Medians, continuity, and vanishing oscillation

Vanishing Mean Oscillation and Regularity in the Calculus of Variations

Toeplitz Operators and Toeplitz Algebra with Symbols of Vanishing Oscillation

We study the C∗-algebra generated by Teoplitz operators with symbols of vanishing (mean) oscillation on the Bergman space of the unit ball. We show that the index calculation for Fredholm operators in this C∗-algebra can be easily and completely reduced to the classic case of Toeplitz operators with symbols that are continuous on the closed unit ball. Moreover, in addition to a number of other ...

Quantiles and Medians

We provide a list of functional equations that characterize quantile functions for collections of bounded and measurable functions. Our central axiom is ordinal covariance. When a probability measure is exogeneously given, we characterize quantiles with respect to that measure through monotonicity with respect to stochastic dominance. When none is given, we characterize those functions which ar...

Geometric medians

We discuss several generalizations of the notion of median to points in $R^d$. They arise in Computational Geometry and in Statistics. The notions are compared with respect to some of their mathematical properties. We also consider computational aspects. The issue of computational complexity raises several intriguing questions.

Range Medians

We study a generalization of the classical median finding problem to batched query case: given an array of unsorted n items and k intervals in the array, the goal is to determine the median in each of the intervals in the array. We give an algorithm that uses O(n log k) comparisons and show a matching lower bound of Ω(n log k) comparisons for this problem.