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International audience; Foreword This document presents a translation for historical perspective of the paper: " B. Courcelle, G. Kahn, and J. Vuillemin. Algorithmes d'´ equivalence et de réduction a des expressions minimales dans une classe d'´ equations récursives simples, in, Jacques Loeckx, editor, Automata, Languages and Programming, volume Abstract In this paper, we describe an algorithm for deciding equivalence in a domain whose objects are defined by uninterpreted fixpoint equations. The algorithm is then applied to finding minimal representations of those objects.

Sario's capacity function of a closed subset γ of the ideal boundary is known to be unique if γ is of positive capacity. The present paper will determine the number of capacity functions of γ in terms of the Heins harmonic dimension when γ has zero capacity, under the assumption that γ is isolated. This includes the special case where γ is the ideal boundary. 1* Capacity functions* Denote by β the ideal boundary of an open Riemann surface R in the sense of Kerekjartό-Stoϊlow. We consider a fixed nonempty closed subset 7 c β which is isolated from S — β — 7. Throughout this paper D will denote a fixed parametric disk about a fixed point ζ e R with a fixed local parameter z and the uniqueness is always referred to this fixed triple (ζ, D, z). Here we do not exclude the case where 7 = β. For a regular region Ω Z) D we denote by yΩ the part of dΩ which is "homologous" to 7. The remainder δΩ = dΩ — yΩ consists of a finite number of analytic Jordan curves δΩj. For a regular exhaustion {RfΛn^o with Ro => D and nonempty τΛo, set yn = ΎE% and δni = dRnJ. Then there exists a unique function py% e H(Rn — ζ) satisfying (a) pΐn I D = log I z - ζ I + hn(z) with hn e H(D) and Λw(ζ) = 0, (b) pϊn I 7n = fcnCr) (const.) and pϊn \ δnj = d ni (const.) so that \ %dpr% = 0, which is called a capacity function of jn (Sario [6]). It is known that kn(y) increases with n and the limit k(y) is independent of the choice of {i2J*= 0. We call β~fc(r) the capacity of 7 and denote it by cap 7. When cap 7 > 0, pTn converges to a functions pr, which

The International Practical Temperature Scale has a lower temperature limit of 90.18 K; measurements below this temperature are commonly made in terms of the various "national" scales. All of these scales differ significantly from the thermodynamic scale and exhibit undesirable variations in the size of the degree as a consequence; the national scales, moreover, are comparison scales, which are widely considered to be less satisfactory than scales defined in terms of fixed points and interpolating instruments. Various proposals for replacing this mixture by a single scale have been or are being developed; these are all based, in the 12 K to 90 K range, on the results of a comparison of four national scales recently carried out at Moscow and Teddington. This paper attempts to analyze the reproducibility of some of these proposed scales and their relation to the current best estimate of the thermodynamic scale. The significant lack of reproducibility found for these proposed scales below 50 K may either be tolerated, or alternatively must be reduced by further restrictions on thermometer characteristics. These proposed scales exhibit systematic variations in the size of the degree with temperature. These variations can be substantially reduced by modification of the mathematical treatment of the experimental data on which the scales are based; one such modification is presented here. Further analysis of the behavour of current and proposed scales will be enormously facilitated if the results of additional, highly accurate comparisons over wide temperature ranges and involving many types of thermometers become available.

J. Leray defined a local fixed point index for functions defined in what he called convexoid spaces. From the standpoint of analysis, the most important example of a convexoid space is a compact subset C c X, X a locally convex topological vector space, such that C = U ?=1 d, where d are compact, convex subsets of X or a homeomorphic image of such a C. In this paper a simple geometric approach is given (see Lemma 2 below) by means of which a fixed point index can be defined for functions with domain in a class of spaces ^ which contains the spaces C mentioned above and also the compact metric ANR's. The usual properties of the fixed point index are established, and it is shown that they axiomatically determine the index for the class of spaces J^~.