The Poisson transform and representations of a free group

Let G be a free group with r generators, 1 < r < ∞. All the eigenfunctions of an operator on G which plays the same role of the Laplace Beltrami operator on semisimple Lie groups are characterized. Furthermore, an analytic family of representations π_z of G on functions on the boundary Ω is considered, defined by $\text{π}{}_{\mathrm{z}}\text{(x)?(ω) = p}{}^{\mathrm{z}}\text{(x, ω)?(x}{}^{\mathrm{?1}}\text{ω)}$, where p(x, ω) is the Poisson kernel relative to the action of G on Ω. It is proved that, for 0 < s = Re z < 1, π_z is uniformly bounded on an appropriate Hilbert space $\text{H}{}_{\mathrm{s}}\text{(Ω)}$. Finally the uniform boundedness of other special representations of G, obtained by considering the free group either as a subgroup of the group of all isometries of a tree or as a subgroup of GL(2, Q_p) is proved.     

Chiral acetylenes as synthetic intermediates—II: Optically active 1,4-dialkylbuta-1,3-dienes: synthesis and conformational studies

The synthesis and the physical properties of some optically active 1,4-dialkylbuta-1,3-dienes are reported. The synthetic processes employed were essentially stereospecific. NMR, UV and CD spectra of these compounds provide information on the main conformation of the conjugated diene system, which was shown to be s-trans planar.     

Refinement of the crystal and molecular structure of 3,5-diamino-1,2,4-dithiazolium bromide. Comparative investigation on the thiouret hydrohalides

The crystal and molecular structure of 3,5-diamino-1,2,4-dithiazolium bromide has been refined from three-dimensional counter data (856 reflections) by least-squares methods, anisotropically for the nonhydrogen and isotropically for the hydrogen atoms, to a finalR=0.036:a=9.948(1),b=12.731(1),c=5.098(1) Å, B=98.70(1)° at 18°C,Z=4,P2₁/n. The S-S (2.075 Å) distance is intermediate between those of the thiouret hydrochloride and hydroiodide. The C-N(terminal) and C-N(bridging) distances are close to their average value (1.322 Å) within their standard deviations. Thev(NH₂) and (NH₂) frequencies of the thiouret hydrohalides are close to those of dithiobiuret while thev(NH) and (NH) bands are, as expected, absent. Thev(CN) andv(CS) bands are shifted to higher and lower frequencies, respectively, while a newv(SS) broad band is observed in the 430–410 cm^–1 region.     

Molecular MNN auger spectra of gaseous tetramethyltin and hexamethylditin

Photoelectron and Auger (MNN) spectra of tetramethyltin and hexamethylditin have been measured and the results compared to those of metallic tin. Energies and intensities of the Auger lines are in agreement with theory based on an Intermediate Coupling scheme since LS coupling theory was inadequate. Relaxation contributions have been evaluated.     

Slow and fast quarks in nucleons

Using light-front dynamics, and the notion of the running effective quark mass, we propose a model of the three-quark nucleon wave function, which accounts simultaneously both for the low and the high energy experimental data. Our wave function is a product of a spinor part, inferred from the QCD sum rules, and a gaussian scalar factor, taken for simplicity. Including all three Ioffe currents we get from one wave function both the negative neutron charge radius 〈r²〉_neutron=-0.108 fm² and the decreasing d/u ratio in the proton for x_Bj increasing from 0.5 to 1.     

Fourier transform infrared investigations of phosphate-silicate coatings on iron

The influence of silica addition and of the annealing on the structure and properties of phosphate coatings are investigated. It has been observed that silica stabilizes phosphates and reduces the transport of iron towards the surface.     

On factorization structures and epidense hulls

Characterizations of epidense subcategories of topological categories and of existence of epidense hulls have been described in [2, 3, 4]. In this paper a similar characterization is given in a much more general setting; for example the category need not have products. The relationship between finite factorization structures and existence of epidense hulls is investigated. It is found to be analogous to the relationship between general factorization structures and epireflective hulls.     

The crystal and molecular structure of chlorobis(N-ethyl-1 3-imidazolidine-2-thione)copper(I)

Summary The crystal structure of the title compound has been determined from x-ray diffractometer data by the heavy-atom method and refined anisotropically by least-squares calculations. Crystals are monoclinic, space groupP 2₁/c, with unit cell dimensions:a=7.321(1),b=14.622(2),c=14.827(2) Å,=92.95(2), Z=4. The finalR index is 4.6%. The copper coordination is trigonal, involving the sulphur atoms of twoN-ethyl-1,3-imidazolidine-2-thione molecules and one chlorine atom. The structure is held together by two intramolecular N-HCl hydrogen bonds and by normal van der Waals interactions.