arXiv subjects
Publications and source records attributed to Fan Xu.
Generalized quantum cluster algebras introduced in [1] are quantum deformation of generalized cluster algebras of geometric types. In this paper, we prove that the Laurent phenomenon holds in these generalized quantum cluster algebras. We also show that upper bounds coincide with the corresponding generalized quantum upper cluster algebras under the "coprimality" condition.
Strange quark matter, which is composed of u, d, and s quarks, could be the true ground of matter. According to this hypothesis, compact stars may actually be strange quark stars, and there may even be stable strange quark dwarfs and strange quark planets. The detection of the binary neutron star merger event GW170817 provides us new clues on the equation of state of compact stars. In this study, the tidal deformability of strange quark planets and strange quark dwarfs are calculated. It is found that the tidal deformability of strange quark objects is smaller than that of normal matter counterparts. For a typical 0.6 M$_\odot$ compact star, the tidal deformability of a strange dwarf is about 1.4 times less than that of a normal white dwarf. The difference is even more significant between strange quark planets and normal matter planets. Additionally, if the strange quark planet is a bare one (i.e., not covered by a normal matter curst), the tidal deformability will be extremely small, which means bare strange quark planets will hardly be distorted by tidal forces. Our study clearly proves the effectiveness of identifying strange quark objects via searching for strange quark planets through gravitational-wave observations.
Recently, the Five-hundred-meter Aperture Spherical radio Telescope (FAST) measured the three-dimensional velocity of PSR J0538+2817 in its associated supernova remnant S147 and found a possible spin-velocity alignment in this pulsar. Here we show that the high velocity and the spin-velocity alignment in this pulsar can be explained by the so-called electromagnetic rocket mechanism. In this framework, the pulsar is kicked in the direction of the spin axis, which naturally explains the spin-velocity alignment. We scrutinize the evolution of this pulsar and show that the pulsar kick can create a highly relativistic jet at the opposite direction of the kick velocity. The lifetime and energetics of the jet is estimated. It is argued that the jet can generate a Gamma-ray Burst (GRB). The long term dynamical evolution of the jet is calculated. It is found that the shock radius of the jet should expand to about 32 pc at present, which is well consistent with the observed radius of the supernova remnant S147 ($32.1\pm4.8$ pc). Additionally, our calculations indicate that the current velocity of the GRB remnant should be about 440 km s$^{-1}$, which is also consistent with the observed blast wave velocity of the remnant of S147 (500 km s$^{-1}$).
For gamma-ray bursts (GRBs) with a plateau phase in the X-ray afterglow, a so called $L-T-E$ correlation has been found which tightly connects the isotropic energy of the prompt GRB ($E_{γ,\rm{iso}}$) with the end time of the X-ray plateau ($T_{a}$) and the corresponding X-ray luminosity at the end time ($L_{X}$). Here we show that there is a clear redshift evolution in the correlation. Furthermore, since the power-law indices of $L_{X}$ and $E_{γ,\rm{iso}}$ in the correlation function are almost identical, the $L-T-E$ correlation is insensitive to cosmological parameters and cannot be used as a satisfactory standard candle. On the other hand, based on a sample including 121 long GRBs, we establish a new three parameter correlation that connects $L_{X}$, $T_{a}$ and the spectral peak energy $E_{\rm{p}}$, i.e. the $L-T-E_{\rm{p}}$ correlation. This correlation strongly supports the so-called Combo-relation established by Izzo et al. (2015). After correcting for the redshift evolution, we show that the de-evolved $L-T-E_{\rm{p}}$ correlation can be used as a standard candle. By using this correlation alone, we are able to constrain the cosmological parameters as $Ω_{m}=0.389^{+0.202}_{-0.141}$ ($1σ$) for the flat $Λ$CDM model, or $Ω_{m}=0.369^{+0.217}_{-0.191}$, $w=-0.966^{+0.513}_{-0.678}$ ($1σ$) for the flat $w$CDM model. Combining with other cosmological probes, more accurate constraints on the cosmology models are presented.
Many machine learning frameworks have been proposed and used in wireless communications for realizing diverse goals. However, their incapability of adapting to the dynamic wireless environment and tasks and of self-learning limit their extensive applications and achievable performance. Inspired by the great flexibility and adaptation of primate behaviors due to the brain cognitive mechanism, a unified cognitive learning (CL) framework is proposed for the dynamic wireless environment and tasks. The mathematical framework for our proposed CL is established. Using the public and authoritative dataset, we demonstrate that our proposed CL framework has three advantages, namely, the capability of adapting to the dynamic environment and tasks, the self-learning capability and the capability of 'good money driving out bad money' by taking modulation recognition as an example. The proposed CL framework can enrich the current learning frameworks and widen the applications.
