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Tengfei Zhao

Publications and source records attributed to Tengfei Zhao.

18 recordsLinked to original sources
arXiv2025

Shallow-water convergence of the intermediate long wave equation in $L^2$

We continue our study on the convergence issue of the intermediate long wave equation (ILW) on both the real line and the circle. In particular, we establish convergence of the scaled ILW dynamics to that of the Korteweg-de Vries equation (KdV) in the shallow-water limit at the $L^2$-level. Together with the recent work by the first three authors and D. Pilod (2024) on the deep-water convergence in $L^2$, this work completes the well-posedness and convergence study of ILW on both geometries within the $L^2$-framework. Our proof equally applies to both geometries and is based on the following two ingredients: the complete integrability of ILW and the normal form method. More precisely, by making use of the Lax pair structure and the perturbation determinant for ILW, recently introduced by Harrop-Griffths, Killip, and Vi\c{s}an (2025), we first establish weakly uniform (in small depth parameters) equicontinuity in $L^2$ of solutions to the scaled ILW, providing a control on the high frequency part of solutions. Then, we treat the low frequency part by implementing a perturbative argument based on an infinite iteration of normal form reductions for KdV.

math.AP↗
arXiv2024

$L^p$ maximal estimates for Weyl sums with $k\ge3$ on $\mathbb{T}$

In this paper, we study the $L^p$ maximal estimates for the Weyl sums $\sum_{n=1}^{N}e^{2\pi i(nx + n^{k}t)}$ with higher-order $k\ge3$ on $\mathbb{T}$, and obtain the positive and negative results. Especially for the case $k=3$, our result is sharp up to the endpoint. The main idea is to investigate the structure of the set where large values of Weyl sums are achieved by making use of the rational approximation and the refined estimate for the exponential sums.

math.NT↗
arXiv2024

Scattering theory for subcritical wave equation with inverse square potential

We consider the scattering theory for the defocusing energy subcritical wave equations with an inverse square potential. By employing the energy flux method we establish energy flux estimates on the light cone. Then by the characteristic line method and radiation theorem, we show that the radial finite-energy solutions scatter to free waves outside of light cones. Using Morawetz estimates we then obtain the scattering theory for radial solutions with finite weighted energy initial data.

math.AP↗
arXiv2024

Unconditional deep-water limit of the intermediate long wave equation in low-regularity

In this paper, we establish the unconditional deep-water limit of the intermediate long wave equation (ILW) to the Benjamin-Ono equation (BO) in low-regularity Sobolev spaces on both the real line and the circle. Our main tool is new unconditional uniqueness results for ILW in $H^s$ when $s_0<s\leq \frac 14$ on the line and $s_0<s< \frac 12$ on the circle, where $s_0 = 3-\sqrt{33/4}\approx 0.1277$. Here, we adapt the strategy of Mo\c{s}incat-Pilod (2023) for BO to the setting of ILW by viewing ILW as a perturbation of BO and making use of the smoothing property of the perturbation term.

math.AP↗
arXiv2024

Robust single-particle cryo-EM image denoising and restoration

Cryo-electron microscopy (cryo-EM) has achieved near-atomic level resolution of biomolecules by reconstructing 2D micrographs. However, the resolution and accuracy of the reconstructed particles are significantly reduced due to the extremely low signal-to-noise ratio (SNR) and complex noise structure of cryo-EM images. In this paper, we introduce a diffusion model with post-processing framework to effectively denoise and restore single particle cryo-EM images. Our method outperforms the state-of-the-art (SOTA) denoising methods by effectively removing structural noise that has not been addressed before. Additionally, more accurate and high-resolution three-dimensional reconstruction structures can be obtained from denoised cryo-EM images.

cs.CV↗
arXiv2023

Fractional Leibniz rule on the torus

We discuss the fractional Leibniz rule for periodic functions on the $d$-dimensional torus, including the endpoint cases. As an application, we present a product estimate, involving distributions of negative regularities.

math.CA↗
arXiv2023

Global well-posedness of the energy-critical stochastic Hartree nonlinear wave equation

We consider the Cauchy problem for the stochastic Hartree nonlinear wave equations (SHNLW) with a cubic convolution nonlinearity and an additive stochastic forcing on the Euclidean space. Our goal in this paper is two-fold. (i) We study the defocusing energy-critical SHNLW on $\mathbb{R}^d$, for $d \geq 5$, and prove that they are globally well-posed with deterministic initial data in the energy space. (ii) Next, we consider the well-posedness of the defocusing energy-critical SHNLW with randomized initial data below the energy space. In particular, when $d=5$, we prove it is almost surely globally well-posed. As a byproduct, by removing the stochastic forcing our result covers the study of the (deterministic) Hartree nonlinear wave equation (HNLW) with randomized initial data below the energy space. The main ingredients in the globalization argument involve the probabilistic perturbation approach by B\'enyi-Oh-Pocovnicu (2015) and Pocovnicu (2017), time integration by parts trick of Oh-Pocovnicu (2016), and an estimate of the Hartree potential energy.

