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Xuemei Li

Publications and source records attributed to Xuemei Li.

At least 19 recordsLinked to original sources
arXiv2026

Construction of two-bubble solutions for the energy-critical NLS in dimension 6

We construct pure two-bubble solutions for the energy-critical focusing nonlinear Schr\"odinger equation in space dimension $N = 6$. They are global in (at least) one time direction and approach a superposition of two stationary states, both centered at the origin. One of the bubbles develops at scale $1$, whereas the length scale of the other converges to $0$ at rate $e^{-|t|}$. The phases of the two bubbles form the right angle. Such solutions were previously constructed in dimension $N \geq 7$. The six-dimension case presents specific difficulties, as the ground state does not belong to $\dot H^{-1}$. This prevents the use of the standard method of removing linear terms in modulation equations via suitable orthogonality conditions, due to loss of coercivity of the energy functional. The main novelty of this work is the introduction of modified modulation parameters to overcome this issue; these can be viewed as an analog of a normal form transformation in the context of modulation analysis. We also establish new coercivity estimates for the linearized energy, whose positive constants depend explicitly on the choice of the orthogonality conditions.

math.AP↗
arXiv2026

Approximate traveling waves for the focusing NLS with an external potential

We construct approximate traveling wave solutions for the $3d$ focusing cubic NLS perturbed by a rapidly decaying external potential. As an application, we demonstrate that there exists a forward-global, $H^1$ bounded, nonscattering solution at any mass-energy higher than the ground state threshold for the unperturbed equation.

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arXiv2026

Construction of two-bubble solutions for the energy-critical Hartree equation

We construct a pure two-bubble solution for the focusing, energy-critical Hartree equation in space dimension $N \geq 7$. The constructed solution is spherically symmetric, global in (at least) the negative time direction and asymptotically behaves as a superposition of two ground states (or bubbles) both centered at the origin, with the ratio of their length scales converging to $0$ and the phases of the two bubbles form the right angle. The main arguments are the modulation analysis, the bootstrap argument and the topological argument. The main novelty with respect to existing constructions of pure two-bubble solutions is the nonlocal interaction, which is more complex to analyze.

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arXiv2025

DM3D: Dynamic Mamba via Offset-Guided Feature Resampling for Point Cloud Understanding

State Space Models (SSMs) model long token sequences of point cloud with linear complexity, but require an unordered point cloud to be serialized. Existing methods mainly address this requirement by designing or learning a better token order. Even a well-constructed order, however, cannot preserve every local relation on an irregular 3D surface: a fixed sequence may still mix points that are close in index but distant in 3D or belong to different object parts. We propose DM3D, a dynamic Mamba architecture that preserves the base token order while adapting local feature support and state propagation. First, according to local feature context, DM3D learns spatial and sequence offsets without constructing a global permutation. Then, spatial offsets adjust the sampling anchors in 3D space, whereas sequence offsets guide feature resampling within a local sequence window, which lets different slots draw from overlapping local supports while retaining their identities. This design preserves the global prior of the original traversal, allowing each token to aggregate a more suitable local context. Second, the state update is modulated by the 3D distance between points at adjacent sequence positions, thereby reducing information propagation when these points are spatially far apart. DM3D reaches 95.2\% accuracy on the ModelNet40, 93.3\% accuracy on the PB\_T50\_RS split of ScanObjectNN, and 84.8\% class mIoU on ShapeNetPart. Extensive experiments on benchmark datasets show that DM3D achieves strong and competitive performance, validating the effectiveness of local feature adaptation for point cloud understanding.

cs.CV↗
arXiv2025

ReCast: Reliability-aware Codebook Assisted Lightweight Time Series Forecasting

Time series forecasting is crucial for applications in various domains. Conventional methods often rely on global decomposition into trend, seasonal, and residual components, which become ineffective for real-world series dominated by local, complex, and highly dynamic patterns. Moreover, the high model complexity of such approaches limits their applicability in real-time or resource-constrained environments. In this work, we propose a novel \textbf{RE}liability-aware \textbf{C}odebook-\textbf{AS}sisted \textbf{T}ime series forecasting framework (\textbf{ReCast}) that enables lightweight and robust prediction by exploiting recurring local shapes. ReCast encodes local patterns into discrete embeddings through patch-wise quantization using a learnable codebook, thereby compactly capturing stable regular structures. To compensate for residual variations not preserved by quantization, ReCast employs a dual-path architecture comprising a quantization path for efficient modeling of regular structures and a residual path for reconstructing irregular fluctuations. A central contribution of ReCast is a reliability-aware codebook update strategy, which incrementally refines the codebook via weighted corrections. These correction weights are derived by fusing multiple reliability factors from complementary perspectives by a distributionally robust optimization (DRO) scheme, ensuring adaptability to non-stationarity and robustness to distribution shifts. Extensive experiments demonstrate that ReCast outperforms state-of-the-art (SOTA) models in accuracy, efficiency, and adaptability to distribution shifts.

