Harmonic functions on hyperbolic graphs

We consider admissible random walks on hyperbolic graphs. For a given harmonic function on such a graph, we prove that asymptotic properties of non-tangential boundedness and non-tangential convergence are almost everywhere equivalent. The proof is inspired by the works of F. Mouton in the cases of Riemannian manifolds of pinched negative curvature and infinite trees. It involves geometric and probabilitistic methods.Comment: 14 page

Wolfe's theorem for weakly differentiable cochains

A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension $m$ in $\mathbb{R}^n$ with the space of flat $m$-cochains, that is, the dual space of flat chains of dimension $m$ in $\mathbb{R}^n$. The main purpose of the present paper is to generalize Wolfe's theorem to the setting of Sobolev differential forms and Sobolev cochains in $\mathbb{R}^n$. A suitable theory of Sobolev cochains has recently been initiated by the second and third author. It is based on the concept of upper norm and upper gradient of a cochain, introduced in analogy with Heinonen-Koskela's concept of upper gradient of a function

Introduction : Text and image in children's literature

Issue theme: Volume 1: Power and Authority in Text and Image: the educational and political dimension of children’s literaturePublisher PDFNon peer reviewe

Numerical simulations of a falling film on the inner surface of a rotating cylinder

A flow in which a thin film falls due to gravity on the inner surface of a vertical, rotating cylinder is investigated. This is performed using two-dimensional (2D) and three-dimensional (3D) direct numerical simulations, with a volume-of-fluid approach to treat the interface. The problem is parameterised by the Reynolds, Froude, Weber and Ekman numbers. The variation of the Ekman number ($Ek$), defined to be proportional of the rotational speed of the cylinder, has a strong effect on the flow characteristics. Simulations are conducted over a wide range of $Ek$ values ($0 \leq Ek \leq 484$) in order to provide detailed insight into how this parameter influences the flow. Our results indicate that increasing $Ek$, which leads to a rise in the magnitude of centrifugal forces, produces a stabilising effect, suppressing wave formation. Key flow features, such as the transition from a 2D to a more complex 3D wave regime, are influenced significantly by this stabilisation, and are investigated in detail. Furthermore, the imposed rotation results in distinct flow characteristics such as the development of angled waves, which arise due to the combination of gravitationally- and centrifugally-driven motion in the axial and azimuthal directions, respectively. We also use a weighted residuals integral boundary layer method to determine a boundary in the space of Reynolds and Ekman numbers that represents a threshold beyond which waves have recirculation regions.Comment: 18 pages, 10 figure

Chromatic numbers of hyperbolic surfaces

Abstract. This article is about chromatic numbers of hyperbolic surfaces. For a metric space, the d-chromatic number is the minimum number of colors needed to color the points of the space so that any two points at distance d are of a different color. We prove upper bounds on the d-chromatic number of any hyperbolic surface which only depend on d. In another direction, we investigate chromatic numbers of closed genus g surfaces and find upper bounds that only depend on g (and not on d). For both problems, we construct families of examples that show that our bounds are meaningful. 1

Chromatic numbers for the hyperbolic plane and discrete analogs

We study colorings of the hyperbolic plane, analogously to the Hadwiger-Nelson problem for the Euclidean plane. The idea is to color points using the minimum number of colors such that no two points at distance exactly $d$ are of the same color. The problem depends on $d$ and, following a strategy of Kloeckner, we show linear upper bounds on the necessary number of colors. In parallel, we study the same problem on $q$-regular trees and show analogous results. For both settings, we also consider a variant which consists in replacing $d$ with an interval of distances