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Ji Li

Publications and source records attributed to Ji Li.

At least 217 records · Page 12Linked to original sources
arXiv2017

Equivalence of Littlewood-Paley square function and area function characterizations of weighted product Hardy spaces associated to operators

Let $L_{1}$ and $L_{2}$ be non-negative self-adjoint operators acting on $L^{2}(X_{1})$ and $L^{2}(X_{2})$, respectively, where $X_{1}$ and $X_{2}$ are spaces of homogeneous type. Assume that $L_{1}$ and $L_{2}$ have Gaussian heat kernel bounds. This paper aims to study some equivalent characterizations of the weighted product Hardy spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ associated to $L_{1}$ and $L_{2}$, for $p \in (0, \infty)$ and the weight $w$ belongs to the product Muckenhoupt class $A_{\infty}(X_{1} \times X_{2})$. Our main result is that the spaces $H^{p}_{w,L_{1},L_{2}}(X_{1}\times X_{2})$ introduced via area functions can be equivalently characterized by Littlewood-Paley $g$-functions, Littlewood-Paley $g^{\ast}_{λ_{1}, λ_{2}}$-functions, and Peetre type maximal functions, without any further assumptions beyond the Gaussian upper bounds on the heat kernels of $L_{1}$ and $L_{2}$. Our results are new even in the unweighted product setting.

math.CA↗
arXiv2017

Commutators, Little BMO and Weak Factorization

In this paper, we provide a direct and constructive proof of weak factorization of $h^1(\mathbb{R})$ (the predual of little BMO space bmo$(\mathbb{R}\times\mathbb{R})$ studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every $f\in h^1(\mathbb{R}\times\mathbb{R})$ there exist sequences $\{α_j^k\}\in\ell^1$ and functions $g_j^k,h^k_j\in L^2(\mathbb{R}^2)$ such that \begin{align*} f=\sum_{k=1}^\infty\sum_{j=1}^\inftyα^k_j\Big(\, h^k_j H_1H_2 g^k_j - g^k_j H_1H_2 h^k_j\Big) \end{align*} in the sense of $h^1(\mathbb{R})$, where $H_1$ and $H_2$ are the Hilbert transforms on the first and second variable, respectively. Moreover, the norm $\|f\|_{h^1(\mathbb{R}\times\mathbb{R})}$ is given in terms of $\|g^k_j\|_{L^2(\mathbb{R}^2)}$ and $\|h^k_j\|_{L^2(\mathbb{R}^2)}$. By duality, this directly implies a lower bound on the norm of the commutator $[b,H_1H_2]$ in terms of $\|b\|_{{\rm bmo}(\mathbb{R}\times\mathbb{R})}$. Our method bypasses the use of analyticity and the Fourier transform, and hence can be extended to the higher dimension case in an arbitrary $n$-parameter setting for the Riesz transforms.

math.CA↗
arXiv2017

Two weight Commutators in the Dirichlet and Neumann Laplacian settings

In this paper we establish the characterization of the weighted BMO via two weight commutators in the settings of the Neumann Laplacian $Δ_{N_+}$ on the upper half space $\mathbb{R}^n_+$ and the reflection Neumann Laplacian $Δ_N$ on $\mathbb{R}^n$ with respect to the weights associated to $Δ_{N_+}$ and $Δ_{N}$ respectively. This in turn yields a weak factorization for the corresponding weighted Hardy spaces, where in particular, the weighted class associated to $Δ_{N}$ is strictly larger than the Muckenhoupt weighted class and contains non-doubling weights. In our study, we also make contributions to the classical Muckenhoupt--Wheeden weighted Hardy space (BMO space respectively) by showing that it can be characterized via area function (Carleson measure respectively) involving the semigroup generated by the Laplacian on $\mathbb{R}^n$ and that the duality of these weighted Hardy and BMO spaces holds for Muckenhoupt $A^p$ weights with $p\in (1,2]$ while the previously known related results cover only $p\in (1,{n+1\over n}]$. We also point out that this two weight commutator theorem might not be true in the setting of general operators $L$, and in particular we show that it is not true when $L$ is the Dirichlet Laplacian $Δ_{D_+}$ on $\mathbb{R}^n_+$.

