A note on the computational complexity of graph vertex partition

A stable set of a graph is a vertex set in which any two vertices are not adjacent. It was proven in [A. Brandstädt, V.B. Le, T. Szymczak, The complexity of some problems related to graph 3-colorability, Discrete Appl. Math. 89 (1998) 59-73] that the following problem is NP-complete: Given a bipartite graph G, check whether G has a stable set S such thatG-Sis a tree. In this paper we prove the following problem is polynomially solvable: Given a graph G with maximum degree 3 and containing no vertices of degree 2, check whether G has a stable set S such thatG-Sis a tree. Thus we partly answer a question posed by the authors in the above paper. Moreover, we give some structural characterizations for a graph G with maximum degree 3 that has a stable set S such that G-S is a tree.     

Cs[Er6C]I12 und Cs2Lu[Lu6C]Cl18: Beispiele für quaternäre reduzierte Halogenide der Lanthanide mit isolierten „Clustern”︁

CS[Er₆C]I₁₂ and cs₂Lu[Lu₆C]CI₁₈: Examples for Quaternary Reduced Halides of the Lanthanides with Isolated “Clusters” Cs[Er₆C]I₁₂ and Cs₂Lu[Lu₆C]Cl₁₈ were obtained as byproducts through metallothermic reductions of ErI₃ and LuCl, with cesium in the presence of carbon in sealed tantalum containers at temperatures ranging from 700 to 940 °C. Cs₂Lu[Lu₆C]Cl₁₈ (isostructural with Cs₂Zr[Zr₆H]Cl₁₈, R 3 , a = 981.7 pm, c = 2723.2 pm, Z = 3, R = 0.082, R, = 0.053) contains octahedral [Lu₆C] clusters which are slightly compressed along the threefold axis and edge-bridged by twelve chloride anions to form [Lu₆C]Cl₁₂ units. Six additional Cl in exo positions of the cluster provide octahedral coordination for the seventh Lu³⁺. Cs⁺ occupies anticuboctahedral interstices within the Cl⁺ layers as a part of the (hexagonal) closest packed arrangement. Cs[Er₆C]I₁₂ (trigonal, R 3, a = 1112.0pm, c = 2063.8pm, Z = 3, R = 0.094, R, = 0.068) exhibits [Er₆C]I₁₂ units as well and shows the structural framework of Sc[Sc₆N]Cl₁₂. Instead of Sc³⁺ in octahedral holes, cesium occupies a regular iodide position within the ccp sheets forming [CsI₃] layers. Both halides are compared with other compounds of the lanthanides containing isolated [M₆X₁₂] clusters. The extreme electron deficiency is discussed.     

The enormity of the zinc deficiency problem and available solutions; an overview

The societal cost of micronutrient deficiency (MND) or the “hidden hunger” is in millions of dollars/year, reducing the GDP of some countries by as much as 11%. Zn is an important micronutrient for both plants and animals. An estimated 17% of the world population, or around 1.1 billion people, are at the risk of zinc (Zn) deficiency. The deficiency has been related to adverse pregnancy outcomes, stunted growth, premature deaths, immune system dysfunctions, neuro-behavioral disorders, and recently with the failure to recover from COVID-19. These health risks associated with Zn deficiency have compelled FAO and WHO to recommend Zn fortification of diet. Correcting Zn deficiency is a challenge due to several reasons. Close to half of the agricultural soils are Zn deficient, and chemical Zn fertilizers are costly and ineffective. Developing Zn-rich crops through plant breeding and genetic engineering is challenging. Zn-dense diet is costly and cannot be implemented in the low-income region most affected by Zn deficiency. Lack of consensus among regulatory bodies on defining and diagnosing Zn deficiency in plants and Humans. Awareness and other sociocultural issues. Among the most important available solutions are zinc biofortification of the cereal crops, use of zinc biofertilizers, development of Zn-efficient crops with reduced phytate content. The use of Zn supplements, dietary modification, and diversification, especially with fish, are proposed as the most accessible and affordable solutions. Awareness programs in areas suffering the most from Zn deficiency are required. Despite the suggestions from FAO and WHO, global efforts to combat Zn deficiency matching those for combating diseases like HIV are not in place. Coordinated efforts of the international community, especially policy-makers, agricultural scientists, dieticians, physicians, and others, are required to address the issue of hidden hunger.     

