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Ji Li

Publications and source records attributed to Ji Li.

At least 127 records · Page 7Linked to original sources
arXiv2023

Optimal lifting of Levi-degenerate hypersurfaces and applications to the Cauchy--Szegö projection

We consider a family of Levi-degenerate finite type hypersurfaces in $\mathbb C^2$, where in general there is no group structure. We lift these domains to stratified Lie groups via a constructive proof, which optimizes the well-known lifting procedure to free Lie groups of general manifolds defined by Rothschild and Stein. This yields an explicit version of the Taylor expansion with respect to the horizontal vector fields induced by the sub-Riemannian structure on these hypersurfaces. Hence, as an application, we establish the Schatten class estimates for the commutator of the Cauchy--Szegö projection with respect to a suitable quasi-metric defined on the hypersurface.

math.CV↗
arXiv2023

Lipschitz and Triebel--Lizorkin spaces, commutators in Dunkl setting

We first study the Lipschitz spaces $Λ_{d}^β$ associated with the Dunkl metric, $β\in(0,1)$, and prove that it is a proper subspace of the classical Lipschitz spaces $Λ^β$ on $\mathbb R^N$, as the Dunkl metric and the Euclidean metric are non-equivalent. Next, we further show that the Lipschitz spaces $Λ^β$ connects to the Triebel--Lizorkin spaces $\dot{ F}^{α,q}_{p,{\rm D}}$ associated with the Dunkl Laplacian $\triangle_{\rm D}$ in $\mathbb R^ N $ and to the commutators of the Dunkl Riesz transform and the fractional Dunkl Laplacian $\triangle_{\rm D}^{-α/2}$, $0<α<\textbf{N}$ (the homogeneous dimension for Dunkl measure), which is represented via the functional calculus of the Dunkl heat semigroup $e^{-t\triangle_{\rm D}}$. The key steps in this paper are a finer decomposition of the underlying space via Dunkl metric and Euclidean metric to bypass the use of Fourier analysis, and a discrete weak-type Calderón reproducing formula in these new Triebel--Lizorkin spaces $\dot{ F}^{α,q}_{p,{\rm D}}$.

math.FA↗
arXiv2023

Beyond spectroscopy. II. Stellar parameters for over twenty million stars in the northern sky from SAGES DR1 and Gaia DR3

We present precise photometric estimates of stellar parameters, including effective temperature, metallicity, luminosity classification, distance, and stellar age, for nearly 26 million stars using the methodology developed in the first paper of this series, based on the stellar colors from the Stellar Abundances and Galactic Evolution Survey (SAGES) DR1 and Gaia EDR3. The optimal design of stellar-parameter sensitive $uv$ filters by SAGES has enabled us to determine photometric-metallicity estimates down to $-3.5$, similar to our previous results with the SkyMapper Southern Survey (SMSS), yielding a large sample of over five million metal-poor (MP; [Fe/H]$\le -1.0$) stars and nearly one million very metal-poor (VMP; [Fe/H]$\le -2.0$) stars. The typical precision is around $0.1$ dex for both dwarf and giant stars with [Fe/H]$>-1.0$, and 0.15-0.25/0.3-0.4 dex for dwarf/giant stars with [Fe/H]$<-1.0$. Using the precise parallax measurements and stellar colors from Gaia, effective temperature, luminosity classification, distance and stellar age are further derived for our sample stars. This huge data set in the Northern sky from SAGES, together with similar data in the Southern sky from SMSS, will greatly advance our understanding of the Milky Way, in particular its formation and evolution.

astro-ph.GA↗
arXiv2023

The Garnett-Jones Theorem on BMO spaces associated with operators and applications

Let $X$ be a metric space with doubling measure, and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ whose heat kernel satisfies the Gaussian upper bound. Let $f$ be in the space $ {\rm BMO}_L(X)$ associated with the operator $L$ and we define its distance from the subspace $L^{\infty}(X)$ under the $ {\rm BMO}_L(X)$ norm as follows: $$ {\rm dist} (f, L^{\infty}):= \inf_{g\in L^{\infty}} \|f -g\|_{{\rm BMO}_L(X)}. $$ In this paper we prove that ${\rm dist} (f, L^{\infty})$ is equivalent to the infimum of the constant $\varepsilon$ in the John-Nirenberg inequality for the space ${\rm BMO}_L(X)$: $$ \sup_B { μ\big(\{ x\in B: |f(x)-e^{-{r_B^2}L}f(x)|>λ\}\big) \over μ(B)} \leq e^{-λ/\varepsilon}\ \ \ \ {\rm for\ large\ } λ. $$ This extends the well-known result of Garnett and Jones \cite{GJ1} for the classical ${\rm BMO}$ space (introduced by John and Nirenberg). As an application, we show that a ${\rm BMO}_L(X)$ function with compact support can be decomposed as the summation of an $L^\infty$-function and the integral of the heat kernel (associated with $L$) against a finite Carleson measure on $X\times[0,\infty)$. The key new technique is a geometric construction involving the semigroup $e^{-tL}$. We also resort to several fundamental tools including the stopping time argument and the random dyadic lattice.

