The Refined Metric Dimension with Applications

Marcinkiewicz exponents and integrals over non‐rectifiable paths

“…The summary length of the vertical segments is infinite. This curve was constructed first in . As shown there, , that is, it does not depend on α .…”

“…Then rectangles p nj are mutually disjoint. We put D 1 :D S S 1 S This curve was constructed first in [30]. As shown there, dm.…”

Metric dimensions, generalized integrations, cauchy transform, and riemann boundary-value problem on nonrectifiable arcs

“…We denote by Δ 1 + s 2 Δ 2 the polygon congruent to Δ 1 ∪ Δ 2 for s 2 = +1 and to Δ 1 \ Δ 2 for s 2 = −1. The mapping G is defined on the set of reducible chains by the equation The refined metric dimension is introduced for closed curves in [15]. Let us note that it is independent of the direction of Γ.…”

“…Recently, the author has introduced two new versions of the metric dimension of Γ, the so-called approximation dimension and refined metric dimension (see [15,17]). It is proved that the replacement of the box dimension in (4) by one of these new characteristics improves this solvability condition.…”

“…The results of articles [15,17] are obtained for closed Jordan curves, and this assumption is essential. In the present paper, we prove their analogs for open Jordan arcs.…”