Let Ω⊂$R$^{$n$}be an arbitrary open set. In this paper it is shown that if a Sobolev function$f$∈$W$^{1,$p$}(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then$f$is weakly zero on ϖΩ in the sense that$f$∈$W$_{0}^{1,$p$}(Ω).
We give a proof of the parabolic/singular Koszul duality for the category O of affine Kac–Moodyalgebras. The main new tool is a relation between moment graphs and finite codimensional affine Schubert varieties. We apply this duality to q-Schur algebras and to cyclotomic rational double affine Heckealgebras.This yields a proof of a conjecture of Chuang–Miyachi relating the level-rank duality with the Ringel–Koszul duality of cyclotomic rational double affine Hecke algebras.
Jintai DingDepartment of Mathematics, University of Cincinnati, OH, USAAlexei MiasnikovDepartment of Mathematics, Stevens Institute of Technology, Hoboken, NJ, USAAlexander UshakovDepartment of Mathematics, Stevens Institute of Technology, Hoboken, NJ, USA
In this paper we analyze the Kahrobaei-Lam-Shpilrain (KLS) key exchange protocols that use extensions by endomorpisms of matrices over a Galois field proposed in \cite{Kahrobaei-Lam-Shpilrain:2014}. We show that both protocols are vulnerable to a simple linear algebra attack.
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