The matrices in question are (Gamma in mathbb {R}^{Ntimes z}) and (Psi in mathbb {R}^{Ntimes N}), where N is the horizon length and z is the state dimension.
The interest of all these techniques is that they preserve certain properties of the matrices in question, such as determinant, rank or inverse, so that these quantities can be calculated after applying the transformation, or that certain matrix operations are algorithmically easier to carry out for some types of matrices.
Convergence properties of these methods are established when the matrix in question is either M-matrix or symmetric matrix.
It is necessary to evaluate the elements of the matrix in question when using inverse methods for the reconstruction of acoustic source strength spectra.
Despite the limitations imposed by the matrix in question, the accuracy results and total error calculations generated during method validation revealed the consistency of quantitation across the entire quantitative range and compare well to the biomarker field (26).
Perry and Lu (1992) suggested that the organic matrix in question might be made from polymers of carbohydrates, for example, cellulose [ 14], and this suggestion was reinforced recently by Fry and colleagues who speculated that the hemicellulose, callose, in horsetail cell walls might be a potential site of silica deposition [ 15].
We derive the dipole matrices of the defects in question and use the corresponding dipole fields to evaluate "effective" tractions along the crack faces and interface to describe the interaction between the main interfacial crack and the defects.
The interaction between the main crack and the defects (e.g. small cracks or inclusions) is described asymptotically by analysing the dipole fields and the corresponding dipole matrices of the defects in question.
As can be readily seen, the difference in the average RMSE between the estimated and true Q-matrix for each approach in question under each of the aforementioned noise levels is marginal (Table 3), suggesting that all methods are equally robust under the considered simulation scenarios with the observed differences being marginal.
(In Sec. 2.1, we used x to denote a given spatial location within the domain of the PDE models. Here x b i denotes the model state at a particular location. In the case of the two-phenotype model, Eqs. (3 - 5), x b i is the 3-vector (g, m, w) giving the density of proliferating and migrating cells and the extracellular matrix at the grid point in question).
Combined with their location embedded within the bone matrix, it is likely that the cells in question are indeed osteocytes.
