103,909 research outputs found
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, , 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 . 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
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
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