Abstract
We consider three strong reducibilities, $s_{1}, s_{2}, Q_{1}$ (where we identify a reducibility $\leqslant _r$ with its index $r$). The first two reducibilities can be viewed as injective ...versions of $s$-reducibility, whereas $Q_1$-reducibility can be viewed as an injective version of $Q$-reducibility. We have, with proper inclusions, $s_{1} \subset s_{2} \subset s$. It is well known that there is no minimal $s$-degree, and there is no minimal $Q$-degree. We show on the contrary that there exist minimal $\varDelta ^{0}_{2}$$s_{2}$-degrees and minimal $\varDelta ^{0}_{2}$$s_{1}$-degrees. On the other hand, both the $\varPi ^{0}_{1}$$s_{2}$-degrees and the $\varPi ^{0}_{1}$$s_{1}$-degrees are downwards dense. By the isomorphism of the $s_1$-degrees with the $Q_1$-degrees induced by complementation of sets, it follows that there exist minimal $\varDelta ^0_2$$Q_1$-degrees, but the c.e. $Q_{1}$-degrees are downwards dense.
In computability theory the most common form of reducibility is Turing reducibility. Earlier, the author found necessary and sufficient conditions for the completeness of computably enumerable sets ...for this reducibility. Later, this criterion was generalized by various authors to new classes of functions, and also investigated for some of the most important reducibilities, stronger than Turing reducibility. In this paper, we discuss one general method for constructing similar criteria for a wide class of reducibility.
Reducibility is an essential characteristic of oxide catalysts in oxidation reactions following the Mars–van Krevelen mechanism. A typical descriptor of the reducibility of an oxide is the cost of ...formation of an oxygen vacancy, which measures the tendency of the oxide to lose oxygen or to donate it to an adsorbed species with consequent change in the surface composition, from M n O m to M n O m–x . The oxide reducibility, however, can be modified in various ways: for instance, by doping and/or nanostructuring. In this review we consider an additional aspect, related to the formation of a metal/oxide interface. This can be realized when small metal nanoparticles are deposited on the surface of an oxide support or when a nanostructured oxide, either a nanoparticle or a thin film, is grown on a metal. In the past decade, both theory and experiment indicate a particularly high reactivity of the oxygen atoms at the boundary region between a metal and an oxide. Oxygen atoms can be removed from interface sites at much lower cost than in other regions of the surface. This can alter completely the reactivity of a solid catalyst. In this respect, reducibility of the bulk material may differ completely from that of the metal/oxide surface. The atomistic study of CO oxidation and water-gas shift reactions are used as examples to provide compelling evidence that the oxidation occurs at specific interface sites, the actual active sites in the complex catalyst. Combining oxide nanostructuring with metal/oxide interfaces opens promising perspectives to turn hardly reducible oxides into reactive materials in oxidation reactions based on the Mars–van Krevelen mechanism.
With the aim of developing an iron ore sinter with a better reducibility, the authors researched the co-firing of green pellets and a sinter mix in a packed bed. First, a physical simulation with a ...simulator for the charging equipment of a sintering machine demonstrated that the green pellets were distributed more strongly to the lowest part of the packed bed because their size is larger than that of the sinter mix. Pot tests based on the simulated pellet distribution proved that it is possible to produce a highly reducible sinter at higher productivity, particularly when a concentrate with a large Blaine surface area is used to produce rigid green pellets. Factors that could reduce product yield were also pointed out, i.e., diffusion of the melt from the sinter mix into the pellets and a larger local heat supply to the sinter mix than is suitable. These factors are thought to hinder the formation of thick and strong bonds in the co-fired sinter bed. A 450 kV class X-ray CT was used to visualize the internal structure of the sinter plugs, and revealed the plug made with the green pellets has higher porosity, which is important for better permeability, but also results in less bond formation, which reduces the yield in a co-fired plug. These findings revealed insights for fully enjoying the advantages of co-firing green pellets and sinter mix, as well as issues to be overcome in order to realize a successful co-firing process.
