381,676 research outputs found
On sets of terms with a given intersection type
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
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
We prove a necessary condition for a four-manifold to be homeomorphic to a -fold irregular dihedral branched cover of a given four-manifold , with a fixed branching set . The branching sets considered are closed oriented surfaces embedded locally flatly in except at one point with a specified cone singularity. The necessary condition obtained is on the rank and signature of the intersection form of and is given in terms of the rank and signature of the intersection form of , the self-intersection number of in 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 and any odd square-free integer , we give two constructions of infinite families of -fold irregular branched covers of . 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
Expansion was invented at the end of the 1970s for calculating principal
typings for -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 -term that is assigned the type. In
the -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
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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