Wachspress quadrilateral patches have been recently studied from the point of view of applications to surface modelling in CAGD [1], [3], [4]. Some more applications for defining barycentric coordinates for arbitrary polygons have also been presented in [5] [9]. The purpose of the present paper is to introduce non-negative Wachspress rational basis functions for surface modelling on pentagonal ...

There has been an upsurge of research on complex networks in recent years. The purpose this paper is to study the mathematical properties random pentagonal chain PE C n with help graph theory. Based id="M2"> , we first obtain expected value expressions Gutman in...

Atomic clusters often show unique, size-dependent properties and have become a fertile ground for the discovery of novel molecular structures and chemical bonding. Here we report an investigation of the B19 2 cluster, which shows chemical bonding reminiscent of that in [10]annulene (C10H10) and [6]circulene (C24H12). Photoelectron spectroscopy reveals a relatively simple spectrum for B19 , with...

A. M. Legendre noted that Euler’s pentagonal number theorem implies that the number of partitions of n into an even number of distinct parts almost always equals the number of partitions of n into an odd number of distinct parts (the exceptions occur when n is a pentagonal number). Subsequently other classes of partitions, including overpartitions, have yielded related Legendre theorems. In thi...

Abstract. I revisit an automated proof of Andrews’ pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews’ result along with a variety of new formulas of similar type. I reveal an interesting relation between the tri-pentagonal theorem and items (19), (20), (94), (98) on the celebrated Slater ...

At the beginning of the series of papers we present systematic approach to exhaust the convex pentagonal tiles of edge-to-edge (EE) tilings. Our procedure is to solve the problem systematically step by step by restricting the candidates to some class. The first task is to classify both of convex pentagons and pentagonal tiling patterns. The classification of the latter is based on the analysis ...

Reported is the new bispidine-derived hexadentate ligand (L = 3-(2-methylpyridyl)-7-(bis-2-methylpyridyl)-3,7-diazabicyclo[3.3.1]nonane) with two tertiary amine and four pyridine donor groups. This ligand can form heterodinuclear and mononuclear complexes and, in the mononuclear compounds discussed here, the ligand may coordinate as a pentadentate ligand, with one of the bispyridinemethane-base...