The paper is devoted to the compactness of the commutators \(aS_\Gamma - S_\Gamma aI\) and \(W_\alpha S_\Gamma - S_\Gamma W_\alpha\), where \(S_\Gamma\) is the Cauchy singular integral operator, \(a\) is a bounded measurable function, \(W_\alpha\) is the shift operator given by \(W_\alpha f = f\circ\alpha\), and \(\alpha\) is a bi-Lipschitz homeomorphism (shift). The cases of the unit circle and the unit interval are considered. We prove that these commutators are compact on rearrangement-invariant spaces with nontrivial Boyd indices if and only if the function a or, respectively, the derivative of the shift a has vanishing mean oscillation.

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.

Energies of the [( n ,l= n -1),1= n =20] and the [( n ,l= n -2),2= n =20] levels have been calculated for several hydrogenlike kaonic atoms throughout the periodic table, using the current world average kaon mass. Calculations were done in the framework of the Klein–Gordon equation, with finite nuclear size and all-order vacuum polarization corrections.

Ab initio calculations have been performed to study the methyl azidoformate (N3COOCH3) and the ethyl azidoformate (N3COOCH2CH3). Several molecular properties, such as conformational equilibrium, optimal geometry, and vibrational frequencies, have been computed for these molecules. Ionization energies based on Koopman's theorem were also computed.