Interior error estimate for periodic homogenization

In a previous article about the homogenization of the classical problem of diff usion in a bounded domain with su ciently smooth boundary we proved that the error is of order $\epsilon^{1/2}$. Now, for an open set with su ciently smooth boundary $C^{1,1}$ and homogeneous Dirichlet or Neuman limits conditions we show that in any open set strongly included in the error is of order $\epsilon$. If the open set $\Omega\subset R^n$ is of polygonal (n=2) or polyhedral (n=3) boundary we also give the global and interrior error estimates

Particle dynamics inside shocks in Hamilton-Jacobi equations

Characteristics of a Hamilton-Jacobi equation can be seen as action minimizing trajectories of fluid particles. For nonsmooth "viscosity" solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that for any convex Hamiltonian there exists a uniquely defined canonical global nonsmooth coalescing flow that extends particle trajectories and determines dynamics inside the shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss relation to the "dissipative anomaly" in the limit of vanishing viscosity.Comment: 15 pages, no figures; to appear in Philos. Trans. R. Soc. series

Homogenization of Maxwell's equations in periodic composites

We consider the problem of homogenizing the Maxwell equations for periodic composites. The analysis is based on Bloch-Floquet theory. We calculate explicitly the reflection coefficient for a half-space, and derive and implement a computationally-efficient continued-fraction expansion for the effective permittivity. Our results are illustrated by numerical computations for the case of two-dimensional systems. The homogenization theory of this paper is designed to predict various physically-measurable quantities rather than to simply approximate certain coefficients in a PDE.Comment: Significantly expanded compared to v1. Accepted to Phys.Rev.E. Some color figures in this preprint may be easier to read because here we utilize solid color lines, which are indistinguishable in black-and-white printin

Turbulence for the generalised Burgers equation

In this survey, we review the results on turbulence for the generalised Burgers equation on the circle: u_t+f'(u)u_x=\nu u_{xx}+\eta,\ x \in S^1=\R/\Z, obtained by A.Biryuk and the author in \cite{Bir01,BorK,BorW,BorD}. Here, f is smooth and strongly convex, whereas the constant 0<\nu << 1 corresponds to a viscosity coefficient. We will consider both the case \eta=0 and the case when \eta is a random force which is smooth in x and irregular (kick or white noise) in t. In both cases, sharp bounds for Sobolev norms of u averaged in time and in ensemble of the type C \nu^{-\delta}, \delta>=0, with the same value of \delta for upper and lower bounds, are obtained. These results yield sharp bounds for small-scale quantities characterising turbulence, confirming the physical predictions \cite{BK07}.Comment: arXiv admin note: substantial text overlap with arXiv:1201.5567, arXiv:1107.4866, arXiv:1208.524

Muon pair creation from positronium in a circularly polarized laser field

We study elementary particle reactions that result from the interaction of an atomic system with a very intense laser wave of circular polarization. As a specific example, we calculate the rate for the laser-driven reaction $e^+e^- \to \mu^+\mu^-$, where the electron and positron originate from a positronium atom or, alternatively, from a nonrelativistic $e^+e^-$ plasma. We distinguish accordingly between the coherent and incoherent channels of the process. Apart from numerical calculations, we derive by analytical means compact formulas for the corresponding reaction rates. The rate for the coherent channel in a laser field of circular polarization is shown to be damped because of the destructive interference of the partial waves that constitute the positronium ground-state wave packet. Conditions for the observation of the process via the dominant incoherent channel in a circularly polarized field are pointed out

Somatoform disorders in the family doctor's practice.

Somatoform disorders – psycho­genic diseases are characterized by pathological physical symptoms that resemble somatic illness. Thus, any organic manifestations, which can be attributed to known diseases are not detected, but there are non-specific functional impairments. Somatoform disorders include somatization disorder, undifferentiated somatoform disorder, hypocho­n­driacal disorder, somatoform dysfunction of the autonomic nervous system and stable somatoform pain disorder. The first part of the article reviewes features of the clinical manifestations of somatization disorder and undifferentiated somatoform disorder. Role of non-benzodiazepine tranquilizers (ADAPTOL) and metabolic drugs (VASONAT) in the treatment of patients with somatoform disorders is discussed. In review article data of neurologists and cardiologists on the effectiveness of anxiolytic drug ADAPTOL and metabolic drug VASONAT in different clinical groups of patients (coronary artery disease, chronic ischemia of the brain), which can significantly improve quality of life, increase exercise tolerance, improve cognitive function and correct mental and emotional disorders are presented

X-ray generation during piezoelectric lighter operation in vacuum

The results of experimental studies on the generation of x-rays when operating a piezoelectric kitchen lighter in a vacuum are presented. For the first time, a new method for increasing the intensity of x-ray radiation in the piezoelectric effect in a high vacuum through the use of an additional electron emitter is proposed and demonstrated. The maximum energy of x-ray bremsstrahlung reaches 14 keV. This means that electrons are accelerated in vacuum in the field of a piezoelectric ceramic to energy of at least 14 ke

Strain Hardening of Polymer Glasses: Entanglements, Energetics, and Plasticity

Simulations are used to examine the microscopic origins of strain hardening in polymer glasses. While stress-strain curves for a wide range of temperature can be fit to the functional form predicted by entropic network models, many other results are fundamentally inconsistent with the physical picture underlying these models. Stresses are too large to be entropic and have the wrong trend with temperature. The most dramatic hardening at large strains reflects increases in energy as chains are pulled taut between entanglements rather than a change in entropy. A weak entropic stress is only observed in shape recovery of deformed samples when heated above the glass transition. While short chains do not form an entangled network, they exhibit partial shape recovery, orientation, and strain hardening. Stresses for all chain lengths collapse when plotted against a microscopic measure of chain stretching rather than the macroscopic stretch. The thermal contribution to the stress is directly proportional to the rate of plasticity as measured by breaking and reforming of interchain bonds. These observations suggest that the correct microscopic theory of strain hardening should be based on glassy state physics rather than rubber elasticity.Comment: 15 pages, 12 figures: significant revision

SINGULAR PERTURBATIONS AND BOUNDARY LAYER THEORY FOR CONVECTION-DIFFUSION EQUATIONS IN A CIRCLE: THE GENERIC NONCOMPATIBLE CASE

We study the boundary layers and singularities generated by a convection-diffusion equation in a circle with noncompatible data. More precisely, the boundary of the circle has two characteristic points where the boundary conditions and the external data $f$ are not compatible. Very complex singular behaviors are observed, and we analyze them systematically for highly noncompatible data. The problem studied here is a simplified model for problems of major importance in fluid mechanics and thermohydraulics and in physics.open4