Hermite‐Hadamard‐type inequalities for interval‐valued coordinated convex functions involving generalized fractional integrals

In this paper, we define interval‐valued left‐sided and right‐sided generalized fractional double integrals. We establish inequalities of Hermite‐Hadamard like for coordinated interval‐valued convex functions by applying our newly defined integrals.     

Kibble–Slepian formula and generating functions for 2D polynomials

We prove a generalization of the Kibble–Slepian formula (for Hermite polynomials) and its unitary analogue involving the 2D Hermite polynomials recently proved in [16]. We derive integral representations for the 2D Hermite polynomials which are of independent interest. Several new generating functions for 2D q-Hermite polynomials will also be given.     

Sobolev Stability of Plane Wave Solutions to the Nonlinear Schrödinger Equation

This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case.     

Classical Backfitting for Smooth-Backfitting Additive Models

Smooth backfitting has been shown to have better theoretical properties than classical backfitting for fitting additive models based on local linear regression. In this article, we show that the smooth backfitting procedure in the local linear case can be alternatively performed as a classical backfitting procedure with a different type of smoother matrices. These smoother matrices are symmetric and shrinking and some established results in the literature are readily applicable. The connections allow the smooth backfitting algorithm to be implemented in a much simplified way, give new insights on the differences between the two approaches in the literature, and provide an extension to local polynomial regression. The connections also give rise to a new estimator at data points. Asymptotic properties of general local polynomial smooth backfitting estimates are investigated, allowing for different orders of local polynomials and different bandwidths. Cases of oracle efficiency are discussed. Computer-generated simulations are conducted to demonstrate finite sample behaviors of the methodology and a real data example is given for illustration. Supplementary materials for this article are available online.     

On the Differential Polynomial of a Graph

We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial B(G; x) :=∑?(G)k=-nB_k(G) x~(n+k), where B_k(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G;x) and its coefficients.In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G.     

一类精细化Eulerian多项式的若干性质

最近,孙华定义了一类新的精细化Eulerian多项式,即$$A_n(p,q)=\sum_{\pi\in \mathfrak{S}_n}p^{{\rm odes}(\pi)}q^{{\rm edes}(\pi)},\ \ n\ge 1,$$ 其中$S_n$表示$\{1,2,\ldots,n\}$上全体$n$阶排列的集合, odes$(\pi)$与edes$(\pi)$分别表示$S_n$中排列$\pi$的奇数位与偶数位上降位数的个数.本文利用经典的Eulerian多项式$A_n(q)$ 与Catalan 序列的生成函数$C(q)$,得到精细化Eulerian 多项式$A_n(p,q)$的指数型生成函数及$A_n(p,q)$的显示表达式.在一些特殊情形,本文建立了$A_n(p,q)$与$A_n(0,q)$或$A_n(p,0)$之间的联系,并利用Eulerian数表示多项式$A_n(0,q)$的系数.特别地,这些联系揭示了Euler数$E_n$与Eulerian数$A_{n,k}$之间的一种新的关系.     

Regularity of weak solutions to rate‐independent systems in one‐dimension

We show that under some appropriate assumptions, every weak solution (e.g. energetic solution) to a given rate‐independent system is of class SBV, or has finite jumps, or is even piecewise C¹. Our assumption is essentially imposed on the energy functional, but not convexity is required.     

CALCULATION FORMULAS FOR PRINCIPAL STRESSES

在前人工作的基础上,运用一元三次方程的理论直接求解应力状态的特征方程,得到了实用的主应力计算公式.     

The a posteriori error estimates of Chebyshev–Petrov–Galerkin methods for second-order equations

In this paper, the a posteriori error estimates of Chebyshev–Petrov–Galerkin approximations are investigated. For simplicity, we choose the Poisson equation with Dirichlet boundary conditions to discuss the a posteriori error estimators, and deduce their efficient and reliable properties. Some numerical experiments are performed to verify the theoretical analysis for the a posteriori error estimators.