This work demonstrates that the distance measuring the likelihood of the graphs of two functions, usually referred as Hausdor distance between functions and widely used in function approximation tasks and signal processing, can be calculated e ciently using grey scale morphological operations even in the case of noncontinuous (discrete as well as nondiscrete) functions. Also we have presented a...

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Σm = {0, ..., m−1} N that are invariant under multiplication by integers. The results apply to the sets {x ∈ Σm : ∀ k, xkx2k · · ·xnk = 0}, where n ≥ 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.

As the identification of surface defects is very important in fruit automatic detection, a new method for the detection of fruit surface defects based on fractal dimension is suggested. In this method, fruit image was collected using computer vision system. The fractal dimension of fruit image was calculated by an improved ‘box dimension’. The fruit fractal dimension reflects the three dimensio...

We prove that the bound on the L norms of the Kakeya type maximal functions studied by Cordoba [2], and by Bourgain [1] are sharp for p > 2. The proof is based on a construction originally due to Schoenberg [5], for which we provide an alternative derivation. We also show that r log(1/r) is the exact Minkowski dimension of the class of Kakeya sets in R, and prove that the exact Hausdorff dimens...

Abstract. In this article, we define the transport dimension of probability measures on Rm using ramified optimal transportation theory. We show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called “the dimensional distance,” on the space of probability measu...

We present the first tree-based regressor whose convergence rate depends only on the intrinsic dimension of the data, namely its Assouad dimension. The regressor uses the RPtree partitioning procedure, a simple randomized variant of k-d trees.

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Σm = {0, ..., m−1}N that are invariant under multiplication by integers. The results apply to the sets {x ∈ Σm : ∀ k, xkx2k · · ·xnk = 0}, where n ≥ 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.

2014 The propagation of a front (we use the model of a forest fire) in a bidimensional random lattice is studied for several different types of interactions. We obtain the corresponding critical concentrations and critical exponents calculated by means of the finite size scaling conjecture. These exponents are expressed in terms of the fractal dimension of the infinite cluster and of the spread...

Orey and Taylor (1974) introduced sets of “fast points” where Brownian increments are exceptionally large, F(λ) := {t ∈ [0, 1] : lim suph→0 |X(t+ h) −X(t)|/ √ 2h| log h|>λ}. They proved that for λ ∈ (0, 1], the Hausdorff dimension of F(λ) is 1 − λ2 a.s. We prove that for any analytic set E ⊂ [0, 1], the supremum of all λ’s for which E intersects F(λ) a.s. equals √ dimP E, where dimP E is the pa...

kA = {a1 + . . .+ ak : ai ∈ A}. We show that for any non-decreasing sequence {αk} ∞ k=1 taking values in [0, 1], there exists a compact set A such that kA has Hausdorff dimension αk for all k ≥ 1. We also show how to control various kinds of dimension simultaneously for families of iterated sumsets. These results are in stark contrast to the Plünnecke-Rusza inequalities in additive combinatoric...