A note on the signature representations of the symmetric groups

For a partition {\lambda} and a prime p, we prove a necessary and sufficient condition for there exists a composition {\delta} such that {\delta} can be obtained from {\lambda} after rearrangement and all the partial sums of {\delta} are not divisible by p.Comment: This is a substantially revised version of previous ones. In the third section, we calculate some explicit p-Kostka number

Numerical approximations of one-point large deviations rate functions of stochastic differential equations with small noise

In this paper, we study the numerical approximation of the one-point large deviations rate functions of nonlinear stochastic differential equations (SDEs) with small noise. We show that the stochastic $\theta$-method satisfies the one-point large deviations principle with a discrete rate function for sufficiently small step-size, and present a uniform error estimate between the discrete rate function and the continuous one on bounded sets in terms of step-size. It is proved that the convergence orders in the cases of multiplicative noises and additive noises are $1/2$ and $1$ respectively. Based on the above results, we obtain an effective approach to numerically approximating the large deviations rate functions of nonlinear SDEs with small time. To the best of our knowledge, this is the first result on the convergence rate of discrete rate functions for approximating the one-point large deviations rate functions associated with nonlinear SDEs with small noise

Convergence analysis of one-point large deviations rate functions of numerical discretizations for stochastic wave equations with small noise

In this work, we present the convergence analysis of one-point large deviations rate functions (LDRFs) of the spatial finite difference method (FDM) for stochastic wave equations with small noise, which is essentially about the asymptotical limit of minimization problems and not a trivial task for the nonlinear cases. In order to overcome the difficulty that objective functions for the original equation and the spatial FDM have different effective domains, we propose a new technical route for analyzing the pointwise convergence of the one-point LDRFs of the spatial FDM, based on the $\Gamma$-convergence of objective functions. Based on the new technical route, the intractable convergence analysis of one-point LDRFs boils down to the qualitative analysis of skeleton equations of the original equation and its numerical discretizations