math.AP↗
arXiv2023

Almost sure scattering for defocusing energy critical Hartree equation on $\R^5$

We consider the defocusing energy-critical Hartree equation $i\pa_tu+Δu=(|\cdot|^{-4}\ast|u|^2)u$ in spatial dimension $d=5$ and prove almost sure scattering with initial data $u_0\in H^s_x(\R^5)$ for any $s\in\R$. The proof relies on the modified interaction Morawetz estimate, the stability theories, the ``Narrowed'' Wiener randomization. We are inspired to consider this problem by the work of Shen-Soffer-Wu \cite{Shen-Soffer-Wu 1}, which treated the analogous problem for the energy-critical Schrödinger equation. The new ingredient in this paper are that we take an alternative proof to give the interaction Morawetz estimate. And the nonlocal nonlinearity term will bring some difficulties.

math.AP↗
arXiv2023

Maximal estimates for the Weyl sums on $\mathbb{T}^{d}$ (with an appendix by Alex Barron)

In this paper, we obtain the maximal estimate for the Weyl sums on the torus $\mathbb{T}^d$ with $d\geq 2$, which is sharp up to the endpoint. We also consider two variants of this problem which include the maximal estimate along the rational lines and on the generic torus. Applications, which include some new upper bound on the Hausdorff dimension of the sets associated to the large value of the Weyl sums, reflect the compound phenomenon between the square root cancellation and the constructive interference. In the Appendix, an alternate proof of Theorem 1.1 inspired by Baker's argument in [1] is given by Barron, which also improves the $N^ε$ loss in Theorem 1.1, and the Strichartz-type estimates for the Weyl sums with logarithmic losses are obtained by the same argument.

math.NT↗
arXiv2022

$L^2$ Schrödinger maximal estimates associated with finite type phases in $\mathbb{R}^2$

In this paper, we establish Schrödinger maximal estimates associated with the finite type phases \begin{equation*} ϕ(ξ_1,ξ_2):=ξ^m_1+ξ^m_2,\;(ξ_1,ξ_2)\in [0,1]^2, \end{equation*} where $m \geq 4$ is an even number. Following [12], we prove an $L^2$ fractal restriction estimate associated with the surfaces \begin{equation*} F^2_m:=\{(ξ_1,ξ_2,ϕ(ξ_1,ξ_2)):\;(ξ_1,ξ_2)\in [0,1]^2\} \end{equation*} as the main result, which also gives results on the average Fourier decay of fractal measures associated with these surfaces. The key ingredients of the proof include the rescaling technique from [16], Bourgain-Demeter's $\ell^2$ decoupling inequality, the reduction of dimension arguments from [17] and induction on scales.

math.CA↗
arXiv2022

Decoupling for finite type phases in higher dimensions

In this paper, we establish an $\ell^2$ decoupling inequality for the hypersurface \[\Big\{(\xi_1,...,\xi_{n-1},\xi_1^m+...+\xi_{n-1}^m): (\xi_1,...,\xi_{n-1}) \in [0,1]^{n-1}\Big\}\]associated with the decomposition adapted to hypersufaces of finite type, where $n\geq 2$ and $m\geq 4$ is an even number. The key ingredients of the proof include an $\ell^2$ decoupling inequality for the hypersurfaces \[\Big\{(\xi_1,...,\xi_{n-1},\phi_1(\xi_1)+...+\phi_s(\xi_s)+\xi_{s+1}^m+...+\xi_{n-1}^m): (\xi_1,...,\xi_{n-1}) \in [0,1]^{n-1}\Big\},\] $0 \leq s \leq n-1$, with $\phi_1,...,\phi_s$ being $m$-nondegenerate.

math.AP↗
arXiv2021

Generating Natural Language Adversarial Examples through An Improved Beam Search Algorithm

