arXiv subjects
Publications and source records attributed to Fang Li.
A cluster automorphism is a $\mathbb{Z}$-algebra automorphism of a cluster algebra $\mathcal A$ satisfying that it sends a cluster to another and commutes with mutations. Chang and Schiffler conjectured that a cluster automorphism of $\mathcal A$ is just a $\mathbb{Z}$-algebra homomorphism of a cluster algebra sending a cluster to another. The aim of this article is to prove this conjecture.
We introduce some modified forms for the degenerate and non-degenerate affine Hecke algebras of type $A$. These are certain subalgebras living inside the inverse limit of cyclotomic Hecke algebras. We construct faithful representations and standard bases for these algebras and give some explicit description of their centers. We show that there are algebra isomorphisms between some generalized Ore localizations of these modified affine Hecke algebras and of the quiver Hecke algebras of type $A$. As an application, we show that the center conjecture for the cyclotomic quiver Hecke algebra of type $A$ holds if and only if the center conjecture for the cyclotomic Hecke algebra of type $A$ holds.
Spectrally uncorrelated biphoton state generated from the spontaneous nonlinear optical process is an important resource for quantum information. Currently such spectrally uncorrelated biphoton state can only be prepared from limited kinds of nonlinear media, thus limiting their wavelengths. In order to explore wider wavelength range, here we theoretically study the generation of spectrally uncorrelated biphoton state from 14 isomorphs of potassium dihydrogen phosphate (KDP) crystal. We find that 11 crystals from the `KDP family' still maintain similar nonlinear optical properties of KDP, such as KDP, DKDP, ADP, DADP, ADA, DADA, RDA, DRDA, RDP, DRDP and KDA, which satisfy 3 kinds of the group-velocity matching conditions for spectrally uncorrelated biphoton state generation from near-infrared to telecom wavelengths. Based on the uncorrelated biphoton state, we investigate the generation of heralded pure-state single photon by detecting one member of the biphoton state to herald the output of the other. The purity of the heralded single photon is as high as 0.98 without using a narrow-band filter; the Hong-Ou-Mandel interference from independent sources can also achieve a visibility of 98%. This study may provide more and better single-photon sources for quantum information processing at near-infrared and telecom wavelengths.
We prove that any skew-symmetrizable cluster algebra is unistructural, which is a conjecture by Assem, Schiffler, and Shramchenko. As a corollary, we obtain that a cluster automorphism of a cluster algebra $\mathcal A(\mathcal S)$ is just an automorphism of the ambient field $\mathcal F$ which restricts to a permutation of cluster variables of $\mathcal A(S)$.
In this paper, we introduce the enough $g$-pairs property for a principal coefficients cluster algebra, which can be understood as a strong version of the sign-coherence of the $G$-matrices. Then we prove that any skew-symmetrizable principal coefficients cluster algebra has the enough $g$-pairs property. As an application, we prove the positivity of denominator vectors for any skew-symmetrizable cluster algebra. In fact, we give complete answers to some long standing conjectures on denominator vectors of cluster variables (see Conjecture 1.1 below), which are proposed by Fomin and Zelevinsky in [Compos. Math. 143(2007), 112-164]. In addition, we prove that the seeds whose clusters contain particular cluster variables form a connected subgraph of the exchange graph of this cluster algebra. Lastly, a criterion to distinguish whether particular cluster variables belong to one common cluster is given.
We introduce and study a class of free boundary models with "nonlocal diffusion", which are natural extensions of the free boundary models in Du and Lin [17] and elsewhere, where "local diffusion" is used to describe the population dispersal, with the free boundary representing the spreading front of the species. We show that this nonlocal problem has a unique solution defined for all time, and then examine its long-time dynamical behavior when the growth function is of Fisher-KPP type. We prove that a spreading-vanishing dichotomy holds, though for the spreading-vanishing criteria significant differences arise from the well known local diffusion model in Du and Lin [17].
Shortcuts to adiabaticity (STA) provide an alternative to adiabatic protocols to guide the dynamics of the system of interest without the requirement of slow driving. We report the controlled speedup via STA of the nonadiabatic dynamics of a Fermi gas, both in the non-interacting and strongly coupled, unitary regimes. Friction-free superadiabatic expansion strokes, with no residual excitations in the final state, are demonstrated in the unitary regime by engineering the modulation of the frequencies and aspect ratio of the harmonic trap. STA are also analyzed and implemented in the high-temperature regime, where the shear viscosity plays a pivotal role and the Fermi gas is described by viscous hydrodynamics.
In Chinese Spallation Neutron Source (CSNS), proton beam is used to hit metal tungsten target, and then high flux neutron are generated for experiments on instruments. For neutron flux spectrum correction, the current of proton beam is for each instrument. A real time monitor system is developed in CSNS, to monitor, broadcast and record each pulse of proton. Each proton pulse charge is measured and marked with high-precision timestamp. Then, the result of measurement will be broadcasted to control room and each neutron instrument. In control room, the proton charge of each pulse is listened and stored in database by agent program for offline use. The high-precision timestamp can be used to proton charge and neutron data alignment in time scale. The architecture of proton beam monitor system is introduced in this paper. And the performance of this system is evaluated in this paper.
In this paper, we prove that positivity of denominator vectors holds for any skew-symmetric cluster algebra.
In this paper, we prove that each matrix in $M_{m\times n}(\mathbb Z_{\geq0})$ is uniformly column sign-coherent with respect to any $n\times n$ skew-symmetrizable integer matrix. Using such matrices, we introduce the definition of irreducible skew-symmatrizable matrix. Based on this, the existence of a maximal green sequence for a skew-symmetrizable matrices is reduced to the existence of a maximal green sequence for irreducible skew-symmetrizable matrices.
