let $gamma(s_n)$ be the minimum number of proper subgroups‎ ‎$h_i, i=1‎, ‎dots‎, ‎l $ of the symmetric group $s_n$ such that each element in $s_n$‎ ‎lies in some conjugate of one of the $h_i.$ in this paper we‎ ‎conjecture that $$gamma(s_n)=frac{n}{2}left(1-frac{1}{p_1}right)‎ ‎left(1-frac{1}{p_2}right)+2,$$ where $p_1,p_2$ are the two smallest primes‎ ‎in the factorization of $ninmathbb{n}$ an...

Let $w$ be a word in the free group on $r$ generators. The expected value of trace independent Haar elements $\mathrm{O}(n)$ gives function ${\cal T}r_{w}^{\mathrm{O}}(n)$ $n$. We show that has convergent Laurent expansion at $n=\infty$ involving maps surfaces and $L^{2}$-Euler characteristics mapping class groups associated to these maps. This can compared known, by now classical, results for ...

a 52-year-old post-menopausal female was admitted to the emergency room due to acute onset of abdominal pain in the right lower quadrant. there was no history of trauma of the abdomen. contrast-enhanced multi-detector computed tomography (mdct) of the abdomen and pelvis revealed a massive right retroperitoneal hematoma and contrast extravasation from a right ovarian artery aneurysm. digital sub...