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I V curves were generated from a group of step potentials (−100 to +100 mV from a holding potential of 0 mV).
Let G be a Carnot group of step two.
The second group of step size parameters is much smaller than the first group.
For comparison, we also choose another group of step size parameters.
Recall that the anisotropic Heisenberg groups are the Carnot group of step two whose group structure is given by (cf. [23]) (2.1).
The resulted curve of under the second group of step size parameters is plotted in Figure 6. Figure 6 Curve of under small step size parameters.
Finally, we give an existence of solution to the sub-Laplace equation on the whole group of Heisenberg type (a specific Carnot group of step two).
Let G be a Carnot group of step two, (Omega subset G) a bounded, open, convex (in the Euclidean sense) set and (varphi in C^2(bar{Omega })).
Brandolini et al. [9] applied these methods to the Dirichlet problems for sub-Laplace equations on the gauge balls in the Heisenberg group which is the simplest Carnot group of step two.
Let G be a Lie group of step two, (Omega subset G) a bounded, open, convex set (in a sense to be defined later) and (varphi in C^2(bar{Omega })).
Recall that the Heisenberg group ℍ n is the Carnot group of step two whose group structure is given by ( x, t ) ∘ ( x ′, t ′ ) = x + x ′, t + t ′ + 2 ∑ j = 1 n x 2 j x ′ 2 j - 1 - x 2 j - 1 x ′ 2 j.
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