3,461 research outputs found

    Metric complements of overt closed sets

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    We show that the set of points of an overt closed subspace of a metric completion of a Bishop-locally compact metric space is located. Consequently, if the subspace is, moreover, compact, then its collection of points is Bishop compact.Comment: 9 pages, 1 figur

    The descriptive theory of represented spaces

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    This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented spaces. Most material is from work-in-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit.Comment: survey of work-in-progres

    Modal Verbs

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    Ellipsis, economy, and the (non)uniformity of traces

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    A number of works have attempted to account for the interaction between movement and ellipsis in terms of an economy condition Max- Elide. We show that the elimination of MaxElide leads to an empirically superior account of these interactions. We show that a number of the core effects attributed to MaxElide can be accounted for with a parallelism condition on ellipsis. The remaining cases are then treated with a generalized economy condition that favors shorter derivations over longer ones. The resulting analysis has no need for the ellipsisspecific economy constraint MaxElide

    Functional versus lexical: a cognitive dichotomy

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    Computing Haar Measures

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    According to Haar's Theorem, every compact group GG admits a unique (regular, right and) left-invariant Borel probability measure ÎŒG\mu_G. Let the Haar integral (of GG) denote the functional ∫G:C(G)∋f↊∫f dÎŒG\int_G:\mathcal{C}(G)\ni f\mapsto \int f\,d\mu_G integrating any continuous function f:G→Rf:G\to\mathbb{R} with respect to ÎŒG\mu_G. This generalizes, and recovers for the additive group G=[0;1)mod  1G=[0;1)\mod 1, the usual Riemann integral: computable (cmp. Weihrauch 2000, Theorem 6.4.1), and of computational cost characterizing complexity class #P1_1 (cmp. Ko 1991, Theorem 5.32). We establish that in fact every computably compact computable metric group renders the Haar integral computable: once asserting computability using an elegant synthetic argument, exploiting uniqueness in a computably compact space of probability measures; and once presenting and analyzing an explicit, imperative algorithm based on 'maximum packings' with rigorous error bounds and guaranteed convergence. Regarding computational complexity, for the groups SO(3)\mathcal{SO}(3) and SU(2)\mathcal{SU}(2) we reduce the Haar integral to and from Euclidean/Riemann integration. In particular both also characterize #P1_1. Implementation and empirical evaluation using the iRRAM C++ library for exact real computation confirms the (thus necessary) exponential runtime
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