Monochromatic k-edge-connection colorings of graphs

A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\ge 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are connected by at least $k$ edge-disjoint monochromatic paths, and $G$ is said to be uniformly monochromatic $k$-edge-connected if every two distinct vertices are connected by at least $k$ edge-disjoint monochromatic paths such that all edges of these $k$ paths are colored with a same color. We use $m{c}_k\left(G\right)$ and $um{c}_k\left(G\right)$ to denote the maximum number of colors that ensures $G$ to be monochromatic $k$-edge-connected and, respectively, $G$ to be uniformly monochromatic $k$-edge-connected. In this paper, we first conjecture that for any $k$-edge-connected graph $G$, $m{c}_k\left(G\right)=e\left(G\right)-e\left(H\right)+\lfloor \frac{k}{2}\rfloor$, where $H$ is a minimum $k$-edge-connected spanning subgraph of $G$. We verify the conjecture for $k=2$. We also prove the conjecture for $G=K_{k+1}$ and $G=K_{k,n}$ with $n\ge k\ge 3$. When $G$ is a minimal $k$-edge-connected graph, we give an upper bound of $m{c}_k\left(G\right)$, i.e., $m{c}_k\left(G\right)\le k-1$. For the uniformly monochromatic $k$-edge-connectivity, we prove that for all $k$, $um{c}_k\left(G\right)=e\left(G\right)-e\left(H\right)+1$, where $H$ is a minimum $k$-edge-connected spanning subgraph of $G$.     

原子在单色辐射作用下长时间内的行为研究

建立了原子在单色辐射作用下的模型，用微扰理论方法求解出原子处于两能级的几率.讨论了原子在单色辐射作用下长时间内的行为，得到了原子因辐射场的影响而变化的速率比能级的衰减速率大的情况下，原子在两能级间振荡，原子和辐射场相互作用越强，辐射场能够与原子发生作用的频率区间越宽,而且原子在单色光长时间辐射作用下高、低能级跃迁的速率不随时间而改变的结论.     

Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey

Nowadays the term monochromatic and heterochromatic (or rainbow, multicolored) subgraphs of an edge-colored graph appeared frequently in literature, and many results on this topic have been obtained. In this paper, we survey results on this subject. We classify the results into the following categories: vertex-partitions by monochromatic subgraphs, such as cycles, paths, trees; vertex partition by some kinds of heterochromatic subgraphs; the computational complexity of these partition problems; some kinds of large monochromatic and heterochromatic subgraphs. We have to point out that there are a lot of results on Ramsey type problem of monochromatic and heterochromatic subgraphs. However, it is not our purpose to include them in this survey because this is slightly different from our topics and also contains too large amount of results to deal with together. There are also some interesting results on vertex-colored graphs, but we do not include them, either. Supported by NSFC, PCSIRT and the “973” program.     

On Some Three-Color Ramsey Numbers

 In this paper we study three-color Ramsey numbers. Let K _i,j denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G ₁, G ₂ and G ₃, if r(G ₁, G ₂)≥s(G ₃), then r(G ₁, G ₂, G ₃)≥(r(G ₁, G ₂)−1)(χ(G ₃)−1)+s(G ₃), where s(G ₃) is the chromatic surplus of G ₃; (ii) (k+m−2)(n−1)+1≤r(K _1,k , K _1,m , K _n )≤ (k+m−1)(n−1)+1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m≥k≥2, there is a constant c such that r(K _k,m , K _k,m , K _n )≤c(n/logn), and r(C _2m , C _2m , K _n )≤c(n/logn) ^m/(m−1) for sufficiently large n. Received: July 25, 2000 Final version received: July 30, 2002 RID="*" ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of education ministry of China, and the foundations of Jiangsu Province Acknowledgments. The authors are grateful to the referee for his valuable comments. AMS 2000 MSC: 05C55     

光谱对小球藻和等鞭金藻生长的影响

采用单色LED光源和荧光灯从效率和速率两个角度研究了光源光谱对海生小球藻和等鞭金藻8701生长的影响,结果表明,连续光谱能够促进较高的最大生长率,蓝光促进生长的效率较高,两者组合能够较好地兼顾效率和速率。为了定量描述光照中的光谱参量,引入光谱吸收系数,根据光源发射光谱和藻体吸收谱计算海藻对光源光谱的量子吸收效率,该系数与实验结果中相应的生长效率成正相关,能够初步以量化形式反映光谱与其生长效率的关系。     

Monochromatic sinks in nearly transitive arc-colored tournaments

Let T be the set of all arc-colored tournaments, with any number of colors, that contain no rainbow 3-cycles, i.e., no 3-cycles whose three arcs are colored with three distinct colors. We prove that if T∈T and if each strong component of T is a single vertex or isomorphic to an upset tournament, then T contains a monochromatic sink. We also prove that if T∈T and T contains a vertex x such that T−x is transitive, then T contains a monochromatic sink. The latter result is best possible in the sense that, for each n≥5, there exists an n-tournament T such that (T−x)−y is transitive for some two distinct vertices x and y in T, and T can be arc-colored with five colors such that T∈T, but T contains no monochromatic sink.     

Conforming finite element computations applied to a spatially periodic,harmonic Navier-Stokes problem

In this paper computations in the two dimensional case of a harmonic Navier-Stokes problem with periodic boundary conditions are presented. This study of an incompressible viscous fluid leads to a non-symmetric linear problem (very low Reynolds number). Moreover unknown functions have complex values (monochromatic dynamic behaviour). Numerical treatment of the incompressibility condition is a generalization of the classical treatment of Stokes problem. A mixed formulation, where discrete pressure plays the role of Lagrange multipliers is used (Uzawa algorithm). Two conforming finite element methods are tested on different meshes. The second one uses a classical refinement in the shape function: the so-called bulb function. All computational tests show that the use of a bulb function on each element gives better results than refinement in the mesh without introducing too many degrees of freedom. Finally numerical results are compared to experimental data.     

A new class of Ramsey-Turán problems

We introduce and study a new class of Ramsey-Turán problems, a typical example of which is the following one:Let ε>0 and G be a graph of sufficiently large order n with minimum degree δ(G)>3n/4. If the edges of G are colored in blue or red, then for all k∈[4,⌊(1/8−ε)n⌋], there exists a monochromatic cycle of length k.     

On a modification of a problem of Bialostocki, Erd?s, and Lefmann

For positive integers m and r, one can easily show there exist integers N such that for every map Δ:{1,2,…,N}→{1,2,…,r} there exist 2m integers

x₁<?<x_m<y₁<?<y_m,