Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle

It is shown that the problem of designing a two-reflector system transforming a plane wave front with given intensity into an output plane front with prescribed output intensity can be formulated and solved as the Monge-Kantorovich mass transfer problem.Comment: 25 pages, 2 figure

Modification of quantum measure in area tensor Regge calculus and positivity

A comparative analysis of the versions of quantum measure in the area tensor Regge calculus is performed on the simplest configurations of the system. The quantum measure is constructed in such the way that it reduces to the Feynman path integral describing canonical quantisation if the continuous limit along any of the coordinates is taken. As we have found earlier, it is possible to implement also the correspondence principle (proportionality of the Lorentzian (Euclidean) measure to $e^{iS}$ ($e^{-S}$), $S$ being the action). For that a certain kind of the connection representation of the Regge action should be used, namely, as a sum of independent contributions of selfdual and antiselfdual sectors (that is, effectively 3-dimensional ones). There are two such representations, the (anti)selfdual connections being SU(2) or SO(3) rotation matrices according to the two ways of decomposing full SO(4) group, as SU(2) $\times$ SU(2) or SO(3) $\times$ SO(3). The measure from SU(2) rotations although positive on physical surface violates positivity outside this surface in the general configuration space of arbitrary independent area tensors. The measure based on SO(3) rotations is expected to be positive in this general configuration space on condition that the scale of area tensors considered as parameters is bounded from above by the value of the order of Plank unit.Comment: 10 pages, plain LaTe

Kolmogorov-Type Theory of Compressible Turbulence and Inviscid Limit of the Navier-Stokes Equations in R3\mathbb{R}^3

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for compressible fluids in $\mathbb{R}^3$. Motivated by the Kolmogorov hypothesis (1941) for incompressible flow, we introduce a Kolmogorov-type hypothesis for barotropic flows, in which the density and the sonic speed normally vary significantly. We then observe that the compressible Kolmogorov-type hypothesis implies the uniform boundedness of some fractional derivatives of the weighted velocity and sonic speed in the space variables in $L^2$, which is independent of the viscosity coefficient $\mu>0$. It is shown that this key observation yields the equicontinuity in both space and time of the density in $L^\gamma$ and the momentum in $L^2$, as well as the uniform bound of the density in $L^{q_1}$ and the velocity in $L^{q_2}$ independent of $\mu>0$, for some fixed $q_1 >\gamma$ and $q_2 >2$, where $\gamma>1$ is the adiabatic exponent. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations for barotropic fluids in $\mathbb{R}^3$. Not only do we offer a framework for mathematical existence theories, but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place, with the bounds that are independent of $\mu>0$, that is in the high Reynolds number limit.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1008.154