Homogeneous Cr Research Articles

A characterization of homogeneous three-dimensional CR manifolds

We characterize homogeneous three-dimensional CR manifolds, in particular Rossi spheres, as critical points of a certain energy functional that depends on the Webster curvature and torsion of the pseudohermitian structure.

Homogeneous CR and Para-CR Structures in Dimensions 5 and 3

Homogeneous CR and Para-CR Structures in Dimensions 5 and 3

Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on CR manifolds

We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented pseudohermitian structures \mathcal{P}_+ . We show that the functionals are continuous with respect to a natural topology on \mathcal{P}_+ . Using an adaptation of the standard Kato–Rellich perturbation theory, we prove that the functionals are (one-sided) differentiable along 1-parameter analytic deformations. We use this differentiability to define the notion of critical pseudohermitian structures, in a generalized sense, for them. We give a necessary (also sufficient in some situations) condition for a pseudohermitian structure to be critical. Finally, we present explicit examples of critical pseudohermitian structures on both homogeneous and non-homogeneous CR manifolds.

Homogeneous CR submanifolds of complex hyperbolic spaces

Homogeneous CR submanifolds of complex hyperbolic spaces

On finitely nondegenerate closed homogeneous CR manifolds

A complex flag manifold $$\textsf {F}{=}{{\textbf {G}}}/{{\textbf {Q}}}$$ decomposes into finitely many real orbits under the action of a real form $${{\textbf {G}}}^\upsigma $$ of $${{\textbf {G}}}$$ . Their embedding into $$\textsf {F}$$ defines on them CR manifold structures. We characterize and list all the closed real orbits which are finitely nondegenerate.

Determination of Activation Energy on Hydrogen Evolution Reaction for Nickel-Based Porous Electrodes during Alkaline Electrolysis

The aim of the present work is to evaluate the activation energy (Ea) at different cathodic overpotentials (η) by potentiodynamic tests which were carried out at different temperatures of Ni-based, NiCr-m, and NiCr-p porous electrodes, during the alkaline electrolysis processes. On the other hand, the electrochemical stability of the electrodes was evaluated by cyclic voltammetry after 1000 cycles of operation and by potentiostatic tests after 10 h at −1.5 V vs. SCE. The electrodes were sintered with a heating rate of 25 °C/min up to a temperature of 1000 °C (Ni-based and NiCr-m) and 1200 °C (NiCr-p) for 60 min. The results showed that the Ea value was lower for the Ni-based system at equilibrium; however, the NiCr-p electrode had a better performance due to higher negative apparent Ea values as a function of η (dEa/dη). The cyclic voltammetry tests suggest that the NiCr-p electrode improves its activity by about 71% in its long-term operation in comparison with Ni-based and NiCr-m. A similar behavior was observed in the potentiostatic test which showed a higher cathodic current density associated with a charge transfer process after 10 h. The higher stability of the NiCr-p is attributed to a homogeneous Cr distribution in the nickel matrix.

Maximal Dimension of Groups of Symmetries of Homogeneous 2-nondegenerate CR Structures of Hypersurface Type with a 1-dimensional Levi Kernel

We prove that for every n ≥ 3 the sharp upper bound for the dimension of the symmetry groups of homogeneous, 2-nondegenerate, (2n + 1)-dimensional CR manifolds of hypersurface type with a 1-dimensional Levi kernel is equal to n2 + 7, and simultaneously establish the same result for a more general class of structures characterized by weakening the homogeneity condition. This supports Beloshapka’s conjecture stating that hypersurface models with a maximal finite dimensional group of symmetries for a given dimension of the underlying manifold are Levi nondegenerate.

Cr-incorporated uranium nitride composite fuels with enhanced mechanical performance and oxidation resistance

Cr-incorporated UN (Cr-UN) composite fuels with different Cr amounts up to 10 wt% are fabricated by spark plasma sintering, and their microstructure and phase heterogeneity are analyzed. Highly densified microstructure and homogeneous Cr distribution are identified for the Cr-incorporated UN pellets, demonstrating a liquid sintering characteristic with a Cr enriched phase on the UN grain boundaries. A ternary phase U2CrN3 forms in the Cr-UN matrix with Cr amounts of 5 wt% and 10 wt%. The SPS densified Cr-UN composite pellets display greatly-improved thermal conductivity and simultaneously high hardness and fracture toughness. The fracture toughness of the Cr-incorporated UN pellet is ∼5.5 MPa-m1/2, representing almost 130% enhancement as compared to that of monolithic UN sintered at the same condition. The Cr-incorporated UN pellets also display enhanced oxidation resistance as evidenced by increased onset temperature for oxidation to 450 oC for the 5 wt% Cr-UN. These results highlight that Cr additive and the formation of a ternary phase can be useful to improve the thermal-mechanical properties and oxidation resistance of UN fuels with well-maintained high fissile element density.

