}{11}\psi (C_n)$ and $\psi (G)<\frac{1}{q?1}\psi (C_n)$. The first result is best possible, since for each $n=4k$, k odd, there exists a group G of order n satisfying $\psi (G)=\frac{7}{11}\psi (C_n)$ and the second result implies that if G is of odd order, then $\psi (G)<\frac{1}{2}\psi (C_n)$. Our results improve the inequality $\psi (G)<\psi (C_n)$ obtained by H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs in 2009, as well as other results obtained by S.M. Jafarian Amiri and M. Amiri in 2014 and by R. Shen, G. Chen and C. Wu in 2015. Furthermore, we obtained some $\psi (G)$-based sufficient conditions for the solvability of G.     

Problème de Helmholtz intérieur : formulation modale couplée à une représentation intégrale

Let Ω be an opened domain of ${\mathbb{R}}^3$ with a boundary Γ. The problem numbered (1) in the text has a unique solution in ${\mathrm{H}}^1(\Omega )$, if $k\in \mathbb{R}$, $R_e(\frac{1}{\zeta })>0$ on a part of Γ the area S of which is different from zero and $g(k,y,?)\in {\mathrm{H}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}(\Gamma )$. ? is the damping of an elastic structure and ζ is the normalised acoustic impedance of the internal wall of the cavity. ? and $\frac{1}{\zeta }$ are small parameters. Thanks to a proper modal expansion and a mean over a narrow band of wave number k, an integral relation between the trace of u on $\Gamma ,\zeta ,?$ and g is built to the first order $?(\frac{1}{\zeta },?)$ in $\frac{1}{\zeta }$ and ?, using the residues theorem. It is not an equivalent equation to the problem, but just a step towards its resolution, which will be published in future papers. To cite this article: D. Brenot, C. R. Acad. Sci. Paris, Ser. I 340 (2005).