‎In this note‎, ‎we consider the concentration function problem for a continuous action of a locally compact group $G$ on a locally compact Hausdorff space $X$‎. ‎We prove a necessary and sufficient condition for the concentration functions of a‎ ‎spread-out irreducible probability measure $mu$ on $G$ to converge to zero.

Let C be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in R 2 with two independent approximation functions; that is if a certain sum converges then the set of all points (x, y) on the curve which satisfy simultaneously the inequalities qx < ψ1(q) and qy < ψ2(q) infinitely often has induced measure 0. This co...

We show that a Radon measure μ in R which is absolutely continuous with respect to the n-dimensional Hausdorff measure H is n-rectifiable if the so called Jones’ square function is finite μ-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all n-rectifiable measures which are absolutely contin...

We consider the problem of improving outcomes for neurosurgery patients by enhancing intraoperative navigation and guidance. Current navigation systems do not accurately account for intraoperative brain deformation. We focus on the brain shift deformation that occurs just after the opening of the skull and dura. The heart of our system is a nonrigid registration technique using a biomechanical ...

Let A be a compact d-rectifiable set embedded in Euclidean space R, d ≤ p. For a given continuous distribution σ(x) with respect to d-dimensional Hausdorff measure on A, our earlier results provided a method for generating N -point configurations on A that have asymptotic distribution σ(x) as N →∞; moreover such configurations are “quasi-uniform” in the sense that the ratio of the covering radi...

In this paper we prove that if φ : C → C is a K-quasiconformal map, with K > 1, and E ⊂ C is a compact set contained in a ball B, then Ċ 2K 2K+1 , 2K+1 K+1 (E) diam(B) 2 K+1 ≥ c−1 ( γ(φ(E)) diam(φ(B)) ) 2K K+1 , where γ stands for the analytic capacity and Ċ 2K 2K+1 , 2K+1 K+1 is a capacity associated to a non linear Riesz potential. As a consequence, if E not K-removable, it has positive capac...

Let A be a given compact subset of the euclidean space. We consider the problem of finding a compact connected set S of minimal 1dimensional Hausdorff measure, among all compact connected sets containing A. We prove that when A is a finite set any minimizer is a finite tree with straight edges, thus recovery the classical Steiner Problem. Analogously, in the case when A is countable, we prove t...

To begin with, let us agree to some basic conventions. We employ the symbols ∆ and ∇ to denote the Laplace operator ∑nk=1 ∂/∂xk and the gradient vector (∂/∂x1, ..., ∂/∂xn) over the Euclidean space R , n ≥ 2. For notational convenience we use X . Y as X ≤ CY for a constant C > 0. We always assume that u is a smooth real-valued function on R, written u ∈ C∞(Rn), and then it generates a conformal ...

In this paper, a new definition of locally convex L-topological vector spaces is given. The relationship between this new definition and the previous definition of locally convex L-topological vector spaces given by Yan and Fang in 1999 is investigated. Moreover, the concept of generalized L-fuzzy semi-norm is introduced. By using a family of generalized L-fuzzy semi-norms, a characterization o...

By reparametrization, one may assume that γ is Lipschitz with constant equal to length(γ). We will also make use of the following simple (and well-known) criterion: a compact set Γ is the image of a rectifiable curve if and only if it is connected and H(Γ) < ∞. Indeed, one may choose γ so that length(γ) ≤ CH(Γ); see, for example, [1, 2]. Here and below, H denotes the one-dimensional Hausdorff m...