Left-invariant CR structures on 3-dimensional Lie groups

The systematic study of CR manifolds originated in two pioneering 1932 papers of Élie Cartan. In the first, Cartan classifies all homogeneous CR 3-manifolds, the most well-known case of which is a one-parameter family of left-invariant CR structures on \(\mathrm {SU}_2= S^3\), deforming the standard ‘spherical’ structure. In this paper, mostly expository, we illustrate and clarify Cartan’s results and methods by providing detailed classification results in modern language for four 3-dimensional Lie groups. In particular, we find that \({\mathrm {SL}_2({\mathbb {R}})}\) admits two one-parameter families of left-invariant CR structures, called the elliptic and hyperbolic families, characterized by the incidence of the contact distribution with the null cone of the Killing metric. Low dimensional complex representations of \({\mathrm {SL}_2({\mathbb {R}})}\) provide CR embedding or immersions of these structures. The same methods apply to all other 3-dimensional Lie groups and are illustrated by descriptions of the left-invariant CR structures for \(\mathrm {SU}_2\), the Heisenberg group, and the Euclidean group.

Higher order Levi forms on homogeneous CR manifolds

We investigate the nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous CR manifolds. Improving previous results, we prove that general orbits of real forms in complex flag manifolds have order less or equal than 3 and the compact ones less or equal 2. Finally we construct by Lie extensions weakly nondegenerate CR vector bundles with arbitrary orders of nondegeneracy.

3-folds CR-embedded in 5-dimensional real hyperquadrics

E. Cartan’s method of moving frames is applied to 3-dimensional manifolds M which are CR-embedded in 5-dimensional real hyperquadrics Q in order to classify M up to CR symmetries of Q given by the action of one of the Lie groups SU(3,1) or SU(2,2). In the latter case, the CR structure of M derives from a shear-free null geodesic congruence on Minkowski spacetime, and the relationship to relativity is discussed. In both cases, we compute which homogeneous CR 3-folds appear in Q.

On transitive contact and CR algebras

We consider locally homogeneous CR manifolds and show that, under a condition only depending on their underlying contact structure, their CR automorphisms form a finite dimensional Lie group

Impact of Al Activators on Structure and Catalytic Performance of Cr Catalysts in Homogeneous Ethylene Oligomerization – A Multitechnique in situ/operando Study

AbstractThe effect of different AlR3 activators (R=methyl, ethyl, isobutyl, n‐octyl) has been studied in comparison to modified methylaluminoxane (MMAO) by operando EPR as well as by in situ UV‐vis, ATR‐IR and XANES/EXAFS spectroscopy during oligomerization of ethylene at 20 bar and 40 °C with a homogeneous Cr complex catalyst formed in situ upon mixing a Cr(acac)3 precursor, a Ph2PN(iPr)PPh2 ligand (PNP) and the activator. Coordination of PNP to Cr(acac)3 is initiated only in the presence of an activator. Highest 1‐octene productivity (detected during operando EPR measurements) was obtained with MMAO which promotes bidentate coordination of the ligand to form an active (PNP)CrII(CH3)2 chelate complex. Rising bulkiness of R in AlR3 leads to only monodentate coordination of PNP to the Cr center by one P atom and increasing reduction to CrI to a maximum extend of around 30 % for AlOct3. This lowers the catalytic performance, which is mainly governed by the mode of PNP coordination rather than by the CrI content.

Mostow’s fibration for canonical embeddings of compact homogeneous $CR$ manifolds

We define a class of compact homogeneous CR manifolds which are bases of Mostow fibrations having total spaces equal to their canonical complex realizations and Hermitian fibers. This is used to establish isomorphisms between their tangential Cauchy–Riemann cohomology groups and the corresponding Dolbeault cohomology groups of the embeddings.

Mostow’s fibration for canonical embeddings of compact homogeneous CR manifolds

We define a class of compact homogeneous CR manifolds which are bases of Mostow fibrations having total spaces equal to their canonical complex realizations and Hermitian fibers. This is used to establish isomorphisms between their tangential Cauchy–Riemann cohomology groups and the corresponding Dolbeault cohomology groups of the embeddings.

Effect of different Zr contents on properties and microstructure of Cu-Cr-Zr alloys

The crystallography and morphology of precipitate particles of Cu-Cr-Zr alloys with varying Zr contents were studied by transmission electron microscopy (TEM) after solution treatments at 950 °C for 1 h and aging treatments at 500 °C for different times ranged from 0.5 h to 24 h. The microhardness and electrical conductivity of Cu-Cr-Zr alloys after various aging process were tested. The results show that the microhardness and electrical conductivity rapidly increased at first, then the microhardness decreased slowly after reaching the peak, while the conductivity continues to increase. Nano-scaled precipitates exhibit two kinds of morphology (coffee bean and ellipse shaped). With increasing Zr content, the Zr-containing precipitation sequence of Cu-Cr-Zr alloys at peak-ageing is Heusler CrCu2Zr → Cu5Zr → Cu4Zr. The Heusler CrCu2Zr phase decomposed into fine and homogeneous Cr and Cu4Zr, resulting in improved alloy properties.

