From Bounded Families of Localized Cosines to Bi-Orthogonal Riesz Bases via Shift-Invariance

The notion of bi-inner product functionals generated by two Bessel sequences {f _n } and {g _n } of functions from L ² was introduced in our earlier work as a vehicle to identify dual frames and bi-orthogonal Riesz bases of L ². The objective was to find conditions under which P is a constant mu<iple of the inner product <f,g> of L ². A necessary and suffici condition derived in is that P is both spatial shift-invariant and phase shift-invariant. A<hough these two shift-invariance properties are, in general, unrelated, it could happen that one is a consequence of the other for certain clases of Bessel sequences {f _n } and {g _n }. In this paper, we show that, indeed, for localized cosines with two-overlapping windoes (i.e., only adjacent window functions are allowed to overlap), spatial shift-invariance of P is already sufficient to guarantee that P is a constant mu<iple of the inner product, while phase shift-invariance is not. Hence, phase shift-invariance of P for two-overlapping localized cosine Bessel sequences is a consequence of spatial shift-invariance, but the converse is not valid. As an application, we also show that two families of localized cosines with uniformly bounded and two-overlapping windows are bi-orthogonal Riesz bases of L ², if and only if P is spatial shift-invariant. In addition, we apply this resu< to generalize a resu< on characterization of dual localized cosine bases in our earlier work in to the mu<ivariate setting. A method for computing the dual windows is also given in this paper.     

Synthesis of a deep cavity calix[4]arene by fourfold Sonogashira cross-coupling reaction and selective fluorescent recognition toward p-nitrophenol

A new tetranaphthyl-calix[4]arene (C4N4) was synthesized by a fourfold Sonogashira cross-coupling reaction, and exhibited high affinity and selectivity for p-nitrophenol (3c) by fluorescence spectroscopy. The NMR, AFM, IR, MALDI-TOF mass spectroscopy, and computational calculations revealed the formation of a host–guest complex driven by π–π stacking interactions.     

Directional motion of water drop on ratchet-like superhydrophobic surfaces

In this article, directional movements of drops on the ratchet-like superhydrophobic surfaces were observed. High-speed CCD images showed the caterpillar-like crawl of a drop on the inclined superhydrophobic surfaces as it rolled along the ridge of ratchet. In contrast, along the opposite direction, the movement of the drop only depended on the end of triple phase contact line while the front of contact line was pinned. The sliding angle (SA) measurements indicated that the ratchet-like superhydrophobic surfaces had directional drop retention traits. Moreover, the reduction of the rise angle ω₁, the height d of the ratchet's ridge and the volume V of the drop can greatly enhance the directional difference of drop retention on the ratchet-like superhydrophobic surfaces. Therefore, it was concluded that the superhydrophobicity and the periodic ratchet-like microstructures were the keys to the directional drop sliding at one-dimensional level. We believe that these findings would be helpful to better understand the ratchet-like effect on the superhydrophobic surfaces and guide some novel engineering applications.     

Weighted norm inequalities for the maximal singular integral operators on spaces of homogeneous type

Weighted norm inequalities with general weights are established for the maximal singular integral operators on spaces of homogeneous type, when the kernel satisfies a Hörmander regularity condition on one variable and a Hölder regularity condition on the other variable.     

关于Zorn的一个问题

最近,G.Schober教授告诉作者M.Zorn教授提出的下述问题,设(t)是[0,1]上的复值函数,是[0,1]的分划,分划直径记作。设p>0,假如(t)满足 其中,则说(t)属于类Z_p.M.Zorn证明,设是z平面上连续曲线,那么为使Riemann积分integral from n=1 (f(z)dz)对一切在C上     

The Simulation and Experiment of In 718 in Warm Laser Shock Processing Without Coating

Journal of Russian Laser Research - A modified constitutive model, combining laser shock processing and dynamic strain aging (DSA), is used to obtain the compressive residual stress field of In 718...     

核满足弱正则条件的多线性算子交换子的加权估计

本文主要研究核满足弱正则条件的算子与BMO函数生成的多线性交换子.建立了多线性交换子在加权Lebesgue空间的一些性质.