The localized longitudinal index theorem for Lie groupoids and the van Est map

We define the “localized index” of longitudinal elliptic operators on Lie groupoids associated with Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of the localized index, is an action of the Hopf algebroid of jets around the unit space, and the characteristic map it induces on Lie algebroid cohomology. This map can be globalized to differentiable groupoid cohomology, giving a definition of the “global index”, that can be computed by localization. This correspondence between the “global” and “localized” index is given by the van Est map for Lie groupoids.     

Asymptotically optimal neighbour sum distinguishing colourings of graphs

Consider a simple graph G = (V,E) and its proper edge colouring c with the elements of the set {1,2,…,k}. The colouring c is said to be neighbour sum distinguishing if for every pair of vertices u, v adjacent in G, the sum of colours of the edges incident with u is distinct from the corresponding sum for v. The smallest integer k for which such colouring exists is known as the neighbour sum distinguishing index of a graph and denoted by . The definition of this parameter, which makes sense for graphs containing no isolated edges, immediately implies that , where Δ is the maximum degree of G. On the other hand, it was conjectured by Flandrin et al. that for all those graphs, except for C₅. We prove this bound to be asymptotically correct by showing that . The main idea of our argument relays on a random assignment of the colours, where the choice for every edge is biased by so called attractors, randomly assigned to the vertices. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 776–791, 2015     

环状硅酸盐图的友好指数集

Let G be a graph with vertex set V(G) and edge set E(G). A labeling f : V(G) →Z2 induces an edge labeling f*: E(G) → Z2 defined by f*(xy) = f(x) + f(y), for each edge xy ∈ E(G). For i ∈ Z2, let vf(i) = |{v ∈ V(G) : f(v) = i}| and ef(i) = |{e ∈ E(G) : f*(e) =i}|. A labeling f of a graph G is said to be friendly if |vf(0)- vf(1)| ≤ 1. The friendly index set of the graph G, denoted FI(G), is defined as {|ef(0)- ef(1)|: the vertex labeling f is friendly}. This is a generalization of graph cordiality. We investigate the friendly index sets of cyclic silicates CS(n, m).     

Forwarding and optical indices of 4-regular circulant networks

An all-to-all routing in a graph G is a set of oriented paths of G, with exactly one path for each ordered pair of vertices. The load of an edge under an all-to-all routing R is the number of times it is used (in either direction) by paths of R, and the maximum load of an edge is denoted by $\pi (G,R)$. The edge-forwarding index $\pi (G)$ is the minimum of $\pi (G,R)$ over all possible all-to-all routings R, and the arc-forwarding index $\overrightarrow{\pi }(G)$ is defined similarly by taking direction into consideration, where an arc is an ordered pair of adjacent vertices. Denote by $w(G,R)$ the minimum number of colours required to colour the paths of R such that any two paths having an edge in common receive distinct colours. The optical index $w(G)$ is defined to be the minimum of $w(G,R)$ over all possible R, and the directed optical index $\overrightarrow{w}(G)$ is defined similarly by requiring that any two paths having an arc in common receive distinct colours. In this paper we obtain lower and upper bounds on these four invariants for 4-regular circulant graphs with connection set $\{\pm 1,\pm s\}$, $1<s<n/2$. We give approximation algorithms with performance ratio a small constant for the corresponding forwarding index and routing and wavelength assignment problems for some families of 4-regular circulant graphs.     

Homoclinic orbits for first order Hamiltonian systems with some twist conditions

In this paper, we study the nonperiodic first-order Hamiltonian system ù = J L(t)u +J H(t, u), where H ∈ C1(R × R2n). With some assumptions on L, the corresponding Hamiltonian operator has only discrete spectrum. By using the index theory for self-adjoint operator equation,we establish the existence of multiple homoclinic orbits for the asymptotically quadratic nonlinearty satisfying some twist conditions between infinity and origin.     

基于状态指标的更新决策方法研究

针对设备在实际运行和执行任务时经常涉及到维修、更新报废等管理活动缺乏定量模型支撑的问题,提出了利用典型部件的预测寿命和非典型部件的历史寿命对设备系统级的健康状态进行综合评估的方法,建立了一种以状态指标为决策序列的动态规划更新决策方法。该方法能综合考虑维修费用和更新费用,确定最优更新决策,降低保障费用,有效的提高了设备的经济效益。最后经过案例分析证明了该方法的有效性,对于提高设备的保障能力和经济效益提供了理论支撑和方法指导。     

折射率正负梯度交替表面的研究

利用不均匀材料提出了一种折射率正负梯度交替表面, 并利用几何光学法和电磁场数值仿真方法对其机理进行了研究与验证, 发现其可以操控电磁波的传播: 在一定的参数下, 其可以将空间中的入射电磁波一直束缚在介质中, 或在介质中传播一定距离后再实现电磁波的释放; 同时该结构具有宽带、极化无关等特性. 利用该结构可引导能流, 减小后向散射截面, 可用作隐身表面.     

增益导引-折射率反导引大模场光纤激光器抽运技术研究进展

介绍了一种新型增益导引-折射率反导引大模场光纤的结构、基本理论以及其在光纤激光器领域的重要应用. 在综合分析了不同抽运条件下的激光输出特性以及光纤的热效应特征后, 得出侧面抽运是增益导引-折射率反导引大模场光纤的最佳抽运方案的结论; 重点介绍了增益导引-折射率反导引大模场光纤侧面抽运的理论模拟以及基于V形槽技术的侧面抽运实验研究过程, 为相关领域的实验研究提供了参考; 最后分析了实验结果与理论差距较大的原因, 并提出了改进的方向.