Invertibility symbol for a Banach algebra generated by two idempotents and a shift

A Banach algebra generated by two idempotentsp, r, identitye and a shiftv which satisfy the conditionspv=vp andrv=v(e–r) is investigated. It is proved that all irreducible representations of algebraA are two- and four-dimensional. The explicit form of these representations is obtained. An Invertibility Symbol is constructed. Some examples are considered.     

A general look at local principles with special emphasis on the norm computation aspect

Local principles associate with every element of a Banach algebra a family of local objects in terms of which the properties of the original element can be studied. In this paper some general relations between three such principles, usually affiliated with the names of Simonenko, Allan/Douglas, and Gohberg/Krupnik, are discussed. Special attention is paid to the question on how the norm of an element can be expressed in terms of the norms of the local objects associated with it. The general theory is illustrated by some concrete results on singular integral operators and Toeplitz operators.     

On the norm of singular integral operator on curves with cusps

Applying the results on singular integral operators with the complex conjugation on curves with cusps (see R. Duduchava, T. Latsabidze, A. Saginashvili, 1992, 1994) the explicit formula for the local norm of the Cauchy singular integral operator on a curve with cusps in a Lebesgue space with an exponential weightL ₂ (, ) is obtained. For curves with angles the formula was already known (see R. Avedanio, N. Krupnik, 1988).     

Submerged Arc Breakdown of Methylene Blue in Aqueous Solutions

Low voltage, low energy submerged pulsed arcs between a pair of carbon or iron electrodes with a pulse repetition rate of 100?Hz, energies of 2.6?C192?mJ and durations of 20, 50 and 100???s were used to remove methylene blue (MB) contamination from 30?ml aqueous solutions. The MB concentration decreased exponentially with rates of 0.0006?C0.0143?s^?1 during processing with the carbon electrode pair. With the iron electrodes, the MB concentration initially decreased faster (0.030?s^?1) than with the carbon electrodes, but later saturated. However when microparticles produced with the iron electrodes were periodically filtered, the high removal rate was maintained. Under these conditions, the volume of water which can be treated per unit energy expenditure was much higher with the submerged arc than with other plasma processes. A kinetic model based on MB degradation by OH· radicals formed by the discharge was formulated. The higher initial MB removal rate with iron electrodes is explained by additional OH· production from Fenton??s reaction between Fe⁺⁺ and H₂O₂ produced by the discharge. This rate is maintained if the eroded iron particles are filtered, but if eroded iron particles accumulate, degradation slows down and stops, possibly because the iron particles catalytically decompose H₂O₂ and hence stops Fenton??s reaction, and either directly or via increased Fe⁺⁺ dissolved from the particles, scavenge the OH· radicals.     

Generation of long bursts of high-repetition-rate arbitrarily shaped optical pulses using time lens

A technique for the generation of long ultrahigh-speed bursts of optical pulses with arbitrary shapes is proposed. A laser pulse is temporally chirped by a time lens and then passes through a filter with a reconfigurable periodic spectral response, which produces time-delayed replicas of the chirped pulse and recombines them. As a result of the temporal interference between the replicas, the chirped pulse is broken up into short pulses with the shape determined by the chosen filter response. It is demonstrated that the filter acts on a long chirped optical pulse as a temporal modulator with a periodic modulation function. The modulation frequency and bandwidth of the modulator can be much higher than for commercially available high-frequency modulators. The additional advantage of this modulator is the arbitrary shape of the modulation function. A 2.4 ns burst of nearly flat-top pulses with a repetition rate of about 400 GHz is obtained in numerical simulations. In addition, the technique proposed can act as a pulse repetition rate multiplier and a pulse compressor. A repetition rate of 1.589 THz and an individual pulse width of 212 fs are achieved in simulations for a 9.7 ns sinusoidally phase modulated pulse burst.     

Convolution equations on finite intervals and factorization of matrix functions

Systems of convolution equations on a finite interval are reduced to the problem of canonical factorization of unimodular matrix-valued functions. The discrete version is considered separately.This research was partially supported by the Israel Science Foundation funded by the Israel Academy of Sciences and Humanities.     

An Explicit Factorization Algorithm

In this paper a new algorithm is proposed for explicit factorization of 2 x 2 matrix functions.     

Pressure effect on the nonradiative process of thioflavin-T

Time-resolved emission techniques were employed to study the nonradiative process of thioflavin-T (ThT) in 1-propanol, 1-butanol, and 1-pentanol as a function of the hydrostatic pressure. Elevated hydrostatic pressure increases the alcohol viscosity, which in turn strongly influences the nonradiative rate of ThT. A diamond-anvil cell was used to increase the pressure up to 2.4 GPa. We found that the nonradiative rate constant, k(nr), decreases with pressure. We further found a remarkable linear correlation between a decrease in k(nr) (increase in the nonradiative lifetime, τ(nr)) and an increase in the solvent viscosity. The viscosity was varied by a factor of 1000 and k(nr) was measured at high pressures, at which the nonradiative rate constant of the molecules decreased from (7 ps)(-1) to (13 ns)(-1), (13 ps)(-1) to (17 ns)(-1) and (17 ps)(-1) to (15 ns)(-1) for 1-propanol, 1-butanol, and 1-pentanol, respectively. The viscosity-dependence of k(nr) is explained by the excited-state rotation rate of the two-ring systems, with respect to each other.     

One-Sided Integral Approximations of the Generalized Poisson Kernel by Trigonometric Polynomials

We consider the generalized Poisson kernel Π _q,α = cos(απ/2)P + sin(απ/2)Q with q ∈ (?1, 1) and α ∈ ?, which is a linear combination of the Poisson kernel \(P(t) = 1/2 + \sum\nolimits_{k = 1}^\infty {{q^k}} \cos kt\)and the conjugate Poisson kernel \(Q(t) = \sum\nolimits_{k = 1}^\infty {{q^k}} \sin kt\). The values of the best integral approximation to the kernel Π _q,α from below and from above by trigonometric polynomials of degree not exceeding a given number are found. The corresponding polynomials of the best one-sided approximation are obtained.