This paper proposes a method for estimation of a class of copula-based semiparametric stationary Markov vector time series models, namely, the two-stage semiparametric pseudo maximum likelihood estimation (2SSPPMLE). These Markov vector time series models are characterized by nonparametric marginal distributions and parametric copula functions of temporal and contemporaneous dependence, while the copulas capture two classes of dependence relationships of Markov time series. We provide simple estimators of marginal distribution and two classes of copulas parameters and establish their asymptotic properties following conclusions in Chen and Fan (2006) and some easily verifiable conditions. Moreover, we obtain the estimation of conditional moment and conditional quantile functions for the bivariate Markov time series model.

It has been established in several types of cancers, that the growth and maintenance of many cancers is due to a (typically small) sub-population of cells with stem-like properties: cells which are capable of indefinite self-renewal and of giving rise to all the different types of cells present in the cancer. The origins of these stem-like cancer cells are not entirely clear, and it has not been established if they originate from healthy stem-cells, healthy self-sustaining subpopulations of progenitor cells or even from mature, fully dierentiated cells by way of dedierentiation. In this paper we investigate some mathematical problems which arise when one considers the possibility of cancer stem-cells arising from healthy progenitor cells (normally possessing limited division potential) which have been perturbed in some way, such as through mutation. For example, glial progenitors have been previously proposed as a possible source of gliomas. We model the progression from stem cell to mature cell where at each cell division we include a probability of a cell either advancing or regressing in maturity. Whereas in normal, healthy cell populations, cells will be more likely to advance in maturity, we suggest that the tendency to advance or regress in maturity may be altered by changes in the cell population such as mutation, and that this may cause the subpopulation of progenitor cells to become self-sustaining, leading to uncontrolled growth of this subpopulation. The conditions, according to our model, under which a population of progenitor cells is self-sustaining are then discussed.

We present a one-dimensional version of a general mesh adaptation technique developed in [1, 2] which is valid for two and three-dimensional problems. The simplicity of the one-dimensional case allows to detail all the necessary steps with very simple computations. We show how the error can be estimated on a piecewise finite element of degree k and how this information can be used to modify the grid using local mesh operations: element division, node elimination and node displacement. Finally, we apply the whole strategy to many challenging singularly perturbed boundary value problems where the one-dimensional setting allows to push the adaptation method to its limits.

Stationary pulse solutions of the cubic-quintic complex Ginzburg-Landau equation are related to heteroclinic orbits in a three-dimensional dynamical systems and they are usually obtained using numerical simulation. The harmonic balance method has severe limitation in computing homoclinic/heteroclinic orbits since the period of such orbits is infinite. In this paper, we present a perturbation-incremental method to find such stationary pulse solutions. With the introduction of a nonlinear transformation, perturbed analytical pulse solutions are obtained in terms of trigonometric functions. Such formulation makes it possible to apply the harmonic balance method to find accurate approximate solutions of the corresponding heteroclinic orbits with arbitrary parametric values. Zero-order analytical solutions from the perturbation step and approximate solutions from the incremental step are compared with that from the bifurcation package AUTO, and they are in good agreement.

We study the existence, uniqueness and continuous dependence on initial data of the solution for a Lotka-Volterra cascade model with one basal species and hierarchal predation. A uniquely solvable, stable, semi-implicit finite-difference scheme is proposed for this system that converges to the true solution uniformly in a finite interval.

An H^1-Galerkin mixed finite element method is applied to the extended Fisher-Kolmogorov equation by employing a splitting technique. The method described in this paper may also be considered as a Petrov-Galerkin method with cubic spline space as trial space and piecewise linear space as test space, since second derivative of a cubic spline is a linear spline. Optimal order error estimates are obtained without any restriction on the mesh. Fully discrete scheme is also discussed and optimal order estimates are obtained. The results are validated with numerical examples.