ROC curves are one of the most widely used displays to evaluate performance of scoring functions. In the paper, we propose a statistical method for directly optimizing the ROC curve. The target is known to be the regression function up to an increasing transformation and this boils down to recovering the level sets of the latter. We propose to use classifiers obtained by empirical risk minimiza...

where the supremum is taken over all cubes Q ⊂ R containing the point x. In [5], L. Diening proved the following remarkable result: if p− > 1, p+ < ∞ and M is bounded on Lp(·), then M is bounded on L (·), where p′(x) = p(x) p(x)−1 . Despite its apparent simplicity, the proof in [5] is rather long and involved. In this paper we extend Diening’s theorem to weighted variable Lebesgue spaces L p(·)...

We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an L-type norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable app...

We are going to give necessary and suucient conditions for two weight norm inequalities for Haar multipliers operators and for square functions. We also give suucient conditions for two weight norm inequalities for the Hilbert transform. 0. Introduction Weighted norm inequalities for singular integral operators appear naturally in many areas of analysis, probability, operator theory ect. The on...

The supremum over all knot sequences of the max-norm of the orthogonal spline projector is studied with respect to the order k of the splines and their smoothness. It is first bounded from below in terms of the max-norm of the orthogonal projector onto a space of incomplete polynomials. Then, for continuous and for differentiable splines, its order of growth is shown to be √ k.

Abstract An estimate for the norm of selfadjoint Toeplitz operators with a radial, bounded and integrable symbol is obtained. This emphasizes fact that such operator strictly less than supremum symbol. Consequences time-frequency localization are also given.

Proof. In view of [Dou98, Proposition 3.11], it suffices to check that the supremum norm invalidates the parallelogram law. To this end, let us choose a very specific function f (x) = cos2(2πx) and g(x) = sin2(2πx). Then we have f (x) + g(x) = 1 and f (x) − g(x) = cos(4πx), hence ‖ f + g‖ = ‖ f − g‖ = 1. On the other hand, we have ‖ f ‖ = ‖g‖ = 1. Consequently ‖ f + g‖ + ‖ f − g‖ = 2 , 4 = 2‖ f...

The paper proposes a novel event-triggered control scheme for nonlinear systems. Specifically, the closed-loop system is associated with pair of auxiliary input and output. output defined as derivative continuous-time function, while disturbance caused by sampling or equivalently integral over period. As result, it forms cyclic mapping from to via dynamics back integral. law constructed make co...