Periodic homogenization of a pseudo-parabolic equation via a spatial-temporal decomposition

Pseudo-parabolic equations have been used to model unsaturated fluid flow in porous media. In this paper it is shown how a pseudo-parabolic equation can be upscaled when using a spatio-temporal decomposition employed in the Peszyn'ska-Showalter-Yi paper [8]. The spatial-temporal decomposition transforms the pseudo-parabolic equation into a system containing an elliptic partial differential equation and a temporal ordinary differential equation. To strengthen our argument, the pseudo-parabolic equation has been given advection/convection/drift terms. The upscaling is done with the technique of periodic homogenization via two-scale convergence. The well-posedness of the extended pseudo-parabolic equation is shown as well. Moreover, we argue that under certain conditions, a non-local-in-time term arises from the elimination of an unknown.Comment: 6 pages, 0 figure

Two-scale convergence for locally-periodic microstructures and homogenization of plywood structures

The introduced notion of locally-periodic two-scale convergence allows to average a wider range of microstructures, compared to the periodic one. The compactness theorem for the locally-periodic two-scale convergence and the characterisation of the limit for a sequence bounded in $H^1(\Omega)$ are proven. The underlying analysis comprises the approximation of functions, which periodicity with respect to the fast variable depends on the slow variable, by locally-periodic functions, periodic in subdomains smaller than the considered domain, but larger than the size of microscopic structures. The developed theory is applied to derive macroscopic equations for a linear elasticity problem defined in domains with plywood structures.Comment: 22 pages, 4 figure

Crossed ladders and Euler’s quartic

We investigate a particular form of the classical “crossed ladders” problem, finding many parametrized solutions, some polynomial, and some involving Fibonacci and Lucas sequences. We establish a connection between this particular form and a quartic equation studied by Euler, giving corresponding solutions to the latter

On weak convergence of locally periodic functions

We prove a generalization of the fact that periodic functions converge weakly to the mean value as the oscillation increases. Some convergence questions connected to locally periodic nonlinear boundary value problems are also considered.Comment: arxiv version is already officia

Some sharp inequalities for integral operators with homogeneous kernel

One goal of this paper is to show that a big number of inequalities for functions in L-p(R+), p >= 1, proved from time to time in journal publications are particular cases of some known general results for integral operators with homogeneous kernels including, in particular, the statements on sharp constants. Some new results are also included, e.g. the similar general equivalence result is proved and applied for 0 < p < 1. Some useful new variants of these results are pointed out and a number of known and new Hardy-Hilbert type inequalities are derived. Moreover, a new Polya-Knopp (geometric mean) inequality is derived and applied. The constants in all inequalities in this paper are sharp

Antibiotic cycling versus mixing: the difficulty of using mathematical models to definitively quantify their relative merits.

Published PDF version deposited in accordance with SHERPA RoMEO guidelines.We ask the question Which antibiotic deployment protocols select best against drug-resistant microbes: mixing or periodic cycling? and demonstrate that the statistical distribution of the performances of both sets of protocols, mixing and periodic cycling, must have overlapping supports. In other words, it is a general, mathematical result that there must be mixing policies that outperform cycling policies and vice versa. As a result, we agree with the tenet of Bonhoefer et al. [1] that one should not apply the results of [2] to conclude that an antibiotic cycling policy that implements cycles of drug restriction and prioritisation on an ad-hoc basis can select against drug-resistant microbial pathogens in a clinical setting any better than random drug use. However, nor should we conclude that a random, per-patient drug-assignment protocol is the de facto optimal method for allocating antibiotics to patients in any general sense