About: Effective dimension

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In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notions of effective dimension) of which the most common is effective Hausdorff dimension. Dimension, in mathematics, is a particular way of describing the size of an object (contrasting with measure and other, different, notions of size). Hausdorff dimension generalizes the well-known integer dimensions assigned to points, lines, planes, etc. by allowing one to distinguish between objects of intermediate size between these integer-dimensional objects. For example, fractal subsets of the plane may have intermediate dimension between 1 and 2, as they are "larger" than lines or curves, and yet "smalle

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rdfs:label	Effective dimension (en)
rdfs:comment	In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notions of effective dimension) of which the most common is effective Hausdorff dimension. Dimension, in mathematics, is a particular way of describing the size of an object (contrasting with measure and other, different, notions of size). Hausdorff dimension generalizes the well-known integer dimensions assigned to points, lines, planes, etc. by allowing one to distinguish between objects of intermediate size between these integer-dimensional objects. For example, fractal subsets of the plane may have intermediate dimension between 1 and 2, as they are "larger" than lines or curves, and yet "smalle (en)
dcterms:subject	Computable analysis Fractals Measure theory Metric geometry Dimension theory
Wikipage page ID	10978612 (xsd:integer)
Wikipage revision ID	1111927880 (xsd:integer)
Link from a Wikipage to another Wikipage	List of fractals by Hausdorff dimension Computability theory Mathematics Computable analysis Hausdorff dimension Algorithmically random sequence Euclidean space Fractal dimension Kolmogorov complexity Fractals Measure theory Metric geometry Dimension Dimension theory Cantor space Martingale (probability theory)
Link from a Wikipage to an external page	https://www.cs.auckland.ac.nz/research/groups/CDMTCS/researchreports/ http://www.cs.iastate.edu/~lutz/papers.html https://web.archive.org/web/20070609152905/http:/math.uni-heidelberg.de/logic/reimann/publications.html
sameAs	Effective dimension Effective dimension Effective dimension Effective dimension
dbp:wikiPageUsesTemplate	dbt:Cite_journal dbt:One_source dbt:Reflist
date	November 2013 (en)
text	The article relies on sources from one research group. (en)
has abstract	In mathematics, effective dimension is a modification of Hausdorff dimension and other fractal dimensions that places it in a computability theory setting. There are several variations (various notions of effective dimension) of which the most common is effective Hausdorff dimension. Dimension, in mathematics, is a particular way of describing the size of an object (contrasting with measure and other, different, notions of size). Hausdorff dimension generalizes the well-known integer dimensions assigned to points, lines, planes, etc. by allowing one to distinguish between objects of intermediate size between these integer-dimensional objects. For example, fractal subsets of the plane may have intermediate dimension between 1 and 2, as they are "larger" than lines or curves, and yet "smaller" than filled circles or rectangles. Effective dimension modifies Hausdorff dimension by requiring that objects with small effective dimension be not only small but also locatable (or partially locatable) in a computable sense. As such, objects with large Hausdorff dimension also have large effective dimension, and objects with small effective dimension have small Hausdorff dimension, but an object can have small Hausdorff but large effective dimension. An example is an algorithmically random point on a line, which has Hausdorff dimension 0 (since it is a point) but effective dimension 1 (because, roughly speaking, it can't be effectively localized any better than a small interval, which has Hausdorff dimension 1). (en)
gold:hypernym	Modification
prov:wasDerivedFrom	wikipedia-en:Effective_dimension?oldid=1111927880&ns=0
page length (characters) of wiki page	12512 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Effective_dimension
is Link from a Wikipage to another Wikipage of	Ludwig Staiger Jack Lutz
is foaf:primaryTopic of	wikipedia-en:Effective_dimension

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