σ2\sigma_2 Yamabe problem on conic 4-spheres

We discuss the constant $\sigma_{2}$ problem for conic 4-spheres. Based on earlier works of Chang-Han-Yang and Han-Li-Teixeira, we are able to find a necessary condition for the existence problem. In particular, when the condition is sharp, we have the uniqueness result similar to that of Troyanov in dimension 2. It indicates that the boundary of the moduli of all conic 4-spheres with constant $\sigma_{2}$ metrics consists of conic spheres with 2 conic points and rotational symmetry.Comment: 20 pages, we makes some changes in the paper posted befor

Some New Constructions of Quantum MDS Codes

It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes of q-ary quantum MDS codes by using generalized Reed-Solomon (GRS) codes and Hermitian construction. The minimum distances of our quantum MDS codes can be larger than q/2+1 Three of these six classes of quantum MDS codes have longer lengths than the ones constructed in [1] and [2], hence some of their results can be easily derived from ours via the propagation rule. Moreover, some known quantum MDS codes of specific lengths can be seen as special cases of ours and the minimum distances of some known quantum MDS codes are also improved as well.Comment: Accepted for publication in IEEE Transactions on Information Theor

Two new classes of quantum MDS codes

Let $p$ be a prime and let $q$ be a power of $p$. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance- separable (MDS) codes with parameters $$[[tq, tq-2d+2, d]]_{q}$$ for any $1 \leq t \leq q, 2 \leq d \leq \lfloor \frac{tq+q-1}{q+1}\rfloor+1$, and $$[[t(q+1)+2, t(q+1)-2d+4, d]]_{q}$$ for any $1 \leq t \leq q-1, 2 \leq d \leq t+2$ with $(p,t,d) \neq (2, q-1, q)$. Our quantum codes have flexible parameters, and have minimum distances larger than $\frac{q}{2}+1$ when $t > \frac{q}{2}$. Furthermore, it turns out that our constructions generalize and improve some previous results.Comment: 14 pages. Accepted by Finite Fields and Their Application

Optimal cyclic (r,δ)(r,\delta) locally repairable codes with unbounded length

Locally repairable codes with locality $r$ ($r$-LRCs for short) were introduced by Gopalan et al. \cite{1} to recover a failed node of the code from at most other $r$ available nodes. And then $(r,\delta)$ locally repairable codes ($(r,\delta)$-LRCs for short) were produced by Prakash et al. \cite{2} for tolerating multiple failed nodes. An $r$-LRC can be viewed as an $(r,2)$-LRC. An $(r,\delta)$-LRC is called optimal if it achieves the Singleton-type bound. It has been a great challenge to construct $q$-ary optimal $(r,\delta)$-LRCs with length much larger than $q$. Surprisingly, Luo et al. \cite{3} presented a construction of $q$-ary optimal $r$-LRCs of minimum distances 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of $q$) via cyclic codes. In this paper, inspired by the work of \cite{3}, we firstly construct two classes of optimal cyclic $(r,\delta)$-LRCs with unbounded lengths and minimum distances $\delta+1$ or $\delta+2$, which generalize the results about the $\delta=2$ case given in \cite{3}. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic $(r,\delta)$-LRCs with unbounded length and larger minimum distance $2\delta$. Furthermore, when $\delta=3$, we give another class of optimal cyclic $(r,3)$-LRCs with unbounded length and minimum distance $6$

Tree Decomposition based Steiner Tree Computation over Large Graphs

In this paper, we present an exact algorithm for the Steiner tree problem. The algorithm is based on certain pre-computed index structures. Our algorithm offers a practical solution for the Steiner tree problems on graphs of large size and bounded number of terminals

A Note On Andrews Inequality

We prove some extensions of Andrews inequality

A sphere theorem for Bach-flat manifolds with positive constant scalar curvature

We show a closed Bach-flat Riemannian manifold with a fixed positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either $L^\infty$ or $L^{\frac{n}{2}}$-norm. Compared with the complete non-compact case done by Kim, we apply a different method to achieve these results. These results generalize a rigidity theorem of positive Einstein manifolds due to M.-A.Singer. As an application, we can partially recover the well-known Chang-Gursky-Yang's $4$-dimensional conformal sphere theorem.Comment: 11 page

On Random Linear Network Coding for Butterfly Network

Random linear network coding is a feasible encoding tool for network coding, specially for the non-coherent network, and its performance is important in theory and application. In this letter, we study the performance of random linear network coding for the well-known butterfly network by analyzing the failure probabilities. We determine the failure probabilities of random linear network coding for the well-known butterfly network and the butterfly network with channel failure probability p.Comment: This paper was submitted to IEEE Communications Letter

Very deep spectroscopy of the bright Saturn Nebula NGC 7009 - II. Analysis of the rich optical recombination spectrum

[Abridged] We present a critical analysis of the rich optical recombination spectrum of NGC 7009, in the context of the bi-abundance nebular model proposed by Liu et al. (2000). The observed relative intensities are compared with the theoretical predictions based on high quality effective recombination coefficients, now available for the recombination line spectrum of a number of heavy element ions. The possibility of plasma diagnostics using the optical recombination lines (ORLs) of heavy element ions is discussed in detail. Plasma diagnostics based on the N II and O II recombination spectra both yield electron temperatures close to 1000 K, which is lower than those derived from the collisionally excited line (CEL) ratios by nearly one order of magnitude. The very low temperatures yielded by the O II and N II ORLs indicate that they originate from very cold regions. The C^{2+}/H^+, N^{2+}/H^+, O^{2+}/H^+ and Ne^{2+}/H^+ ionic abundance ratios derived from ORLs are consistently higher, by about a factor of 5, than the corresponding values derived from CELs. In calculating the ORL ionic abundance ratios, we have used the newly available high quality effective recombination coefficients, and adopted an electron temperature of 1000 K, as given by the ORL diagnostics and as a consequence presumably representing the physical conditions prevailing in the regions where the heavy element ORLs arise. A comparison of the results of plasma diagnostics and abundance determinations for NGC 7009 points to the existence of "cold", metal-rich (i.e. H-deficient) inclusions embedded in the hot, diffuse ionized gas, first postulated by Liu et al. (2000).Comment: Accepted for publication in MNRAS (50 pages of main text; 13 pages of appendix; in total 55 figures and 28 tables

Adaptive Euler-Maruyama method for SDEs with non-globally Lipschitz drift: Part II, infinite time interval

This paper proposes an adaptive timestep construction for an Euler-Maruyama approximation of the ergodic SDEs with a drift which is not globally Lipschitz over an infinite time interval. If the timestep is bounded appropriately, we show not only the stability of the numerical solution and the standard strong convergence order, but also that the bound for moments and strong error of the numerical solution are uniform in T, which allow us to introduce the adaptive multilevel Monte Carlo. Numerical experiments support our analysis.Comment: 36 pages, 3 figures. arXiv admin note: text overlap with arXiv:1609.0810