We mainly introduce an abstract pattern to study cluster algebras. Cluster algebras, generalized cluster algebras and Laurent phenomenon algebras are unified in the language of generalized Laurent phenomenon algebras (briefly, GLP algebras) from the perspective of Laurent phenomenon. In this general framework, we firstly prove that each positive and d-vector-positive GLP algebra has the proper Laurent monomial property and thus its cluster monomials are linearly independent. Skew-symmetric cluster algebras are verified to be d-vector-positive, which gives the affirmation of Conjecture [conjd] in [FZ3] in this case. And since the positivity of skew-symmetric cluster algebras is well-known, the new proof is obtained for the linearly independence of cluster monomials of skew-symmetric cluster algebras. For a class of GLP algebras which are pointed, g-vectors of cluster monomials are defined. We verify that for any positive GLP algebra pointed at $t_0$, the g-vectors ${\bf g}_{1;t}^{t_0},\cdots, {\bf g}_{n;t}^{t_0}$ form a $\mathbb Z$-basis of $\mathbb Z^n$, and different cluster monomials have different g-vectors. As a direct application, we verify that Conjecture [conjg] in [FZ3] holds for skew-symmetrizable cluster algebras and acyclic sign-skew-symmetric cluster algebras.
For a skew-symmetrizable cluster algebra $\mathcal A_{t_0}$ with principal coefficients at $t_0$, we prove that each seed $Σ_t$ of $\mathcal A_{t_0}$ is uniquely determined by its {\bf C-matrix}, which was proposed by Fomin and Zelevinsky in \cite{FZ3} as a conjecture. Our proof is based on the fact that the positivity of cluster variables and sign-coherence of $c$-vectors hold for $\mathcal A_{t_0}$, which was actually verified in \cite{GHKK}. More discussion is given in the sign-skew-symmetric case so as to obtain a conclusion as weak version of the conjecture in this general case.
Optimal performance of thermal machines is reached by suppressing friction. Friction in quantum thermodynamics results from fast driving schemes that generate nonadiabatic excitations. The far-from-equilibrium dynamics of quantum devices can be tailored by shortcuts to adiabaticity to suppress quantum friction. We experimentally demonstrate friction-free superadiabatic strokes with a trapped unitary Fermi gas as a working substance and establish the equivalence between the superadiabatic mean work and its adiabatic value.
We report an observation of a dynamical super Efimovian expansion in a two-component strongly interacting Fermi gas by engineering time dependent external harmonic trap frequencies. When trap frequency is followed as $[1/4t^2+1/t^2λ\log(t/t_*)]^{1/2}$, where $t_*$ and $λ$ are two control parameters, and the change is faster than a critical value, the expansion of such the quantum gas shows a novel dynamics due to its spatial and dynamical scaling symmetry. A clear double-log periodicity, which is a hallmark of the super Efimov effect, is emergent for the cloud size in the expansion. The universality of such scaling dynamics is verified both in the non-interacting limit and in the unitarity limit. Observing super-Efmovian evolution represents a paradigm in probing universal properties and allows in a new way to study many-body nonequilibrium dynamics with experiments.
The GANs promote an adversarive game to approximate complex and jointed example probability. The networks driven by noise generate fake examples to approximate realistic data distributions. Later the conditional GAN merges prior-conditions as input in order to transfer attribute vectors to the corresponding data. However, the CGAN is not designed to deal with the high dimension conditions since indirect guide of the learning is inefficiency. In this paper, we proposed a network ResGAN to generate fine images in terms of extremely degenerated images. The coarse images aligned to attributes are embedded as the generator inputs and classifier labels. In generative network, a straight path similar to the Resnet is cohered to directly transfer the coarse images to the higher layers. And adversarial training is circularly implemented to prevent degeneration of the generated images. Experimental results of applying the ResGAN to datasets MNIST, CIFAR10/100 and CELEBA show its higher accuracy to the state-of-art GANs.
Using the unfolding method given in \cite{HL}, we prove the conjectures on sign-coherence and a recurrence formula respectively of ${\bf g}$-vectors for acyclic sign-skew-symmetric cluster algebras. As a following consequence, the conjecture is affirmed in the same case which states that the ${\bf g}$-vectors of any cluster form a basis of $\mathbb Z^n$. Also, the additive categorification of an acyclic sign-skew-symmetric cluster algebra $\mathcal A(Σ)$ is given, which is realized as $(\mathcal C^{\widetilde Q},Γ)$ for a Frobenius $2$-Calabi-Yau category $\mathcal C^{\widetilde Q}$ constructed from an unfolding $(Q,Γ)$ of the acyclic exchange matrix $B$ of $\mathcal A(Σ)$.
In this paper, we gives a complete classification of the global dynamics of two- species Lotka-Volterra competition models with nonlocal dispersals: where K, P represent nonlocal operators, under the assumptions that the nonlo- cal operators are symmetric, the models admit two semi-trivial steady states and 0<bc<1. In particular, when both semi-trivial steady states are locally stable, it is proved that there exist infinitely many steady states and the solution with non- negative and nontrivial initial data converges to some steady state. Furthermore, we generalize these results to the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals.
In this paper, we mainly consider the long-time behavior of solutions for the Cahn-Hilliard-Navier-Stokes system with dynamic boundary conditions and two polynomial growth nonlinearities of arbitrary order. We prove the existence of a finite dimensional global attractor for the Cahn-Hilliard-Navier-Stokes system with dynamic boundary conditions by using the $\ell$-trajectories method.