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}).

Error reconciliation is an important technique for learning with error (LWE) and ring-LWE (RLWE)-based constructions. In this paper, we present a comparison analysis on two error reconciliation-based RLWE key exchange protocols: Ding et al. in 2012 (DING12) and Bos et al. in 2015 (BCNS15). We take them as examples to explain the core idea of error reconciliation, building key exchange over RLWE problem, implementation and real-world performance, and compare them comprehensively. We also analyse a LWE key exchange 'Frodo' that uses an improved error reconciliation mechanism in BCNS15. To the best of our knowledge, our work is the first to present at least 128-bit classic (80-bit quantum) and 256-bit classic (> 200-bit quantum) secure parameter choices for DING12 with efficient portable C/C++ implementations. Benchmark shows that our efficient implementation is 11× faster than BCNS15 and one key exchange execution only costs 0.07 ms on a four-year-old middle range CPU. Error reconciliation is 1.57× faster than BCNS15.

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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