We study rational streams (over a field) from a coalgebraic perspective. Exploiting the finality of the set of streams, we present an elementary and uniform proof of the equivalence of four notions of representability of rational streams: by finite dimensional linear systems; by finite stream circuits; by finite weighted stream automata; and by finite dimensional subsystems of the set of streams.

We give a proper reparametrization theorem for a set of rational parametric equations which is proper for all but one of its parameters. We also give an algorithm to determine whether a set of rational parametric equations belongs to this class, and if it does, we reparametrize it such that the new parametric equations are proper.

Our aim is to study the set of K-rational expressions describing rational series. More precisely we are concerned with the definition of quotients of this set by coarser and coarser congruences which lead to an extension – in the case of multiplicities – of some classical results stated in the Boolean case. In particular, multiplicity analogues of the well known theorems of Brzozowski and Antim...

Rational double points are the simplest surface singularities. In this essay we will be mainly concerned with the geometry of the exceptional set corresponding to the resolution of a rational double point. We will derive the classification of rational double points in terms of Dynkin diagrams.

Following the first part of our research, we prove in this paper that a sub-hyperbolic semi-rational map with infinite post-critical set is combinatorially and locally holomorphically equivalent to a rational map if and only if it has no Thurston obstruction. Moreover, the rational map is unique up to holomorphic conjugation.

We consider the family of rational relations on words, i.e., relations recognized by multitape automata, which have rational trace on any finite intersection of rational relations. We prove that this family consists of finite unions of relations which are of two types: stars of tuples possibly with an extra prefix and suffix and relations with all but one singleton set components.

We develop an obstruction theory for homotopy of homomorphisms f, g : M → N between minimal differential graded algebras. We assume that M = ΛV has an obstruction decomposition given by V = V0⊕V1 and that f and g are homotopic on ΛV0. An obstruction is then obtained as a vector space homomorphism V1 → H(N ). We investigate the relationship between the condition that f and g are homotopic and th...

In this paper, we show that the Chvátal-Gomory closure of any compact convex set is a rational polytope. This resolves an open question of Schrijver [17] for irrational polytopes1, and generalizes the same result for the case of rational polytopes [17], rational ellipsoids [8] and strictly convex bodies [7].

Introduction. Diophantus noted that the rational numbers 1/16, 33/16, 17/4 and 105/16 have the following property: the product of any two of them increased by 1 is a square of a rational number (see [2, 3]). Let n be an integer. A set of positive integers {a1, a2, . . . , am} is said to have the property D(n) if aiaj + n is a perfect square for all 1 ≤ i < j ≤ m. Such a set is called a Diophant...