4,348 research outputs found

    The contributions of domain-general and numerical factors to third-grade arithmetic skills and mathematical learning disability

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    Explanations of the marked individual differences in elementary school mathematical achievement and mathematical learning disability (MLD or dyscalculia) have involved domain-general factors (working memory, reasoning, processing speed and oral language) and numerical factors that include single-digit processing efficiency and multi-digit skills such as number system knowledge and estimation. This study of third graders (N = 258) finds both domain-general and numerical factors contribute independently to explaining variation in three significant arithmetic skills: basic calculation fluency, written multi-digit computation, and arithmetic word problems. Estimation accuracy and number system knowledge show the strongest associations with every skill and their contributions are both independent of each other and other factors. Different domain-general factors independently account for variation in each skill. Numeral comparison, a single digit processing skill, uniquely accounts for variation in basic calculation. Subsamples of children with MLD (at or below 10th percentile, n = 29) are compared with low achievement (LA, 11th to 25th percentiles, n = 42) and typical achievement (above 25th percentile, n = 187). Examination of these and subsets with persistent difficulties supports a multiple deficits view of number difficulties: most children with number difficulties exhibit deficits in both domain-general and numerical factors. The only factor deficit common to all persistent MLD children is in multi-digit skills. These findings indicate that many factors matter but multi-digit skills matter most in third grade mathematical achievement

    First Order Theories of Some Lattices of Open Sets

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    We show that the first order theory of the lattice of open sets in some natural topological spaces is mm-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., Rn\mathbb{R}^n, n1n\geq1, and the domain PωP\omega) this theory is mm-equivalent to first order arithmetic

    N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces

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    We establish a correspondence between generalized quiver gauge theories in four dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are enumerated and their numbers are compared. The correspondence is particularly striking for genus zero torsion-free congruence subgroups as exemplified by those which arise in Moonshine. We analyze in detail the case of index 24, where modular elliptic K3 surfaces emerge: here, the elliptic j-invariants can be recast as dessins d'enfant which dictate the Seiberg-Witten curves.Comment: 42+1 pages, 5 figures; various helpful comments incorporate

    Topological representation zeta functions of unipotent groups

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    Inspired by work surrounding Igusa's local zeta function, we introduce topological representation zeta functions of unipotent algebraic groups over number fields. These group-theoretic invariants capture common features of established pp-adic representation zeta functions associated with pro-pp groups derived from unipotent groups. We investigate fundamental properties of the topological zeta functions considered here. We also develop a method for computing them under non-degeneracy assumptions. As an application, among other things, we obtain a complete classification of topological representation zeta functions of unipotent algebraic groups of dimension at most six.Comment: 27 page

    Eligibility and inscrutability

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    The philosophy of intentionality asks questions such as: in virtue of what does a sentence, picture, or mental state represent that the world is a certain way? The subquestion I focus upon here concerns the semantic properties of language: in virtue of what does a name such as ‘London’ refer to something or a predicate such as ‘is large’ apply to some object? This essay examines one kind of answer to this “metasemantic”1 question: interpretationism, instances of which have been proposed by Donald Davidson, David Lewis, and others. I characterize the “twostep” form common to such approaches and briefl y say how two versions described by David Lewis fi t this pattern. Then I describe a fundamental challenge to this approach: a “permutation argument” that contends, by interpretationist lights, there can be no fact of the matter about lexical content (e.g., what individual words refer to). Such a thesis cannot be sustained, so the argument threatens a reductio of interpretationism. In the second part of the article, I will give what I take to be the best interpretationist response to the inscrutability paradox: David Lewis’s appeal to the differential “eligibility” of semantic theories. I contend that, given an independently plausible formulation of interpretationism, the eligibility response is an immediate consequence of Lewis’s general analysis of the theoretical virtue of simplicity. In the fi nal sections of the article, I examine the limitations of Lewis’s response. By focusing on an alternative argument for the inscrutability of reference, I am able to describe conditions under which the eligibility result will deliver the wrong results. In particular, if the world is complex enough and our language suffi ciently simple, then reference may be determinately secured to the wrong things

    Jet bundles on Gorenstein curves and applications

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    In the last twenty years a number of papers appeared aiming to construct locally free replacements of the sheaf of principal parts for families of Gorenstein curves. The main goal of this survey is to present to the widest possible audience of mathematical readers a catalogue of such constructions, discussing the related literature and reporting on a few applications to classical problems in Enumerative Algebraic Geometry.Comment: Minor revisions, improved expositio
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