Asymptotic analysis of the p-Laplacian flow in an exterior domain

We consider the Dirichlet problem for the p  -Laplacian evolution equation, ut=Δpu$u_t={\mathrm{\Delta }}_{p}u$, where p>2$p>2$, posed in an exterior domain in RN${\mathbf{R}}^N$, with zero Dirichlet boundary condition and with integrable and nonnegative initial data. We are interested in describing the influence of the holes of the domain on the large time behaviour of the solutions. Such behaviour varies depending on the relative values of N and p  . We must distinguish between the behaviour near infinity of space (outer analysis), and near the holes (inner analysis). We obtain that the outer analysis is given in all cases by certain self-similar solutions and the inner analysis is given by quasi-stationary states. Logarithmic corrections to exact self-similarity appear in the critical case N=p$N=p$, which is mathematically more interesting. In this first paper we treat only the cases N>p$N>p$ and N=p$N=p$, the case N<p$N<p$ will be considered in a companion work.     

Tree Quantum Field Theory

We propose a new formalism for quantum field theory (QFT) which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. This formalism is constructive, i.e., it computes correlation functions through convergent rather than divergent expansions. It applies both to Fermionic and Bosonic theories. It is compatible with the renormalization group, and it allows to define non-perturbatively differential renormalization group equations. It accommodates any general stable polynomial Lagrangian. It can equally well treat noncommutative models or matrix models such as the Grosse–Wulkenhaar model. Perhaps most importantly it removes the space-time background from its central place in QFT, paving the way for a non-perturbative definition of field theory in non-integer dimension.     

A generalization of the charge equilibration method for nonmetallic materials

Assigning effective atomic charges that properly reproduce the electrostatic fields of molecules is a crucial step in the construction of accurate interatomic potentials. We propose a new approach to calculate these charges, which as previous approaches are, is based on the idea of charge equilibration. However, we only allow charge to flow between covalently bonded neighbors by using the concept of so-called split charges. The semiempirical fit parameters in our approach do not only reflect atomic properties (electronegativity and atomic hardness) but also bond-dependent properties. The new method contains two popular but hitherto disjunct approaches as limiting cases. We apply our methodology to a set of molecules containing the elements silicon, carbon, oxygen, and hydrogen. Effective charges derived from electrostatic potential surfaces can be predicted more than twice as accurately as with previous works, at the expense of one additional fit parameter per bond type controlling the polarizability between two bonded atoms. Additional bond-type parameters can be introduced, but barely improve the results. An increase in accuracy of only 30% over existing techniques is achieved when predicting Mulliken charges. However, this could be improved with additional bond-type parameters.     

飞秒激光光刻波导偏振器

飞秒激光诱导折射率变化提供了一种灵活的三维光子器件制作手段.飞秒激光光刻的II类波导具有偏振导光特性,可以作为波导偏振器,但是对于要保留的偏振分量损耗太大.本文阐述了一种利用飞秒激光在熔融石英中制作的新型低损耗波导偏振器.它由中间的一根I类波导及两侧的两根II类纳米光栅轨迹构成.基于飞秒激光诱导的纳米光栅的偏振依赖散射特性,II类纳米光栅轨迹能够对I类波导的倏逝场进行调制.偏振方向垂直于纳米光栅的模式相对于偏振方向平行于纳米光栅的模式有更大的散射损耗,因此导通的是偏振方向平行于纳米光栅的模式.研究了消光比随I类波导与II类纳米光栅轨迹之间的间距的变化关系,选择一个最佳间距来进一步研究消光比随II类纳米光栅轨迹长度的变化关系.在间距6 μm,II类纳米光栅轨迹扫描长度6 mm处实现了最大15.91 dB的消光比.通过增加II类轨迹的长度或者数量,很容易得到更高的消光比.     

The effect of copper ions addition on structural and optical properties of zinc borate glasses

Glasses in the ternary system xCuO?(100 ? x)[55B₂O₃·45ZnO] (0 ≤ x ≤ 20 mol%) have been prepared by melting at 1200 °C and rapidly cooling at room temperature. The effect of copper ions addition in 55B₂O₃·45ZnO glass matrix together with the matrix effect on paramagentic behavior has been investigated using X-ray diffraction (XRD), Fourier transform infrared (FTIR) spectroscopy, differential thermal analysis (DTA), electron paramagnetic resonance (EPR), ultraviolet–visible (UV–VIS) spectroscopy and density measurements. The increase of the number of non-bridging oxygen (NBO) atoms as a function of CuO content in these glasses leads to the decrease of glass polymerization which reduces the stability of the glasses and favors the association of copper ions in clusters. This leads to the major changes of structural and optical properties of the studied glasses as can be seen from the data obtained by FTIR and EPR spectroscopies.     

