On partitions of K2,3-free graphs under degree constraints

Suppose that $s$, $t$ are two positive integers, and $?$ is a set of graphs. Let $g(s,t;?)$ be the least integer $g$ such that any $?$-free graph with minimum degree at least $g$ can be partitioned into two sets which induced subgraphs have minimum degree at least $s$ and $t$, respectively. For a given graph $H$, we simply write $g(s,t;H)$ for $g(s,t;?)$ when $?=\left\{H\right\}$. In this paper, we show that if $s,t\ge 2$, then $g(s,t;K_{2,3})\le s+t$ and $g(s,t;\{K_3,C_8,K_{2,3}\})\le s+t?1$. Moreover, if $?$ is the set of graphs obtained by connecting a single vertex to exactly two vertices of $K_4?e$, then $g(s,t;?)\le s+t$ on $?$-free graphs with at least five vertices, which generalize a result of Liu and Xu (2017).     

Partition functions and a generalized coloring‐flow duality for embedded graphs

Let G be a finite group and a class function. Let be a directed graph with for each vertex a cyclic order of the edges incident to it. The cyclic orders give a collection F of faces of H. Define the partition function , where denotes the product of the κ‐values of the edges incident with v (in cyclic order), where the inverse is taken for edges leaving v. Write , where the sum runs over irreducible representations λ of G with character and with for every λ. When H is connected, it is proved that , where 1 is the identity element of G. Among the corollaries, a formula for the number of nowhere‐identity G‐flows on H is derived, generalizing a result of Tutte. We show that these flows correspond bijectively to certain proper G‐colorings of a covering graph of the dual graph of H. This correspondence generalizes coloring‐flow duality for planar graphs.     

An ultrafilter with property

Let be a -supercompact cardinal. We show that carries a normal ultrafilter with a property introduced by Menas. With it we give a transparent proof of Kamo's theorem that carries a normal ultrafilter with the partition property.

     

混凝土断裂的连续-非连续方法

采用有限元形函数作为单位分解函数,位移间断用富集节点的附加自由度表示,建立了允许在单元内部位移非连续的局部富集公式以表征混凝土的开裂区域．富集基函数由节点形函数和节点形函数与间断函数的乘积的并集构成．非连续位移的扩展路径完全与网格结构无关．不同于以非协调应变为基础的嵌入非连续模型,对单元的类型没有限制而且间断位移可以贯穿单元边界．局部富集思想与扩展有限元类似,但富集点自由度保持节点位移的物理意义不变,使相邻单元无需进行富集运算．在变分公式中引入混凝土粘结本构定律,推导了考虑断裂过程区非线性影响的基本方程．对混凝土粘结裂纹扩展的数值模拟说明了该计算方法的有效性．     

基于混合范数约束的非均匀稀疏水声信道估计方法*

水声信道具有明显的簇状稀疏特性,即稀疏的信道冲激响应大部分为零或接近零的小值系数,而非零值系数是以簇的形式非均匀分布于时延域。为此本文提出了一种基于非均匀混合范数约束仿射投影算法的水声信道估计方法。该方法首先根据信道簇状结构对其进行非均匀分组,基于此将范数约束规则加入仿射投影算法中,具体方法为：对簇状部分施加范数约束,有效提高系数间的相关性,而簇状结构与其他零值抽头之间利用范数约束实现了整体的稀疏特性。数值仿真以及深海远程水声通信实验数据处理结果表明了本文所提出的水声信道估计算法相较现有稀疏信道估计方法能够实现更快的收敛速度以及更高精度的信道估计结果。     

Effect of inorganic salt on partition of high‐polarity parishins in two‐phase solvent systems and separation by high‐speed counter‐current chromatography from Gastrodia elata Blume

