This paper proposes the Gramian-preserving frequency transformation for linear discrete-time state-space systems. In this frequency transformation, we replace each delay element of a discrete-time system with an allpass system that has a balanced realization. This approach can generate transformed systems that have the same controllability/observability Gramians as those of the original system....

We show that a dissipative, ergodic measure preserving transformation of a σ-finite, non-atomic measure space always has many non-proportional, absolutely continuous, invariant measures and is ergodic with respect to each one of these. §0 Introduction Let (X,B, m, T ) be an invertible, ergodic measure preserving transformation of a σ-finite measure space, then there are no other σ-finite, m-abs...

The failure of heterogeneous materials with microstructures is a complex process damage nucleation, growth and localisation. This spans multiple length scales challenging to simulate numerically due its high computational cost. One option use domain decomposed multi-scale methods dynamical refinement. If needed, these refine coarse regions into fine-scale representation explicitly model the in ...