Densest translational lattice packing of non-convex polygons

A translational lattice packing of k polygons P₁,P₂,P₃,…,P_k is a (non-overlapping) packing of the k polygons which is replicated without overlap at each point of a lattice i₀v₀+i₁v₁, where v₀ and v₁ are vectors generating the lattice and i₀ and i₁ range over all integers. A densest translational lattice packing is one which minimizes the area |v₀×v₁| of the fundamental parallelogram. An algorithm and implementation is given for densest translational lattice packing. This algorithm has useful applications in industry, particularly clothing manufacture.     

Relation between spectral classes of a self-similar Cantor set

A self-similar Cantor set is completely decomposed as a class of the lower (upper) distribution sets. We give a relationship between the distribution sets in the distribution class and the subsets in a spectral class generated by the lower (upper) local dimensions of a self-similar measure. In particular, we show that each subset of a spectral class is exactly a distribution set having full measure of a self-similar measure related to the distribution set using the strong law of large numbers. This gives essential information of its Hausdorff and packing dimensions. In fact, the spectral class by the lower (upper) local dimensions of every self-similar measure, except for a singular one, is characterized by the lower or upper distribution class. Finally, we compare our results with those of other authors.     

Cookie-Cuter集的多重Fractal分解的Packing维数

直线上的Ｃｏｏｋｉｅ－Ｃｕｔｔｅｒ集是一个扩张动力系统的不变集，并且是一个Ｆｒａｃｔａｌ集合．本文得到Ｃｏｏｋｉｅ－Ｃｕｔｔｅｒ集的多重Ｆｒａｃｔａｌ分解的Ｐａｃｋｉｎｇ维数，从而证明其是Ｔａｙｌｏｒ意义下的Ｆｒａｃｔａｌ集合     

Packing Steiner trees: polyhedral investigations

LetG=(V, E) be a graph andT⊆V be a node set. We call an edge setS a Steiner tree forT ifS connects all pairs of nodes inT. In this paper we address the following problem, which we call the weighted Steiner tree packing problem. Given a graphG=(V, E) with edge weightsw _e , edge capacitiesc _e ,e∈E, and node setT ₁,…,T _N , find edge setsS ₁,…,S _N such that eachS _k is a Steiner tree forT _k , at mostc _e of these edge sets use edgee for eache∈E, and the sum of the weights of the edge sets is minimal. Our motivation for studying this problem arises from a routing problem in VLSI-design, where given sets of points have to be connected by wires. We consider the Steiner tree packing problem from a polyhedral point of view and define an associated polyhedron, called the Steiner tree packing polyhedron. The goal of this paper is to (partially) describe this polyhedron by means of inequalities. It turns out that, under mild assumptions, each inequality that defines a facet for the (single) Steiner tree polyhedron can be lifted to a facet-defining inequality for the Steiner tree packing polyhedron. The main emphasis of this paper lies on the presentation of so-called joint inequalities that are valid and facet-defining for this polyhedron. Inequalities of this kind involve at least two Steiner trees. The classes of inequalities we have found form the basis of a branch & cut algorithm. This algorithm is described in our companion paper (in this issue).     

新型大颗粒离子交换树脂的表征和催化活性

用红外、核磁及热分析技术表征由溶胶-凝胶法制备、具有离子交换性能的大颗粒树脂催化剂，证实了树脂催化剂由含苯基及含硅的化合物构成，热稳定性高于商品离子交换树脂近100℃。对该树脂催化剂进行磺化反应的最佳条件为：温度，25℃；浓度，氯磺酸：三氯甲烷-1：4（Vo1）；时间，12h。其缩醛化反应的催化活性与商品大孔离子交换树脂相当。     

119Sn electric field gradients in model clusters of chalcogenide glasses

Calculations of ¹¹⁹Sn electric field gradients (EFG) have been performed using the Extended Hückel approximation on characteristic molecular clusters simulating possible types of sites in chalcogenide glasses. The motivation for these calculations derives from theoretical concepts on varying near neighbor relationships in these types of glasses, and from recent ¹¹⁹Sn Mössbauer experiments on Sn-doped Ge_x(Se or S)_1–x bulk glasses which reveal three types (A, B and C) of chemically inequivalent sites, with distinct values and composition dependences for their isomer shifts and quadrupole splittings. The model clusters chosen for the calculations were the ethane-like (Ge₂Se₃)_n quasi-one-dimensional chains of varying lengths which have been proposed as possible sources of the B site. In addition, calculations were also carried out on several additional types of clusters, in order to help in interpreting the results for the chains. We find that the magnitude of the quadrupole splitting in isolated linear ethane-like chains is very small, and almost independent of the particular site along the chain at which Sn replaces Ge. It therefore seems unlikely that such isolated linear clusters would be the source of the B sites. These sites are more likely to be related to distortions of the ethane-like clusters into non-linear configurations, as well as interactions with neighboring clusters, as forced by the constraints of the packing in the structure of the glass.     

Packing Structures of Thiophene Dimers and Their Effects on Excitation Energies of Thiophene Dimers

IntroductionPi(π) conjugated polymers have received exten-sive experimental and theoretical attention because oftheir fascinating electronic and optical properties[1—3].Recently, many new experiment results[4—6]have re-vealed that the inter-chain inter…     

Functionals on the spaces of convex bodies

In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.     

A first principles explanation for the experimentally observed increase in A-term band broadening in small domain silica monoliths and other chromatographic supports

The present computational study illustrates how the existence of a residual lower limit on the variance of the skeleton and through-pore size of monolithic columns can be expected to severely compromise the possibility to prepare well-performing small domain monolithic columns. Adopting rather conservative estimates for the minimal standard deviation on the pore and the skeleton size (0.2 and 0.04 microm, respectively), the presented calculations show that, if such a fixed lower limit on the size variance exists, it will be impossible to decrease the A-term band broadening below a given critical value, no matter how small the domain size is made. From a given critical domain size value on, any attempt to further decrease the domain size without being able to co-reduce the size variance can be expected to be counterproductive and leads to an increase instead of to a further decrease of the plate heights.