A constructive description of Hölder classes on closed Jordan curves

Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that ¦f(z) ? R_n(z)¦ ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ ₁ ⁺ /n^(z) and ρ ₁ ^- /n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) .     

Some features of solar wind protons, α particles and heavy ions behaviour: The Prognoz 7 and Prognoz 8 experimental results

The paper describes the results of investigations of the solar wind ions, carried out on board the high apogee Prognoz 7 and Prognoz 8 Earth's satellites with the aid of an SKS instrument (USSR) and a Monitor instrument (USSR-SSR). Behaviour of proton and that of components on the front of Earth's bow and interplanetary shock waves were compared by means of the energoanalysis and energy-mass analysis techniques. In several long-term periods of observation the solar wind heavy ions — oxygen, silicon and iron were determined. It enabled us to estimate the solar corona chemical composition and electron temperature.Presented at the 5th General Assembly IAGA/IAMAP August 5–17, 1985, Prague, Czechoslovakia.     

Analysis of the 3-dimensional electron distribution in silicon using directional Compton profile measurements

A method of reconstructing the electron momentum density (p), and its Fourier transform,B(t), from a series of directional Compton profiles is described. It is based on a double Fourier inversion technique and an expansion in lattice harmonic functions. The effect of random errors has been analysed, and the implications for the data collection discussed. We have used the reconstruction technique to obtain (p) andB(t) for silicon from six directional Compton profiles measured with 412 KeV gamma-radiation. The experimental result is in good agreement with earlier measurements and with available solid state theories. A recent Wannier function calculation for silicon using orthogonalised bond orbitals provides a useful tool for identifying the physical origins of the observed anisotropies. A comparison between the information presented in position and momentum space shows that the ease of interpretation depends upon the degree to which the various interactions give rise to localised features in each representation.     

Exact solution of the Heisenberg XXZ model of spin s

A generalization of the Heisenberg XXZ model to the case of arbitrary spin with anisotropy of light-plane type is investigated. The base state of the system, the spectrum of excitations over it, and their scattering matrix are found. The magnetic susceptibility of the model for small magnetic fields and the low-temperature heat capacity are computed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 145, pp. 109–133, 1985.     

The likelihood ratio for densities with singularities

Let t_n be the Bayesian estimator of the parameter constructed from independent observations in the case of infinite information and probability distribution density with some singularities. It is shown that under certain conditions on the behavior of the densities near their singularities, the normalizing factor (n) ensuring that _n=(n) (t_n-) has a nontriviallimiting distribution for n is regularly varying in Karamata's sense. The limiting distribution of _n is determined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 74, pp. 66–82, 1977.I would like to acknowledge the guidance of I. A. Ibragimov in the course of this research.     

Study of diffusion creep accompanying superplastic deformation of the alloy ZhS6KP

The development of diffusion creep (DC) accompanying superplastic deformation (SPD) of the highly doped nickel alloy ZhS6KP is studied based on an investigation of the redistribution of dispersed intragrain deposits of the phase of Ni₃(Al, Ti). Deposit-free zones (DFZ) are formed in the alloy, held at the temperature of SPD and accompanying deformation at the grain boundaries. The contribution of DC to the deformation was determined from a comparative analysis of the width of the DFZ in the deformed and undeformed parts of the samples for different grain sizes and rates of DC taking into account diffusion accommodation. It was established that for the optimum rate of SPD the upper limit of the contribution of DC to deformation for 2-m grains does not exceed 11%. It is concluded, based on the distribution of DFZ, that the effect of DC accompanying SPD is determined by diffusion fluxes, associated with the local concentration of stress accompanying the development of grain-boundary slipping.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 40–45, June, 1986.In conclusion, we thank Yu. M. Mishin for a useful discussion of the results.     

Electric and tensoelectric study of radiation defects in GaAs

The electric and tensoelectric properties are studied for GaAs crystals which have been irradiated by electrons (2.3 Mev) at 300°K with integrated fluxes of up to 2·10¹⁵ –1·10¹⁹ cm^–2. On the basis of the electrical neutrality equation, including seven energy levels (E1–E5, H0, H1) of the radiation defects, the specific resistivity and the strain sensitivity coefficient are quantitatively analyzed as a function of exposure. The pressure coefficients for the E1–E5 levels with respect to the _c ⁶ point are determined to be (0, 9.6, 11.0, 11.6, 11.6)·10^–6 eV/bar, respectively, at 300°K.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 81–87, October, 1986.     

Gallium arsenide avalanche-type S-diode based on a n+-π-ν-n structure

The results are presented of an investigation of the electrical characteristics of avalanche S-diodes based on a new type of structure (⁺---n), which is obtained by diffusing iron in GaAs with n = 1.10¹⁷ cm^–3. The diodes have regions of S-type negative differential resistance (NDR), for both bias polarities and have a number of other features by comparison with diodes based on the --n structures with fused contacts. We discuss the mechanism for the formation of the NDR region of the voltage-current characteristics of the diode.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 54–58, April, 1986.     

Vacuum polarization by a Coulomb center and strong magnetic field. modification of Coulomb's law

The proper time method is used to examine the effect of polarization of the electron-positron vacuum on the nature of the interaction of a charged particle with the field of a nucleus and a magnetic field. It is shown that the interaction of an electron with a nucleus becomes anisotropic in the presence of a magnetic field. The dielectric constants of the vacuum and the effective charge of the Coulomb center are calculated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 38–45, November, 1986.