Let $Q$ be the affine quiver of type $\widetilde{A}_{2n-1,1}$ and $\mathcal{A}_{q}(Q)$ be the quantum cluster algebra associated to the valued quiver $(Q,(2,2,\dots,2))$. We prove some cluster multiplication formulas, and deduce that the cluster variables associated with vertices of $Q$ satisfy a quantum analogue of the constant coefficient linear relations. We then construct two bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-bases $\mathcal{B}$ and $\mathcal{S}$ of $\mathcal{A}_{q}(Q)$ consisting of positive elements, and prove that $\mathcal{B}$ is an atomic basis.
We develop and prove the analogs of some results shown in [Berenstein A., Fomin S., Zelevinsky A., Duke Math. J. 126 (2005), 1-52] concerning lower and upper bounds of cluster algebras to the generalized cluster algebras of geometric type. We show that lower bounds coincide with upper bounds under the conditions of acyclicity and coprimality. Consequently, we obtain the standard monomial bases of these generalized cluster algebras. Moreover, in the appendix, we prove that an acyclic generalized cluster algebra is equal to the corresponding generalized upper cluster algebra without the assumption of the existence of coprimality.
Lotus leaves floating on water usually experience short-wavelength edge wrinkling that decays toward the center, while the leaves growing above water normally morph into a global bending cone shape with long rippled waves near the edge. Observations suggest that the underlying water (liquid substrate) significantly affects the morphogenesis of leaves. To understand the biophysical mechanism under such phenomena, we develop mathematical models that can effectively account for inhomogeneous differential growth of floating and free-standing leaves, to quantitatively predict formation and evolution of their morphology. We find, both theoretically and experimentally, that the short-wavelength buckled configuration is energetically favorable for growing membranes lying on liquid, while the global buckling shape is more preferable for suspended ones. Other influencing factors such as stem/vein, heterogeneity and dimension are also investigated. Our results provide a fundamental insight into a variety of plant morphogenesis affected by water foundation and suggest that such surface instabilities can be harnessed for morphology control of biomimetic deployable structures using substrate or edge actuation.
Let $A$ be the path algebra of a finite acyclic quiver $Q$ over a finite field. We realize the quantum cluster algebra with principal coefficients associated to $Q$ as a sub-quotient of a certain Hall algebra involving the category of morphisms between projective $A$-modules.
This paper studies caching in (K+L-1) x K partially connected wireless linear networks, where each of the K receivers locally communicates with L out of the K+L-1 transmitters, and caches are at all nodes. The goal is to design caching and delivery schemes to reduce the transmission latency, by using normalized delivery time (NDT) as the performance metric. For small transmitter cache size (any L transmitters can collectively store the database just once), we propose a cyclic caching strategy so that each of every L consecutive transmitters caches a distinct part of each file; the delivery strategy exploits coded multicasting and interference alignment by introducing virtual receivers. The obtained NDT is within a multiplicative gap of 2 to the optimum in the entire cache size region, and optimal in certain region. For large transmitter cache size (any L transmitters can collectively store the database for multiple copies), we propose a modified caching strategy so that every bit is repeatedly cached at consecutive transmitters; the delivery strategy exploits self-interference cancellation and interference neutralization. By combining these schemes, the NDT is optimal in a larger region. We also extend our results to linear networks with heterogeneous receiver connectivity and partially connected circular networks.
This paper studies the computation-communication tradeoff in a heterogeneous MapReduce computing system where each distributed node is equipped with different computation capability. We first obtain an achievable communication load for any given computation load and any given function assignment at each node. The proposed file allocation strategy has two steps: first, the input files are partitioned into disjoint batches, each with possibly different size and computed by a distinct node; then, each node computes additional files from its non-computed files according to its redundant computation capability. In the Shuffle phase, coded multicasting opportunities are exploited thanks to the repetitive file allocation among different nodes. Based on this scheme, we further propose the computation-aware and the shuffle-aware function assignments. We prove that, by using proper function assignments, our achievable communication load for any given computation load is within a constant multiplicative gap to the optimum in an equivalent homogeneous system with the same average computation load. Numerical results show that our scheme with shuffle-aware function assignment achieves better computation-communication tradeoff than existing works in some cases.
