15243737
Dual renormings of Banach spaces
We prove that a Banach space admitting an equivalent WUR norm is an Asplund space. Some related dual renormings are also presented.
122468852
Remarks on symmetric smooth norms
46561034
Arithmetical Interpretations of Dynamic Logic
An arithmetical interpretation of dynamic propositional logic (DPL) is a mapping f satisfying the following: (1) fassociates with each formula A of DPL a sentencef(A) of Peano arithmetic (PA) and with each program a a formula f(a) of PA with one free variable describing formally a supertheory of PA; (2) f commutes with logical connectives; (3) f([aK]A) is the sentence saying that f(A) is provable in the theory f(a); (4) for each axiom A of DPL, f(A) is provable in PA (and consequently, for each A provable in DPL, f(A) is provable in PA). The arithmetical completeness theorem is proved saying that a formula A of DPL is provable in DPL iff for each arithmetical interpretationf,f(A) is provable in PA. Various modifications of this result are considered. Introduction. The aim of the present paper is to apply techniques of arithmetical interpretations (or: provability interpretations) of modal propositional logic (as developed and studied by deJongh, Sambin, Solovay, Smoryn'ski, Guaspari and others, see [12], [1 1], [10]) to dynamic logic. Dynamic logic (or: logic of programs) is a generalization of modal logic extensively studied in computer science (Pratt, Fischer, Ladner, Segerbergh, Parikh, see [9], [4], [8]). Both various systems of modal logic and systems of dynamic logic have well-behaving Kripke semantics (necessity means validity in all alternative states-worlds) and complete axiomatizations (for modal logic see [6] and [12], for dynamic logic see [8]). The idea behind arithmetical interpretations of modal logic is to interpret necessity as formal provability (in arithmetic). Now, dynamic logic is a modal logic with many modalities of necessity (corresponding to programs) so that the question arises whether one could interpret these modalities by formal provability in various theories (and obtain some completeness results). This question will be answered positively. What is the meaning of our results? First, the question just formulated seems to be a very natural question and it seems not to be trivial to answer it. Second, our result could elucidate from an unexpected point of view the saying of Kowalski [5]: "algorithm logic + control". We shall see that programs (algorithms)i.e. their logical components-may be modelled by some formal theories. Third, it has proved to be useful to investigate modal calculi having various modalities, not just the necessity, when studying self-reference in Peano arithmetic ((see [10], [11]). And we have in dynamic logic an example of such a calculus having both Received October 15, 1981. ? 1983, Association for Symbolic Logic 0022-4812/83/4803-0021 /$02.00
10820860
Epistemic Entrenchment and Arthmetical Hierarchy (Abstract)
In a rotary coupling, such as the coupling between a piston pin and the holes for the same bored in the bosses of a piston for an internal combustion engine, a device for improving the lubrication, comprising a number of turns of capillary circumferential groove hollowed in one of the coupling surfaces, said groove having preferably a depth ranging from 5 to 20 mu m, a pitch ranging from 0.25 to 1 mm and an eccentricity not exceeding 2 mu m.