Strongly Even‐Cycle Decomposable Graphs

A graph is strongly even‐cycle decomposable if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length. We prove that several fundamental composition operations that preserve the property of being Eulerian also yield strongly even‐cycle decomposable graphs. As an easy application of our theorems, we give an exact characterization of the set of strongly even‐cycle decomposable cographs.     

Graphs with No Induced Five‐Vertex Path or Antipath

We prove that a graph G contains no induced ‐vertex path and no induced complement of a ‐vertex path if and only if G is obtained from 5‐cycles and split graphs by repeatedly applying the following operations: substitution, split unification, and split unification in the complement, where split unification is a new class‐preserving operation introduced here.     

On the Number of k‐Dominating Independent Sets

We study the existence and the number of k‐dominating independent sets in certain graph families. While the case namely the case of maximal independent sets—which is originated from Erd?s and Moser—is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of k‐dominating independent sets in n‐vertex graphs is between and if , moreover the maximum number of 2‐dominating independent sets in n‐vertex graphs is between and . Graph constructions containing a large number of k‐dominating independent sets are coming from product graphs, complete bipartite graphs, and finite geometries. The product graph construction is associated with the number of certain Maximum Distance Separable (MDS) codes.     

Embedded Surfaces for Symplectic Circle Actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces.More precisely, it is shown that(1) if(M, ω) admits a Hamiltonian S~1-action, then there exists a two-sphere S in M with positive symplectic area satisfying c1(M, ω), [S] 0,and(2) if the action is non-Hamiltonian, then there exists an S~1-invariant symplectic2-torus T in(M, ω) such that c1(M, ω), [T] = 0. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott,Lupton-Oprea, and Ono: Suppose that(M, ω) is a smooth closed symplectic manifold satisfying c1(M, ω) = λ· [ω] for some λ∈ R and G is a compact connected Lie group acting effectively on M preserving ω. Then(1) if λ 0, then G must be trivial,(2) if λ = 0, then the G-action is non-Hamiltonian, and(3) if λ 0, then the G-action is Hamiltonian.     

Ehrhart Polynomials of 3-Dimensional Simple Integral Convex Polytopes

The author gives an explicit formula on the Ehrhart polynomial of a 3-dimensional simple integral convex polytope by using toric geometry.     

半参数线性混合效应模型的联合变量选择

多数基于线性混合效应模型的变量选择方法分阶段对固定效应和随机效应进行选择,方法繁琐、易产生模型偏差,且大部分非参数和半参数的线性混合效应模型只涉及非参数部分的光滑度或者固定效应的选择,并未涉及非参变量或随机效应的选择。本文用B样条函数逼近非参数函数部分,从而把半参数线性混合效应模型转化为带逼近误差的线性混合效应模型。对随机效应的协方差矩阵采用改进的乔里斯基分解并重新参数化线性混合效应模型,接着对该模型的极大似然函数施加集群ALASSO惩罚和ALASSO惩罚两类惩罚,该法能实现非参数变量、固定效应和随机效应的联合变量选择,基于该法得出的估计量也满足相合性、稀疏性和Oracle性质。文章最后做了个数值模拟,模拟结果表明,本文提出的估计方法在变量选择的准确性、参数估计的精度两个方面均表现较好。     

Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems

In this paper we introduce the theory of dominant solutions at infinity for nonoscillatory discrete symplectic systems without any controllability assumption. Such solutions represent an opposite concept to recessive solutions at infinity, which were recently developed for such systems by the authors. Our main results include: (i) the existence of dominant solutions at infinity for all ranks in a given range depending on the order of abnormality of the system, (ii) construction of dominant solutions at infinity with eventually the same image, (iii) classification of dominant and recessive solutions at infinity with eventually the same image, (iv) limit characterization of recessive solutions at infinity in terms of dominant solutions at infinity and vice versa, and (v) Reid’s construction of the minimal recessive solution at infinity. These results are based on a new theory of genera of conjoined bases for symplectic systems developed for this purpose in this paper.     

基于S-R和分解定理的几何非线性问题的数值计算分析

为了探究几何非线性问题的数值求解方法,采用理论推导、MATLAB编程计算、有限元模拟相结合的方法,基于S-R和分解定理及更新拖带坐标描述法,运用插值型无单元Galerkin方法对几何非线性问题的增量变分方程进行了推导,并通过四点Gauss积分法和不动点迭代法对其进行求解.最后以平面悬臂梁的大变形问题为例进行求解计算,发现与ANSYS的计算结果拟合相似度很高,说明了所采用的几何非线性力学理论及数值计算方法的正确性和合理性,为求解几何非线性问题提供了一种新的依据.     

An analysis on the efficiency of Euler's method for computing the matrix pth root

It is shown that the matrix sequence generated by Euler's method starting from the identity matrix converges to the principal pth root of a square matrix, if all the eigenvalues of the matrix are in a region including the one for Newton's method given by Guo in 2010. The convergence is cubic if the matrix is invertible. A modification version of Euler's method using the Schur decomposition is developed. Numerical experiments show that the modified algorithm has the overall good numerical behavior.