Consider a neighborhood graph, for example a k-nearest neighbor graph, that is constructed on sample points drawn according to some density p. Our goal is to re-weight the graph’s edges such that all cuts and volumes behave as if the graph was built on a different sample drawn from an alternative density p. We introduce the f -adjusted graph and prove that it provides the correct cuts and volum...

We consider the oscillatory property of the following p(t)-Laplacian equations −(|u′|p(t)−2u′)′ = 1/tθ(t)g(t,u), t > 0. Since there is no Picone-type identity for p(t)Laplacian equations, it is an unsolved problem that whether the Sturmian comparison theorems for p(x)-Laplacian equations are valid or not. We obtain sufficient conditions of the oscillatory of solutions for p(t)-Laplacian equations.