Sull'omotopia uniforme

Summary Following in the classical theory's footsteps, it is possible to construct a ? rational homotopy theory ?. To this purpose, we take all the uniform maps f: I _Q ⁿ →S (where I_Q is the closed unit interval of the rational line equipped with the standard metric uniformity) as paths of a uniform space S. This brings us to the definition of the rational homotopy groups Q_n(S, U_Q, x) and enables us to consider the related exact sequences. All these objects are uniform invariants. The main problem which arises now is to find a suitable space S* in order that its classical homotopy groups Π_n(S*,x) are isomorphic to the classical ones of S. We are able to reach an answer to this question if S is a metrizable uniform space. Since considering the completion Ŝ of S serves no useful purpose (as it is shown by a simple example), we prove that the required space is the ? rational path completion ? S* of S, with S⊆S*⊆Ŝ. We finally recall that the rational uniform homotopy is a special case of regular homotopy, which has been defined and widely investigated in[1].

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Entrata in Redazione il 5 gennaio 1978.

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Lavoro svolto nell'ambito del gruppo GNSAGA del CNR.     

Relationship between the Jung constant and acertain projection constant in Banach spaces

Summary For a Banach spaceX the parameter λ₁(X) is the infimum of the numbers a with the following property: for everyZ )X with dimZ/X=1 there exists a projectionP:Z→X with |P|≤a. We give an upper bound for λ₁(X) in terms of the Jung constantJ(X). Some known and some new results are then deduced. A characteristic property ofJ ₁-spaces is also proved.

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Riassunto Scopo di questa nota è mostrare la stretta relazione che intercorre tra la costante di JungJ(X) di uno spazio di BanachX e la costante λ₁(X). Si stabilisce una limitazione superiore di λ₁, come funzione diJ, dalla quale si possono dedurre alcuni nuovi risultati e altri conosciuti. è inoltre dimostrata una caratterizzazione deiJ ₁-spazi.

     

Thiophene series. XV The effect of leaving groups on the reactivity of nitro-activated thiophene derivatives with sodium thiophenoxide

The effect of leaving groups on the reactivity of 3-X-2-nitrothiophene (1), 2-X-3-nitrothiophene (II) and 2-X-5-nitrothiophene (III) (X = Cl, Br, I, OC₆H₄NO₂(p), SO₂C₆ H₅) with sodium thiophenoxide has been examined. The results show that the reactivity ratio kIII/kII is always greater than unity and is relatively uninfluenced by changing leaving groups compared to the ratio kI/kII. The ratio kI/kII is greater or smaller than unity, according to following patterns of leaving groups Examination of the reactivity of the three series of compounds having Cl, Br, I and C₆H₄NO₂(p) as leaving groups showed the absence of an “element effect” which indicates that in the transition states of these displacement reactions there is little breaking of the bond to the group being displaced.