Let G be a graph and p ∈ [1,∞]. The parameter fp(G) is the least integer k such that for all m and all vectors (rv)v∈V (G) ⊆ R , there exist vectors (qv)v∈V (G) ⊆ R k satisfying ‖rv − rw‖p = ‖qv − qw‖p, for all vw ∈ E(G). It is easy to check that fp(G) is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour [9], there are a finite number of excluded m...

We give a result on existence of periodic orbits for autonomous differential systems with arbitrary finite dimension. It is based on a Poincaré-Bendixson property enjoyed by a new class of monotone systems introduced in L. A. Sanchez, Cones of rank 2 and the PoincaréBendixson property for a new class of monotone systems, Journal of Differential Equations 216 (2009), 1170-1190. A concrete applic...

It is proved that for any given integer k > 2 and a k-monotone function g, there exists a k-monotone function g̃ ≤ g of the form c(a − x) + and passing through a fixed point in the support of g. The result is motivated by the problem of the existence of the LSE of a k-monotone density.

We consider the problem of distribution-free testing of the class of monotone monomials and the class of monomials over n variables. While there are very efficient testers for a variety of classes of functions when the underlying distribution is uniform, designing distribution-free testers (which must work under an arbitrary and unknown distribution) tends to be more challenging. When the under...

Monotone Lévy processes with additive increments are defined and studied. It is shown that these processes have natural Markov structure and their Markov transition semigroups are characterized using the monotone Lévy-Khintchine formula. 17 Monotone Lévy processes turn out to be related to classical Lévy processes via Attal's " remarkable transformation. " A monotone analogue of the family of e...

Given a monotone graphical property Q, how large should d(n) be to ensure that if (Hn) is any sequence of graphs satisfying |Hn| = n and (Hn)¿d(n), then almost every induced subgraph of Hn has property Q? We prove essentially best possible results for the following monotone properties: (i) k-connected for :xed k, (ii) Hamiltonian. c © 2002 Elsevier Science B.V. All rights reserved.

Let k ≥ 2 be a fixed integer and P be a property of k-uniform hypergraphs. In other words, P is a (typically infinite) family of k-uniform hypergraphs and we say a given hypergraph H satisfies P if H ∈ P . For a given constant η > 0 a k-uniform hypergraph H on n vertices is η-far from P if no hypergraph obtained from H by changing (adding or deleting) at most ηn edges in H satisfies P . More pr...

The single crossing property plays a crucial role in monotone comparative statics (Milgrom and Shannon (1994)), yet in some important applications the property cannot be directly assumed or easily derived. Difficulties often arise because the property cannot be aggregated: the sum of two functions with the single crossing property need not have the same property. We obtain the precise condition...

Let C be an n-dimensional integral box, and π be a monotone property defined over the elements of C. We consider the problems of incrementally generating jointly the families Fπ and Gπ of all minimal subsets satisfying property π and all maximal subsets not satisfying property π, when π is given by a polynomial-time satisfiability oracle. Problems of this type arise in many practical applicatio...