Parishins are high‐polarity and major bioactive constituents in Gastrodia elata Blume. In this study, the effect of several inorganic salts on the partition of parishins in two‐phase solvent systems was investigated. Adding ammonium sulfate, which has a higher solubility in water, was found to significantly promote the partition of parishins in the upper organic polar solvents. Based on the results, a two‐phase solvent system composed of butyl alcohol/acetonitrile/near‐saturated ammonium sulfate solution/water (1.5:0.5:1.2:1, v/v/v/v) was used for the purification of parishins by high‐speed counter‐current chromatography. Fractions obtained from high‐speed counter‐current chromatography were subjected to semi‐preparative high‐performance liquid chromatography to remove salt and impurities. As a result, parishin E (6.0 mg), parishin B (7.8 mg), parishin C (3.2 mg), gastrodin (15.3 mg), and parishin A (7.3 mg) were isolated from water extract of Gastrodia elata Blume (400 mg). These results demonstrated that adding inorganic salt that has high solubility in water to the two‐phase solvent system in high‐speed counter‐current chromatography was a suitable approach for the purification of high‐polarity compounds.     

Melodamide A from Melodorum fruticosum — Quantification using HPLC and one‐step‐isolation by centrifugal partition chromatography

Melodamide A, a phenolic amide from the leaves of Melodorum fruticosum Lour., has previously shown pronounced anti‐inflammatory activity. In order to rapidly isolate larger quantities for biological testing, a fast, one‐step isolation method by centrifugal partition chromatography was developed within this study. Fractionation of the dichloromethane extract was performed with a two‐phase solvent system consisting of n‐hexane, ethyl acetate, methanol, and water (3:7:5:5, v/v), leading to the isolation of melodamide A with a purity of >90% and a yield of 6.7 w% within 32 min. The developed method can also be used in dual mode for the enrichment of further constituents like flavonoids or chalcones. In order to support the centrifugal partition chromatography method development, additionally, a high‐performance liquid chromatography method was established and validated to determine quantities of melodamide A in plant material and crude extracts. Analysis of M. fruticosum leaves and a dichloromethane extract obtained from this plant material showed a total melodamide A content of 0.19 ± 0.008 and 8.9 ± 0.249 w%, respectively.     

Identifying stochastic basin hopping by partitioning with graph modularity

It has been known that noise in a stochastically perturbed dynamical system can destroy what was the original zero-noise case barriers in the phase space (pseudobarrier). Noise can cause the basin hopping. We use the Frobenius-Perron operator and its finite rank approximation by the Ulam-Galerkin method to study transport mechanism of a noisy map. In order to identify the regions of high transport activity in the phase space and to determine flux across the pseudobarriers, we adapt a new graph theoretical method which was developed to detect active pseudobarriers in the original phase space of the stochastic dynamic. Previous methods to identify basins and basin barriers require a priori knowledge of a mathematical model of the system, and hence cannot be applied to observed time series data of which a mathematical model is not known. Here we describe a novel graph method based on optimization of the modularity measure of a network and introduce its application for determining pseudobarriers in the phase space of a multi-stable system only known through observed data.     

Greedy splitting algorithms for approximating multiway partition problems

Given a system (V,T,f,k), where V is a finite set, is a submodular function and k2 is an integer, the general multiway partition problem (MPP) asks to find a k-partition ={V₁,V₂,...,V_k} of V that satisfiesfor all i and minimizes f(V₁)+f(V₂)+···+f(V_k), where is a k-partition of hold. MPP formulation captures a generalization in submodular systems of many NP-hard problems such as k-way cut, multiterminal cut, target split and their generalizations in hypergraphs. This paper presents a simple and unified framework for developing and analyzing approximation algorithms for various MPPs.Mathematics Subject Classification (1991): 20E28, 20G40, 20C20Acknowledgement This research is partially supported by the Scientific Grant-in-Aid from Ministry of Education, Science, Sports and Culture of Japan. The authors would like to thank the anonymous referees for their valuable comments and suggestions.     

Generators

In this note we indicate the importance of the notion of “generator” for the classification problem in ergodic theory, and we give a simple new construction of a finite generator for an ergodic transformation of finite entropy This revised version was published online in June 2006 with corrections to the Cover Date.