Super-expanders and warped cones

For a Banach space $X$, we show that any family of graphs quasi-isometric to levels of a warped cone $\mathcal O_\Gamma Y$ is an expander with respect to $X$ if and only if the induced $\Gamma$-representation on $L^2(Y;X)$ has a spectral gap. This provides examples of graphs that are an expander with respect to all Banach spaces of non-trivial type.Comment: 15 pages; to appear in Ann. Inst. Fourier; exposition rewritten, main result slightly generalised to accommodate local spectral gap

Straightening warped cones

We provide the converses to two results of J. Roe (Geom. Topol. 2005): first, the warped cone associated to a free action of an a-T-menable group admits a fibred coarse embedding into a Hilbert space, and second, a free action yielding a warped cone with property A must be amenable. We construct examples showing that in both cases the freeness assumption is necessary. The first equivalence is obtained also for other classes of Banach spaces, in particular for $L^p$-spaces.Comment: Final authors' version of the article published by JTA. Changes since v2: the proof of Lem. 3.8 (now Prop. 3.10) is split between several lemmata, the proof of Thm 4.2 simplified and more detaile

Two conjectures on coarse conjugacy by Geller and Misiurewicz

In their study of coarse entropy, W. Geller and M. Misiurewicz introduced the notion of coarse conjugacy: a version of conjugacy appropriate for dynamics on metric spaces observed from afar. They made two conjectures on coarse conjugacy generalising their results. We disprove both of these conjectures. We investigate the impact of extra assumptions on the validity of the conjectures: We show that the result of Geller and Misiurewicz towards one of the conjectures can be considered optimal, and we prove the other conjecture under an assumption complementary to that from the referenced work.Comment: 13 page

On Weakly Associative Lattices and Near Lattices

The main aim of this article is to introduce formally two generalizations of lattices, namely weakly associative lattices and near lattices, which can be obtained from the former by certain weakening of the usual well-known axioms. We show selected propositions devoted to weakly associative lattices and near lattices from Chapter 6 of [15], dealing also with alternative versions of classical axiomatizations. Some of the results were proven in the Mizar [1], [2] system with the help of Prover9 [14] proof assistant.Damian Sawicki - Institute of Informatics, University of Białystok, PolandAdam Grabowski - Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Garrett Birkhoff. Lattice Theory. Providence, Rhode Island, New York, 1967.B.A. Davey and H.A. Priestley. Introduction to Lattices and Order. Cambridge University Press, 2002.Ervin Fried and George Grätzer. Some examples of weakly associative lattices. Colloquium Mathematicum, 27:215–221, 1973. doi:10.4064/cm-27-2-215-221.Adam Grabowski. Mechanizing complemented lattices within Mizar system. Journal of Automated Reasoning, 55:211–221, 2015. doi:10.1007/s10817-015-9333-5.Adam Grabowski and Markus Moschner. Managing heterogeneous theories within a mathematical knowledge repository. In Andrea Asperti, Grzegorz Bancerek, and Andrzej Trybulec, editors, Mathematical Knowledge Management Proceedings, volume 3119 of Lecture Notes in Computer Science, pages 116–129. Springer, 2004. doi:10.1007/978-3-540-27818-4_9. 3rd International Conference on Mathematical Knowledge Management, Bialowieza, Poland, Sep. 19–21, 2004.Adam Grabowski and Damian Sawicki. On two alternative axiomatizations of lattices by McKenzie and Sholander. Formalized Mathematics, 26(2):193–198, 2018. doi:10.2478/forma-2018-0017.Adam Grabowski and Christoph Schwarzweller. Translating mathematical vernacular into knowledge repositories. In Michael Kohlhase, editor, Mathematical Knowledge Management, volume 3863 of Lecture Notes in Computer Science, pages 49–64. Springer, 2006. doi:https://doi.org/10.1007/11618027 4. 4th International Conference on Mathematical Knowledge Management, Bremen, Germany, MKM 2005, July 15–17, 2005, Revised Selected Papers.Adam Grabowski, Artur Korniłowicz, and Christoph Schwarzweller. Equality in computer proof-assistants. In Ganzha, Maria and Maciaszek, Leszek and Paprzycki, Marcin, editor, Proceedings of the 2015 Federated Conference on Computer Science and Information Systems, volume 5 of ACSIS-Annals of Computer Science and Information Systems, pages 45–54. IEEE, 2015. doi:10.15439/2015F229.George Grätzer. General Lattice Theory. Academic Press, New York, 1978.George Grätzer. Lattice Theory: Foundation. Birkhäuser, 2011.Dominik Kulesza and Adam Grabowski. Formalization of quasilattices. Formalized Mathematics, 28(2):217–225, 2020. doi:10.2478/forma-2020-0019.William McCune. Prover9 and Mace4. 2005–2010.William McCune and Ranganathan Padmanabhan. Automated Deduction in Equational Logic and Cubic Curves. Springer-Verlag, Berlin, 1996.Ranganathan Padmanabhan and Sergiu Rudeanu. Axioms for Lattices and Boolean Algebras. World Scientific Publishers, 2008.Piotr Rudnicki and Josef Urban. Escape to ATP for Mizar. In First International Workshop on Proof eXchange for Theorem Proving-PxTP 2011, 2011.Stanisław Zukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215–222, 1990.292778

Elective lung resection increases spatial QRS-T angle and QTc interval

Background: Lung resection changes intra-thoracic anatomy, which may affect electrocardiographic results. While postoperative cardiac arrhythmias have been recognized after lung resection, no study has documented changes in vectorcardiographic variables in patients undergoing this surgery. The purpose of this study was to analyse changes in spatial QRS-T angle (spQRS-T) and corrected QT interval (QTc) after lung resection.Methods: Adult patients undergoing elective lung resection under general anaesthesia were studied. The patients were allocated into four groups: those undergoing (1) left lobectomy (LL); (2) left pneumonectomy (LP); (3) right lobectomy (RL); and (4) right pneumonectomy (RP). The spQRS-T angle and QTc interval were measured one day before surgery (baseline) and 24, 48 and 72 h after surgery.Results: Seventy-one adult patients (47 men and 24 women) aged 47–80 (65 ± 7) years were studied. In the study group as a whole, lung resection was associated with significant increases in spQRS-T (p < 0.001) and QTc (p < 0.05 at 24 and 48 h and p < 0.01 at 72 h). The greatest changes were noted in patients undergoing LP. Postoperative atrial fibrillation (AF) was noted in 6.4% of patients studied, in whom the widest spQRS-T angle and the most prolonged QTc intervals were also noted.Conclusions: Lung resection widens the spQRS-T angle and prolongs the QTc interval, especially in patients undergoing LP. While postoperative AF was a relatively rare complication after lung resection in this study, it was associated with the widest spQRS-T angles and most prolonged QTc intervals