In this paper, we consider the Hölder continuity of the integrated density of states (IDS). Applying Avila’s almost reducible result and KAM technique, we proved that there exists a dense subset of Liouvillean frequencies α , for which the IDS of the analytic quasi-periodic Schrödinger operator is ( χ - log )-Hölder continuous for any χ >1 , provided that the subcritical strip of the operator satisfies h_0 > 2β (α ) . We also proved the χ -Hölder continuity of the IDS for a dense subset of Liouvillean frequencies for operators with 0< χ < 1/2 , if the subcritical strip satisfies h_0 > 8β/ 1-2χ .

We establish the absolute continuity of the integrated density of states (IDS) for quasi-periodic Schr\"odinger operators with a large trigonometric potential and Diophantine frequency. This partially solves Eliasson's open problem in 2002. Furthermore, this result can be extended to a class of quasi-periodic long-range operators on $\ell^2(\Z^d)$. Our proof is based on stratified quantitative almost reducibility results of dual cocycles. Specifically, we prove that a generic analytic one-parameter family of cocycles, sufficiently close to constant coefficients, is reducible except for a zero Hausdorff dimension set of parameters. This result affirms Eliasson's conjecture in 2017.

In this paper,we consider the reducibility of three-dimensional skew symmetric systems.We obtain a reducibility result if the base frequency is high-dimensional weak Liouvillean and the parameter is sufficiently small.The proof is based on a modified KAM theory for 3-dimensional skew symmetric systems.

Synthetic Aperture Radar (SAR) images often suffer from inherent speckle noise that can significantly degrades their quality, and makes it difficult to identify important targets or extract useful information from them. To solve this problem, a novel SAR image despeckling model called HM-SIDLR is proposed, which can suppress noise and preserve image edges from low-rank residues simultaneously. Specifically, our method considers that there is some structural information discarded as noise in residues. Therefore, we construct a hierarchical model to extract more edge details which can compensate for the over-smoothing problem caused by removing most of the speckle noise in the low-rank part. So as to learn the information from the edge subspace, a tunable prior knowledge matrix is designed, allowing for differentiation between clean and noisy pixels. Experimental results with the Virtual SAR dataset indicate that HM-SIDLR can achieve comparable despeckling performance to the state-of-the-art (SOTA) results.

The low-rank models have gained remarkable performance in the field of remote sensing image denoising. Nonetheless, the existing low-rank-based methods view residues as noise and simply discard them. This causes denoised results to lose many important details, especially the edges. In this paper, we propose a new denoising method named EPLRR-RSID, which focuses on edge preservation to improve the image quality of the details. Specifically, we considered the low-rank residues as a combination of useful edges and noisy components. In order to better learn the edge information from the low-rank representation (LRR), we designed multi-level knowledge to further distinguish the edge part and the noise part from the residues. Furthermore, a manifold learning framework was introduced in our proposed model to better obtain the edge information, as it can find the structural similarity of the edge part while suppressing the influence of the non-structural noise part. In this way, not only the low-rank part is better learned, but also the edge part is precisely preserved. Extensive experiments on synthetic and several real remote sensing datasets showed that EPLRR-RSID has superior advantages over the compared state-of-the-art (SOTA) approaches, with the mean edge protect index (MEPI) values reaching at least 0.9 and the best values in the no-reference index BRISQUE, which represents that our method improved the image quality by edge preserving.

Robust unsupervised feature learning is a critical yet tough task for synthetic aperture radar (SAR) automatic target recognition (ATR) with limited labeled data. The developing contrastive self-supervised learning (CSL) method, which learns informative representations by solving an instance discrimination task, provides a novel method for learning discriminative features from unlabeled SAR images. However, the instance-level contrastive loss can magnify the differences between samples belonging to the same class in the latent feature space. Therefore, CSL can dispel these targets from the same class and affect the downstream classification tasks. In order to address this problem, this paper proposes a novel framework called locality preserving property constrained contrastive learning (LPPCL), which not only learns informative representations of data but also preserves the local similarity property in the latent feature space. In LPPCL, the traditional InfoNCE loss of the CSL models is reformulated in a cross-entropy form where the local similarity of the original data is embedded as pseudo labels. Furthermore, the traditional two-branch CSL architecture is extended to a multi-branch structure, improving the robustness of models trained with limited batch sizes and samples. Finally, the self-attentive pooling module is used to replace the global average pooling layer that is commonly used in most of the standard encoders, which provides an adaptive method for retaining information that benefits downstream tasks during the pooling procedure and significantly improves the performance of the model. Validation and ablation experiments using MSTAR datasets found that the proposed framework outperformed the classic CSL method and achieved state-of-the-art (SOTA) results.

