Search results for "Lip"
showing 10 items of 8306 documents
Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups
2020
This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $\mathbb{H}^n$, $n\in \mathbb{N}$. For $1\leq k\leq n$, we show that every intrinsic $L$-Lipschitz graph over a subset of a $k$-dimensional horizontal subgroup $\mathbb{V}$ of $\mathbb{H}^n$ can be extended to an intrinsic $L'$-Lipschitz graph over the entire subgroup $\mathbb{V}$, where $L'$ depends only on $L$, $k$, and $n$. We further prove that $1$-dimensional intrinsic $1$-Lipschitz graphs in $\mathbb{H}^n$, $n\in \mathbb{N}$, admit corona decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that…
Elliptic equations and maps of bounded length distortion
1988
On considere l'equation elliptique d'ordre 2: L(u)=Σ i,f=1 n ∂ 1 (a ij ∂ ju )=0 ou les coefficients a ij sont des fonctions C 1 dans un domaine D de R n
Vertical versus horizontal Sobolev spaces
2020
Let $\alpha \geq 0$, $1 < p < \infty$, and let $\mathbb{H}^{n}$ be the Heisenberg group. Folland in 1975 showed that if $f \colon \mathbb{H}^{n} \to \mathbb{R}$ is a function in the horizontal Sobolev space $S^{p}_{2\alpha}(\mathbb{H}^{n})$, then $\varphi f$ belongs to the Euclidean Sobolev space $S^{p}_{\alpha}(\mathbb{R}^{2n + 1})$ for any test function $\varphi$. In short, $S^{p}_{2\alpha}(\mathbb{H}^{n}) \subset S^{p}_{\alpha,\mathrm{loc}}(\mathbb{R}^{2n + 1})$. We show that the localisation can be omitted if one only cares for Sobolev regularity in the vertical direction: the horizontal Sobolev space $S_{2\alpha}^{p}(\mathbb{H}^{n})$ is continuously contained in the vertical Sobolev sp…
Amorphous ultra-wide bandgap ZnOx thin films deposited at cryogenic temperatures
2020
Crystalline wurtzite zinc oxide (w-ZnO) can be used as a wide band gap semiconductor for light emitting devices and for transparent or high temperature electronics. The use of amorphous zinc oxide (a-ZnO) can be an advantage in these applications. In this paper we report on X-ray amorphous a-ZnOx thin films (~500 nm) deposited at cryogenic temperatures by reactive magnetron sputtering. The substrates were cooled by a nitrogen flow through the copper substrate holder during the deposition. The films were characterized by X-ray diffraction (XRD), Raman, infrared, UV-Vis-NIR spectroscopies, and ellipsometry. The a-ZnOx films on glass and Ti substrates were obtained at the substrate holder temp…
Correlative study of structural and optical properties of ZnSe under severe plastic deformation
2019
The effect of plastic deformation on the optical and structural properties of ZnSe crystals has been investigated. The optical properties have been monitored by cathodoluminescence measurements as a function of the deformation degree. Remarkable differences in the defect-related emissions from the most severely deformed areas have been encountered. Deformation of the crystal lattice of ZnSe, associated with slip phenomena, has been studied by means of Electron Backscattered Diffraction and micro-Raman spectroscopy. The relation between the deformation and the optical properties of the ZnSe crystals has been described.
Band gap of corundumlike α−Ga2O3 determined by absorption and ellipsometry
2017
The electronic structure near the band gap of the corundumlike $\ensuremath{\alpha}$ phase of ${\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ has been investigated by means of optical absorption and spectroscopic ellipsometry measurements in the ultraviolet (UV) range (400--190 nm). The absorption coefficient in the UV region and the imaginary part of the dielectric function exhibit two prominent absorption thresholds with wide but well-defined structures at 5.6 and 6.3 eV which have been ascribed to allowed direct transitions from crystal-field split valence bands to the conduction band. Excitonic effects with large Gaussian broadening are taken into account through the Elliott-Toyozawa model, which y…
High spatial resolution strain measurements at the surface of duplex stainless steels
2007
International audience; The determination of local strain fields at the surface of materials is of major importance for understanding their reactivity. In the present paper, lithography is used to fabricate grid points at the microscale and to map strain gradients within grains and between grains. This method was applied to duplex stainless steels which exhibit heterogeneous strain distributions under straining conditions. The influence of various parameters (the specimen microstructure, the density of slip bands, the number of systems activated and the grid geometry) on the strain value was discussed.
Determination of refractive index of submicron-thick films using resonance shift in a four-layer slab waveguide
2017
The measurement of refractive index of very thin films at the order of ten to hundred nanometers is cumbersome and usually requires employing sophisticated techniques such as the spectral ellipsometry. In this paper we describe a simple contact method for measuring the refractive index of thin films. Here we have used the prism-coupling technique for characterizing samples prepared as four-layer slab waveguides. The waveguide resonance condition can be calculated by solving simple analytic transcendental equations for the slab waveguide. Then the captured mode position as a function of cladding thickness is used for probing the refractive index of cladding layer. We used indium-tin-oxide la…
Three X-ray Flares Near Primary Eclipse of the RS CVn Binary XY UMa
2016
We report on an archival X-ray observation of the eclipsing RS CVn binary XY UMa ($\rm P_{orb}\approx$ 0.48d). In two $\emph{Chandra}$ ACIS observations spanning 200 ks and almost five orbital periods, three flares occurred. We find no evidence for eclipses in the X-ray flux. The flares took place around times of primary eclipse, with one flare occurring shortly ($<0.125\rm P_{orb}$) after a primary eclipse, and the other two happening shortly ($<0.05\rm P_{orb}$) before a primary eclipse. Two flares occurred within roughly one orbital period ($\Delta \phi\approx1.024\rm P_{orb}$) of each other. We analyze the light curve and spectra of the system, and investigate coronal length scales both…
Strain rate dependence for evolution of steady state grain sizes: Insights from high-strain experiments on ice
2019
Abstract Understanding of the microstructural evolution and equilibrium grain size development during steady state tertiary flow is essential in order to improve our knowledge of ice and rock deformation. This contribution presents results from in situ transmitted light deformation experiments of natural glacier ice, with the development of the microstructure in a tertiary flow regime. We conducted one relative slower ( 1 × 10 − 6 1/s) and two relative faster-strain rate ( 2 × 10 − 6 1/s) pure shear experiments at −10 °C, up to a shortening of ∼57%. Microstructure development was followed by time-lapse observations, and two new microstructure-based indicators, the ‘seeding rate’ and the ‘mi…