Selective Construction of Trefoil knots and a Molecular Borromean Ring Induced by Steric Hindrance of Thioether Ligands

Two Cp*−Rh^III based trefoil knots were obtained in high yield under ambient conditions via the coordination-driven self-assembly of semi-rigid thioether dipyridyl ligand 1,4-bis[(pyridin-4-ylthio)methyl]benzene ( L₁ ), ligand chloranilic acid (H₂− CA ) and 6,11-dihydroxytetracene-5,12-dione (H₂- TtDo ) with Cp*Rh^III metal corner units, respectively. Furthermore, using the bulkier 4,4′-{[(2,5-dimethyl-1,4-phenylene)bis(methylene)]bis(sulfanediyl)}dipyridine ( L₂ ) in the place of ligand L₁ in the construction process resulted in the formation of a teranuclear metallacycle and a template-free Borromean ring in high yields thanks to significantly altered intermolecular forces between the constituent ligands induced by the sterically-hindering methyl groups of L₂ , as demonstrated via a detailed X-ray crystallographic analysis and NMR spectroscopy.     

基本解方法与边界节点法求解Helmholtz方程的比较研究

基本解方法和边界节点法是基于径向基函数的两种重要无网格边界离散数值技术。针对Helmholtz方程,本文比较研究这两种数值方法在不同计算区域问题上的计算精度、插值矩阵对称性、病态性及计算成本。数值试验结果表明,两种方法都可以有效求解边界数据准确的Helmholtz问题。在数值离散过程中,两种方法都可以通过调整配置点的位...     

Adaptive Surface Reconstruction Based on Tensor Product Algebraic Splines

Surface reconstruction from unorganized data points is a challenging problem in Computer Aided Design and Geometric Modeling. In this paper, we extend the mathematical model proposed by Juttler and Felis (Adv. Comput. Math., 17 (2002), pp. 135-152) based on tensor product algebraic spline surfaces from fixed meshes to adaptive meshes. We start with a tensor product algebraic B-spline surface defined on an initial mesh to fit the given data based on an optimization approach. By measuring the fitting errors over each cell of the mesh, we recursively insert new knots in cells over which the errors are larger than some given threshold, and construct a new algebraic spline surface to better fit the given data locally. The algorithm terminates when the error over each cell is less than the threshold. We provide some examples to demonstrate our algorithm and compare it with Jiittler's method. Examples suggest that our method is effective and is able to produce reconstruction surfaces of high quality.AMS subject classifications: 65D17     

On Fundamental Group of a Certain Class of Welded Knots

In this paper, a certain class of welded knots K_(2n) is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K_(2n),n ∈ Z~+, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.     

One‐step boundary knot method for discontinuous coefficient elliptic equations with interface jump conditions

This study makes the first attempt to apply the boundary knot method (BKM), a meshless collocation method, to the solution of linear elliptic problems with discontinuous coefficients, also known as the elliptic interface problems. The additional jump conditions are usually required to be prescribed at the interface which is used to maintain the well‐posedness of the considered problem. To solve the problem efficiently, the original governing equation is first transformed into an equivalent inhomogeneous modified Helmholtz equation in the present numerical formulation. Then the computational domain is divided into several subdomains, and the solution on each subdomain is approximated using the BKM approach. Unlike the conventional two‐step BKM, this study presents a one‐step BKM formulation which possesses merely one influence matrix for inhomogeneous problems. Several benchmark examples with various discontinuous coefficients have been tested to demonstrate the accuracy and efficiency of the present BKM scheme. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1509–1534, 2016     

Excluded Minors and the Ribbon Graphs of Knots

In this article we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus, the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterizations of minor closed families of ribbon graphs. Our main result is an excluded minor characterization of the family of ribbon graphs that represent knot and link diagrams.     

微液滴在仿蛛丝结构TiO2纤维表面的定向运动与粘附

制备了具有响应性的仿蛛丝周期纺锤节TiO₂纤维。通过改变纤维表面微观结构、紫外光照和超声波等手段,仿蛛丝结构TiO₂纤维表面浸润性会发生响应性变化。纤维表面的这种浸润性变化,不仅实现了水滴从纺锤节两端到中心处的定向运动,而且可以使得水滴被纤维纺锤节粘附。当仿蛛丝结构TiO₂纤维表面为亲水状态时,无论纺锤节光滑还是粗糙,水都是从纺锤节两端向中心方向运动;当仿蛛丝TiO₂纤维表面疏水时,水滴会被具有粗糙表面的纺锤节两端粘附。     

Influences of flexible defect on the interplay of supercoiling and knotting of circular DNA

Knots are discovered in biophysical systems, such as DNA and proteins. Knotted portions in knotted DNA are significantly bent and their corresponding bending angles are comparable with or larger than the sharp bending angle resulting in flexible defects. The role of flexible defects in the interplay of supercoiling and knotting of circular DNA were predicted by a Monte Carlo simulation. In knotted DNA with a particular knot type, a flexible defect noticeably enhances the supercoiling of the knotted DNA and the decreasing excitation energy makes the knotted portion more compact. A reduction in twist rigidity and unwinding of flexible defects are incorporated into the numerical simulations, so that interplay of supercoiling and knotting of circular DNA is studied under torsional conditions. Increasing unwinding not only results in a wider linking number distribution, but also leads to a drift of the distribution to lower values. A flexible defect has obvious effects on knotting probability. The summation of equilibrium distribution probability for nontrivial knotted DNA with different contour length does not change with excitation energy monotonically and has a maximum at an intermediate value of excitation energy around 5 kBT. In the phase space of knot length and gyration radius of knotted DNA, knot length does not anticorrelate with its gyration radius, which is attributed to the flexible defect in the knotted portion, which leads to the release of bending energy and inhibited the competition between entropy and bending energy.     

Knotted pictures of the GHZ states on the surface of a trivial torus

By means of the torus knot theory method, this paper presents the complete process of obtaining the knotted pictures of eight GHZ states on the surface of a trivial torus from the knotted pictures of eight basic three-qubit states on the surface of a trivial torus. Thus, we obtain eight knotted pictures 121 linkage on the ordinary plane.