On A Paper Of Saha And Ray
On Conditions for Unrectifiability of a Metric Space
We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.
On dimensions of semimetrized measure spaces
On hypersurfaces of limite type
On indecomposable sets with applications
In this note we show the characteristic function of every indecomposable set F in the plane is BVequivalent to the characteristic function a closed set See Formula in PDF See Formula in PDF . We show by example this is false in dimension three and above. As a corollary to this result we show that for everyϵ > 0 a set of finite perimeter Scan be approximated by a closed subset See Formula in PDF See Formula in PDF with finitely many indecomposable components and with the property that See...
On multiplicity functions and Lebesgue area
On Nikodym-type sets in high dimensions
We prove that the complement of a higher-dimensional Nikodym set must have full Hausdorff dimension.
On sets under certain transformations in RN.
On some properties of the Cantor set and the construction of a class of sets with Cantor set properties
On the 1/2 Problem of Besicovitch: quasi-arcs do not contain sharp saw-teeth.
In this paper we give an alternative proof of our recent result that totally unrectifiable 1-sets which satisfy a measure-theoretic flatness condition at almost every point and sufficiently small scales, satisfy Besicovitch's 1/2-Conjecture which states that the lower spherical density for totally unrectifiable 1-sets should be bounded above by 1/2 at almost every point. This is in contrast to rectifiable 1-sets which actually possess a density equal to unity at almost every point. Our present method...
On the connection between Hausdorff measures and generalized capacity
On the continuity of the Faber mapping
On the Euler Characteristic in Spaces with a Separability Property.
On the Hausdorff dimension of some fractal sets
On the measurability of sets of pairs of intersecting nonisotropic straight lines of type beta in the simply isotropic space
The measurable sets of pairs of intersecting non-isotropic straight lines of type and the corresponding densities with respect to the group of general similitudes and some its subgroups are described. Also some Crofton-type formulas are presented.
On the minima of functionals with linear growth
On the Relations Between 2D and 3D Fractal Dimensions: Theoretical Approach and Clinical Application in Bone Imaging
The inner knowledge of volumes from images is an ancient problem. This question becomes complicated when it concerns quantization, as the case of any measurement and in particular the calculation of fractal dimensions. Trabecular bone tissues have, like many natural elements, an architecture which shows a fractal aspect. Many studies have already been developed according to this approach. The question which arises however is to know to which extent it is possible to get an exact determination of the...
On the visibility of invisible sets.
On transformations of sets in
Optimal transportation for multifractal random measures and applications
In this paper, we study optimal transportation problems for multifractal random measures. Since these measures are much less regular than optimal transportation theory requires, we introduce a new notion of transportation which is intuitively some kind of multistep transportation. Applications are given for construction of multifractal random changes of times and to the existence of random metrics, the volume forms of which coincide with the multifractal random measures.