Motivated by diffusion processes on metric graphs and ramified spaces, we consider an abstract setting for interface problems with coupled dynamic boundary conditions belonging to a quite general class. Beside well-posedness, we discuss positivity, L∞-contractivity and further invariance properties. We show that the parabolic problem with dynamic boundary conditions enjoy these properties if an...

The lower bounds for the a posteriori prediction error of a nonlinear predictor realized as a neural network are provided. These are obtained for a priori adaptation and a posteriori error networks with sigmoid nonlinearities trained by gradient-descent learning algorithms. A contractivity condition is imposed on a nonlinear activation function of a neuron so that the a posteriori prediction er...

In a recent paper, we investigated factorization properties of Hermite subdivision schemes by means of the so–called Taylor factorization. This decomposition is based on a spectral condition which is satisfied for example by all interpolatory Hermite schemes. Nevertheless, there exist examples of Hermite schemes, especially some based on cardinal splines, which fail the spectral condition. For ...

An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite-dimensional manifold.An abstract ...

Strongly contracting dynamical systems have numerous properties (e.g., incremental ISS), find widespread applications in controls and learning), their study is receiving increasing attention. This work starts with the simple observation that, given a strongly system, its adjoint system also contracting, same rate, respect to dual norm, under time reversal. As main implication of this contractiv...

Let N be a simply connected nilpotent Lie group and let S = N o (R+)d be a semidirect product, (R+)d acting on N by diagonal automorphisms. Let (Qn, Mn) be a sequence of i.i.d. random variables with values in S. Under natural conditions, including contractivity in the mean, there is a unique stationary measure ν on N for the Markov process Xn = MnXn−1 + Qn. We prove that for an appropriate homo...