Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems

In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point).      

Commuting foliations

The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations using Nambu structures, i.e., integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures and show some basic results about commuting Nambu structures.     

Liquid–liquid phase transition and anomalous diffusion in simulated liquid GeO2

We perform molecular dynamics (MD) simulation of diffusion in liquid GeO₂ at the temperatures ranged from 3000 to 5000 K and densities ranged from 3.65 to 7.90 g/cm³. Simulations were done in a model containing 3000 particles with the new interatomic potentials for liquid and amorphous GeO₂, which have weak Coulomb interaction and Morse-type short-range interaction. We found a liquid–liquid phase transition in simulated liquid GeO₂ from a tetrahedral to an octahedral network structure upon compression. Moreover, such phase transition accompanied with an anomalous diffusion of particles in liquid GeO₂ that the diffusion constant of both Ge and O particles strongly increases with increasing density (e.g. with increasing pressure) and it shows a maximum at the density around 4.95 g/cm³. The possible relation between anomalous diffusion of particles and structural phase transition in the system has been discussed.     

Structural properties of amorphous TiO<Subscript>2</Subscript> nanoparticles

Structural properties of amorphous TiO₂ spherical nanoparticles have been studied in models with different sizes of 2 nm, 3 nm, 4 nm and 5 nm under non-periodic boundary conditions. We use the pairwise interatomic potentials proposed by Matsui and Akaogi. Models have been obtained by cooling from the melt via molecular dynamics (MD) simulation. Structural properties of an amorphous nanoparticle obtained at 350 K have been analyzed in detail through the partial radial distribution functions (PRDFs), coordination number distributions, bond-angle distributions and interatomic distances. Moreover, we show the radial density profile in a nanoparticle. Calculations show that size effects on structure of a model are significant and that if the size is larger than 3 nm, amorphous TiO₂ nanoparticles have a distorted octahedral network structure with the mean coordination number Z_Ti-O ≈6.0 and Z_O-Ti ≈3.0 like those observed in the bulk. Surface structure and surface energy of nanoparticles have been obtained and presented.     

A note on degenerate corank-one singularities of integrable Hamiltonian systems

We prove that, in a neighborhood of a corank-1 singularity of an analytic integrable Hamiltonian system with n degrees of freedom, there is a locally-free analytic symplectic \Bbb T^n-1 {\Bbb T}^{n-1} -action which preserves the moment map, under some mild conditions. This result allows one to classify generic degenerate corank-one singularities of integrable Hamiltonian systems. It can also be applied to the study of (non)integrability of perturbations of integrable systems.     

Structural properties of amorphous Fe2O3 nanoparticles

We have investigated the microstructure of amorphous Fe₂O₃ nanoparticles by using molecular dynamics (MD) simulations. Non-periodic boundary conditions with Born-Mayer type pair potentials were used to simulate a spherical model of different diameters of 2, 3, 4 and 5 nm. Structural properties of an amorphous model obtained at 350 K have been analyzed in detail through the partial radial distribution functions (PRPFs), coordination number distributions, bond-angle distributions and interatomic distances. Calculations showed that structural characteristics of the model are in qualitative agreement with the experimental data. The observation of a large amount of structural defects as the particle size is decreased suggested that surface structure strongly depends on the size of nanoparticles. In addition, surface structure of amorphous Fe₂O₃ nanoparticles have been studied and compared with that observed in the core and in the bulk counterpart. Radial density profiles and stoichiometry in morphous Fe₂O₃ nanoparticles were also found and discussed.     

Another Note on Focus-Focus Singularities

We show a natural relation between the monodromy formula for focus-focus singularities of integrable Hamiltonian systems and a formula of Duistermaat–Heckman, and extend the main results of our previous note ( ¹-action, monodromy, and topological classification) to the case of degenerate focus-focus singularities. We also consider the non-Hamiltonian case, local normal forms, etc.     

Approximation of classes of smooth functions of several variables

The paper deals with imbedding theorems and harmonic approximation for classes of periodic functions of several variables. Necessary and sufficient conditions for imbedding are found for various classes of smooth functions. The paper contains asymptotic estimates for fractional order derivatives of multidimensional Dirichlet and Favard kernels, for Fourier best approximation and for the Kolmogorov diameter of smooth function classes.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 207–226, 1984.     

Topological classification of integrable geodesic flows on orientable two-dimensional Riemannian manifolds with additional integral depending on momenta linearly or quadratically

Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 3, pp. 42–56, July–September, 1993.