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arXiv2025

Boundary Regularity for Fully Nonlinear Parabolic equations on $C^{1,\mathrm{Dini}}$ Domains

We establish the boundary pointwise Lipschitz regularity on exterior $C^{1,\mathrm{Dini}}$ domains and the Hopf lemma on interior $C^{1,\mathrm{Dini}}$ domains for fully nonlinear parabolic equations by a unified perturbation method. In fact, above two regularity hold for more general solution sets, i.e., the Pucci's class $S^*(\lambda, \Lambda, f)$. Furthermore, based on the boundary pointwise Lipschitz regularity, we obtain the global ${W}^{2,\delta}$ regularity on exterior $C^{1,\mathrm{Dini}}$ domains for any $0<\delta<1$, which is new even for the harmonic functions.

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arXiv2025

The existence of quasi-periodic invariant tori and double Hopf bifurcation of van der Pol's oscillator with delayed feedback

The double Hopf bifurcation and the existence of quasi-periodic invariant tori in a delayed van der Pol's oscillator are considered by regarding the damped coefficient and the delay as bifurcation parameters. Applying the center manifold reduction and the normal form method, we derive the normal form near the critical point and analyse the existence of invariant 2-tori and 3-tori for the truncated normal form. Furthermore, the effect of higher-order terms on these invariant tori are investigated by a KAM theorem, and it is proved that in a sufficiently small neighborhood of the bifurcation point, the delayed van der Pol's oscillator has quasi-periodic invariant 2-tori and 3-tori for most of the parameter set where its truncated normal form possesses quasi-periodic invariant 2-tori and 3-tori, respectively.

math.DS↗
arXiv2025

Global dynamics above the ground state for the energy-critical Hartree equation with radial data

Based on the concentration-compactness-rigidity argument in \cite{KenM:NLS,KenM:NLW} and the non-degeneracy of the ground state in \cite{LLTX:Nondeg,LLTX:g-Hart,LTX:Nondeg}, long time dynamics for the focusing energy-critical Hartree equation with radial data have been classified when the energy $E(u_0)\leq E(W)$ in \cite{LiMZ:crit Hart,LLTX:g-Hart,MWX:Hart,MXZ:crit Hart:f rad}, where $W$ is the ground state. In this paper, we continue the study on the dynamics of the radial solutions with the energy $E(u_0)$ at most slightly larger than that of the ground states. This is an extension of the results \cite{KriNS:NLW rad, KriNS:NLW non,NakR,NakS:NLKG,NakS:book,NakS:NLS,NakS:NLKG:non,Roy} on NLS, NLW and NLKG, which were pioneered by K. Nakanishi and W. Schlag in \cite{NakS:NLKG, NakS:book} in the study of nonlinear Klein-Gordon equation in the subcritical case. The argument is an adaptation of the works in \cite{KriNS:NLW rad, KriNS:NLW non,NakR,Roy}, the proof uses an analysis of the hyperbolic dynamics near the ground state and the variational structure far from them. The key components that allow to classify the solutions are the hyperbolic (ejection) dynamical behavior near the ground state and the one-pass lemma.

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arXiv2025

Minding Fuzzy Regions: A Data-driven Alternating Learning Paradigm for Stable Lesion Segmentation

Deep learning has achieved significant advancements in medical image segmentation, but existing models still face challenges in accurately segmenting lesion regions. The main reason is that some lesion regions in medical images have unclear boundaries, irregular shapes, and small tissue density differences, leading to label ambiguity. However, the existing model treats all data equally without taking quality differences into account in the training process, resulting in noisy labels negatively impacting model training and unstable feature representations. In this paper, a data-driven alternating learning (DALE) paradigm is proposed to optimize the model's training process, achieving stable and high-precision segmentation. The paradigm focuses on two key points: (1) reducing the impact of noisy labels, and (2) calibrating unstable representations. To mitigate the negative impact of noisy labels, a loss consistency-based collaborative optimization method is proposed, and its effectiveness is theoretically demonstrated. Specifically, the label confidence parameters are introduced to dynamically adjust the influence of labels of different confidence levels during model training, thus reducing the influence of noise labels. To calibrate the learning bias of unstable representations, a distribution alignment method is proposed. This method restores the underlying distribution of unstable representations, thereby enhancing the discriminative capability of fuzzy region representations. Extensive experiments on various benchmarks and model backbones demonstrate the superiority of the DALE paradigm, achieving an average performance improvement of up to 7.16%.

