On the packing chromatic number of some lattices

For a positive integer k, a k-packing in a graph G is a subset A of vertices such that the distance between any two distinct vertices from A is more than k. The packing chromatic number of G is the smallest integer m such that the vertex set of G can be partitioned as V₁,V₂,…,V_m where V_i is an i-packing for each i. It is proved that the planar triangular lattice T and the three-dimensional integer lattice Z³ do not have finite packing chromatic numbers.     

有序介孔SiO2的制备及其在色谱填料方面的应用

利用水热合成方法,在低浓度盐酸下用三嵌段共聚物EO106PO70EO106（F127）作为模板剂,在100℃,130℃和150℃的条件下,制备了3种笼型介孔二氧化硅.通过粉末N2气吸附-脱附实验表征其孔径（PD）及比表面积（BET）、扫描电镜（SEM）观察其表面形貌、透射电子显微镜（TEM）分析其微观结构;通过表征结果分析,该材料具有笼型介孔结构.同时,将该介孔材料应用在色谱分离方面,对氯苯,苯乙烯,1,2-二苯乙烯,联苯4种物质混合物进行分离,与其他多种二氧化硅材料比较,150℃的条件下合成的KIT-5-150分离效果最好.     

Valid constraints for the Point Packing in a Square problem

We consider the problem of placing n$n$ points in the unit square in such a way as to maximize their minimum pairwise distance m$m$. Starting from two properties of the optimal solution presented by Locatelli and Raber in [Discrete Applied Mathematics 122 (1–3) (2002) 139–166], and using the known theoretical lower and upper bounds, we derive some constraints for tightening the original formulation of the problem.     

Constraints in container loading – A state-of-the-art review

Container loading is a pivotal function for operating supply chains efficiently. Underperformance results in unnecessary costs (e.g. cost of additional containers to be shipped) and in an unsatisfactory customer service (e.g. violation of deadlines agreed to or set by clients). Thus, it is not surprising that container loading problems have been dealt with frequently in the operations research literature. It has been claimed though that the proposed approaches are of limited practical value since they do not pay enough attention to constraints encountered in practice.     

An infinite family of subcubic graphs with unbounded packing chromatic number

Recently, Balogh et al. (2018) answered in negative the question that was posed in several earlier papers whether the packing chromatic number is bounded in the class of graphs with maximum degree 3. In this note, we present an explicit infinite family of subcubic graphs with unbounded packing chromatic number.     

Optimal packings of two to four equal circles on any flat torus

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.     

Synthesis of Poly（p-phenyleneethynylene）s with Crosslinkable End or Side Groups and Its Solid State Structure and Optical Properties Before and after Crosslinking

Two crosslinkable poly(p-phenyleneethynylene)s(PPEs): poly[2,5-di(2'-ethyl-hexyloxy)-1,4-phenylenee-thynylene] with end-capped vinyl(PPE1) and poly[2,5-di(allyloxy)-1,4-phenyleneethynylene-2,5-di(2'-ethyl-hexyloxy)-1,4-phenyleneethynylene](PPE2) were synthesized. Via the thermal addition reactions of vinyl end groups of PPE1 and allyloxy side groups of PPE2, crosslinked polymers C-PPE1 and C-PPE2 were obtained, respectively. The two polymers were characterized by wide-angle X-ray diffraction(WXRD), ultravio...