Today's mobile data traffic is dominated by content-oriented traffic. Caching popular contents at the network edge can alleviate network congestion and reduce content delivery latency. This paper provides a comprehensive and unified study of caching and delivery techniques in wireless radio access networks (RANs) with caches at all edge nodes (ENs) and user equipments (UEs). Three cache-aided RAN architectures are considered: RANs without fronthaul, with dedicated fronthaul, and with wireless fronthaul. It first reviews in a tutorial nature how caching facilitates interference management in these networks by enabling interference cancellation (IC), zero-forcing (ZF), and interference alignment (IA). Then, two new delivery schemes are presented. One is for RANs with dedicated fronthaul, which considers centralized cache placement at the ENs but both centralized and decentralized placement at the UEs. This scheme combines IA, ZF, and IC together with soft-transfer fronthauling. The other is for RANs with wireless fronthaul, which considers decentralized cache placement at all nodes. It leverages the broadcast nature of wireless fronthaul to fetch not only uncached but also cached contents to boost transmission cooperation among the ENs. Numerical results show that both schemes outperform existing results for a wide range of system parameters, thanks to the various caching gains obtained opportunistically.
Let $\A$ be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev's theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of $\A$ via its derived Hall algebra and Bridgeland Hall algebra of $m$-cyclic complexes.
Let $Q$ be a finite quiver without loops and $\mathcal{Q}_α$ be the Lusztig category for any dimension vector $α$. The purpose of this paper is to prove that all Frobenius eigenvalues of the $i$-th cohomology $\mathcal{H}^i(\mathcal{L})|_x$ for a simple perverse sheaf $\mathcal{L}\in \mathcal{Q}_α$ and $x\in \mathbb{E}_α^{F^n}=\mathbb{E}_α(\mathbb{F}_{q^n})$ are equal to $(\sqrt{q^n})^{i}$ as a conjecture given by Schiffmann (\cite{Schiffmann2}). As an application, we prove the existence of a class of Hall polynomials.
Super-resolution fluorescence microscopy, with a resolution beyond the diffraction limit of light, has become an indispensable tool to directly visualize biological structures in living cells at a nanometer-scale resolution. Despite advances in high-density super-resolution fluorescent techniques, existing methods still have bottlenecks, including extremely long execution time, artificial thinning and thickening of structures, and lack of ability to capture latent structures. Here we propose a novel deep learning guided Bayesian inference approach, DLBI, for the time-series analysis of high-density fluorescent images. Our method combines the strength of deep learning and statistical inference, where deep learning captures the underlying distribution of the fluorophores that are consistent with the observed time-series fluorescent images by exploring local features and correlation along time-axis, and statistical inference further refines the ultrastructure extracted by deep learning and endues physical meaning to the final image. Comprehensive experimental results on both real and simulated datasets demonstrate that our method provides more accurate and realistic local patch and large-field reconstruction than the state-of-the-art method, the 3B analysis, while our method is more than two orders of magnitude faster. The main program is available at https://github.com/lykaust15/DLBI
Instability patterns of rolling up a sleeve appear more intricate than the ones of walking over a rug on floor, both characterized as uniaxially compressed soft-film/stiff-substrate systems. This can be explained by curvature effects. To investigate pattern transitions on a curved surface, we study a soft shell sliding on a rigid cylinder by experiments, computations and theoretical analyses. We reveal a novel post-buckling phenomenon involving multiple successive bifurcations: smooth-wrinkle-ridge-sagging transitions. The shell initially buckles into periodic axisymmetric wrinkles at the threshold and then a wrinkle-to-ridge transition occurs upon further axial compression. When the load increases to the third bifurcation, the amplitude of the ridge reaches its limit and the symmetry is broken with the ridge sagging into a recumbent fold. It is identified that hysteresis loops and the Maxwell equal-energy conditions are associated with the co-existence of wrinkle/ridge or ridge/sagging patterns. Such a bifurcation scheme is inherently general and independent of material constitutive models.
This paper aims to characterize the synergy of distributed caching and wireless fronthaul in a fog radio access network (Fog-RAN) where all edge nodes (ENs) and user equipments (UEs) have a local cache and store contents independently at random. The network operates in two phases, a file-splitting based decentralized cache placement phase and a fronthaul-aided content delivery phase. We adopt normalized delivery time (NDT) to characterize the asymptotic latency performance with respect to cache size and fronthaul capacity. Both an achievable upper bound and a theoretical lower bound of NDT are obtained, and their multiplicative gap is within 12. In the proposed delivery scheme, we utilize the fronthaul link, by exploiting coded multicasting, to fetch both non-cached and cached contents to boost EN cooperation in the access link. In particular, to fetch contents already cached at ENs, an additional layer of coded multicasting is added on the coded messages desired by UEs in the fronthaul link. Our analysis shows that the proposed delivery scheme can balance the delivery latency between the fronthaul link and access link, and is approximately optimum under decentralized caching.
The objective of the present paper is to prove cluster multiplication theorem in the quantum cluster algebra of type $A_{2}^{(2)}$. As corollaries, we obtain bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-bases established in [6], and naturally deduce the positivity of the elements in these bases. One bar-invariant basis as the triangular basis of this quantum cluster algebra is also explicitly described.