A signed digraph S is the digraph D by assigning signs 1 or −1 to each arc of D. The base of S is the minimum number k such that there is a pair walks which have the same initial and terminal point with length k, but different signs. In this paper we show that for n ≥ 5 the upper bound of the base of a primitive non-powerful signed tournament Sn, which is the signed digraph by assigning 1 or −1...

This note should be seen as an appendum to the work of E. Kowalski and the second author [KM1] and deals with the problem of bounding, unconditionally, the order of vanishing at the critical point in a family of L-functions. This problem is illustrated in the particular case of L-functions of weight-2 primitive modular forms of prime level. Recall the notation from [KM1]: For q a prime number, ...

In this paper, we rigorously derive the governing equations describing motion of a stable stratified fluid, from mathematical point view. particular, prove that scaled Boussinesq strongly converge to viscous primitive with density stratification as aspect ratio goes zero, and rate convergence is same order ratio. Moreover, in obtain result, also establish global well-posedness strong solutions ...

let $r$ be an infinite ring. here we prove that if $0_r$ belongs to ${x_1x_2cdots x_n ;|; x_1,x_2,dots,x_nin x}$ for every infinite subset $x$ of $r$, then $r$ satisfies the polynomial identity $x^n=0$. also we prove that if $0_r$ belongs to ${x_1x_2cdots x_n-x_{n+1} ;|; x_1,x_2,dots,x_n,x_{n+1}in x}$ for every infinite subset $x$ of $r$, then $x^n=x$ for all $xin r$.