We define a subdivision network ¿S of a given network ¿; by inserting a new vertex in every edge, so that each edge is replaced by two new edges with conductances that fulfill electrical conditions on the new network. In this work, we firstly obtain an expression for the Green kernel of the subdivision network in terms of the Green kernel of the base network. Moreover, we also obtain the effective resistance and the Kirchhoff index of the subdivision network in terms of the corresponding parameters on the base network. Finally, as an example, we carry out the computations in the case of a wheel.
We present a geometric approach to the classification of monogenic invariant subspaces, alternative to the classical algebraic one, which allows us to obtain several matricial canonical forms for each class. Some applications are derived: canonical coordinates of a vector with regard to an endomorphism, and a canonical form for uniparametric linear control systems, not necessarily controllable, with regard to linear changes of state variables. Moreover, the pointwise construction
can be extended to differentiable families of changes of basis when differentiable families of equivalent monogenic subspaces are considered.
Given a general local differentiable family of pairs of matrices, we obtain
a local differentiable family of feedbacks solving the pole assignment
problem, that is to say, shifting the spectrum into a prefixed one. We point
out that no additional hypothesis is needed. In fact, simple approaches
work in particular cases (controllable pairs, constancy of the dimension of
the controllable subspace, and so on). Here the general case is proved by
means of Arnold’s techniques: the key point is to reduce the construction to
a versal deformation of the central pair; in fact to a quite singular
miniversal one for which the family of feedbacks can be explicitly
constructed. As a direct application, a differentiable family of stabilizing
feedbacks is obtained, provided that the central pair is stabilizable.