Related papers: Sharp edge, vertex, and mixed Cheeger type inequal…

We investigate the rigidity problem for the sharp spectral gap on Finsler manifolds of weighted Ricci curvature bound $\text{Ric}_{\infty} \geq K > 0$. Our main results show that if the equality holds, the manifold necessarily admits a…

We prove a Cheeger inequality for the first positive Steklov eigenvalue. It involves two isoperimetric constants.

We introduce notions of Cheeger constants for graphons and graphings. We prove Cheeger and Buser inequalities for these. On the way we prove co-area formulae for graphons and graphings.

We prove a new generalization of the higher-order Cheeger inequality for partitioning with buffers. Consider a graph $G=(V,E)$. The buffered expansion of a set $S \subseteq V$ with a buffer $B \subseteq V \setminus S$ is the edge expansion…

In this paper, we study the Steklov eigenvalue of a Riemannian manifold (M, g) with smooth boundary. For compact M , we establish a Cheeger-type inequality for the first Steklov eigenvalue by the isocapacitary constant. For non-compact M ,…

We construct two minimal Cheeger sets in the Euclidean plane, i.e. unique minimizers of the ratio "perimeter over area" among their own measurable subsets. The first one gives a counterexample to the so-called weak regularity property of…

In this note we show the convergence of the fundamental solutions of the parabolic equations assuming the Cheeger-Gromov convergence of the underlying manifolds and the uniform $L^1$-bound of the solutions. We also prove a local integral…

Approximate Bayesian computation has emerged as a standard computational tool when dealing with the increasingly common scenario of completely intractable likelihood functions in Bayesian inference. We show that many common Markov chain…

Kernel methods are successful approaches for different machine learning problems. This success is mainly rooted in using feature maps and kernel matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel matrix, while for…

To solve linear PDEs on metric graphs with standard coupling conditions (continuity and Kirchhoff's law), we develop and compare a spectral, a second-order finite difference, and a discontinuous Galerkin method. The spectral method yields…

We prove a lower bound on the sharp Poincar\'e-Sobolev embedding constants for general open sets, in terms of their inradius. We consider the following two situations: planar sets with given topology; open sets in any dimension, under the…

The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure…

This paper introduces a high-order-accurate strategy for integration of singular kernels and edge-singular integral densities that appear in the context of boundary integral equation formulations of the problem of acoustic scattering. In…

We study inhomogeneous continuous-time weakly ergodic Markov chains with a finite state space. We introduce the notion of a Markov chain with the regular structure of an infinitesimal matrix and study the sharp upper bounds on the rate of…

The goal of the paper is to sharpen and generalise bounds involving the Cheeger's isoperimetric constant $h$ and the first eigenvalue $\lambda_{1}$ of the Laplacian. A celebrated lower bound of $\lambda_{1}$ in terms of $h$,…

Let $d\geq 2$ be an integer and let $2d/(d-1) < q \leq \infty$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big\|\widehat{f\sigma}\big\|_{L^q_{{\rm rad}}L^2_{{\rm…

We state and prove a Cheeger-like inequality for coexact 1-forms on closed orientable Riemannian manifolds.

In terms of the number of triangles, it is known that there are more than exponentially many triangulations of surfaces, but only exponentially many triangulations of surfaces with bounded genus. In this paper we provide a first geometric…

We introduce a new framework that yields spectral bounds on norms of functions of transition maps for finite, homogeneous Markov chains. The techniques employed work for bounded semigroups, in particular for classical as well as for quantum…

We define the spectral gap of a Markov chain on a finite state space as the second-smallest singular value of the generator of the chain, generalizing the usual definition of spectral gap for reversible chains. We then define the relaxation…