The research of adversarial attacks in the text domain attracts many interests in the last few years, and many methods with a high attack success rate have been proposed. However, these attack methods are inefficient as they require lots of queries for the victim model when crafting text adversarial examples. In this paper, a novel attack model is proposed, its attack success rate surpasses the benchmark attack methods, but more importantly, its attack efficiency is much higher than the benchmark attack methods. The novel method is empirically evaluated by attacking WordCNN, LSTM, BiLSTM, and BERT on four benchmark datasets. For instance, it achieves a 100\% attack success rate higher than the state-of-the-art method when attacking BERT and BiLSTM on IMDB, but the number of queries for the victim models only is 1/4 and 1/6.5 of the state-of-the-art method, respectively. Also, further experiments show the novel method has a good transferability on the generated adversarial examples.

cs.CL↗
arXiv2021

A More Compact Object Detector Head Network with Feature Enhancement and Relational Reasoning

Modeling implicit feature interaction patterns is of significant importance to object detection tasks. However, in the two-stage detectors, due to the excessive use of hand-crafted components, it is very difficult to reason about the implicit relationship of the instance features. To tackle this problem, we analyze three different levels of feature interaction relationships, namely, the dependency relationship between the cropped local features and global features, the feature autocorrelation within the instance, and the cross-correlation relationship between the instances. To this end, we propose a more compact object detector head network (CODH), which can not only preserve global context information and condense the information density, but also allows instance-wise feature enhancement and relational reasoning in a larger matrix space. Without bells and whistles, our method can effectively improve the detection performance while significantly reducing the parameters of the model, e.g., with our method, the parameters of the head network is 0.6 times smaller than the state-of-the-art Cascade R-CNN, yet the performance boost is 1.3% on COCO test-dev. Without losing generality, we can also build a more lighter head network for other multi-stage detectors by assembling our method.

cs.CV↗
arXiv2019

A remark on the scattering theory for the 2d radial focusing INLS

We consider the scattering results of the radial solutions below the ground state to the focusing inhomogeneous nonlinear Schrödinger equation $$i\partial_tu+Δu +|x|^{-b}|u|^{p}u=0$$ in two dimension, where $0<b<1$ and $2-b<p<\infty$. We use a modified version of Arora-Dodson-Murphy's approach [1] to give a new proof that extends the scattering results of [10] and avoids concentration compactness.

math.AP↗
arXiv2019

On the dimension of divergence sets of Schrödinger equation with complex time

This article studies the pointwise convergence for the fractional Schrödinger operator $P^{t}_{a,γ}$ with complex time in one spatial dimension. Through establishing $L^2$-maximal estimates for initial datum in $H^{s}(\mathbb{R})$, we see that the solution converges to the initial data almost everywhere with $s>\frac14 a(1-\frac1γ)_+$ when $0 \frac{1}{2}(1-\frac{1}γ)_{+}$ when $a=1$. By constructing counterexamples, we show that this result is almost sharp up to the endpoint. These results extends the results of P. Sjölin, F. Soria and A. Baily. Second, we study the Hausdorff dimension of the set of the divergent points, by showing some $L^1$-maximal estimates with respect to general Borel measure. Our results reflect the interaction between dispersion effect and dissipation effect, arising from the fractional Schrödinger type operator $P^{t}_{a,γ}$ with the complex time.

math.AP↗
arXiv2018

Scattering for 3d cubic focusing NLS on the domain outside a convex obstacle revisited

In this article, we consider the focusing cubic nonlinear Schrödinger equation(NLS) in the exterior domain outside of a convex obstacle in $\mathbb{R}^3$ with Dirichlet boundary conditions. We revisit the scattering result below ground state of Killip-Visan-Zhang by utilizing Dodson and Murphy's argument and the dispersive estimate established by Ivanovici and Lebeau, which avoids using the concentration compactness. We conquer the difficulty of the boundary in the focusing case by establishing a local smoothing effect of the boundary. Based on this effect and the interaction Morawetz estimates, we prove the solution decays at a large time interval, which meets the scattering criterions.

math.AP↗
arXiv2018

The global well-posedness and scattering for the $5$D defocusing conformal invariant NLW with radial initial data in a critical Besov space

In this paper, we obtain the global well-posedness and scattering for the radial solution to the defocusing conformal invariant nonlinear wave equation with initial data in the critical Besov space $\dot{B}^3_{1,1}\times\dot{B}^2_{1,1}(\mathbb{R}^5)$. This is the five dimensional analogue of \cite{dodson-2016}, which is the first result on the global well-posedness and scattering of the energy subcritical nonlinear wave equation without the uniform boundedness assumption on the critical Sobolev norms employed as a substitute of the missing conservation law with respect to the scaling invariance of the equation. The proof is based on exploiting the structure of the radial solution, developing the Strichartz-type estimates and incorporation of the strategy in \cite{dodson-2016}, where we also avoid a logarithm-type loss by employing the inhomogeneous Strichartz estimates.

math.AP↗