Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest.     

Proper minimal points of nonconvex sets in Banach spaces in terms of the limiting normal cone

In this paper, proper minimal elements of a given nonconvex set in a real ordered Banach space are defined utilizing the limiting (Mordukhovich) normal cone. The newly defined points are called limiting proper minimal (LPM) points. It is proved that each LPM is a proper minimal in the sense of Borwein under some assumptions. The converse holds in Asplund spaces. The relation of LPM points with Benson, Henig, super and proximal proper minimal points are established. Under appropriate assumptions, it is proved that the set of robust elements is a subset of the set of LPM points, and the set of LPM points is dense in that of minimal points. Another part of the paper is devoted to scalarization-based and distance function-based characterizations of the LPM points. The paper is closed by some results about LPM solutions of a set-valued optimization problem via variational analysis tools. Clarifying examples are given in addition to the theoretical results.     

Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest. Received 19 June 2001; in revised form 17 January 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

A branch-and-cut-and-price approach for the pickup and delivery problem with shuttle routes

The Pickup and Delivery Problem with Shuttle routes (PDPS) is a special case of the Pickup and Delivery Problem with Time Windows (PDPTW) where the trips between the pickup points and the delivery points can be decomposed into two legs. The first leg visits only pickup points and ends at some delivery point. The second leg is a direct trip – called a shuttle – between two delivery points. This optimization problem has practical applications in the transportation of people between a large set of pickup points and a restricted set of delivery points.     

平面C-Bézier曲线的奇拐点分析

本文完全地讨论了平面C-曲线和平面C-Bezier曲线的奇拐点和凸性性质:曲线段为且必为下列情形之一:有一各拐点,两个拐点,一个尖点,一个二重结点,处处为凸;并给出了相应的用控制多边形相对位置表示的充分必要条件.     

Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve

Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterize those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only periodic points are the Weierstraß points and the singularities. Our main tool is the Hodge-theoretic characterization of Teichmüller curves. We deduce from it a finiteness result for the Mordell-Weil group of the family of Jacobians over a Teichmüller curve. The link to the classification of periodic points is provided by interpreting them as sections of the family of curves over a covering of the Teichmüller curve.     

Converting triangulations to quadrangulations

We study the problem of converting triangulated domains to quadrangulations, under a variety of constraints. We obtain a variety of characterizations for when a triangulation (of some structure such as a polygon, set of points, line segments or planar subdivision) admits a quadrangulation without the use of Steiner points, or with a bounded number of Steiner points. We also investigate the effect of demanding that the Steiner points be added in the interior or exterior of a triangulated simple polygon and propose efficient algorithms for accomplishing these tasks. For example, we give a linear-time method that quadrangulates a triangulated simple polygon with the minimum number of outer Steiner points required for that triangulation. We show that this minimum can be at most n/3, and that there exist polygons that require this many such Steiner points. We also show that a triangulated simple n-gon may be quadrangulated with at most n/4 Steiner points inside the polygon and at most one outside. This algorithm also allows us to obtain, in linear time, quadrangulations from general triangulated domains (such as triangulations of polygons with holes, a set of points or line segments) with a bounded number of Steiner points.     

Isodistant points in competitive network facility location

An isodistant point is any point on a network which is located at a predetermined distance from some node. For some competitive facility location problems on a network, it is verified that optimal (or near-optimal) locations are found in the set of nodes and isodistant points (or points in the vicinity of isodistant points). While the nodes are known, the isodistant points have to be determined for each problem. Surprisingly, no algorithm has been proposed to generate the isodistant points on a network. In this paper, we present a variety of such problems and propose an algorithm to find all isodistant points for given threshold distances associated with the nodes. The number of isodistant points is upper bounded by nm, where n and m are the number of nodes and the number of edges, respectively. Computational experiments are presented which show that isodistant points can be generated in short run time and the number of such points is much smaller than nm. Thus, for networks of moderate size, it is possible to find optimal (or near-optimal) solutions through the Integer Linear Programming formulations corresponding to the discrete version of such problems, in which a finite set of points are taken as location candidates.     

射影极小曲面的杜慕兰变换(Ⅳ)

<正> 在论文(Ⅰ)里曾经提到一个射影极小曲面容有这样的二线(?)W,每个是以原曲面和它的一个第二 D 变换曲面为其二焦曲面的.本文的目的在于阐明:在直线空间 S_5里,每个线(?)W 的对应的拉勃拉斯叙列内接于原曲面8和其—D变换曲面(?)的戈德叙列     

Triangular Bézier sub-surfaces on a triangular Bézier surface

This paper considers the problem of computing the Bézier representation for a triangular sub-patch on a triangular Bézier surface. The triangular sub-patch is defined as a composition of the triangular surface and a domain surface that is also a triangular Bézier patch. Based on de Casteljau recursions and shifting operators, previous methods express the control points of the triangular sub-patch as linear combinations of the construction points that are constructed from the control points of the triangular Bézier surface. The construction points contain too many redundancies. This paper derives a simple explicit formula that computes the composite triangular sub-patch in terms of the blossoming points that correspond to distinct construction points and then an efficient algorithm is presented to calculate the control points of the sub-patch.     

