Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions $d \geq 2$. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green's function in weighted spaces.Comment: added applications, e.g. two-scale expansion, variance estimate of RV

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted Oscillating Brownian motion.For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations

Two-Dimensional Mathematical Modelling of a Dam-Break Wave in a Narrow Steep Stream

The paper deals with hydraulic aspects of a wave, emerging as a result of a potential dam break of the upper storage reservoir of the pumpedstorage hydropower plant Kolarjev vrh. A two-dimensional depth-averaged mathematical approach was used. The upper storage reservoir and its dam failure were modelled with the mathematical model PCFLOW2D, which is based on the Cartesian coordinate numerical mesh.\ud The results of PCFLOW2D were used as the upper boundary condition for the mathematical model PCFLOW2D-ORTHOCURVE, based on the orthogonal curvilinear numerical mesh. The model PCFLOW2D-ORTHOCURVE provided a tool for the analysis of flood wave flow in a steep, narrow and geometrically diversified stream channel. The classic Manning’s equation fails to give good results for streams with steep bed\ud slopes and therefore, a different equation should be used. The application of the Rickenmann’s equation was chosen, presented in a form similar to Manning’s equation. For the purpose of the example given here, the equation was somewhat simplified and adapted to the data available. The roughness coefficient used at each calculation cell depended on the slope of that cell. The results of numerical calculations\ud were compared to measurements carried out on a physical model in the scale of 1 : 200. Regarding the complexity of the flow phenomenon a rather good correlation of maximum depth was established: only at one gauge the difference in water depth was up to 27% while at the other four it was 7% of water depth on average

Finite-temperature scalar fields and the cosmological constant in an Einstein universe

We study the back reaction effect of massless minimally coupled scalar field at finite temperatures in the background of Einstein universe. Substituting for the vacuum expectation value of the components of the energy-momentum tensor on the RHS of the Einstein equation, we deduce a relationship between the radius of the universe and its temperature. This relationship exhibit a maximum temperature, below the Planck scale, at which the system changes its behaviour drastically. The results are compared with the case of a conformally coupled field. An investigation into the values of the cosmological constant exhibit a remarkable difference between the conformally coupled case and the minimally coupled one.Comment: 7 pages, 2 figure

Shock wave focusing using geometrical shock dynamics

A finite-difference numerical method for geometrical shock dynamics has been developed based on the analogy between the nonlinear ray equations and the supersonic potential equation. The method has proven to be an efficient and inexpensive tool for approximately analyzing the focusing of weak shock waves, where complex nonlinear wave interactions occur over a large range of physical scales. The numerical results exhibit the qualitative behavior of strong, moderate, and weak shock focusing observed experimentally. The physical mechanisms that are influenced by aperture angle and shock strength are properly represented by geometrical shock dynamics. Comparison with experimental measurements of the location at which maximum shock pressure occurs shows good agreement, but the maximum pressure at focus is overestimated by about 60%. This error, though large, is acceptable when the speed and low cost of the method is taken into consideration. The error is primarily due to the under prediction of disturbance speed on weak shock fronts. Adequate resolution of the focal region proves to be particularly important to properly judge the validity of shock dynamics theory, under-resolution leading to overly optimistic conclusions