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Let $m>1$ and $\mathfrak {d} \neq 0$ be integers such that $v_{p}(\mathfrak {d}) \neq m$ for any prime p. We construct a matrix $A(\mathfrak {d})$ of size $(m-1) \times (m-1)$ depending on only of $\mathfrak {d}$ with the following property: For any tame $ \mathbb {Z}/m \mathbb {Z}$-number field K of discriminant $\mathfrak {d}$, the matrix $A(\mathfrak {d})$ represents the Gram matrix of the integral trace-zero form of K. In particular, we have that the integral trace-zero form of tame cyclic number fields is determined by the degree and discriminant of the field. Furthermore, if in addition to the above hypotheses, we consider real number fields, then the shape is also determined by the degree and the discriminant.
This chapter presents results from a longitudinal investigation of the form and function of mothers’ questions to their one–, two–, and three–year–old children in a challenging task context across a diverse sample of 64 families in Norway. We examine the implications of mothers’ questions for children’s concurrent task performance and later language development. The findings suggest that mothers vary quite a bit in their use of questions. Moreover, the mothers show a decrease in their use of questions that are direct in their informational intent and/or simpler in their form over time, and an increase in questions that are indirect in intent and complex in their form (wh-questions). Mothers who more often ask wh–questions at child age two years have children with higher language skills at age four years, whereas use of simpler questions at child age two is negatively related to children’s concurrent task success and later language skills. Together with the existing literature, this study suggests that questions are not just a mechanism for cognitive development because they allow children to obtain the information they need, but also that parental questions scaffold children’s language and possibly cognitive development more general by guiding their exploration.
Children struggle to identify the defining properties of shapes through elementary school (Satlow & Newcombe, 1998). Perhaps preschoolers struggle to learn these properties because they have little conception of what constitutes relevant shape knowledge or how to go about acquiring it. We outline empirical support for the idea that children’s questions show us not only what a child knows in a given domain, but how that child is approaching the act of knowledge acquisition itself. In this way, studying children’s questions provides information that cannot be gleaned from studies which simply test children’s domain–specific knowledge. To illustrate this point, we examine existing research into children’s questions about animals, to show how this idea applies in relation to an extensive body of research on children’s questioning behavior. Second, we review our own study of children’s questions about shapes, to see how preschoolers’ approach to knowledge acquisition differs in a domain composed of pure nominal kinds rather than natural kinds. Our results suggest that, in contrast to the animal domain, three–year–olds demonstrate very little awareness of relevant information in the shape domain. We conclude by considering why this might be the case and suggesting some ways it could be rectified.
The use of a shoal analysis and patch estimation system (SHAPES) to analyse hydro-acoustic data is discussed. SHAPES was used to detect and characterise sardine schools from four echo integration surveys off the coast of South Africa. Methods of school detection and characterisation, assumptions made and corrections applied to school variables are presented. Measurements of school morphology and internal school features are discussed and compared to other relevant works. Significant relationships were established, both between measures of school morphology and internal density structure. Several variables showed significant differences between surveys. Composite factors derived from principle component analyses showed that morphological variables are the most important school descriptors. No differences in the loading of composite factors between surveys are evident. This suggests an inherent spatial and temporal stability in sardine school characteristics. This fact may prove beneficial for comparisons of school structure between co-occurring fish species and possibly aid fish school identification in the future.
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