Using the piecewise monotone property, we give a full description of non-monotonicity height of PM functions with a single fort on compact interval. This method is also available for general PM functions with finitely many forts, as well as those functions defined on the whole real line. Finally, we obtain a sufficient and necessary condition for the finite non-monotonicity height by characteri...

Shape constrained densities are encountered in many nonparametric estimation problems. The classes of monotone or convex (and monotone) densities can be viewed as special cases of the classes of k−monotone densities. A density g is said to be k−monotone if (−1)g is nonnegative, nonincreasing and convex for l = 0, . . . , k−2 if k ≥ 2, and g is simply nonincreasing if k = 1. These classes of sha...

We survey a number of minor-monotone graph parameters and their relationship to the complexity of routing on graphs. In particular we compare the interval routing parameters κslir(G) and κsir(G) with Colin de Verdière’s graph invariant μ(G) and its variants λ(G) and κ(G). We show that for all the known characterizations of θ(G) with θ(G) being μ(G), λ(G) or κ(G), that θ(G) ≤ 2κslir(G) − 1 and θ...

In this paper, we consider several property testing problems and ask how the query complexity depends on the distance parameter ǫ. We achieve new lower bounds in this setting for the problems of testing whether a function is monotone and testing whether the function has low Fourier degree. For monotonicity testing, our lower bound matches the recent upper bound of Chakrabarty and Seshadhri [12].

We shall consider slightly more general problems. Namely, we shall investigate the existence of continuous: piecewise monotone, piecewise strictly monotone, and piecewise linear n-th roots of interval maps which have a continuous n-th root. Here by an n-th root of f we mean a map g such that f = g (g is the n-th iterate of g). A continuous map f : I → J , where I, J are closed intervals, is pie...

In this work we are concerned with maximality of monotone operators representable by certain convex functions in non-reflexive Banach spaces. We also prove that these maximal monotone operators satisfy a Brønsted-Rockafellar type property. 2000 Mathematics Subject Classification: 47H05, 49J52, 47N10.

A monotone drawing of a graph G is a straight-line drawing of G such that every pair of vertices is connected by a path that is monotone with respect to some direction. Trees, as a special class of graphs, have been the focus of several papers and, recently, He and He [6] showed how to produce a monotone drawing of an arbitrary n-vertex tree that is contained in a 12n× 12n grid. In this paper, ...