Duality in parameter space and approximation of measures for mixing repellers

For one-dimensional expanding mapsT with an invariant measure we consider, in a parameter space, the envelope _n of the real lines associated to any couple of points of the orbit, connected byn iterations ofT. If the map hass inverses and is piecewise linear, then the sets _n are just the union ofs ⁿ points and converge to the invariant Cantor set ofT. A correspondence between all the sets and their measures is established and allows one to associate the atomic measure on ₁ to the completly continuous measure on the Cantor set. If the map is nonlinear, hyperbolic, and hass inverses, the sets _n are homeomorphic to the Cantor set; they converge to the Cantor set ofT and their measures converge to the measure of the Cantor set whenn. The correspondence between the sets _n allows one to define converging approximation schemes for the map an its measure: one replaces each of thes ⁿ disjoint sets with a point in a convenient neighborhood and a probability equal to its measure and transforms it back to the original set ₁. All the approximations with linear Cantor systems previously proposed are recovered, the converging proprties being straightforward in the present scheme. Moreover, extensions to higher dimensionality and to nondisconnected repellers arte possible and are briefly examined.     

Asymptotic behavior of energy band associated with a negative energy level

The mathematical foundation of the tight binding approximation is given. If ₀ is a negative energy level of a real potentialq, then there exists an energy band for a one-dimensional chain with period 2T of the same atoms which lies near ₀. We study this band whenT tends to infinity.On leave of absence from the Department of Physics, Leningrad State University, Leningrad, USSR.     

Hereditary optimal control problems: Numerical method based upon a padé approximation

In this paper, we consider a particular approximation scheme which can be used to solve hereditary optimal control problems. These problems are characterized by variables with a time-delayed argumentx(t – ). In our approximation scheme, we first replace the variable with an augmented statey(t) x(t - ). The two-sided Laplace transform ofy(t) is a product of the Laplace transform ofx(t) and an exponential factor. This factor is approximated by a first-order Padé approximation, and a differential relation fory(t) can be found. The transformed problem, without any time-delayed argument, can then be solved using a gradient algorithm in the usual way. Four problems are solved to illustrate the validity and usefulness of this technique.This research was supported in part by the National Aeronautics and Space Administration under NASA Grant NCC-2-106.     

One-dimensional harmonic lattice caricature of hydrodynamics: Second approximation

The long-time behavior of an infinite chain of coupled harmonic oscillators is studied. In addition to a limiting hydrodynamic (Euler-type) equation, the next approximation is investigated. The corresponding equation is derived.     

Surface properties of finite classical Coulomb systems: Debye-Hückel approximation and computer simulations

We report analytical and numerical studies of surface correlations in finite, homogeneously polarizable, classical Coulomb systems placed in an insulating or conducting environment. Their purpose is to understand the phenomenological, shape-dependent laws of electrostatics, from the point of view of statistical mechanics; we focus on the knowledge of the dielectric susceptibility of the system, a quantity proportional to the equilibrium fluctuation of the system's instantaneous polarization per unit volume. This goal has been achieved for a system in a conducting state. The picture is that the shape-dependent part of the susceptibilities results from the action of unbounded observables (the second moments of the instantaneous polarization of the system) on long-range surface correlations and that the relations of electrostatics are verified by means of shape-dependent thermodynamic limits. This picture is supported (i) by exact solutions and asymptotic analysis of the Debye-Hückel approximation of multicomponent plasmas in disks and spheres with insulating and conducting environment and also in ellipses in a vacuum, and (ii) by computer simulations of a one-component plasma in a disk with different environments, notably a conducting environment with permeable and impermeable wall. These observations have revealed for the first time the reason why the susceptibility of a conducting disk in a conductor with impermeable walls diverges linearly with the radius of the disk: this is due to the occurrence of long-range radial correlations in the conductor. These findings are quantitatively interpreted in terms of a novel canonical Debye-Huckel approximation as contrasted to the ordinary grand canonical version. Lastly a fresh look at the problem of the surface correlations of a conductor in a vacuum, which places the observer close to the surface of the conductor but in the vacuum, is presented and applied to the disk, the ellipse, the cylinder, the sphere, and the wedge.     

Infrared asymptotic freedom of a hierarchical φ 3 6 lattice theory

For the weakly coupled lattice ₃ ⁶ theory in a hierarchical model approximation a nonperturbative renormalization group analysis in the spirit of Gawedzki and Kupiainen is performed to study the flow of the effective actions. We deduce a domain of attraction to the tricritical (Gaussian) fixed point. The two relevant coupling constants of the problem are controlled by analytic continuation to complex domains, tracing their images under the renormalization group iterations.     

Ab initio calculations of small hydrides including electron correlation

Five different structures of CH₅ ⁺ and one structure of CH₅ ^– are calculated using a gaussian basis both in the SCF approximation and with the inclusion of electron correlation in the independent electron pair approximation (IEPA). While on SCF level the C _sstructure of CH₅ ⁺ has to lowest energy, the energy difference between the C _sand C _2vstructures becomes negligible if correlation is included. In contrast to this the approach of a proton to CH₄ at large and intermediate distances is most favorable towards a corner of the CH₄ tetrahedron which means a structure. The decomposition of CH₅ ⁺ into CH₃ ⁺ and H₂ requires 20kcal/mol on SCF level and 40 kcal/mol if correlation is included.     

A comparative quantum-chemical analysis of electronic structures and spectroscopic parameters of cycloalkanes and cyclopolysilanes

A comparative quantum-chemical analysis of the electronic structures and spectroscopic parameters of the cycloalkanes C₃H₆, C₄H₈, C₅H₁₀, and C₆H₁₂ and their silicon analogs Si₃H₆, Si₄H₈, Si₅H₁₀ and Si₆H₁₂ was performed in the framework of the SCF MO LCAO method in the INDO approximation. Qualitative interpretation of “abnormal” ionization potentials and energies of electronic absorption spectra of cyclopolysilanes has been given. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1105–1108, June, 1997.     

New method for calculating the adsorption of noble gases on amorphous surfaces

A new approach to calculating the equilibrium characteristics of the adsorption of noble gases on the amorphous surfaces of adsorbents was developed and applied to the Ar−TiO₂(rutile) system. Intermolecular adsorbate-adsorbate interactions are taken into account for the nearest neighbors in the quasi-chemical approximation. The lattice energy parameters of all interactions of the model are determined from the Lennard-Jones potential (12-6). The formation of amorphous TiO₂(rutile) surface includes completion of the surface layers and partial removal of the surface oxygen ions. The quality of the amorphization procedure was confirmed by the experimentally measured heats and isotherms of adsorption of the system under study. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1109–1118, June, 1997.