Suppose in an arithmetic unverse we have two predicates φ and ψ for natural numbers, satisfying a base case φ(0) → ψ(0) and an induction step that, for generic n, the hypothesis φ(n) → ψ(n) allows one to deduce φ(n+ 1) → ψ(n+ 1). Then it is already true in that arithmetic universe that (∀n)(φ(n) → ψ(n)). This is substantially harder than in a topos, where cartesian closedness allows one to form...

Let C be a full-dimensional pointed closed convex cone in R obtained by taking the conic hull of a strictly convex set. Given A ∈ Qm×n1 , B ∈ Qm×n2 and b ∈ Q, a simple conic mixed-integer set (SCMIS) is a set of the form {(x, y) ∈ Z1 ×R2 | Ax+By− b ∈ C}. In this paper, we give a complete characterization of the closedness of convex hulls of SCMISs. Under certain technical conditions on the cone...

In this paper we reconsider the question of when the continuous linear image of a closed convex cone is closed in Euclidean space. In particular, we show that although it is not true that the closedness of the image is preserved under small perturbations of the linear mappings it is “almost” true that the closedness of the image is preserved under small perturbations, in the sense that, for “al...

The closedness type regularity conditions have proven during the last decade to be viable alternatives to their more restrictive interiority type counterparts, in both convex optimization and different areas where it was successfully applied. In this review article we deand reconstruct some closedness type regularity conditions formulated by means of epigraphs and subdifferentials, respectively...

Mourgues and Ressayre [10] showed that any real closed field is isomorphic to a truncation closed subfield of a Hahn field over R. The papers [5, 6, 7, 4] continue the study of truncation closedness. The results are typically that truncation closedness is robust in the sense of being preserved under a variety of extensions. One significance of truncation closedness is that it enables transfinit...

Some research works have mentioned the similarity of autopoiesis with (M,R) systems proposed by Rosen, from the perspective of closedness of the systems. However, there are some difference between the aspects of closedness required for autopoiesis and (M,R) systems. This paper aims at clarifying these differences to investigate the possibility of algebraic description of living systems, based o...

In this paper, we obtain several new generalized KKM-type theorems under a new coercivity condition and the condition of intersectionally closedness which improves condition of transfer closedness. As applications, we obtain new versions of equilibrium problem, minimax inequality, coincidence theorem, fixed point theorem and an existence theorem for an 1-person game.

Defining a system in distinction from its environment is a fundamental but elusive problem in artificial life as well as in real-world complex systems. While many notions of closure gives a qualitative and absolute criteria for the system–environment distinction, the concept of “informational closure” proposed by Bertschinger et al. (Bertschinger et al., 2006, Proc. GWAL-7, p.9, IOS Press) give...