24,341 research outputs found
Applying Ragin's Crisp and Fuzzy Set QCA to Large Datasets: Social Class and Educational Achievement in the National Child Development Study
The paper explores the use of Charles Ragin's Qualitative Comparative Analysis (QCA) in both its crisp and fuzzy set versions in the study of the relations between social class origin, sex, 'ability' and subsequent educational achievement. The work reported is part of a larger ongoing project which is employing QCA to compare these relations within two birth cohorts. Here data are used from the British National Child Development Study, i.e. from children born in 1958. The paper has a methodological focus, bringing out the strengths but also the difficulties that arise when employing QCA with a large dataset of this type. In particular, the problem of calibrating membership in fuzzy sets in a context where detailed case knowledge is not available is illustrated. It is also shown how the use of gradually increasing thresholds with Ragin's fs/QCA software can bring out the relative importance of various factors in accounting for achievement. The QCA-based analysis suggests that the processes of educational attainment can, at best, only be seen as partly falling under a 'meritocratic' description. It is also hoped that this paper will serve as a useful introduction to the potential of QCA for readers not yet familiar with it.QCA, Social Class, Educational Attainment, Gender, Fuzzy Sets, Meritocracy.
The moduli space of matroids
In the first part of the paper, we clarify the connections between several
algebraic objects appearing in matroid theory: both partial fields and
hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are
compatible with the respective matroid theories. Moreover, fuzzy rings are
ordered blueprints and lie in the intersection of tracts with ordered
blueprints; we call the objects of this intersection pastures.
In the second part, we construct moduli spaces for matroids over pastures. We
show that, for any non-empty finite set , the functor taking a pasture
to the set of isomorphism classes of rank- -matroids on is
representable by an ordered blue scheme , the moduli space of
rank- matroids on .
In the third part, we draw conclusions on matroid theory. A classical
rank- matroid on corresponds to a -valued point of
where is the Krasner hyperfield. Such a point defines a
residue pasture , which we call the universal pasture of . We show that
for every pasture , morphisms are canonically in bijection with
-matroid structures on .
An analogous weak universal pasture classifies weak -matroid
structures on . The unit group of can be canonically identified with
the Tutte group of . We call the sub-pasture of generated by
``cross-ratios' the foundation of ,. It parametrizes rescaling classes of
weak -matroid structures on , and its unit group is coincides with the
inner Tutte group of . We show that a matroid is regular if and only if
its foundation is the regular partial field, and a non-regular matroid is
binary if and only if its foundation is the field with two elements. This
yields a new proof of the fact that a matroid is regular if and only if it is
both binary and orientable.Comment: 83 page
Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning
Several logical operators are defined as dual pairs, in different types of
logics. Such dual pairs of operators also occur in other algebraic theories,
such as mathematical morphology. Based on this observation, this paper proposes
to define, at the abstract level of institutions, a pair of abstract dual and
logical operators as morphological erosion and dilation. Standard quantifiers
and modalities are then derived from these two abstract logical operators.
These operators are studied both on sets of states and sets of models. To cope
with the lack of explicit set of states in institutions, the proposed abstract
logical dual operators are defined in an extension of institutions, the
stratified institutions, which take into account the notion of open sentences,
the satisfaction of which is parametrized by sets of states. A hint on the
potential interest of the proposed framework for spatial reasoning is also
provided.Comment: 36 page
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