Three-dimensional vertex model in statistical mechanics from Baxter-Bazhanov model

We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube, and the Wu-Kadanoff-Wegner duality between the cube- and vertex-type tetrahedron equations is obtained explicitly for the Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by considering the symmetry property of the weight function, which corresponds to the three-dimensional Baxter-Bazhanov model. The vertex-type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube. We write down the symmetry relations of the weight functions under the actions of the symmetry groupG of the cube. The six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected, with the six spaces in which the vertextype tetrahedron equation is defined.     

Solutions of the Functional Tetrahedron Equation Connected with the Local Yang-Baxter Equation for the Ferro-Electric Condition

A local (or modified) Yang–Baxter equation (LYBE) gives a functional map from the parameters of the weights in the left-hand side to the parameters of the corresponding weights in the right-hand side of the LYBE. Such maps solve the functional tetrahedron equation. In this Letter, all the maps associated with LYBE's of the ferro-electric type with a single parameter in each weight matrix are classified.     

Gd4I6CN: Ein Carbidnitrid mit Ketten aus Gd6(C2)-Oktaedern und Gd6N2-Tetraederdoppeln

Gd₄I₆CN: A Carbide Nitride with Chains of Gd₆(C₂) Octahedra and Gd₆N₂ Double Tetrahedra The compound α-Gd₄I₆CN is prepared by reaction of Gd, GdI₃, C, and GdN (1:2:1:1 mole ratio) at 1 170 K in sealed Ta tubes. It is obtained as brown red, transparent needles which are air and moisture sensitive. The structure of α-Gd₄I₆CN contains Gd₆ octahedra centered by C₂ groups and double tetrahedra centered by N atoms. The units are alternatingly connected via common edges to form chains (Gd₂Gd_4/2C₂) (Gd_2/2Gd_2/2N)₂ parallel [001]. The linear chains are surrounded by I atoms above all free edges of the metal polyhedra and linked according to (Gd₂Gd_4/2C₂) (Gd₂Gd_4/2N₂)I_4/2I₈I₂ in the a – b plane. We also found β-Gd₄I₆CN, which is formed in a monotropic transition from the α-form. In the structure the chains of Gd octahedra and tetrahedra as described for α-Gd₄I₆CN are more densely packed. The structure of Y₆I₉C₂N is composed by chains of pairs of Y-octahedra and Y-tetrahedra, respectively. The octahedra are centered by C₂ groups, the tetrahedra by N-atoms. We also synthesized the compounds Gd₄Br₆CN und La₄I₆CN by tempering at 1 220 K. They are isotypic with α-Gd₄I₆CN.     

四面体表面上的正则曲线

给出四面体表面上正则曲线的长度特征 .     

LiCl熔盐分子动力学模拟的Delaunay单纯形分析

利用Delaunay单纯形理论,对以LiCl为代表的熔盐系的动力学模拟构型进行了众多参数的实现及其统计.重点分析了熔化过程中Delaunay单纯形体积、面积、四面体系数及其Kirie单纯形随温度变化的趋势.借助逾渗理论和染色方法,熔化的过程可以认为是大面积Delaunay单纯形出现、成串直至逾渗的过程,而液体流动性的根源则是大体积Delaunay单纯形的出现和增加.同时,四面体系数的分布进一步证实Kirie单纯形的变化趋势.     

Study on the local structure and EPR spectroscopy of compressed CuX4 (X = N,S, Cl and Br) tetrahedral cluster

The spin Hamiltonian parameters g and A for copper(II) in CuX₄ (X = N, S, Cl and Br) tetrahedrons are theoretically investigated within a uniform model by utilizing the perturbation method and cluster approach for 3d⁹ ions suffering D_2d symmetry in view of host structures and angular distortions in terms of a single parameter β. For all the studied polymers, the isotropy of g tensors g_av and the hyperfine structure constant component A_‖ are found, respectively, to undergo an increasing and decreasing trend with the enlarging average X–Cu–X angle β, while the obtained cubic crystal field parameter D_q and isotropic Fermi contact constant κ are experiencing a declining and growing tendency with the rising Cu–X bond length, respectively. The critical angle β_c is concluded locating in the range of 109.471°–120° in which the g-isotropy value would reach the maximum, while the hyperfine structure factor might vanish that needs to be verified in future X-ray and EPR experiments. This work would be beneficial to the establishment of complete EPR and structure diagram for CuX₄ complexes and the synthesis of new compounds possessing novel physical and chemical properties.     

