Aromaticity of corazulenic fullerenes

A novel class of non-classical fullerenes, having pentagon–heptagon pairs, as in azulene, is modeled. The various coverings, sometimes alternating azulenic and benzenic units, are designed by some new sequences of map operations or generalized operations. The hypothetical azulenic fullerenes are characterized by PM3 semiempirical data and POAV1 strain energy SE. Their aromaticity is discussed in the light of several criteria. The HOMA index of aromaticity enabled evaluation of global and local aromaticity of the designed fullerenes.     

Step length problem for trimming curve approximation in tessellating trimmed surfaces

A trimmed parametric surface is mainly composed of a surface together with trimming curves lying in D, the parametric space of the surface. By investigating the interrelation between surface tessellation and trimming curve approximation, we point out some problems on trimming curve approximation in existing trimmed surface tessellation algorithms. Counter examples are presented to show that a valid approximation of trimming curves in D together with the refinement imposed by surface tessellation does not necessarily generate a valid linear approximation in 3D space. To assure the 3D derivation tolerance, we propose two novel step-length estimation methods such that a piecewise linear interpolant of the trimming curve based on the proposed step lengths will result in a valid linear approximation in 3D space. The first method exploits the triangle inequality and takes the derivation tolerance in 3D space into account to compute the effective step length. Our second method is based on segmenting the trimming curve into subcurves first and then approximates each subcurve according to the derivation tolerance in 3D space. Moreover, several empirical tests are given to demonstrate the correctness of our step length estimations.     

Retro–Leapfrog and structure elucidation

Operations on maps are topological-geometrical tools used for transforming a given polyhedral tessellation. Investigation of fullerene structure often needs information on the original map which transformed into a larger molecular structure. Operations leading to the previous, smaller structures are called Retro-operations. They appear particularly useful in studies of structure elucidation or stability of series of fullerenes. The paper presents the first structure affiliation of the well-known C₆₀ fullerene to a family of Leapfrog fullerenes with relatedness being established by map operation. Thus, the tessellation of C₆₀ is described as an Archimedean, joint Sumanene-hexagon covering, in tetrahedral disposition. The other members of family show essentially the same covering and predicted good stability. Related Leapfrog fullerenes showing a disjoint Sumanene covering are also given.     

单轴拉伸过程中双相不锈钢局部微观变形行为的细观力学有限元分析

针对双相不锈钢中奥氏体相和铁素体相分别展开了纳米压痕实验,并通过有限元反演得到两相各自的拉伸应力-应变关系,利用Voronoi Tessellation法生成代表性的微结构体积单元,对双相不锈钢的单轴拉伸行为进行了有限元仿真和模拟,研究了双相不锈钢在拉伸过程中的局部应力、应变分布和演化规律．结果表明,利用Voronoi Tessellation法建立单元模型,结合本文通过纳米压痕实验获取的两相力学性能参数,可以很好地模拟双相不锈钢的整体单拉行为,奥氏体比铁素体软,拉伸载荷下双相不锈钢的应变集中在奥氏体中,应力集中在铁素体中;局部应力应变的分布特征与两相分布特征和晶粒形状有关,最大应变值主要集中在奥氏体晶粒狭长且尖锐的区域,而最大应力则主要发生在铁素体晶粒狭长和尖锐的区域;对于奥氏体和铁素体晶粒占比相当的双相不锈钢,其虽然可以具有较为综合的宏观力学性能,但是其微观应力集中的区域和应力最大值相对较大．研究成果为进一步揭示双相不锈钢局部失效机理奠定了基础．     

No acute tetrahedron is an 8-reptile

An $r$-gentiling is a dissection of a shape into $r\ge 2$ parts which are all similar to the original shape. An $r$-reptiling is an $r$-gentiling of which all parts are mutually congruent. The complete characterization of all reptile tetrahedra has been a long-standing open problem. This note concerns acute tetrahedra in particular. We find that no acute tetrahedron is an $r$-gentile or $r$-reptile for any $r<10$. The proof is based on showing that no acute spherical diangle can be dissected into less than ten acute spherical triangles.     

Tessellating polyominos in the plane

Let the R² space be divided into unit squares where a polyomino is a finite, connected set of unit squares. In this paper, we give a necessary and sufficient condition on tessellating polyominos by observing an unexpected relation between such tessellations and systems of arithmetic progressions.     

Asymptotic behaviour of the phase in non-smooth obstacle scattering

In this paper, we study the asymptotic behaviour of the scattering phases(λ) of the Dirichlet Laplacian associated with obstacle , where Ω is a bounded open subset of ℝ ⁿ (n≥2) with non-smooth boundary ∂Ω and connected complement Ω _e =ℝ ⁿ . We can prove that if Ω satisfies a certain geometrical condition, then

Exploratory data analysis for planar tessellations: Structural analysis and point process methods

This paper presents methods for the exploratory analysis of particular geometrical data, namely planar tessellations. At first, two non-stochastic methods are suggested which may help to classify tessellations and to understand their structure. The first one consists of approximating a given tessellation by a Dirichlet tessellation. The other one uses the nodes of a given tessellation and tests the possibility of reconstructing it by a fixed rule of connecting nodes by edges. Furthermore, in order to obtain information on the spatial behaviour of a tessellation, we suggest the use of the methods of point process statistics. In particular, pair correlation and mark correlation functions describe spatial correlations in tessellations.     

Remarks to the topological characterization of cellular systems

In this note we discuss and solve an open problem (a conjecture) posed in a paper “On the combinatorial characterization of quasicrystals” published earlier in the Journal of Geometry and Physics. The conjecture proved is valid for the majority of space filling cellular systems (polycrystals, nanotubes, and fullerenes).     

Partitioning 3-homogeneous latin bitrades

A latin bitrade is a pair of partial latin squares that define the difference between two arbitrary latin squares and of the same order. A 3-homogeneous bitrade has three entries in each row, three entries in each column, and each symbol appears three times in . Cavenagh [2] showed that any 3-homogeneous bitrade may be partitioned into three transversals. In this paper we provide an independent proof of Cavenagh’s result using geometric methods. In doing so we provide a framework for studying bitrades as tessellations in spherical, euclidean or hyperbolic space. Additionally, we show how latin bitrades are related to finite representations of certain triangle groups.