The hypocoercivity index for the short time behavior of linear time-invariant ODE systems

We consider the class of conservative-dissipative ODE systems, which is a subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize asymptotically stable, conservative-dissipative ODE systems via the hypocoercivity (theory) of their system matrices. Our main result is a concise characterization of the hypocoercivity index (an algebraic structural property of matrices with positive semi-definite Hermitian part introduced in Achleitner, Arnold, and Carlen (2018)) in terms of the short time behavior of the propagator norm for the associated conservative-dissipative ODE system

Wave impedance matrices for cylindrically anisotropic radially inhomogeneous elastic solids

Impedance matrices are obtained for radially inhomogeneous structures using the Stroh-like system of six first order differential equations for the time harmonic displacement-traction 6-vector. Particular attention is paid to the newly identified solid-cylinder impedance matrix ${\mathbf Z} (r)$ appropriate to cylinders with material at $r=0$, and its limiting value at that point, the solid-cylinder impedance matrix ${\mathbf Z}_0$. We show that ${\mathbf Z}_0$ is a fundamental material property depending only on the elastic moduli and the azimuthal order $n$, that ${\mathbf Z} (r)$ is Hermitian and ${\mathbf Z}_0$ is negative semi-definite. Explicit solutions for ${\mathbf Z}_0$ are presented for monoclinic and higher material symmetry, and the special cases of $n=0$ and 1 are treated in detail. Two methods are proposed for finding ${\mathbf Z} (r)$, one based on the Frobenius series solution and the other using a differential Riccati equation with ${\mathbf Z}_0$ as initial value. %in a consistent manner as the solution of an algebraic Riccati equation. The radiation impedance matrix is defined and shown to be non-Hermitian. These impedance matrices enable concise and efficient formulations of dispersion equations for wave guides, and solutions of scattering and related wave problems in cylinders.Comment: 39 pages, 2 figure

Optimal Unambiguous State Discrimination of two density matrices and its link with the Fidelity

Recently the problem of Unambiguous State Discrimination (USD) of mixed quantum states has attracted much attention. So far, bounds on the optimum success probability have been derived [1]. For two mixed states they are given in terms of the fidelity. Here we give tighter bounds as well as necessary and sufficient conditions for two mixed states to reach these bounds. Moreover we construct the corresponding optimal measurement strategies. With this result, we provide analytical solutions for unambiguous discrimination of a class of generic mixed states. This goes beyond known results which are all reducible to some pure state case. Additionally, we show that examples exist where the bounds cannot be reached.Comment: 10 page