An O(log k log^2 n)-competitive Randomized Algorithm for the k-Sever Problem

In this paper, we show that there is an O(log k log^2 n)-competitive randomized algorithm for the k-sever problem on any metric space with n points, which improved the previous best competitive ratio O(log^2 k log^3 n log log n) by Nikhil Bansal et al. (FOCS 2011, pages 267-276).Comment: arXiv admin note: text overlap with arXiv:1410.495

On weakly S-embedded subgroups and weakly τ\tau-embedded subgroups

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be weakly S-embedded in $G$ if there exists $K\unlhd G$ such that $HK$ is S-quasinormal in $G$ and $H\cap K\leq H_{seG}$, where $H_{seG}$ is the subgroup generated by all those subgroups of $H$ which are S-quasinormally embedded in $G$. We say that $H$ is weakly $\tau$-embedded in $G$ if there exists $K\unlhd G$ such that $HK$ is S-quasinormal in $G$ and $H\cap K\leq H_{\tau G}$, where $H_{\tau G}$ is the subgroup generated by all those subgroups of $H$ which are $\tau$-quasinormal in $G$. In this paper, we study the properties of the weakly S-embedded subgroups and the weakly $\tau$-embedded subgroups, and use them to determine the structure of finite groups

On the π\piF\mathfrak{F}-norm and the H\mathfrak{H}-F\mathfrak{F}-norm of a finite group

Let $\mathfrak{H}$ be a Fitting class and $\mathfrak{F}$ a formation. We call a subgroup $\mathcal{N}_{\mathfrak{H},\mathfrak{F}}(G)$ of a finite group $G$ the $\mathfrak{H}$-$\mathfrak{F}$-norm of $G$ if $\mathcal{N}_{\mathfrak{H},\mathfrak{F}}(G)$ is the intersection of the normalizers of the products of the $\mathfrak{F}$-residuals of all subgroups of $G$ and the $\mathfrak{H}$-radical of $G$. Let $\pi$ denote a set of primes and let $\mathfrak{G}_\pi$ denote the class of all finite $\pi$-groups. We call the subgroup $\mathcal{N}_{\mathfrak{G}_\pi,\mathfrak{F}}(G)$ of $G$ the $\pi\mathfrak{F}$-norm of $G$. A normal subgroup $N$ of $G$ is called $\pi\mathfrak{F}$-hypercentral in $G$ if either $N=1$ or $N>1$ and every $G$-chief factor below $N$ of order divisible by at least one prime in $\pi$ is $\mathfrak{F}$-central in $G$. Let $Z_{\pi\mathfrak{F}}(G)$ denote the $\pi\mathfrak{F}$-hypercentre of $G$, that is, the product of all $\pi\mathfrak{F}$-hypercentral normal subgroups of $G$. In this paper, we study the properties of the $\mathfrak{H}$-$\mathfrak{F}$-norm, especially of the $\pi\mathfrak{F}$-norm of a finite group $G$. In particular, we investigate the relationship between the $\pi'\mathfrak{F}$-norm and the $\pi\mathfrak{F}$-hypercentre of $G$

Minimal degree H(curl) and H(div) conforming finite elements on polytopal meshes

We construct H(curl) and H(div) conforming finite elements on convex polygons and polyhedra with minimal possible degrees of freedom, i.e., the number of degrees of freedom is equal to the number of edges or faces of the polygon/polyhedron. The construction is based on generalized barycentric coordinates and the Whitney forms. In 3D, it currently requires the faces of the polyhedron be either triangles or parallelograms. Formula for computing basis functions are given. The finite elements satisfy discrete de Rham sequences in analogy to the well-known ones on simplices. Moreover, they reproduce existing H(curl)-H(div) elements on simplices, parallelograms, parallelepipeds, pyramids and triangular prisms. Approximation property of the constructed elements is also analyzed, by showing that the lowest-order simplicial Nedelec- Raviart-Thomas elements are subsets of the constructed elements on arbitrary polygons and certain polyhedra.Comment: 32 page

Finite groups in which SS-permutability is a transitive relation

A subgroup $H$ of a finite group $G$ is said to be SS-permutable in $G$ if $H$ has a supplement $K$ in $G$ such that $H$ permutes with every Sylow subgroup of $K$. A finite group $G$ is called an SST-group if SS-permutability is a transitive relation on the set of all subgroups of $G$. The structure of SST-groups is investigated in this paper

On Π\Pi-supplemented subgroups of a finite group

A subgroup $H$ of a finite group $G$ is said to satisfy $\Pi$-property in $G$ if for every chief factor $L/K$ of $G$, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi(HK/K\cap L/K)$-number. A subgroup $H$ of $G$ is called to be $\Pi$-supplemented in $G$ if there exists a subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq I\leq H$, where $I$ satisfies $\Pi$-property in $G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that some primary subgroups of $G$ are $\Pi$-supplemented in $G$. The main result we proved improves a large number of earlier results.Comment: arXiv admin note: text overlap with arXiv:1301.636

New Epitaxial Thin Film Models and numerical approximation

This paper concerns new continuum phenomenological model for epitaxial thin film growth with three different forms of the Ehrlich-Schwoebel current. Two of these forms were first proposed by Politi and Villain [1996] and then studied by Evans, Thiel and Bartelt [2006]. The other one is completely new. Following the techniques used in Li and Liu [2003], we present rigorous analysis of the well-posedness, regularity and time stability for the new model. We also studied both the global and the local behavior of the surface roughness in the growth process. The new model differs from other known models in that it features a linear convex part and a nonlinear concave part, and thus by using a convex-concave time splitting scheme, one can naturally build unconditionally stable semi-implicit numerical discretizations with linear implicit parts, which is much easier to implement than conventional models requiring nonlinear implicit parts. Despite this fundamental difference in the model, numerical experiments show that the nonlinear morphological instability of the new model agrees well with results of other models published in Li and Liu [2003], which indicates that the new model correctly captures the essential morphological states in the thin film growth process.Comment: 28 pages, 20 figure

On HC-subgroups of a finite group

A subgroup $H$ of a finite group $G$ is said to be an $\mathscr{H}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g \cap N_T(H)\leq H$ for all $g\in G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that certain subgroups of $G$ of arbitrary prime power order are $\mathscr{H}C$-subgroups of $G$

Ground State Solutions of Kirchhoff-type Fractional Dirichlet Problem with pp-Laplacian

We consider the Kirchhoff-type $p$-Laplacian Dirichlet problem containing the left and right fractional derivative operators. By using the Nehari method in critical point theory, we obtain the existence theorem of ground state solutions for such Dirichlet problem.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1607.0158

PR Product: A Substitute for Inner Product in Neural Networks

In this paper, we analyze the inner product of weight vector w and data vector x in neural networks from the perspective of vector orthogonal decomposition and prove that the direction gradient of w decreases with the angle between them close to 0 or {\pi}. We propose the Projection and Rejection Product (PR Product) to make the direction gradient of w independent of the angle and consistently larger than the one in standard inner product while keeping the forward propagation identical. As a reliable substitute for standard inner product, the PR Product can be applied into many existing deep learning modules, so we develop the PR Product version of fully connected layer, convolutional layer and LSTM layer. In static image classification, the experiments on CIFAR10 and CIFAR100 datasets demonstrate that the PR Product can robustly enhance the ability of various state-of-the-art classification networks. On the task of image captioning, even without any bells and whistles, our PR Product version of captioning model can compete or outperform the state-of-the-art models on MS COCO dataset. Code has been made available at:https://github.com/wzn0828/PR_Product.Comment: ICCV2019 ora