Kaimanovich (2003) [9] introduced the concept of augmented tree on the symbolic space of a selfsimilar set. It is hyperbolic in the sense of Gromov, and it was shown by Lau and Wang (2009) [12] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in detail a class of simple augmented trees and the Lipschitz equivalence of such trees. The main purpose is to use this to study the Lipschitz equivalence problem of the totally disconnected self-similar sets which has been undergoing some extensive development recently.

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We consider the solution of large-scale Lyapunov and Stein equations. For Stein equations, the well-known Smith method will be adapted, with A_k = A^{2^k} not explicitly computed but in the recursive form A_k = A^{2^k}, and the fast growing but diminishing components in the approximate solutions truncated. Lyapunov equations will be first treated with the Cayley transform before the Smith method is applied. For algebraic equations with numerically low-ranked solutions of dimension <i>n</i>, the resulting algorithms are of an efficient <i>O</i>(<i>n</i>) computational complexity and memory requirement per iteration and converge essentially quadratically. An application in the estimation of a lower bound of the condition number for continuous-time algebraic Riccati equations is presented, as well as some numerical results.

We investigate variational principles for the approximation of quantum dynamics that apply for approximation manifolds that do not have complex linear tangent spaces. The first one, dating back to McLachlan (1964) minimizes the residuum of the time-dependent Schrödinger equation, while the second one, originating from the lecture notes of Kramer--Saraceno (1981), imposes the stationarity of an action functional. We characterize both principles in terms of metric and a symplectic orthogonality conditions, consider their conservation properties, and derive an elementary a-posteriori error estimate. As an application, we revisit the time-dependent Hartree approximation and frozen Gaussian wave packets.

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