拟线性热方程的函数分离变量解

用群状结构法研究拟线性热方程的分离变量解,对于允许和型分离变量解的二阶拟线性热方程给出了一个完整的分类.说明了一些带有函数类型反应项的方程具有函数分离变量解,推广了前人的结论.     

Estimates of approximation numbers and applications

In this study, the estimates of approximation numbers of embedding operators in weighted spaces have been analyzed. These estimates depend on orders of differential operators, dimensions of function spaces and weighted functions. This fact implies that the associated embedding operators belong to Schatten class of compact operators. By using these estimates, the discreetness of spectrum and completion of root elements relating to principal nonselfedjoint degenerate differential operators is obtained.     

Approximate Derivative-Dependent Functional Variable Separation for the Generalized Diffusion Equations with Perturbation

As an extension to the derivative-dependent functional variable separation approach, the approximate derivative-dependent functional variable separation approach is proposed, and it is applied to study the generalized diffusion equations with perturbation. Complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is obtained. As a result, the corresponding approximate derivative-dependent functional separable solutions to some resulting perturbed equations are derived by way of examples.     

一类连续可分离背包问题的直接算法

对于一类带有单个线性约束以及盒约束的一般连续可分离二次背包问题给出了一种直接的算法,根据模型特有的结构,通过调节线性约束的拉格朗日乘子λ 的取值范围,以及在算法求解过程中通过判断目标函数一次项中的变量是否在盒约束范围内,来逐步确定所有变量的最优值, 并通过该算法得到的实验结果与其他算法的比较,说明了这种算法的可行性和有效性．     

一类多量子比特完全可分态的可分表示及应用

可分量子态能够写成乘积态的凸组合.通过可分量子态的可分表达式可以更好地理解量子态的数学结构.给出了一类多量子比特完全可分量子态的可分分解表示.此外,通过该类可分态的完全可分表示得到了一类阶数为2k的正交矩阵.     

A note on separation of variables solutions of generalized nonlinear diffusion equations

In this letter we show how more cases of the generalized nonlinear diffusion equation that contain separable solutions to those found by Zhang et al. (2002, 2003) via the generalized conditional symmetries method, can be found. Importantly we demonstrate with an example that can be used to describe water flow in unsaturated soil, and which can provide new insights when plant-root uptake is affected by water speed.     

Novel CuFe2O4@SiO2‐OP2O5H magnetic nanoparticles: Preparation,characterization and first catalytic application to the synthesis of 1,8‐dioxo‐octahydroxanthenes

New functionalized magnetic core–shell nanoparticles, CuFe₂O₄@SiO₂‐OP₂O₅H, were prepared by grafting of phosphorus pentoxide on CuFe₂O₄@SiO₂ nanoparticles and characterized using Fourier transform infrared spectroscopy, scanning electron microscopy, energy‐dispersive X‐ray analysis, inductively coupled plasma optical emission spectrometry and vibrating sample magnetometry. The catalytic activity of CuFe₂O₄@SiO₂‐OP₂O₅H as a novel catalyst was evaluated in the synthesis of 1,8‐dioxo‐octahydroxanthenes under solvent‐free conditions. The results showed that the catalyst has high activity and the desired products are obtained in high yields within short reaction times. The catalyst is readily recovered using magnetic decantation and can be used at least four times without noticeable deterioration in catalytic activity.     

Solving bilinear tensor least squares problems and application to Hammerstein identification

Bilinear tensor least squares problems occur in applications such as Hammerstein system identification and social network analysis. A linearly constrained problem of medium size is considered, and nonlinear least squares solvers of Gauss–Newton‐type are applied to numerically solve it. The problem is separable, and the variable projection method can be used. Perturbation theory is presented and used to motivate the choice of constraint. Numerical experiments with Hammerstein models and random tensors are performed, comparing the different methods and showing that a variable projection method performs best.     

Triton X-100 functionalized Cu(II) dihydrazone based complex immobilized on Fe3O4@dopa: A highly efficient catalyst for oxidation of alcohols,alkanes, and sulfides and epoxidation of alkenes

Here, we have presented a protocol for green synthesis, characterization, and catalytic application of TX100/Fe₃O₄@dopa@CuL ( FDCTX ) magnetically separable nanoparticles. Fe₃O₄@dopa@CuL ( FDC ) was synthesized using a four-step procedure: (i) synthesis of a dihydrazone derivative, (ii) reaction of the dihydrazone derivative with copper perchlorate salt to generate a copper complex of the dihydrazone derivative, (iii) immobilization of the complex onto Fe₃O₄@dopa to generate FDC , and (iv) coating of FDC with surfactant Triton X-100. The as-synthesized homogeneous complex was well characterized using UV–Vis., Fourier-transform infrared (FT-IR), electrospray ionization–mass spectrometry, and single-crystalX-ray techniques. Single-crystalX-ray analysis revealed the tetranuclear framework of the complex. The heterogeneous nanoparticles ( FDCTX ) were characterized using FT-IR, powder X-ray diffraction, scanning electron microscopy, transmission electron microscopy, energy-dispersiveX-ray spectroscopy, magnetic hysteresis, and dynamic light scattering techniques. Finally, both the homogeneous and heterogeneous catalysts were utilized for efficient oxidation of alcohols, alkanes, and sulfides and epoxidation of alkenes. A most probable mechanism for the oxidation reaction is proposed at the end of the article.