Let Ψ : Z → Z be an affine-linear map, and Ψ̇ be its linear part. A famous and difficult open conjecture of Dickson predicts the distribution of prime points in the affine sublattice Ψ(Z) of Z. Assuming the Gowers Inverse conjecture GI(s) and the Möbius and nilsequences conjecture MN(s) for some finite s > 1, GreenTao [GT3] verified Dickson’s conjecture for Ψ of complexity s. Let Φ : Z → Z be an...

The concepts of paracontracting, pseudocontracting and nonexpanding operators have been shown to be useful in proving convergence of asynchronous or parallel iteration algorithms. The purpose of this paper is to give characterizations of these operators when they are linear and finite-dimensional. First we show that pseudocontractivity of stochastic matrices with respect to ‖ · ‖∞ is equivalent...

In this paper, we first consider the disjoint union of two fuzzy metric spaces and non-existence of fuzzy contractive maps. We prove that there exists a natural fuzzy metric on the disjoint union of two fuzzy metric spaces. We also show that there is a fuzzy metric space which does not admit any fuzzy contractive map. Second, we introduce the notion of a fuzzy diameter for fuzzy metric spaces a...

We introduce the max-atom problem (MAP): solving (in Z) systems of inequations of the form max(x, y)+k z, where x, y, z are variables and k ∈ Z. Our initial motivation for MAP was reasoning on delays in circuits using SAT Modulo Theories [10], viewing MAP as a natural extension of Difference Logic, i.e., inequations of the form x+ k y. Here we show that MAP is PTIME-equivalent to several rather...

Contractive digraphs and splicing machines, notions that seem to be new in the theory of directed graphs, are introduced. The existence and uniqueness of the central object, the attractor of a contractive digraph, is proved, and the attractor is shown to be the unique fixed point of a natural operator defined on sets of vertices of the digraph. The attractor of a contractive digraph is then sho...

Streams, or infinite lists, have many applications in functional programming, and are naturally defined using recursive equations. But how do we ensure that such equations make sense, i.e. that they actually produce well-defined streams? In this article we present a new approach to this problem, based upon the topological notion of contractive functions on streams. In particular, we give a soun...

Streams, or infinite lists, have many applications in functional programming, and are naturally defined using recursive equations. But how do we ensure that such equations make sense, i.e. that they actually produce well-defined streams? In this article we present a new approach to this problem, based upon the topological notion of contractive functions on streams. In particular, we give a soun...

In 2007, Haung and Zhang introduced the notion of cone metric spaces. In this paper, we define an ordered space E, and we discuss some properties and examples. Also, normed ordered space is introduced. We recall properties of R, and we discuss their extension to E. We introduce the notion of E-metric spaces and characterize cone metric space. Afterwards, we get generalizations of notions of con...