In this study, influence of crystal structure and chemical composition for Silico-Ferrite of Calcium and Aluminum (SFCA) on its reducibility are examined. Eight types of powder samples containing ...SFCA and SFCA-I were prepared using chemical reagents by the heat treatments in air. The samples were heated up to 800°C in the different atmospheres of CO-CO2-H2-H2O systems. The reducibility of the samples was evaluated using the peak intensity ratio identified by XRD before and during the reduction experiment. The intensity of SFCA peak is not decreased up to 700°C in CO-CO2 atmosphere, whereas the intensity become weak in CO-CO2-H2-H2O atmosphere. The intensity of SFCA-I peak is decreased above 500oC in all atmospheric condition and the reduction reactions are enhanced by the addition of H2-H2O gas. Decrease in intensity of SFCA peak is independent of Fe composition, whereas that of SFCA-I is decreased with decreasing Fe concentration. The difference in reducibility is attributed to the difference in the crystal structure of multi-component CF. SFCA and SFCA-I are composed of pyroxene and spinel units. Since the pyroxene unit contains more gangue minerals than spinel unit, it implies that the pyroxene unit shows low reducibility than the spinel units. Since SFCA-I contains more the spinel units than SFCA, SFCA-I is easily reduced than SFCA.
In the present article, we study universal, minimal, and complete numberings of families of arithmetic sets. We show that, for every
and every nontrivial
-computable family
, there exists a
...-computable numbering that is not universal with respect to positive reducibilities and is complete with respect to each element
. For finite families of computably enumerable sets, we obtain necessary and sufficient conditions for existence of numberings that are complete, computable, and not universal with respect to positive reducibilities. For every infinite
-computable family
and every element
, we construct a
-computable numbering that is complete with respect to
and minimal with respect to classical and positive reducibilities.
The reducibility of C0(2) operators Zhu, Senhua; Yang, Yixin; Li, Ran ...
Journal of mathematical analysis and applications,
02/2020, Volume:
482, Issue:
2
Journal Article
Peer reviewed
In this paper, we give a necessary and sufficient condition for the reducibility of a C0(2) operator by using its characteristic function. Moreover, we obtain the number of reducing subspaces of a ...C0(2) operator. As an application, we will restudy the reducibility of the truncated Toeplitz operator Az2.
In this paper we prove reducibility of a class of first order, quasi-linear, quasi-periodic time dependent PDEs on the torus∂tu+ζ⋅∂xu+a(ωt,x)⋅∂xu=0,x∈Td,ζ∈Rd,ω∈Rν. As a consequence we deduce a ...stability result on the associated Cauchy problem in Sobolev spaces. By the identification between first order operators and vector fields this problem can be formulated as the problem of finding a change of coordinates which conjugates a weakly perturbed constant vector field on Tν+d to a constant diophantine flow. For this purpose we generalize Moser's straightening theorem: considering smooth perturbations we prove that the corresponding straightening torus diffeomorphism is smooth, under the assumption that the perturbation is small only in some given Sobolev norm and that the initial frequency belongs to some Cantor-like set. In view of applications in KAM theory for PDEs we provide also tame estimates on the change of variables.
The influence of chromium and aluminium doping on the over-reduction during activation of iron-oxide-based water-gas shift catalysts was investigated using Mössbauer spectroscopy for the first time. ...In situ Mössbauer spectra of catalysts exposed to industrially relevant gas compositions were recorded with increasingly reducing R factors R = CO*H2/CO2*H2O. Whereas α-Fe and cementite formed during exposure of a non-doped iron-oxide catalyst to process conditions with an R factor of 2.09, such phases were only observed at R = 4.60 for a chromium-doped catalyst, showing that chromium stabilizes the catalyst. Over-reduction was enhanced to R = 2.88 in a chromium-copper co-doped catalyst. α-Fe was already observed at R = 1.64 in an aluminium-doped catalyst, while cementite formation occurred at R = 2.09, showing that over-reduction was enhanced, the presence of aluminium delaying carburization. Co-doping copper in the aluminium-doped catalyst showed cementite formation at R = 2.09, the same as a non-doped catalyst.
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•In situ Mössbauer spectroscopy of Fe3O4 water-gas shift catalysts.•Active phase formation studied using industrially relevant gas feed compositions.•Impact of Cr and Al dopants and Cu promotion in fresh α-Fe2O3 investigated.•Cr preferred dopant against over-reduction, Al less effective.•Cu enhances slightly over-reduction of the active phase.
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