In the previous studies, it was shown that the ditopological texture spaces provide a unified setting for the study of topology, bitopology and fuzzy topology on Hutton algebras. Following that, a suitable uniformity theory in textures was constructed under the name di-uniformity, and this represented a generalization of classical quasi-uniformity. Therefore, by defining regular Cauchy difilters on a texture, the concept of dicompleteness as a natural counterpart of "completeness" in the sense of quasi-uniformity has been introduced for di-uniform texture spaces.

Under certain conditions there are considerably more quasi-lattices than Banach lattices. For example, (within reason) nearly every finite dimensional ordered Banach space is a quasi-lattice, while there is essentially only one n-dimensional Banach lattice for each n.

We will show how certain properties carry over from Banach lattices to quasi-lattices and discuss some still open problems.

(Technical details are kept to a minimum, the talk should be accessible to students and non-analysts who know what a Banach space is).

Great progress has been made for high order finite difference methods during the last 20 years. It is now rather straightforward to construct stable and high order accurate multi-block finite difference schemes in a building-block-like manner. In this talk we will discus new developments, new extensions, new fields of applications, share experience on the full scale use of the technique, and speculate about future trends. We present the theory by using simple model problems, and exemplify with complex multi-physics applications.

First we walk through the basic technique for a simple model problem where all the ingredients such as well-posedness, summation-by-parts operators, penalty techniques for boundary and interface conditions, and the energy method are explained. Next, we discuss the application of the theory to coupling techniques for conjugate heat transfer, earthquake simulations, fluid- solid interaction and simulation of potential propagation in neuronal branches. Finally we discuss new time-integration procedures, new boundary conditions for the Navier-Stokes equations and a new variance reduction technique for stochastic partial differential equations.