Edge-pancyclicity of Möbius cubes

The Möbius cube Mn is a variant of the hypercube Qn and has better properties than Qn with the same number of links and processors. It has been shown by Fan [J. Fan, Hamilton-connectivity and cycle-embedding of Möbius cubes, Inform. Process. Lett. 82 (2002) 113-117] and Huang et al. [W.-T. Huang, W.-K. Chen, C.-H. Chen, Pancyclicity of Möbius cubes, in: Proc. 9th Internat. Conf. on Parallel and Distributed Systems (ICPADS'02), 17-20 Dec. 2002, pp. 591-596], independently, that Mn contains a cycle of every length from 4 to n². In this paper, we improve this result by showing that every edge of Mn lies on a cycle of every length from 4 to n² inclusive.     

Hamilton-connectivity and cycle-embedding of the Möbius cubes

The recently introduced interconnection network, the Möbius cube, is an important variant of the hypercube. This network has several attractive properties compared with the hypercube. In this paper, we show that the n-dimensional Möbius cube M_n is Hamilton-connected when n?3. Then, by using the Hamilton-connectivity of M_n, we also show that any cycle of length l (4?l?2ⁿ) can be embedded into M_n with dilation 1 (n?2). It is a fact that the n-dimensional hypercube Q_n does not possess these two properties.     

Apollonius tenth problem via radius adjustment and Möbius transformations

The Apollonius Tenth Problem, as defined by Apollonius of Perga circa 200 B.C., has been useful for various applications in addition to its theoretical interest. Even though particular cases have been handled previously, a general framework for the problem has never been reported. Presented in this paper is a theory to handle the Apollonius Tenth Problem by characterizing the spatial relationship among given circles and the desired Apollonius circles. Hence, the given three circles in this paper do not make any assumption regarding on the sizes of circles and the intersection/inclusion relationship among them. The observations made provide an easy-to-code algorithm to compute any desired Apollonius circle which is computationally efficient and robust.     

Constructing the nearly shortest path in crossed cubes

A graph G is panconnected if each pair of distinct vertices u,v∈V(G) are joined by a path of length l for all d_G(u,v)?l?|V(G)|-1, where d_G(u,v) is the length of a shortest path joining u and v in G. Recently, Fan et. al. [J. Fan, X. Lin, X. Jia, Optimal path embedding in crossed cubes, IEEE Trans. Parall. Distrib. Syst. 16 (2) (2005) 1190-1200, J. Fan, X. Jia, X. Lin, Complete path embeddings in crossed cubes, Inform. Sci. 176 (22) (2006) 3332-3346] and Xu et. al. [J.M. Xu, M.J. Ma, M. Lu, Paths in Möbius cubes and crossed cubes, Inform. Proc. Lett. 97 (3) (2006) 94-97] both proved that n-dimensional crossed cube, CQ_n, is almost panconnected except the path of length d_CQn(u,v)+1 for any two distinct vertices u,v∈V(CQ_n). In this paper, we give a necessary and sufficient condition to check for the existence of paths of length d_CQn(u,v)+1, called the nearly shortest paths, for any two distinct vertices u,v in CQ_n. Moreover, we observe that only some pair of vertices have no nearly shortest path and we give a construction scheme for the nearly shortest path if it exists.     

Conditional fault diameter of crossed cubes

The conditional connectivity and the conditional fault diameter of a crossed cube are studied in this work. The conditional connectivity is the connectivity of an interconnection network with conditional faults, where each node has at least one fault-free neighbor. Based on this requirement, the conditional connectivity of a crossed cube is shown to be 2n−2$2n-2$. Extending this result, the conditional fault diameter of a crossed cube is also shown to be D(C_Qn)+3$D\left(C{Q}_n\right)+3$ as a set of 2n−3$2n-3$ node failures. This indicates that the conditional fault diameter of a crossed cube is increased by three compared to the fault-free diameter of a crossed cube. The conditional fault diameter of a crossed cube is approximately half that of the hypercube. In this respect, the crossed cube is superior to the hypercube.     

Node-pancyclicity and edge-pancyclicity of crossed cubes

Crossed cubes are important variants of the hypercubes. It has been proven that crossed cubes have attractive properties in Hamiltonian connectivity and pancyclicity. In this paper, we study two stronger features of crossed cubes. We prove that the n-dimensional crossed cube is not only node-pancyclic but also edge-pancyclic for n?2. We also show that the similar property holds for hypercubes.     

