In this paper, we provide a number of new estimates on the stability and convergence both hybrid discontinuous Galerkin (HDG) weak (WG) methods. By using standard Brezzi theory mixed methods, carefully define appropriate norms for various discretization variables then establish that error hold uniformly with respect to stabilization parameters. As result, by taking limit parameters, show HDG me...

It is often suggested that the place cells of the hippocampus (and more recently, the grid cells of the entorhinal cortex) furnish a cognitive map. However, this can only be part of the story: (1) the ‘‘hippocampal chart’’ provided by place cells and grid cells differs radically when a rat is placed in different environments and so a higher level organization is needed to link these charts into...

Wnts and bone morphogenetic proteins (BMPs) are signaling elements that are crucial for a variety of events in animal development. In Drosophila, Wingless (Wg, a Wnt ligand) and Decapentaplegic (Dpp, a BMP homolog) are thought to function through distinct signal transduction pathways and independently direct the patterning of the wing. However, recent studies suggest that Mothers against Dpp (M...

Intestinal stem cell (ISC) self-renewal and proliferation are directed by Wnt/β-catenin signaling in mammals, whereas aberrant Wnt pathway activation in ISCs triggers the development of human colorectal carcinoma. Herein, we have utilized the Drosophila midgut, a powerful model for ISC regulation, to elucidate the mechanisms by which Wingless (Wg)/Wnt regulates intestinal homeostasis and develo...

A weak Galerkin (WG) finite element method without stabilizers was introduced in Ye and Zhang (2020) on polytopal mesh. Then it improved (2021) with order one superconvergence. The goal of this paper is to develop a new stabilizer free WG This has convergence rates two orders higher than the optimal for corresponding solution both an energy norm L2 norm. numerical examples are tested low high e...

The development of the Drosophila leg requires both Decapentaplegic (Dpp) and Wingless (Wg), two signals that establish the proximo-distal (PD) axis by activating target genes such as Distalless (Dll). Dll expression in the leg depends on a Dpp- and Wg-dependent phase and a maintenance phase that is independent of these signals. Here, we show that accurate Dll expression in the leg results from...

BACKGROUND Secreted signaling proteins of the Wingless (Wg)/Wnt, Hedgehog and bone morphogenetic protein (BMP)/Decapentaplegic (Dpp) families function as morphogens to control growth and pattern formation during development. Although these proteins have been shown to act directly on distant cells in the developing limbs of the fruit fly Drosophila, little is known about how ligand gradients for...

Drosophila Wingless (Wg) is a morphogen that determines cell fate during development. Previous studies have shown that endocytic pathways regulate Wg trafficking and signaling. Here, we showed that loss of vamp7, a gene required for vesicle fusion, dramatically increased Wg levels and decreased Wg signaling. Interestingly, we found that levels of Dally-like (Dlp), a glypican that can interact w...