图的直径与Betti亏数

对于任意的正数Ｍ以及正整数ｄ≥４,存在直径为ｄ的ｉ－边连通无环图Ｇ使得ζ（Ｇ）≥Ｍ,其中ζ（Ｇ）是Ｇ的Ｂｅｔｔｉ亏数,ｉ＝１,２,３。     

Comparison of certain value-at-risk estimation methods for the two-parameter Weibull loss distribution

The Weibull distribution is one of the most important distributions that is utilized as a probability model for loss amounts in connection with actuarial and financial risk management problems. This paper considers the Weibull distribution and its quantiles in the context of estimation of a risk measure called Value-at-Risk (VaR). VaR is simply the maximum loss in a specified period with a pre-assigned probability level. We attempt to present certain estimation methods for VaR as a quantile of a distribution and compare these methods with respect to their deficiency (Def) values. Along this line, the results of some Monte Carlo simulations, that we have conducted for detailed investigations on the efficiency of the estimators as compared to MLE, are provided.     

Structural properties and electrical behaviour in the polycrystalline lanthanum-deficiency La1−x□xMnO3 manganites

Electrical conductance and X-ray diffraction (XRD) measurements of lanthanum-deficiency La_1−x□_xMnO₃ (x=0.05, 0.10 and 0.20) polycrystalline samples were performed to examine the effect of the internal pressure at B-site on the conduction mechanism. The structural study reveals that all samples crystallize in the rhombohedral system. The electronic conduction appears to be thermally activated at high temperature, which indicates the presence of semiconductor behaviour. The increase of the x content converts 3x Mn³⁺ to 3x Mn⁴⁺ ions with smaller ionic radius, which reduces the internal pressure and leads to the increase of the one-electron bandwidth W. This increase causes the appearance of metallic behaviour at low temperature for x=0.10 and 0.20 content.     

嵌入在克莱茵瓶上的图的最大亏格

设$\phi: G\rightarrow S$是图$G$在曲面$S$上的2 -胞腔嵌入. 若$G$的所有面都是依次相邻, 即嵌入图$G$的对偶图有哈密顿圈, 则将$\phi$称为一个面依次相邻的嵌入. 该文研究了在克莱茵瓶上有面依次相邻嵌入的图的最大亏格.     

The Symmetry of Singular Hamiltonian Differential Operators and Properties of Deficiency Indices

The symmetry of singular Hamiltonian differential operators is proved under the standard ＂definiteness condition＂, which is strictly weaker than the densely definite condition used by A. M. Krall. Meanwhile, some properties of deficiency indices are given.     

Stability of index for linear relations and its applications

This paper is concerned with the stability of linear relations in Banach and Hilbert spaces. Several important results about the stability of closedness and the stability of indices of operators in Banach spaces are extended to the linear relations in Banach spaces. Moreover, the stability of deficiency indices of dissipative linear relations in Hilbert spaces is studied. As an important application, we discuss the stability of deficiency indices of dissipative linear relations generated by second-order difference expressions.     

图的最大亏格、支配数和围长

一个连图G的最大亏格γM（G）=（β（G）-ξ（G）/2,其中β（G）=E（G）-V（G 1是G的圈秩,ξ（G）是G的Betti亏数,本文利用G的支配数和围长给出了G的Betti亏数ξ（G）的一个上界,从而也给出了最大亏格γ（M（G）的一个下界,而且它是可达的,对于某些图类,该下界比黄元秋（2000）所给下界更好。