Sobolev classes of Banach space-valued functions and quasiconformal mappings

We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincaré inequality. J. H. supported by NSF grant DMS9970427. P. K. supported by the Academy of Finland, project 39788. N. S. supported in part by Enterprise Ireland. J. T. T. supported by an NSF Postdoctoral Research Fellowship.     

Triple-bond covalent radii

A system of additive covalent radii is proposed for sigma(2) pi(4) triple bonds involving elements from Be to E 112 (eka-mercury). Borderline cases with weak multiple bonding are included. Only the elements in Group 1, the elements Zn-Hg in Group 12 and Ne in Group 18 are then totally excluded. Gaps are left at late actinides and some lanthanides. The standard deviation for the 324 included data points is 3.2 pm.     

A very short uranium-uranium bond: the predicted metastable U(2)2+

Quantum chemical calculations, based on multiconfigurational wave functions and including relativistic effects, show that the U(2)2+ system has a large number of low-lying electronic states with S of 0 to 2 and Lambda ranging from zero to ten. These states share a very small bond length of about 2.30 A, compared to 2.43 A in neutral U2. The Coulomb explosion to 2 U+ lowers the energy by only 1.6 eV and is separated by a broad barrier.     

HgH4 and HgH6: further candidates for high-valent mercury compounds

Mercury tetrahydride (D4h) is calculated to have similar bond lengths and vibrational frequencies as the already known HgH2 and to lie energetically 200 kJ mol-1 above HgH2 + H2, in a local well, about 40 kJ mol-1 below a transition state.     

Uncertainty and complementarity in axiomatic quantum mechanics

In this work an investigation of the uncertainty principle and the complementarity principle is carried through. A study of the physical content of these principles and their representation in the conventional Hilbert space formulation of quantum mechanics forms a natural starting point for this analysis. Thereafter is presented more general axiomatic framework for quantum mechanics, namely, a probability function formulation of the theory. In this general framework two extra axioms are stated, reflecting the ideas of the uncertainty principle and the complementarity principle, respectively. The quantal features of these axioms are explicated. The sufficiency of the state system guarantees that the observables satisfying the uncertainty principle are unbounded and noncompatible. The complementarity principle implies a non-Boolean proposition structure for the theory. Moreover, nonconstant complementary observables are always noncompatible. The uncertainty principle and the complementarity principle, as formulated in this work, are mutually independent. Some order is thus brought into the confused discussion about the interrelations of these two important principles. A comparison of the present formulations of the uncertainty principle and the complementarity principle with the Jauch formulation of the superposition principle is also given. The mutual independence of the three fundamental principles of the quantum theory is hereby revealed.     

Line width transitions in the deuteron magnetic resonance of polycrystalline ND4Cl,ND4Br,and ND4I

The line width of the deuteron magnetic resonance in polycrystalline ND₄Cl, ND₄Br, and ND₄I has been measured from 300°K down to 115°K. Below 200, 172, and 140°K in ND₄Cl, ND₄Br, and ND₄I, respectively, the line rapidly broadens. In addition, the I ? II transition of ND₄I causes a change of line width. The observed line widths agree with those calculated by the present writers from theVan Vleck second moment formula assuming a Gaussian line shape. For this agreement, in the phases II and III of ND₄Cl and in the phase I of ND₄I the deuteron-halogen interactions should be taken into account, whereas in the phases II and III of ND₄Br and ND₄I they should be omitted.     

On Banaschewski functions in lattices

Presented by Ralph Freese     

Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc

Let be a holomorphic self-map of the unit disc . For every , there is a measure on (sometimes called Aleksandrov measure) defined by the Poisson representation . Its singular part measures in a natural way the ``affinity' of for the boundary value . The affinity for values inside is provided by the Nevanlinna counting function of . We introduce a natural measure-valued refinement of and establish that the measures are obtained as boundary values of the refined Nevanlinna counting function . More precisely, we prove that is the weak limit of whenever converges to non-tangentially outside a small exceptional set . We obtain a sharp estimate for the size of in the sense of capacity.

     

How many hydrogen atoms can be bound to a metal? Predicted MH12 species

Quantum chemical calculations predict the existence of new molecular species with general formula MH12, where M is a group 6 atom. The previous MHn species had n values up to 9. The new systems with n = 12 would be a new record for metal hydrides.