An HDG Method for Dirichlet Boundary Control of Convection Dominated Diffusion PDE

We first propose a hybridizable discontinuous Galerkin (HDG) method to approximate the solution of a \emph{convection dominated} Dirichlet boundary control problem. Dirichlet boundary control problems and convection dominated problems are each very challenging numerically due to solutions with low regularity and sharp layers, respectively. Although there are some numerical analysis works in the literature on \emph{diffusion dominated} convection diffusion Dirichlet boundary control problems, we are not aware of any existing numerical analysis works for convection dominated boundary control problems. Moreover, the existing numerical analysis techniques for convection dominated PDEs are not directly applicable for the Dirichlet boundary control problem because of the low regularity solutions. In this work, we obtain an optimal a priori error estimate for the control under some conditions on the domain and the desired state. We also present some numerical experiments to illustrate the performance of the HDG method for convection dominated Dirichlet boundary control problems

Inhibition of cancer cell invasion and metastasis by genistein

Genistein is a small, biologically active flavonoid that is found in high amounts in soy. This important compound possesses a wide variety of biological activities, but it is best known for its ability to inhibit cancer progression. In particular, genistein has emerged as an important inhibitor of cancer metastasis. Consumption of genistein in the diet has been linked to decreased rates of metastatic cancer in a number of population-based studies. Extensive investigations have been performed to determine the molecular mechanisms underlying genistein’s antimetastatic activity, with results indicating that this small molecule has significant inhibitory activity at nearly every step of the metastatic cascade. Reports have demonstrated that, at high concentrations, genistein can inhibit several proteins involved with primary tumor growth and apoptosis, including the cyclin class of cell cycle regulators and the Akt family of proteins. At lower concentrations that are similar to those achieved through dietary consumption, genistein can inhibit the prometastatic processes of cancer cell detachment, migration, and invasion through a variety of mechanisms, including the transforming growth factor (TGF)-β signaling pathway. Several in vitro findings have been corroborated in both in vivo animal studies and in early-phase human clinical trials, demonstrating that genistein can both inhibit human cancer metastasis and also modulate markers of metastatic potential in humans, respectively. Herein, we discuss the variety of mechanisms by which genistein regulates individual steps of the metastatic cascade and highlight the potential of this natural product as a promising therapeutic inhibitor of metastasis