$H(2)$-Unknotting Number of a Knot

An $H(2)$-move is a local move of a knot which is performed by adding a half-twisted band. It is known an $H(2)$-move is an unknotting operation. We define the $H(2)$-unknotting number of a knot $K$ to be the minimum number of $H(2)$-moves needed to transform K into a trivial knot. We give several methods to estimate the $H(2)$-unknotting number of a knot. Then we give tables of $H(2)$-unknotting numbers of knots with up to 9 crossings.     

H（2）-unknotting Number of a Knot

An H（2）-move is a local move of a knot which is performed by adding a half-twisted band. It is known an H（2）-move is an unknotting operation. We define the H（2）-unknotting number of a knot K to be the minimum number of H（2）-moves needed to transform K into a trivial knot. We give several methods to estimate the H（2）-unknotting number of a knot. Then we give tables of H（2）-unknotting numbers of knots with up to 9 crossings.     

Computing Canonical Polygonal Schemata with Generalized Maps

This paper shows that a well-known algorithm proposed to compute the canonical polygonal schema of a surface can be transferred onto a 2-dimensional generalized map. We show that transformation rules on polygonal schemata can be achieved in O(1) with generalized maps, which can help optimizing existing algorithms.     

多边形链图中的完美匹配数

多边形链图的完美匹配数（即多边形碳氢链状聚合物的Ｋｅｋｕｌｅ结构数）是数学化学研究的重要内容之一。我们给出了一个求该数的简洁算法，并证明该数是一个多项式。做为应用，对于一类特殊的多边形链图，给出了具体的表达式。     

Cauchy多边形数定理的一个简短证明

本文对下述事实给出一个简单的证明:每个自然数是m+2个m+2边形数之和. 设m≥1,一个m+2边形数是形如 Pm(k)=m/2(k2-k)+k,(k=0,1,2,…)的数.Fermat[3]断言:每一个自然数是m+2个m+2边形数之和.对于m=2,Lagrange[5]证明了每一个自然数是4个平方数P2(k)=k2之和.对于m=1,Gauss [4]证明了每一个自然数是3个三角数P1(k)=1/2(k2+k)之和,或等价的,每一个满足n≡3(mod 8)的正整数n都是3个奇数平方之和,Cauchy[1]对所有的m≥3证明了Fermat的断言,Legendre[6]进一步细化和推广了这一结果.对于m≥3且n≤120m,Pepin [8]给出了将n写成m+2个m+2边形数之和的显示表达的表,其中至少有m-2个取值于0或1.     

区间数判断矩阵的一致性及权重计算

定义区间数判断矩阵的一致性可接受等概念,利用一组约束等式提取原区间数判断矩阵的全部一致性信息,并在此基础上建立非线性规划模型,提出一种新的权重计算方法,最后以算例说明其运用。     

Genus of the Torus Knot

In this paper,we give a metheod to obtain genus of the torus knot by some regular projection of the torus knot and have proved that the penus of the torus knot of type p,q is(p-1)(q-1).     

Polygonal heat conductors with a stationary hot spot

We consider a convex polygonal heat conductor whose inscribed circle touches every side of the conductor. Initially, the conductor has constant temperature and, at every time, the temperature of its boundary is kept at zero. The hot spot is the point at which temperature attains its maximum at each given time. It is proved that, if the hot spot is stationary, then the conductor must satisfy two geometric conditions. In particular, we prove that these geometric conditions yield some symmetries provided the conductor is either pentagonal or hexagonal. This research was partially supported by Grants-in-Aid for Scientific Research (B) (# 12440042) and (B) (# 15340047) of Japan Society for the Promotion of Science, and by a Grant of the Italian MURST.     

Polygonal invariant curves for a planar piecewise isometry

We investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones. For this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components. The union of all periodic components is an invariant pentagon with unequal sides. Transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange. The restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region. The proofs are based on exact calculations in a rational cyclotomic field. We use the system to shed some light on a conjecture that PWIs can possess transitive invariant curves that are not smooth.

     

Polygonal approximation of flows

The analysis of the qualitative behavior of flows generated by ordinary differential equations often requires quantitative information beyond numerical simulation which can be difficult to obtain analytically. In this paper we present a computational scheme designed to capture qualitative information using ideas from the Conley index theory. Specifically we design an combinatorial multivalued approximation from a simplicial decomposition of the phase space, which can be used to extract isolating blocks for isolated invariant sets. These isolating blocks can be computed rigorously to provide computer-assisted proofs. We also obtain local conditions on the underlying simplicial approximation that guarantees that the chain recurrent set can be well-approximated.     

