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Let $n\geq 1,0<\rho<1, \max\{\rho,1-\rho\}\leq \delta\leq 1$ and $$m_1=\rho-n+(n-1)\min\{\frac 12,\rho\}+\frac {1-\delta}{2}.$$ If the amplitude $a$ belongs to the H\"{o}rmander class $S^{m_1}_{\rho,\delta}$ and $\phi\in \Phi^{2}$ satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator $T_{\phi,a}$ defined by \begin{align*} T_{\phi,a}f(x)=\int_{\mathbb{R}^{n}}e^{i\phi(x,\xi)}a(x,\xi)\widehat{f}(\xi)d\xi, \end{align*} is bounded from the local Hardy space $h^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$. As a corollary, we can also obtain the corresponding $L^p(\mathbb{R}^n)$-boundedness when $1

In this article, we study maximal functions related to hypersurfaces with vanishing Gaussian curvature in $\mathbb{R}^3$. Firstly, we characterize the $L^p\rightarrow L^q$ boundedness of local maximal operators along homogeneous hypersurfaces. Moreover, weighted $L^p$-estimates are obtained for the corresponding global operators. Secondly, for a class of hypersurfaces that lack a homogeneous structure and pass through the origin, we attempt to look for other geometric properties instead of height of hypersurfaces to characterize the optimal $L^p$-boundedness of the corresponding global maximal operators.

In this paper, we establish the $L^{p}(\mathbb{R}^{d})$-boundedness of the variation operator and the $\delta$-jump operator for generalized spherical means, and we also show the necessary conditions for the $L^{p}(\mathbb{R}^{d})$-boundedness of these operators. These results are almost optimal when $d=2$., Comment: 25 pages, 2 figures

With the rise of "Metaverse" and "Web 3.0", Non-Fungible Token (NFT) has emerged as a kind of pivotal digital asset, garnering significant attention. By the end of March 2024, more than 1.7 billion NFTs have been minted across various blockchain platforms. To effectively locate a desired NFT, conducting searches within a vast array of NFTs is essential. The challenge in NFT retrieval is heightened due to the high degree of similarity among different NFTs, regarding regional and semantic aspects. In this paper, we will introduce a benchmark dataset named "NFT Top1000 Visual-Text Dataset" (NFT1000), containing 7.56 million image-text pairs, and being collected from 1000 most famous PFP1 NFT collections2 by sales volume on the Ethereum blockchain. Based on this dataset and leveraging the CLIP series of pre-trained models as our foundation, we propose the dynamic masking fine-tuning scheme. This innovative approach results in a 7.4\% improvement in the top1 accuracy rate, while utilizing merely 13\% of the total training data (0.79 million vs. 6.1 million). We also propose a robust metric Comprehensive Variance Index (CVI) to assess the similarity and retrieval difficulty of visual-text pairs data. The dataset will be released as an open-source resource. For more details, please refer to: https://github.com/ShuxunoO/NFT-Net.git., Comment: 11 pages,12figures to be published in ACM Multimedia 2024, see https://openreview.net/forum?id=xUtNrKH8iB&noteId=xUtNrKH8iB

Imitation learning attracts much attention for its ability to allow robots to quickly learn human manipulation skills through demonstrations. However, in the real world, human demonstrations often exhibit random behavior that is not intended by humans. Collecting high-quality human datasets is both challenging and expensive. Consequently, robots need to have the ability to learn behavioral policies that align with human intent from imperfect demonstrations. Previous work uses confidence scores to extract useful information from imperfect demonstrations, which relies on access to ground truth rewards or active human supervision. In this paper, we propose a transition-based method to obtain fine-grained confidence scores for data without the above efforts, which can increase the success rate of the baseline algorithm by 40.3$\%$ on average. We develop a generalized confidence-based imitation learning framework for guiding policy learning, called Confidence-based Inverse soft-Q Learning (CIQL), as shown in Fig.1. Based on this, we analyze two ways of processing noise and find that penalization is more aligned with human intent than filtering., Comment: Our code see https://github.com/XizoB/CIQL

We propose a simple method of synchronous combined modulations to generate the exact analytic solutions for a spin-orbit (SO) coupled ultracold atom held in a non-Hermitian double-well potential. Based on the obtained analytical solutions, we mainly study the parity-time ($\mathcal{PT}$) symmetry of this system and the system stability for both balanced and unbalanced gain-loss between two wells. Under balanced gain and loss, the effect of the proportional constants between synchronous combined modulations and the SO-coupling strength on the $\mathcal{PT}$-symmetry breaking is revealed analytically. Surprisingly, we find when the Zeeman field is present, the stable spin-flipping tunneling between two wells can not occur in the non-Hermitian SO-coupled ultracold atomic system, but the stable spin-conserving tunneling can be performed. Under unbalanced gain and loss, the unique set of parameter conditions that can cause the system to stabilize is found. The results may provide a possibility for the exact control of $\mathcal{PT}$-symmetry breaking and quantum spin dynamics in a non-Hermitian SO-coupled system.

In this paper, we study the $L^{p}$-improving property for the maximal operators along a large class of curves of finite type in the plane with dilation set $E \subset [1,2]$. The $L^{p}$-improving region depends on the order of finite type and the fractal dimension of $E$. In particular, various impacts of non-isotropic dilations are also considered.

Exact solutions for spin-orbit (SO) coupled cold atomic systems are very important and rare in physics. In this paper, we propose a simple method of combined modulations to generate the analytic exact solutions for an SO-coupled boson held in a driven double well. For the cases of synchronous combined modulations and the spin-conserving tunneling, we obtain the general analytical accurate solutions of the system respectively. For the case of spin-flipping tunneling under asynchronous combined modulations, we get the special exact solutions in simple form when the driving parameters satisfy certain conditions. Based on these obtained exact solutions, we reveal some intriguing quantum spin dynamical phenomena, for instance, the arbitrary population transfer (APT) with and/or without spin-flipping, the controlled coherent population conservation (CCPC), and the controlled coherent population inversion (CCPI). The results may have potential applications in the preparation of accurate quantum entangled states and quantum information processing., Comment: 8 pages, 3 figures

For decreasing sequences $\{t_{n}\}_{n=1}^{\infty}$ converging to zero, we obtain the almost everywhere convergence results for sequences of Schr\"{o}dinger means $e^{it_{n}\Delta}f$, where $f \in H^{s}(\mathbb{R}^{N}), N\geq 2$. The convergence results are sharp up to the endpoints, and the method can also be applied to get the convergence results for the fractional Schr\"{o}dinger means and nonelliptic Schr\"{o}dinger means.