On Some Integral Equations in Hilbert Space with an Application to the Theory of Elasticity

A Volterra type integral equation in a Hilbert space with an additional linear operator L and a spectral parameter depending on time is considered. If the parameter does not belong to the spectrum of L unconditional solvability of the considered problem is proved. In the case where the initial value of the parameter coincides with some isolated point of the spectrum of the operator L sufficient conditions for solvability are established. The obtained results are applied to the partial integral equations associated with a contact problem of the theory of elasticity.     

Determination of refractive indexes of dielectric films by the use of immersion liquids

Examples of using immersion liquids in ellilipsometry and multiple-beam interferometry are given. The refractive index of very thin (∼ 5.0 nm) silica films on silicon and on films of silicon nitride are determined by the ellipsometry method. The method of precise determination of refractive index of monolayer dielectric films is described when a precision equal to ± 3 · 10⁻⁴ was reached. A method of determination film thicknesses near the substrate surface in a twolayer system is suggested. The construction of different prisms is given by means of which the immersion liquids are inserted. The selection of the equations for the determination of refractive index in the multiple-beam interferometry is discussed. Two versions of the use of immersion liquids for the determination of refractive indexes of dielectric films by interferometry method are given.     

Description of the Real von Neumann Algebras with Abelian Skew-Symmetric Part

In this note we describe (up to isomorphism) the real von Neiman algebras R with Abelian skew-symmetric part , i.e., such that xy-yx=0 for any .     

Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov     

Local and 2-local derivations on noncommutative Arens algebras

The paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra L ^ω(M,τ) associated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every 2-local derivation on L ^ω(M,τ) is a spatial derivation, and if M is a finite von Neumann algebra, then each local derivation on L ^ω(M,τ) is also a spatial derivation and every 2-local derivation on M is in fact an inner derivation.     

Jordan operator algebras

A survey of recent results in classification of JW-algebras (weakly closed Jordan algebras of self-adjoint operators in the Hilbert space) is given along with connections of JW-algebras with their enveloping W^*-algebras. It is shown how these results are applied in the proofs of analogs of many important results in the theory of W^*-algebras.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 27, pp. 67–98, 1985.     

Optical characteristics of thin rutile films on fused quartz substrates

Polycrystalline rutile films are synthesized on fused quartz substrates by the method of thermal oxidation of a titanium metal layer in air at 800°C. The optical parameters of the TiO ₂ films are determined for a wavelength of λ = 0.6328 μm by the method of laser zero ellipsometry. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 15–20, May, 2006.