arXiv subjects
Publications and source records attributed to Ming Ding.
This paper is to prove that a fermionic Novikov algebra equipped with an invariant non-degenerate symmetric bilinear form is a Novikov algebra.
Todays heterogeneous networks comprised of mostly macrocells and indoor small cells will not be able to meet the upcoming traffic demands. Indeed, it is forecasted that at least a 100x network capacity increase will be required to meet the traffic demands in 2020. As a result, vendors and operators are now looking at using every tool at hand to improve network capacity. In this epic campaign, three paradigms are noteworthy, i.e., network densification, the use of higher frequency bands and spectral efficiency enhancement techniques. This paper aims at bringing further common understanding and analysing the potential gains and limitations of these three paradigms, together with the impact of idle mode capabilities at the small cells as well as the user equipment density and distribution in outdoor scenarios. Special attention is paid to network densification and its implications when transitioning to ultra-dense small cell deployments. Simulation results show that network densification with an average inter site distance of 35 m can increase the cell- edge UE throughput by up to 48x, while the use of the 10GHz band with a 500MHz bandwidth can increase the network capacity up to 5x. The use of beamforming with up to 4 antennas per small cell base station lacks behind with cell-edge throughput gains of up to 1.49x. Our study also shows how network densifications reduces multi-user diversity, and thus proportional fair alike schedulers start losing their advantages with respect to round robin ones. The energy efficiency of these ultra-dense small cell deployments is also analysed, indicating the need for energy harvesting approaches to make these deployments energy- efficient. Finally, the top ten challenges to be addressed to bring ultra-dense small cell deployments to reality are also discussed.
Raptor codes have been widely used in many multimedia broadcast/multicast applications. However, our understanding of Raptor codes is still incomplete due to the insufficient amount of theoretical work on the performance analysis of Raptor codes, particularly under maximum-likelihood (ML) decoding, which provides an optimal benchmark on the system performance for the other decoding schemes to compare against. For the first time, this paper provides an upper bound and a lower bound, on the packet error performance of Raptor codes under ML decoding, which is measured by the probability that all source packets can be successfully decoded by a receiver with a given number of successfully received coded packets. Simulations are conducted to validate the accuracy of the analysis. More specifically, Raptor codes with different degree distribution and pre-coders, are evaluated using the derived bounds with high accuracy.
We prove a multiplication theorem of a quantum Caldero-Chapoton map associated to valued quivers which extends the results in \cite{DX}\cite{D}. As an application, when $Q$ is a valued quiver of finite type or rank 2, we obtain that the algebra $\mathcal{AH}_{|k|}(Q)$ generated by all cluster characters (see Definition \ref{def}) is exactly the quantum cluster algebra $\mathcal{EH}_{|k|}(Q)$ and various bases of the quantum cluster algebras of rank 2 can naturally be deduced.
We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types. Under the specialization of $q$ and coefficients to 1, these bases are generic bases of finite and affine cluster algebras.
We extend the definition of a quantum analogue of the Caldero-Chapoton map defined in \cite{rupel}. When $Q$ is a quiver of finite type, we prove that the algebra $\mathcal{AH}_{|k|}(Q)$ generated by all cluster characters (see Definition \ref{def}) is exactly the quantum cluster algebra $\mathcal{EH}_{|k|}(Q)$.
We prove a multiplication theorem for quantum cluster algebras of acyclic quivers. The theorem generalizes the multiplication formula for quantum cluster variables in \cite{fanqin}. We apply the formula to construct some $\mathbb{ZP}$-bases in quantum cluster algebras of finite and affine types. Under the specialization $q$ and coefficients to $1$, these bases are the integral bases of cluster algebra of finite and affine types (see \cite{CK1} and \cite{DXX}).
In \cite{CK2005} and \cite{SZ}, the authors constructed the bases of cluster algebras of finite types and of type $\widetilde{A}_{1,1}$, respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras of affine types.
We construct bar-invariant $\mathbb{Z}[q^{\pm 1/2}]-$bases of the quantum cluster algebra of the Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the cluster algebra of the Kronecker quiver in the sense of \cite{sherzel},\cite{calzel} and \cite{gls} respectively. As a byproduct, we prove the positivity of the elements in these bases.
Y. Palu has generalized the cluster multiplication formulas to 2-Calabi-Yau categories with cluster tilting objects (\cite{Palu2}). The aim of this note is to construct a variant of Y. Palu's formula and deduce a new version of the cluster multiplication formula (\cite{XiaoXu}) for acyclic quivers in the context of cluster categories.
We define an analogue of the Caldero-Chapoton map (\cite{CC}) for the cluster category of finite dimensional nilpotent representations over a cyclic quiver. We prove that it is a cluster character (in the sense of \cite{Palu}) and satisfies some inductive formulas for the multiplication between the generalized cluster variables (the images of objects of the cluster category under the map). Moreover, we construct a $\mathbb{Z}$-basis for the algebras generated by all generalized cluster variables.
By using the Ringel-Hall algebra approach, we investigate the structure of the Lie algebra $L(Λ)$ generated by indecomposable constructible sets in the varieties of modules for any finite dimensional $\mathbb{C}$-algebra $Λ.$ We obtain a geometric realization of the universal enveloping algebra $R(Λ)$ of $L(Λ).$ This generalizes the main result of Riedtmann in \cite{R}. We also obtain Green's theorem in \cite{G} in a geometric form for any finite dimensional $\mathbb{C}$-algebra $Λ$ and use it to give the comultiplication formula in $R(Λ).$
The canonical bases of cluster algebras of finite types and rank 2 are given explicitly in \cite{CK2005} and \cite{SZ} respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras for affine types $\widetilde{A}_{n,n},\widetilde{D}$ and $\widetilde{E}$. Moreover, we give an inductive formula for computing the multiplication between two generalized cluster variables associated to objects in a tube.