Polybasic polyhedra: structure of polyhedra with edge vectors of support size at most 2

It has widely been recognized that submodular set functions and base polyhedra associated with them play fundamental and important roles in combinatorial optimization problems. In the present paper, we introduce a generalized concept of base polyhedron. We consider a class of pointed convex polyhedra in R^V whose edge vectors have supports of size at most 2. We call such a convex polyhedron a polybasic polyhedron. The class of polybasic polyhedra includes ordinary base polyhedra, submodular/supermodular polyhedra, generalized polymatroids, bisubmodular polyhedra, polybasic zonotopes, boundary polyhedra of flows in generalized networks, etc. We show that for a pointed polyhedron PR^V, the following three statements are equivalent:

(1) P is a polybasic polyhedron.

(2) Each face of P with a normal vector of the full support V is obtained from a base polyhedron by a reflection and scalings along axes.

(3) The support function of P is a submodular function on each orthant of R^V.

This reveals the geometric structure of polybasic polyhedra and its relation to submodularity.     

Poly[[chlorido(1,10‐phenanthroline‐κ2N,N′)copper(II)]‐μ3‐1,1,3,3‐tetracyano‐2‐ethoxypropenido‐κ3N:N′:N′′]: coordination polymer sheets linked into bilayers by hydrogen bonds

In the title compound, [Cu(C₉H₅N₄O)Cl(C₁₂H₈N₂)]_n or [Cu(tcnoet)Cl(phen)]_n, where phen is 1,10‐phenanthroline and tcnoet is 1,1,3,3‐tetracyano‐2‐ethoxypropenide, the axially elongated (4 + 2) coordination polyhedron around the Cu^II centre contains N atoms from three different tcnoet ligands. The resulting coordination polymer takes the form of sheets which are linked in pairs by a single C—H...N hydrogen bond to form bilayers. The bond lengths provide evidence for significant bond fixation in the phen ligand and extensive electronic delocalization in the tcnoet ligand, where the two –C(CN)₂ units are rotated, in conrotatory fashion, out of the plane of the central C₃O fragment.     

Proline adducts of metallo-organic,sextuply-stranded lanterns as homogeneous catalysts for asymmetric catalysis

Two homochiral metal–organic, sextuply-stranded, lantern-like complexes, Ce-L-Pro1 and Ce-D-Pro1, were synthesized via self-assembly by incorporating a L or D-pyrrolidine moiety within the respective ligands. The Ce-Pro1 complexes are comprised of chiral pyrrolidine-containing ligands and have lantern-like cavities. They function as asymmetric catalysts that promote cyanosilylation reactions with good conversions and enantioselectivity.     

The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature

This is a continuation of the Cambridge Tract ``Harmonic maps between Riemannian polyhedra', by J. Eells and the present author. The variational solution to the Dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary, and thereby uniquely determined. The domain space is a compact admissible Riemannian polyhedron with boundary, while the target can be, for example, a simply connected complete geodesic space of nonpositive Alexandrov curvature; alternatively, the target may have upper bounded curvature provided that the maps have a suitably small range. Essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain.

     

A semi-analytic method for calculating non-probabilistic reliability index based on interval models

This paper proves: (1) non-probabilistic reliability index of a structure exists merely at one of intersection points at which normalized failure surfaces of the structure intersects the straight lines passing not only through origin of an normalized infinite space but also through vertices of a symmetric convex polyhedron with its sym-center at the origin, and (2) the non-probabilistic reliability index equals to absolute value of the coordinate components of a particular intersection point. Based on a reduction of the feasible region, a semi-analytical method for calculating the reliability index is developed. The method proves to be simple and of practical significance, and has several advantages over the existing unconstrained multivariate nonlinear optimization approach.     

On octahedral fulleroids

The discovery of the first fullerene has raised an interest in the study of other candidates for modelling of carbon molecules. As a generalization of the fullerenes, fulleroids are defined as cubic convex polyhedra with all the faces of size five or greater. In this paper, we give necessary and sufficient condition for the existence of fulleroids with only pentagonal and n-gonal faces and with the symmetry group isomorphic to the full symmetry group of the regular octahedron. The existence is proved by finding infinite series of examples. The nonexistence is proved using symmetry invariants.     

Regular Polyhedra With Translational Symmetries

In this note we present a group theoretical approach to polyhedra with a translation subgroup in their automorphism group.     

一类解析多面体的奇异积分

本文利用Ｒ．Ｈａｒｖｅｙ和Ｊ．Ｐｏｒｋｉｎｇ的方法首先定义广义式的Ｃａｕｃｈｙ主值，利用同伦公式，借助积分变换技巧研究Ｗｅｉｌ型积分的边界性质，得到Ｐｌｅｍｅｌｊ公式．它有别于通常研究边界性质的方法．本文引入细复广义权和Ｃｈｏｑｕｅｔ型复广义权的概念，讨论了某些与复广义权相关的函数的拟连续性与细拟处处连续的关系．     

Multidimensional Polyhedral Gamut Representation of Reflective Objects and Calculation of Metamer Ensembles

The spectral reflectance of most reflective objects, such as natural objects and color hardcopy, is relatively smooth and can be approximated by several principal components with high accuracy. Although the subspace spanned by those principal components represents a space in which reflective objects exist, it does not provide the limit of the object distribution or the gamut. In this paper we propose to represent the gamut of reflective objects as a convex polyhedron in the subspace spanned by several principal components. The concept of the polyhedral gamut representation based on the spectral reflectance database and its application to the calculation of metamer ensembles are described. The color-mismatch volume caused by different illuminant and/or observer for a metamer ensemble is also calculated and compared with the theoretical ones.     

激光扫描转镜工作角偏差检测新方法

提出一种检测激光扫描转镜工作角偏差的新方法，该法通过两个激光扫描转镜相应面相互比较和检测，仅用一台自准直仪，可同时检测两个扫描转镜，且两个转镜具有相同的检测精度。介绍了该法的检测原理，导出了计算工作角偏差的公式，给出了检测结果的标准差，最后还给出了检测实例。该法检测仪器单一，检测精度高，不仅适于激光扫描转镜的检测，也可用于多面棱体的检定。