Effective Bounds on Strong Unicity in L1-Approximation

In this paper we present another case study in the general project of Proof Mining which means the logical analysis of prima facie non-effective proofs with the aim of extracting new computationally relevant data. We use techniques based on monotone functional interpretation (developed in [17]) to analyze Cheney's simplification [6] of Jackson's original proof [9] from 1921 of the uniqueness of the best L1-approximation of continuous functions f in C[0, 1] by polynomials p in Pn of degre

Proof mining with the bounded functional interpretation

In this doctoral thesis, we will see how the bounded functional interpretation of Ferreira and Oliva [13] can be used and contribute to the Proof Mining program, a program which aims to extract computational information from mathematical theorems using proof-theoretic techniques. We present a method for the elimination of sequential weak compactness arguments from the quantitative analysis of certain mathematical results. This method works as a “macro” and allowed us to obtain quantitative versions of important results of F. E. Browder [6], R. Wittmann [51] and H. H. Bauschke [2] in fixed point theory in Hilbert spaces. Although Browder’s and Wittmann’s theorems were previously analyzed by Kohlenbach using the monotone functional interpretation, it was not clear why such analyses did not require the use of functionals defined by bar recursion. This phenomenon is now fully understood, by a theoretical justification for the elimination of sequential weak compactness in the context of the bounded functional interpretation. Bauschke’s theorem is an important generalization of Wittmann’s theorem and its original proof is also analyzed here. The analyses of these results also required a quantitative version of a projection argument which turned out to be simpler when guided by the bounded functional interpretation than when using the monotone functional interpretation. In the context of the theory of monotone operators, results due to Boikanyo/Moro¸sanu [5] and Xu [52] for the strong convergence of variants of the proximal point algorithm were analyzed and bounds on the metastablility property of these iterations obtained. These results are the first applications of the bounded functional interpretation to the proof mining of concrete mathematical results

Measuring robustness of community structure in complex networks

The theory of community structure is a powerful tool for real networks, which can simplify their topological and functional analysis considerably. However, since community detection methods have random factors and real social networks obtained from complex systems always contain error edges, evaluating the robustness of community structure is an urgent and important task. In this letter, we employ the critical threshold of resolution parameter in Hamiltonian function, $\gamma_C$, to measure the robustness of a network. According to spectral theory, a rigorous proof shows that the index we proposed is inversely proportional to robustness of community structure. Furthermore, by utilizing the co-evolution model, we provides a new efficient method for computing the value of $\gamma_C$. The research can be applied to broad clustering problems in network analysis and data mining due to its solid mathematical basis and experimental effects.Comment: 6 pages, 4 figures. arXiv admin note: text overlap with arXiv:1303.7434 by other author

Using concept lattices to mine functional dependencies

Concept Lattices have been proved to be a valuable tool to represent the knowlegde in a database. In this paper we show how functional dependencies in databases can be extracted using Concept Lattices, not preprocessing the original database, but providing a new closure operator. We also prove that this method generalizes the previous methods and closure operators that are being used to find association rules in binary databases.Postprint (published version

Proof mining in metric fixed point theory and ergodic theory

In this survey we present some recent applications of proof mining to the fixed point theory of (asymptotically) nonexpansive mappings and to the metastability (in the sense of Terence Tao) of ergodic averages in uniformly convex Banach spaces.Comment: appeared as OWP 2009-05, Oberwolfach Preprints; 71 page

A functional interpretation for nonstandard arithmetic

We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Goedel and Shoenfield can be used to rewrite proofs performed in these systems into standard ones. These functional interpretations show in particular that our nonstandard systems are conservative extensions of extensional Heyting and Peano arithmetic in all finite types, strengthening earlier results by Moerdijk, Palmgren, Avigad and Helzner. We will also indicate how our rewriting algorithm can be used for term extraction purposes. To conclude the paper, we will point out some open problems and directions for future research and mention some initial results on saturation principles