Recent zbMATH articles in MSC 57Qhttps://www.zbmath.org/atom/cc/57Q2021-02-12T15:23:00+00:00WerkzeugSufficient condition for tangential transversality.https://www.zbmath.org/1452.490142021-02-12T15:23:00+00:00"Apostolov, Stoyan R."https://www.zbmath.org/authors/?q=ai:apostolov.stoyan-r"Krastanov, Mikhail I."https://www.zbmath.org/authors/?q=ai:krastanov.mikhail-ivanov"Ribarska, Nadezhda K."https://www.zbmath.org/authors/?q=ai:ribarska.nadezhda-kTransversality coming from mathematical analysis and differential topology is also an extremely natural and convenient concept in some parts of variational analysis (see [\textit{A. D. Ioffe}, J. Optim. Theory Appl. 174, No. 2, 343--366 (2017; Zbl 1382.49014)]). The extension to subtransversality [\textit{D. Drusvyatskiy} et al., Found. Comput. Math. 15, No. 6, 1637--1651 (2015; Zbl 1338.49057)] for two closed sets \(A,B\) at \(x_0 \in A\cap B\), given by \(d(x,A\cap B)\le K (d(x,A)+d(x,B))\) for some \(K>0\), some neighborhood \(U\) of \(x_0\) and all \(x \in U\), is very useful for deriving necessary optimality conditions of the Pontryagin maximum principle. In the article, suffficient conditions of tangential transversality introduced by \textit{M. Bivas} et al. [J. Math. Anal. Appl. 481, No. 1, Article ID 123445, 21 p. (2020; Zbl 1432.49017)] are investigated and applied to abstract optimal control in B-spaces. Tangential transversality is sufficient for subtransversality [Bivas et al., loc. cit.]. The well-known sufficient condition for tangential transversality using compactly epi-Lipschitz (massive) sets is weakened to a symmetric condition with respect to the sets \(A,B\) taking uniform tangent sets of corresponding tangent cones for proving tangential transversality of \(A,B\) at \(x_0\) (Th. 3.1 and Th. 3.2). The result yields an abstract version of the well-known Aubin condition from [\textit{F. Clarke}, Necessary conditions in dynamic optimization. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1093.49017)] which is then applied to an abstract optimal control problem in Banach spaces. They introduce as specialization of the above symmetric condition so-called jointly massive sets \(A,B\) -- splitting the massiveness between \(A\) and \(B\) -- which ensure tangential transversality too. The proofs are given in detail for new results.
For better understanding the ideas behind this paper, study first [Bivas et al., loc. cit.].
Reviewer: Armin Hoffmann (Ilmenau)Invariants and TQFT's for cut cellular surfaces from finite 2-groups.https://www.zbmath.org/1452.570252021-02-12T15:23:00+00:00"Bragança, Diogo"https://www.zbmath.org/authors/?q=ai:braganca.diogo"Picken, Roger"https://www.zbmath.org/authors/?q=ai:picken.roger-fSummary: In this brief sequel to a previous article \textit{D. Bragança} and \textit{R. Picken} [Bol. Soc. Port. Mat. 74, 17--44 (2016; Zbl 07301091)], we recall the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colorings of the 1- and 2-cells with elements of a finite 2-group, subject to a ``fake flatness'' condition for each 2-cell. These invariants, which extend Yetter's invariants to this class of surfaces, are also described in a TQFT setting. A result from the previous article concerning the commuting fraction of a group is generalized to the 2-group context.On the center of the group of quasi-isometries of the real line.https://www.zbmath.org/1452.200362021-02-12T15:23:00+00:00"Chakraborty, Prateep"https://www.zbmath.org/authors/?q=ai:chakraborty.prateepSummary: Let \(QI ( \mathbb{R} )\) denote the group of all quasi-isometries \(f : \mathbb{R} \rightarrow \mathbb{R} \). Let \(Q_+\)(and \(Q_-)\) denote the subgroup of \(QI ( \mathbb{R} )\) consisting of elements which are identity near \(- \infty \) (resp. \(+ \infty )\). We denote by \(QI^+(\mathbb{R} )\) the index 2 subgroup of \(QI ( \mathbb{R} )\) that fixes the ends \(+ \infty, - \infty \). We show that \(QI^+(\mathbb{R}) \cong Q_+ \times Q_-\). Using this we show that the center of the group \(QI ( \mathbb{R} )\) is trivial.