Convergence analysis of a weak Galerkin finite element method on a Shishkin mesh for a singularly perturbed fourth-order problem in 2D

We consider the singularly perturbed fourth-order boundary value problem $\varepsilon ^{2}\Delta ^{2}u-\Delta u=f$ on the unit square $\Omega \subset \mathbb{R}^2$, with boundary conditions $u = \partial u / \partial n = 0$ on $\partial \Omega$, where $\varepsilon \in (0, 1)$ is a small parameter. The problem is solved numerically by means of a weak Galerkin(WG) finite element method, which is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on finite element partitions consisting of polygons of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with $N^2$ elements is constructed ,convergence of the method is proved in a discrete $H^2$ norm for the corresponding WG finite element solutions and numerical results are presented

On slope genera of knotted tori in 4-space

In this note, we investigate genera for the slopes of a knotted torus in the 4-sphere analogous to the genus of a classical knot. We compare various formulations of this notion, and use this notion to study the extendable subgroup of the mapping class group of the knotted torus.Comment: 24 pages, accepted for publication by Pacific Journal of Mathematic

Astragaloside IV inhibits pathological functions of gastric cancer-associated fibroblasts through regulation of HOXA6/ZBTB12 axis

[email protected], [email protected] Cancer-associated fibroblasts (CAFs) play critical roles in the tumor microenvironment and exert tumor-promoting or tumor-retarding effects on cancer development. Astragaloside IV has been suggested to rescue the pathological impact of CAFs in gastric cancer. This study aimed to investigate the potential mechanism of astragaloside IV in the regulation of CAF pathological functions in gastric cancer development. Homeobox A6 (HOXA6), and Zinc Finger and BTB Domain Containing 12 (ZBTB12) are highly expressed in gastric CAFs compared with normal fibroblasts (NFs) based on the GSE62740 dataset. We found that astragaloside IV-stimulated CAFs suppressed cell growth, migration, and invasiveness of gastric cancer cells. HOXA6 and ZBTB12 were downregulated after astragaloside IV treatment in CAFs. Further analysis revealed that HOXA6 or ZBTB12 knockdown in CAFs also exerted inhibitory effects on the malignant phenotypes of gastric cells. Additionally, HOXA6 or ZBTB12 overexpression in CAFs enhanced gastric cancer cell malignancy, which was reversed after astragaloside IV treatment. Moreover, based on the hTFtarget database, ZBTB12 is a target gene that may be transcriptionally regulated by HOXA6. The binding between HOXA6 and ZBTB12 promoter in 293T cells and CAFs was further confirmed. HOXA6 silencing also induced the downregulation of ZBTB12 mRNA and protein in CAFs. Astragaloside IV was demonstrated to regulate the expression of ZBTB12 by mediating the transcriptional activity of HOXA6. Our findings shed light on the therapeutic value of astragaloside IV for gastric cancer

Cell responses to two kinds of nanohydroxyapatite with different sizes and crystallinities

Xiaochen Liu1, Minzhi Zhao1, Jingxiong Lu2, Jian Ma4, Jie Wei2, Shicheng Wei1,31Center for Biomedical Materials and Tissue Engineering, Academy for Advanced Interdisciplinary Studies, Peking University, Beijing, 2Key Laboratory for Ultrafine Materials of Ministry of Education, East China University of Science and Technology, Shanghai, 3Department of Oral and Maxillofacial Surgery, School of Stomatology, Peking University, Beijing, 4Hospital of Stomatology, Tongji University, Shanghai, ChinaIntroduction:Hydroxyapatite (HA) is the principal inorganic constituent of human bone. Due to its good biocompatibility and osteoconductivity, all kinds of HA particles were prepared by different methods. Numerous reports demonstrated that the properties of HA affected its biological effects.Methods: Two kinds of nanohydroxyapatite with different sizes and crystallinities were obtained via a hydrothermal treatment method under different temperatures. It was found that at a temperature of 140°C, a rod-like crystal (n-HA1) with a diameter of 23 ± 5 nm, a length of 47 ± 14 nm, and crystallinity of 85% ± 5% was produced, while at a temperature of 80°C, a rod-like crystal (n-HA2) with a diameter of 16 ± 3 nm, a length of 40 ± 10 nm, and crystallinity of 65% ± 3% was produced. The influence of nanohydroxyapatite size and crystallinity on osteoblast viability was studied by MTT, scanning electron microscopy, and flow cytometry.Results: n-HA1 gave a better biological response than n-HA2 in promoting cell growth and inhibiting cell apoptosis, and also exhibited much more active cell morphology. Alkaline phosphatase activity for both n-HA2 and n-HA1 was obviously higher than for the control, and no significant difference was found between n-HA1 and n-HA2. The same trend was observed on Western blotting for expression of type I collagen and osteopontin. In addition, it was found by transmission electron microscopy that large quantities of n-HA2 entered into the cell and damaged the cellular morphology. Release of tumor necrosis factor alpha from n-HA2 was markedly higher than from n-HA1, indicating that n-HA2 might trigger a severe inflammatory response.Conclusion: This work indicates that not all nanohydroxyapatite should be considered a good biomaterial in future clinical applications.Keywords: nanohydroxyapatite, osteoblast-like cells, cell viability, cell differentiatio

VITATECS: A Diagnostic Dataset for Temporal Concept Understanding of Video-Language Models

The ability to perceive how objects change over time is a crucial ingredient in human intelligence. However, current benchmarks cannot faithfully reflect the temporal understanding abilities of video-language models (VidLMs) due to the existence of static visual shortcuts. To remedy this issue, we present VITATECS, a diagnostic VIdeo-Text dAtaset for the evaluation of TEmporal Concept underStanding. Specifically, we first introduce a fine-grained taxonomy of temporal concepts in natural language in order to diagnose the capability of VidLMs to comprehend different temporal aspects. Furthermore, to disentangle the correlation between static and temporal information, we generate counterfactual video descriptions that differ from the original one only in the specified temporal aspect. We employ a semi-automatic data collection framework using large language models and human-in-the-loop annotation to obtain high-quality counterfactual descriptions efficiently. Evaluation of representative video-language understanding models confirms their deficiency in temporal understanding, revealing the need for greater emphasis on the temporal elements in video-language research.Comment: 23 pages, 6 figures, 18 tables, data is available at https://github.com/lscpku/VITATEC