math.CA↗
arXiv2023

The Riesz Transform and Fractional Integral Operators in the Bessel Setting

Fix $λ>0$. Consider the Bessel operator $\triangle_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ on $\mathbb{R_+}$, where $\mathbb{R_+}:=(0,\infty)$ and $dm_λ:=x^{2λ}dx$ with $dx$ the Lebesgue measure. We provide a deeper study of the Bessel Riesz transform and fractional integral operator via the related Besov and Triebel--Lizorkin spaces associated with $\triangle_λ$. Moreover, we investigate some possible characterization of the commutator of fractional integral operator, which was missing in the literature of the Bessel setting.

math.CA↗
arXiv2023

Sharp estimates for imaginary powers of Bessel operators

Let $L f(x):=-\frac{d^2}{dx^2}f(x)-\frac{ r}{x}\frac{d}{dx}f(x),\quad x>0, r>0$ be the Bessel operator on $((0,\infty), |\cdot|, x^rdx)$. In this paper, we prove the sharp weak type $(1,1)$ estimate for the imaginary power $L^{iα}, α\in \mathbb R$, of the Bessel operator.

math.CA↗
arXiv2023

Planning Automated Driving with Accident Experience Referencing and Common-sense Inferencing

Although a typical autopilot system far surpasses humans in term of sensing accuracy, performance stability and response agility, such a system is still far behind humans in the wisdom of understanding an unfamiliar environment with creativity, adaptivity and resiliency. Current AD brains are basically expert systems featuring logical computations, which resemble the thinking flow of a left brain working at tactical level. A right brain is needed to upgrade the safety of automated driving vehicle onto next generation by making intuitive strategical judgements that can supervise the tactical action planning. In this work, we present the concept of an Automated Driving Strategical Brain (ADSB): a framework of a scene perception and scene safety evaluation system that works at a higher abstraction level, incorporating experience referencing, common-sense inferring and goal-and-value judging capabilities, to provide a contextual perspective for decision making within automated driving planning. The ADSB brain architecture is made up of the Experience Referencing Engine (ERE), the Common-sense Referencing Engine (CIE) and the Goal and Value Keeper (GVK). 1,614,748 cases from FARS/CRSS database of NHTSA in the period 1975 to 2018 are used for the training of ERE model. The kernel of CIE is a trained model, COMET-BART by ATOMIC, which can be used to provide directional advice when tactical-level environmental perception conclusions are ambiguous; it can also use future scenario models to remind tactical-level decision systems to plan ahead of a perceived hazard scene. GVK can take in any additional expert-hand-written rules that are of qualitative nature. Moreover, we believe that with good scalability, the ADSB approach provides a potential solution to the problem of long-tail corner cases encountered in the validation of a rule-based planning algorithm.

cs.RO↗
arXiv2023

Evaluation functions and composition operators on Banach spaces of holomorphic functions

Let $B(Ω)$ be the Banach space of holomorphic functions on a bounded connected domain $Ω$ in $\mathbb C^n$, which contains the ring of polynomials on $Ω$. In this paper, we first establish a criterion for $B(Ω)$ to be reflexive via evaluation functions on $B(Ω)$, that is, $B(Ω)$ is reflexive if and only if the evaluation functions span the dual spaces $(B(Ω))^{*} $. Moreover, under suitable assumptions on $Ω$ and $B(Ω)$, we establish a characterization of the composition operator $C_φ$ to be a Fredholm operator on $B(Ω)$ via the property of the holomorphic self-map $φ:Ω\toΩ$. Our new approach utilizes the symbols of composition operators to construct a linearly independent function sequence, which bypasses the use of boundary behavior of reproducing kernels as those may not be applicable in our general setting.

math.CV↗
arXiv2022

The Kohn-Laplacian and Cauchy-Szegö projection on Model Domains

We study the Kohn-Laplacian and its fundamental solution on some model domains in $\mathbb C^{n+1}$, and further discuss the explicit kernel of the Cauchy-Szegö projections on these model domains using the real analysis method. We further show that these Cauchy-Szegö kernels are Calderón-Zygmund kernels under the suitable quasi-metric.

math.CV↗
arXiv2022

ClipCrop: Conditioned Cropping Driven by Vision-Language Model

Image cropping has progressed tremendously under the data-driven paradigm. However, current approaches do not account for the intentions of the user, which is an issue especially when the composition of the input image is complex. Moreover, labeling of cropping data is costly and hence the amount of data is limited, leading to poor generalization performance of current algorithms in the wild. In this work, we take advantage of vision-language models as a foundation for creating robust and user-intentional cropping algorithms. By adapting a transformer decoder with a pre-trained CLIP-based detection model, OWL-ViT, we develop a method to perform cropping with a text or image query that reflects the user's intention as guidance. In addition, our pipeline design allows the model to learn text-conditioned aesthetic cropping with a small cropping dataset, while inheriting the open-vocabulary ability acquired from millions of text-image pairs. We validate our model through extensive experiments on existing datasets as well as a new cropping test set we compiled that is characterized by content ambiguity.