Ethylene oligomerization in zeolite-grafted Cr(III)-diphosphinoamine catalysts using triisobutylaluminium as cocatalyst: Change from dimerization to trimerization due to confinement effect

Zeolite-supported Cr(III)-diphosphinoamine (Cr(III)-PNP) catalysts were prepared through grafting PNP on HY and NaY zeolites followed by complexing with CrCl3(THF)3 for ethylene oligomerization. The structure of supported Cr(III)-PNP catalysts was characterized by scanning electron microscopy, X-ray diffraction, nitrogen adsorption and desorption, thermogravimetric analyses and Fourier transform infrared, and the influence of the supported pattern on reactivity for ethylene oligomerization were investigated. The results revealed that the complex of Cr(III)-PNP was grafted on silicon hydroxyls in the pore channel of HY zeolite to decrease pore size but to maintain pore structure. Comparing with homogeneous Cr(III)-PNP producing 1-butene as main product, HY-supported catalyst had higher activity and selectivity toward 1-hexene increased from 4.07% to 73.24% using triisobutylaluminium as cocatalyst. The increase is attributed to confinement effect of the pore channel, which increases the stability of the chromacycloheptane intermediate to 1-hexene. The confinement effect for ethylene oligomerization was revealed in experiment.

Intensification of heterogeneous TiO2 photocatalysis using an innovative micro–meso-structured-reactor for Cr(VI) reduction under simulated solar light

Process intensification of heterogeneous TiO2 photocatalysis using a new type of static mixer micro–meso-structured-reactor, based on the NETmix reactor, was evaluated to promote the hexavalent chromium reduction to its trivalent state, in the presence of sacrificial agents. The photocatalytic micro–meso-structured reactor comprises a back acrylic slab where chambers and channels are imprinted and a frontal borosilicate slab with especially high UV transparency, irradiated by simulated solar light. An UV transparent cellulose acetate (CA) sheet coated with TiO2-P25 by a simple spray method was assembled between the two slabs of the photoreactor. The reactor geometry provides an efficient exposure of TiO2-P25 CA sheets to radiation, with an illuminated surface per unit of volume inside the reactor of 470m2m−3 and 1.36g TiO2 per liter of liquid inside the reactor. Beyond that, it provides a good degree of mixing using a laminar flow, maximizing heterogeneous and homogeneous Cr(VI) reduction reactions. Firstly, different organic species were tested as hole scavengers in order to avoid the Cr(III) re-oxidation. The effect of the degree of mixing inside the photoreactor, according to the Reynolds number, on the Cr(VI) photocatalytic reduction was then evaluated. Following, operating conditions such as TiO2 mass, tartaric acid concentration, pH value, temperature and initial Cr(VI) concentration were analyzed. Results indicate high reduction rates (1mmolCr(VI)m−3illuminated volumes−1) and a photonic efficiency of 3.96% for Re=830, 30mg TiO2 and a Cr(VI):tartaric acid molar ratio of 1:90 at pH 3.0 and 25°C. Finally, the CA sheets coated with thin films of TiO2-P25 showed to be suitable for continuous Cr(VI) reduction, without losing efficiency during 3 consecutive photocatalytic cycles.

Lie algebras of infinitesimal CR automorphisms of weighted homogeneous and homogeneous CR-generic submanifolds of CN

We consider the significant class of homogeneous CR manifolds represented by some weighted homogeneous polynomials and we derive some plain and useful features which enable us to set up a fast effective algorithm to compute homogeneous components of their Lie algebras of infinitesimal CR automorphisms. This algorithm mainly relies upon a natural gradation of the sought Lie algebras, and it also consists in treating separately the related graded components. While classical methods are based on constructing and solving some associated PDE systems which become time consuming as soon as the number of variables increases, the new method presented here is based on plain techniques of linear algebra. Furthermore, it benefits from a divide-and-conquer strategy to break down the computations into some simpler subcomputations. Also, we consider the new and effective concept of comprehensive Gr?bner systems which provides us some powerful tools to treat the computations in the much complicated parametric case. The designed algorithm is also implemented in the Maple software, what required also implementing a recently designed algorithm of Kapur et al.

Selective Oxidation of Glycerol with 3% H2O2 Catalyzed by LDH-Hosted Cr(III) Complex

A series of layered double hydroxides (LDHs) –hosted sulphonato-salen Cr(III) complexes were prepared and characterized by various physico-chemical measurements, such as Fourier transform infrared spectroscopy (FTIR), ultraviolet-visible spectroscopy (UV-Vis), powder X-ray diffraction (XRD), transmission electron microscope (TEM), scanning electron microscope (SEM) and elemental analysis. Additionally, their catalytic performances were investigated in the selective oxidation of glycerol (GLY) using 3% H2O2 as an oxidant. It was found that all the LDH-hosted Cr(III) complexes exhibited significantly enhanced catalytic performance compared to the homogeneous Cr(III) complex. Additionally, it was worth mentioning that the metal composition of LDH plates played an important role in the catalytic performances of LDH-hosted Cr(III) complex catalysts. Under the optimal reaction conditions, the highest GLY conversion reached 85.5% with 59.3% of the selectivity to 1,3-dihydroxyacetone (DHA). In addition, the catalytic activity remained after being recycled five times.