On conformal invariants in problems of constructive function theory on sets of the real line

This is a survey of some recent results by the author and his collaborators in the constructive theory of functions of a real variable. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory in the complex plane

Magnetotransport studies of SiGe-based p-type heterostructures: problems of the effective mass determination

The Shubnikov–de Haas oscillations method of the effective mass extraction was illustrated by the magnetotransport properties investigation of two-dimensional hole gas in Si₁₋xGex (x = 0.13, 0.36, 0.95, 0.98) QWs. We have found that for certain samples our data cannot be fitted to standard theoretical curves in which the scattering of charge carriers is described by conventional Dingle factor. It is demonstrated that reasons of deviations of the experiment from the theory are as follows; (i) influence of the spin splitting on amplitude of SdH oscillations maxima; (ii) extra broadening of the Landau levels attributed to existence of inhomogeneous distribution of the carrier concentration; (iii) the influence of the concurrent existence of short and long-range scattering potentials; (iv) the population of second energy level in the quantum well. The ways to calculate the effective masses m* of holes in all cases are presented and values of m* are found for studied heterostructures

Quasi-Monte Carlo rules for numerical integration over the unit sphere S2\mathbb{S}^2

We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a $(0,m,2)$-net given in the unit square $[0,1]^2$ to the sphere $\mathbb{S}^2$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on $\mathbb{S}^2$. And finally, we prove an upper bound on the spherical cap $L_2$-discrepancy of order $N^{-1/2} (\log N)^{1/2}$ (where $N$ denotes the number of points). This slightly improves upon the bound on the spherical cap $L_2$-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm. Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the $(0,m,2)$-nets lifted to the sphere $\mathbb{S}^2$ have spherical cap $L_2$-discrepancy converging with the optimal order of $N^{-3/4}$

Point sets on the sphere S2\mathbb{S}^2 with small spherical cap discrepancy

In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of spherical Fibonacci lattices converges with order $N^{-1/2}$. Such point sets are therefore useful for numerical integration and other computational simulations. The proof uses an area-preserving Lambert map. A detailed analysis of the level curves and sets of the pre-images of spherical caps under this map is given

Approximation of harmonic functions on compact sets in ℂ

The direct theorem of the theory of approximation of harmonic functions is established in the case of functions defined on a compact set, the complement of which with respect to ℂ is a John domain

The Nikol'skii-Timan-Dzjadyk Theorem for Functions on Compact Sets of the Real Line.

The Nikol'skii-Timan-Dzjadyk theorem concerning the polynomial approximation of functions on the interval [-1, 1] is generalized to the case of the approximation of functions given on a compact set on the real line which can consist of an infinite number of intervals

Germanium quantum well with two subbands occupied: kinetic properties

Multisubband transport of the p-type Si₀.₄Ge₀.₆/Ge/Si₀.₄Ge₀.₆ heterostructure has been investigated by means of magnetotransport measurements at low temperatures and high magnetic fields. Two frequency Shubnikov–de Haas oscillations indicate occupation of two subbands. This allows us to determine the densities and mobilities of the charge carriers on each subband. Shubnikov–de Haas oscillations reveal two 2D conduction subbands with carrier effective masses of 0.112m₀ and 0.131m₀. The quantum Hall ferromagnetic states which results from the crossing of two Landau levels with opposite spin and different subband was observed in SiGe systems for the first time

The overheating effects in germanium quantum well with two subbands occupied

The charge carrier overheating effect was studied in the p-type Si₀.₄Ge₀.₆/Ge/Si₀.₄Ge₀.₆ heterostructure with two subband occupy. The temperature dependences of hole-phonon relaxation time τh-ph at weak magnetic fields demonstrated transition of the 2D system from regime of “partial inelasticity” characterized by dependence τ⁻¹h-ph ∝ T² to regime of small-angle scattering, described by dependence τ-1h-ph ∝ T⁵ with temperature increase. But in higher magnetic fields the dependence τ⁻¹h-ph ∝ T³ manifests itself on dependences τh-ph(Th-ph). The possible explanations of such dependences are discussed.Ефект перегріву носіїв заряду вивчався в гетероструктурі Si₀.₄Ge₀.₆/Ge/Si₀.₄Ge₀.₆ p-типу з двома зайнятими підзонами. В слабких магнітних полях температурні залежності часу дірково-фононної релаксації h-ph τ демонструють перехід двовимірної системи з режиму «часткової непружності», яка характеризується залежністю τ⁻¹h-ph ∝ T² до режиму малокутового розсіювання, що описується залежністю τ-1h-ph ∝ T⁵ з підвищенням температури. В більш високих магнітних полях залежність τ⁻¹h-ph ∝ T³ змінюється на -ph -ph ( ) τh hT . Обговорюються можливі варіанти пояснення таких спостережень