Electronic Research Announcements

April 2018 , Volume 25

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Zermelo deformation of finsler metrics by killing vector fields
Patrick Foulon and Vladimir S. Matveev
2018, 25: 1-7 doi: 10.3934/era.2018.25.001 +[Abstract](708) +[HTML](440) +[PDF](299.36KB)
Abstract:

We show how geodesics, Jacobi vector fields, and flag curvature of a Finsler metric behave under Zermelo deformation with respect to a Killing vector field. We also show that Zermelo deformation with respect to a Killing vector field of a locally symmetric Finsler metric is also locally symmetric.

Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds
Mohammadreza Molaei
2018, 25: 8-15 doi: 10.3934/era.2018.25.002 +[Abstract](608) +[HTML](364) +[PDF](398.47KB)
Abstract:

We consider a discrete dynamical system on a pseudo-Riemannian manifold and we determine the concept of a hyperbolic set for it. We insert a condition in the definition of a hyperbolic set which implies to the unique decomposition of a part of tangent space (at each point of this set) to two unstable and stable subspaces with exponentially increasing and exponentially decreasing dynamics on them. We prove the continuity of this decomposition via the metric created by a torsion-free pseudo-Riemannian connection. We present a global attractor for a diffeomorphism on an open submanifold of the hyperbolic space \begin{document}$H^2(1)$\end{document} which is not a hyperbolic set for it.

Signatures, sums of hermitian squares and positive cones on algebras with involution
Vincent Astier and Thomas Unger
2018, 25: 16-26 doi: 10.3934/era.2018.25.003 +[Abstract](627) +[HTML](363) +[PDF](379.4KB)
Abstract:

We provide a coherent picture of our efforts thus far in extending real algebra and its links to the theory of quadratic forms over ordered fields in the noncommutative direction, using hermitian forms and "ordered" algebras with involution.

On the torsion in the center conjecture
Vitali Kapovitch, Anton Petrunin and Wilderich Tuschmann
2018, 25: 27-35 doi: 10.3934/era.2018.25.004 +[Abstract](532) +[HTML](354) +[PDF](336.41KB)
Abstract:

We present a condition for towers of fiber bundles which implies that the fundamental group of the total space has a nilpotent subgroup of finite index whose torsion is contained in its center. Moreover, the index of the subgroup can be bounded in terms of the fibers of the tower.

Our result is motivated by the conjecture that every almost nonnegatively curved closed \begin{document}$ m $\end{document}-dimensional manifold \begin{document}$ M $\end{document} admits a finite cover \begin{document}$ \tilde M $\end{document} for which the number of leafs is bounded in terms of \begin{document}$ m $\end{document} such that the torsion of the fundamental group \begin{document}$ π_1 \tilde M $\end{document} lies in its center.

On the norm continuity of the hk-fourier transform
Juan H. Arredondo, Francisco J. Mendoza and Alfredo Reyes
2018, 25: 36-47 doi: 10.3934/era.2018.25.005 +[Abstract](495) +[HTML](332) +[PDF](364.71KB)
Abstract:

In this work we study the Cosine Transform operator and the Sine Transform operator in the setting of Henstock-Kurzweil integration theory. We show that these related transformation operators have a very different behavior in the context of Henstock-Kurzweil functions. In fact, while one of them is a bounded operator, the other one is not. This is a generalization of a result of E. Liflyand in the setting of Lebesgue integration.

Explicit geodesics in Gromov-Hausdorff space
Samir Chowdhury and Facundo Mémoli
2018, 25: 48-59 doi: 10.3934/era.2018.25.006 +[Abstract](398) +[HTML](266) +[PDF](397.37KB)
Abstract:

We provide an alternative, constructive proof that the collection \begin{document}${\mathcal{M}}$\end{document} of isometry classes of compact metric spaces endowed with the Gromov-Hausdorff distance is a geodesic space. The core of our proof is a construction of explicit geodesics on \begin{document}${\mathcal{M}}$\end{document}. We also provide several interesting examples of geodesics on \begin{document}${\mathcal{M}}$\end{document}, including a geodesic between \begin{document}${\mathbb{S}}^0$\end{document} and \begin{document}${\mathbb{S}}^n$\end{document} for any \begin{document}$n\geq 1$\end{document}.

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