Complete path embeddings in crossed cubes

Crossed cubes are popular variants of hypercubes. In this paper, we study path embeddings between any two distinct nodes in crossed cubes. We prove two important results in the n-dimensional crossed cube: (a) for any two nodes, all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1; (b) for any two integers n ? 2 and l with , there always exist two nodes x and y whose distance is l, such that no path of length l + 1 can be embedded between x and y with dilation 1. The obtained results are optimal in the sense that the dilations of path embeddings are all 1. The results are also complete, because the embeddings of paths of all possible lengths between any two nodes are considered.     

Edge-pancyclicity and path-embeddability of bijective connection graphs

An n-dimensional Bijective Connection graph (in brief BC graph) is a regular graph with 2ⁿ nodes and n2ⁿ⁻¹ edges. The n-dimensional hypercube, crossed cube, Möbius cube, etc. are some examples of the n-dimensional BC graphs. In this paper, we propose a general method to study the edge-pancyclicity and path-embeddability of the BC graphs. First, we prove that a path of length l with dist(X_n, x, y) + 2 ? l ? 2ⁿ − 1 can be embedded between x and y with dilation 1 in X_n for x, y ∈ V(X_n) with x ≠ y in X_n, where X_n (n ? 4) is a n-dimensional BC graph satisfying the three specific conditions and V(X_n) is the node set of X_n. Furthermore, by this result, we can claim that X_n is edge-pancyclic. Lastly, we show that these results can be applied to not only crossed cubes and Möbius cubes, but also other BC graphs except crossed cubes and Möbius cubes. So far, the research on edge-pancyclicity and path-embeddability has been limited in some specific interconnection architectures such as crossed cubes, Möbius cubes.     

Node-pancyclicity and edge-pancyclicity of hypercube variants

Twisted cubes, crossed cubes, Möbius cubes, and locally twisted cubes are some of the widely studied hypercube variants. The 4-pancyclicity of twisted cubes, crossed cubes, Möbius cubes, locally twisted cubes and the 4-edge-pancyclicity of twisted cubes, crossed cubes, Möbius cubes are proven in [C.P. Chang, J.N. Wang, L.H. Hsu, Topological properties of twisted cube, Inform. Sci. 113 (1999) 147-167; C.P. Chang, T.Y. Sung, L.H. Hsu, Edge congestion and topological properties of crossed cubes, IEEE Trans. Parall. Distr. 11 (1) (2000) 64-80; J. Fan, Hamilton-connectivity and cycle embedding of the Möbius cubes, Inform. Process. Lett. 82 (2002) 113-117; X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Appl. Math. Lett. 17 (2004) 919-925; J. Fan, N. Yu, X. Jia, X. Lin, Embedding of cycles in twisted cubes with edge-pancyclic, Algorithmica, submitted for publication; J. Fan, X. Lin, X. Jia, Node-pancyclic and edge-pancyclic of crossed cubes, Inform. Process. Lett. 93 (2005) 133-138; M. Xu, J.M. Xu, Edge-pancyclicity of Möbius cubes, Inform. Process. Lett. 96 (2005) 136-140], respectively. It should be noted that 4-edge-pancyclicity implies 4-node-pancyclicity which further implies 4-pancyclicity. In this paper, we outline an approach to prove the 4-edge-pancyclicity of some hypercube variants and we prove in particular that Möbius cubes and locally twisted cubes are 4-edge-pancyclic.     

Fault-free Hamiltonian cycles in crossed cubes with conditional link faults

The crossed cube, which is a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, assuming that each node is incident with at least two fault-free links, we show that an n-dimensional crossed cube contains a fault-free Hamiltonian cycle, even if there are up to 2n − 5 link faults. The result is optimal with respect to the number of link faults tolerated. We also verify that the assumption is practically meaningful by evaluating its occurrence probability, which is very close to 1.     

Möbius变换下四次有理抛物-PH曲线的C2 Hermite插值

目的 曲线插值问题在机器人设计、机械工业、航天工业等诸多现代工业领域都有广泛的应用，而已知端点数据的Hermite插值是计算机辅助几何设计中一种常用的曲线构造方法，本文讨论了一种偶数次有理等距曲线，即四次抛物-PH曲线的C² Hermite插值问题。方法 基于M bius变换引入参数，利用复分析的方法构造了四次有理抛物-PH曲线的C² Hermite插值，给出了具体插值算法及相应的Bézier曲线表示和控制顶点的表达式。结果 通过给出"合理"的端点插值数据，以数值实例表明了该算法的有效性，所得12条插值曲线中，结合最小绝对旋转数和弹性弯曲能量最小化两种准则给出了判定满足插值条件最优曲线的选择方法，并以具体实例说明了与其他插值方法的对比分析结果。结论 本文构造了M bius变换下的四次有理抛物-PH曲线的C² Hermite插值，在保证曲线次数较低的情况下，达到了连续性更高的插值条件，计算更为简单，插值效果明显，较之传统奇数次PH曲线具有更加自然的几何形状，对偶数次PH曲线的相关研究具有一定意义。     

Darboux cyclides and webs from circles

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order ?4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Möbius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.     

