Comparison theorems for manifolds with mean convex boundary

Let M-n be an n-dimensional Riemannian manifold with boundary partial derivative M. Assuming that Ricci curvature is bounded from below by (n - 1)k, for k is an element of R, we give a sharp estimate of the upper bound of rho(x) = d(x, partial derivative M), in terms of the mean curvature bound of the boundary. When partial derivative M is compact, the upper bound is achieved if and only if M is isometric to a disk in space form. A Kahler version of estimation is also proved. Moreover, we prove a Laplacian comparison theorem for distance function to the boundary of Kahler manifold and also estimate the first eigenvalue of the real Laplacian.SCI(E)[email protected]

A simple finite-difference modification for improving accuracy near a corner in heat flow problems

Traditional numerical methods of solution of heat flow problems are subject to inaccuracies near sharp corners, where the derivatives of the exact solution may "become unbounded (Jeffreys1). 2Previous methods of overcoming this difficulty (Motz, Woods3

Effects of turbulent dust grain motion to interstellar chemistry

Theoretical studies have revealed that dust grains are usually moving fast through the turbulent interstellar gas, which could have significant effects upon interstellar chemistry by modifying grain accretion. This effect is investigated in this work on the basis of numerical gas-grain chemical modeling. Major features of the grain motion effect in the typical environment of dark clouds (DC) can be summarised as follows: 1) decrease of gas-phase (both neutral and ionic) abundances and increase of surface abundances by up to 2-3 orders of magnitude; 2) shifts of the existing chemical jumps to earlier evolution ages for gas-phase species and to later ages for surface species by factors of about ten; 3) a few exceptional cases in which some species turn out to be insensitive to this effect and some other species can show opposite behaviors too. These effects usually begin to emerge from a typical DC model age of about 10^5 yr. The grain motion in a typical cold neutral medium (CNM) can help overcome the Coulomb repulsive barrier to enable effective accretion of cations onto positively charged grains. As a result, the grain motion greatly enhances the abundances of some gas-phase and surface species by factors up to 2-6 or more orders of magnitude in the CNM model. The grain motion effect in a typical molecular cloud (MC) is intermediate between that of the DC and CNM models, but with weaker strength. The grain motion is found to be important to consider in chemical simulations of typical interstellar medium.Comment: 20 pages, 10 figures and 2 table

Rapid adiabatic preparation of injective PEPS and Gibbs states

We propose a quantum algorithm for many-body state preparation. It is especially suited for injective PEPS and thermal states of local commuting Hamiltonians on a lattice. We show that for a uniform gap and sufficiently smooth paths, an adiabatic runtime and circuit depth of $O(\operatorname{polylog}N)$ can be achieved for $O(N)$ spins. This is an almost exponential improvement over previous bounds. The total number of elementary gates scales as $O(N\operatorname{polylog}N)$. This is also faster than the best known upper bound of $O(N^2)$ on the mixing times of Monte Carlo Markov chain algorithms for sampling classical systems in thermal equilibrium.Comment: 5 (+12) pages, 2 figure