Nonlinear evolution equations for describing waves in bubbly liquids with viscosity and heat transfer consideration

Nonlinear evolution equations of the fourth order and its partial cases are derived for describing nonlinear pressure waves in a mixture liquid and gas bubbles. Influence of viscosity and heat transfer is taken into account. Exact solutions of nonlinear evolution equation of the fourth order are found by means of the simplest equation method. Properties of nonlinear waves in a liquid with gas bubbles are discussed.     

Berezin transform in polynomial bergman spaces

Fix a smooth weight function Q in the plane, subject to a growth condition from below. Let K_m,n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n ? 1 of finite L²‐norm with respect to the measure e^?mQ dA. Here dA is normalized area measure, and m is a positive real scaling parameter. The (polynomial) Berezin measure for the point z₀ is a probability measure that defines the (polynomial) Berezin transform for continuous . We analyze the semiclassical limit of the Berezin measure (and transform) as m → + ∞ while n = m τ + o(1), where τ is fixed, positive, and real. We find that the Berezin measure for z₀ converges weak‐star to the unit point mass at the point z₀ provided that Δ Q(z₀) > 0 and that z₀ is contained in the interior of a compact set , defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane. For points , the Berezin measure cannot converge to the point mass at z₀. In the model case Q(z) = |z|², when is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z₀ relative to . Our results have applications to the study of the eigenvalues of random normal matrices. The auxiliary results include weighted L²‐estimates for the equation when f is a suitable test function and the solution u is restricted by a polynomial growth bound at ∞. © 2009 Wiley Periodicals, Inc.     

Model reduction in topology optimisation analysing the inner structure of sensitivity matrices

Topology optimisation models usually contain a great number of design variables and correspondingly lead to large matrices (pseudo load matrix and sensitivity matrix) which appear in sensitivity analysis. We apply singular value decomposition (SVD) to these matrices to analyse their inner structure. Based on the obtained information, we perform model reduction by transformation of the design variables into a lower-dimensional space. Numerical examples illustrate the advocated theoretical concept. Reasonable results are obtained, based on only a fraction of all design variables. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the electron transfer through Geobacter sulfurreducens PilA protein

Geobacter sulfurreducens pili composed of the Type IV pili structural peptide PilA have been implicated as efficient electronic conductors. Though investigated experimentally, no detailed theoretical studies have been performed to date that provide quantitative estimation of the transmission spectrum, electron transfer (ET) paths, efficiency of current generation, and other factors needed for understanding possible mechanisms of conductivity. In the present work, we calculate from first principles the possibilities of electron tunneling through 3 PilA fragments which structure was identified recently by NMR. The results indicate that positively charged amino acids, arginines and lysines form electrostatic traps in the middle of the peptide preventing ET at low bias voltages (<～6 V). At higher biases the traps are filled with electrons making possible sequential electron tunneling through the central part of the protein. In addition, leucines and phenylalanines form ET loops facilitating electron stabilization within the protein and sequential ET. Our results indicate that ET through the PilA protein cannot occur by coherent ET, but suggest a sequential (incoherent) mechanism. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 1706–1717     

Elliptic solutions for a family of fifth order nonlinear evolution equations

The family of fifth order nonlinear evolution equations is studied. Some traveling wave elliptic solutions are found. The classification of these exact solutions is given.     

Octonions and Singular Solutions of Hessian Elliptic Equations

Algebra of Octonions is used to construct singular viscosity solutions of fully nonlinear Hessian elliptic equations. These equations are written in the form of an Isaacs equation.