Abstract. With help of suitable anisotropic Minkowski’s contents and Hausdorff measures some results are obtained concerning removability of singularities for solutions of partial differential equations with anisotropic growth in the vicinity of the singular set.

Let A be a compact set in Rp of Hausdorff dimension d. For s ∈ (0, d), the Riesz s-equilibrium measure μs is the unique Borel probability measure with support in A that minimizes Is(μ) := " 1 |x − y|s dμ(y)dμ(x) over all such probability measures. If A is strongly (Hd , d)-rectifiable, then μs converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s...

This paper examines the constructive Hausdorff and packing dimensions of weak truth-table degrees. The main result is that every infinite sequence S with constructive Hausdorff dimension dimH(S) and constructive packing dimension dimP(S) is weak truth-table equivalent to a sequence R with dimH(R) ≥ dimH(S)/dimP(S)− ǫ, for arbitrary ǫ > 0. Furthermore, if dimP(S) > 0, then dimP(R) ≥ 1− ǫ. The re...

In computer vision, the partial Hausdorff distances (PHDs) are used to compare images but with strong limitations of using fixed fractions. In this paper, we formulate an adaptive and robust Hausdorff distance (ARHD) with nonparametric and robust statistical methods. The new distance is estimated using the empirical distribution of the distance variable based on the distance map of the template...

In this paper, the concept of fuzzy automata normed linear structure spaces is introduced and suitable examples are provided. ;The ;concepts of fuzzy automata $alpha$-open sphere, fuzzy automata $mathscr{N}$-locally compact spaces, fuzzy automata $mathscr{N}$-Hausdorff spaces are also discussed. Some properties related with to fuzzy automata normed linear structure spaces and fuzzy automata $ma...

Using the domain-theoretic model for geometric computation, we define the partial Delaunay triangulation and the partial Voronoi diagram of N partial points in R2 and show that these operations are domain-theoretically computable and effectively computable with respect to Hausdorff distance and Lebesgue measure. These results are obtained by showing that the map which sends three partial points...

We investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients Sn(x) = ∑n j=1 aj(x), where x = [a1(x), a2(x), · · · ] is the continued fraction expansion of an irrational x ∈ (0, 1). Precisely, for an increasing function φ : N → N, one is interested in the Hausdorff dimension of the sets Eφ = { x ∈ (0, 1) : lim n→∞ Sn(x) φ(n) = 1 } . Several cases...

We show that if a real x ∈ 2 is strongly Hausdorff Hh-random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π...

In this paper, we presented a novel approach for automatic 3D face verification from range data. The method consists of range data registration and comparison. There are two steps in registration procedure: the coarse step conducting the normalization by exploiting a priori knowledge of the human face and facial features, and the fine step aligning the input data with the model stored in the da...

We continue the study of effective Hausdorff dimension as it was initiated by LUTZ. Whereas he uses a generalization of martingales on the Cantor space to introduce this notion we give a characterization in terms of effective s-dimensional Hausdorff measures, similar to the effectivization of Lebesgue measure by MARTIN-LÖF. It turns out that effective Hausdorff dimension allows to classify sequ...