On stationary solutions of a nonlinear system of reactor dynamic equations with distributed parameters

We analyze a nonlinear stationary model of reactor dynamics with distributed parameters. We find sufficient conditions for the existence of bifurcation points in this system and study the behavior of solutions in a neighborhood of the bifurcation points. We prove the existence of countably many bifurcation points in the case of a homogeneous medium and obtain constructive estimates for the distance between the bifurcation points.     

一种基于二部图的试题知识点标注方法

针对在线教育中试题知识点自动标注问题,本文提出了一种基于二部图的试题知识点标注方法。首先,为了合理划分知识点粒度,本文构建了知识点知识图谱来融合知识点;其次,基于教材等语料抽取知识点与特征词二部图和试题与特征词二部图,并利用TF-IDF公式计算知识点与特征词、试题与特征词之间的边权,以构建试题-知识点二部图模型;再次,提出词频加权的相似性度量方法计算试题和知识点之间的相似度,将相似度最高的知识点作为试题知识点标签。最后,以某在线教育平台提供的高中历史试题为实验数据集进行数值实验,实验结果表明该方法的效果显著优于朴素贝叶斯、K最近邻、随机森林以及支持向量机。     

Efficiency in constrained continuous location

We present a geometrical characterization of the efficient, weakly efficient and strictly efficient points for multi-objective location problems in presence of convex constraints and when distances are measured by an arbitrary norm. These results, established for a compact set of demand points, generalize similar characterizations previously obtained for uncontrained problems. They are used to show that, in planar problems, the set of constrained weakly efficient points always coincides with the closest projection of the set of unconstrained weakly efficient points onto the feasible set. This projection property which are known previously only for strictly convex norms, allows to easily construct all the weakly efficient points and provides a useful localization property for efficient and strictly efficient points.     

Geodesic Ham-Sandwich Cuts

Let P be a simple polygon with m vertices, k of which are reflex, and which contains r red points and b blue points in its interior. Let n = m + r + b. A ham-sandwich geodesic is a shortest path in P between two points on the boundary of P that simultaneously bisects the red points and the blue points. We present an O(n log k)-time algorithm for finding a ham-sandwich geodesic. We also show that this algorithm is optimal in the algebraic computation tree model when parameterizing the running time with respect to n and k.     

On component-size bounded Steiner trees

A Steiner tree is a tree interconnecting a given set of points in a metric space such that all leaves are given points. A (full) component of a Steiner tree is a subtree which results from splitting the Steiner tree at some given points. A k-size Steiner tree is a Steiner tree in which every component has at most k given points. The k-Steiner ratio is the largest lower bound for the ratio between lengths of a minimum Steiner tree and a minimum k-size Steiner tree for the same set of points. In this paper, we determine the 3-Steiner ratio in weighted graphs.     

There Are Integral Heptagons, no Three Points on a Line, no Four on a Circle

We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erdős.     

An improved sectional-Anosov closing lemma

We prove that all nonwandering points of a sectional-Anosov flow on a compact 3-manifold can be approximated by periodic points or by points for which the omega-limit set is a singularity. This improves the closing lemma in Morales (Mich. Math. J. 56(1):29?C53, 2008). We also describe a sectional-Anosov flow for which the recurrent points are not dense in the nonwandering set.     

Stochastic processes with imbedded marked point processes (pmp) and thcir application in queneing

In generalization of special cases of the literature a class of stochastic processes (PMP) is defined with an imbedded stochastic marked point process of “basic points” which must not be renewal points. A theorem (“intensity conservation principle”) has been proved concerning a relation between stationary distribution of PMP at arbitrary points in time and distributions and intensities connected with the basic points. This relationship simultaneously yields a general method for determination of stationary quantities at arbitrary points in time by means of the corresponding “imbedded” quantities. Some applications to concrete queueing systems have been demonstrated, where arrival or departure epochs of customers are used as basic points. Under weaker independence assumptions as till now done in the literature, new relations are given.     

Newton interpolation at Leja points

The Newton form is a convenient representation for interpolation polynomials. Its sensitivity to perturbations depends on the distribution and ordering of the interpolation points. The present paper bounds the growth of the condition number of the Newton form when the interpolation points are Leja points for fairly general compact sets K in the complex plane. Because the Leja points are defined recursively, they are attractive to use with the Newton form. If K is an interval, then the Leja points are distributed roughly like Chebyshev points. Our investigation of the Newton form defined by interpolation at Leja points suggests an ordering scheme for arbitrary interpolation points.Research supported in part by NSF under Grant DMS-8704196 and by U.S. Air Force Grant AFSOR-87-0102.On leave from University of Kentucky, Department of Mathematics, Lexington, KY 40506, U.S.A.     

Triangulation sous contraintes en dimension quelconque

It is not always possible to insert a facet of any dimension in a triangulation of a finite set of points in R^d for d > 2 without the adjunction of Steiner points. We propose here a simple method consisting in the inclusion of a triangulation of the constrained facet by adjunction of points upon it. These points are defined by using the intersections of the constrained facet with the simplices of the given triangulation.