Robust development of biological organisms in the presence of genetic and epi-genetic perturbations is important for time spans short relative to evolutionary time. Gradients of receptor bound signaling morphogens are responsible for patterning formation and development. A variety of inhibitors for reducing ectopic signaling activities are known to exist and their specific role in down-regulating the undesirable ectopic activities reasonably well understood. However, how a developing organism manages to adjust inhibition/stimulation in response to genetic and/or environmental changes remains to be uncovered. The need to adjust for ectopic signaling activities requires the presence of one or more feedback mechanisms to stimulate the needed adjustment. As the ultimate effect of many inhibitors (including those of the nonreceptor type) is to reduce the availability of signaling morphogens for binding with signaling receptors, a negative feedback on signaling morphogen synthesis rate based on a root-mean-square measure of the spatial distribution of signaling concentration offers a simple approach to robusness and has been demonstrated to be effective in a proof-of-concept implementation. In this paper, we complement the previous investigation of feedback in steady state by examining the effect of one or more feedback adjustments during the transient phase of the biological development.

An arbitrary resolution video coding method based on compressive sampling is proposed. In this method, a video is coded using compressive measurements. The compressive measurements are made on videos of high resolution. The measurements may be used to reconstruct the video at the same resolution as the original video, and any subset of the measurements can be used to reconstruct video at lower resolution with a lower complexity. Video coding with arbitrary solution has important application in mobile video transmission.

We propose a mathematical model and immersed boundary method for the growth and breakup of diatom chains. Diatom chains are treated as zero thickness open curves thanks to their small aspect ratio. The growth of the chain is modelled by adding small pieces of diatoms at the two end points while the breakup is done by removing a small piece in the middle of the chain. Numerical experiments are carried out to investigate the effects of growth and breakup on the sedimentation rate of diatom chains. Simulations of multiple diatom chains show that sedimentation rate is highly dependent on diatoms' spatial distribution. The results can be used to explain the observations that diatoms often form chain-like structures in natural habitats.

Since the successful construction of the so-called double negative metamaterials in 2000, there has been a growing interest in studying metamaterials across many disciplinaries. In this paper, we present a survey of recent progress in metamaterials and its applications from the mathematical point of view. Due to the great amount of papers published in this area, here we mainly discuss those issues interested to us. Our main goal is to attract more mathematicians to study this fascinating subject.

This paper is devoted to the presentation and analysis of some linear and quadratic finite volume (FV) schemes for elliptic problems with singular solutions due to the non-smoothness of the domain. Our FV schemes are constructed over specially-designed graded triangular meshes. We provide sharp parameter selection criteria for the graded mesh, such that both the linear and quadratic FV schemes achieve the optimal convergence rate approximating singular solutions in $H^1$. In addition, we show that on the same mesh, a linear FV scheme obtains the optimal rate of convergence in $L^2$. Numerical tests are provided to verify the analysis.

In this paper, we discuss the time optimal control problems governed by heat equation. The variational discretization concept is introduced for the approximation of the control, and the semi-discrete finite element method is applied for the controlled heat equation. We prove optimal a priori error estimate for the optimal time $T$, and quasi-optimal estimates for the optimal control $u$, the related state $y$ and adjoint state $p$.

MRICE-E model is a new integrated assessment model (IAM) applied on evaluating climate change and the loss of economic welfare. Its economic module for China adopts a dynamic, nonlinear and multi-sectional CGE model. In this paper, we are concerned with the effects of perturbations in input-output coefficients in the CGE model. In the analytical framework, some concepts such as the Lyapunov exponent and the condition number from dynamic system and numerical linear algebra are employed to measure the errors brought by perturbations of the I-O coefficients. We finally derive the upper bound estimation of errors growth through time. To reduce the effects of the possible perturbations, some suggestions about categorization of the industrial sectors are given in the end.

Convection-diffusion equations provide the basis for describing heat and mass transfer phenomena as well as processes of continuum mechanics. An unsteady problem is considered for a convection and a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the fractional power of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The results of numerical experiments are presented for a model two-dimensional problem.

In this paper we design a modified version of FVCOM by adopting hybrid meshes and shifting the placement of velocity variables from the centroids of elements to the middle points of edges. A simplified version of geostrophic equation is solved to test the new scheme, and illustrates a nearly uniform error distribution.