Inductive Lusternik-Schnirelmann category in a model category

We introduce the notion of inductive category in a model category and prove that it agrees with the Ganea approach given by Doeraene. This notion also coincides with the topological one when we consider the category of (well-) pointed topological spaces.Comment: 14 page

Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk

In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for a class of Hamiltonian systems at a fixed energy level.Comment: 59 pages, 3 figures. To appear on Nonlinear Analysis Series A: Theory, Methods & Application

The boundary value problem for discrete analytic functions

This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal. We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S. Smirnov from 2010. This was proved earlier by R. Courant-K. Friedrichs-H. Lewy and L. Lusternik for square lattices, by D. Chelkak-S. Smirnov and implicitly by P.G. Ciarlet-P.-A. Raviart for rhombic lattices. In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A. Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory.Comment: 22 pages, 6 figures. Several changes: Theorem 1.2 generalized, several assertions added, minor correction in the proofs of Lemma 2.5, 3.3, Example 3.6, Corollary 5.

Module sectional category of products

Adapting a result of Félix–Halperin–Lemaire concerning the Lusternik–Schnirelmann category of products, we prove the additivity of a rational approximation for Schwarz’s sectional category with respect to products of certain fibrations.J.C. is supported by the Polish National Science Centre Grant 2016/21/ P/ST1/03460 within the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665778 and by the Belgian Interuniversity Attraction Pole (IAP) within the framework “Dynamics, Geometry and Statistical Physics” (DYGEST P7/18). L.V. is partially supported by Portuguese Funds through FCT – Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013

The type numbers of closed geodesics

A short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesicsComment: 29 pages, an appendix to the Russian translation of "The calculus of variations in the large" by M. Mors

Secondary LS category of measured laminations

In the author's Ph.D., a version of the tangential LS category for foliated spaces depending on a transverse invariant measure, called the measured category, was introduced. Unfortunately, the measured category vanishes easily. When it is zero, the rate of convergence to zero of the quantity involved in the definition, by taking arbitrarily large homotopies, gives a new invariant, called the secondary measured category. Several versions of classical results are proved for the secondary measured category. It is also shown that the secondary measured category is a transverse invariant related to the growth of (pseudo)groups. The equality between secondary category and the growth of a group is done in the case of free suspensions by Rohlin groups.Comment: 14 pages. arXiv admin note: substantial text overlap with arXiv:1112.500

Measurable versions of the LS category on laminations

We give two new versions of the LS category for the set-up of measurable laminations defined by Berm\'udez. Both of these versions must be considered as "tangential categories". The first one, simply called (LS) category, is the direct analogue for measurable laminations of the tangential category of (topological) laminations introduced by Colman Vale and Mac\'ias Virg\'os. For the measurable lamination that underlies any lamination, our measurable tangential category is a lower bound of the tangential category. The second version, called the measured category, depends on the choice of a transverse invariant measure. We show that both of these "tangential categories" satisfy appropriate versions of some well known properties of the classical category: the homotopy invariance, a dimensional upper bound, a cohomological lower bound (cup length), and an upper bound given by the critical points of a smooth function.Comment: 22 page

Les premières étapes du calcul symbolique.

Lusternik L. A. Les premières étapes du calcul symbolique.. In: Revue d'histoire des sciences, tome 25, n°3, 1972. pp. 201-206