math.AP↗
arXiv2017

Product Hardy, BMO spaces and iterated commutators associated with Bessel Schrödinger operators

In this paper we establish the product Hardy spaces associated with the Bessel Schrödinger operator introduced by Muckenhoupt and Stein, and provide equivalent characterizations in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions, and Littlewood--Paley theory, which are consistent with the classical product Hardy space theory developed by Chang and Fefferman. Moreover, in this specific setting, we also provide another characterization via the Telyakovskií transform, which further implies that the product Hardy space associated with this Bessel Schrödinger operator is isomorphic to the subspace of suitable "odd functions" in the standard Chang--Fefferman product Hardy space. Based on the characterizations of these product Hardy spaces, we study the boundedness of the iterated commutator of the Bessel Riesz transforms and functions in the product BMO space associated with Bessel Schrödinger operator. We show that this iterated commutator is bounded above, but does not have a lower bound.

math.CA↗
arXiv2017

Carleson measures, BMO spaces and balayages associated to Schrodinger operators

Let $Ł$ be a Schrödinger operator of the form $Ł=-Δ+V$ acting on $L^2(\mathbb R^n)$, $n\geq3$, where the nonnegative potential $V$ belongs to the reverse Hölder class $B_q$ for some $q\geq n.$ Let ${\rm BMO}_{\mathcal{L}}(\RR)$ denote the BMO space associated to the Schrödinger operator $Ł$ on $\RR$. In this article we show that for every $f\in {\rm BMO}_{\mathcal{L}}(\RR)$ with compact support, then there exist $g\in L^{\infty}(\RR)$ and a finite Carleson measure $μ$ such that $$ f(x)=g(x) + S_{μ, {\mathcal P}}(x) $$ with $\|g\|_{\infty} +\||μ\||_{c}\leq C \|f\|_{{\rm BMO}_{\mathcal{L}}(\RR)},$ where $$ S_{μ, {\mathcal P}}=\int_{{\mathbb R}^{n+1}_+} {\mathcal P}_t(x,y) dμ(y, t), $$ and ${\mathcal P}_t(x,y)$ is the kernel of the Poisson semigroup $\{e^{-t\sqrtŁ}\}_{t> 0} $ on $L^2(\mathbb R^n)$. Conversely, if $μ$ is a Carleson measure, then $S_{μ, {\mathcal P}}$ belongs to the space ${\rm BMO}_{\mathcal{L}}(\RR)$. This extends the result for the classical John--Nirenberg BMO space by Carleson \cite{C} (see also \cite{U,GJ,W}) to the BMO setting associated to Schrödinger operators.

math.AP↗
arXiv2017

Hardware-Driven Nonlinear Activation for Stochastic Computing Based Deep Convolutional Neural Networks

Recently, Deep Convolutional Neural Networks (DCNNs) have made unprecedented progress, achieving the accuracy close to, or even better than human-level perception in various tasks. There is a timely need to map the latest software DCNNs to application-specific hardware, in order to achieve orders of magnitude improvement in performance, energy efficiency and compactness. Stochastic Computing (SC), as a low-cost alternative to the conventional binary computing paradigm, has the potential to enable massively parallel and highly scalable hardware implementation of DCNNs. One major challenge in SC based DCNNs is designing accurate nonlinear activation functions, which have a significant impact on the network-level accuracy but cannot be implemented accurately by existing SC computing blocks. In this paper, we design and optimize SC based neurons, and we propose highly accurate activation designs for the three most frequently used activation functions in software DCNNs, i.e, hyperbolic tangent, logistic, and rectified linear units. Experimental results on LeNet-5 using MNIST dataset demonstrate that compared with a binary ASIC hardware DCNN, the DCNN with the proposed SC neurons can achieve up to 61X, 151X, and 2X improvement in terms of area, power, and energy, respectively, at the cost of small precision degradation.In addition, the SC approach achieves up to 21X and 41X of the area, 41X and 72X of the power, and 198200X and 96443X of the energy, compared with CPU and GPU approaches, respectively, while the error is increased by less than 3.07%. ReLU activation is suggested for future SC based DCNNs considering its superior performance under a small bit stream length.

