Function spaces with dominating mixed smoothness

We study several techniques whichare well known in the case of Besov and TriebelLizorkin spaces and extend them to spaces with dominating mixed smoothness. We use the ideas of Triebel to prove three important decomposition theorems. We deal withsocalled atomic, subatomic and wavelet decompositions. All these theorems have much in common. fRoughly speaking, they say that a function belongs to some function space if, and only if, it can be decomposed into the sum of products of coefficients and corresponding building blocks, where the coefficients belong to an appropriate sequence space. These decomposition theorems estabilisha veryusefulconnection between function and sequence spaces. We use them in the study of the decay of entropy numbers of compact embeddings between two function spaces of dominating mixed smoothness reducingthis problem to the same question on the sequence space level. The considered scales cover many important specific spaces (Sobolev, Zygmund, Besov) and we get generalisations of respective assertions of Belinsky, Dinh Dung and Temlyakov

Big Data of Materials Science - Critical Role of the Descriptor

Statistical learning of materials properties or functions so far starts with a largely silent, non-challenged step: the choice of the set of descriptive parameters (termed descriptor). However, when the scientific connection between the descriptor and the actuating mechanisms is unclear, causality of the learned descriptor-property relation is uncertain. Thus, trustful prediction of new promising materials, identification of anomalies, and scientific advancement are doubtful. We analyse this issue and define requirements for a suited descriptor. For a classical example, the energy difference of zincblende/wurtzite and rocksalt semiconductors, we demonstrate how a meaningful descriptor can be found systematically.Comment: Accepted to Phys. Rev. Let