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On the completeness theorem of many-sorted equational logic and the equivalence between Hall algebras and Binabou theories

100%

Vidal J. , Tur J.

Reports on Mathematical Logic

2006

tom Nr 40

127-158

The completeness theorem of equational logic of Birkhoff asserts the coincidence of the model-theoretic and proof- theoretic consequence relations. Goguen and Meseguer, giving a sound and adequate system of inference rules for many-sorted deduction, founded ultimately on the congruences on Hall algebras, generalized the completeness theorem of Birkhoff to the completeness theorem of many-sorted equational logic. In this paper, after simplifying the specification of Hall algebras as given by Goguen-Meseguer, we obtain another many-sorted equational calculus from which we prove that the inference rules of abstraction and concretion due to Goguen-Meseguer are derived rules. Finally, after defining the Bienabou algebras for a set of sorts S we prove that the category of Bienabou algebras for S is equivalent to the category of Hall algebras for S and isomorphic to the category of Bienabou theories for S, i.e., the many-sorted counterpart of the category of Lawvere theories, hence that Hall algebras and Bienabou theories are equivalent.

Precise determination of HPGe detector efficiency for gamma spectrometry measurements of environmental samples with variable geometry and density

80%

Barrera M. , Casas-Ruiz M. , Alonso J. J. , Vidal J.

tom Vol. 62, No. 1

47--59

A methodology to determine the full energy peak efficiency (FEPE) for precise gamma spectrometry measurements of environmental samples with high-purity germanium (HPGe) detector, valid when this efficiency depends on the energy of the radiation E, the height of the cylindrical sample H, and its density ρ is introduced. The methodology consists of an initial calibration as a function of E and H and the application of a self-attenuation factor, depending on the density of the sample ρ, in order to correct for the different attenuation of the generic sample in relation to the measured standard. The obtained efficiency can be used in the whole range of interest studied, E = 120–2000 keV, H = 1–5 cm, and ρ = 0.8–1.7 g/cm3, being its uncertainty below 5%. The efficiency has been checked by the measurement of standards, resulting in a good agreement between experimental and expected activities. The described methodology can be extended to similar situations when samples show geometric and compaction differences.