15 research outputs found
A journey through computability, topology and analysis
This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and descriptive set theory. We will especially focus on Weihrauch reducibility, as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of the open and clopen Ramsey theorems. In particular, since there is not a canonical way to phrase these theorems as multi-valued functions, we identify 8 different multi-valued functions (5 corresponding to the open Ramsey theorem and 3 corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch and arithmetic Weihrauch reducibility. We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. These notions turn out to be extremely relevant when exploring the Weihrauch degree of the problem DS of computing descending sequences in ill-founded linear orders. They allow us to show that DS, and the Weihrauch equivalent problem BS of finding bad sequences through non-well quasi-orders, while being very "hard" to solve, are rather weak in terms of uniform computational strength. We then generalize DS and BS by considering Gamma-presented orders, where Gamma is a Borel pointclass or Delta11, Sigma11, Pi11. We study the obtained DS-hierarchy and BS-hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level. Finally, we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and of Salem sets. We first work in the hyperspace K([0,1]) of compact subsets of [0,1] and show that the closed Salem sets form a Pi03-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in K([0,1]^d). We also generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still Pi03-complete when we endow K(R^d) with the Fell topology. A similar result holds also for the Vietoris topology. We conclude by showing how these results can be used to characterize the Weihrauch degree of the functions computing the Hausdorff and Fourier dimensions
On the descriptive complexity of Salem sets
In this paper we study the notion of Salem set from the point of view of
descriptive set theory. We first work in the hyperspace of
compact subsets of and show that the closed Salem sets form a
-complete family. This is done by characterizing the
complexity of the family of sets having sufficiently large Hausdorff or Fourier
dimension. We also show that the complexity does not change if we increase the
dimension of the ambient space and work in . We then
generalize the results by relaxing the compactness of the ambient space, and
show that the closed Salem sets are still -complete when
we endow the hyperspace of all closed subsets of with the Fell
topology. A similar result holds also for the Vietoris topology.Comment: Extended Lemma 3.1, fixed Lemma 5.3 and improved the presentation of
the results. To appear in Fundamenta Mathematica
Effective aspects of Hausdorff and Fourier dimension
In this paper, we study Hausdorff and Fourier dimension from the point of
view of effective descriptive set theory and Type-2 Theory of Effectivity.
Working in the hyperspace of compact subsets of , with
or , we characterize the complexity of the family
of sets having sufficiently large Hausdorff or Fourier dimension. This, in
turn, allows us to show that family of all the closed Salem sets is
-complete. One of our main tools is a careful analysis of the
effectiveness of a classical theorem of Kaufman. We furthermore compute the
Weihrauch degree of the functions computing Hausdorff and Fourier dimension of
closed sets.Comment: 36 page
The open and clopen Ramsey theorems in the Weihrauch lattice
We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to ATR_0 from the point of view of reverse mathematics, there is not a canonical way to phrase them as multivalued functions. We identify eight di\ufb00erent multivalued functions (\ufb01ve corresponding to the open Ramsey theorem and three corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. In particular one of our functions turns out to be strictly stronger than any previously studied multivalued functions arising from statements around ATR_0
The Weihrauch lattice at the level of : the Cantor-Bendixson theorem
This paper continues the program connecting reverse mathematics and
computable analysis via the framework of Weihrauch reducibility. In particular,
we consider problems related to perfect subsets of Polish spaces, studying the
perfect set theorem, the Cantor-Bendixson theorem and various problems arising
from them. In the framework of reverse mathematics these theorems are
equivalent respectively to and
, the two strongest subsystems of second
order arithmetic among the so-called big five. As far as we know, this is the
first systematic study of problems at the level of
in the Weihrauch lattice.
We show that the strength of some of the problems we study depends on the
topological properties of the Polish space under consideration, while others
have the same strength once the space is rich enough.Comment: 35 page
Algebraic properties of the first-order part of a problem
In this paper we study the notion of first-order part of a computational
problem, first introduced by Dzhafarov, Solomon, and Yokoyama, which captures
the "strongest computational problem with codomain that is
Weihrauch reducible to ". This operator is very useful to prove separation
results, especially at the higher levels of the Weihrauch lattice. We explore
the first-order part in relation with several other operators already known in
the literature. We also introduce a new operator, called unbounded finite
parallelization, which plays an important role in characterizing the
first-order part of parallelizable problems. We show how the obtained results
can be used to explicitly characterize the first-order part of several known
problems
On the descriptive complexity of Salem sets
In this paper we study the notion of Salem set from the point of view of
descriptive set theory. We first work in the hyperspace of
compact subsets of and show that the closed Salem sets form a
-complete family. This is done by characterizing the
complexity of the family of sets having sufficiently large Hausdorff or Fourier
dimension. We also show that the complexity does not change if we increase the
dimension of the ambient space and work in . We then
generalize the results by relaxing the compactness of the ambient space, and
show that the closed Salem sets are still -complete when
we endow with the Fell topology. A similar result
holds also for the Vietoris topology. We apply our results to characterize the
Weihrauch degree of the functions computing the Hausdorff and Fourier
dimensions
Finding descending sequences through ill-founded linear orders
In this work we investigate the Weihrauch degree of the problem
of finding an infinite descending sequence through a given ill-founded linear
order, which is shared by the problem of finding a bad sequence
through a given non-well quasi-order. We show that , despite being
hard to solve (it has computable inputs with no hyperarithmetic solution), is
rather weak in terms of uniform computational strength. To make the latter
precise, we introduce the notion of the deterministic part of a Weihrauch
degree. We then generalize and by considering
-presented orders, where is a Borel
pointclass or , ,
. We study the obtained -hierarchy and
-hierarchy of problems in comparison with the (effective) Baire
hierarchy and show that they do not collapse at any finite level
Old and New NICE Guidelines for the Evaluation of New Onset Stable Chest Pain: A Real World Perspective
Stable chest pain is a common clinical presentation that often requires further investigation using noninvasive or invasive testing, resulting in a resource-consuming problem worldwide. At onset of 2016, the National Institute for Health and Care Excellence (NICE) published an update on its guideline on chest pain. Three key changes to the 2010 version were provided by the new NICE guideline. First, the new guideline recommends that the previously proposed pretest probability risk score should no longer be used. Second, they also recommend that a calcium score of zero should no longer be used to rule out coronary artery disease (CAD) in patients with low pretest probability. Third, the new guideline recommends that all patients with new onset chest pain should be investigated with a coronary computed tomographic angiography (CTA) as a first-line investigation. However, in real world the impact of implementation of CTA for the evaluation of new onset chest pain remains to be evaluated, especially regarding its cost effectiveness. The aim of the present report was to discuss the results of the studies supporting new NICE guideline and its comparison with European and US guidelines