On Wavelet Decomposition of the Singular Splines

One of the approaches to the problem of approximating functions with a singularity is the creation of an approximating apparatus based on splines with the same feature. For the wavelet decomposition of spline spaces it is important that the property of the embedding of these spaces is associated with embedding grids. The purpose of this paper is to consider ways of constructing spaces of splines with a predefined singularity and obtain their wavelet decomposition. Here the concept of generalized smoothness is used, within which the mentioned singularity is generalized smooth. This approach leads to the construction of a system of embedded spaces on embedded grids. A spline-wavelet decomposition of mentioned spaces is presented. Reconstruction formulas are done

Vascularized Multi-Functional Materials and Structures

Here we draw attention to the development of smart materials with embedded vasculatures that provide multiple functionality: volumetric cooling, self-healing, mechanical strength, etc. Vascularization is achieved by using tree-shaped (dendritic) and grid-shaped flow architectures. As length scales become smaller, dendritic vascularization provides dramatically superior volumetric bathing and transport properties than the use of bundles of parallel microchannels. Embedded grids of channels provide substantially better volumetric bathing when the channels have multiple diameters that are selected optimally and put in the right places. Two novel dendritic architectures are proposed: trees matched canopy to canopy, and trees that alternate with upside down trees. Both have optimized length scales and layouts. Flow architectures are derived from principle, in accordance with constructal theory, not by mimicking nature.