cs.CV↗
arXiv2017

Characterizations of $H^1_{Δ_N}(\mathbb{R}^n)$ and $\rm BMO_{Δ_N}(\mathbb{R}^n)$ via Weak Factorizations and Commutators

This paper provides a deeper study of the Hardy and $\rm BMO$ spaces associated to the Neumann Laplacian $Δ_N$. For the Hardy space $H^1_{Δ_N}(\mathbb{R}^n)$ (which is a proper subspace of the classical Hardy space $H^1(\mathbb{R}^n)$) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space ${\rm BMO}_{Δ_N}(\mathbb{R}^n)$ (which contains the classical $\rm BMO(\mathbb{R}^n)$) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neumann Laplacian on $L^\infty(\mathbb{R}^n)$ functions and in terms of the behavior of the commutator with the Riesz transforms. The results obtained extend many of the fundamental results known for $H^1(\mathbb{R}^n)$ and $\rm BMO(\mathbb{R}^n)$.

math.CA↗
arXiv2017

Dynamics of the localized nonlinear waves in spin-1 Bose-Einstein condensates with time-space modulation

We investigate the dynamics of the localized nonlinear matter wave in spin-1 Bose-Einstein condensates with trapping potentials and nonlinearities dependent on time and space. We solve the three coupled Gross-Pitaevskii equation by similarity transformation and obtain two families of exact matter wave solutions in terms of Jacobi elliptic functions and Mathieu equation. The localized states of the spinor matter wave describe the dynamics of vector breathing solitons, moving breathing solitons, quasibreathing solitons and resonant solitons. The results of stability show that one order vector breathing solitons, quasibreathing solitons, resonant solitons, and the moving breathing solitons ψ_{\pm1} are all stable but the moving breathing solitons ψ_0 is unstable. We also present the experimental parameters to realize these phenomena in the future experiments.

nlin.PS↗
arXiv2017

SC-DCNN: Highly-Scalable Deep Convolutional Neural Network using Stochastic Computing

With recent advancing of Internet of Things (IoTs), it becomes very attractive to implement the deep convolutional neural networks (DCNNs) onto embedded/portable systems. Presently, executing the software-based DCNNs requires high-performance server clusters in practice, restricting their widespread deployment on the mobile devices. To overcome this issue, considerable research efforts have been conducted in the context of developing highly-parallel and specific DCNN hardware, utilizing GPGPUs, FPGAs, and ASICs. Stochastic Computing (SC), which uses bit-stream to represent a number within [-1, 1] by counting the number of ones in the bit-stream, has a high potential for implementing DCNNs with high scalability and ultra-low hardware footprint. Since multiplications and additions can be calculated using AND gates and multiplexers in SC, significant reductions in power/energy and hardware footprint can be achieved compared to the conventional binary arithmetic implementations. The tremendous savings in power (energy) and hardware resources bring about immense design space for enhancing scalability and robustness for hardware DCNNs. This paper presents the first comprehensive design and optimization framework of SC-based DCNNs (SC-DCNNs). We first present the optimal designs of function blocks that perform the basic operations, i.e., inner product, pooling, and activation function. Then we propose the optimal design of four types of combinations of basic function blocks, named feature extraction blocks, which are in charge of extracting features from input feature maps. Besides, weight storage methods are investigated to reduce the area and power/energy consumption for storing weights. Finally, the whole SC-DCNN implementation is optimized, with feature extraction blocks carefully selected, to minimize area and power/energy consumption while maintaining a high network accuracy level.

cs.CV↗
arXiv2017

LAMOST Spectroscopic Survey of the Galactic Anticentre (LSS-GAC): the second release of value-added catalogues

