We define a multidimensional rearrangement, which is related to classical inequalities for functions that are monotone in each variable. We prove the main measure theoretical results of the new theory and characterize the functional properties of the associated weighted Lorentz spaces. Mathematics Subject Classification 2000: 46E30, 46B25.

This note contains a new algorithm for computing a function f(k) introduced by Erdős to measure the minimal gap size in the sequence of integers at least one of whose prime factors exceeds k. This algorithm enables us to show that f(k) is not monotone, verifying a conjecture of Ecklund and Eggleton.

For noninvertible maps, mainly subshifts of finite type and piecewise monotone interval maps, we investigate what happens if we follow backward trajectories, random in the sense that at each step every preimage can be chosen with equal probability. In particular, we ask what happens if we try to compute the entropy this way. It tuns out that instead of the topological entropy we get the metric ...

We consider a compound testing problem within the Gaussian sequence model in which the null and alternative are specified by a pair of closed, convex cones. Such cone testing problem arise in various applications, including detection of treatment effects, trend detection in econometrics, signal detection in radar processing, and shape-constrained inference in non-parametric statistics. We provi...

We propose a new solution concept to address the problem of sharing a surplus among the agents generating it. The problem is formulated in the preferences-endowments space. The solution is defined recursively, incorporating notions of consistency and fairness and relying on properties satisfied by the Shapley value for Transferable Utility (TU) games. We show a solution exists, and call it an O...

We consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on d × d × n-dimensional tripartite mixed states. We show that this assisted entanglement, when measured in terms of the generalized concurrence (named G-concurrence) is (tightly) bounded by an entanglement monotone, which we call the G-concurren...

We propose a new solution concept to address the problem of sharing a surplus among the agents generating it. The problem is formulated in the preferences-endowments space. The solution is defined recursively, incorporating notions of consistency and fairness and relying on properties satisfied by the Shapley value for Transferable Utility (TU) games. We show a solution exists, and call it the ...

We propose a new solution concept to address the problem of sharing a surplus among the agents generating it. The sharing problem is formulated in the preferences-endowments space. The solution is defined in a recursive manner incorporating notions of consistency and fairness and relying on properties satisfied by the Shapley value for Transferable Utility (TU) games. We show a solution exists,...