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In this short note, we prove that singular Reeb vector fields associated with generic -contact forms on three dimensional manifolds with compact embedded critical surfaces have either (at least) or an infinite number of escape orbits, where denotes the number of connected components of the critical set. In case where the first Betti number of a connected component of the critical surface is positive, there exist infinitely many escape orbits. A similar result holds in the case of -Beltrami vector fields that are not -Reeb. The proof is based on a more detailed analysis of the main result in [19]. [ABSTRACT FROM AUTHOR]

This article presents an analysis of the marginalization of the federative entities of Mexico. The study is part of a larger project whose objective is to identify the fundamental economic sectors of Mexico. The sociodemographic comparison of the States of Mexico is formulated as a ranking problem, regarding their marginality level. The research framework is based on the multiple criteria decision-making applying a hierarchical process to analyze the marginality with various dimensions. The data used to evaluate the marginality in Mexico are indicators constructed by the Population and Housing Census information in 2020. [ABSTRACT FROM AUTHOR]

In this article, motivated by the study of symplectic structures on manifolds with boundary and the systematic study of b$b$‐symplectic manifolds started in Guillemin, Miranda, and Pires Adv. Math. 264 (2014), 864–896, we prove a slice theorem for Lie group actions on b$b$‐symplectic manifolds. [ABSTRACT FROM AUTHOR]

In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a |$b$| -symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [ 34 ] and [ 35 ] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and |$b$| -symplectic forms in [ 34 ]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set |$Z$| of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on |$Z$|⁠. [ABSTRACT FROM AUTHOR]

In this article, we construct a compact Riemannian manifold of high dimension on which the time-dependent Euler equations are Turing complete. More precisely, the halting of any Turing machine with a given input is equivalent to a certain global solution of the Euler equations entering a certain open set in the space of divergence-free vector fields. In particular, this implies the undecidability of whether a solution to the Euler equations with an initial datum will reach a certain open set or not in the space of divergence-free fields. This result goes one step further in Tao's programme to study the blow-up problem for the Euler and Navier–Stokes equations using fluid computers. As a remarkable spin-off, our method of proof allows us to give a counterexample to a conjecture of Moore dating back to 1998 on the non-existence of analytic maps on compact manifolds that are Turing complete. [ABSTRACT FROM AUTHOR]