A fast iterative method to find the matrix geometric mean of two HPD matrices

The purpose of this research is to present a novel scheme based on a quick iterative scheme for calculating the matrix geometric mean of two Hermitian positive definite (HPD) matrices. To do this, an iterative scheme with global convergence is constructed for the sign function using a novel three‐step root‐solver. It is proved that the new scheme is convergent and shown to have global convergence behavior for this target, when square matrices having no pure imaginary eigenvalues. Next, the constructed scheme is used and extended through a well‐known identity for the calculation of the matrix geometric mean of two HPD matrices. Ultimately, several experiments are collected to show its usefulness.     

第二类端点奇异Fredholm积分方程的分数阶退化核方法

本文针对第二类端点奇异Fredholm积分方程构造基于分数阶Taylor展开的退化核方法，设计了两种计算格式，一是在全区间上使用分数阶Taylor展开式近似核函数，二是在包含奇点的小区间上采用分数阶插值，在剩余区间上采用分段二次多项式插值逼近核函数.讨论了两种退化核方法收敛的条件，并给出了混合插值法的收敛阶估计.数值算例表明对于非光滑核函数分数阶退化核方法有着良好的计算效果，且混合二次插值法比全区间上的分数阶退化核方法有着更广泛的适用范围.     

Maximizing the mean subtree order

This article focuses on properties and structures of trees with maximum mean subtree order in a given family; such trees are called optimal in the family. Our main goal is to describe the structure of optimal trees in and , the families of all trees and caterpillars, respectively, of order . We begin by establishing a powerful tool called the Gluing Lemma, which is used to prove several of our main results. In particular, we show that if is an optimal tree in or for , then every leaf of is adjacent to a vertex of degree at least . We also use the Gluing Lemma to answer an open question of Jamison and to provide a conceptually simple proof of Jamison's result that the path has minimum mean subtree order among all trees of order . We prove that if is optimal in , then the number of leaves in is and that if is optimal in , then the number of leaves in is . Along the way, we describe the asymptotic structure of optimal trees in several narrower families of trees.     

Fractional powers of the noncommutative Fourier's law by the S‐spectrum approach

Let e_?, for ? = 1,2,3, be orthogonal unit vectors in and let be a bounded open set with smooth boundary ?Ω. Denoting by a point in Ω, the heat equation, for nonhomogeneous materials, is obtained replacing the Fourier law, given by the following: into the conservation of energy law, here a, b, are given functions. With the S‐spectrum approach to fractional diffusion processes we determine, in a suitable way, the fractional powers of T. Then, roughly speaking, we replace the fractional powers of T into the conservation of energy law to obtain the fractional evolution equation. This method is important for nonhomogeneous materials where the Fourier law is not simply the negative gradient. In this paper, we determine under which conditions on the coefficients a, b, the fractional powers of T exist in the sense of the S‐spectrum approach. More in general, this theory allows to compute the fractional powers of vector operators that arise in different fields of science and technology. This paper is devoted to researchers working in fractional diffusion and fractional evolution problems, partial differential equations, and noncommutative operator theory.     

考虑溯源系统应用的优质农产品双渠道供应链决策研究

优质农产品溯源系统的引入,既能提升有效供给、缩短流通时间,又能降低流通中的数量损耗和价值损失,由此带来了双渠道供应链系统在投资、销售、定价等方面的系列决策问题。考虑制造商开辟电子渠道、从溯源系统降低流通损耗和提高质量出发,分别构建制造商主导型的双渠道供应链决策模型。通过对比分析应用前后各成员在投资、定价及销售方面的最优策略,得到相关的核心边界值。数值仿真结果表明:核心边界值主要与双重损耗的改善程度及其他参数相关,各成员及系统接受溯源系统的应用成本不同,应用成本对各渠道销量的影响程度亦不相同,但各成员调整定价的应用成本边界值是一致的。     

Synthesis,crystal structure,DNA/bovine serum albumin binding and antitumor activity of two transition metal complexes with 4‐acylpyrazolone derivative

Two new complexes, namely [Cu₆L₆] ( 1 ) and [Zn(HL)₂] ( 2 ) (H₂L = N‐(1‐phenyl‐3‐methyl‐4‐propenylidene‐5‐pyrazolone)‐2‐furancarboxylic acid hydrazide), have been synthesized and characterized. Single crystal X‐ray analysis indicates that complex 1 has a hexanuclear structure and complex 2 exhibits a mononuclear structure. The DNA/bovine serum albumin (BSA) binding properties of complexes 1 and 2 were investigated by absorption spectroscopy and fluorescence quenching. Both complexes could effectively intercalate to DNA with calculated quenching constants of 2.6 × 10⁵ and 1.25 × 10⁵ M^?1, respectively. The quenching mechanism of the intrinsic fluorescence of BSA by the complexes was found to be a static one. The cytotoxicities of 1 and 2 were investigated in two human tumor cell lines, human esophageal cancer cells (Eca‐109) and cervical cancer cells (HeLa). Complex 1 exhibits higher antitumor activity than 2 . Furthermore, 1 can inhibit HeLa cells by inducing apoptosis and G0/G1 phase cell cycle arrest. All results demonstrate that 1 and 2 both have DNA/BSA binding capacity and antitumor activity.     

MAO‐free and extremely active catalytic system for ethylene tetramerization

The original Sasol catalytic system for ethylene tetramerization is composed of a Cr source, a PNP ligand, and MAO (methylaluminoxane). The use of expensive MAO in excess has been a critical concern in commercial operation. Many efforts have been made to replace MAO with non‐coordinating anions (e.g., [B(C₆F₅)₄]^?); however, most of such attempts were unsuccessful. Herein, an extremely active catalytic system that avoids the use of MAO is presented. The successive addition of two equivalent [H(OEt₂)₂]⁺[B(C₆F₅)₄]^? and one equivalent CrCl₃(THF)₃ to (acac)AlEt₂ and subsequent treatment with a PNP ligand [CH₃(CH₂)₁₆]₂C(H)N(PPh₂)₂ ( 1 ) yielded a complex presumably formulated as [ 1 ‐CrAl (acac)Cl₃(THF)]²⁺[B(C₆F₅)₄]^?₂, which exhibited high activity when combined with iBu₃Al (1120 kg/g‐Cr/h; ~4 times that of the original Sasol system composed of Cr (acac)₃, iPrN(PPh₂)₂, and MAO). Via the introduction of bulky trialkylsilyl substituents such as –SiMe₃, –Si(nBu)₃, or –SiMe₂(CH₂)₇CH₃ at the para‐position of phenyl groups in 1 (i.e., by using [CH₃(CH₂)₁₆]₂C(H)N[P(C₆H₄‐p‐SiR₃)₂]₂ instead of 1 ), the activities were dramatically improved, i.e., tripled (2960–3340 kg/g‐Cr/h; more than 10 times that of the original Sasol system). The generation of significantly less PE (<0.2 wt%) even at a high temperature is another advantage achieved by the introduction of bulky trialkylsilyl substituents. NMR studies and DFT calculations suggest that increase of the steric bulkiness on the alkyl‐N and P‐aryl moieties restrict the free rotation around (alkyl)N–P (aryl) bonds, which may cause the generation of more robust active species in higher proportion, leading to extremely high activity along with the generation of a smaller amount of PE.