arXiv subjects

Lei Ouyang

Publications and source records attributed to Lei Ouyang.

8 recordsLinked to original sources

arXiv2026

Krein-Feller operators on Riemannian manifolds: compactness of embedding and Hodge's theorem

For a bounded open set Omega in a complete oriented Riemannian n-manifold and a positive finite Borel measure mu with support contained in Omega, we define an associated Krein-Feller operators (or Laplacian) Delta_mu by assuming the Poincar'e inequalities for the measure mu. We obtain sufficient conditions for the operator to have compact resolvent and in this case, we prove the Hodge theorem for functions, which states that there exists an orthonormal basis of L^2(Omega,mu) consisting of eigenfunctions of Delta_mu, the eigenspaces are finite-dimensional, and the eigenvalues of -Delta_mu are real, countable, and increasing to infinity. One of these sufficient conditions is that the lower L^infty-dimension dim_infty(mu) of mu is greater than n-2. We prove that the compactness of embedding for functions also hold for measures without compact support, provided the manifold is of bounded geometry. The main idea of our proof is to use Toponogov's and Rauch's comparison theorems to extend a classical compact embedding theorem of Maz'ja to Riemannian manifolds. For a compact Riemannian manifold, using the above results, we also obtain sufficient conditions for Hodge Laplacian on k-forms, to have compact resolvent. Our result extends the classical Hodge theorem to Krein-Feller operators. We study the condition dim_infty(mu)>n-2 for self-similar and self-conformal measures. Results in this paper extend analogous ones by Hu et al. in J. Funct. Anal., which are established for measures on R^n.

math.FA↗

arXiv2026

WS-IMUBench: Can Weakly Supervised Methods from Audio, Image, and Video Be Adapted for IMU-based Temporal Action Localization?

IMU-based Human Activity Recognition (HAR) has enabled a wide range of ubiquitous computing applications, yet its dominant clip classification paradigm cannot capture the rich temporal structure of real-world behaviors. This motivates a shift toward IMU Temporal Action Localization (IMU-TAL), which predicts both action categories and their start/end times in continuous streams. However, current progress is strongly bottlenecked by the need for dense, frame-level boundary annotations, which are costly and difficult to scale. To address this bottleneck, we introduce WS-IMUBench, a systematic benchmark study of weakly supervised IMU-TAL (WS-IMU-TAL) under only sequence-level labels. Rather than proposing a new localization algorithm, we evaluate how well established weakly supervised localization paradigms from audio, image, and video transfer to IMU-TAL under only sequence-level labels. We benchmark seven representative weakly supervised methods on seven public IMU datasets, resulting in over 3,540 model training runs and 7,080 inference evaluations. Guided by three research questions on transferability, effectiveness, and insights, our findings show that (i) transfer is modality-dependent, with temporal-domain methods generally more stable than image-derived proposal-based approaches; (ii) weak supervision can be competitive on favorable datasets (e.g., with longer actions and higher-dimensional sensing); and (iii) dominant failure modes arise from short actions, temporal ambiguity, and proposal quality. Finally, we outline concrete directions for advancing WS-IMU-TAL (e.g., IMU-specific proposal generation, boundary-aware objectives, and stronger temporal reasoning). Beyond individual results, WS-IMUBench establishes a reproducible benchmarking template, datasets, protocols, and analyses, to accelerate community-wide progress toward scalable WS-IMU-TAL.

cs.CV↗

arXiv2024

Iterated relation systems on Riemannian manifolds

For fractals on Riemannian manifolds, the theory of iterated function systems often does not apply well directly, as fractal sets are often defined by relations that are multivalued or non-contractive. To overcome this difficulty, we introduce the notion of iterated relation systems. We study the attractor of an iterated relation system and formulate a condition under which such an attractor can be identified with that of an associated graph-directed iterated function system. Using this method, we obtain dimension formulas for the attractor of an iterated relation system under the graph open set condition or the graph finite type condition. This method improves the one in [Ngai-Xu, J. Geom. Anal. {\bf 33} (2023), 262], which relies on knowing the specific structure of the attractor.

math.DS↗

arXiv2024

Differential equations defined by Kre\uın-Feller operators on Riemannian manifolds

We study linear and semi-linear wave, heat, and Schrödinger equations defined by Kre\uın-Feller operator $-Δ_μ$ on a complete Riemannian $n$-manifolds $M$, where $μ$ is a finite positive Borel measure on a bounded open subset $Ω$ of $M$ with support contained in $\overlineΩ$. Under the assumption that $\underline{\operatorname{dim}}_{\infty}(μ)>n-2$, we prove that for a linear or semi-linear equation of each of the above three types, there exists a unique weak solution. We study the crucial condition $\dim_(μ)>n-2$ and provide examples of measures on $\mathbb{S}^2$ and $\mathbb{T}^2$ that satisfy the condition. We also study weak solutions of linear equations of the above three classes by using examples on $\mathbb{S}^1$

math.FA↗

arXiv2024

Hodge Theorem for Krein-Feller operators on compact Riemannian manifolds

For an open set in a compact smooth oriented Riemannian n-manifold and a positive finite Borel measure with support contained in the closure of the open set, we define an associated Krein-Feller operator on k-forms by assuming the Poincare inequality. Krein-Feller operators on Euclidean space have been studied extensively in fractal geometry. Using results established by the authors, we obtain sufficient conditions for Krein-Feller operator to have compact resolvent. Under these conditions, we prove the Hodge theorem for forms, which states that there exists an orthonormal basis of L2 consisting of eigenforms of Krein-Feller operator, the eigenspaces are finite-dimensional, and the eigenvalues of Krein-Feller operator are real, countable, and increasing to infinity. One of these sufficient conditions is that the dimension of the measure is greater than n-2. Our result extends the classical Hodge theorem to Krein-Feller operators.

