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How incomputable is the separable Hahn-Banach theorem?
[article]
2009
arXiv
pre-print
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Extending work of Brattka, we show that a natural multi-valued function associated with the Hahn-Banach Extension Theorem is Sep-complete. ...
We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. ...
Moreover we prove that Sep ∼ =c HB, establishing the degree of incomputability of the separable Hahn-Banach Theorem. ...
arXiv:0808.1663v3
fatcat:7nu2sqnj3bfabnlpjkdsawfpsy
How Incomputable is the Separable Hahn-Banach Theorem?
2008
Electronical Notes in Theoretical Computer Science
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We use these notions to provide a method to determine the computational complexity of the Hahn-Banach Extension Theorem. ...
In particular, we show how to use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. ...
Moreover we prove that Sep ∼ = c HB, establishing the degree of incomputability of the separable Hahn-Banach Theorem. ...
doi:10.1016/j.entcs.2008.12.009
fatcat:c5wcvh2wq5co7f2gfogjaf76om
TAMING THE INCOMPUTABLE, RECONSTRUCTING THE NONCONSTRUCTIVE AND DECIDING THE UNDECIDABLE IN MATHEMATICAL ECONOMICS
2012
New Mathematics and Natural Computation
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The Hahn-Banach theorem, in other words, has a constructive version, but only on subspaces of separable normed spaces. ...
contraction mapping theorem, …x point theorems (I shall only consider the Banach Contraction Mapping theorem and the Brouwer and Schauder …x point theorems) and the Hahn-Banach (and associated separating ...
doi:10.1142/s1793005712400017
fatcat:iuupodfe2jflnjtwdqhejvwtoi
ON THE FOUNDATIONS OF MATHEMATICAL ECONOMICS
2012
New Mathematics and Natural Computation
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He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications ...
of these arguments for such "crown jewels" of mathematical economics as the existence of general equilibrium and the second welfare theorem. ...
The usual caveat holds. ...
doi:10.1142/s1793005712400029
fatcat:u4is4hiyevebhmkxo3jwaxxbxu
Detection and Characterisation of Multipartite Quantum Entanglement
[article]
2013
arXiv
pre-print
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In particular, it is completely unclear how several concepts from the bipartite case can be generalised in a meaningful way to multipartite scenarios. ...
The research field of quantum entanglement theory is comparatively new. ...
The entanglement witness theorem is a direct consequence of the
Hahn-Banach-theorem, which states the following: Given two disjoint convex
sets, at least one of which is closed, then there exists a ...
arXiv:1305.7401v1
fatcat:7x33xohmtjahnnywse7vqwnu6e
Detection and characterisation of multipartite entanglement
2012
unpublished
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In particular, it is completely unclear how several concepts from the bipartite case can be generalised in a meaningful way to multipartite scenarios. ...
The research field of quantum entanglement theory is comparatively new. ...
The entanglement witness theorem is a direct consequence of the
Hahn-Banach-theorem, which states the following: Given two disjoint convex
sets, at least one of which is closed, then there exists a ...
doi:10.25365/thesis.26062
fatcat:ig7l2d77wrhr7o7wt2cdsfvm34
Entanglement and correlations in composite quantum systems
2012
unpublished
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Moreover, the classification of multipartite entanglement is also addressed, where a systematic approach for discriminating between different classes is given. ...
The consequences of entanglement are far-reaching. On a fundamental level, it allows to demonstrate that there is no local-realistic alternative to quantum theory. ...
CS and MH acknowledge financial support from the Austrian FWF, Project P21947N16. BCH acknowledges the fellowship MOEL 428. ...
doi:10.25365/thesis.23533
fatcat:m7aszusjhrdyde4dsv7l3fbaj4