Recent zbMATH articles in MSC 08https://www.zbmath.org/atom/cc/082022-05-16T20:40:13.078697ZWerkzeugCorrection to: ``Infinitary addition, real numbers, and taut monads''https://www.zbmath.org/1483.180052022-05-16T20:40:13.078697Z"Janelidze, George"https://www.zbmath.org/authors/?q=ai:janelidze.george"Street, Ross"https://www.zbmath.org/authors/?q=ai:street.ross-hFrom the text: In the original publication of the article [the authors, ibid. 26, No. 5, 1047--1064 (2018; Zbl 1480.18006)], Eq. (3.24) was published incorrectly. The corrected equation is given in this correction article.
\[
a_0=\sum_{i\in\mathbb{N}}b_i \Rightarrow ((\exists_{a_1,a_2,}\dots \forall_{n\in\mathbb{N}} a_n=b_n+a_{n+1})\ \&\ \forall_{c_0,c_1,}\dots((\forall{n\in\mathbb{N}}c_n=b_n+c_{n+1})\Rightarrow a_0\leq c_0)),
\]
The original article has been corrected.Topological obstructions for robustly transitive endomorphisms on surfaceshttps://www.zbmath.org/1483.370412022-05-16T20:40:13.078697Z"Lizana, C."https://www.zbmath.org/authors/?q=ai:lizana.cristina"Ranter, W."https://www.zbmath.org/authors/?q=ai:ranter.wagnerThe authors show that every robustly transitive surface endomorphism displaying critical points is a partially hyperbolic endomorphism and that the only surfaces that might admit robustly transitive endomorphisms are either the torus \(\mathbb{T}^2\) or the Klein bottle \(\mathbb{K}^2\). They also prove that the action of a transitive endomorphism admitting a dominated splitting in the first homology group of the surface has at least one eigenvalue with modulus larger than one.
Reviewer: Miguel Paternain (Montevideo)Convexity in topological betweenness structureshttps://www.zbmath.org/1483.540172022-05-16T20:40:13.078697Z"Anderson, Daron"https://www.zbmath.org/authors/?q=ai:anderson.daron"Bankston, Paul"https://www.zbmath.org/authors/?q=ai:bankston.paul"McCluskey, Aisling"https://www.zbmath.org/authors/?q=ai:mccluskey.aisling-eA betweenness structure is a pair \(\langle X,[\cdot,\cdot,\cdot] \rangle\), where \(X\) is a set and \([\cdot,\cdot,\cdot]\subset X^{3}\) is a ternary relation satisfying that
\begin{itemize}
\item[(B1)] Inclusivity: \((\forall\ xy )\) \(([x,y,y] \wedge [x,x,y])\)
\item[(B2)] Symmetry: \((\forall\ xzy )\) \(([x,z,y] \rightarrow [y,z,x])\)
\item[(B3)] Uniqueness: \((\forall\ xz)\) \(([x,z,x]\rightarrow x=z)\)
\end{itemize}
Given a betweenness structure \(\langle X,[\cdot,\cdot,\cdot] \rangle\), an interval is defined as \([a,b]=\{c\in X:[a,c,b]\}\), a convex subset of \(X\) is a subset \(C\) of \(X\) such that \(a,b\in C\), implies \([a,b]\subset C\). The span of a subset \(A\) of \(X\) is defined as \([A]=\bigcup\{[a,b]:a,b\in A\}\). The convex hull of \(A\) is defined as \([A]^{\omega}=\bigcup\{[A]^{n}:n\in\omega\}\), where \([A]^{0}=A\) and \([A]^{n+1}=[[A]^{n}]\).
With a detailed analysis of examples, in this paper the authors show how the notion of betweenness is related to several important concepts in mathematics. In particular if besides a betweenness structure, \(X\) has a topology, it is possible to define interesting relations between the two structures, starting by asking that intervals are closed. In this sense, the authors define local convexity, upper (and lower) semi-continuity of betweenness, and a type of internal continuity of the betweenness. They obtain results connecting the convexity and the topology in compact connected Hausdorff spaces which are aposyndetic or hereditary unicoherent. In particular, they study how the span and the convex hull interact with the topological closure and interior operators.
Reviewer: Alejandro Illanes (Ciudad de México)When symmetries are not enough: a hierarchy of hard constraint satisfaction problemshttps://www.zbmath.org/1483.681412022-05-16T20:40:13.078697Z"Gillibert, Pierre"https://www.zbmath.org/authors/?q=ai:gillibert.pierre"Jonušas, Julius"https://www.zbmath.org/authors/?q=ai:jonusas.julius"Kompatscher, Michael"https://www.zbmath.org/authors/?q=ai:kompatscher.michael"Mottet, Antoine"https://www.zbmath.org/authors/?q=ai:mottet.antoine"Pinsker, Michael"https://www.zbmath.org/authors/?q=ai:pinsker.michaelBatalin-Vilkovisky quantization of fuzzy field theorieshttps://www.zbmath.org/1483.811062022-05-16T20:40:13.078697Z"Nguyen, Hans"https://www.zbmath.org/authors/?q=ai:nguyen.hans"Schenkel, Alexander"https://www.zbmath.org/authors/?q=ai:schenkel.alexander"Szabo, Richard J."https://www.zbmath.org/authors/?q=ai:szabo.richard-jSummary: We apply the modern Batalin-Vilkovisky quantization techniques of Costello and Gwilliam to noncommutative field theories in the finite-dimensional case of fuzzy spaces. We further develop a generalization of this framework to theories that are equivariant under a triangular Hopf algebra symmetry, which in particular leads to quantizations of finite-dimensional analogues of the field theories proposed recently through the notion of `braided \(L_{\infty}\)-algebras'. The techniques are illustrated by computing perturbative correlation functions for scalar and Chern-Simons theories on the fuzzy 2-sphere, as well as for braided scalar field theories on the fuzzy 2-torus.