The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecrea...

Let m,n ≥ 1 be integers. Define Tm,n to be the transportation polytope consisting of the m × n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case Bn = Tn,n is the much-studied Birkhoffvon Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determ...

let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.

We prove that every integer ≥ exp(524) is a sum of seven non negative cubes. 1 History and statements In his 1770’s ”Meditationes Algebraicae”, E.Waring asserted that every positive integer is a sum of nine non-negative cubes. A proof was missing, as was fairly common at the time, the very notion of proof being not so clear. Notice that henceforth, we shall use cubes to denote cubes of non-nega...