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How incomputable is the separable Hahn-Banach theorem?
[article]

2009
*
arXiv
*
pre-print

Extending work of Brattka, we show that a natural multi-valued function associated with

arXiv:0808.1663v3
fatcat:7nu2sqnj3bfabnlpjkdsawfpsy
*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*. ...##
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How Incomputable is the Separable Hahn-Banach Theorem?

2008
*
Electronical Notes in Theoretical Computer Science
*

We use these notions to provide a method to determine

doi:10.1016/j.entcs.2008.12.009
fatcat:c5wcvh2wq5co7f2gfogjaf76om
*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*. ...##
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TAMING THE INCOMPUTABLE, RECONSTRUCTING THE NONCONSTRUCTIVE AND DECIDING THE UNDECIDABLE IN MATHEMATICAL ECONOMICS

2012
*
New Mathematics and Natural Computation
*

*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*...

##
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ON THE FOUNDATIONS OF MATHEMATICAL ECONOMICS

2012
*
New Mathematics and Natural Computation
*

He sees their approach as consistent with

doi:10.1142/s1793005712400029
fatcat:u4is4hiyevebhmkxo3jwaxxbxu
*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. ...##
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Detection and Characterisation of Multipartite Quantum Entanglement
[article]

2013
*
arXiv
*
pre-print

In particular, it

arXiv:1305.7401v1
fatcat:7x33xohmtjahnnywse7vqwnu6e
*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 ...##
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Detection and characterisation of multipartite entanglement

2012
unpublished

In particular, it

doi:10.25365/thesis.26062
fatcat:ig7l2d77wrhr7o7wt2cdsfvm34
*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 ...##
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Entanglement and correlations in composite quantum systems

2012
unpublished

Moreover,

doi:10.25365/thesis.23533
fatcat:m7aszusjhrdyde4dsv7l3fbaj4
*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. ...