The Structure of Lie Algebras and the Classification Problem for Partial Differential Equations

The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. Since the procedure which we use, relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. This material is dispersed in the literature and is not fully available in English. After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial differential equations under study.Comment: 45 page

Note: The effect of viscosity on the rate of diffusion-limited association of nanoparticles

A study is conducted to investigate the effect of viscosity on the rate of diffusion-limited association of nanoparticles. The study focuses on the diffusion-limited association of suspended nanoparticles, to keep the analysis mathematically transparent. It is demonstrated that the reduction of association rate constant as compared with the definition in another equation is physically related to hydrodynamic slowdown of the particle mobility in the contact region

Inhibition of the receptor-mediated virion attachment to a lipid membrane

The forefront of the anti-viral defence is sometimes aimed at virion attachment to a host membrane. This step or, more specifically, virion contacts with cellular membrane receptors (or, e.g., glycolipids) can be inhibited by antibodies (or specially chosen or designed compounds) via their association with virions. In this case, the full-scale attachment of virions to a host membrane occurs via a subtle interplay of the formation and rupture of multiple virion-inhibitor and virion-receptor bonds. We present a kinetic model describing this interplay and illustrating general trends in the process under consideration

Viral capsids: Kinetics of assembly under transient conditions and kinetics of disassembly

The available kinetic models of assembly of viral protein capsids are focused primarily on the situations in vitro where the amount of protein is fixed. In vivo, however, the viral protein synthesis and capsid assembly occur under transient conditions in parallel with viral genome replication. Herein, a kinetic model describing the latter case of capsid assembly is proposed with emphasis on the period corresponding to the initial stage of viral genome replication. The analysis is aimed at small icosahedral capsids. With biologically reasonable values of model parameters, the model predicts rapid exponential growth of the populations of monomers and fully assembled capsids during the transient period of genome replication. Under the subsequent steady-state conditions with respect to replication, the monomer population is predicted to be nearly constant while the number of fully assembled capsids increases linearly. The kinetics of capsid disassembly, described briefly as well under conditions of negligible monomer concentration, exhibit a short induction period when the number of proteins in a capsid is only slightly smaller than in the beginning, followed by more rapid protein detachment. According to calculations, the latter kinetics may strongly depend on protein degradation

Two-Dimensional Flow Nanometry of Biological Nanoparticles for Accurate Determination of Their Size and Emission Intensity

Biological nanoparticles (BNPs) are of high interest due to their key role in various biological processes and use as biomarkers. BNP size and molecular composition are decisive for their functions, but simultaneous determination of both properties with high accuracy remains challenging, which is a severe limitation. Surface-sensitive microscopy allows one to precisely determine fluorescence or scattering intensity, but not the size of individual BNPs. The latter is better determined by tracking their random motion in bulk, but the limited illumination volume for tracking this motion impedes reliable intensity determination. We here show that attaching BNPs (specifically, vesicles and functionalized gold NPs) to a supported lipid bilayer, subjecting them to a hydrodynamic flow, and tracking their motion via surface-sensitive imaging enable to determine their diffusion coefficients and flow-induced drift velocities and to accurately quantify both BNP size and emission intensity. For vesicles, the high accuracy is demonstrated by resolving the expected radius-squared dependence of their fluorescence intensity.Comment: 28 pages, 5 figure

On separable Fokker-Planck equations with a constant diagonal diffusion matrix

We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the drift coefficients $B_1(\vec x),B_2(\vec x),B_3(\vec x)$ providing separability of the corresponding Fokker-Planck equations and carry out variable separation in the latter. It is established, in particular, that the necessary condition for the Fokker-Planck equation to be separable is that the drift coefficients $\vec B(\vec x)$ must be linear. We also find the necessary condition for R-separability of the Fokker-Planck equation. Furthermore, exact solutions of the Fokker-Planck equation with separated variables are constructedComment: 20 pages, LaTe

On separable Schr\"odinger equations

We classify (1+3)-dimensional Schr\"odinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables. As a result, we get eleven classes of the electromagnetic vector potentials of the electromagnetic field $A(t, \vec x)=(A_0(t, \vec x)$, $\vec A(t, \vec x))$ providing separability of the corresponding Schr\"odinger equations. It is established, in particular, that the necessary condition for the Schr\"odinger equation to be separable is that the magnetic field must be independent of the spatial variables. Next, we prove that any Schr\"odinger equation admitting variable separation into second-order ordinary differential equations can be reduced to one of the eleven separable Schr\"odinger equations mentioned above and carry out variable separation in the latter. Furthermore, we apply the results obtained for separating variables in the Hamilton-Jacobi equation.Comment: 30 pages, LaTe

Effect of lattice strain on hydrogen diffusion in Pd: A density functional theory study

The density functional theory is used to study the effect of lattice strain on hydrogen diffusion in Pd. The activation energy for this process is found to increase dramatically with increasing compressive lattice strain. In particular, the activation energy is close to double for an isotropic compression of 5% both in the alpha and beta phases. For tensile strain, the activation energy is instead decreased. This finding has important consequences for the interpretation of various kinetic processes occurring with participation of hydrogen and other interstitial atoms in macroscopic solid samples and nanoparticles

Cabrera-Mott kinetics of oxidation of metal nanowires

The Cabrera-Mott model, implying that oxidation of a metal is limited by the field-facilitated activated jumps of metal ions at the metal-oxide interface, was originally proposed to interpret growth of thin oxide films on planar metal surfaces. Recently, the model was used to describe oxidation of spherical nanoparticles with small radius of curvature. Here, we analyze oxidation of nanowires. The increase of the oxide thickness with increasing time for a nanowire is shown to be slower than that for a nanoparticle with the same radius, but faster than in the case of a planar surface

New approach to interpretation of airborne magnetic and electromagnetic data

Journal ArticleWe present a new technique for underground imaging based on the idea of space-frequency filtering and downward continuation of the observed airborne magnetic and electromagnetic data. The technique includes two major methods. The first method is related to the downward analytical continuation and is based on the calculation of the total normalized gradient of the observed field. The second method is based on Wiener filtering and takes into account a priori information about typical AEM anomaly shape from a possible target