Sentence examples for changes of basis from inspiring English sources

We can summarize the changes of basis by introducing the square matrices M and L, defined as M m n = ∫ 0 1 T m (y ) C n (y ) d y y (1 − y ), and L m n = ∫ 0 1 C m (y ) T n (y ) y (1 − y )   d y, being both related as M T = L−1 (i.e., the transpose matrix of M equals the inverse matrix of L), as is expected.

This relation between and or and is actually established by using the change-of-basis matrix W1/2 defined by Eq. (10).

The change-of-basis matrix W1/2 is selected so that the scalar product in (10) with l denoting the SH degree.

In many applications, a change of basis will allow different lattices to be utilized, but this is not always the case, because a change of basis is not always desirable or even possible.

The challange of dealing with the "moving" horizon in this new basis, however, might make implementing such a change of basis impractical.

Therefore, if we perform a change of basis before we represent the BRDF we should see slow variation along one axis, potentially leading to better compression.

Some particular examples are considered, in order to demonstrate the capability of the proposed algorithm, as well as its precision with a change of basis functions.

By exploiting some properties of the incomplete gamma function, it transpires that the change of basis can be achieved without dealing with any infinite expansions.

Because features in common BRDFs, including specular and retroreflective peaks, are aligned with the transformed coordinate axes, the change of basis reduces storage requirements for a large class of BRDFs.

Effects of (i) change of basis set from minimal to double zeta, (ii) change in secondary structure from β-pleated to α-helical, (iii) presence of solvation shell and (iv) binding of H+ and Li+ ions to peptide group on the resulting solution as well as on electronic structure and conduction properties of polypeptides are investigated.

The idea is to avoid any change of basis in the process of polynomial differentiation.