Three-dimensional integrable models and associated tangle invariants

In this paper we show that the Boltzmann weights of the three-dimensional Baxter-Bazhanov model give representations of the braid group if some suitable spectral limits are taken. In the trigonometric case we classify all possible spectral limits which produce braid group representations. Furthermore, we prove that for some of them we get cyclotomic invariants of links and for others we obtain tangle invariants generalizing the cyclotomic ones.     

Bodies of constant width in arbitrary dimension

We give a number of characterizations of bodies of constant width in arbitrary dimension. As an application, we describe a way to construct a body of constant width in dimension n, one of its (n – 1)‐dimensional projection being given. We give a number of examples, like a four‐dimensional body of constant width whose 3D‐projection is the classical Meissner's body. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Importance of H-Bonding Interactions in Organic Ammonium Tetrathiotungstates

Summary. Four new organic ammonium tetrathiotungstates (N–Me–enH₂)[WS₄] (1), (N,N′-dm-1,3-pnH₂)[WS₄] (2), (1,4-bnH₂)[WS₄] (3), and (mipaH)₂[WS₄] (4), (N–Me–enH₂ = N-methylethylenediammonium, N,N′-dm-1,3-pnH₂ = N,N′-dimethyl-1,3-propanediammonium, 1,4-bnH₂ = 1,4-butanediammonium, and mipaH = monoisopropylammonium) were synthesized by the base promoted cation exchange reaction and characterized by elemental analysis, infrared, Raman, UV-Vis and ¹H NMR spectroscopy as well as single crystal X-ray crystallography. The structures of 1–4 consist of [WS₄]²⁻ tetrahedra which are linked to the organic ammonium cations via N–H⋯S hydrogen bonding. The strength and number of the S⋯H interactions affect the W–S bond lengths as evidenced by distinct short and long W–S bonds. The IR spectra exhibit splitting of the W–S vibrations, which can be attributed to the distortion of the [WS₄]²⁻ tetrahedron. From a comparative study of several known tetrathiotungstates it is observed that a difference of more than 0.033 ? between the longest and shortest W–S bonds in a tetrathiotungstate will result in the splitting of the asymmetric stretching vibration of the W–S bond.     

Seed‐Mediated Synthesis of Gold Tetrahedra in High Purity and with Tunable,Well‐Controlled Sizes

We report a facile synthesis of Au tetrahedra in high purity and with tunable, well‐controlled sizes via seed‐mediated growth. The success of this synthesis relies on the use of single‐crystal, spherical Au nanocrystals as the seeds and manipulation of the reaction kinetics to induce an unsymmetrical growth pattern for the seeds. In particular, the dropwise addition of a precursor solution with a syringe pump, assisted by cetyltrimethylammonium chloride and bromide at appropriate concentrations, was found to be critical to the formation of Au tetrahedra in high purity. Their sizes could be readily tuned in the range of 30–60 nm by simply varying the amount of precursor added to the reaction solution. The current strategy not only enables the synthesis of Au tetrahedra with tunable and controlled sizes but also provides a facile and versatile approach to reducing the symmetry of nanocrystals made of a face‐centered cubic lattice.     

O6-Alkylguanine DNA Alkyltransferase Mediated Disassembly of a DNA Tetrahedron

Tetrahedron DNA structures were formed by the assembly of three-way junction ( TWJ ) oligonucleotides containing O⁶-2′-deoxyguanosine-alkylene-O⁶-2′-deoxyguanosine (butylene and heptylene linked) intrastrand cross-links (IaCLs) lacking a phosphodiester group between the 2′-deoxyribose residues. The DNA tetrahedra containing TWJs were shown to undergo an unhooking reaction by the human DNA repair protein O⁶-alkylguanine DNA alkyltransferase (hAGT) resulting in structure disassembly. The unhooking reaction of hAGT towards the DNA tetrahedra was observed to be moderate to virtually complete depending on the protein equivalents. DNA tetrahedron structures have been explored as drug delivery platforms that release their payload in response to triggers, such as light, chemical agents or hybridization of release strands. The dismantling of DNA tetrahedron structures by a DNA repair protein contributes to the armamentarium of approaches for drug release employing DNA nanostructures.     

Some useful integer arrays for the teaching of generalization skills

Teachers of first-year college mathematics and engineering courses must often spend considerable time reviewing material originally taught in high school. Instead of this being a mere exercise in repetition, this article suggests that such a review can enrich and revitalize by unifying some of the subjects that need to be re-taught. In the example presented, the subjects in question are absolute values, graphs and solutions of equations, and domains of definition. These are unified by the problem of finding an analytic expression for a square and triangle and their interiors. In the course of the development, basic notions such as the additive property of areas and convexity are introduced. The approach presented in the article was tried with secondary school teachers participating in professional development workshops and with students at a technical college; the teachers and students responded enthusiastically to the material.     

柴胡舒肝丸的毛细管电泳指纹图谱及其黄芩苷含量的测定

建立了柴胡舒肝丸(Chaihu Shugan Pill, CHSGP)的毛细管区带电泳指纹图谱(capillary electrophoresis fingerprint, CEFP),并采用内标法测定了黄芩苷的含量。以50 mmol/L硼砂-150 mmol/L磷酸二氢钠-50 mmol/L磷酸氢二钠(1:1:1, v/v/v)(含5 mmol/L庚烷磺酸钠)为背景电解质(BGE)溶液,采用未涂层石英毛细管(总长度75 cm,有效分离长度63 cm,内径75 μm),以色谱指纹图谱分离量指数(RF)为目标函数优化实验条件,在紫外检测波长265 nm、运行电压11 kV条件下,以黄芩苷峰为参照物峰,确定了22个共有指纹峰,建立了CHSGP的CEFP,通过对20批样品聚类分析确定用其中13批生成对照CEFP(RCEFP),以此RCEFP为标准用系统指纹定量法鉴别20批柴胡舒肝丸质量。结果其中的4批化学成分数量和分布比例不合格,4批含量明显偏低,其他12批完全合格。采用内标法测定黄芩苷的含量,在5～200 mg/L范围内线性良好(r=0.9999),平均回收率(n=9)为98.2%。该法具有较好的精密度和重现性,为柴胡舒肝丸的质量控制提供了一种新的参考。     

On a Construction of Fast Direct Solvers

Fast direct solvers for the Poisson equation with homogeneous Dirichlet and Neumann boundary conditions on special triangles and tetrahedra are constructed. The domain given is extended by symmetrization or skew symmetrization onto a rectangle or a rectangular parallelepiped and a fast direct solver is used there. All extendable domains are found. Eigenproblems are also considered.     

正四面体生成的一般Sierpinski块的Hausdorff维数与Hausdorff测度的估计

该文引入正四面体生成的一般Sierpinski块Er(0＜r≤0.5)的概念及其构造.通过求出Er计盒维数得到其Hausdorff维数,并得到了它们的Hausdorff测度的较好估计,其主要结果改进了现有文献的相关结果.     

Nonobtuse Tetrahedral Partitions that Refine Locally Towards Fichera-Like Corners

Linear tetrahedral finite elements whose dihedral angles are all nonobtuse guarantee the validity of the discrete maximum principle for a wide class of second order elliptic and parabolic problems. In this paper we present an algorithm which generates nonobtuse face-to-face tetrahedral partitions that refine locally towards a given Fichera-like corner of a particular polyhedral domain. The first author was supported by the Swedish Foundation for Strategic Research, the second author was supported by Grant No. 49051 of the Academy of Finland, the third author was supported by Grant No. A 1019201 of the Academy of Sciences of the Czech Republic and by Institutional Research Plan AV0Z 10190503.