arXiv subjects
Publications and source records attributed to Ji Li.
We study weighted Besov and Triebel--Lizorkin spaces associated with Hermite expansions and obtain (i) frame decompositions, and (ii) characterizations of continuous Sobolev-type embeddings. The weights we consider generalize the Muckhenhoupt weights.
Let $M^{(k)}$, $k=1,2,\ldots, n$, be the boundary of an unbounded polynomial domain $Ω^{(k)}$ of finite type in $\mathbb C ^2$, and let $\Box_b^{(k)}$ be the Kohn Laplacian on $M^{(k)}$. In this paper, we study multivariable spectral multipliers $m(\Box_b^{(1)},\ldots, \Box_b^{(n)})$ acting on the Shilov boundary $\widetilde{M}=M^{(1)} \times\cdots\times M^{(n)}$ of the product domain $Ω^{(1)}\times\cdots\times Ω^{(n)}$. We show that if a function $F(λ_1, \ldots ,λ_n)$ satisfies a Marcinkiewicz-type differential condition, then the spectral multiplier operator $m(\Box_b^{(1)}, \ldots, \Box_b^{(n)})$ is a product Calderón--Zygmund operator of Journé type.
Pre-trained large-scale language models have increasingly demonstrated high accuracy on many natural language processing (NLP) tasks. However, the limited weight storage and computational speed on hardware platforms have impeded the popularity of pre-trained models, especially in the era of edge computing. In this work, we propose an efficient transformer-based large-scale language representation using hardware-friendly block structure pruning. We incorporate the reweighted group Lasso into block-structured pruning for optimization. Besides the significantly reduced weight storage and computation, the proposed approach achieves high compression rates. Experimental results on different models (BERT, RoBERTa, and DistilBERT) on the General Language Understanding Evaluation (GLUE) benchmark tasks show that we achieve up to 5.0x with zero or minor accuracy degradation on certain task(s). Our proposed method is also orthogonal to existing compact pre-trained language models such as DistilBERT using knowledge distillation, since a further 1.79x average compression rate can be achieved on top of DistilBERT with zero or minor accuracy degradation. It is suitable to deploy the final compressed model on resource-constrained edge devices.
We study the dynamics of a circadian oscillator model which was proposed by Tyson, Hong, Thron and Novak. This model indicates a molecular mechanism for the circadian rhythm in Drosophila. After giving a detailed study of the equilibria, we further investigate the effects of the rates of mRNA degradation and synthesis. When the rate of mRNA degradation is rather fast, we prove that there are no periodic orbits in this model. When the rate of mRNA degradation is slow enough, this model is transformed into a slow-fast system. Then based on the geometric singular perturbation theory, we prove the existence of canard explosion, relaxation oscillations, homoclinic/heteroclinic orbits and saddle-node bifurcations as the rates of mRNA degradation and synthesis change. Finally, we give the biological interpretation of the obtained results and point out that this model can be transformed into a Liénard-like equation, which could be helpful to investigate the dynamics of the general case.
We characterize the compactness of commutators in the Bloom setting. Namely, for a suitably non-degenerate Calderón--Zygmund operator $T$, and a pair of weights $ σ, ω\in A_p$, the commutator $ [T, b]$ is compact from $ L ^{p} (σ) \to L ^{p} (ω)$ if and only if $ b \in VMO _{ν}$, where $ ν= (σ/ ω) ^{1/p}$. This extends the work of the first author, Holmes and Wick. The weighted $VMO$ spaces are different from the classical $ VMO$ space. In dimension $ d =1$, compactly supported and smooth functions are dense in $ VMO _{ν}$, but this need not hold in dimensions $ d \geq 2$. Moreover, the commutator in the product setting with respect to little VMO space is also investigated.
In this paper, we study the quantitative weighted bounds for the $q$-variational singular integral operators with rough kernels. The main result is for the sharp truncated singular integrals itself $$ \|V_q\{T_{Ω,\varepsilon}\}_{\varepsilon>0}\|_{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} \|Ω\|_{ L^\infty}(w)_{A_p}^{1+1/q}\{w\}_{A_p},$$ where the quantity $(w)_{A_p}$, $\{w\}_{A_p}$ will be recalled in the introduction; we do not know whether this is sharp, but it is the best known quantitative result for this class of operators, since when $q=\infty$, it coincides with the best known quantitative bounds by Di Pilino--Hytönen--Li or Lerner. In the course of establishing the above estimate, we obtain several quantitative weighted bounds which are of independent interest. We hereby highlight two of them. The first one is $$ \|V_q\{ϕ_k\ast T_Ω\}_{k\in\mathbb Z}\|_{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} \|Ω\|_{ L^\infty}(w)_{A_p}^{1+1/q}\{w\}_{A_p},$$ where $ϕ_k(x)=\frac1{2^{kn}}ϕ(\frac x{2^k})$ with $ϕ\in C^\infty_c(\mathbb R^n)$ being any non-negative radial function, and the sharpness for $q=\infty$ is due to Lerner; the second one is $$ \|\mathcal{S}_q\{T_{Ω,\varepsilon}\}_{\varepsilon>0}\|_{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} \|Ω\|_{ L^\infty}(w)_{A_p}^{1/q}\{w\}_{A_p},$$ and the sharpness for $q=\infty$ follows from the Hardy--Littlewood maximal function.
