Raman microscopy of the molybdate minerals koechlinite, iriginite and lindgrenite

A series of molybdate bearing minerals including wulfenite, powellite, lindgrenite and iriginite have been analysed by Raman microscopy. These minerals are closely related and often have related paragenesis. Raman microscopy enables the selection of individual crystals of these minerals for spectroscopic analysis even though several of the minerals can be found in the same matrix because of the paragenetic relationships between the minerals. The molybdenum bearing minerals lindgrenite, iriginite and koechlinite were studied by scanning electron microscopy and compositionally analysed by EDX methods using an electron probe before Raman spectroscopic analyses. The Raman spectra are assigned according to factor group analysis and related to the structure of the minerals. These minerals have characteristically different Raman spectra.

On Ordinal VC-Dimension and Some Notions of Complexity

We generalize the classical notion of Vapnik–Chernovenkis (VC) dimension to ordinal VC-dimension, in the context of logical learning paradigms. Logical learning paradigms encompass the numerical learning paradigms commonly studied in Inductive Inference. A logical learning paradigm is defined as a set W of structures over some vocabulary, and a set D of first-order formulas that represent data. The sets of models of ϕ in W, where ϕ varies over D, generate a natural topology W over W. We show that if D is closed under boolean operators, then the notion of ordinal VC-dimension offers a perfect characterization for the problem of predicting the truth of the members of D in a member of W, with an ordinal bound on the number of mistakes. This shows that the notion of VC-dimension has a natural interpretation in Inductive Inference, when cast into a logical setting. We also study the relationships between predictive complexity, selective complexity—a variation on predictive complexity—and mind change complexity. The assumptions that D is closed under boolean operators and that W is compact often play a crucial role to establish connections between these concepts. We then consider a computable setting with effective versions of the complexity measures, and show that the equivalence between ordinal VC-dimension and predictive complexity fails. More precisely, we prove that the effective ordinal VC-dimension of a paradigm can be defined when all other effective notions of complexity are undefined. On a better note, when W is compact, all effective notions of complexity are defined, though they are not related as in the noncomputable version of the framework.