Liouville theorem for weighted p-Lichnerowicz equation on smooth metric measure space

In this paper, let $(M^n,g,d\mu )$ be n-dimensional noncompact metric measure space which satisfies Poincaré inequality with some Ricci curvature condition. We obtain a Liouville theorem for positive weak solutions to weighted p-Lichnerowicz equation

${△}_{p,f}v+c{v}^{\sigma }=0,$

where $c\le 0,m>n\ge 1,1<p<\frac{m?1+\sqrt{(m?1)(m+3)}}{2},\sigma \le p?1$ are real constants.     

The character and the wave front set correspondence in the stable range

We relate the distribution characters and the wave front sets of unitary representation for real reductive dual pairs of type I in the stable range.     

The behavior of differential forms under purely inseparable extensions

Let F be a field of characteristic 2. In this paper we give a complete computation of the kernel of the homomorphism $H_2^{m+1}(F)?H_2^{m+1}(L)$ induced by scalar extension, where $L/F$ is a purely inseparable extension (of any degree), $H_2^{m+1}(F)$ is the cokernel of the Artin–Schreier operator $?:{\mathrm{\Omega }}_F^m?{\mathrm{\Omega }}_F^m/\mathrm{d}{\mathrm{\Omega }}_F^{m?1}$ given by: $x\frac{\mathrm{d}{x}_1}{x_1}\wedge ?\wedge \frac{\mathrm{d}{x}_m}{x_m}?(x^2?x)\frac{\mathrm{d}{x}_1}{x_1}\wedge ?\wedge \frac{\mathrm{d}{x}_m}{x_m}+\mathrm{d}{\mathrm{\Omega }}_F^{m?1}$, where ${\mathrm{\Omega }}_F^m$ is the space of absolute m-differential forms over F and d is the differential operator. Other related results are included.     

Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

Controlled synthesis of ZnO nanoparticles from a Zn(II) coordination polymer: Structural characterization,optical properties and photocatalytic activity

A zinc coordination polymer derived from pyridine-2,6-dicarboxylate (PDC), {[Zn₂(PDC)₂]}_n, was successfully prepared via conventional, sonication and microwave-irradiation methods. The composition and characteristics of the obtained coordination polymers (CPs) were investigated by elemental analysis, TGA/DTA, X-ray diffraction and spectroscopic techniques. The so obtained CPs were heat-treated in the air at 600 °C for 2 h to produce ZnO of nanosized particles (NPs). It is of interest to note that the synthesis approach of the precursor greatly affects both the nanoparticle size and the structure of the resulting ZnO NPs. Moreover, the smallest particle size was associated with the sample derived from the ultrasonically prepared precursor. TEM analysis revealed that all samples have sphere-like morphologies. Structural analysis of the prepared ZnO samples was conducted and compared using Rietveld analysis of their PXRD patterns. Optical band gap calculations based on analysis of the UV–vis spectra of ZnO samples using Tauc's power law were achieved. The highest band gap of 3.63 eV was observed for ZnO sample obtained from the ultrasonically prepared precursor. Furthermore, the photocatalytic activity of ZnO NPs for the removal of Eosin Y color was monitored. The highest removal efficiency was recorded for ZnO originated from the ultrasonically synthesized precursor. Enhancement of removal efficiency that reached 98% was attained in only a period of 8 min. Its recycling test showed that it can be reused without structural changes over four cycling experiments.     

On universal subsets of Banach spaces

We prove that to most of the known hypercyclic operators A on separable Banach spaces there exist compact (compact convex, compact connected) subsets K of E such that each compact (compact convex, compact connected) subset of E can be approximated with respect to Hausdorff's distance by for suitable . Received July 8, 1997, in final form October 17, 1997     

Congruence criteria for finite subsets of complex projective and complex hyperbolic spaces

We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP ⁿ . For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP ⁿ and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by congruence classes or if distances π/2 are allowed. Finally we do the same kind of investigation also for the complex hyperbolic space ℂH ⁿ . Most of the results are completely analogous, however, there are also some interesting differences. Received: 15 January 1996     

A dual graph-norm refinement indicator for finite volume approximations of the Euler equations

Summary. We generalise and apply a refinement indicator of the type originally designed by Mackenzie, Süli and Warnecke in [15] and [16] for linear Friedrichs systems to the Euler equations of inviscid, compressible fluid flow. The Euler equations are symmetrized by means of entropy variables and locally linearized about a constant state to obtain a symmetric hyperbolic system to which an a posteriori error analysis of the type introduced in [15] can be applied. We discuss the details of the implementation of the refinement indicator into the DLR--Code which is based on a finite volume method of box type on an unstructured grid and present numerical results. Received May 15, 1995 / Revised version received April 17, 1996