cs.CV↗
arXiv2025

Hopf and double Hopf bifurcations in a delayed lateral vibration model of footbridges induced by pedestrians

In this paper, we investigate the dynamical behaviors of a delayed lateral vibration model of footbridges proposed based on the facts that pedestrians will reduce their walking speed or stop walking when the response of the footbridge becomes sufficiently large, and that the bridge velocity can not be changed at once when the pedestrians begin to walk on the bridge. By analyzing the distribution of roots of the associated characteristic equation, we find that there are only two types of bifurcations in this model: Hopf bifurcation and double Hopf bifurcation, and give the condition on the stability of the trivial solution. By using the center manifold theorem and bifurcation theory of delayed differential equations, we obtain the dynamical behavior in these bifurcations, specially including the stability of periodic solutions and invariant tori bifurcating from the trivial solution in these bifurcations. Finally, we prove that this model exhibits quasi-periodic vibrations by KAM theorems, besides periodic vibrations.

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arXiv2024

Bridging the Modality Gap: Dimension Information Alignment and Sparse Spatial Constraint for Image-Text Matching

Many contrastive learning based models have achieved advanced performance in image-text matching tasks. The key of these models lies in analyzing the correlation between image-text pairs, which involves cross-modal interaction of embeddings in corresponding dimensions. However, the embeddings of different modalities are from different models or modules, and there is a significant modality gap. Directly interacting such embeddings lacks rationality and may capture inaccurate correlation. Therefore, we propose a novel method called DIAS to bridge the modality gap from two aspects: (1) We align the information representation of embeddings from different modalities in corresponding dimension to ensure the correlation calculation is based on interactions of similar information. (2) The spatial constraints of inter- and intra-modalities unmatched pairs are introduced to ensure the effectiveness of semantic alignment of the model. Besides, a sparse correlation algorithm is proposed to select strong correlated spatial relationships, enabling the model to learn more significant features and avoid being misled by weak correlation. Extensive experiments demonstrate the superiority of DIAS, achieving 4.3\%-10.2\% rSum improvements on Flickr30k and MSCOCO benchmarks.

cs.CV↗
arXiv2024

Dynamic properties of a class of van der Pol-Duffing oscillators

In this paper, we study the existence of bifurcation of a van der Pol-Duffing oscillator with quintic terms and its quasi-periodic solutions by means of qualitative and bifurcation theories. Firstly, we analyze the autonomous system and find that it has two kinds of local bifurcations and a global bifurcation: pitchfork bifurcation, Hopf bifurcation, homoclinic bifurcation. It is worth noting that the disappearance of the homoclinic orbit is synchronized with the emergence of a large limit cycle. Then, by discussing the stability of equilibria at infinity and the orientation of the trajectory, the existence and stability of limit circles of the autonomous system are analyzed by combining the Poincar\'{e}-Bendixson theorem and the index theory. The global phase portrait and the numerical simulation of the autonomous system in different parameter values are given. Finally, the existence of periodic and quasi-periodic solutions to periodic forced system is proved by a KAM theorem.

math.DS↗
arXiv2024

U-Mixer: An Unet-Mixer Architecture with Stationarity Correction for Time Series Forecasting

Time series forecasting is a crucial task in various domains. Caused by factors such as trends, seasonality, or irregular fluctuations, time series often exhibits non-stationary. It obstructs stable feature propagation through deep layers, disrupts feature distributions, and complicates learning data distribution changes. As a result, many existing models struggle to capture the underlying patterns, leading to degraded forecasting performance. In this study, we tackle the challenge of non-stationarity in time series forecasting with our proposed framework called U-Mixer. By combining Unet and Mixer, U-Mixer effectively captures local temporal dependencies between different patches and channels separately to avoid the influence of distribution variations among channels, and merge low- and high-levels features to obtain comprehensive data representations. The key contribution is a novel stationarity correction method, explicitly restoring data distribution by constraining the difference in stationarity between the data before and after model processing to restore the non-stationarity information, while ensuring the temporal dependencies are preserved. Through extensive experiments on various real-world time series datasets, U-Mixer demonstrates its effectiveness and robustness, and achieves 14.5\% and 7.7\% improvements over state-of-the-art (SOTA) methods.

cs.LG↗
arXiv2023

SecureBERT and LLAMA 2 Empowered Control Area Network Intrusion Detection and Classification

