Arrays, mathematical concept

The definition of crystal as a periodic array of identical motifs corresponds to the mathematical concept of lattice. Under this assumption, crystallography has developed powerful methods for allowing crystal structure determination of different materials, from simple metals (where the asymmetric imit may be the single atom) to protein crystals (containing thousands of atoms per cell) [62]. 

To describe all the properties of the resulting array in a precise framework, several mathematical concepts must be introduced. They allow for selection of the parallelepipeds that are suitable for a good partitioning. The complete description of the techniques for projection and compression is available elsewhere [11, 12. 13]. 

Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrbdinger inteijected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrodinger s wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. 

As early as 1829, the observation of grain boundaries was reported. But it was more than one hundred years later that the structure of dislocations in crystals was understood. Early ideas on strain-figures that move in elastic bodies date back to the turn of this century. Although the mathematical theory of dislocations in an elastic continuum was summarized by [V. Volterra (1907)], it did not really influence the theory of crystal plasticity. X-ray intensity measurements [C.G. Darwin (1914)] with single crystals indicated their mosaic structure (j.e., subgrain boundaries) formed by dislocation arrays. Prandtl, Masing, and Polanyi, and in particular [U. Dehlinger (1929)] came close to the modern concept of line imperfections, which can move in a crystal lattice and induce plastic deformation. 

Decompositions of three-way arrays into these two different models require different data analysis methods therefore, finding out if the internal structure of a three-way data set is trilinear or nontrilinear is essential to ensure the selection of a suitable chemometric method. In the previous paragraphs, the concept of trilinearity was tackled as an exclusively mathematical problem. However, the chemical information is often enough to determine whether a three-way data set presents this feature. How to link chemical knowledge with the mathematical structure of a three-way data set can be easily illustrated with a real example. 

The definition (7.32) masks the local and non-local contributions from bodies to the flow. A more systematic approach to characterising the Eulerian mean velocity is to decompose the flow into (i) a far field flow contribution - far from each body but still within the group of bodies - and (ii) a near field flow contribution - local to each body. This concept, originally described qualitatively by Kowe et al. [353], is strictly valid for dilute arrays since it formally requires the bodies to be widely separated, so that there is a separation of lengthscales between the near and far field, scaling approximately as 0(a) and 0(LS) respectively. The decomposition is defined formally here for potential flows. The far field flow, u, is defined mathematically as the sum of the dipolar and source contributions from the bodies, by assuming the bodies shrink to zero, so that (from (7.31))... 

This chapter introduces concepts of two-way and multi-way algebra. This includes definition of arrays and their subparts, a number of useful matrix products and the different concepts of two-way and three-way rank. Also linearity, bilinearity and trilinearity are defined. It is important to remember about rank that the problem-defined rank (chemical rank, pseudorank) has to be chosen by the data analyst using certain criteria where the mathematically defined ranks are extremes that are not useful for modeling of noisy data. 

Section 2.2 introduced a large array of monstrous mathematical beings that exhibit pathological properties defying the traditional concept of dimension. In Section 2.3, we saw that various alternatives to the traditional topological dimension have been devised by mathematicians. In spite of their multiplicity, these dimensions have tended to make the mathematical monsters somewhat less terrifying. 

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