A DNA trapping and extraction microchip using periodically crossed electrophoresis in a micropillar array

This paper presents an integrated deoxyribose nucleic acid (DNA) trapping and extraction microchip based on the electrophoresis using periodically crossed electric fields in the micropillar array. The extraction microchip, integrated with a micropillar array, microchannels, nano-gap entropic barriers, loading and unloading windows, has been fabricated by a 3-mask microfabrication process. Using the electric field crossed at 120°, the microchip is designed to trap the DNA molecules, whose reorientation time is longer than the period of the crossed field, within the micropillars distributed at 60° direction. In the fabricated extraction microchip, three different DNA molecules, including λ DNA (48.5 kbp), micrococcus DNA (115 kbp) and T4 DNA (168.9 kbp) show the reorientation times of 4.80 ± 0.44 s, 7.12 ± 0.75 s and 9.71 ± 0.30 s, respectively, at the crossed electric field of 6.25 V/cm. Among three DNA molecules, T4 DNA could not come out of the micropillar array for the electric field of 6.25 V/cm crossed at the period of 10 s. We have demonstrated that the present DNA extraction microchip separates DNA molecules larger than a critical value, which can be adjusted by the period of the electric field across the micropillar array.     

Blind digital watermarking of rational Bézier and B-spline curves and surfaces with robustness against affine transformations and Möbius reparameterization

We present a blind watermarking scheme for rational Bézier and B-spline curves and surfaces which is shape-preserving and robust against the affine transformations and Möbius reparameterization that are commonly used in geometric modeling operations in CAD systems. We construct a watermark polynomial with real coefficients of degree four which has the watermark as the cross-ratio of its complex roots. We then multiply the numerator and denominator of the original curve or surface by this polynomial, increasing its degree by four but preserving its shape. Subsequent affine transformations and Möbius reparameterization leave the cross-ratio of these roots unchanged. The watermark can be extracted by finding all the roots of the numerator and denominator of the curve or surface: the cross-ratio of the four common roots will be the watermark. Experimental results confirm both the shape-preserving property and its robustness against attacks by affine transformations and Möbius reparameterization.     

形式背景下的学习路径与技能评估

在认知学习过程中,学习者可能学习并掌握某些技能,但知识状态却无法发生改变.在此情形下,根据学习者知识状态的改变不足以对其技能进行准确评估,因此,文中基于技能函数,运用形式概念分析的方法寻找学习路径并进行技能评估.首先,介绍后继状态、有效技能和良好技能函数的概念.然后,基于形式背景,在两种情形下讨论技能函数满足良好性的条件,得到满足良好性条件下可进行逐步有效学习和有效评估的结果,并设计获取良好技能背景、良好技能函数及寻找学习路径的算法.最后,在两个数据集上进行实验分析,验证文中算法的有效性,并且得出如下结论:基于良好技能函数得到的学习路径图,不仅可有效指导学习者进行学习,还可根据学习者知识状态的变化评估其是否掌握相应的有效技能.     

Domain wall arrays, fronts, and bright and dark solitons in a generalized derivative nonlinear Schrödinger equation

We obtain some exact solutions of a generalized derivative nonlinear Schrödinger equation, including domain wall arrays (periodic solutions in terms of elliptic functions), fronts, and bright and dark solitons. In certain parameter domains, fundamental bright and dark solitons are chiral, and the propagation direction is determined by the sign of the self-steepening parameter. Moreover, we also find the chirping reversal phenomena of fronts, and bright and dark solitons, and discuss two different ways to produce the chirping reversal.     

Global stability analysis in Cohen-Grossberg neural networks with delays and inverse Hölder neuron activation functions

In this paper, a novel class of Cohen-Grossberg neural networks with delays and inverse Hölder neuron activation functions are presented. By using the topological degree theory and linear matrix inequality (LMI) technique, the existence and uniqueness of equilibrium point for such Cohen-Grossberg neural networks is investigated. By constructing appropriate Lyapunov function, a sufficient condition which ensures the global exponential stability of the equilibrium point is established. Two numerical examples are provided to demonstrate the effectiveness of the theoretical results.