Littlewood-Paley operators with variable kernels

In this paper we study a certain directional Hilbert transform and the bound-edness on some mixed norm spaces. As one of applications, we prove the Lp-boundedness of the Littlewood-Paley operators with variable kernels. Our results are extensions of some known theorems.     

Symbolic-numeric Gaussian cubature rules

It is well known that Gaussian cubature rules are related to multivariate orthogonal polynomials. The cubature rules found in the literature use common zeroes of some linearly independent set of products of basically univariate polynomials. We show how a new family of multivariate orthogonal polynomials, so-called spherical orthogonal polynomials, leads to symbolic-numeric Gaussian cubature rules in a very natural way. They can be used for the integration of multivariate functions that in addition may depend on a vector of parameters and they are exact for multivariate parameterized polynomials. Purely numeric Gaussian cubature rules for the exact integration of multivariate polynomials can also be obtained.We illustrate their use for the symbolic-numeric solution of the partial differential equations satisfied by the Appell function F₂, which arises frequently in various physical and chemical applications. The advantage of a symbolic-numeric formula over a purely numeric one is that one obtains a continuous extension, in terms of the parameters, of the numeric solution. The number of symbolic-numeric nodes in our Gaussian cubature rules is minimal, namely m for the exact integration of a polynomial of homogeneous degree 2m−1.In Section 1 we describe how the symbolic-numeric rules are constructed, in any dimension and for any order. In Sections 2, 3 and 4 we explicit them on different domains and for different weight functions. An illustration of the new formulas is given in Section 5 and we show in Section 6 how numeric cubature rules can be derived for the exact integration of multivariate polynomials. From Section 7 it is clear that there is a connection between our symbolic-numeric cubature rules and numeric cubature formulae with a minimal (or small) number of nodes.     

The Changes of Microstructure and Physical Properties of Isotactic Polypropylene/β Nucleation Agent/Polyolefin Elastomer Induced by Annealing Following Processing

Blends of isotactic polypropylene (iPP)/β nucleation agent (β-NA)/polyolefin elastomer (POE) were prepared by injection molding. The microstructure and mechanical properties of these blends before and after being annealed at various temperatures and times were studied. It was found that annealing simultaneously increased the tensile strength and impact strength. As known, the degree of orientation decreased from the skin layer to the core layer. The orientation of all layers decreased with the increase of annealing temperature and time. The results showed that annealing gave rise to chain rearrangement in both the crystalline and POE phases which, we suggest, played a crucial role in determining the mechanical properties of the blends.     

齐型空间上的Morrey空间的算子有界性及其应用

作者建立了一大类次线性算子和交换子在齐型空间上的Morrey空间中的有界性．应用这些结果,作者研究了具有不连续系数的超抛物方程解的局部正则性．