math.FA↗

arXiv2021

Spectrometer-free optical hydrogen sensing based on Fano-like spatial distribution of transmission in a metal-insulator-metal plasmonic Doppler grating

Optical nanosensors are promising for hydrogen sensing because they are small, free from spark generation, and feasible for remote optical readout. Conventional optical nanosensors require broadband excitation and spectrometers, rendering the devices bulky and complex. An alternative is spatial intensity-based optical sensing, which only requires an imaging system and a smartly designed platform to report the spatial distribution of analytical optical signals. Here, we present a spatial intensity-based hydrogen sensing platform based on Fano-like spatial distribution of the transmission in a Pd-Al2O3-Au metal-insulator-metal plasmonic Doppler grating (MIM-PDG). The MIM-PDG manifests the Fano resonance as an asymmetric spatial transmission intensity profile. The absorption of hydrogen changes the spatial Fano-like transmission profiles, which can be analyzed with a "spatial" Fano-resonance model and the extracted Fano resonance parameters can be used to establish analytical calibration lines. While gratings sensitive to hydrogen absorption are suitable for hydrogen sensing, we also found hydrogen insensitive gratings, which provide an unperturbed reference signal and may find applications in nanophotonic devices, that require a stable optical response under fluctuating hydrogen atmosphere. The MIM-PDG platform is a spectrometer-free and intensity-based optical sensor that requires only an imaging system, making it promising for cellphone-based optical sensing applications.

physics.optics↗

arXiv2021

Spatially Resolving the Enhancement Effect in Surface-Enhanced Coherent Anti-Stokes Raman Scattering by Plasmonic Doppler Gratings

In this work, we introduce the platform of plasmonic Doppler grating (PDG) to experimentally investigate the enhancement effect of plasmonic gratings in the input and output beams of nonlinear surface-enhanced coherent anti-Stokes Raman scattering (SECARS). PDGs are designable azimuthally chirped gratings that provide broadband and spatially dispersed plasmonic enhancement. Therefore, they offer the opportunity to observe and compare the overall enhancement from different combinations of enhancement in individual input and output beams. We first confirm PDG's capability of spatially separating the input and output enhancement in linear surface-enhanced fluorescence and Raman scattering. We then investigate spatially resolved enhancement in nonlinear SECARS, where coherent interaction of the pump, Stokes, and anti-Stokes beams is enhanced by the plasmonic gratings. By mapping the SECARS signal and analyzing the azimuthal angle-dependent intensity, we characterize the enhancement at individual frequencies. Together with theoretical analysis, we show that while simultaneous enhancement in the input and output beams is important for SECARS, the enhancement in the pump and anti-Stokes beams plays a more critical role in the overall enhancement than that in the Stokes beam. This work provides an insight into the enhancement mechanism of plasmon-enhanced spectroscopy, which is important for the design and optimization of plasmonic gratings. The PDG platform may also be applied to study enhancement mechanisms in other nonlinear light-matter interactions or the impact of plasmonic gratings on the fluorescence lifetime.

physics.optics↗

arXiv2019

Designable spectrometer-free index sensing using plasmonic Doppler gratings

Typical nanoparticle-based plasmonic index sensors detect the spectral shift of localized surface plasmon resonance (LSPR) upon the change of environmental index. Therefore, they require broadband illumination and spectrometers. The sensitivity and flexibility of nanoparticle-based index sensors are usually limited because LSPR peaks are usually broad and the spectral position cannot be freely designed. Here, we present a fully designable index sensing platform using a plasmonic Doppler grating (PDG), which provides broadband and azimuthal angle-dependent grating periodicities. Different from LSPR, the PDG index sensor is based on the momentum matching between photons and surface plasmons via the lattice momentum of the grating. Therefore, index change is translated into the variation of in-plane azimuthal angle for photon-to-plasmon coupling, which manifests as directly observable dark bands in the reflection image. The PDG can be freely designed to optimally match the range of index variation for specific applications. In this work, we demonstrate PDG index sensors for large (n = 1.00 to 1.52) and small index variation (n = 1.3330 to 1.3650). The tiny and nonlinear index change of water-ethanol mixture has been clearly observed and accurately quantified. Since the PDG is a dispersive device, it enables on-site and single-color index sensing without a spectrometer and provides a promising spectroscopic platform for on-chip analytical applications.

physics.app-ph↗