Let G be a finitely generated group equipped with a finite symmetric generating set and the associated word length function | · |. We study the behavior of the probability of return for random walks driven by symmetric measures μ that are such that ∑ ρ(|x|)μ(x) < ∞ for increasing regularly varying or slowly varying functions ρ, for instance, s 7→ (1+s)α, α ∈ (0, 2], or s 7→ (1 + log(1 + s))ǫ, ǫ...

If G is a finitely presented group, then H2 (∞) (G,A) vanishes for all injective Banach spaces iff the regularized homological area function satisfies the linear isoperimetric inequality. This contrasts with the known result that G is word hyperbolic iff the homological area function satisfies the linear isoperimetric inequality. A closed 3-manifold group G is hyperbolic iff H2 (∞) (G,A) vanish...

Let V and D denote, respectively, total variation and divergence. We study lower bounds of D with V fixed. The theoretically best (i.e. largest) lower bound determines a function L = L(V ), Vajda’s tight lower bound, cf. Vajda, [?]. The main result is an exact parametrization of L. This leads to Taylor polynomials which are lower bounds for L, and thereby extensions of the classical Pinsker ine...

In this paper we give some asymptotic estimates for the best lower bound C(d,k,p) of the Jensen’s functional for when the polynomial f has a concentration d at low degrees k measured by the lpnorm under the Lp-norm, p&gt;2.

We study properties of functions convex with respect to a given family X of vector fields, a notion that appears natural in Carnot-Carathéodory metric spaces. We define a suitable subdifferential and show that a continuous function is X -convex if and only if such subdifferential is nonempty at every point. For vector fields of Carnot type we deduce from this property that a generalized Fenchel...

This paper considers Riemannian metrics on two dimensional disks where all geodesics are minimizing. A sharp reverse isoperimetric inequality is proven. This in turn yields near optimal bounds for the area of disks as well as near optimal upper bounds on the first nonzero Neumann eigenvalue of the Laplacian in terms only of the radius.

The theory and applications of dynamic derivatives on time scales have recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-α derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen’s inequality on time scales via the diamond-α integral and present som...

In this note we prove a version of the classical Schwarz lemma for the first eigenvalue of the Laplacian with Dirichlet boundary data. A key ingredient in our proof is an isoperimetric inequality for the first eigenfunction, due to Payne and Rayner, which we reinterpret as an isoperimetric inequality for a (singular) conformal metric on a bounded domain in the plane.

We work in the density model of random groups. We prove that they satisfy an isoperimetric inequality with sharp constant 1− 2d depending upon the density parameter d. This implies in particular a property generalizing the ordinary C ′ small cancellation condition, which could be termed “macroscopic cancellation”. This also sharpens the evaluation of the hyperbolicity constant δ. As a consequen...

The aim of this study was to investigate social inequality in physical inactivity among adolescents from 1991 to 2014 and to describe any changes in inequality during this period. The analyses were based on data from the Danish part of the HBSC study, which consists of seven comparable cross-sectional studies of nationally representative samples of 11-15-year old adolescents. The available data...