Factorization of matrix functions and characteristic properties of the circle

In the present paper we discuss two problems on factorizations of matrix-valued functions with respect to a simple closed rectifiable curve . These two problems are related and we show that in both of them circular contours play a remarkable role.     

On the norm of polynomials of two adjoint projections

Letf(X, Y) be a polynomial of two non-commuting variables and letP be an arbitrary nontrivial projection operator in Hilbert space. The class of all polynomialsf(X, Y) for which f(P, P ^*) depends only onf and P are described. In the case when such a dependence exists the explicit formula is obtained. Some applications to singular integral operators are given.     

Reconfigurable generation of high-repetition-rate optical pulse sequences based on time-domain phase-only filtering

We propose and demonstrate a fiber-based phase-only filtering technique for programmable optical pulse shaping, in which the filtering operation is implemented in the time domain by means of an electro-optical (EO) phase modulator. The technique has been applied for generating customized ultrahigh-repetition-rate optical pulse sequences (>40 GHz) from single input pulses by driving the EO phase modulator with a periodic electronic waveform (RF tone). The generated output pulses are replicas of the input pulse and both the repetition rate and the envelope profile of the generated sequences can be controlled and tuned electronically using this approach.     

Space and time evolution of plasma potential in T-10 undervariation of main gas influx

The HIBP method was used in the experiments on the T-10 tokamak to measure the space potential distribution as a function of gas feed. A high electric field was observed in the region where the ionization of neutrals takes place. The correlation of the potential with the gas flux dynamics and the MHD activity was also found     

Extension theorems for Fredholm and invertibility symbols

This paper is a continuation of [GK3] where the theory of Invertibility Symbol in Banach algebras was developed. In the present paper we generalize these results for the case when the Invertibility Symbol is defined on a subalgebra of the Banach algebras. The difficulty which arises here in this more general case is connected with the fact that some elements of the subalgebra may have the inverses which do not belong to the subalgebra. This generalization of the theory allows us to study the Fredholm Symbols of linear operators. Applications to subalgebras generated by two idempotents and to algebras generated by singular integral operators are presented.     

Local principle of computation of the factor-norms of singular integral operators

V. I. Lenin Kishinev State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 2, pp. 57–58, April–June, 1988.     

On the continuity of the spectrum in certain Banach algebras

In this paper we describe some classes of linear operatorsTL(H) (mainly Toeplitz, Wiener-Hopf and singular integral) on a Hilbert spacesH such that the spectrum (T, L(H)) is continuous at the pointsT from these classes. We also describe some subalgebras of the algebras for which the spectrum (x,) becomes continuous at the pointsx when (x,) is restricted to the subalgebra . In particular, we show that the spectrum (x,) is continuous in Banach algebras with polynomial identities. Examples of such algebras are given.This research was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

Decomposition of Dissolved Methylene Blue in Water Using a Submerged Arc Between Titanium Electrodes

Low voltage, low energy submerged pulsed arcs between Ti electrodes with a pulse repetition rate of 100 Hz, energies of 2.6–192 mJ and durations of 10–40 μs, followed by aging in the dark, were used to decompose 10 mg/l methylene blue (MB) contamination in 40 ml aqueous solutions, with and without the addition of 0.5 % H₂O₂. The impact of the arc treatment on the MB removal ratio (C₀–C_ta)/C₀ was considered as a function of aging time t_a, where C₀ and C_ta are the MB concentrations initially and after t_a (the time needed to complete removal of MB after cessation of exposure of the arc). Particles eroded from the electrodes during the discharge enabled MB decomposition during aging. The particles were studied by XRD, XPS and Raman analysis, and titanium oxides and peroxides were found. MB decomposition during aging is explained by the formation of a surface layer of titanium peroxide that forms by the interaction of titanium dioxide with H₂O₂, which produce radicals which oxidize the MB. The 99.6 % MB removal yield (G_99.6 = 90 g/kWhr) of the submerged pulsed arc process with Ti electrodes and addition of 0.5 % H₂O₂ was more than 60 times larger than obtained at 50 % removal with other plasma methods.