Ranks of binary relations

This paper clarifies the notion of rank of a binary relation. Indeed several definitions cohabit in the literature and two of them have never been distinguished. The main results concern the comparison of these two notions. We show that the two definitions are really different, even under strong restrictive conditions about the algebraic structure of the semigroup.     

On the optimization of approximate integration by Monte Carlo methods

We solve the problem of optimization of monte Carlo methods for approximate integration over an arbitrary absolutely continuous measure. We propose a convenient model of Monte Carlo methods which uses the notion of transition probability.     

Meromorphic functions in the solution of the two-dimensional problem of elasticity

We introduce meromorphic functions that follow logically from the solutions of doubly periodic problems of two-dimensional elasticity (a rectangular period lattice). We establish their properties and their connection with the known meromorphic functions. We give convenient formulas for numerical implementation. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, pp. 53–58.     

Partially ordered baer involutive algebras

A survey is given of the basic results of the theory of AW^*-algebras, the properties of the rings of measurable and locally measurable operators associated with von Neumann algebras and AW^*-algebras are described, Baer^*-algebras admitting partial orderings compatible with their algebraic structures are investigated.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 27, pp. 99–128, 1985.     

Mg 2p shallow core-level and local atomic structure of i-ZnMgRE quasicrystals

We present a detailed analysis of the Mg 2p shallow core-levels measured on icosahedral single-grain ZnMgY, ZnMgHo, and ZnMgEr quasicrystals during a photoelectron microscopy study. The synchrotron radiation photoemission measurements were performed on in situ cleaved samples at a pressure of 10⁻¹⁰ mbar and at low temperature, typically 90-150 K. The Mg 2p photoemission lines are essentially broadened as compared to those of the Mg 2p spin-orbit doublet recorded on the Zn₂Mg crystalline Laves phase. The broadening is associated to the coordination shifts of the Mg 2p level due to the inequivalent magnesium sites in the quasicrystalline lattice. The coordination shifts are calculated on the basis of i-ZnMg(Ho, Y) atomic structure data, recently determined from the pair distribution function analysis. The coordination shifts obtained are up to 0.2 eV. The Mg 2p experimental spectral intensity is nicely reproduced by a superposition of coordination-shifted Mg 2p spin-orbit doublets.     

Multiphonon renormalization of the spectrum of two-dimensional excitons in a magnetic field

The renormalization of the dispersion law for two-dimensional excitons in a magnetic field is studied by the Green's-function method taking into account multiphonon processes.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 3–7, April, 1987.