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Metric spaces
"... These slides: available on my web pageExecutive summary Magnitude is a realvalued invariant of metric spaces. It seems not to have been previously investigated. Conjecturally, it captures a great deal of geometric information. It arose from a general study of ‘size ’ in mathematics. Plan 1. Where d ..."
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These slides: available on my web pageExecutive summary Magnitude is a realvalued invariant of metric spaces. It seems not to have been previously investigated. Conjecturally, it captures a great deal of geometric information. It arose from a general study of ‘size ’ in mathematics. Plan 1. Where does magnitude come from? 2. The magnitude of a finite space 3. The magnitude of a compact space
Quantitative Concept Analysis
 In Florent Domenach, Dmitry
, 2012
"... Abstract. Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisf ..."
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Abstract. Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy all attributes in the second, and vice versa. Many applications, though, provide contexts with quantitative information, telling not just whether an object satisfies an attribute, but also quantifying this satisfaction. Contexts in this form arise as rating matrices in recommender systems, as occurrence matrices in text analysis, as pixel intensity matrices in digital image processing, etc. Such applications have attracted a lot of attention, and several numeric extensions of FCA have been proposed. We propose the framework of proximity sets (proxets), which subsume partially ordered sets (posets) as well as metric spaces. One feature of this approach is that it extracts from quantified contexts quantified concepts, and thus allows full use of the available information. Another feature is that the categorical approach allows analyzing any universal properties that the classical FCA and the new versions may have, and thus provides structural guidance for aligning and combining the approaches.
© Author(s) 2013. CC Attribution 3.0 License. Atmospheric Chemistry and Physics
, 2013
"... pen A ccess Quantifying aerosol mixing state with entropy and diversity measures ..."
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pen A ccess Quantifying aerosol mixing state with entropy and diversity measures
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"... All intext references underlined in blue are linked to publications on ResearchGate, ..."
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All intext references underlined in blue are linked to publications on ResearchGate,
Bicompletions of Distance Matrices To Samson Abramsky on the occasion of his 60th birthday
"... Abstract. In the practice of information extraction, the input data are usually arranged into pattern matrices, and analyzed by the methods of linear algebra and statistics, such as principal component analysis. In some applications, the tacit assumptions of these methods lead to wrong results. The ..."
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Abstract. In the practice of information extraction, the input data are usually arranged into pattern matrices, and analyzed by the methods of linear algebra and statistics, such as principal component analysis. In some applications, the tacit assumptions of these methods lead to wrong results. The usual reason is that the matrix composition of linear algebra presents information as flowing in waves, whereas it sometimes flows in particles, which seek the shortest paths. This waveparticle duality in computation and information processing has been originally observed by Abramsky. In this paper we pursue a particle view of information, formalized in distance spaces, which generalize metric spaces, but are slightly less general than Lawvere’s generalized metric spaces. In this framework, the task of extracting the ’principal components ’ from a given matrix of data boils down to a bicompletion, in the sense of enriched category theory. We describe the bicompletion construction for distance matrices. The practical goal that motivates this research is to develop a method to estimate the hardness of attack constructions in security. 1
ORIGINAL RESEARCH ARTICLE
, 2012
"... doi: 10.3389/fmicb.2012.00293 Millimeterscale patterns of phylogenetic and trait diversity in a salt marsh microbial mat ..."
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doi: 10.3389/fmicb.2012.00293 Millimeterscale patterns of phylogenetic and trait diversity in a salt marsh microbial mat