Irregularly shaped space-filling truncated octahedra

For any parent tetrahedron ABCD, centroids of selected sub-tetrahedra form the vertices of an irregularly shaped space-filling truncated octahedron. To reflect these properties, such a figure will be called an ISTO. Each edge of the ISTO is parallel to and one-eighth the length of one of the edges of tetrahedron ABCD and the volume of the ISTO is 3/16-th the volume of the tetrahedron. The ISTO is symmetric about the centroid of tetrahedron ABCD and each face is symmetric about a centre and has an opposite face that is parallel and congruent. The area of the faces of the ISTO is not proportional to that of the generating tetrahedron.     

A sharp polynomial estimate of positive integral points in a 4‐dimensional tetrahedron and a sharp estimate of the Dickman‐de Bruijn function

The estimate of integral points in right‐angled simplices has many applications in number theory, complex geometry, toric variety and tropical geometry. In [24], [25], [27], the second author and other coworkers gave a sharp upper estimate that counts the number of positive integral points in n dimensional () real right‐angled simplices with vertices whose distance to the origin are at least . A natural problem is how to form a new sharp estimate without the minimal distance assumption. In this paper, we formulate the Number Theoretic Conjecture which is a direct correspondence of the Yau Geometry conjecture. We have proved this conjecture for . This paper gives hope to prove the new conjecture in general. As an application, we give a sharp estimate of the Dickman‐de Bruijn function for .     

Crystal Structure and Antitumor Activity of Tri[2-[N-（4′-methyl-benzylsulfonyl）amino]ethyl]-amine

The crystal structure of the title compound (C27H38N4O7S3, Mr = 626.79) has been determined by single-crystal X-ray diffraction. The crystal is of triclinic, space group Pīwith a = 9.411(1), b = 11.645(2), c = 14.672(2) (A。), α = 91.80(1), β = 95.36(1), γ =104.56(1)o, V = 1547.0 (A。)3, Z = 2, Dc = 1.346 g/cm3, λ = 0.71073 (A。), μ(MoKα) = 0.289 mm－1 and F(000) = 664. The structure was refined to R = 0.0406 and wR = 0.1177 for 4103 observed reflections with I > 2σ(I). X-ray diffraction analysis reveals that the title compound is a practically distorted tetrahedron and each molecule contains one lattice H2O by hydrogen bond. The antitumor activity of the title compound against HL-60 human leukemia cells has also been studied by MTT method.     

层错四面体对单晶铜层裂行为影响的分子动力学研究

层错四面体是一种典型的三维空位型缺陷,广泛存在于受辐照后的面心立方金属材料中,对材料的力学性能有显著的影响.目前,关于层错四面体对辐照材料层裂行为的影响还缺乏深入系统的研究.本文使用分子动力学方法模拟了含有层错四面体的单晶铜在不同冲击速度下的层裂行为,对整个冲击过程中的自由表面速度及微结构演化等进行了深入的分析.研究发现,层错四面体在冲击波作用下会发生坍塌,并进一步诱导材料产生位错、层错等缺陷.在中低速度加载下,层错四面体坍塌引起的缺陷快速向周围扩展,为孔洞提供了更宽的形核区域,促进了孔洞的异质成核,造成材料层裂强度大幅度减小.当冲击速度较高时,层错四面体坍塌导致的局部缺陷对材料的层裂强度不再有明显影响.