A New Variational Formulation for a Kind of Reaction-diffusion Problem in Broken Sob olev Space

In this paper, a new variational formulation for a reaction-diffusion problem in broken Sobolev space is proposed. And the new formulation in the broken Sobolev space will be proved that it is well-posed and equivalent to the standard Galerkin variational formulation. The method will be helpful to easily solve the original partial differential equation numerically. And the method is novel and interesting, which can be used to deal with some complicated problem, such as the low regularity problem, the differential-integral problem and so on.     

On Some Properties of $c$-Frames

In this paper we discuss about $c$-frames, namely continuous frames. Since, $c$-frames are generalizations of discrete frames, we generalize some results of discrete frames to continuous version. We explain some results about relations of projections in Hilbert spaces and $c$-frames to characterize these frames. Also, we will specify (precisely) the synthesis and frame operators of Bochner integrable $c$-frames. Finally, we classify Hilbert-Schmidt operators by $c$-frames and express some new identities for Parseval $c$-frames.     

Some Topological Properties of Charming Spaces

In this paper, we mainly discuss the class of charming spaces. First, we show that there exists a charming space such that the Tychonoff product is not a charming space. Then we discuss some properties of charming spaces and give some characterizations of some class of charming spaces. Finally, we show that the Suslin number of an arbitrary charming rectifiable space is countable.     

Multilinear Calderón-Zygmund Operators and Their Commutators with BMO Functions in Herz-Morrey Spaces with Variable Smoothness and Integrability

In this paper, we obtain that multilinear Calderón-Zygmund operators and their commutators with BMO functions are bounded on products of Herz-Morrey spaces with variable smoothness and integrability. The vector-valued setting of multilinear Calderón-Zygmund operators is also considered.     

Error estimates for space–time discontinuous Galerkin formulation based on proper orthogonal decomposition

In this study, proper orthogonal decomposition (POD) method is applied to diffusion–convection–reaction equation, which is discretized using space–time discontinuous Galerkin (dG) method. We provide estimates for POD truncation error in dG-energy norm, dG-elliptic projection, and space–time projection. Using these new estimates, we analyze the error between the dG and the POD solution, and the error between the exact and the POD solution. Numerical results, which are consistent with theoretical convergence rates, are presented.     

Computation of waveguide scattering matrix near thresholds

A waveguide occupies infinite strip with one or several narrows on a two-dimensional (2D) plane and is governed by the Helmholtz equation with Dirichlet boundary condition. On the waveguide continuous spectrum, which coincides with a half-axis, a scattering matrix is defined. At each point of the continuous spectrum this matrix has finite size, which changes at thresholds. The thresholds form a sequence of positive numbers increasing to infinity. Approximate calculation of the scattering matrix in a threshold vicinity requires special treatment. We discuss and compare two methods of numerical approximation to the scattering matrix near a threshold.