A single-step HSS method for non-Hermitian positive definite linear systems

In this paper, based on the Hermitian and skew-Hermitian splitting (HSS) iteration method, a single-step HSS (SHSS) iteration method is introduced to solve the non-Hermitian positive definite linear systems. Theoretical analysis shows that, under a loose restriction on the iteration parameter, the SHSS method is convergent to the unique solution of the linear system. Furthermore, we derive an upper bound for the spectral radius of the SHSS iteration matrix, and the quasi-optimal parameter is obtained to minimize the above upper bound. Numerical experiments are reported to the efficiency of the SHSS method; numerical comparisons show that the proposed SHSS method is superior to the HSS method under certain conditions.     

Maximum Order of Triangle‐Free Graphs with a Given Rank

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced triangle‐free graphs with a given rank and characterize all such graphs achieving the maximum order.     

矩阵方程AX=B的中心对称定秩解及其最佳逼近

本文研究了矩阵方程AX=B的中心对称解.利用矩阵对的广义奇异值分解和广义逆矩阵,获得了该方程有中心对称解的充要条件以及有解时,最大秩解、最小秩解的一般表达式,并讨论了中心对称最小秩解集合中与给定矩阵的最佳逼近解.     

钙蛋白酶-10 基因SNP-19 与多囊卵巢综合征相关性研究

目的 探讨钙蛋白酶-10 基因（CAPN-10）单核苷酸多态性-19（SNP-19）与多囊卵巢综合征（PCOS）的相关性。 方法 采用聚合酶链反应- 限制性片段长度多态性（PCR-RFLP）技术,对107 例PCOS患者（PCOS 组）及111 例健康体检者（对照组）的CAPN-10 SNP-19 进行分析,并测定其性激素和生化指标。PCOS组再根据BMI与胰岛素抵抗（IR）稳态模型指数（HOMA-IR）分为肥胖组与非肥胖组、IR 组与非IR 组,分析CAPN-10 SNP-19 与其关系。结果 与对照组相比,PCOS 组BMI、空腹血糖（FPG）、TG、TC、LDL、黄体生成素（LH）、黄体生成素/ 卵泡刺激素（LH/FSH）、睾酮（T）均升高（均P＜0.01）。PCOS 组CAPN-10 SNP-19 1等位基因及1/2基因型频率均高于对照组（均P＜0.05）;且CAPN-10 SNP-19 1/2基因型PCOS 发病风险较1/1、2/2 基因型增加约2倍（OR=1.9,95％CI=1.1~3.3,P＜0.05）。CAPN-10 SNP-19 1、2等位基因频率PCOS 肥胖组与非肥胖组、IR 组与非IR组比较均无统 计学差异（均P＞0.05）;PCOS 肥胖组和IR组CAPN-10 SNP-19 1/2基因型频率分别高于非肥胖组和非IR 组,但差异均无统计学意义（均P＞0.05）。结论 CAPN-10 SNP-19 与PCOS 遗传易感性有关,可能与PCOS 肥胖及IR 无关。     

Fast enclosure for the minimum norm least squares solution of the matrix equation AXB = C

Fast algorithms for enclosing the minimum norm least squares solution of the matrix equation AXB = C are proposed. To develop these algorithms, theories for obtaining error bounds of numerical solutions are established. The error bounds obtained by these algorithms are verified in the sense that all the possible rounding errors have been taken into account. Techniques for accelerating the enclosure and obtaining smaller error bounds are introduced. Numerical results show the properties of the proposed algorithms. Copyright © 2015 John Wiley & Sons, Ltd.     

Biembedding Steiner Triple Systems in Surfaces Using the Bose Construction

A uniform framework is presented for biembedding Steiner triple systems obtained from the Bose construction using a cyclic group of odd order, in both orientable and nonorientable surfaces. Within this framework, in the nonorientable case, a formula is given for the number of isomorphism classes and the particular biembedding of Ducrocq and Sterboul (preprint 18pp., 1978) is identified. In the orientable case, it is shown that the biembedding of Grannell et al. (J Combin Des 6 ( 7 ), 325–336) is, up to isomorphism, the unique biembedding of its type. Automorphism groups of the biembeddings are also given.     

Hochschild and cyclic (co)homology of superadditive categories

We define the Hochschild and cyclic (co)homology groups for superadditive categories and show that these (co)homology groups are graded Morita invariants. We also show that the Hochschild and cyclic homology are compatible with the tensor product of superadditive categories.     

The adversary degree-associated reconstruction number of double-brooms

A vertex-deleted subgraph of a graph G is a card. A dacard specifies the degree of the deleted vertex along with the card. The adversary degree-associated reconstruction number $\mathrm{adrn}(G)$ is the least k such that every set of k dacards determines G. We determine $\mathrm{adrn}(D_{m,n,p})$, where the double-broom $D_{m,n,p}$ with $p\ge 2$ is the tree with $m+n+p$ vertices obtained from a path with p vertices by appending m leaves at one end and n leaves at the other end. We determine $\mathrm{adrn}(D_{m,n,p})$ for all $m,n,p$. For $2\le m\le n$, usually $\mathrm{adrn}(D_{m,n,p})=m+2$, except $\mathrm{adrn}(D_{m,m+1,p})=m+1$ and $\mathrm{adrn}(D_{m,m+2,p})=m+3$. There are exceptions when $(m,n)=(2,3)$ or $p=4$. For $m=1$ the usual value is 4, with exceptions when $p\in \{2,3\}$ or $n=2$.     

Switching adaptive controllers to control fractional‐order complex systems with unknown structure and input nonlinearities

This article investigates the chaos control problem for the fractional‐order chaotic systems containing unknown structure and input nonlinearities. Two types of nonlinearity in the control input are considered. In the first case, a general continuous nonlinearity input is supposed in the controller, and in the second case, the unknown dead‐zone input is included. In each case, a proper switching adaptive controller is introduced to stabilize the fractional‐order chaotic system in the presence of unknown parameters and uncertainties. The control methods are designed based on the boundedness property of the chaotic system's states, where, in the proposed methods the nonlinear/linear dynamic terms of the fractional‐order chaotic systems are assumed to be fully unknown. The analytical results of the mentioned techniques are proved by the stability analysis theorem of fractional‐order systems and the adaptive control method. In addition, as an application of the proposed methods, single input adaptive controllers are adopted for control of a class of three‐dimensional nonlinear fractional‐order chaotic systems. And finally, some numerical examples illustrate the correctness of the analytical results. © 2014 Wiley Periodicals, Inc. Complexity 21: 211–223, 2015