Thermodynamic Limit for Directed Polymers and Stationary Solutions of the Burgers Equation

The first goal of this paper is to prove multiple asymptotic results for a time-discrete and space-continuous polymer model of a random walk in a random potential. These results include: existence of deterministic free energy density in the infinite-volume limit for every fixed asymptotic slope, concentration inequalities for free energy implying a bound on its fluctuation exponent, and straightness estimates implying a bound on the transversal fluctuation exponent. The culmination of this program is almost sure existence and uniqueness of polymer measures on one-sided infinite paths with given endpoint and slope, and interpretation of these infinite-volume Gibbs measures as thermodynamic limits. Moreover, we prove that marginals of polymer measures with the same slope and different endpoints are asymptotic to each other. The second goal of the paper is to develop ergodic theory of the Burgers equation with positive viscosity and random kick forcing on the real line without any compactness assumptions. Namely, we prove a one force–one solution principle, using the infinite-volume polymer measures to construct a family of stationary global solutions for this system, and proving that each of those solutions is a one-point pullback attractor on the initial conditions with the same spatial average. This provides a natural extension of the same program realized for the inviscid Burgers equation with the help of action minimizers that can be viewed as zero temperature limits of polymer measures. © 2018 Wiley Periodicals, Inc.     

Laboratory Experimental Facility for Studying Metamaterial Sound-absorbing Duct Linings with Grazing Sound Propagation and How the Type of Source Influences their Efficiency

Acoustical Physics - The article describes an experimental facility for studying and testing sound-absorbing linings made of acoustic metamaterials designed to reduce the noise generated by a...     

On Diffraction of a Sawtooth Nonlinear Wave by a Narrow Circular Aperture in a Screen

Acoustical Physics - The paper presents the experimental and theoretical studies of diffraction effects in intense acoustic beams diffracted by a narrow circular aperture in a screen. The...     

Extremal Problems and Quadratic Differentials

We study extremal problems for domains with free poles that belong to ray systems of points. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 3, pp. 298–303, July–September, 2005.     

Operation of a magnetic pulse compressor with electrodynamic acceleration of a liner

Experimental data for the acceleration of plate liners with a magnetic flux electromagnetic compressor are reported. The compressor is designed for peaking the power of current pulses with an initial amplitude of 3 MA. The compression zone of the liner has the form of a stripline, which makes it possible to use one current source both for the acceleration of the liner and for the generation of a magnetic flux in the compression zone. The acceleration dynamics and the deformation of plate liners during an operating cycle of the compressor are studied. In the experiments conducted, the compressor is powered by a capacitor battery. The current amplitude in the output capacitive load rises by 1.6 times, while the pulse narrows twofold. Experimental data are compared with numerical calculation data for the liner material flow, which are found to adequately describe processes taking place in such a compressor.     

Application of a separating transformation to estimates of inner radii of open sets

We obtain solutions of new extremal problems of the geometric theory of functions of a complex variable related to estimates for the inner radii of nonoverlapping domains. Some known results are generalized to the case of open sets. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1313–1321, October, 2007.     

Sharp estimates for inner radii of systems of nonoverlapping domains and open sets

We study extremal problems of the geometric theory of functions of a complex variable. Sharp upper estimates are obtained for the product of inner radii of disjoint domains and open sets with respect to equiradial systems of points. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1601–1618, December, 2007.     

Quadratic differentials in problems of the product of inner radii of nonoverlapping domains

We study applications of the theory of quadratic differentials to the solution of new extremal problems of nonoverlapping domains with free poles on rays and generalize known results to a certain class of open sets. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 4, pp. 435–442, October–December, 2007.