二部图形式的Erd\H{O}s-S\''{o}s猜想

二部图形式的Erd\H{O}s-S\''{o}s猜想     

具有特殊坐标曲线网的B\''ezier曲面

在计算机辅助几何设计中, B\''ezier曲面是一类重要的参数曲面.在微分几何中,坐标曲线网也是重要的研究内容.本文中,我们对具有特殊坐标曲线网(如正交曲线网、曲率曲线网、共轭曲线网等)的B\''ezier曲面进行研究.此外,我们还构造了满足能量约束的特殊B\''ezier曲面,给出了基于控制结构的充分条件并给出具体实例.     

扰动Fej\'{e}r点上复Hermite插值的联合逼近

$D$是复平面中由闭Jordan曲线$\Ga$围成的单连区域. 考虑在$\Ga$上扰动Fej\''er点的 Hermite插值一致逼近、平均逼近和联合逼近于函数$f\in A^{(q)}(\o D)$. 该文中的逼近阶一般说来是不可再改进的, 区域的边界限制条件到目前为止是最少的. 以往的全部同类结果都包括在该文中作为特殊情形, 由于该文方法上的改进, 简化和省去了以往 某些证明过程.     

平面三次Pythagorean-Hodograph Hyperbolic曲线

本文基于Pythagorean-hodograph (PH)曲线和代数双曲线的良好几何特性,构造了Pythagorean-Hodograph Hyperbolic (PH-H)曲线,并给出了PH-H曲线的定义以及相应性质.同时,分别利用Hyperbolic基函数和Algebraic Hyperbolic (AH) B\''ezier基函数,得到了平面三次AH B\''ezier曲线为PH曲线的两个不同的充要条件.此外,三次PH-H曲线也被用于求解具有确定解的$G^1$ Hermite插值问题.文中给出了具体实例来说明我们的方法.     

一类广义Li\'enard方程周期正解的存在性

本文证明一类广义Li\''enard方程周期正解的存在性及渐近稳定性. 我们讨论的非自治函数可满足超线性条件,克服了连续定理讨论超线性条件的困难. 文章的最后,我们给出两个例子和数值解以及对应的相图和时间序列图来验证我们的结论.     

受L\''evy噪声扰动的Logistic方程的全局吸引性

本文考虑了受L\''evy噪声扰动的Logistic方程. 在合适的条件下, 我们得到了解的全局存在性与唯一性; 我们证明了当初始值小于环境的容纳量时, 唯一的正的平衡态具有全局吸引性.     

非齐度量测度空间上Morrey-Herz空间上的Calder\'{o}n-Zygmund算子及其交换子[英文]

设$(\mathcal{X},d,\mu)$ 是一个同时满足上双倍条件 和几何双倍条件的非齐度量测度空间, 本文中, 引进一类非齐度量测度空间上的Morrey-Herz 空间, 利用非齐度量测度空间的特征, 特别是$\eta$-弱逆倍条件, 证明 Calder\''{o}n-Zygmund算子及其交换子在Morrey-Herz空间上的有界性.     

三调和三角B\''ezier曲面设计

偏微分方程曲面设计,是由给定边界条件出发构造满足偏微分方程的曲面.本文基于三调和方程,提出三类边界条件,分别通过求解线性方程组,给出三调和三角形B\''ezier曲面的设计方法.证明了在这些边界条件下,生成曲面的唯一性,并分别给出具体曲面设计算法.通过实例验证了本文结论的有效性,并对三种边界条件进行对比分析.     

完全图中的正常染色的路和圈

令$K_{n}^{c}$表示$n$ 个顶点的边染色完全图.
令 $\Delta^{mon}
(K_{n}^{c})$表示$K^c_{n}$的顶点上关联的同种颜色的边的最大数目.
如果$K_{n}^{c}$中的一个圈（路）上相邻的边染不同颜色,则称它为正常染色的.
B. Bollob\'{a}s和P. Erd\"{o}s (1976) 提出了如下猜想：若 $\Delta^{{mon}}
(K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$, 则$K_{n}^{c}$中含有一个正常染
色的Hamilton圈. 这个猜想至今还未被证明.我们研究了上述条件下的正常染色的路和圈.     

