This work describes periodic matrices in the general linear group over the real numbers field and over the maximal Abelian extension ℚ
ab
of the rational numbers field. It is shown that for the case ...of real numbers the general question is reduced to the 2×2 matrices. A simple periodicity criterion is provided for them. We demonstrate a geometric interpretation of the results. The main result is an algorithm that tests periodicity of a matrix and, if the matrix is periodic, finds its Jordan form.
In this paper, we consider the 𝕋-space structure of the relatively free Grassmann algebra 𝔽
(3)
without unity over an infinite field of prime and zero characteristic. Our work is focused on ...𝕋-spaces 𝕎
n
generated by all so-called
n
-words. A question about connections between 𝕎
r
and 𝕎
n
for different natural numbers
r
and
n
is investigated. The proved theorem on these connections allows us to construct the diagrams of inclusions, which, to some extent, clarify the structure of the algebra: the basic 𝕋-spaces produce infinite strictly descending chains of inclusions in the algebra 𝔽
(3)
.
In the paper, we continue the study of the relatively free Grassmann algebra
without unit over an infinite field of characteristic 2, which was initiated in previous works of the author. The main ...attention is paid here to the relationship between the
-spaces of n-words
, i.e., the
-spaces generated by all monomials in
containing each of their variables with multiplicity
n
. The results of this note will enable one to form a more complete picture of possible inclusions between the
-spaces of
r
-and
n
-words for
r > n
.
In this paper, we consider the ð-space structure of the relatively free Grassmann algebra ð½(3) without unity over an infinite field of prime and zero characteristic. Our work is focused on ...ð-spaces ðn generated by all n-words. A question about connections between ðr and ðn for different natural numbers r and n is investigated. The proved theorem on these connections allows us to construct the diagrams of inclusions that, to some extent, clarify the structure of the algebra: the basic ð-spaces produce infinite strictly descending chains of inclusions in the algebra ð½(3).
In this paper, we consider the 𝕋-space structure of the relatively free Grassmann algebra 𝔽
(3)
without unity over an infinite field of prime and zero characteristic. Our work is focused on ...𝕋-spaces 𝕎
n
generated by all
n
-words. A question about connections between 𝕎
r
and 𝕎
n
for different natural numbers
r
and
n
is investigated. The proved theorem on these connections allows us to construct the diagrams of inclusions that, to some extent, clarify the structure of the algebra: the basic 𝕋-spaces produce infinite strictly descending chains of inclusions in the algebra 𝔽
(3)
.
In this paper, we consider the ð-space structure of the relatively free Grassmann algebra ð½ (3) without unity over an infinite field of prime and zero characteristic. Our work is focused on ...ð-spaces ðn generated by all so-called n-words. A question about connections between ðr and ðn for different natural numbers r and n is investigated. The proved theorem on these connections allows us to construct the diagrams of inclusions, which, to some extent, clarify the structure of the algebra: the basic ð-spaces produce infinite strictly descending chains of inclusions in the algebra ð½(3).
In this work, we consider the operations over Abelian integers of rank $n$. By definition, such numbers are elements of the complex field and have the form of polynomials with integer coefficients ...from the $n$th degree primitive root of 1. In contrast, the degrees of such polynomials are not greater than Euler's totient function $\varphi(n)$. We provide an example to show that there are infinitely many Abelian integers inside any zero-centered circle on the complex plane. In this work, for considered operations we give in particular the algorithm of calculation of the inverse for the Abelian integer of rank $n$. It allows us to analyze not only the rings of such numbers but also the fields of Abelian integers. Natural arithmetics for such algebraic structures leads us to study the polynomials with integer Abelian coefficients. Thus, in the presented work we also investigate the problem of finding roots of such polynomials. As a result, we provide an algorithm that finds the integer Abelian roots of the polynomials over the ring of Abelian integers. This algorithm is based on the proposed statement that all roots of the polynomial are bounded by some domain. The computer calculations confirm the statistical truth of the statement.
Cobalt catalysts as used in the Fischer-Tropsch synthesis (FTS) are relatively expensive (as compared to iron) and need to have a high metal dispersion and long life to be able to offer a good ...balance between cost and performance. The oxidation of nano-sized metallic cobalt to cobalt oxide during Fischer-Tropsch synthesis has long been postulated as a major deactivation mechanism. However, to date there is no consistent picture. This paper presents an extensive overview of the literature on this topic of deactivation by means of oxidation for unsupported as well as silica-, alumina- and titania-supported cobalt catalysts. Furthermore, it presents results on the deactivation of an industrial Co/Al2O3 catalyst as obtained by pseudo in situ X-ray diffraction, magnetic measurements and X-ray absorption near-edge spectroscopy. These analyses were performed to study the oxidation state of spent industrial Co/Al2O3 catalyst samples withdrawn from a slurry reactor operating under realistic FTS conditions, and it was concluded that oxidation can be ruled out as a major deactivation mechanism. Finally, these data together with all relevant literature were used to create a common view on the oxidation behaviour of metallic cobalt during FTS. The apparent discrepancies in literature on the oxidation behaviour of cobalt are most likely due to the lack of direct characterisation of the cobalt oxidation state and due to the comparison of catalysts with varying cobalt crystallites sizes, compared at different reactor partial pressures of hydrogen and water (PH2O/PH2). It was shown that the oxidation of cobalt can be prevented by selecting the correct combination of the reactor partial pressures of hydrogen and water (PH2O/PH2) and the cobalt crystallite size.
An X-ray study of titaniaand alumina-supported rhodium catalysts for the hydrogenation of hydrocarbons is performed. Rhodium is shown to be present in the form of highly dispersed oxide, independent ...of the type of support. The atomic structure of alumina changes, while no change in the real structure of titanium oxide is observed.
