In a recent paper, Pertti Mattila asked which gauge functions φ have the property that for any Borel set A ⊂ R2 with Hausdorff measure Hφ(A) > 0, the projection of A to almost every line has positive length. We show that finiteness of ∫ 1 0 φ(r) r2 dr, which is known to be sufficient for this property, is also necessary for regularly varying φ. Our proof is based on a random construction adapte...

Let y = h(x) be defined for 0 < x < oo and assume values in 0 ^ y ^ + co. Let S be any linear set of points and p an arbitrary positive number. Cover S by a countable number of open intervals h , I2, • • • of lengths xx, x2 , • • • each of which is less than p, and denote by mp(S; h) the lower bound of h(xi) + h(x2) + • • • for all such coverings of S. Then m(S; h) = limp|0mp(/S; h) is called t...

Abstract. With a map f : Ω → R,Ω ⊂ R, that belongs to the John Ball classAp,q(Ω) where n − 1 < p < n and q ≥ p/(p − 1) one can associate a set valued map F whose values F (x) ⊂ R are subsets ofR describing the topological character of the singularity of f at x ∈ Ω. Šverak conjectured that Hn−1(F (S)) = 0, where S is the set of points at which f is not continuous andHn−1 is the Hausdorff measure...

This paper deals with the problem of estimating the level sets L(c) = {F (x) ≥ c}, with c ∈ (0, 1), of an unknown distribution function F on R2+. A plug-in approach is followed. That is, given a consistent estimator Fn of F , we estimate L(c) by Ln(c) = {Fn(x) ≥ c}. In our setting, noncompactness property is a priori required for the level sets to estimate. We state consistency results with res...

Abstract The present paper deals with some existence results for the Darboux problem of partial fractional random differential equations finite delay. arguments are based on a fixed point theorem stochastic domain combined measure noncompactness. An illustration is given to show applicability our results.