Abstract In the late 1940s and early 1950s, Lorenzen developed his operative logic mathematics, a form of constructive mathematics. Nowadays this is mostly seen as precursor better-known dialogical (Notable exceptions are works Schroeder-Heister 2008; Coquand Neuwirth 2017; Kahle Oitavem 2020.), one might assume that same philosophical motivations were present in both works. However, we want to...

This paper examines the recursive definition of an increasing sequence of nested sets by means of a control set whose countably many successive redefinitions leads to a contradictory result that compromises ω-order and the Axiom of Infinity. 1. Recursion and successiveness A recursive definition usually starts with a first definition (basic clause) which is followed by an infinite (usually ω-or...

We analyze the performance of a data-assimilation algorithm based on a linear feedback control when used with observational data that contains measurement errors. Our model problem consists of dynamics governed by the two-dimension incompressible Navier–Stokes equations, observational measurements given by finite volume elements or nodal points of the velocity field and measurement errors which...

It is a ubiquitous opinion among mathematicians that a real number is just a point in the (real) line. If this rough definition is not enough, then a mathematician may provide a formal definition of the real numbers in the set theoretical and axiomatic fashion, i.e. via Cauchy sequences or Dedekind cuts, or as the collection of axioms (up to isomorphism) characterizing exactly the set of real n...

Macroscopic quantities beyond effective elastic tensors are presented that can be used to assess the local state of stress within a composite in the linear elastic regime. These are presented in a general homogenization context. It is shown that the gradient of the effective elastic property can be used to develop a lower bound on the maximum pointwise equivalent stress in the fine-scale limit....

We give an example of a co-compact Kleinian group F which contains a subgroup F 0 having the property that E~/Fo is contractible but not simply connected at infinity.

The integrability of quasihomogeneous systems is considered, and the properties of the first integrals and the inverse integrating factors of such systems are shown. By solving the systems of ordinary differential equations which are established by using the vector fields of the quasihomogeneous systems, one can obtain an inverse integrating factor of the systems. Moreover, the integrability of...