We present the second release of value-added catalogues of the LAMOST Spectroscopic Survey of the Galactic Anticentre (LSS-GAC DR2). The catalogues present values of radial velocity $V_{\rm r}$, atmospheric parameters --- effective temperature $T_{\rm eff}$, surface gravity log$g$, metallicity [Fe/H], $α$-element to iron (metal) abundance ratio [$α$/Fe] ([$α$/M]), elemental abundances [C/H] and [N/H], and absolute magnitudes ${\rm M}_V$ and ${\rm M}_{K_{\rm s}}$ deduced from 1.8 million spectra of 1.4 million unique stars targeted by the LSS-GAC since September 2011 until June 2014. The catalogues also give values of interstellar reddening, distance and orbital parameters determined with a variety of techniques, as well as proper motions and multi-band photometry from the far-UV to the mid-IR collected from the literature and various surveys. Accuracies of radial velocities reach 5kms$^{-1}$ for late-type stars, and those of distance estimates range between 10 -- 30 per cent, depending on the spectral signal-to-noise ratios. Precisions of [Fe/H], [C/H] and [N/H] estimates reach 0.1dex, and those of [$α$/Fe] and [$α$/M] reach 0.05dex. The large number of stars, the contiguous sky coverage, the simple yet non-trivial target selection function and the robust estimates of stellar radial velocities and atmospheric parameters, distances and elemental abundances, make the catalogues a valuable data set to study the structure and evolution of the Galaxy, especially the solar-neighbourhood and the outer disk.

astro-ph.GA↗
arXiv2016

Implementation of dual time stepping and GMRES of implicit gas-kinetic scheme for unsteady flow simulations

A dual time-stepping strategy of gas-kinetic scheme for the simulation of unsteady flows is introduced in this work. The dual time-stepping strategy is always used in the approaches of unsteady flows, and the ability of dual time-stepping to accelerate the computation with acceptable error tolerance is evident. In our paper, we adopt the techniques of dual time-stepping methods for implicit gas-kinetic scheme to simulate the unsteady flows, which is very popular for the numerical methods based on the Navier-Stokes equations. It is carried out by (a) solving the gas-kinetic scheme in finite volume method; (b) obtaining the inviscid flux Jacobian by Roe scheme; (c) involving the computation of viscous flux Jacobian which is not mentioned in the previous implicit gas-kinetic schemes; (d) approximating the linear system of pseudo steady state by generalized minimal residual algorithm (GMRES). The explicit gas-kinetic scheme has been proved to be an accurate approach for both the steady and unsteady flows, and the implicit gas-kinetic scheme is also be developed to accelerate the convergence of steady state. The dual time-stepping method proposed in our study is of great importance to the computations of unsteady flows. Several numerical cases are performed to evaluate the behavior of dual time stepping strategy of gas-kinetic scheme. The incompressible flow around blunt bodies (stationary circular cylinder and square cylinder) and the transonic buffet on the NACA0012 airfoil are simulated to demonstrate the overall performance of the proposed method which is applicable to the fluid flows from laminar to turbulent and from incompressible to compressible.

physics.flu-dyn↗
arXiv2016

Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups

The Marcinkiewicz multipliers are $L^{p}$ bounded for $1<p<\infty $ on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ (Müller, Ricci and Stein \cite{MRS}). This is surprising in the sense that these multipliers are invariant under a two parameter group of dilations on $\mathbb{C}^{n}\times \mathbb{R}$, while there is \emph{no} two parameter group of \emph{automorphic} dilations on $\mathbb{H} ^{n}$. The purpose of this paper is to establish a theory of the flag Lipschitz space on the Heisenberg group $\mathbb{H}^{n}\simeq \mathbb{C}^{n}\times \mathbb{R}$ in the sense `intermediate' between the classical Lipschitz space on the Heisenberg group $\mathbb{H}^{n}$ and the product Lipschitz space on $\mathbb{C}^{n}\times \mathbb{R}$. We characterize this flag Lipschitz space via the Littelewood-Paley theory and prove that flag singular integral operators, which include the Marcinkiewicz multipliers, are bounded on these flag Lipschitz spaces.