XRD and EPR structural investigation of some zinc borate glasses doped with iron ions

Glasses in the system xFe₂O₃·(100?x) [45ZnO·55B₂O₃] (0≤x≤10 mol%) have been prepared by melting at 1200 °C and rapidly cooling at room temperature. The obtained samples were submitted to an additional thermal treatment at 570 °C for 12 h in order to relax the glass structure as well as to improve the local order. The as cast and heat treated samples were investigated using X-ray diffraction (XRD) and electron paramagnetic resonance (EPR) measurements. The XRD patterns of all the studied samples show their vitreous nature. Structural modifications occurring in the heat treated samples compared to the untreated ones have been pointed out. EPR spectra of untreated and heat treated samples revealed resonance absorptions centered at g≈2.0, g≈4.3 and g≈6.4. The compositional variation of the line intensity and linewidth of the absorptions from g≈4.3 and g≈2.0 have been interpreted in terms of the variation in the concentration of the Fe³⁺ ions and the interaction between the iron ions. The EPR spectra of the untreated samples containing 5 mol% Fe₂O₃ have been studied at different temperatures (110–290 K). The line intensity of the resonance signals decreases with increase in temperature whereas the linewidth is found to be independent of temperature. It was also found that the temperature variation of reciprocal line intensity obeys the Boltzmann law.     

Mechanistic aspects of hydrosilylation catalyzed by (ArN=)Mo(H)(Cl)(PMe3)3

The reaction of (ArN=)MoCl(2)(PMe(3))(3) (Ar = 2,6-diisopropylphenyl) with L-Selectride gives the hydrido-chloride complex (ArN=)Mo(H)(Cl)(PMe(3))(3) (2). Complex 2 was found to catalyze the hydrosilylation of carbonyls and nitriles as well as the dehydrogenative silylation of alcohols and water. Compound 2 does not show any productive reaction with PhSiH(3); however, a slow H/D exchange and formation of (ArN=)Mo(D)(Cl)(PMe(3))(3) (2(D)) was observed upon addition of PhSiD(3). Reactivity of 2 toward organic substrates was studied. Stoichiometric reactions of 2 with benzaldehyde and cyclohexanone start with dissociation of the trans-to-hydride PMe(3) ligand followed by coordination and insertion of carbonyls into the Mo-H bond to form alkoxy derivatives (ArN=)Mo(Cl)(OR)(PMe(2))L(2) (3: R = OCH(2)Ph, L(2) = 2 PMe(3); 5: R = OCH(2)Ph, L(2) = η(2)-PhC(O)H; 6: R = OCy, L(2) = 2 PMe(3)). The latter species reacts with PhSiH(3) to furnish the corresponding silyl ethers and to recover the hydride 2. An analogous mechanism was suggested for the dehydrogenative ethanolysis with PhSiH(3), with the key intermediate being the ethoxy complex (ArN=)Mo(Cl)(OEt)(PMe(3))(3) (7). In the case of hydrosilylation of acetophenone, a D-labeling experiment, i.e., a reaction of 2 with acetophenone and PhSiD(3) in the 1:1:1 ratio, suggests an alternative mechanism that does not involve the intermediacy of an alkoxy complex. In this particular case, the reaction presumably proceeds via Lewis acid catalysis. Similar to the case of benzaldehyde, treatment of 2 with styrene gives trans-(ArN=)Mo(H)(η(2)-CH(2)═CHPh)(PMe(3))(2) (8). Complex 8 slowly decomposes via the release of ethylbenzene, indicating only a slow insertion of styrene ligand into the Mo-H bond of 8.     

An unexpected mechanism of hydrosilylation by a silyl hydride complex of molybdenum

Carbonyl hydrosilylation catalyzed by (ArN)Mo(H)(SiH(2)Ph)(PMe(3))(3) (3) is unusual in that it does not involve the expected Si-O elimination from intermediate (ArN)Mo(SiH(2)Ph)(O(i)Pr)(PMe(3))(2) (7). Instead, 7 reversibly transfers β-CH hydrogen from the alkoxide ligand to metal.     

Propagators for Noncommutative Field Theories

In this paper we provide exact expressions for propagators of noncommutative Bosonic or Fermionic field theories after adding terms of the Grosse-Wulkenhaar type to the usual free action in order to ensure Langmann–Szabo covariance. We emphasize the new Fermionic case and we give in particular all necessary bounds for the multiscale analysis and renormalization of the noncommutative Gross-Neveu model. Communicated by Raimar Wulkenhaar Dedicated to the memory of Daniel Arnaudon Submitted: December 6, 2005; Accepted: March 21, 2006