Semivariation in LpL^p-spaces

summary:Suppose that $X$ and $Y$ are Banach spaces and that the Banach space $X\hat\otimes_\tau Y$ is their complete tensor product with respect to some tensor product topology $\tau$. A uniformly bounded $X$-valued function need not be integrable in $X\hat\otimes_\tau Y$ with respect to a $Y$-valued measure, unless, say, $X$ and $Y$ are Hilbert spaces and $\tau$ is the Hilbert space tensor product topology, in which case Grothendieck's theorem may be applied. In this paper, we take an index $1 \le p < \infty$ and suppose that $X$ and $Y$ are $L^p$-spaces with $\tau_p$ the associated $L^p$-tensor product topology. An application of Orlicz's lemma shows that not all uniformly bounded $X$-valued functions are integrable in $X\hat\otimes_{\tau_p} Y$ with respect to a $Y$-valued measure in the case $1\le p < 2$. For $2 < p <\infty$, the negative result is equivalent to the fact that not all continuous linear maps from $\ell^1$ to $\ell^p$ are $p$-summing, which follows from a result of S. Kwapien

ELECTRONIC STRUCTURE OF FINITE-LENGTH CARBON NANOTUBES: CROSSOVER FROM FULLERENES TO NANOTUBES

Finite-size effects in armchair (n,n) carbon nanotubes are studied as a function of the tube length by using the generalized tight-binding calculations. End structures of the tubes considered here are caps formed from hemispherical pieces of Ih fullerenes and hydrogen-free open ends. It has been clarified that the dimensionality in electronic structures of the finite-length nanotubes only depends on an aspect ratio of the tube diameter to the length of the cylindrical region. The aspect ratio where the finite-length tubes exhibit one-dimensional properties is found to be about four. The results corroborate that the nanotubes with their length of 10–100 nm experimentally observed could be regarded as one-dimensional electron systems

Energetics and electronic structure of semiconducting nanotubes adsorbed on SiO2 surfaces

First-principle calculations are performed to explore the energetics and electronic structures of semiconducting nanotubes adsorbed on the (0 0 0 1) surfaces of α-quartz (SiO2). We find that the absorption energy of nanotube on SiO2 is View the MathML source, strongly depends on the adsorption site and the inter-unit spacing of the nanotube and the SiO2. The electronic structure of this hybrid structure is semiconducting so that the electronic states near the energy gap are caused by those of the nanotubes. A detailed analysis of the electron states of the nanotubes reveals that the energy gaps between valence and conduction bands are slightly modulated by adsorption

Sodium enrichment on glass surface during heating of heavy-metal-containing glasses under a reductive atmosphere

AbstractSodium enrichment occurs on the surface of the glass that is in contact with the gas phase during the heat treatment of lead-containing glass under a reductive atmosphere. This technique was previously found to promote lead recovery in waste-glass treatment, and may be potentially applied to glasses containing other heavy metal oxides. Thus, the efficiencies of sodium enrichment were compared among glasses with different heavy metal species (PbO, CuO, and ZnO) in the heat-treatment under a CO-containing atmosphere. The sodium enrichment efficiencies in the treatment of the PbO- and CuO-containing glasses were higher than that in the treatment of the ZnO containing glass. This was because the efficiencies were related to the reduction of the heavy metal oxides. The mass ratio of Na to Si on the glass surface increased as the PbO concentration decreased via reduction of PbO. The sodium-rich phase was separated together with a copper-concentrated phase that was generated via the reduction of CuO. However, ZnO in the glass is thermodynamically more difficult to reduce in the CO-containing atmosphere used in the study, resulting in the lower efficiency of the sodium enrichment

Lp-valued measures without finite X-semivariation for 2 < p < ∞

We show that for 1 ≤ p < ∞, the property that every Lp-valued vector measure has finite X-semivariation in Lp(μ, X) is equivalent to the property that every continuous linear map from 1 to X is p-summing. For 2 < p < ∞, we explicitly construct an Lp([0, 1])-valued measure without finite Lp-semivariation.Generalitat ValencianaMinisterio de Educación y CienciaUniversidad Politécnica de ValenciaCentre for Mathematics and its Applications at the Australian National Universit

Energetics and electronic structure of nanoscale rotors consisting of triptycene and hydrocarbon molecules

Using the density functional theory with generalized gradient approximation, we studied the energetics and electronic structures of nanoscale rotors consisting of tryptycene and hydrocarbon molecules with respect to their mutual orientation. Energy barriers for the rotational motion of an attached hydrocarbon molecule range from 40 to 200 meV, depending on the attached molecular species and arrangements. The electronic structure of the nanoscale molecular rotors does not depend on the rotational angle of the attached hydrocarbon molecules

Energetics and electronic structures of polymeric all-benzene hollow-cages and planar networks

We studied the energetics and electronic structure of polymerized benzene hollow-cages and sheet using the density functional theory with the generalized gradient approximation. The energetics and electronic structure of the polymeric benzene cages and sheet depend on their size and dimensionality. Because of the symmetric network topology as well as the constituent benzene units, the cages possess highly bunched states around the Fermi level. The energy gap between the highest occupied and the lowest unoccupied states of the cages is approximately proportional to their curvature, owing to the decrease of the strain. The polymerized benzene sheet is a direct gap semiconductor with the gap of 2.4 eV between the less dispersive states of the highest branch of the valence and the lowest branch of the conduction bands