与马猜想有关的一类不定方程

设$p$是奇素数, $b,t,r\in{\rm N}$. 1992 年, 马少麟猜想丢番图方程 $x^2=2^{2b+2}p^{2t}-2^{b+2}p^{t+r}+1$有唯一的正整数解$(x,b,p,t,r)=(49,3,5,1,2)$, 并且证明了这个猜想蕴含McFarland关于乘子为$-1$ 的阿贝尔差集的猜想.在[Ma S L, MaFarland''conjecture on Abelian difference sets with multiplier-1[J]. {\it Designs, Codes and Cryptography,} 1992, 1:321--332.]中, 马少麟证明了: 若$t\geq r$,则丢番图方程$x^2=2^{2b+2}p^{2t}-2^{b+2}p^{t+r}+1$没有正整数解. 本文证明了: 若$a>1$是奇数,$t\geq r$, 那么丢番图方程$x^2=2^{2b+2}a^{2t}-2^{b+2}a^{t+r}+1$的正整数解由$t=r=1, x+a\sqrt{2^{b+2}(2^b-1)}=(2^{b+1}-1+\sqrt{2^{b+2}(2^b-1)})^{n}$给出, 其中$n$为奇数.作者也证明了: 若$p$是奇素数, 则$(x,b,p,t,r)=(7,3,5,1,2)$是丢番图方程$x^4=2^{2b+2}p^{2t}-2^{b+2}p^{t+r}+1$的唯一正整数解.     

图的平面Turán数和平面anti-Ramsey数

在所有顶点数为$n$且不包含图$G$作为子图的平面图中,具有最多边数的图的边数称为图$G$的平面Turán数,记为$ex_{_\mathcal{P}}（n,G）$。给定正整数$n$以及平面图$H$,用$\mathcal{T}_n （H）$来表示所有顶点数为$n$且不包含$H$作为子图的平面三角剖分图所组成的图集合。设图集合$\mathcal{T}_n （H）$中的任意平面三角剖分图的任意$k$边染色都不包含彩虹子图$H$,则称满足上述条件的$k$的最大值为图$H$的平面anti-Ramsey数,记作$ar_{_\mathcal{P}}（n,H）$。两类问题的研究均始于2015年左右,至今已经引起了广泛关注。全面地综述两类问题的主要研究成果,以及一些公开问题。     

完全$r$部图乘积上的Graham猜想

图G的Pebbling数f(G)是最小的正整数n,使得不论n个Pebble如何放置在G的顶点上,总可以通过一系列的Pebbling移动把1个Pebble移到任意一点上,其中Pebbling移动是从一个顶点处移走两个Pebble而把其中一个移到与其相邻的一个顶点上.Graham猜测对于任意的连通图G和H有f(G×H)≤f(G)f(H).本文证明对于一个完全γ部图和一个具有2-Pebbleing性质的图来说,Graham猜想成立.作为一个推论,当G和H均为完全γ部图时,Graham猜想成立.     

4一致反超图的最小边数问题

混合超图是在超图的基础上添加一个反超边得到的图．超边和反超边的区别主要体现在着色要求上．在着色中,要求每一超边至少要有两个点着不同的颜色,而每一反超边至少有两个点着相同的颜色．最大最小颜色数分别称为混合超图的上色数和下色数。本文主要研究反超图,即只含反超边的超图。讨论了上色数为3的4一致超图的最小边数问题．给出了上色数为3的4一致反超图的最小边数的一个上界和一个下界．     

About the upper chromatic number of a co-hypergraph

A mixed hypergraph consists of two families of subsets: the edges and the co-edges. In a coloring every co-edge has at least two vertices of the same color, and every edge has at least two vertices of different colors. The largest and smallest possible number of colors in a coloring is termed the upper and lower chromatic numbers, respectively. In this paper we investigate co-hypergraphs i.e., the hypergraphs with only co-edges, with respect to the property of coloring. The relationship between the lower bound of the size of co-edges and the lower bound of the upper chromatic number is explored. The necessary and sufficient conditions for determining the upper chromatic numbers, of a co-hypergraph are provided. And the bounds of the number of co-edges of some uniform co-hypergraphs with certain upper chromatic numbers are given.     

