Monotonicity Theorems

More Monotonicity Theorems for Partitions

Consider the collection of all integer partitions, whose part sizes lie in a given set. Such a set is called monotone if the generating function has weakly increasing coefficients. The monotone subsets are classified, assuming an open conjecture.

Monotonicity Theorems for Laplace Beltrami Operator on Riemannian Manifolds

Abstract. For free boundary problems on Euclidean spaces, the monotonicity formulas of Alt-Caffarelli-Friedman and Caffarelli-Jerison-Kenig are cornerstones for the regularity theory as well as the existence theory. In this article we establish the analogs of these results for the LaplaceBeltrami operator on Riemannian manifolds. As an application we show that our monotonicity theorems can be e...

Stochastic Monotonicity and Realizable Monotonicity

We explore and relate two notions of monotonicity, stochastic and realizable, for a system of probability measures on a common finite partially ordered set (poset) S when the measures are indexed by another poset A. We give counterexamples to show that the two notions are not always equivalent, but for various large classes of S we also present conditions on the poset A that are necessary and s...

Fixed Point Theorems for Nonlinear Operators with and without Monotonicity in Partially Ordered Banach Spaces

Monotonicity Testing

We present improved algorithms for testing monotonicity of functions. Namely, given the ability to query an unknown function f : E' -+ E, where E and E are finite ordered sets, the test always accepts a monotone f and rejects f with high probability if it is e-far from being monotone (i.e., if every monotone function differs from f on more than an E fraction of the domain). For any E > 0, the q...