Points, Plasticity, and the Logic of Contraction in Alain Badiou and Catherine Malabou

This article presents Catherine Malabou and Alain Badiou as theorists of contraction (a kind of reduction or tightening that accompanies every process of transformation) and its related operations of self-reflexivity and infinite iteration. Trading on these commonalities, the article hopes to draw out Malabou’s and Badiou’s respective formalist commitments. On Badiou’s side, it sharpens the question of what is at stake in something as regulated as a “procedure”; on Malabou’s, it recognizes formal stakes to plasticity that often go unrecognized because of her penchant for biology. The article then concludes with a broad comparison of these two thinkers in terms of their accounts of potential and imagines the critiques each might leverage against the other. Where Malabou might well regard Badiou to be too tightly constraining the shape of the future, Badiou is likely to find in Malabou one more instance of a naïve democratic materialism.

An Infinite Natural Product

We study a countably infinite iteration of the natural product between ordinals.We present an “effective” way to compute this countable natural product; in the non trivial cases the result depends only on the natural sum of the degrees of the factors, where the degree of a nonzero ordinal is the largest exponent in its Cantor normal form representation. Thus we are able to lift former results about infinitary sums to infinitary products. Finally, we provide an order-theoretical characterization of the infinite natural product; this characterization merges in a nontrivial way a theorem by Carruth describing the natural product of two ordinals and a known description of the ordinal product of a possibly infinite sequence of ordinals.

No Need for Infinite Iteration

As part of his argument for a “Copernican revolution” in social ontology, Hans Bernhard Schmid (2005) argues that the individualistic approach to social ontology is critically flawed. This article rebuts his claim that the notion of mutual belief necessarily entails infinite iteration of beliefs about the intentions of others, and argues that collective action can arise from individual contributions without such iteration. What matters is whether or when there are grounds for belief, and while extant groups and social structures may be relevant to some forms of collective action, this does not show that all forms of collective action depends on such such pre-established collectivity.

CONVERGENCE OF AN ITERATIVE NONLINEAR SCHEME FOR DENOISING OF PIECEWISE CONSTANT IMAGES

In this paper we present a new efficient iterative nonlinear scheme for recovering of a piecewise constant image from an observed image containing additive noise. We apply an adaptive neighborhood filter which comes from robust statistics and completely rejects outliers being greater than a certain constant. We prove that the iterated application of the scheme leads to a piecewise constant image. This observation generalizes the known results on convergence of nonlinear diffusion schemes to a constant steady-state. Moreover, we show that the partition of the image determining the piecewise constant steady-state after an infinite iteration process can already be found after a finite number of iteration steps. This result can be used for a fast approximation of the piecewise constant image by a mean value procedure. We examine the relations of our scheme to average and bilateral filtering, diffusion filtering and wavelet shrinkage. Numerical experiments illustrate the performance of the algorithm.

SUBWORD COMPLEXITY OF PROFINITE WORDS AND SUBGROUPS OF FREE PROFINITE SEMIGROUPS

We study free profinite subgroups of free profinite semigroups of the same rank using, as main tools, iterated continuous endomorphisms, subword complexity, and the associated entropy. Main results include a general scheme to produce such subgroups and a proof that the complement of the minimal ideal in a free profinite semigroup on more than one generator is closed under all implicit operations that do not lie in the minimal ideal and even under their infinite iteration.

A Semantic Approach to Fairness

In the semantic framework of metric process theory, we undertake a general investigation of fairness of processes from two points of view: (1) intrinsic fairness of processes, and (2) fair operations on processes. Regarding (1), we shall define a “fairification” operation on processes called Fair such that for every (generally unfair) process p the process Fair(p) is fair, and contains precisely those paths of p that are fair. Its definition uses systematic alternation of random choices. The second part of this paper treats the notion of fair operations on processes: suppose given an operator on processes (like merge, or infinite iteration), we want to define a fair version of it. For the operation of infinite iteration we define a fair version, again by a “fair scheduling” technique.

Examples for the Theory of Infinite Iteration of Summability Methods

Garten and Knopp [7] introduced the notion of infinite iteration of Césaro (C1 ) averages, which they called H∞ summability. Flehinger [6] (apparently unaware of [7]) produced the first nontrivial example of an H∞ summable sequence: the sequence ﹛ai ﹜ ∞i=1 where at is 1 or 0 as the lead digit of the integer i is one or not. Duran [2] has provided an elegant treatment of H∞ summability as a special case of summability with respect to an ergodic semigroup of transformations.