In this paper we study the boundedness and compactness characterizations of the commutator of Calderón-Zygmund operators $T$ on spaces of homogeneous type $(X,d,μ)$ in the sense of Coifman and Weiss. More precisely, We show that the commutator $[b, T]$ is bounded on weighted Morrey space $L_ω^{p,κ}(X)$ ($κ\in(0,1), ω\in A_{p}(X), 1<p<\infty$) if and only if $b$ is in the BMO space. Moreover, the commutator $[b, T]$ is compact on weighted Morrey space $L_ω^{p,κ}(X)$ ($κ\in(0,1), ω\in A_{p}(X), 1<p<\infty$) if and only if $b$ is in the VMO space.
We consider double canard cycles including two canards in singularly perturbed planar systems with two canard points. Previous work studied the complex oscillations including relaxation oscillations and canard cycles in singularly perturbed planar systems with one-parameter layer equations, which have precisely one canard point, two jump points or one canard point and one jump point. Based on the normal form theory, blow-up technique and Melnikov theory, we investigate double canard cycles induced by two Hopf breaking mechanisms at two non-degenerate canard points. Finally, we apply the obtained results to a class of cubic Lienard equations with quadratic damping.
In natural language processing (NLP), the "Transformer" architecture was proposed as the first transduction model replying entirely on self-attention mechanisms without using sequence-aligned recurrent neural networks (RNNs) or convolution, and it achieved significant improvements for sequence to sequence tasks. The introduced intensive computation and storage of these pre-trained language representations has impeded their popularity into computation and memory-constrained devices. The field-programmable gate array (FPGA) is widely used to accelerate deep learning algorithms for its high parallelism and low latency. However, the trained models are still too large to accommodate to an FPGA fabric. In this paper, we propose an efficient acceleration framework, Ftrans, for transformer-based large scale language representations. Our framework includes enhanced block-circulant matrix (BCM)-based weight representation to enable model compression on large-scale language representations at the algorithm level with few accuracy degradation, and an acceleration design at the architecture level. Experimental results show that our proposed framework significantly reduces the model size of NLP models by up to 16 times. Our FPGA design achieves 27.07x and 81x improvement in performance and energy efficiency compared to CPU, and up to 8.80x improvement in energy efficiency compared to GPU.
The goal of this paper is to develop a high order numerical method based on Kinetic Inviscid Flux (KIF) method and Flux Reconstruction (FR) framework. The KIF aims to find a balance between the excellent merits of Gas-Kinetic Scheme (GKS) and the lower computational costs. The idea of KIF can be viewed as an inviscid-viscous splitting version of the gas-kinetic scheme, and Shu and Ohwada have made the fundamental contribution. The combination of Totally Thermalized Transport (TTT) scheme and Kinetic Flux Vector Splitting (KFVS) method are achieved in KIF. Using a coefficient which is related to time step $δt$ and averaged collision time $τ$, KIF can adjust the weights of TTT and KFVS flux in the simulation adaptively. By doing the inviscid-viscous splitting, KIF is very suitable and easy to integrate into the existing framework. The well understood FR framework is used widely for the advantages of robustness, economical costs and compactness. The combination of KIF and FR is originated by three motivations. The first purpose is to develop a high order method based on the gas kinetic theory. The second reason is to keep the advantages of GKS. The last aim is that the designed method should be more efficient. In present work, we use the KIF method to replace the Riemann flux solver applied in the interfaces of elements. The common solution at the interface is computed according to the gas kinetic theory, which makes the combination of KIF and FR scheme more reasonable and available. The accuracy and performance of present method are validated by several numerical cases. The Taylor-Green vortex problem has been used to verify its potential to simulate turbulent flows.
We investigate the dynamics of a three-dimensional system modeling a molecular mechanism for the circadian rhythm in Drosophila. We first prove the existence of a compact attractor in the region with biological meaning. Under the assumption that the dimerization reactions are fast, in this attractor we reduce the three-dimensional system to a simpler two-dimensional system on the persistent normally hyperbolic slow manifold.
