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

Guido Gherardi, Alberto Marcone
2009 arXiv   pre-print
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?

Guido Gherardi, Alberto Marcone
2008 Electronical Notes in Theoretical Computer Science  
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

K. VELA VELUPILLAI
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  ... 
doi:10.1142/s1793005712400017 fatcat:iuupodfe2jflnjtwdqhejvwtoi

ON THE FOUNDATIONS OF MATHEMATICAL ECONOMICS

J. BARKLEY ROSSER
2012 New Mathematics and Natural Computation  
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]

Andreas Gabriel
2013 arXiv   pre-print
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

Andreas Gabriel
2012 unpublished
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

Christoph Ari Spengler
2012 unpublished
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