Strichartz Estimates for Non-Elliptic Schrödinger Equations on Compact Manifolds

In this note we consider the Schrödinger equation on compact manifolds equipped with possibly degenerate metrics. We prove Strichartz estimates with a loss of derivatives. The rate of loss of derivatives depends on the degeneracy of metrics. For the non-degenerate case we obtain, as an application of the main result, the same Strichartz estimates as that in the elliptic case. This extends Strichartz estimates for Riemannian metrics proved by Burq-Gérard-Tzvetkov to the non-elliptic case and improves the result by Salort for the degenerate case. We also investigate the optimality of the result for the case on 𝕊³ × 𝕊³.     

基于改进U-Net++的CT影像肺结节分割算法

卷积神经网络的语义分割模型未有效利用特征权重信息,导致在医学图像复杂场景中分割边界出现欠分割现象.针对该问题,基于融合自适应加权聚合策略提出一种改进的U-Net++网络,并将其应用于电子计算机断层扫描影像肺结节分割.该模型首先在卷积神经网络中提取出不同深度特征语义级别的信息,再结合权重聚合模块,自适应地学习各层特征的权...     

带有加权Lipschitz函数的交换子的有界性

研究了由加权Lipschitz函数b和Calderón-Zygmund奇异积分算子T生成的交换子Tb在一些加权空间上的有界性,涉及到加权Hardy空间,加权Herz空间及和加权Herz型Hardy空间.同时也得到了其相应的端点估计.     

问题变分不等式的一类各向异性Crouzeix-Raviart型有限元逼近

本文研究了Signorini变分不等式问题的一类各向异性Crouzeix-Raviart型非协调有限元逼近。通过一些新的技巧,得到了相应的最优误差估计。     

Perpetual cutoff method and CDE′(K,N) condition on graphs

By using the perpetual cutoff method, we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CD{E}^{\prime }(K,N). This generalizes a main result of F. Münch who considers the case of CD(K, \infty) curvature. Hence, we answer a question raised by Münch. For that purpose, we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded Laplacian \text{Δ} and perpetual cutoff semigroup P_t^W in our setting.     

单位球上加权Bergman空间上加权复合算子的本性范数

本文给出了单位球上加权Bergman空间上的加权复合算子的本性范数,并刻画了这类加权复合算子的有界性和紧性.     

A NUMERICALLY STABLE BLOCK MODIFIED GRAM-SCHMIDT ALGORITHM FOR SOLVING STIFF WEIGHTED LEAST SQUARES PROBLEMS

Recently, Wei in proved that perturbed stiff weighted pseudoinverses and stiff weighted least squares problems are stable, if and only if the original and perturbed coefficient matrices A and A^- satisfy several row rank preservation conditions. According to these conditions, in this paper we show that in general, ordinary modified Gram-Schmidt with column pivoting is not numerically stable for solving the stiff weighted least squares problem. We then propose a row block modified Gram-Schmidt algorithm with column pivoting, and show that with appropriately chosen tolerance, this algorithm can correctly determine the numerical ranks of these row partitioned sub-matrices, and the computed QR factor R^- contains small roundoff error which is row stable. Several numerical experiments are also provided to compare the results of the ordinary Modified Gram-Schmidt algorithm with column pivoting and the row block Modified Gram-Schmidt algorithm with column pivoting.     

A posteriori error estimates for FEM with violated Galerkin orthogonality

This article investigates Petrov‐Galerkin discretizations of operator equations with linearly stable operators, where the residual does not belong to the annihilator W of the discrete test space W_h. Conforming and nonconforming methods are considered separately, and for the treatment of the nonconforming situation the concept of elliptic lifting is introduced. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 241–259, 2002; DOI 10.1002/num.1005