Unitarily invariant norm and $Q$-norm estimations for the Moore--Penrose inverse of multiplicative perturbations of matrices

Let $B$ be a multiplicative perturbation of $A\in\mathbb{C}^{m\times n}$ given by $B=D_1^* A D_2$, where $D_1\in\mathbb{C}^{m\times m}$ and $D_2\in\mathbb{C}^{n\times n}$ are both nonsingular. New upper bounds for $\Vert B^\dag-A^\dag\Vert_U$ and $\Vert B^\dag-A^\dag\Vert_Q$ are derived, where $A^\dag,B^\dag$ are the Moore-Penrose inverses of $A$ and $B$, and $\Vert \cdot\Vert_U,\Vert \cdot\Vert_Q$ are any unitarily invariant norm and $Q$-norm, respectively. Numerical examples are provided to illustrate the sharpness of the obtained upper bounds.     

Finite group action and Gromov-Witten invariants in symplectic four-manifolds

Let a finite group act semi-freely on a closed symplectic four-manifold with a 2-dimensional fixed point set. Then we show that the relative Gromov-Witten invariants are the same as the invariants on the quotient set-up with respect to the fixed point set.     

Vector invariants for the two-dimensional modular representation of a cyclic group of prime order

In this paper, we study the vector invariants of the 2-dimensional indecomposable representation V₂ of the cyclic group, Cp, of order p over a field F of characteristic p, FCp[mV₂]. This ring of invariants was first studied by David Richman (1990) [20] who showed that the ring required a generator of degree m(p−1), thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p=2. This conjecture was proved by Campbell and Hughes (1997) in [3]. Later, Shank and Wehlau (2002) in [24] determined which elements in Richman's generating set were redundant thereby producing a minimal generating set.We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants FCp[mV₂]. In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for FCp[mV₂]. Further, our results provide a procedure for finding an explicit decomposition of F[mV₂] into a direct sum of indecomposable Cp-modules. Finally, noting that our representation of Cp on V₂ is as the p-Sylow subgroup of SL₂(Fp), we describe a generating set for the ring of invariants F[mV₂]^SL₂(Fp) and show that (p+m−2)(p−1) is an upper bound for the Noether number, for m>2.     

Consistent nonnegative estimates of variance components

In this paper, the estimation of variance components in the linear mixed model with two random effects is investigated. The class of combination estimates based on the quadratic invariant statistics and consistent nonnegative estimates are obtained. Furthermore, it is shown that the consistent nonnegative estimate dominates ANOVA estimate under some conditions.     

Lagrangean duals and exact solution to the capacitated p-center problem

In this work, we address the capacitated p-center problem (CpCP). We study two auxiliary problems, discuss their relation to CpCP, and analyze the lower bounds obtained with two different Lagrangean duals based on each of these auxiliary problems. We also compare two different strategies for solving exactly CpCP, based on binary search and sequential search, respectively. Various data sets from the literature have been used for evaluating the performance of the proposed algorithms.     

顾客为子树结构的树上反中心选址问题

顾客为子树结构的树上反中心选址问题是在树T上寻找一点(位于顶点处或在边的内部),使得该点与子树结构的顾客之间的最小赋权带加数距离尽可能地大.给出了该问题的一个有效算法,其时间复杂度为O(cn+sum from j=1 to m n_j),其中n_j为各子树T_j的顶点个数,c为不同的子树权重个数,n为树的顶点数.     

一类三维系统的次调和解与不变环面的分支

讨论一类三维自治系统的闭轨在周期扰动下的分支问题.利用Poincare映射与积分流形定理,得到扰动系统存在次调和解和不变环面的条件,以及次调和解的鞍结点分支.     

Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits.More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x~2+ y~2+ 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincar′e disc.     

On the Darboux Integrability of the Hindmarsh–Rose Burster

We study the Hindmarsh–Rose burster which can be described by the differential system = y-x~3+ bx~2+ I-z,  = 1-5 x2~-y, z = μ(s(x-x_0)-z),where b, I, μ, s, x_0 are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist.     

Multipliers on Vector-valued <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-spaces for Hypergroups

In this article, we extend the well known Wendel’s theorem to the context of vector-valued L¹-spaces on hypergroups. In this regard, various cases have been studied.