On generalized Kneser hypergraph colorings

In 2002, the second author presented a lower bound for the chromatic numbers of hypergraphs , “generalized r-uniform Kneser hypergraphs with intersection multiplicities s.” It generalized previous lower bounds by K?í? (1992/2000) for the case s=(1,…,1) without intersection multiplicities, and by Sarkaria (1990) for . Here we discuss subtleties and difficulties that arise for intersection multiplicities si>1:

(1)

In the presence of intersection multiplicities, there are two different versions of a “Kneser hypergraph,” depending on whether one admits hypergraph edges that are multisets rather than sets. We show that the chromatic numbers are substantially different for the two concepts of hypergraphs. The lower bounds of Sarkaria (1990) and Ziegler (2002) apply only to the multiset version.

(2)

The reductions to the case of prime r in the proofs by Sarkaria and by Ziegler work only if the intersection multiplicities are strictly smaller than the largest prime factor of r. Currently we have no valid proof for the lower bound result in the other cases.

We also show that all uniform hypergraphs without multiset edges can be represented as generalized Kneser hypergraphs.     

The intersection map of subgroups and certain classes of finite groups

The goals of this paper are twofold. One is to look at the behavior of the collections of permutable subgroups and S-permutable subgroups under the intersection map into a fixed subgroup of a group. The other is to locally analyze the intersection map in connection with -, -, and -groups. In particular, we generalize Theorem 1 of Bauman [Arch. Math. (Basel) 25:337–340, 1974] to - and -groups.      

Helly EPT graphs on bounded degree trees: Characterization and recognition

The edge-intersection graph of a family of paths on a host tree is called an $EPT$ graph. When the tree has maximum degree $h$, we say that the graph is $[h,2,2]$. If, in addition, the family of paths satisfies the Helly property, then the graph is Helly $[h,2,2]$. In this paper, we present a family of $EPT$ graphs called gates which are forbidden induced subgraphs for $[h,2,2]$ graphs. Using these we characterize by forbidden induced subgraphs the Helly $[h,2,2]$ graphs. As a byproduct we prove that in getting a Helly $EPT$-representation, it is not necessary to increase the maximum degree of the host tree. In addition, we give an efficient algorithm to recognize Helly $[h,2,2]$ graphs based on their decomposition by maximal clique separators.     

寻找课堂的“催化剂”，激发元素化学趣味性

针对元素化学课程特点,本文提出重视教案设计、精选国内外化学元素视频、航空航天特色结合找交集、日常生活巧结合、诺贝尔大奖结合讲"课程思政"和紧跟时代的考试试题设计等教学手段,"催化"学生学习化学的兴趣,激发课堂活力。     

Exact hyperplane covers for subsets of the hypercube

Alon and Füredi (1993) showed that the number of hyperplanes required to cover ${\{0,1\}}^n?\{0\}$ without covering 0 is n. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering ${\{0,1\}}^n$ while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left open.     

Qualitative behavior of solutions of Liénard-type systems with state-dependent impulses

State-dependent impulsive differential equations can be used to describe phenomena in which the velocity of an object suddenly changes when the object enters a predetermined state. This study examines the effect of state-dependent impulses on the rotation of all orbits of the Liénard system, which plays an important role in many research areas, including electrical circuits, diverse engineering fields, economics, ecology, and physiology. Determining whether all orbits of the Liénard system other than the origin, which is the only fixed point, rotate around the origin, is the basis of other properties of the orbit and has become a significant research subject; however, very few studies have examined the effects of state-dependent impulses on this behavior, and the main theorem presented in this paper addresses this. This rotation problem is reduced to establishing whether all orbits intersect the vertical isocline, which is discussed in detail. To facilitate the understanding of the proof of the main theorem, an overview is presented before providing the actual proof. The main theorem and some lemmas are proved using phase plane analysis. The application of the main theorem to Euler’s equations is also described.     

光电经纬仪实时引导的实现

在由四台光电经纬仪与雷达组成的测量网络中,为了提高光电经纬仪实时捕获跟踪空间目标的能力,建立基于目标位置估计的实时引导系统.利用滑窗式最小二乘法将光电经纬仪实时交汇数据和雷达的实测数据进行互联,解算出目标的瞬时位置,经时延补偿后获得目标的位置估计,经坐标转换后实现光电经纬仪的实时引导.实时引导系统充分利用了所有测量站的测量信息,可提高引导数据的引导准确度和可靠性.试验结果表明,该方法快速提供的平滑引导数据,不仅能准确将目标引入光电经纬仪的视场,亦可为其他测量设备提供参考数据.     

Intersection of three planes revisited – an algebraic approach

Given three planes in space, a complete characterization of their intersection is provided. Special attention is paid to the case when the intersection set does not exist of one point only. Besides the vector cross product, the tool of generalized inverse of a matrix is used extensively.