Polygonal chain approximation: a query based approach

In this paper we present a new, query based approach for approximating polygonal chains in the plane. We give a few results based on this approach, some of more general interest, and propose a greedy heuristic to speed up the computation. Our algorithms are simple, based on standard geometric operations, and thus suitable for efficient implementation. We also show that the query based approach can be used to obtain a subquadratic time exact algorithm with infinite beam criterion and Euclidean distance metric if some condition on the input path holds. Although in a special case, this is the first subquadratic result for path approximation with Euclidean distance metric.     

Polygonal smoothing of the empirical distribution function

We present two families of polygonal estimators of the distribution function: the first family is based on the knowledge of the support while the second addresses the case of an unknown support. Polygonal smoothing is a simple and natural method for regularizing the empirical distribution function \(F_n\) but its properties have not been studied deeply. First, consistency and exponential type inequalities are derived from well-known convergence properties of \(F_n\). Then, we study their mean integrated squared error (MISE) and we establish that polygonal estimators may improve the MISE of \(F_n\). We conclude by some numerical results to compare these estimators globally, and also together with the integrated kernel distribution estimator.     

Asymptotics for the Heat Content of a Planar Region with a Fractal Polygonal Boundary

Let k 3 be an integer. For 0<s<1, let D_s R² be the setthat is constructed iteratively as follows. Take a regular openk-gon with sides of unit length, attach regular open k-gonswith sides of length s to the middles of the edges, and so on.At each stage of the iteration the k-gons that are added area factor s smaller than the previous generation and are attachedto the outer edges of the family grown so far. The set D_s isdefined to be the interior of the closure of the union of allthe k-gons. It is easy to see that there must exist some s_k> 0 such that no k-gons overlap if and only if 0 < s s_k. We derive an explicit formula for s_k. The set D_s is open, bounded, connected and has a fractal polygonalboundary. Let denote the heat content of D_s at time t when D_s initially has temperature 0and D_s is kept at temperature 1. We derive the complete short-timeexpansion of up to terms that are exponentially small in 1/t. It turns out that there arethree regimes, corresponding to 0<s<1/(k–1), s=1/(k–1),and 1/(k–1)<s s_k. For s 1/(k–1) the expansionhas the form where p_s is a log (1/s²)-periodic function, d_s=log (k–1)/log(1/s) is a similarity dimension, A_s and B are constants relatedto the edges and vertices, respectively, of D_s, and r_s is anerror exponent. For s=1/(k–1), the t^1/2-term carries anadditional log t. 1991 Mathematics Subject Classification: 11D25,11G05, 14G05.     

Motion Planning for a Convex Polygon in a Polygonal Environment

We study the motion-planning problem for a convex m -gon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the three-dimensional space of all free placements of P in Q ) in time that is near-quadratic in mn , which is nearly optimal in the worst case. The algorithm is also conceptually simple. Previous solutions were incomplete, more expensive, or produced only part of the free configuration space. Combining our solution with parametric searching, we obtain an algorithm that finds the largest placement of P in Q in time that is also near-quadratic in mn . In addition, we describe an algorithm that preprocesses the computed free configuration space so that reachability queries can be answered in polylogarithmic time. Received September 9, 1997, and in revised form September 24, 1998.     

The Canonical Genus of a Classical and Virtual Knot

A diagram D of a knot defines the corresponding Gauss Diagram G _D . However, not all Gauss diagrams correspond to the ordinary knot diagrams. From a Gauss diagram G we construct closed surfaces F _G and S _G in two different ways, and we show that if the Gauss diagram corresponds to an ordinary knot diagram D, then their genus is the genus of the canonical Seifert surface associated to D. Using these constructions we introduce the virtual canonical genus invariant of a virtual knot and find estimates on the number of alternating knots of given genus and given crossing number.     

Knot modules and the Nakanishi index

We examine the structure of the knot module of and show that the Nakanishi index of this knot is 2. The Nakanishi indices of and are also determined by means of the Fox-Smythe row class. Finally, we point out that the Nakanishi index is not additive over knot composition.

     

Polygonal interface problems for the biharmonic operator

We study some boundary value problems on two-dimensional polygonal topological networks, where on each face, the considered operator is the biharmonic operator. The transmission conditions we impose along the edges are inspired by the models introduced by H. Le Dret [13] and Destuynder and Nevers [9]. The boundary conditions on the external edges are the classical ones. This class of problem contains the boundary value problems for the biharmonic equation in a plane polygon (see [3, 11, 12, 18]). Conforming to the classical results cited above, we prove that the weak solution of our problem admits a decomposition into a regular part and a singular part, the latter being a linear combination of singular functions depending on the domain and the considered boundary value problem. Finally, we give the exact formula for the coefficients of these singularities.