Phase Transitions on Markovian Bipartite Graphs—an Application of the Zero-range Process

We analyze the existence and the size of the giant component in the stationary state of a Markovian model for bipartite multigraphs, in which the movement of the edge ends on one set of vertices of the bipartite graph is a zero-range process, the degrees being static on the other set. The analysis is based on approximations by independent variables and on the results of Molloy and Reed for graphs with prescribed degree sequences. The possible types of phase diagrams are identified by studying the behavior below the zero-range condensation point. As a specific example, we consider the so-called Evans interaction. In particular, we examine the values of a critical exponent, describing the growth of the giant component as the value of the dilution parameter controlling the connectivity is increased above the critical threshold. Rigorous analysis spans a large portion of the parameter space of the model exactly at the point of zero-range condensation. These results, supplemented with conjectures supported by Monte Carlo simulations, suggest that the phenomenological Landau theory for percolation on graphs is not broken by the fluctuations.     

Asymptotic paths for subsolutions of quasilinear elliptic equations

Letu be an entire lower semicontinuous subsolution to the quasilinear elliptic equation divA(x,u)=0 in ⁿ . It is shown that ifu is not bounded above, then there exists a path going to infinity along whichu tends to infinity. The result extends works of Talpur, Fuglede, and others. Growth aspects of subsolutions are also studied.     

A general route to 3-unsubstituted 1,5-diaryl-2,4-pentanediones and -4-Methoxy-2-pentanones

2-Isoxazolines are converted to corresponding 3-hydroxyketones which are used as precursors to produce new 3-unsubstituted 1,5-diaryl-2,4-pentanediones and -4-methoxy-2-pentanones.     

Thermoluminescence glow curves of CaF2: Dy crystals irradiated by soft X-rays

Applying a deconvolution of the thermoluminescence glow curves, parameters of single glow peaks of CaF₂: Dy TLD 200 dosemeters irradiated by soft X-rays were determined. A dependence of the height ratio of low temperature (T393, 413 and 473 K) single peaks on energy of absorbed photons was measured in a region of 1–22.2 keV. Standard radionuclides¹⁰⁹Cd,²³⁸Pu,⁵⁵Fe and iodine laser produced aluminium plasma (T _e 500 eV) were used as soft X-ray sources. The ratios of the heights of different single peaks are discussed with respect to high local doses. The decreasing ratio of the heights of the first and third and/or second and third peak with increasing photon energy allows to determine reversely a mean photon energy of absorbed soft X-ray radiation.     

Density functional calculations of electronic g-tensors for semiquinone radical anions. The role of hydrogen bonding and substituent effects

A recently developed density functional approach has been used to carry out a systematic computational study of electronic g-tensors for a series of 1,4-semiquinone radical anions. Good agreement with high-field EPR data in frozen 2-propanol is achieved only after taking into account the significant reduction of g-tensor anisotropy caused by hydrogen bonding to solvent molecules. The comparison of various model systems for the first solvation shell suggests two hydrogen bonds from 2-propanol molecules to each of the carbonyl groups of the radical anions, and one additional hydrogen bond to each of the methoxy groups in ubiquinone systems. 2-Propanol makes stronger hydrogen bonds than water and thus influences g-tensor anisotropy more strongly. Substituent effects at the semiquinone are reproduced quantitatively by the calculations. The g-tensor anisotropy is influenced significantly by the conformations of methyl and methoxy substituents, with opposite contributions. Analyses and interpretations of the interrelations between structure, bonding, and spectroscopic data are provided. The relevance of the computational results for the EPR spectroscopy of semiquinone radical anions in photosynthetic reaction centers is discussed.     

Effect of molecular size on the parity-non-conserving contributions to the nuclear magnetic resonance shielding constant

We present density-functional theory studies on the effects of molecular size on the parity-violating contribution to the nuclear magnetic shielding constant. We focus on models with different backbone and side chain lengths, as well as the details of geometry optimization for certain helical polysilylenes and investigate the parity-violating contribution to the shielding constant of the ²⁹Si nucleus of the backbone. Our calculations show that the molecular geometry has a large influence on the magnitude of the parity-violating shielding contribution, a result that is in line with the previous studies on much smaller molecules. In addition, we find convergence in the magnitude of the PV effect with respect to system size, when using geometries that preserve the helical Si backbone structure optimized for the largest of the present systems. This can be interpreted in terms of the non-size-extensive nature of the parity-violating operator influencing the leading-order effect on nuclear magnetic shielding, as opposed to the size-extensive interaction affecting the energy difference between enantiomers. Our molecules are truncated models of large polysilylene systems, for which a difference in the ²⁹Si chemical shift between enantiomers has been observed to be 0.06 ppm (Fujiki in Macromol Rapid Commun 22, 669–674, 2001). As expected based on earlier first principles studies of small molecules, we do not find support for the difference to be of the parity-violating origin. Instead, the predicted parity-violation-induced splitting of the ²⁹Si resonance is found to converge at values around 10^?8 ppm with increasingly large Si backbone.     

Sobolev Spaces with Zero Boundary Values on Metric Spaces

We generalize the definition of the first order Sobolev spaces with zero boundary values to an arbitrary metric space endowed with a Borel regular measure. We show that many classical results extend to the metric setting. These include completeness, lattice properties and removable sets.