The \bar \partial -Neumann operator and commutators of the Bergman projection and multiplication operators

We prove that compactness of the canonical solution operator to 84_Article_IEq2.gif" alt="$$ \bar \partial $$" align="middle" border="0"> restricted to (0, 1)-forms with holomorphic coefficients is equivalent to compactness of the commutator 84_Fig1_HTML.gif" alt="MediaObjects/10587_2008_84_Fig1_HTML.gif"> defined on the whole L _(0,1)²(Ω), where 84_Article_IEq3.gif" alt="$$ \bar M $$" align="middle" border="0"> is the multiplication by 84_Article_IEq4.gif" alt="$$ \bar z $$" align="middle" border="0"> and 84_Article_IEq5.gif" alt="$$ \bar \partial $$" align="middle" border="0"> is the orthogonal projection of L _(0,1)²(Ω) to the subspace of (0, 1) forms with holomorphic coefficients. Further we derive a formula for the 84_Article_IEq6.gif" alt="$$ \bar z $$" align="middle" border="0">-Neumann operator restricted to (0, 1) forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by z and 84_Article_IEq7.gif" alt="$$ \bar \partial $$" align="middle" border="0">. Partially supported by the FWF grant P19147-N13.     

Crystal structure of bis(2,3,5-trimethylpyridine N-oxide) 2,4,6-trinitrophenolate: Cation complex tied by the extremely strong OHO hydrogen bonding

The salt of bis(2,3,5-trimethylpyridine N-oxide) 2,4,6-trinitrophenolate, C₂₂H₂₅N₅O₉, crystallizes in monoclinic space group C2/c with a=12.411(2)Å, b=15.326(3)Å, c=13.559(3)Å, 84tr7823l62/xxlarge946.gif" alt="beta" align="MIDDLE" BORDER="0">=113.13(1)°, V=2371.8(8)ų, Z=4, D _c=1.410 mg/m³, F(000)=1056. The molecule of 2,3,5-trimethylpyridine N-oxide is protonated. Two pyridine N-oxide molecules are linked by extremely strong OHO hydrogen bonding (2.378(4)Å) to form a cation complex. CHO hydrogen bonding and 84tr7823l62/xxlarge960.gif" alt="pgr" align="BASELINE" BORDER="0">–84tr7823l62/xxlarge960.gif" alt="pgr" align="BASELINE" BORDER="0"> interactions also play important role in stablizing the structure of the salt.     

Molecular dynamics in polyethylene and ethylene-1-butene copolymer investigated by NMR methods

1H and 13C NMR spectra and 1H spin-lattice relaxation times T₁ and T_1ρ have been employed to study the structure and molecular dynamics in polyethylene and ethylene-1-butene copolymer in the temperature range from 100 to 370 K. Results are interpreted in terms of α, β and γ -relaxation, as well as methyl group rotation. The activation energies for all motions were established. The incorporation of 1-butene into ethylene chain leads to an increase of mobility in amorphous and crystalline phases as well as appearance the 13C resonance characteristic to the monoclinic structure in addition to the orthorhombic observed in both polymers. The crystallinity degree derived from T_1ρ in studied polymers is close to that determined using DSC method.     

Structural Characters and Isolated Stability of Phosphorus Polyanions

The optimized geometries at the RHF/6-311 G^** level, the relatively stable energy at the MPW1PW91/6-311 G^** level and the structural characters of anions have been acquired, indicating the stability is related to the chemical bonding of μ2-P atoms and the distribution of negative charges. The configurations of cage units P8^4- and P9^5- are stable due to the less torsion, but their Es values are relatively higher than that of P7^3- with more μ2-P atoms and the isolated stability is lower than that of P7^3-. They potentially play an important role as intermediate in chemical reaction of producing complicated polyphosphides. Based on the related electronic properties, a stable polyanion must have low valence electron concentration, no (μ2-P)-(μ2-P) bond and a little dispersive charge. The earmark IR frequencies of cage units have been assigned to the vibration models in the end.     

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.     

Generalizing inplace multiplicity identities for integer compositions

In a recent paper, the authors gave two new identities for compositions, or ordered partitions, of integers. These identities were based on closely-related integer partition functions which have recently been studied. In the process, we also extensively generalized both of these identities. Since then, we asked whether one could generalize one of these results even further by considering compositions in which certain parts could come from t kinds (rather than just two kinds, which was the crux of the original result). In this paper, we provide such a generalization. A straightforward bijective proof is given and generating functions are provided for each of the types of compositions which arise. We close by briefly mentioning some arithmetic properties satisfied by the functions which count such compositions.