Numerous studies have proved their effective strength in detecting Control Area Network (CAN) attacks. In the realm of understanding the human semantic space, transformer-based models have demonstrated remarkable effectiveness. Leveraging pre-trained transformers has become a common strategy in various language-related tasks, enabling these models to grasp human semantics more comprehensively. To delve into the adaptability evaluation on pre-trained models for CAN intrusion detection, we have developed two distinct models: CAN-SecureBERT and CAN-LLAMA2. Notably, our CAN-LLAMA2 model surpasses the state-of-the-art models by achieving an exceptional performance 0.999993 in terms of balanced accuracy, precision detection rate, F1 score, and a remarkably low false alarm rate of 3.10e-6. Impressively, the false alarm rate is 52 times smaller than that of the leading model, MTH-IDS (Multitiered Hybrid Intrusion Detection System). Our study underscores the promise of employing a Large Language Model as the foundational model, while incorporating adapters for other cybersecurity-related tasks and maintaining the model's inherent language-related capabilities.

cs.CR↗
arXiv2023

Dynamics of radial threshold solutions for generalized energy-critical Hartree equation

In this paper, we study long time dynamics of radial threshold solutions for the focusing, generalized energy-critical Hartree equation and classify all radial threshold solutions. The main arguments are the spectral theory of the linearized operator, the modulational analysis and the concentration compactness rigidity argument developed by T. Duyckaerts and F. Merle to classify all threshold solutions for the energy critical NLS and NLW in \cite{DuyMerle:NLS:ThresholdSolution, DuyMerle:NLW:ThresholdSolution}, later by D. Li and X. Zhang in \cite{LiZh:NLS, LiZh:NLW} in higher dimensions. The new ingredient here is to solve the nondegeneracy of positive bubble solutions with nonlocal structure in $\dot H^1(\R^N)$ (i.e. the spectral assumption in \cite{MiaoWX:dynamic gHartree}) by the nondegeneracy result of positive bubble solution in $L^{\infty}(\R^N)$ in \cite{LLTX:Nondegeneracy} and the Moser iteration method in \cite{DiMeVald:book}, which is related to the spectral analysis of the linearized operator with nonlocal structure, and plays a key role in the construction of the special threshold solutions, and the classification of all threshold solutions.

math.AP↗
arXiv2023

MPR-Net:Multi-Scale Pattern Reproduction Guided Universality Time Series Interpretable Forecasting

Time series forecasting has received wide interest from existing research due to its broad applications and inherent challenging. The research challenge lies in identifying effective patterns in historical series and applying them to future forecasting. Advanced models based on point-wise connected MLP and Transformer architectures have strong fitting power, but their secondary computational complexity limits practicality. Additionally, those structures inherently disrupt the temporal order, reducing the information utilization and making the forecasting process uninterpretable. To solve these problems, this paper proposes a forecasting model, MPR-Net. It first adaptively decomposes multi-scale historical series patterns using convolution operation, then constructs a pattern extension forecasting method based on the prior knowledge of pattern reproduction, and finally reconstructs future patterns into future series using deconvolution operation. By leveraging the temporal dependencies present in the time series, MPR-Net not only achieves linear time complexity, but also makes the forecasting process interpretable. By carrying out sufficient experiments on more than ten real data sets of both short and long term forecasting tasks, MPR-Net achieves the state of the art forecasting performance, as well as good generalization and robustness performance.

cs.LG↗
arXiv2023

Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations

In this paper, we show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{equation*} -{\Delta u}\sts{x} -{\bm\alpha}\sts{N,\lambda} \int_{\R^N} { \frac{ u^{p}\sts{y}}{\pabs{\,x-y\,}{\lambda}} }\diff{y}\, u^{p-1}\sts{x} =0,\quad x\in \R^N \end{equation*} where $N\geq 3$, $0<\lambda<N$, $p=\frac{2N-\lambda}{N-2}$ and ${\bm\alpha}\sts{N,\lambda}$ is a normalized constant such that $ u(x)=\left(1+|x|^2\right)^{-\frac{N-2}{2} }$ is a bubble solution of the equation \eqref{NLH}. It solves an open nondegeneracy problem in \cite{MWX:Hartree, GMYZ2022cvpde} and generalizes the partial nondegeneracy results in \cite{DY2019dcds, GWY2020na, LTX2021} to the full range $0<\lambda<N$. The key observation is that by use of the stereographic projection $\mathcal{S}$, the weighted pushforward map $\mathcal{S}_*$ is one-to-one map between the null space of the linearized operator and the spherical harmonic function subspace $\mathcal{H}_1^{N+1}$ of degree one.

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