一个R2上含双曲函数核的Hilbert型不等式

通过引入参数，构造了一个全平面上的、含双曲函数的非齐次核函数。利用正切函数的有理分式展开，建立了最佳常数因子与正切函数高阶导数相关联的Hilbert型积分不等式。 作为应用，通过赋予参数不同的值，建立了一些有意义的特殊结果。     

Hamiltonian cycles in 7-tough (P3 ∪ 2P1)-free graphs

The toughness of a noncomplete graph G is the maximum real number t such that the ratio of $|S|$ to the number of components of $G?S$ is at least t for every cutset S of G. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant $t_0$ such that every graph with toughness at least $t_0$ is hamiltonian, is still open for general graphs. A graph is called $(P_3\cup 2P_1)$-free if it does not contain any induced subgraph isomorphic to $P_3\cup 2P_1$, the disjoint union of $P_3$ and two isolated vertices. In this paper, we confirm Chvátal's toughness conjecture for $(P_3\cup 2P_1)$-free graphs by showing that every 7-tough $(P_3\cup 2P_1)$-free graph on at least three vertices is hamiltonian.     

Synthesis,cytotoxicity, apoptosis and cell cycle arrest of a monoruthenium(II)-substituted Dawson polyoxotungstate

By 5-h reaction of cis-[Ru^IICl₂(DMSO)₄] (M2) with K₁₀[α₂-P₂W₁₇O₆₁] (M3) in ice-cooled, HCl-acidic aqueous solution, a water-soluble 1:2-type diamagnetic ruthenium(II) complex of formula K₁₈[Ru^II(DMSO)₂(P₂W₁₇O₆₁)₂]·35H₂O (M1) was unexpectedly obtained as an analytically pure, homogeneous tan-colored solid, in which two DMSO ligands are coordinated to the ruthenium(II) atom. The cytotoxic potential of the complex was tested on C33A, DLD-1, and HepG-2 cancer cells and human normal embryonic lung fibroblasts cell MRC-5; the viability of the treated cells was evaluated by MTT assay. The mode of cell death was assessed by morphological study of DNA damage and apoptosis assays. Compound M1 induced cell death in a dose-dependent manner, and the mode of cell death was essentially apoptosis though necrosis was also noticed. Cell cycle analysis by flow cytometry indicated that M1 caused cell cycle arrest and accumulated cells in S phase.     

A note on 3‐partite graphs without 4‐cycles

Let $C_4$ be a cycle of order 4. Write $ex\left(n,n,n,C_4\right)$ for the maximum number of edges in a balanced 3‐partite graph whose vertex set consists of three parts, each has $n$ vertices that have no subgraph isomorphic to $C_4$. In this paper, we show that $ex\left(n,n,n,C_4\right)\ge \frac{3}{2}n\left(p+1\right)$, where $n=\frac{p\left(p?1\right)}{2}$ and $p$ is a prime number. Note that $ex\left(n,n,n,C_4\right)\le \left(\frac{3\sqrt{2}}{2}+o\left(1\right)\right)n^{\frac{3}{2}}$ from Tait and Timmons's works. Since for every integer $m$, one can find a prime $p$ such that $m\le p\le \left(1+o\left(1\right)\right)m$, we obtain that $\underset{n\to \infty }{\lim }\frac{ex\left(n,n,n,C_4\right)}{\frac{3\sqrt{2}}{2}{n}^{\frac{3}{2}}}=1$.     

Spontaneous translation of nanodroplet over a heterogeneous surface due to thermal cycles: role of solid–liquid interfacial fluctuations

ABSTRACT

We study the molecular-scale features of the solid surface that result in the spontaneous motion of a nanodroplet due to the periodic variation of temperature. We first employ a thermodynamic model to predict the variation of solid–fluid interfacial properties that can result in the above motion. The model identifies a composite (surface couple) made of two surfaces that are characterised by a large difference between the entropic parts of the solid–liquid interfacial free energies. In order to understand the molecular-scale features of the two surfaces that may form a surface couple, we performed grand canonical Monte Carlo simulations of Lennard Jones fluid and crystalline surfaces made of Lennard Jones-like atoms. We then used the cumulant expansions of the perturbation formulas to divide the interfacial entropy into two parts: The one that is directly affected by the solid–fluid attraction (direct part), and the other (indirect part) that is indirectly affected by the solid–fluid attraction via the alteration of interfacial fluctuations. Our results indicate that two surfaces form a surface couple if the differences between their chemical natures lead to large differences in the indirect part of the interfacial entropy, while the direct part remains relatively unaffected.     

Exceptional Cycles in the Bounded Derived Categories of Quivers

An exceptional n-cycle in a Hom-finite triangulated category with Serre functor has been recently introduced by Broomhead,Pauksztello and Ploog.When n=1,it is a spherical object.We explicitly determine all the exceptional cycles in the bounded derived category D^b(kQ)of a finite quiver Q without oriented cycles.In particular,if Q is an Euclidean quiver,then the length type of exceptional cycles in D^b(kQ)is exactly the tubular type of Q;if Q is a Dynkin quiver of type E_m(m=6,7,8),or Q is a wild quiver,then there are no exceptional cycles in D^b(kQ);and if Q is a Dynkin quiver of type An or D_n,then the length of an exceptional cycle in D^b(kQ)is either h or h/2,where h is the Coxeter number of Q.