381,676 research outputs found

    On sets of terms with a given intersection type

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    We are interested in how much of the structure of a strongly normalizable lambda term is captured by its intersection types and how much all the terms of a given type have in common. In this note we consider the theory BCD (Barendregt, Coppo, and Dezani) of intersection types without the top element. We show: for each strongly normalizable lambda term M, with beta-eta normal form N, there exists an intersection type A such that, in BCD, N is the unique beta-eta normal term of type A. A similar result holds for finite sets of strongly normalizable terms for each intersection type A if the set of all closed terms M such that, in BCD, M has type A, is infinite then, when closed under beta-eta conversion, this set forms an adaquate numeral system for untyped lambda calculus. A number of related results are also proved

    Describability via ubiquity and eutaxy in Diophantine approximation

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    We present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical ubiquity techniques, as well as the mass and the large intersection transference principles, thereby leading to a thorough description of the properties in terms of Hausdorff measures and large intersection classes associated with general gauge functions. The sets issued from eutaxic sequences of points and optimal regular systems may naturally be described within this framework. The discussed applications include the classical homogeneous and inhomogeneous approximation, the approximation by algebraic numbers, the approximation by fractional parts, the study of uniform and Poisson random coverings, and the multifractal analysis of L{\'e}vy processes.Comment: 94 pages. Notes based on lectures given during the 2012 Program on Stochastics, Dimension and Dynamics at Morningside Center of Mathematics, the 2013 Arithmetic Geometry Year at Poncelet Laboratory, and the 2014 Spring School in Analysis held at Universite Blaise Pasca

    On the Classification of Irregular Dihedral Branched Covers of Four-Manifolds

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    We prove a necessary condition for a four-manifold YY to be homeomorphic to a pp-fold irregular dihedral branched cover of a given four-manifold XX, with a fixed branching set BB. The branching sets considered are closed oriented surfaces embedded locally flatly in XX except at one point with a specified cone singularity. The necessary condition obtained is on the rank and signature of the intersection form of YY and is given in terms of the rank and signature of the intersection form of XX, the self-intersection number of BB in XX and classical-type invariants of the singularity. Secondly, we show that, for an infinite class of singularities, the necessary condition is sharp. That is, if the singularity is a two-bridge slice knot, every pair of values of the rank and signature of the intersection form which the necessary condition allows is in fact realized by a manifold dihedral cover. In a slightly more general take on this problem, for an infinite class of simply-connected four-manifolds XX and any odd square-free integer p3˘e1p\u3e1, we give two constructions of infinite families of pp-fold irregular branched covers of XX. The first construction produces simply-connected manifolds as the covering spaces, while the second produces simply-connected stratified spaces with one singular stratum. The branching sets in the first of these constructions have two singularities of the same type. In the second construction, there is one singularity, whose type is the connected sum of a knot with itself

    Realisability Semantics for Intersection Types and Expansion Variables

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    Expansion was invented at the end of the 1970s for calculating principal typings for λ\lambda-terms in type systems with intersection types. Expansion variables (E-variables) were invented at the end of the 1990s to simplify and help mechanise expansion. Recently, E-variables have been further simplified and generalised to also allow calculating type operators other than just intersection. There has been much work on denotational semantics for type systems with intersection types, but none whatsoever before now on type systems with E-variables. Building a semantics for E-variables turns out to be challenging. To simplify the problem, we consider only E-variables, and not the corresponding operation of expansion. We develop a realisability semantics where each use of an E-variable in a type corresponds to an independent degree at which evaluation occurs in the λ\lambda-term that is assigned the type. In the λ\lambda-term being evaluated, the only interaction possible between portions at different degrees is that higher degree portions can be passed around but never applied to lower degree portions. We apply this semantics to two intersection type systems. We show these systems are sound, that completeness does not hold for the first system, and completeness holds for the second system when only one E-variable is allowed (although it can be used many times and nested). As far as we know, this is the first study of a denotational semantics of intersection type systems with E-variables (using realisability or any other approach)

    Incidence combinatorics of resolutions

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    We introduce notions of combinatorial blowups, building sets, and nested sets for arbitrary meet-semilattices. This gives a common abstract framework for the incidence combinatorics occurring in the context of De Concini-Procesi models of subspace arrangements and resolutions of singularities in toric varieties. Our main theorem states that a sequence of combinatorial blowups, prescribed by a building set in linear extension compatible order, gives the face poset of the corresponding simplicial complex of nested sets. As applications we trace the incidence combinatorics through every step of the De Concini-Procesi model construction, and we introduce the notions of building sets and nested sets to the context of toric varieties. There are several other instances, such as models of stratified manifolds and certain graded algebras associated with finite lattices, where our combinatorial framework has been put to work; we present an outline in the end of this paper.Comment: 20 pages; this is a revised version of our preprint dated Nov 2000 and May 2003; to appear in Selecta Mathematica (N.S.
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