A stability criterion for high-frequency oscillations

We show that a simple Levi compatibility condition determines stability of WKB solutions to semilinear hyperbolic initial-value problems issued from highly-oscillating initial data with large amplitudes. The compatibility condition involves the hyperbolic operator, the fundamental phase associated with the initial oscillation, and the semilinear source term; it states roughly that hyperbolicity is preserved around resonances. If the compatibility condition is satisfied, the solutions are defined over time intervals independent of the wavelength, and the associated WKB solutions are stable under a large class of initial perturbations. If the compatibility condition is not satisfied, resonances are exponentially amplified, and arbitrarily small initial perturbations can destabilize the WKB solutions in small time. The amplification mechanism is based on the observation that in frequency space, resonances correspond to points of weak hyperbolicity. At such points, the behavior of the system depends on the lower order terms through the compatibility condition. The analysis relies, in the unstable case, on a short-time Duhamel representation formula for solutions of zeroth-order pseudo-differential equations. Our examples include coupled Klein-Gordon systems, and systems describing Raman and Brillouin instabilities.Comment: Final version, to appear in M\'em. Soc. Math. F

Numerical studies of entangled PPT states in composite quantum systems

We report here on the results of numerical searches for PPT states with specified ranks for density matrices and their partial transpose. The study includes several bipartite quantum systems of low dimensions. For a series of ranks extremal PPT states are found. The results are listed in tables and charted in diagrams. Comparison of the results for systems of different dimensions reveal several regularities. We discuss lower and upper bounds on the ranks of extremal PPT states.Comment: 18 pages, 4 figure

A Bramble-Pasciak conjugate gradient method for discrete Stokes equations with random viscosity

We study the iterative solution of linear systems of equations arising from stochastic Galerkin finite element discretizations of saddle point problems. We focus on the Stokes model with random data parametrized by uniformly distributed random variables and discuss well-posedness of the variational formulations. We introduce a Bramble-Pasciak conjugate gradient method as a linear solver. It builds on a non-standard inner product associated with a block triangular preconditioner. The block triangular structure enables more sophisticated preconditioners than the block diagonal structure usually applied in MINRES methods. We show how the existence requirements of a conjugate gradient method can be met in our setting. We analyze the performance of the solvers depending on relevant physical and numerical parameters by means of eigenvalue estimates. For this purpose, we derive bounds for the eigenvalues of the relevant preconditioned sub-matrices. We illustrate our findings using the flow in a driven cavity as a numerical test case, where the viscosity is given by a truncated Karhunen-Lo\`eve expansion of a random field. In this example, a Bramble-Pasciak conjugate gradient method with block triangular preconditioner outperforms a MINRES method with block diagonal preconditioner in terms of iteration numbers.Comment: 19 pages, 1 figure, submitted to SIAM JU

Low rank positive partial transpose states and their relation to product vectors

It is known that entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. In a previous paper we presented a classification of rank four entangled PPT states which we believe to be complete. In the present paper we continue our investigations of the low rank entangled PPT states. We use perturbation theory in order to construct rank five entangled PPT states close to the known rank four states, and in order to compute dimensions and study the geometry of surfaces of low rank PPT states. We exploit the close connection between low rank PPT states and product vectors. In particular, we show how to reconstruct a PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.Comment: 29 pages, 4 figure