cs.CV↗
arXiv2022

Dirichlet problem for Schrödinger operators on Heisenberg groups

We investigate the Dirichlet problem associated to the Schrödinger operator $\mathcal L=-Δ_{\mathbb{H}^n}+V$ on Heisenberg group $\mathbb H^n$: \begin{align*} \begin{cases} \partial_{ss}u(g,s)-\mathcal L u(g,s)=0\,,\quad &{\rm in \,\ } \mathbb{H}^n\times\mathbb{R}^+,\\ u(g,0)=f \,,\quad &{\rm on \,\ } \mathbb{H}^n \end{cases} \end{align*} with $f$ in $L^p(\mathbb{H}^n)$ ($1< p<\infty$) and in $H^1_{\mathcal L}(\mathbb{H}^n)$, i.e., the Hardy space associated with $\mathcal L$. Here $Δ_{\mathbb{H}^n}$ is the sub-Laplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse Hölder class $B_{Q/2}$ with $Q$ the homogeneous dimension of $\mathbb{H}^n$. The new approach is to establish a suitable weak maximum principle, which is the key to solve this problem under the condition $V\in B_{Q/2}$. This result is new even back to $\mathbb R^n$ (the condition will become $V\in B_{n/2}$) since the previous known result requires $V\in B_{(n+1)/2}$ which went through a Liouville type theorem.

math.AP↗
arXiv2022

Sympletic tracking methods for insertion devices: a Robinson wiggler example

Modern synchrotron light sources are often characterized with high-brightness synchrotron radiation from insertion devices. Inevitably, insertion devices introduce nonlinear distortion to the beam motion. Symplectic tracking is crucial to study the impact, especially for the low- and medium-energy storage rings. This paper uses a Robinson wiggler as an example to illustrate an universally applicable analytical representation of the magnetic field and to summarizes four different symplectic tracking methods.

physics.acc-ph↗
arXiv2022

The Two Weight Inequality for Poisson Semigroup on Manifold with Ends

Let $M = \mathbb R^m \sharp \mathcal R^n$ be a non-doubling manifold with two ends $\mathbb R^m \sharp \mathcal R^n$, $m > n \ge 3$. Let $Δ$ be the Laplace--Beltrami operator which is non-negative self-adjoint on $L^2(M)$. Then $Δ$ and its square root $\sqrtΔ$ generate the semigroups $e^{-tΔ}$ and $e^{-t\sqrtΔ}$ on $L^2(M)$, respectively. We give testing conditions for the two weight inequality for the Poisson semigroup $e^{-t\sqrtΔ}$ to hold in this setting. In particular, we prove that for a measure $μ$ on $M_{+}:=M\times (0,\infty)$ and $σ$ on $M$: $$ \|\mathsf{P}_σ(f)\|_{L^2(M_{+};μ)} \lesssim \|f\|_{L^2(M;σ)}, $$ with $\mathsf{P}_σ(f)(x,t):= \int_M \mathsf{P}_t(x,y)f(y) \,dσ(y)$ if and only if testing conditions hold for the Poisson semigroup and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in these testing conditions.

math.CA↗
arXiv2022

Detection of DC electric forces with zeptonewton sensitivity by single-ion phonon laser

Detecting extremely small forces helps exploring new physics quantitatively. Here we demonstrate that the phonon laser made of a single trapped $^{40}$Ca$^{+}$ ion behaves as an exquisite sensor for small force measurement. We report our successful detection of small electric forces regarding the DC trapping potential with sensitivity of 2.41$\pm$0.49 zN/$\sqrt{\rm Hz}$, with the ion only under Doppler cooling, based on the injection-locking of the oscillation phase of the phonon laser in addition to the classical squeezing applied to suppress the measurement uncertainty. We anticipate that such a single-ion sensor would reach a much better force detection sensitivity in the future once the trapping system is further improved and the fluorescence collection efficiency is further enhanced.

physics.atom-ph↗
arXiv2022

The Schrödinger equation in $L^p$ spaces for operators with heat kernel satisfying Poisson type bounds

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 $n$. In this paper, we study sharp endpoint $L^p$-Sobolev estimates for the solution of the initial value problem for the Schrödinger equation, $i \partial_t u + L u=0 $ and show that for all $f\in L^p(X), 1<p<\infty,$ \begin{eqnarray*} \left\| e^{itL} (I+L)^{-{σn}} f\right\|_{p} \leq C(1+|t|)^{σn} \|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ σ\geq \big|{1\over 2}-{1\over p}\big|, \end{eqnarray*} where the semigroup $e^{-tL}$ generated by $L$ satisfies a Poisson type upper bound. This extends the previous result in \cite{CDLY1} in which the semigroup $e^{-tL}$ generated by $L$ satisfies the exponential decay.

math.AP↗