Uniform estimates and uniqueness of stationary solutions to the drift–diffusion model for semiconductors

We study the stationary problem of the drift–diffusion model with a mixed boundary condition. For this problem, the existence of solutions was established in general settings, while the uniqueness was investigated only in some special cases which do not entirely cover situations that semiconductor devices are used in integrated circuits. In this paper, we prove the uniqueness in a physically relevant situation. The key to the proof is to derive two-sided uniform estimates for the densities of the electron and hole. We establish a new technique to show the lower bound. This together with the Moser iteration method leads to the upper bound.     

A posteriori error estimates of mixed DG finite element methods for linear parabolic equations

In this article, we analyse a posteriori error estimates of mixed finite element discretizations for linear parabolic equations. The space discretization is done using the order λ?≥?1 Raviart–Thomas mixed finite elements, whereas the time discretization is based on discontinuous Galerkin (DG) methods (r?≥?1). Using the duality argument, we derive a posteriori l ^∞(L ²) error estimates for the scalar function, assuming that only the underlying mesh is static.     

Cloaking via impedance boundary condition for the 2-D Helmholtz equation

We consider control problems for the 2-D Helmholtz equation in an unbounded domain with partially coated boundary. Dirichlet boundary condition is given on one part of the boundary and the impedance boundary condition is imposed on another its part. The role of control in control problem under study is played by boundary impedance. Quadratic tracking–type functionals for the field play the role of cost functionals. Solvability of control problems is proved. The uniqueness and stability of optimal solutions with respect to certain perturbations of both cost functional and incident field are established.     

Robust optimization with applications to game theory

In this article, we investigate robust optimization equilibria with two players, in which each player can neither evaluate his opponent's strategy nor his own cost matrix accurately while may estimate a bounded set of the strategy or cost matrix. We obtain a result that solving this equilibria can be formulated as solving a second-order cone complementarity problem under an ellipsoid uncertainty set or a mixed complementarity problem under a box uncertainty set. We present some numerical results to illustrate the behaviour of robust optimization equilibria.     

Differentiation of teaching and learning mathematics: an action research study in tertiary education

Diversity and differentiation within our classrooms, at all levels of education, is nowadays a fact. It has been one of the biggest challenges for educators to respond to the needs of all students in such a mixed-ability classroom. Teachers’ inability to deal with students with different levels of readiness in a different way leads to school failure and all the negative outcomes that come with it. Differentiation of teaching and learning helps addressing this problem by respecting the different levels that exist in the classroom, and by responding to the needs of each learner. This article presents an action research study where a team of mathematics instructors and an expert in curriculum development developed and implemented a differentiated instruction learning environment in a first-year engineering calculus class at a university in Cyprus. This study provides evidence that differentiated instruction has a positive effect on student engagement and motivation and improves students’ understanding of difficult calculus concepts.     

On polygonal approximation of brownian motion in stochastic integral

For Brownian motion B denote by B^δ its polygonal approximation corresponding to a partition δ of [0,1]. It is proved that if E(f¹|X_t|^p dt<∞ for some p>2 then converges to in mean as the mesh |Δ|→0 provided the symmetric (Stratonovich) stochastic integral is determined (in the sense given in [4])     

On optimality of regular projective estimators for semimartingale models,part ii: asymptotically linear estimators

In this paper we consider estimators that (asymptotically) admit a so called linear representation. Using a parametrization of the model, that has been defined in a previous paper [1], and a certain notion of smoothness of the parametrization, it is possible to define a concept of optimality for these estimators and to characterize the optimal estimators. In contrast with the situation in [1], only the compensator is fully parametrized by the parameter we want to estimate. Embedding the problem under consideration in the previously developed framework then requires the introduction of several nuisance parameters, that are needed to describe certain stochastic integrals with respect to the compensator of the jump measure     

Primal mixed formulations for the coupling of FEM and BEM. Part I: linear problems

We apply the boundary integral equation method and a primal mixed finite element approach to study the weak solvability and Galerkin approximations of linear interior transmission problems arising in potential theory and elastostatics. The existence and uniqueness of solution of the resulting weak formulations and of the associated discrete schemes are derived by using the classical theory for variational problems with constraints. Suitable finite element subspaces of Lagrange type satisfying the compatibility conditions are utilized for defining the Galerkin scheme. The error analysis and corresponding rates of convergence are also provided.     

Strong Convergence of a Projection-Type Method for Mixed Variational Inequalities in Hilbert Spaces

Abstract

In this article, a projection-type method for mixed variational inequalities is proposed in Hilbert spaces. The proposed method has the following nice features: (i) The algorithm is well defined whether the solution set of the problem is nonempty or not, under some mild assumptions; (ii) If the solution set is nonempty, then the sequence generated by the method is strongly convergent to the solution, which is closest to the initial point; (iii) The existence of the solutions to variational inequalities can be verified through the behavior of the generated sequence. The results presented in this article generalize and improve some known results.