math.FA↗
arXiv2016

Weighted estimates for powers and Smoothing estimates of Schrödinger operators with inverse-square potentials

Let $\mathcal{L}_a$ be a Schrödinger operator with inverse square potential $a|x|^{-2}$ on $\mathbb{R}^d, d\geq 3$. The main aim of this paper is to prove weighted estimates for fractional powers of $\mathcal{L}_a$. The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to $\mathcal{L}_a$. As an application, we obtain smoothing estimates regarding the propagator $e^{it\mathcal{L}_a}$.

math.AP↗
arXiv2016

Estimating stellar atmospheric parameters, absolute magnitudes and elemental abundances from the LAMOST spectra with Kernel-based Principal Component Analysis

Accurate determination of stellar atmospheric parameters and elemental abundances is crucial for Galactic archeology via large-scale spectroscopic surveys. In this paper, we estimate stellar atmospheric parameters -- effective temperature T_{\rm eff}, surface gravity log g and metallicity [Fe/H], absolute magnitudes M_V and M_{Ks}, α-element to metal (and iron) abundance ratio [α/M] (and [α/Fe]), as well as carbon and nitrogen abundances [C/H] and [N/H] from the LAMOST spectra with amultivariate regressionmethod based on kernel-based principal component analysis, using stars in common with other surveys (Hipparcos, Kepler, APOGEE) as training data sets. Both internal and external examinations indicate that given a spectral signal-to-noise ratio (SNR) better than 50, our method is capable of delivering stellar parameters with a precision of ~100K for Teff, ~0.1 dex for log g, 0.3 -- 0.4mag for M_V and M_{Ks}, 0.1 dex for [Fe/H], [C/H] and [N/H], and better than 0.05 dex for [α/M] ([α/Fe]). The results are satisfactory even for a spectral SNR of 20. The work presents first determinations of [C/H] and [N/H] abundances from a vast data set of LAMOST, and, to our knowledge, the first reported implementation of absolute magnitude estimation directly based on the observed spectra. The derived stellar parameters for millions of stars from the LAMOST surveys will be publicly available in the form of value-added catalogues.

astro-ph.SR↗
arXiv2016

On weak-star convergence in product Hardy spaces on spaces of homogeneous type

A classical theorem of Jones and Journé on weak-star convergence in the Hardy space $H^1$ was generalised to the multiparameter setting by Pipher and Treil. We prove the analogous result when the underlying space is a product space of homogeneous type. The main tools we use for this setting are from recent work in papers by Chen, Li and Ward and by Han, Li and Ward.

math.FA↗
arXiv2016

Dirac monopoles with polar-core vortex induced by spin-orbit coupling in spinor Bose-Einstein condensates

We report Dirac monopoles with polar-core vortex induced by spin-orbit coupling in ferromagnetic Bose-Einstein condensates, which are attached to two nodal vortex lines along the vertical axis. These monopoles are more stable in the time scale of experiment and can be detected through directly imaging vortex lines. When the strength of spin-orbit coupling increases, Dirac monopoles with vortex can be transformed into those with square lattice. In the presence of spin-orbit coupling, increasing the strength of interaction can induce a cyclic phase transition from Dirac monopoles with polar-core vortex to those with Mermin-Ho vortex. The spin-orbit coupled Bose-Einstein condensates not only provide a new unique platform for investigating exotic monopoles and relevant phase transitions, but also can preserve stable monopoles after a quadrupole field is turned off.

cond-mat.quant-gas↗
arXiv2016

A Moser type inequality for Bessel Laplace equations and applications

In this paper, we study Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and related the harmonic function theory introduced by Muckenhoupt--Stein. We establish the Moser type inequality for these harmonic functions, which is missing in this setting before. We then apply it to give a direct proof for the equivalence of characterizations of the Hardy spaces associated to Bessel operator via non-tangential maximal function and radial maximal function defined in terms of the Poisson semigroup.

math.AP↗