Poincar\'e and log-Sobolev inequalities for mixtures

This work studies mixtures of probability measures on $\mathbb{R}^n$ and gives bounds on the Poincar\'e and the log-Sobolev constant of two-component mixtures provided that each component satisfies the functional inequality, and both components are close in the $\chi^2$-distance. The estimation of those constants for a mixture can be far more subtle than it is for its parts. Even mixing Gaussian measures may produce a measure with a Hamiltonian potential possessing multiple wells leading to metastability and large constants in Sobolev type inequalities. In particular, the Poincar\'e constant stays bounded in the mixture parameter whereas the log-Sobolev may blow up as the mixture ratio goes to $0$ or $1$. This observation generalizes the one by Chafa\"i and Malrieu to the multidimensional case. The behavior is shown for a class of examples to be not only a mere artifact of the method.Comment: 13 page

Euler class groups, and the homology of elementary and special linear groups

We prove homology stability for elementary and special linear groups over rings with many units improving known stability ranges. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative local rings with infinite residue fields, we show that the obstruction to further stability is given by Milnor-Witt K-theory. As an application we construct Euler classes of projective modules with values in the cohomology of the Milnor Witt K-theory sheaf. For d-dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the Euler class is necessary and sufficient for a projective module P of rank d to split off a rank 1 free direct summand. Along the way we obtain a new presentation of Milnor-Witt K-theory.Comment: 64 pages. Revised Section 5. Comments welcome

Gluing Riemannian manifolds with curvature operators at least k

We glue two manifolds which have curvature operators at least k (in the sense of eigenvalues) along their common boundary. We show that if the sum of the second fundamental forms of the boundary is positive semidefinite, then the curvature operator of the resulting manifold is at least k up to an arbitrarily small error term. Similar results hold for Ricci, scalar, bi, isotropic and flag curvature, respectively

Macroscopic limit of the Becker-D\"oring equation via gradient flows

This work considers gradient structures for the Becker-D\"oring equation and its macroscopic limits. The result of Niethammer [17] is extended to prove the convergence not only for solutions of the Becker-D\"oring equation towards the Lifshitz-Slyozov-Wagner equation of coarsening, but also the convergence of the associated gradient structures. We establish the gradient structure of the nonlocal coarsening equation rigorously and show continuous dependence on the initial data within this framework. Further, on the considered time scale the small cluster distribution of the Becker--D\"oring equation follows a quasistationary distribution dictated by the monomer concentration

Direct numerical simulation of open-channel flow over a fully-rough wall at moderate relative submergence

Direct numerical simulation of open-channel flow over a bed of spheres arranged in a regular pattern has been carried out at bulk Reynolds number and roughness Reynolds number (based on sphere diameter) of approximately 6900 and 120, respectively, for which the flow regime is fully-rough. The open-channel height was approximately 5.5 times the diameter of the spheres. Extending the results obtained by Chan-Braun et al. (J. Fluid Mech., vol. 684, 2011, 441) for an open-channel flow in the transitionally-rough regime, the present purpose is to show how the flow structure changes as the fully-rough regime is attained and, for the first time, to enable a direct comparison with experimental observations. The results indicate that, in the vicinity of the roughness elements, the average flow field is affected both by Reynolds number effects and by the geometrical features of the roughness, while at larger wall-distances this is not the case, and roughness concepts can be applied. The flow-roughness interaction occurs mostly in the region above the virtual origin of the velocity profile, and the effect of form-induced velocity fluctuations is maximum at the level of sphere crests. The spanwise length scale of turbulent velocity fluctuations in the vicinity of the sphere crests shows the same dependence on the distance from the wall as that observed over a smooth wall, and both vary with Reynolds number in a similar fashion. Moreover, the hydrodynamic force and torque experienced by the roughness elements are investigated. Finally, the possibility either to adopt an analogy between the hydrodynamic forces associated with the interaction of turbulent structures with a flat smooth wall or with the surface of the spheres is also discussed, distinguishing the skin-friction from the form-drag contributions both in the transitionally-rough and in the fully-rough regimes.Comment: 46 pages, 26 figure

Charge conservation in RHIC and contributiuons to local parity violation observables

Relativistic heavy ion collisions provide laboratory environments from which one can study the creation of a novel state of matter, the quark gluon plasma. The existence of such a state is postulated to alter the mechanism and evolution of charge production, which then becomes manifest in charge correlations. We study the separation of balancing charges at kinetic freeze-out by analyzing recent result on balancing charge correlations for Au+Au collisions at $\sqrt{s_{NN}}=200 {GeV}$. We find that balancing charges are emitted from significantly smaller regions in central collisions compared to peripheral collisions. The results indicate that charge diffusion is small and that the centrality dependence points to a change of the production mechanism. In addition we calculate the contributions from charge-balance correlations to STAR's local parity violation observable. We find that local charge conservation, when combined with elliptic flow, explains much of STAR's measurement.Comment: 11 pages, 10 figure