Related papers: Packing measures and dimensions on cartesian produ…
We study continuity and discontinuity of the upper and lower (modified) box-counting, Hausdorff, packing, (modified) correlation measure-dimension mappings under the weak, setwise and TV topology on the space of Borel measures respectively…
We examine Frostman-type characterisations and other extremal measure criteria for a range of fractal dimensions of sets. In particular we derive properties of the less familiar modified lower box dimension and upper correlation dimension.…
A construction of product measures is given for an arbitrary sequence of measure spaces via outer measure techniques without imposing any condition on the underlying measure spaces. This approach concludes finally the problem of the…
We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also…
Alternative novel measures of the distance between any two partitions of a n-set are proposed and compared, together with a main existing one, namely 'partition-distance' D(.,.). The comparison achieves by checking their restriction to…
Besicovitch showed that if a set is null for the Hausdorff measure associated to a given dimension function, then it is still null for the Hausdorff measure corresponding to a smaller dimension function. We prove that this is not true for…
In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the…
Basic properties of Hausdorff content, dimension, and measure of subsets of metric spaces are discussed, especially in connection with Lipschitz mappings and topological dimension.
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. In the context of a branch-and-bound framework for solving these packing problems to optimality, it is…
In this work we study the Hausdorff dimension of measures whose weight distribution satisfies a markov non-homogeneous property. We prove, in particular, that the Hausdorff dimensions of this kind of measures coincide with their lower…
The orthogonality properties of certain subspaces associated with bivariate Bernstein-Szeg\H{o} measures are considered. It is shown that these spaces satisfy more orthogonality relations than expected from the relations that define them.…
Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure…
We study a generalization of Mor\'an's sum sets, obtaining information about the $h$-Hausdorff and $h$-packing measures of these sets and certain of their subsets.
We investigate the box dimensions of inhomogeneous self-similar sets. Firstly, we extend some results of Olsen and Snigireva by computing the upper box dimensions assuming some mild separation conditions. Secondly, we investigate the more…
We investigate a quasisymmetrically invariant counterpart of the topological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension, gives a lower bound on the topological Hausdorff dimension of…
We discuss the problem of determining the dimension of self-similar sets and measures on $\mathbf{R}$. We focus on the developments of the last four years. At the end of the paper, we survey recent results about other aspects of…
We introduce a concept of porosity for measures and study relations between dimensions and porosities for two classes of measures: measures on $R^n$ which satisfy the doubling condition and strongly porous measures on $R$.
We study certain infinite-dimensional probability measures in connection with frame analysis. Earlier work on frame-measures has so far focused on the case of finite-dimensional frames. We point out that there are good reasons for a sharp…
A number of topics involving metrics and measures are discussed, including some of the special structure associated with ultrametrics.
Notions of (pointwise) tangential dimension are considered, for measures of R^n. Under regularity conditions (volume doubling), the upper resp. lower dimension at a point x of a measure can be defined as the supremum, resp. infimum, of…