LIMIT CYCLES OF THE GENERALIZED POLYNOMIAL LINARD DIFFERENTIAL SYSTEMS

Using the averaging theory of first and second order we study the maximum number of limit cycles of generalized Linard differential systems{x = y + εh_l~1(x) + ε~2h_l~2(x),y=-x- ε(f_n~1(x)y~(2p+1) + g_m~1(x)) + ∈~2(f_n~2(x)y~(2p+1) + g_m~2(x)),which bifurcate from the periodic orbits of the linear center x = y,y=-x,where ε is a small parameter.The polynomials h_l~1 and h_l~2 have degree l;f_n~1and f_n~2 have degree n;and g_m~1,g_m~2 have degree m.p ∈ N and[·]denotes the integer part function.     

The Hardness of 3-Uniform Hypergraph Coloring

We prove that coloring a 3-uniform 2-colorable hypergraph with c colors is NP-hard for any constant c. The best known algorithm [20] colors such a graph using O(n^1/5) colors. Our result immediately implies that for any constants k ≥ 3 and c₂ > c₁ > 1, coloring a k-uniform c₁-colorable hypergraph with c₂ colors is NP-hard; the case k = 2, however, remains wide open. This is the first hardness result for approximately-coloring a 3-uniform hypergraph that is colorable with a constant number of colors. For k ≥ 4 such a result has been shown by [14], who also discussed the inherent difference between the k = 3 case and k ≥ 4. Our proof presents a new connection between the Long-Code and the Kneser graph, and relies on the high chromatic numbers of the Kneser graph [19,22] and the Schrijver graph [26]. We prove a certain maximization variant of the Kneser conjecture, namely that any coloring of the Kneser graph by fewer colors than its chromatic number, has ‘many’ non-monochromatic edges. * Research supported by NSF grant CCR-9987845. † Supported by an Alon Fellowship and by NSF grant CCR-9987845. ‡ Work supported in part by NSF grants CCF-9988526 and DMS 9729992, and the State of New Jersery.     

Coloring mixed hypertrees

A mixed hypergraph is a triple (V,C,D) where V is its vertex set and C and D are families of subsets of V, called C-edges and D-edges, respectively. For a proper coloring, we require that each C-edge contains two vertices with the same color and each D-edge contains two vertices with different colors. The feasible set of a mixed hypergraph is the set of all k's for which there exists a proper coloring using exactly k colors. A hypergraph is a hypertree if there exists a tree such that the edges of the hypergraph induce connected subgraphs of the tree.We prove that feasible sets of mixed hypertrees are gap-free, i.e., intervals of integers, and we show that this is not true for precolored mixed hypertrees. The problem to decide whether a mixed hypertree can be colored by k colors is NP-complete in general; we investigate complexity of various restrictions of this problem and we characterize their complexity in most of the cases.     

关于完全二部图$K_{4, n}$和$K_{n, n}$的点可区别 $IE$-全染色的一些注记

Let G be a simple graph.An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color.Let C(u) be the set of colors of vertex u and edges incident to u under f.For an IE-total coloring f of G using k colors,if C(u)=C(v) for any two different vertices u and v of V(G),then f is called a k-vertex-distinguishing IE-total-coloring of G,or a k-VDIET coloring of G for short.The minimum number of colors required for a VDIET coloring of G is denoted by χ ie vt (G),and it is called the VDIET chromatic number of G.We will give VDIET chromatic numbers for complete bipartite graph K4,n (n≥4),K n,n (5≤ n ≤ 21) in this article.     

子立方无爪图的单射边色数至多为6

设$G$是一个图. 图$G$的一个单射边染色是指图$G$的一个边染色, 使得距离为$2$的两条边或者在同一个三角形中的两条边染不同的颜色. 图$G$的单射边色数是指图$G$的任意单射边染色所需要的最少颜色数. 关于单射边色数有一个猜想: 任意一个子立方图的单射边色数都不超过$6$. 在本文中, 我们证明了这个猜想对子立方无爪图是成立的, 并且给出图例说明上界$6$是紧的. 同时, 我们的证明隐含了求解这类图不超过$6$种颜色的单射边染色方案的一个线性时间算法.