A result about the density of iterated line intersections in the plane

Let S be a finite set of points in the plane and let be the set of intersection points between pairs of lines passing through any two points in S. We characterize all configurations of points S such that iteration of the above operation produces a dense set. We also discuss partial results on the characterization of those finite point-sets with rational coordinates that generate all of through iteration of .     

无符号拉普拉斯谱半径的新上界

如果一个图存在定向满足其最大出度△~＋不超过最大度△的一半,则通过估计图的半边路径（semi-edge walk）的个数,得到了该图的无符号拉普拉斯谱半径的一个新上界.进而根据D.Goncalves对平面图边分解的结果,得到了平面图无符号拉普拉斯谱半径的一个新上界.     

Asymptotic stability at infinity for differentiable vector fields of the plane

Let be a differentiable (but not necessarily C¹) vector field, where σ>0 and . Denote by R(z) the real part of z∈C. If for some ?>0 and for all , no eigenvalue of DpX belongs to , then: (a) for all , there is a unique positive semi-trajectory of X starting at p; (b) it is associated to X, a well-defined number I(X) of the extended real line [−∞,∞) (called the index of X at infinity) such that for some constant vector v∈R² the following is satisfied: if I(X) is less than zero (respectively greater or equal to zero), then the point at infinity ∞ of the Riemann sphere R²∪{∞} is a repellor (respectively an attractor) of the vector field X+v.     

Linear coloring of planar graphs with large girth

A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number of the graph G is the smallest number of colors in a linear coloring of G. In this paper we prove that every planar graph G with girth g and maximum degree Δ has if G satisfies one of the following four conditions: (1) g≥13 and Δ≥3; (2) g≥11 and Δ≥5; (3) g≥9 and Δ≥7; (4) g≥7 and Δ≥13. Moreover, we give better upper bounds of linear chromatic number for planar graphs with girth 5 or 6.     

A generalization of Tutte’s theorem on Hamiltonian cycles in planar graphs

In 1956, W.T. Tutte proved that a 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. We prove that a planar graph G has a cycle containing a given subset X of its vertex set and any two prescribed edges of the subgraph of G induced by X if |X|≥3 and if X is 4-connected in G. If X=V(G) then Sanders’ result follows.     

Total colorings and list total colorings of planar graphs without intersecting 4-cycles

Suppose that G is a planar graph with maximum degree Δ and without intersecting 4-cycles, that is, no two cycles of length 4 have a common vertex. Let χ^″(G), and denote the total chromatic number, list edge chromatic number and list total chromatic number of G, respectively. In this paper, it is proved that χ^″(G)=Δ+1 if Δ≥7, and and if Δ(G)≥8. Furthermore, if G is a graph embedded in a surface of nonnegative characteristic, then our results also hold.     

A quadratic system with two parallel straight-line-isoclines

In this paper, a quadratic system with two parallel straight-line-isoclines is considered. This system corresponds to the system of class II in the classification of Ye Yanqian [Ye Yanqian et al., Theory of Limit Cycles, Transl. Math. Monogr., vol. 66, American Mathematical Society, Providence, RI, 1986]. Using the field rotation parameters of the constructed canonical system and geometric properties of the spirals filling the interior and exterior domains of its limit cycles, we prove that the maximum number of limit cycles in a quadratic system with two parallel straight-line-isoclines and two finite singular points is equal to two. Besides, we obtain the same result in a different way: applying the Wintner–Perko termination principle for multiple limit cycles and using the methods of global bifurcation theory developed in [V.A. Gaiko, Global Bifurcation Theory and Hilbert’s Sixteenth Problem, Kluwer, Boston, 2003].     

Undirected power graphs of semigroups

The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,b∈S are adjacent if and only if a≠b and a ^m =b or b ^m =a for some positive integer m. In this paper we characterize the class of semigroups S for which G(S) is connected or complete. As a consequence we prove that G(G) is connected for any finite group G and G(G) is complete if and only if G is a cyclic group of order 1 or p ^m . Particular attention is given to the multiplicative semigroup ℤ _n and its subgroup U _n , where G(U _n ) is a major component of G(ℤ _n ). It is proved that G(U _n ) is complete if and only if n=1,2,4,p or 2p, where p is a Fermat prime. In general, we compute the number of edges of G(G) for a finite group G and apply this result to determine the values of n for which G(U _n ) is planar. Finally we show that for any cyclic group of order greater than or equal to 3, G(G) is Hamiltonian and list some values of n for which G(U _n ) has no Hamiltonian cycle.     

New families of perfect nonlinear polynomial functions

A new class of almost perfect nonlinear (APN) polynomial functions has been recently introduced. We give some generalizations of these functions and deduce new families of perfect nonlinear (PN) functions. We show that these PN functions are CCZ-inequivalent to the known perfect nonlinear functions.     

关于纯位移边界条件的平面弹性问题Locking-Free有限元格式

In this paper,we discusse the locking phenomenon of the finite element method for the pure displacement boundary value problem in the planar elasticity as Lame constantλ→∞,The locking-free scheme of Crouziex-Raviart element was proposed and analyzed by Brenner et al.[2]and [3].We firstly present the derivation of Brenner‘s scheme,then propose and analyse a locking-free scheme,then propose and analyse a locking-free scheme of noncon-forming rectangle finite element.