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18
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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Cited by 14 (1 self)
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
A quadratic rate of asymptotic regularity for CAT(0)spaces
, 2005
"... In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the KrasnoselskiMann iterations of nonexpansive mappings in CAT(0)spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hy ..."
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Cited by 10 (4 self)
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In this paper we obtain a quadratic bound on the rate of asymptotic regularity for the KrasnoselskiMann iterations of nonexpansive mappings in CAT(0)spaces, whereas previous results guarantee only exponential bounds. The method we use is to extend to the more general setting of uniformly convex hyperbolic spaces a quantitative version of a strengthening of Groetsch’s theorem obtained by Kohlenbach using methods from mathematical logic (socalled “proof mining”).
Strongly Uniform Bounds from SemiConstructive Proofs
, 2004
"... In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and nor ..."
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Cited by 10 (7 self)
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In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters raging over metrically bounded (not necessarily compact!) spaces. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semiintuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain nonconstructive principles. Contrary to
Rates of Asymptotic Regularity for Halpern Iterations of Nonexpansive Mappings
, 2008
"... In this paper we obtain new effective results on the Halpern iterations of nonexpansive mappings using methods from mathematical logic or, more specifically, prooftheoretic techniques. We give effective rates of asymptotic regularity for the Halpern iterations of nonexpansive selfmappings of nonemp ..."
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Cited by 4 (3 self)
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In this paper we obtain new effective results on the Halpern iterations of nonexpansive mappings using methods from mathematical logic or, more specifically, prooftheoretic techniques. We give effective rates of asymptotic regularity for the Halpern iterations of nonexpansive selfmappings of nonempty convex sets in normed spaces. The paper presents another case study in the project of proof mining, which is concerned with the extraction of effective uniform bounds from (primafacie) ineffective proofs.
Nonexpansive iterations in uniformly convex Whyperbolic spaces
, 2008
"... We propose the class of uniformly convex Whyperbolic spaces with monotone modulus of uniform convexity (UCWhyperbolic spaces for short) as an appropriate setting for the study of nonexpansive iterations. UCWhyperbolic spaces are a natural generalization both of uniformly convex normed spaces and ..."
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Cited by 2 (0 self)
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We propose the class of uniformly convex Whyperbolic spaces with monotone modulus of uniform convexity (UCWhyperbolic spaces for short) as an appropriate setting for the study of nonexpansive iterations. UCWhyperbolic spaces are a natural generalization both of uniformly convex normed spaces and CAT(0)spaces. Furthermore, we apply proof mining techniques to get effective rates of asymptotic regularity for Ishikawa iterations of nonexpansive selfmappings of closed convex subsets in UCWhyperbolic spaces. These effective results are new even for uniformly convex Banach spaces.