In set theory, the foundation axiom (or regularity)(F) says that the relation ∈ is well-founded, that is, there is no infinite descending ∈-sequence : · · · ∈ x 2 ∈ x 1 ∈ x 0. If we identify ∈ in a set with ← in a graph, the set is identified with a graph. The set 1 is 1 = {φ}, that is, φ ∈ 1. It corresponds to a graph x 1 ← x 0 with nodes x 0 , x 1. In terms of graphs and nodes, the well-found...

In this paper we prove that for every bumpy Finsler metric F on every rationally homological n-dimensional sphere Sn with n ≥ 2, there exist always at least two distinct prime closed geodesics.

We consider the p-Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite p and investigate the limit problem as p →∞.

We consider the p–Laplacian operator on a domain equipped with a Finsler metric. After deriving and recalling relevant properties of its first eigenfunction for p > 1, we investigate the limit problem as p → 1.

In this paper, we prove that for every Finsler metric on S there exist at least two distinct prime closed geodesics. For the case of the two-sphere, this solves an open problem posed by D. V. Anosov in 1974.

We define the Liouville distribution on the tangent bundle of a pseudo-Finsler manifold and prove that it is integrable. Also, we find geometric properties of both leaves of Liouville distribution and the vertical distribution.

Motivated in part by the bi-gravity approach to massive gravity, we introduce and study multimetric Finsler geometry. For case of an arbitrary number dimensions, some general properties geometry terms its Riemannian ingredients, while two-dimensional case, derive all Cartan equations as well explicitly find Holmes–Thompson measure.

The aim of this paper is to construct horizontal Chern forms of a holomorphic vector bundle using complex Finsler structures. Also, some properties of these forms are studied.

We show the existence of at least two geometrically distinct closed geodesics on an n-dimensional sphere with a bumpy and non-reversible Finsler metric for n > 2.