In this paper, we first of all define the distance measure entitled generalized Hausdorff distance between two trapezoidal generalized fuzzy numbers (TGFNs) that has been introduced by Chen [10]. Then using a other distance and combining with generalized Hausdorff distance, we define the similarity measure. The basic properties of the above mentioned similarity measure are proved in detail. Fin...

We use means of formal language theory to estimate the Hausdorff measure of sets of a certain shape in Cantor space. These sets are closely related to infinite iterated function systems in fractal geometry. Our results are used to provide a series of simple examples for the noncoincidence of limit sets and attractors for infinite iterated function systems. ∗A preliminary version appeared as On ...

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a boundedly finite measure. We prove that this space with the ...

The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the n-dimensional Euclidean space, and this was taken further by Hausdorff [Hau19] who generalized the notion of s-dimensional measure to include non-intege...

In this paper we analyze Cantor type sets constructed by the removal of open intervals whose lengths are the terms of the p-sequence, {k−p}∞k=1. We prove that these Cantor sets are s-sets, by providing sharp estimates of their Hausdorff measure and dimension. Sets of similar structure arise when studying the set of extremal points of the boundaries of the so-called random stable zonotopes.

Hausdorff distance is a deformation tolerant measure between two sets of points. The main advantage of this measure is that it does not need an explicit correspondence between the points of the two sets. This paper presents the application to automatic face recognition of a novel supervised Hausdorff-based measure. This measure is designed to minimize the distance between sets of the same class...

Let λ(X) denote Lebesgue measure. If X ⊆ [0, 1] and r ∈ (0, 1) then the r-Hausdorff capacity of X is denoted by H(X) and is defined to be the infimum of all ∑ ∞ i=0 λ(Ii) r where {Ii}i∈ω is a cover of X by intervals. The r Hausdorff capacity has the same null sets as the r-Hausdorff measure which is familiar from the theory of fractal dimension. It is shown that, given r < 1, it is possible to ...

We study the Hausdorff dimension and the pointwise dimension of measures that are not necessarily ergodic. In particular, for conformal expanding maps and hyperbolic diffeomorphisms on surfaces we establish explicit formulas for the pointwise dimension of an arbitrary invariant measure in terms of the local entropy and of the Lyapunov exponents. These formulas are obtained with a direct approac...

I t1 1 ⊗ · · · ⊗ tn n γ. This problem will be called the weak multilinear Hausdorff problem of moments for μk. Comparison with classical results will allow us to relate the weak multilinear Hausdorff problem with the multivariate Hausdorff problem. A solution to the strong multilinear Hausdorff problem of moments will be provided by exhibiting necessary and sufficient conditions for the existen...

It is known that a real valued measure (1) on the a-ring of Baire sets of a locally compact Hausdorff space, or (2) on the Borel sets of a complete separable metric space is regular. Recently Dinculeanu and Kluvanek used regularity of non-negative Baire measures to prove that any Baire measure with values in a locally convex Hausdorff topological vector space (TVS) is regular. Subsequently a di...