5,285 research outputs found
Computability and Representations of the Zero Set
In this note we give a new representation for closed sets under which the robust zero set of a function is computable. We call this representation the component cover representation. The computation of the zero set is based on topological index theory, the most powerful tool for finding robust solutions of equations
Bounded time computation on metric spaces and Banach spaces
We extend the framework by Kawamura and Cook for investigating computational
complexity for operators occurring in analysis. This model is based on
second-order complexity theory for functions on the Baire space, which is
lifted to metric spaces by means of representations. Time is measured in terms
of the length of the input encodings and the required output precision. We
propose the notions of a complete representation and of a regular
representation. We show that complete representations ensure that any
computable function has a time bound. Regular representations generalize
Kawamura and Cook's more restrictive notion of a second-order representation,
while still guaranteeing fast computability of the length of the encodings.
Applying these notions, we investigate the relationship between purely metric
properties of a metric space and the existence of a representation such that
the metric is computable within bounded time. We show that a bound on the
running time of the metric can be straightforwardly translated into size bounds
of compact subsets of the metric space. Conversely, for compact spaces and for
Banach spaces we construct a family of admissible, complete, regular
representations that allow for fast computation of the metric and provide short
encodings. Here it is necessary to trade the time bound off against the length
of encodings
A Galois connection between Turing jumps and limits
Limit computable functions can be characterized by Turing jumps on the input
side or limits on the output side. As a monad of this pair of adjoint
operations we obtain a problem that characterizes the low functions and dually
to this another problem that characterizes the functions that are computable
relative to the halting problem. Correspondingly, these two classes are the
largest classes of functions that can be pre or post composed to limit
computable functions without leaving the class of limit computable functions.
We transfer these observations to the lattice of represented spaces where it
leads to a formal Galois connection. We also formulate a version of this result
for computable metric spaces. Limit computability and computability relative to
the halting problem are notions that coincide for points and sequences, but
even restricted to continuous functions the former class is strictly larger
than the latter. On computable metric spaces we can characterize the functions
that are computable relative to the halting problem as those functions that are
limit computable with a modulus of continuity that is computable relative to
the halting problem. As a consequence of this result we obtain, for instance,
that Lipschitz continuous functions that are limit computable are automatically
computable relative to the halting problem. We also discuss 1-generic points as
the canonical points of continuity of limit computable functions, and we prove
that restricted to these points limit computable functions are computable
relative to the halting problem. Finally, we demonstrate how these results can
be applied in computable analysis
On the computability of some positive-depth supercuspidal characters near the identity
This paper is concerned with the values of Harish-Chandra characters of a
class of positive-depth, toral, very supercuspidal representations of -adic
symplectic and special orthogonal groups, near the identity element. We declare
two representations equivalent if their characters coincide on a specific
neighbourhood of the identity (which is larger than the neighbourhood on which
Harish-Chandra local character expansion holds). We construct a parameter space
(that depends on the group and a real number ) for the set of
equivalence classes of the representations of minimal depth satisfying some
additional assumptions. This parameter space is essentially a geometric object
defined over \Q. Given a non-Archimedean local field \K with sufficiently
large residual characteristic, the part of the character table near the
identity element for G(\K) that comes from our class of representations is
parameterized by the residue-field points of . The character values
themselves can be recovered by specialization from a constructible motivic
exponential function. The values of such functions are algorithmically
computable. It is in this sense that we show that a large part of the character
table of the group G(\K) is computable
Effective zero-dimensionality for computable metric spaces
We begin to study classical dimension theory from the computable analysis
(TTE) point of view. For computable metric spaces, several effectivisations of
zero-dimensionality are shown to be equivalent. The part of this
characterisation that concerns covering dimension extends to higher dimensions
and to closed shrinkings of finite open covers. To deal with zero-dimensional
subspaces uniformly, four operations (relative to the space and a class of
subspaces) are defined; these correspond to definitions of inductive and
covering dimensions and a countable basis condition. Finally, an effective
retract characterisation of zero-dimensionality is proven under an effective
compactness condition. In one direction this uses a version of the construction
of bilocated sets.Comment: 25 pages. To appear in Logical Methods in Computer Science. Results
in Section 4 have been presented at CCA 201
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