We prove necessary conditions for the Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces \(L^{p(\cdot)}_w\) (weighted Lebesgue spaces with variable exponent). Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever \(w\) is a Khvedelidze weight, and the variable exponent \(p(t)\) satisfies the estimate \(|p(\tau)-p(t)|\le A/(-\log|\tau-t|)\).

We prove criteria for the invertibility of the binomial functional operator
\[
A=aI-bW_\alpha
\]
in the Lebesgue spaces \(L^p(0,1)\), \( 1 < p < \infty\), where \(a\) and \(b\) are continuous functions on \((0,1)\), \(I\) is the identity operator, \(W_\alpha\) is the shift operator, \(W_\alpha f=f\circ\alpha\), generated by a non-Carleman shift \(\alpha:[0,1]\to[0,1]\) which has only two fixed points \(0\) and \(1\). We suppose that \(\log\alpha'\) is bounded and continuous on \((0,1)\) and that \(a,b,\alpha'\) slowly oscillate at \(0\) and \(1\). The main difficulty connected with slow oscillation is overcome by using the method of limit operators.

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The relativistic version of the binary-encounter Bethe (BEB) model is used to calculate cross sections for K-shell ionization of atoms by electron impact. The BEB model requires only two atomic constants, the binding energy and kinetic energy of the K electrons. These constants are listed for carbon to antimony. Comparisons with available experimental data on N, O, Na, Al, Cl, Ca, Cu, Se and Sb show good agreement. The K-shell ionization cross sections for C, Mg, P, S, Cr, As and Cd are tabulated.

In this paper we extend results on Fredholmness of singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces \(X(\Gamma)\) with nontrivial Boyd indices \(\alpha_X,\beta_X\) [K98] to the weighted case. Suppose a weight \(w\) belongs to the Muckenhoupt classes \(A_{\frac{1}{\alpha_X}}(\Gamma)\) and \(A_{\frac{1}{\beta_X}}(\Gamma)\). We prove that these conditions guarantee the boundedness of the Cauchy singular integral operator \(S\) in the weighted rearrangement-invariant space \(X(\Gamma,w)\). Under a ``disintegration condition'' we construct a symbol calculus for the Banach algebra generated by singular integral operators with matrix-valued piecewise continuous coefficients and get a formula for the index of an arbitrary operator from this algebra. We give nontrivial examples of spaces, for which this ``disintegration condition'' is satisfied. One of such spaces is a Lebesgue space with a general Muckenhoupt weight over an arbitrary Carleson curve.

Let \(\Gamma\) be an oriented Jordan smooth curve and \(\alpha\) a diffeomorphism of $\Gamma$ onto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertibility of the binomial functional operator \(A=aI-bW\) where $a$ and $b$ are continuous functions, \(I\) is the identity operator, \(W\) is the shift operator, \(Wf=f\circ\alpha\), on a reflexive rearrangement-invariant space \(X(\Gamma)\) with Boyd indices \(\alpha_X,\beta_X\) and Zippin indices \(p_X,q_X\) satisfying inequalities
\[
0<\alpha_X=p_X\le q_X=\beta_X<1.
\]

A propose for this article is to present information on the emergence of the knowledge field of sociology of work and labor studies in Portugal. The period of 1974-1984 produced an interesting bibliography in the fields of social stratification, labor conflicts and social history. However, and since the mid-1980s, significant changes have emerged. There took place an increased theoretical diversity and also there was a major preoccupation with the development of a more professional teaching of the discipline. At this time, the first graduated specialists in Sociology appeared in the labor market since 1980, working in municipalities, industrial firms, public health institutions, schools, or even in research and development (R&D) units. The Portuguese association in this field (APSIOT) organized many scientific meetings, debates with unionists, managers and politicians, beside the regular publication of review “Organizações e Trabalho” (Organization and Work) since 1989. It started the diffusion of a professional culture of sociologists who associate science and their occupation concerning themselves not only with scientific quality, but also with technical competence and social responsibility. At the same time, one could assist to the participation of sociologists in the social and organizational molding of new technologies in order to promote alternative production systems.

We construct a presymbol for the Banach algebra \(\operatorname{alg}(\Omega, S)\) generated by the Cauchy singular integral operator \(S\) and the operators of multiplication by functions in a Banach subalgebra \(\Omega\) of \(L^\infty\). This presymbol is a homomorphism \(\operatorname{alg}(\Omega,S)\to\Omega\oplus\Omega\) whose kernel coincides with the commutator ideal of \(\operatorname{alg}(\Omega,S)\). In terms of the presymbol, necessary conditions for Fredholmness of an operator in \(\operatorname{alg}(\Omega,S)\) are proved. All operators are considered on reflexive rearrangement-invariant spaces with nontrivial Boyd indices over the unit circle.

We prove the inverse closedness of the Banach algebra \(\mathfrak{A}_p\) of functional operators with non-Carleman shifts, which have only two fixed points, in the Banach algebra of all bounded linear operators on \(L^p\). We suppose that \(1 \le p \le \infty\) and the generators of the algebra \(\mathfrak{A}_p\) have essentially bounded data. An invertibility criterion for functional operators in \(\mathfrak{A}_p\) is obtained in terms of
the invertibility of a family of discrete operators on \(l^p\). An effective invertibility criterion is established for binomial difference operators with \(l^\infty\) coefficients on the spaces \(l^p\). Using the reduction to binomial difference operators, we give effective criteria of invertibility for binomial functional operators on the spaces \(L^p\).

Ferroelectric and high dielectric permittivity films are currently being investigated in view of their use as gate dielectrics in MIS structures. Along with the suppression of tunnelling currents at small gate thickness, they provide a memory function to MIS structures, which can be used in non-volatile memory applications. In this work we report fabrication and characterization of novel metal-ferroelectric-amorphous silicon structures. The structures consist of glass/ITO substrates coated with PZT 20/80 films (sol-gel) followed by an active layer (i-a-SiC:H, deposited by plasma enhanced chemical vapor deposition (PECVD)). A strong capacitance hysteresis is observed in C-V curves in electron accumulation region (VG > 0), accompanied with a large increase in the capacitance of ferroelectric-semiconductor structures at low frequencies. Threshold voltage for electron accumulation is about 10 V being dependent on the ferroelectric polarization switching. © 2002 Elsevier Science B.V. All rights reserved.

PbZrxTi1-xO3 (PZT) films are currently being investigated in view of their large switching polarization and piezoelectric coefficients useful for various applications. Besides, PZT films possess large photosensitivity, which, in combination with the above listed properties, can be a base for future microelectronic applications including photostrictive actuators and optical storage devices. In this work, PZT thin films of several compositions (x=0.2 and 0.45) were deposited on Pt-coated Si and ITO/glass substrates via modified sol-gel technique. Microstructures of the films were evaluated using XRD, SEM and AFM. The optical transmission measurements on PZT films deposited on ITO/glass revealed a high transparency over 80% and a band gap of about 3.4 eV. The observed photocurrent exhibited a maximum and was attributed to band-to-band optical transitions.