Orthogonal exponentials, translations, and Bohr completions

We are concerned with an harmonic analysis in Hilbert spaces L²(μ), where μ is a probability measure on . The unifying question is the presence of families of orthogonal (complex) exponentials e_λ(x)=exp(2πiλx) in L²(μ). This question in turn is connected to the existence of a natural embedding of L²(μ) into an L²-space of Bohr almost periodic functions on . In particular we explore when L²(μ) contains an orthogonal basis of e_λ functions, for λ in a suitable discrete subset in ; i.e, when the measure μ is spectral. We give a new characterization of finite spectral sets in terms of the existence of a group of local translation. We also consider measures μ that arise as fixed points (in the sense of Hutchinson) of iterated function systems (IFSs), and we specialize to the case when the function system in the IFS consists of affine and contractive mappings in . We show in this case that if μ is then assumed spectral then its partitions induced by the IFS at hand have zero overlap measured in μ. This solves part of the Łaba–Wang conjecture. As an application of the new non-overlap result, we solve the spectral-pair problem for Bernoulli convolutions advancing in this way a theorem of Ka-Sing Lau. In addition we present a new perspective on spectral measures and orthogonal Fourier exponentials via the Bohr compactification.     

A duality principle for groups

The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for L²(Rd) if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for II₁ factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when G is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well-known problem about the classification of free group von Neumann algebras.     

The quantitative Faber–Krahn inequality for the Robin Laplacian

We prove a quantitative form of the Faber–Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set.     

The incompressible limit of the full Navier–Stokes–Fourier system on domains with rough boundaries

We study the incompressible limit of the full Navier–Stokes–Fourier system on condition that the boundary of the spatial domain oscillates with the amplitude and wave length proportional to the Mach number. Assuming the fluid satisfies the complete slip boundary conditions on the oscillating boundary, we identify the asymptotic limit, and, in particular, establish strong (pointwise) convergence of the velocities towards a solenoidal vector field.     

Quantum beats in the 3γ annihilation decay of Positronium observed by perturbed angular distribution

We have applied conventional Time Differential Perturbed Angular Correlation (TDPAC) method to observe the anisotropy oscillations in the 3γ annihilation decay of polarized Positronium in a weak magnetic field. The effect, as predicted theoretically and experimentally demonstrated by Barishevsky et al. [V.G. Barishevsky, O.N. Metelitsa, V.V. Tikhomirov, Oscillations of the positronium decay γ-quantum angular distribution in a magnetic field, J. Phys. B: At. Mol. Opt. Phys.22 (1989) 2835], is induced by the coherent admixture of the m = 0 states of ortho-Positronium (o-Ps) and para-Positronium (p-Ps) in interaction with the magnetic field.The following experimental characteristics are to be considered:

(i)

the oscillation frequency corresponds to the difference in energy of the Ps atom levels in magnetic field and is proportional with H²;

(ii)

in a fixed geometry the modulation depth (oscillations amplitude) depends on the mean positron polarization;

(iii)

privileged angles of the polarization vector, magnetic field and detectors are required for optimizing the observed oscillations amplitude.

The normalized difference spectrum function (R(t)) obtained from time spectra measured in vacuum and in different gaseous atmospheres (Ar, H₂, N₂) have the oscillations amplitude constant and we conclude that the Ps atoms are not fully thermalized over a time interval of about 400 ns.The R(t) functions obtained for o-Ps annihilation decays, in dry air or Ar-O mixture, have the oscillations amplitude time dependent due, probably, to the paramagnetism of the Oxygen molecules.     

Defects in the p-n junction and current drift in selenium rectifiers

Drift is considered in relation to a defective p-n junction having localized conduction channels.     

Probe studies of silicon p-n junctions

The results of single-probe studies of thick and thin silicon p-n junctions are described. The electric fields in a diode with a p⁺-n junction and in a stabilitron with a p-n junction in the forward and reverse directions are studied. Measurements are made at various currents and temperatures. The p-n junctions are analyzed on the basis of the probe measurements.     

Generalized twist for the torsion of micropolar cylinders

Summary A problem of Truesdell concerning the theory of torsion is studied in the framework of micropolar elasticity.

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Sommario Un problema di Truesdell relativo alla teoria di torsione viene studiato nell'ambito dell'elasticità micropolare.

     

A generalized theory of linear micropolar thermoelasticity

Summary The linear theory of thermoelasticity established by Green and Lindsay with the help of an entropy production inequality proposed by Green and Laws is generalized to the case of a homogeneous micropolar continuum. The basic equations are derived using invariance conditions under superposed rigid body motions.

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Sommario La teoria lineare della termoelasticità stabilita da Green e Lindsay sulla base della disuguaglianza della produzione dell'entropia proposta da Green e Laws è generalizzata al caso di un continuo micropolare omogeneo. Le equazioni fondamentali vengono derivate usando condizioni di invarianza rispetto a moti rigidi sovrapposti.