For linear differential systems, the Sacker–Sell spectrum (dichotomy spectrum) and the contractible set are the same. However, we claim that this is not true for the linear difference equations. A counterexample is given. For the convenience of research, we study the relations between the dichotomy spectrum and the contractible set under the framework on time scales. In fact, by a counterexample, we show that the contractible set could be different from dichotomy spectrum on time scales established by Siegmund [J. Comput. Appl. Math., 2002]. Furthermore, we find that there is no bijection between them. In particular, for the linear difference equations, the contractible set is not equal to the dichotomy spectrum. To counter this mismatch, we propose a new notion called generalized contractible set and we prove that the generalized contractible set is exactly the dichotomy spectrum. Our approach is based on roughness theory and Perron's transformation. In this paper, a new method for roughness theory on time scales is provided. Moreover, we provide a time-scaled version of the Perron's transformation. However, the standard argument is invalid for Perron's transformation. Thus, some novel techniques should be employed to deal with this problem. Finally, an example is given to verify the theoretical results.

In the present work, LaNi0.5Ti0.5O3, LaNi0.5Ti0.45Co0.05O3, and LaNi0.45Co0.05Ti0.5O3 perovskites were synthesized using the modified Pechini method. After reduction, the studied perovskites changed crystal structure from the perovskite crystal structure to a cubic symmetry, with space group $Pm\bar{3}m$. The reduction partially decomposed the samples to Ni0 (Co free perovskite), Ni0–Co0, La2O3, La2TiO5, and non-stoichiometric La2NiO4, depending on H2 content of the reductive gases. The degree of reduction of nickel from LaNi0.5Ti0.5O3 reduced with 1.8% H2/Ar and 10% H2/Ar was equal to 36.5% and 95.3%, respectively, while that from LaNi0.5Ti0.45Co0.05O3 or LaNi0.45Co0.05Ti0.5O3, including cobalt, reduced with 10% H2/Ar, was equal to 71.9% and 93.9%, respectively. LaNi0.5Ti0.45Co0.05O3 showed Ni3+ and Co3+ amounts higher than the other perovskites. By increasing H2 content in the reductive mixture from 1.8% to 10%, sintering of metallic nickel was not observed. Moreover, Ni0 displayed weaker metal–support interaction than that observed for Co0, where the support was composed of La containing oxides. LaNi0.5Ti0.5O3 perovskite was used as a catalyst for steam reforming of methane. Syngas production was attributed to the number of Ni sites determined using Rietveld Refinement of X-ray diffraction pattern of this catalyst after the reaction.

In this paper we consider the following nonlinear quasi-periodic system:

$$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$

$A$

$d\times d$

$\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$

$\unicode[STIX]{x1D716}$

$h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$

$x\rightarrow 0$

$P,g$

$h$

$t$

$\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$

$\unicode[STIX]{x1D6FC}$

$\unicode[STIX]{x1D716}$

$$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$

$h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$

$\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$

$\unicode[STIX]{x1D716}$

Let ${\cal T}$ be any of the three canonical truth theories CT− (compositional truth without extra induction), FS− (Friedman–Sheard truth without extra induction), or KF− (Kripke–Feferman truth without extra induction), where the base theory of ${\cal T}$ is PA (Peano arithmetic). We establish the following theorem, which implies that ${\cal T}$ has no more than polynomial speed-up over PA.

Theorem.${\cal T}$is feasibly reducible to PA, in the sense that there is a polynomial time computable function f such that for every${\cal T}$-proof π of an arithmetical sentence ϕ, f (π) is a PA-proof of ϕ.

Thermal reduction in wolframite (composition: Fe1-xMnxWO4 (0.08<x<0.87)) from different deposits in Argentina, was analysed using a temperature-programmed reduction (TPR) technique. X-ray powder diffraction analysis (XRPD), vibrational spectroscopy (IR and Raman), scanning electron microscopy (SEM) and electron microprobe analysis (EDAX) were used to assess the relationship between reactivity degree in a reducing atmosphere (H2 10% – N2 90%) and chemical composition of wolframite.Similar studies were carried out with other related minerals, especially the ones found in natural deposits as alteration or association products. Stability of the solid solution serial members is correlated with redox properties of divalent cations.

We consider systems of quasi-periodic linear differential equations associated to a ‘resonant’ frequency vector ${\it\omega}$, namely, a vector whose coordinates are not linearly independent over $\mathbb{Z}$. We give sufficient conditions that ensure that a small analytic perturbation of a constant system is analytically conjugate to a ‘resonant cocycle’. We also apply our results to the non-resonant case: we obtain sufficient conditions for reducibility.

Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let ${{A}_{\mathbb{R}}}\left( K \right)$ denote the real Banach algebra of all real symmetric continuous functions on $K$ that are analytic in the interior ${{K}^{\circ }}$ of $K$, endowed with the supremum norm. We characterize all unimodular pairs $\left( f,\,g \right)$ in ${{A}_{\mathbb{R}}}{{\left( K \right)}^{2}}$ which are reducible. In addition, for an arbitrary compact $K$ in $\mathbb{C}$, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of $A\left( K \right)$ is 1. Finally, we also characterize all compact real symmetric sets $K$ such that ${{A}_{\mathbb{R}}}\left( K \right)$, respectively ${{C}_{\mathbb{R}}}\left( K \right)$, has Bass stable rank 1.

The Australian Pilbara blend lump ore, which entered the Chinese market in 2007, is aspecial blending ore. To master its behavior and effect in a BF adequately, themetallurgical properties and high temperature reactivity of the Pilbara blend lump ore andsix other kinds of lump ore were researched. The reasonable proportion and mixing patternof the Pilbara blend lump ore was studied by carrying out softening and dropping propertytests, and this provides technological support for practical production.

A stochastic process is locally stationary if its covariance function can be expressed as the product of a positive function multiplied by a stationary covariance. In this paper, we characterize nonstationary stochastic processes that can be reduced to local stationarity via a bijective deformation of the time index, and we give the form of this deformation under smoothness assumptions. This is an extension of the notion of stationary reducibility. We present several examples of nonstationary covariances that can be reduced to local stationarity. We also investigate the particular situation of exponentially convex reducibility, which can always be achieved for a certain class of separable nonstationary covariances.

We determine all the complex polynomials f(X) such that, for two suitable distinct, nonconstant rational functions g(t) and h(t), the equality f(g(t)) = f(h(t)) holds. This extends former results of Tverberg, and is a contribution to the more general question of determining the polynomials f(X) over a number field K such that f(X) − λ has at least two distinct K-rational roots for infinitely many λ ∈ K.