The $(\varphi,k)$ rectifiable subsets of $n$-space

Subsets of Rectifiable curves in Hilbert Space-The Analyst’s TSP

We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do so by extending and improving results of Peter Jones and Kate Okikiolu for sets in Rd . Their results formed the basis of quantitative rectifiability in Rd . We prove a quantitative version of the f...

Hypersurfaces of the Heisenberg Group Are N-rectifiable

Definition 1. Let N ′ be a Carnot group and N be a subgroup of N ′ with Hausdorff dimension k. A subset E of another Carnot Group (M,dM ) is said to be N -rectifiable if there exists U , a positive measure subset of N , and a Lipschitz map f : U → M such that H k M (E \ f(U)) = 0. We say E is countably N -rectifiable if there exist a countable number of pairs of subsets Ui and Lipschitz maps fi...

m-Ary Hypervector Space: Convergent Sequences and Bundle Subsets.

In this paper, we have generalized the definition of vector space by considering the group as a canonical $m$-ary hypergroup, the field as a krasner $(m,n)$-hyperfield and considering the multiplication structure of a vector by a scalar as hyperstructure. Also we will be consider a normed $m$-ary hypervector space and introduce the concept of convergence of sequence on $m$-ary hypernormed space...

Rectifiable and Flat G Chains in a Metric Space

A new class of ( H k , 1 ) - rectifiable subsets of metric spaces ∗

The main motivation of this paper arises from the study of Carnot–Carathéodory spaces, where the class of 1-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree k, which are Hö...