We consider the damped and driven two-dimensional Euler equations in the plane with weak solutions having finite energy and enstrophy. We show that these (possibly non-unique) solutions satisfy the energy and enstrophy equality. It is shown that this system has a strong global and a strong trajectory attractor in the Sobolev space H. A similar result on the strong attraction holds in the spaces...

In the definition of the graph parameters μ(G) and ν(G), introduced by Colin de Verdière, and in the definition of the graph parameter ξ(G), introduced by Barioli, Fallat, and Hogben, a transversality condition is used, called the Strong Arnol’d Hypothesis. In this paper, we define the Strong Arnol’d Hypothesis for linear subspaces L ⊆ R with respect to a graph G = (V,E), with V = {1, 2, . . . ...

Let I be any index set. We consider the Banach algebra Ce + l(I) with the Hadamard product, and prove that its Bass and topological stable ranks are both equal to 1. We also characterize divisors, maximal ideals, closed ideals and closed principal ideals. For I = N we also characterize all prime z-ideals in this Banach algebra.

let $r$ be a commutative ring. in this paper, by using algebraic properties of $r$, we study the hase digraph of prime ideals of $r$.

In this paper, we define the notion of minimal prime ideals hoops and investigate some properties them. Then by using annihilators, study relation between annihilators. Also, introduce zero divisors elements prove that set all is a union hoop. Finally, notions maximal hoop, two new as p-ideal m-ideal. them every semi-simple hoop an m-ideal it.