We prove a covering lemma for rectangles in Rn which has connections to a problem of Zygmund and its solution in three dimensions by Cordoba. One of the objectives of this note is to revive interest in a collection of problems in differentiation theory related to a conjecture of Zygmund. We shall also give another proof of the sharp mapping properties near L of the strong maximal operator by pr...

Given a row contraction of operators on Hilbert space and a family of projections on the space which stabilize the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries which satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in ...

Fuzzy Mathematical Morphology aims to extend the binary morphological operators to grey-level images. In order to define the basic morphological operations fuzzy erosion, dilation, opening and closing, we introduce a general method based upon fuzzy implication and inclusion grade operators, including as particular case, other ones existing in related literature In the definition of fuzzy erosio...

We introduce a new class of strongly degenerate nonlinear parabolic PDEs ( p − 2 ) Δ ∞ , X N + u Y t m ⋅ ∇ ∂ = 0 ∈ R × 1 combining the classical PDE Kolmogorov and normalized -Laplace operator. characterize solutions in terms an asymptotic mean value property results are connected to analysis certain tug-of-war games with noise. The functions for introduced approximate stated when parameter tha...

Frames which can be generated by the action of some operators (e.g. translation, dilation, modulation, ...) on a single element $f$ in a Hilbert space, called coherent frames. In this paper, we introduce a class of continuous frames in a Hilbert space $mathcal{H}$ which is indexed by some locally compact group $G$, equipped with its left Haar measure. These frames are obtained as the orbits of ...

The concept of strictness of morphological operators is discussed, and it is shown that the ordinary morphological operators have extreme strictness which leads to sensitivity to noise and digital artifacts. Based on this observation new morphological operators that generalize the ordinary morphological operators, are de ned. The generalized operators have controllable strictness and so excessi...