The non-commutative torus $C^*(\mathbb{Z}^n,\omega )$ is realized as the $C^*$-algebra of sections of a locally trivial $C^*$-algebra bundle over $\widehat{S_{\omega }}$ with fibres isomorphic to $C^*(\mathbb{Z}^n/S_{\omega }, \omega _1)$ for a totally skew multiplier $\omega _1$ on $\mathbb{Z}^n/S_{\omega }$. D. Poguntke [9] proved that $A_{\omega }$ is stably isomorphic to $C(\widehat{S_{\omega }}) \otimes C^*(\mathbb{Z}^n/S_{\omega }, \omega _1) \cong C(\widehat{S_{\omega }}) \otimes A_{\varphi } \otimes M_{kl}(\mathbb{C})$ for a simple non-commutative torus $A_{\varphi }$ and an integer $kl$. It is well-known that a stable isomorphism of two separable $C^*$-algebras is equivalent to the existence of equivalence bimodule between them. We construct an $A_{\omega }$-$C(\widehat{S_{\omega }}) \otimes A_{\varphi }$-equivalence bimodule.
We generalize Jiroušek's (\emph {right}) \emph {composition operator} in such a way that it can be applied to distribution functions with values in a "semifield", and introduce (parenthesized) \emph {compositional expressions}, which in some sense generalize Jiroušek's "generating sequences" of compositional models. We say that two compositional expressions are \emph {equivalent} if their evaluations always produce the same results whenever they are defined. Our first result is that a set system H is star-like with centre X \emph {if and only if} every two compositional expressions with "base scheme" H and "key" X are equivalent. This result is stronger than Jiroušek's result which states that, if H is star-like with centre X, then every two generating sequences with base scheme H and key X are equivalent. Then, we focus on \emph {canonical expressions}, by which we mean compositional expressions θ such that the sequence of the sets featured in θ and arranged in order of appearance enjoys the "running intersection property". Since every compositional expression, whose base scheme is a star-like set system with centre X and whose key is X, is a canonical expression, we investigate the equivalence between two canonical expressions with the same base scheme and the same key. We state a graphical characterization of those set systems H such that every two canonical expressions with base scheme H and key X are equivalent, and also provide a graphical algorithm for their recognition. Finally, we discuss the problem of detecting conditional independences that hold in a compositional model.d
Recently, the weak Triebel-Lizorkin space was introduced by Grafakos and He, which includes the standard Triebel-Lizorkin space as a subset. The latter has a wide applications in aspects of analysis. In this paper, the authors firstly give equivalent quasi-norms of weak Triebel-Lizorkin spaces in terms of Peetre's maximal functions. As an application of those equivalent quasi-norms, an atomic decomposition of weak Triebel-Lizorkin spaces is given., Wenchang Li, Jingshi Xu., and Seznam literatury
In this note all vectors and ε-vectors of a system of m ≤ n linearly independent contravariant vectors in the n-dimensional pseudo-Euclidean geometry of index one are determined. The problem is resolved by finding the general solution of the functional equation F(Au 1 , Au 2 , . . . , Au m ) = (det A) λ · A · F(u 1 , u 2 , . . . , u m ) with λ = 0 and λ = 1, for an arbitrary pseudo-orthogonal matrix A of index one and given vectors u 1 , u 2 , . . . , u m .
There are four kinds of scalars in the n-dimensional pseudo-Euclidean geometry of index one. In this note, we determine all scalars as concomitants of a system of m ≤ n linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation F(Au 1 , Au 2 , . . . , Au m ) = ϕ (A) · F(u 1 , u 2 , . . . , u m ) using two homomorphisms ϕ from a group G into the group of real numbers R0 = (R \ {0} , ·).
In this note, there are determined all biscalars of a system of s ≤ n linearly independent contravariant vectors in n-dimensional pseudo-Euclidean geometry of index one. The problem is resolved by finding a general solution of the functional equation F(Au1 , Au2 , . . . , Aus ) = (sign(det A))F(u1 , u2 ,...,us ) for an arbitrary pseudo-orthogonal matrix A of index one and the given vectors u1 , u2 ,...,us .
This paper deals with the reconstruction of the now longer preserved gallery of coats of arms at Roupov Castle (District of Klatovy, Western Bohemia) based on manuscripts XVII.A.8 and XVII. E. 28 a from the Czech National Library. Information from individual manuscripts was combined to form an image of probably the largest Czech family coat of arms gallery at the end of the 16th century containing a collection of coats of arms from 270 noblemen and noblewomen. The gallery probands are Jan Nezdický of Roupov († before 1607) and his two wives – Dorota Bezdružická of Kolovraty and Benigna of Švamberk. The paper draws attention to the utilization of hitherto neglected manuscript sources for research into displays of self-awareness among the privileged classes and it attempts to show the way in which the nobility used genealogical and heraldic means for representative purposes. Not least, these manuscripts are often the only source of information on genealogical and heraldic artefacts which are no longer in existence.