Anisotropic analysis of VEM for time-harmonic Maxwell equations in inhomogeneous media with low regularity

It has been extensively studied in the literature that solving Maxwell equations is very sensitive to the mesh structure, space conformity and solution regularity. Roughly speaking, for almost all the methods in the literature, optimal convergence for low-regularity solutions heavily relies on conforming spaces and highly-regular simplicial meshes. This can be a significant limitation for many popular methods based on polytopal meshes in the case of inhomogeneous media, as the discontinuity of electromagnetic parameters can lead to quite low regularity of solutions near media interfaces, and potentially worsened by geometric singularities, making many popular methods based on broken spaces, non-conforming or polytopal meshes particularly challenging to apply. In this article, we present a virtual element method for solving an indefinite time-harmonic Maxwell equation in 2D inhomogeneous media with quite arbitrary polytopal meshes, and the media interface is allowed to have geometric singularity to cause low regularity. There are two key novelties: (i) the proposed method is theoretically guaranteed to achieve robust optimal convergence for solutions with merely $\mathbf{H}^{\theta}$ regularity, $\theta\in(1/2,1]$; (ii) the polytopal element shape can be highly anisotropic and shrinking, and an explicit formula is established to describe the relationship between the shape regularity and solution regularity. Extensive numerical experiments will be given to demonstrate the effectiveness of the proposed method

Virtual Element Methods Without Extrinsic Stabilization

Virtual element methods (VEMs) without extrinsic stabilization in arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM and a conforming VEM in arbitrary dimension under the mesh assumption that all the faces of each polytope are simplices. The key is to construct local $H({\rm div})$-conforming macro finite element spaces such that the associated $L^2$ projection of the gradient of virtual element functions is computable, and the $L^2$ projector has a uniform lower bound on the gradient of virtual element function spaces in $L^2$ norm. Optimal error estimates are derived for these VEMs. Numerical experiments are provided to test the VEMs without extrinsic stabilization.Comment: 25 pages, 8 figure

Scalable manifold learning by uniform landmark sampling and constrained locally linear embedding

As a pivotal approach in machine learning and data science, manifold learning aims to uncover the intrinsic low-dimensional structure within complex nonlinear manifolds in high-dimensional space. By exploiting the manifold hypothesis, various techniques for nonlinear dimension reduction have been developed to facilitate visualization, classification, clustering, and gaining key insights. Although existing manifold learning methods have achieved remarkable successes, they still suffer from extensive distortions incurred in the global structure, which hinders the understanding of underlying patterns. Scalability issues also limit their applicability for handling large-scale data. Here, we propose a scalable manifold learning (scML) method that can manipulate large-scale and high-dimensional data in an efficient manner. It starts by seeking a set of landmarks to construct the low-dimensional skeleton of the entire data, and then incorporates the non-landmarks into the learned space based on the constrained locally linear embedding (CLLE). We empirically validated the effectiveness of scML on synthetic datasets and real-world benchmarks of different types, and applied it to analyze the single-cell transcriptomics and detect anomalies in electrocardiogram (ECG) signals. scML scales well with increasing data sizes and embedding dimensions, and exhibits promising performance in preserving the global structure. The experiments demonstrate notable robustness in embedding quality as the sample rate decreases.Comment: 33 pages, 10 figure

Geometric Decomposition and Efficient Implementation of High Order Face and Edge Elements

This paper delves into the world of high-order curl and div elements within finite element methods, providing valuable insights into their geometric properties, indexing management, and practical implementation considerations. It first explores the decomposition of Lagrange finite elements. The discussion then extends to H(div)-conforming finite elements and H(curl)-conforming finite element spaces by choosing different frames at different sub-simplex. The required tangential continuity or normal continuity will be imposed by appropriate choices of the tangential and normal basis. The paper concludes with a focus on efficient indexing management strategies for degrees of freedom, offering practical guidance to researchers and engineers. It serves as a comprehensive resource that bridges the gap between theory and practice in the realm of high-order curl and div elements in finite element methods, which are vital for solving vector field problems and understanding electromagnetic phenomena.Comment: 25 pages, 8 figure

Comparison of the efficacy of oral contraceptives and levonorgestrel intrauterine system in intermenstrual bleeding caused by uterine niche

This study aimed to compare the effectiveness of oral contraceptives and a levonorgestrel intrauterine system in treating intermenstrual bleeding due to uterine niche. We retrospectively analyzed 72 patients with intermenstrual bleeding due to uterine niche from January 2017 to December 2021, of whom 41 were treated with oral contraceptives and 31 with a levonorgestrel intrauterine system. Post-treatment follow-ups at 1, 3, and 6 months were conducted to compare the efficiency and adverse effects between the two groups. In the oral contraceptive group, the effectiveness rate was higher than 80% at 1- and 3-months post-treatment and higher than 90% at 6 months. In the levonorgestrel intrauterine system group, the effectiveness rates were 58.06%, 54.84%, and 61.29% at 1, 3, and 6 months of treatment, respectively. Oral contraceptives were more effective than the levonorgestrel intrauterine system in treating intermenstrual bleeding caused by uterine niche (p < 0.05)

Enhanced photodynamic therapy through multienzyme-like MOF for cancer treatment

Overcoming resistance to apoptosis is a major challenge in cancer therapy. Recent research has shown that manipulating mitochondria, the organelles critical for energy metabolism in tumor cells, can increase the effectiveness of photodynamic therapy and trigger apoptosis in tumor cells. However, there is currently insufficient research and effective methods to exploit mitochondrial damage to induce apoptosis in tumor cells and improve the effectiveness of photodynamic therapy. In this study, we present a novel nanomedicine delivery and therapeutic system called PyroFPSH, which utilizes a nanozymes-modified metal-organic framework as a carrier. PyroFPSH exhibits remarkable multienzyme-like activities, including glutathione peroxidase (GPx) and catalase (CAT) mimicry, allowing it to overcome apoptosis resistance, reduce endogenous glutathione levels, and continuously generate reactive oxygen species (ROS). In addition, PyroFPSH can serve as a carrier for the targeted delivery of sulfasalazine, a drug that can induce mitochondrial depolarization in tumor cells, thereby reducing oxygen consumption and energy supply in the mitochondria of tumor cells and weakening resistance to other synergistic treatment approaches. Our experimental results highlight the potential of PyroFPSH as a versatile nanoplatform in cancer treatment. This study expands the biomedical applications of nanomaterials as platforms and enables the integration of various novel therapeutic strategies to synergistically improve tumor therapy. It deepens our understanding of multienzyme-mimicking active nanocarriers and mitochondrial damage through photodynamic therapy. Future research can further explore the potential of PyroFPSH in clinical cancer treatment and improve its drug loading capacity, biocompatibility and targeting specificity. In summary, PyroFPSH represents a promising therapeutic approach that can provide new insights and possibilities for cancer treatment