This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo $p$ of Shimura varieties. Motivated by \cite{Vollaard-Wedhorn} and \cite{Rapoport-Terstiege-Wilson}, we classify the cases where the basic locus is (in a natural way) the union of classical Deligne-Lusztig sets associated to Coxeter elements. We show that if this is satisfied, then the Newton strata and Ekedahl-Oort strata have many nice properties.

InthispaperwedevelopaconservativelocaldiscontinuousGalerkin(LDG) method for the Schro ̈ dinger-Korteweg-de Vries (Sch-KdV) system, which arises in var- ious physical contexts as a model for the interaction of long and short nonlinear waves. Conservative quantities in the discrete version of the number of plasmons, energy of the oscillations and the number of particles are proved for the LDG scheme of the Sch-KdV system. Semi-implicit time discretization is adopted to relax the time step constraint from the high order spatial derivatives. Numerical results for accuracy tests of stationary traveling soliton, and the collision of solitons are shown. Numerical ex- periments illustrate the accuracy and capability of the method.

We consider$Thurston maps$, i.e., branched covering maps$f:S$^{2}→$S$^{2}that are$post-critically finite$. In addition, we assume that$f$is$expanding$in a suitable sense. It is shown that each sufficiently high iterate$F$=$f$^{$n$}of$f$is$semi-conjugate$to$z$^{$d$}:$S$^{1}→$S$^{1}, where$d$= deg F. More precisely, for such an$F$we construct a$Peano curve γ$:$S$^{1}→$S$^{2}(onto), such that$F$∘$γ$($z$) =$γ$($z$^{$d$}) (for all$z$∈$S$^{1}).

In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero

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