On the spectral properties of Witten-Laplacians, their range projections and Brascamp-Lieb's inequality

A study is made of a recent integral identity of B. Helffer and J. Sjöstrand, which for a not yet fully determined class of probability measures yields a formula for the covariance of two functions (of a stochastic variable); in comparison with the Brascamp-Lieb inequality, this formula is a more flexible and in some contexts stronger means for the analysis of correlation asymptotics in statistical mechanics. Using a fine version of the Closed Range Theorem, the identity's validity is shown to be equivalent to some explicitly given spectral properties of Witten-Laplacians on Euclidean space, and the formula is moreover deduced from the obtained abstract expression for the range projection. As a corollary, a generalised version of Brascamp-Lieb's inequality is obtained. For a certain class of measures occuring in statistical mechanics, explicit criteria for the Witten-Laplacians are found from the Persson-Agmon formula, from compactness of embeddings and from the Weyl calculus, which give results for closed range, strict positivity, essential self-adjointness and domain characterisations.Supported by TMR grant FMRX-CT960001 of the European Commision, PDE and QM at Université de Paris-Sud, France; partly by the Danish Natural Sciences Research Council, grant 9700987.     

Realizations for Schur upper triangular operators centered at an arbitrary point

Reproducing kernel spaces introduced by L. de Branges and J. Rovnyak provide isometric, coisometric and unitary realizations for Schur functions, i.e. for matrix-valued functions analytic and contractive in the open unit disk. In our previous paper [12] we showed that similar realizations exist in the nonstationary setting, i.e. when one considers upper triangular contractions (which appear in time-variant system theory as transfer functions of dissipative systems) rather than Schur functions and diagonal operators rather than complex numbers. We considered in [12] realizations centered at the origin. In the present paper we study realizations of a more general kind, centered at an arbitrary diagonal operator. Analogous realizations (centered at a point of the open unit disk) for Schur functions were introduced and studied in [3] and [4].     

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations

With “hat” denoting the Banach envelope (of a quasi-Banach space) we prove that if 0<p<1, 0<q<1, ℝ, while if 0<p<1, 1≤q<+∞, ∝, and if 1≤p<+∞, 0<q<1, ℝ. Applications to questions regarding the global interior regularity of solutions to Poisson type problems for the three-dimensional Lamé system in Lipschitz domains are presented.     

An end point estimate for maximal commutators

In this paper we prove that the maximal commutator of singular integral operator [b, T]^* satisfies the inequality:

Kähler structures on complex torus

Consider the complex torus T _C under the natural action of the compact real torus T. In this paper, we study T-invariant Kähler structures ω on T_C. For each ω, we consider the corresponding line bundleL on T _C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ? whose curvature is ω. We construct a canonical T-invariant L ²-structure on the sections ofL,and let H _ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H _ω is a unitary T-representation, and we study the multiplicity of the (l-dimensional) irreducible unitary T-representations in H_ω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω.     

Effects of Different Dispersion Methods on the Microscopical Morphology of TiO2 Film

Nanocrystalline porous TiO2 films were prepared on conducting glass supports （ITO） by processed commercial TiO2 nanometre powder （P25）. Three methods of physical dispersing for TiO2 powder, i.e. grinding, magnetic stirring, sonicleaning, were used to disperse TiO2 nanometre powder. Surface morphologies of TiO2 films were observed by optic-microscope and SEM. It is found that the surface morphologies of TiO2 films are determined not only by the dispersing methods but also by the percentage of TiO2 powder in the dispersing system. Different film morphologies can be obtained under the same preparation condition but with different dispersing methods. A lot of cracks exist on the film surface for which the TiO2 slurry is dispersed by grinding. Magnetic stirring leads to some white points and micro-holes on the film surface. Only a few of micro-holes can be observed on the film surface, in which the TiO2 slurry is dispersed by sonicleaning. Different surface morphologies can also be found with different thicknesses of TiO2 films. Different film thicknesses are due to different percentages of TiO2 powder in the slurry. The related mechanism leading to different features of the surface morphologies for the TiO2 films is discussed.     

二次风率对分级进风燃烧室内湍流燃烧与NOx生成的影响

本文在分级进风燃烧室的热态实验装置上,测量了燃烧室内湍流燃烧的温度场和组分浓度场,研究了分级进风的流量比率即二次风率对燃烧及NOx生成的影响.得到了四组不同二次风率下燃烧室内气体温度和O2、CO2、CO与NO浓度的分布.     

On Two Theorems about Symplectic Reflection Algebras

We give a new proof and an improvement of two Theorems of J. Alev, M.A. Farinati, T. Lambre and A.L. Solotar [1] : the first one about Hochschild cohomology spaces of some twisted bimodules of the Weyl Algebra W, and the second one about Hochschild cohomology spaces of the smash product G * W (G a finite subgroup of SP (2n)) and, as a consequence, we then give a new proof of a Theorem of P. Etingof and V. Ginzburg (Invent Math 147:243–348, 2002), which shows that the Symplectic Reflection Algebras are deformations of G * W (and, in fact, all possible ones). Dedicated to my friend J.-C. Cortet.     

Almost hermitian structures with parallel torsion

The characteristic connection of an almost hermitian structure is a hermitian connection with totally skew-symmetric torsion. The case of parallel torsion in dimension 6 is of particular interest. In this work, we give a full classification of the algebraic types of the torsion form, and, on the basis of this, undertake a systematic investigation into the possible geometries. Numerous naturally reductive spaces are constructed and classified, and examples on nilmanifolds given.