Synthetic aperture radar imaging is a powerful remote sensing technology, but its images are often degraded by speckle noise, reducing quality. In recent years, speckle reduction has become a hot topic in SAR research, with various methods emerging. However, traditional low-rank SAR denoising causes edge blurring and loss of important features. To address this, we propose a new approach using prior knowledge to distinguish noise pixels from edges. By separating low-rank residuals and effectively extracting edge information, our method outperforms others, generating noise-suppressed SAR images while retaining features. Experiments demonstrate the effectiveness of our approach.

Sarnaku2019s Mu00F6bius disjointness conjecture states that Mu00F6bius function is disjoint to any zero entropy dynamics. We prove that Mu00F6bius disjointness conjecture holds for one-frequency analytic quasi-periodic cocycles which are almost reducible, which extends (Liu and Sarnak in Duke Math J 164(7):1353u20131399, 2015; Wang in Invent Math 209:175u2013196, 2017) to the noncommutative case. The proof relies on quantitative version of almost reducibility.

Sarnak’s Möbius disjointness conjecture states that Möbius function is disjoint to any zero entropy dynamics. We prove that Möbius disjointness conjecture holds for one-frequency analytic quasi-periodic cocycles which are almost reducible, which extends (Liu and Sarnak in Duke Math J 164(7):1353–1399, 2015; Wang in Invent Math 209:175–196, 2017) to the noncommutative case. The proof relies on quantitative version of almost reducibility.

We prove that for quasi-periodically forced harmonic oscillator, if the forcing frequency ω∈R2﹨Q2 is non-resonant and the forcing is partial Gevrey smooth, then there exist response solutions. This generalized the result by [22] where the forcing term is real analytic. The proof is based on modified KAM theory for lower dimensional invariant tori.

In this paper, we prove that for any d-frequency analytic quasiperiodic Schrödinger operator, if the frequency is weak Liouvillean, and the potential is small enough, then the corresponding operator has absolutely continuous spectrum. Moreover, in the case d=2, we even establish the existence of ac spectrum under small potential and some super-Liouvillean frequency, and this result is optimal due to a recent counterexample of Avila and Jitomirskaya [6].

We prove that an analytic quasiperiodically forced circle flow with a not super-Liouvillean base frequency and which is close enough to some constant rotation is C ∞ rotations reducible, provided its fibered rotation number is Diophantine with respect to the base frequency. As a corollary, we obtain that among such systems, the linearizable ones and those displaying mode-locking are locally dense for the C ∞ -topology.

We study order-preserving C^1-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures of nearly all family members are one-rectifiable, that is, absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs.

We show that a generic quasiperiodically forced circle homeomorphism is mode-locked: the rotation number in the fibres is rationally related to the rotation number in the base and it is stable under small perturbations of the system. As a consequence, this implies that for a generic parameter family of quasiperiodically forced circle homeomorphisms satisfying a twist condition, the graph of the rotation number as a function of the parameter is a devil's staircase.

We study the boundedness of solutions for non-linear quasi-periodic differential equations with Liouvillean frequencies. We proved that if the forcing is quasi-periodic in time with two frequencies which is not super-Liouvillean, then all solutions of the equation are bounded. The proof is based on action-angle variables and modified KAM theory.

We prove that the non-linear quasi-periodically forced harmonic oscillator with two frequencies ( 1 , α ) (1,\alpha ) has at least one response solution if the forcing is small. No arithmetic condition on the frequency is assumed and the smallness of the non-linear forcing does not depend on α \alpha . The result strengthens the existing results in the literature where the frequency is assumed to be Diophantine. The proof is based on a modified KAM theory for the lower dimensional tori.

A new scheme for key distribution based on variant properties of chaos synchronization in cascaded semiconductor lasers with phase-modulation feedback. The security of the scheme relies on the practical difficulty of completely observing chaotic signals.

For commuting smooth quasiperiodically forced circle diffeomorphisms, we show that if the base frequencies and the fibred rotation numbers jointly satisfy some simultaneous Diophantine condition and if the diffeomorphisms are in some C ∞ C^{\infty } neighborhood of the corresponding rotations, then they are simultaneously C ∞ C^{\infty } -linearizable.

For any analytic quasiperiodically forced circle diffeomorphisms (ω,〈pq,ω〉+εf), where f is fixed and ε is small, we show that if ω is Diophantine and the fibred rotation number of the diffeomorphism remains constant in a unilateral neighborhood of ε=0 (i.e., there is a unilateral phase-locking at ε=0), then the diffeomorphism has at least one analytic q-invariant torus, provided ε is small enough.

We study the reducibility problems for quasiperiodic cocycles in linear Lie groups with one frequency, irrespective of any Diophantine condition on the base dynamics. Under a non-degeneracy condition, a positive measure diagonalizable result is obtained for quasiperiodic \({GL(d,\mathbb R)}\) cocycles which are close to constants. It generalizes previous works by Avila–Fayad–Krikorian and Hou–You, and our approach is based on periodic approximation and KAM schemes.