Sentence examples for dimensional case is from inspiring English sources

Finally, some numerics that confirm the predicted behavior of the method are presented and a discussion on the two dimensional case is given, together with few 2D numerical experiments.

Hence, this work proposes a framework for measuring the circularity which offers several advantages: it is not affected by the overlapping, incompleteness of borders (in this work only for two dimensional case is dealt with), invariance of the resolution, and accuracy of the border detection.

The mobility model for the discrete two dimensional case is given by.

A sufficient and necessary condition for existence of solution for the boundary blow-up problem in one dimensional case is obtained.

To avoid this problem, we consider here a simple and efficient algorithm based in the N R iteration which, in the one dimensional case, is equivalent to the steepest ascend/descend method with an adaptive step size defined by ∈ k = 1 | g ′′ ( θ ( k ) ) |, i.e.: θ ( k + 1 ) = θ ( k ) ± g ′ ( θ ( k ) ) | g ″ ( θ ( k ) ) |, (40).

The number of coupled ordinary differential equations was determined by the number of grid points employed for the system, which for the one dimensional case is 32 unless otherwise noted.

Numerical experiments displaying the accuracy of the numerical solution for the two dimensional case are provided.

Numerical examples, up to the full three dimensional case, are presented which validate the method and assess its efficiency in 1D, 2D and 3D.

A breakthrough in this one dimensional case was made by Atkinson Everitt [19] who proved there is no eigenvalue if (E ge 4K^2/pi ^2) and that there are examples with eigenvalues arbitrarily close to this bound.

One of the big differences between one and higher dimensional cases is that for (nge 2) the Green function is usually not symmetric.

An improvement in the finite-dimensional case is given.