We consider weighted $L^p(w)$ boundedness ($1<p<\infty $ and $w$ a Muckenhoupt $A_p$ weight) of the Calderón commutator $\mathcal C_Ω$ associated with rough homogeneous kernel, under the condition $Ω\in L^q(\mathbb S^{n-1})$ for $q_0<q\leq\infty$ with $q_0$ a fixed constant depending on $w$. Comparing to the previous related known results (assuming $Ω\in L^\infty(\mathbb S^{n-1})$), our result for $Ω\in L^q(\mathbb S^{n-1})$ with $q$ in the range $(q_0,\infty)$ is new. We also obtain a quantitative weighted bound for this $\mathcal C_Ω$ on $L^p(w)$, which is the best known quantitative result for this class of operators.
The Degasperis-Procesi equation is the integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the Desgasperis-Procesi (DP) equation on the real line. %extending our previous work on their spectral stability \cite{LLW}. The main difficulty stems from the fact that the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the $L^2$-norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. The remedy is to observe that, given a sufficiently smooth initial condition satisfying a measurable constraint, the $L^\infty$ orbital norm of the perturbation is bounded above by a function of its $L^2$ orbital norm, yielding the orbital stability in the $L^2\cap L^\infty$ space.
In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood--Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel--Pólya inequalities for flag quadratic forms; (2) introducing the maximal function and area function via flag Poisson kernels and flag version of harmonic functions; (3) developing an atomic decomposition via the finite speed propagation and area function in terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the multi-parameter Hardy space with the flag structure.
Let $(X,d,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, i.e. $d$ is a quasi metric on $X$ and $μ$ is a positive measure satisfying the doubling condition. Suppose that $u$ and $v$ are two locally finite positive Borel measures on $(X,d,μ)$. Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón--Zygmund operator $T$ from $L^{2}(u)$ to $L^{2}(v)$ in terms of the $A_{2}$ condition and two testing conditions. For every cube $B\subset X$, we have the following testing conditions, with $\mathbf{1}_{B}$ taken as the indicator of $B$ \begin{equation*} \Vert T(u\mathbf{1}_{B})\Vert _{L^{2}(B, v)}\leq \mathcal{T}\Vert 1_{B}\Vert _{L^{2}(u)}, \end{equation*} \begin{equation*} \Vert T^{\ast }(v\mathbf{1}_{B})\Vert _{L^{2}(B, u)}\leq \mathcal{T}\Vert 1_{B}\Vert _{L^{2}(v)}. \end{equation*} The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.
We present a sample of $\sim$ 140,000 primary red clump (RC) stars of spectral signal-to-noise ratios higher than 20 from the LAMOST Galactic spectroscopic surveys, selected based on their positions in the metallicity-dependent effective temperature--surface gravity and color--metallicity diagrams, supervised by high-quality $Kepler$ asteroseismology data. The stellar masses and ages of those stars are further determined from the LAMOST spectra, using the Kernel Principal Component Analysis method, trained with thousands of RCs in the LAMOST-$Kepler$ fields with accurate asteroseismic mass measurements. The purity and completeness of our primary RC sample are generally higher than 80 per cent. For the mass and age, a variety of tests show typical uncertainties of 15 and 30 per cent, respectively. Using over ten thousand primary RCs with accurate distance measurements from the parallaxes of Gaia DR2, we re-calibrate the $K_{\rm s}$ absolute magnitudes of primary RCs by, for the first time, considering both the metallicity and age dependencies. With the the new calibration, distances are derived for all the primary RCs, with a typical uncertainty of 5--10 per cent, even better than the values yielded by the Gaia parallax measurements for stars beyond 3--4 kpc. The sample covers a significant volume of the Galactic disk of $4 \leq R \leq 16$ kpc, $|Z| \leq 5$ kpc, and $-20 \leq ϕ\leq 50^{\circ}$. Stellar atmospheric parameters, line-of-sight velocities and elemental abundances derived from the LAMOST spectra and proper motions of Gaia DR2 are also provided for the sample stars. Finally, the selection function of the sample is carefully evaluated in the color-magnitude plane for different sky areas. The sample is publicly available.
Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p'_0)$-estimates of order $m$ for some $1\leq p_0 < 2$. In this paper we prove {\it sharp} endpoint $L^p$-Sobolev bound for the Schrödinger group $e^{itL}$, that is for every $p\in (p_0, p'_0)$ there exists a constant $C=C(n,p)>0$ independent of $t$ such that \begin{eqnarray*} \left\| (I+L)^{-{s}}e^{itL} f\right\|_{p} \leq C(1+|t|)^{s}\|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ s\geq n\big|{1\over 2}-{1\over p}\big|. \end{eqnarray*} As a consequence, the above estimate holds for all $1<p<\infty$ when the heat kernel of $L$ satisfies a Gaussian upper bound. This extends classical results due to Feffermann and Stein, and Miyachi for the Laplacian on the Euclidean spaces ${\mathbb R}^n$. We also give an application to obtain an endpoint estimate for $L^p$-boundedness of the Riesz means of the solutions of the Schrödinger equations.
A knot theoretic algorithm is proposed to model `fragile topology' of quantum physics.