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In this paper we study a Dirichlet-type differential inclusion involving the Finsler-Laplace operator on a complete Finsler manifold. Depending on the positive $\lambda$ parameter of the inclusion, we establish non-existence, as well as existence and multiplicity results by applying non-smooth variational methods. The main difficulties are given by the problem's highly nonlinear nature due to the general Finslerian setting, as well as the nonsmooth context., Comment: 6 pages

We propose a train rescheduling algorithm which applies a standardized feature selection based on pairwise conflicts in order to serve as input for the reinforcement learning framework. We implement an analytical method which identifies and optimally solves every conflict arising between two trains, then we design a corresponding observation space which features the most relevant information considering these conflicts. The data obtained this way then translates to actions in the context of the reinforcement learning framework. We test our preliminary model using the evaluation metrics of the Flatland Challenge. The empirical results indicate that the suggested feature space provides meaningful observations, from which a sensible scheduling policy can be learned., Comment: 20th Jubilee World Symposium on Applied Machine Intelligence and Informatics (SAMI), Poprad, Slovakia, March 2-5, 2022

We present the isometry between the 2-dimensional Funk model and the Finsler-Poincar\'e disk. Then, we introduce the Finslerian Poincar\'e upper half plane model, which turns out to be also isometrically equivalent to the previous models. As application, we state the gapless character of the first eigenvalue for the aforementioned three spaces., Comment: IEEE 15th International Symposium on Applied Computational Intelligence and Informatics, Timisoara, Romania

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Krist\'aly [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey-Sobolev inequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature. A byproduct of this method is a Hardy-Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler-Laplace operator. Our results are also new in the Riemannian setting., Comment: 18 pages; to appear in Comm. Contemp. Math

We establish a bipolar Hardy inequality on complete, not necessarily reversible Finsler manifolds. We show that our result strongly depends on the geometry of the Finsler structure, namely on the reversibility constant $r_F$ and the uniformity constant $l_F$. Our result represents a Finslerian counterpart of the Euclidean multipolar Hardy inequality due to Cazacu and Zuazua (2013) and the Riemannian case considered by Faraci, Farkas and Krist\'aly (2018).

We establish Hardy inequalities involving a weight function on complete, not necessarily reversible Finsler manifolds. We prove that the superharmonicity of the weight function provides a sufficient condition to obtain Hardy inequalities. Namely, if $\rho$ is a nonnegative function and $-\boldsymbol{\Delta} \rho \geq 0$ in weak sense, where $\boldsymbol{\Delta}$ is the Finsler-Laplace operator defined by $ \boldsymbol{\Delta} \rho = \mathrm{div}(\boldsymbol{\nabla} \rho)$, then we obtain the generalization of some Riemannian Hardy inequalities given in D'Ambrosio and Dipierro (Ann. Inst. H. Poincar\'e, 2013). By extending the results obtained, we prove a weighted Caccioppoli-type inequality, a Gagliardo-Nirenberg inequality and a Heisenberg-Pauli-Weyl uncertainty principle on complete Finsler manifolds. Furthermore, we present some Hardy inequalities on Finsler-Hadamard manifolds with finite reversibility constant, by defining the weight function with the help of the distance function. Finally, we extend a weighted Hardy-inequality to a class of Finsler manifolds of bounded geometry.

Given a complete non-compact Riemannian manifold $(M,g)$ with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries $G$ of $(M,g)$ that characterizes the coerciveness of $G$ in the sense of Skrzypczak and Tintarev (Arch. Math., 2013). Furthermore, under these conditions, compact Sobolev-type embeddings \`a la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embedding properties as their Riemannian companions; sharpness of such constructions are shown by means of the Funk model. As an application, a quasilinear PDE on Randers spaces is studied by using the above compact embeddings and variational arguments., Comment: 24 pages

Let $(M,g)$ be an $n$-dimensional $(n\geq 3)$ compact Riemannian manifold with Ric$_{(M,g)}\geq (n-1)g$. If $(M,g)$ supports an AB-type critical Sobolev inequality with Sobolev constants close to the optimal ones corresponding to the standard unit sphere $(\mathbb S^n,g_0)$, we prove that $(M,g)$ is topologically close to $(\mathbb S^n,g_0)$. Moreover, the Sobolev constants on $(M,g)$ are precisely the optimal constants on the sphere $(\mathbb S^n,g_0)$ if and only if $(M,g)$ is isometric to $(\mathbb S^n,g_0)$; in particular, the latter result answers a question of V.H. Nguyen., Comment: 7 pages