On the electric current from electrodes in a magnetized semiconductor film

The problem on the electric current from electrodes in a magnetizedsemiconductor film is reduced to the skew derivative problemfor the Laplace equation outside cuts in a plane. The problemfor the Laplace equation is studied under different conditionsat infinity, which have a certain physical meaning. With thehelp of potential theory, the skew derivative problem is reducedto a Fredholm integral equation of the second kind, which isuniquely solvable. The Neumann problem for the Laplace equationin the exterior of cuts in a plane is a particular case of ourproblem.     

Continuous singular measures with absolutely continuous convolution squares

We prove for every non-abelian compact connected group there is a continuous, singular, central measure with in for all . We also construct such measures on some families of non-abelian compact totally disconnected groups. These results settle an open question of Ragozin.

     

A new approach to reduction of the Neumann problem in acoustic scattering to a non-hypersingular integral equation

The Neumann problem for the propagative Helmholtz equation inthe exterior of several bodies (obstacles) is studied in twoand three dimensions by a special modification of the boundaryintegral equation method. This modification can be called the'method of interior boundaries', because additional boundariesare introduced inside scattering bodies. The solution of theproblem is obtained in the form of a single layer potentialon the whole boundary. The density in the potential satisfiesthe uniquely solvable Fredholm equation of the second kind andcan be computed by standard codes. In fact our method holdsfor any positive wave numbers.     

Metal-insulator crossover in high T c cuprates: A gauge field approach

A metal-insulator crossover appears in the experimental data for in-plane resistivity of underdoped cuprates and a range of superconducting cuprates in the presence of a strong magnetic field suppressing superconductivity. We propose an explanation for this phenomenon based on a gauge field theory approach to the t-J model. In this approach, based on a formal spin-charge separation, the low energy effective action describes gapful spinons (with a theoretically derived doping dependence of the gap m _s ² ∼ δ| ln δ|) and holons with finite Fermi surface (ɛF ∼ tδ) interacting via a gauge field whose basic effect on the spinons is to bind them into overdamped spin waves, shifting their gap by a damping term linear in T, which causes the metal-insulator crossover. The presence of a magnetic field perpendicular to the plane acts by increasing the damping, in turn producing a big positive transverse in-plane magnetoresistance at low T, as experimentally observed.     

Algebras of invariant functions on the Shilov boundaries of Siegel domains

Let be a bounded symmetric domain and the Shilov boundary of . Let be the Shilov boundary of the Siegel domain realization of . We consider the case when is the exceptional non-tube type domain of the type . We prove that is not a Gelfand pair and thus resolve an open question of G. Carcano.