On the numerical solution of ill‐conditioned linear systems by regularization and iteration

We propose to reduce the (spectral) condition number of a given linear system by adding a suitable diagonal matrix to the system matrix, in particular by shifting its spectrum. Iterative procedures are then adopted to recover the solution of the original system. The case of real symmetric positive definite matrices is considered in particular, and several numerical examples are given. This approach has some close relations with Riley's method and with Tikhonov regularization. Moreover, we identify approximately the aforementioned procedure with a true action of preconditioning.     

Linear independence of a finite set of time-frequency shifts

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L² function. Firstly, background and motivation for the conjecture are provided. Secondly, the main progress of this linear independence in the past twenty five years is reviewed. Finally, the partial results of the high dimensional case and other cases for the conjecture are briefly presented.     

(Sm0.5Gd0.2Nd0.3)2(Hf0.3Ce0.7)2O7复合氧化物的热物理性能

为寻求新型热障涂层用陶瓷材料,本文采用高温固相烧结法制备了(Sm_0.5Gd_0.2Nd_0.3)₂(Hf_0.3Ce_0.7)₂O₇复合氧化物。利用XRD分析了其晶体结构,SEM分析其显微组织,膨胀仪测试其热膨胀性能,激光热导仪测试其热扩散系数。结果表明,成功制备了具有单一萤石晶体结构的(Sm_0.5Gd_0.2Nd_0.3)₂(Hf_0.3Ce_0.7)₂O₇复合氧化物。其显微组织结构致密,晶界清晰无其他相存在。由于复杂的元素组成和较大的原子量,其热导率明显低于7YSZ和Sm₂Ce₂O₇。其较低的热膨胀系数则归因于B位离子较小的离子半径,但其热膨胀系数依然满足热障涂层的要求。     

基于不确定理论的铁路货运需求预测

合理预测铁路货运需求是铁路管理部门建设、运营等决策基础。为应对铁路货运需求的复杂变化,基于Pearson相关性分析方法筛选出铁路货运需求的七个具有关键影响的因素,并结合不确定理论建立不确定多元线性回归模型,相应的铁路货运预测结果由传统单一值变成可能的需求区间范围,更加符合处于不确定环境下的铁路货运需求实际情况。选取国家统计局2004~2016年相关数据进实证研究,并与回归模型以及BP模型的预测结果对比分析,实验表明不确定多元线性回归的预测结果更加精确。     

Spectral estimates for saddle point matrices arising in weak constraint four‐dimensional variational data assimilation

We consider the large sparse symmetric linear systems of equations that arise in the solution of weak constraint four‐dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3 × 3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2 × 2 block structure, or further to symmetric positive definite systems. In this article, we analyse how sensitive the spectra of these matrices are to the number of observations of the underlying dynamical system. We also obtain bounds on the eigenvalues of the matrices. Numerical experiments are used to confirm the theoretical analysis and bounds.