CZ:Featured article/Current: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Chunbum Park
imported>Chunbum Park
No edit summary
Line 1: Line 1:
== '''[[Hausdorff dimension]]''' ==
== '''[[National Institute of Standards and Technology]]''' ==
''by  [[User:Melchior Grutzmann|Melchior Grutzmann]] (and [[User:Brandon Piercy|Brandon Piercy]] and [[User:Hendra I. Nurdin|Hendra I. Nurdin]])
''by  [[User:Paul Wormer|Paul Wormer]], [[User:Milton Beychok|Milton Beychok]] and [[User:John R. Brews|John R. Brews]]


----
----


In [[mathematics]], the '''Hausdorff dimension''' is a way of defining a possibly fractional exponent for all figures in a [[metric space]] such that the dimension describes partially the amount to that the set fills the space around it.  For example, a [[plane (geometry)|plane]] would have a Hausdorff dimension of 2, because it fills a 2-parameter subset. However, it would not make sense to give the [[Sierpiński triangle]] [[fractal]] a dimension of 2, since it does not fully occupy the 2-dimensional realm. The Hausdorff dimension describes this mathematically by measuring the size of the set. For self-similar sets there is a relationship to the number of self-similar subsets and their scale.
{{Image|NIST Blue Logo.png|right|168px}}
The '''National Institute of Standards and Technology''' (NIST) is a  [[United States]] federal agency within the [[U.S. Department of Commerce]].<ref>[http://www.nist.gov/public_affairs/general_information.cfm NIST General Information], from the NIST website.</ref> The institute was founded in 1901 with the aim to advance measurement science, standards, and technology. NIST was known between 1901–1988 as the '''[[National Bureau of Standards]]''' (NBS).


=== Informal definition ===
NIST has an operating budget of about $1.6 [[Parts-per notation#Summary of large number names|billion]]<ref name=Budget>[http://nist.gov/public_affairs/budget/2010budgetpiechart.cfm NIST Resources Fiscal Year 2010], from the NIST website.</ref> and operates in two locations: [[Gaithersburg, Maryland]], and [[Boulder, Colorado]]. NIST employs a staff of about 2,900 scientists, engineers, technicians, and support and administrative personnel. About 2,600 associates and facility users from academia, industry and other government agencies complement the staff.<ref>[http://www.nist.gov/hrmd/perks.cfm Why Work at NIST?], from the NIST website.</ref>
Intuitively, the dimension of a set is the number of independent parameters one has to pick in order to fix a point. This is made rigorously with the notion of ''d''-dimensional (topological) [[manifold]] which are particularly regular sets. The problem with the classical notion is that you can easily break up the digits of a real number to map it bijectively to two (or ''d'') real numbers.  The example of space filling curves shows that it is even possible to do this in a continuous (but non-bijective) way.


The notion of Hausdorff dimension refines this notion of dimension such that the dimension can be any non-negative number.
==History==


Benoît Mandelbrot discovered<ref>B.B. Mandelbrot: ''The fractal geometry of nature'', Freemann '''(1983)''', ISBN 978-0-716-711-865</ref> that many objects in nature are not strictly classical smooth bodies, but best approximated as fractal sets, i.e. subsets of '''R'''<sup>''N''</sup> whose Hausdorff dimension is strictly greater than its topological dimension.
Article 1, Section 8 of the [[United States Constitution]] grants the [[U.S. Congress]] the power to '''''"To coin money, regulate the value thereof, and of foreign coin, and fix the standard of weights and measures"'''''. In June 1836, almost fifty years after the U. S. Constitution was ratified, the [[U.S. Senate]] and the [[U.S. House of Representatives]] adopted a joint resolution establishing a [[U.S. Office of Weights and Measures]] within the [[U.S. Department of the Treasury]]. From that date until March 1901, the Office of Weights and Measures was administered mostly by the [[U.S. Coast Survey]], later renamed as the [[U.S. Coast and Geodetic Survey]] (USC&GS), within the [[U.S. Department of the Treasury]].<ref>There were some time periods during which the [[U.S. Army]] and/or the [[U.S. Navy]] administered the USC&GS</ref> [[Ferdinand Rudolph Hassler]], a professor of mathematics, served as the head of U.S. Coast Survey as well as the Office of Weights and Measures from 1836 to 1843.<ref>[http://www.dean.usma.edu/math/about/history/hassler.htm Ferdinand Rudolph Hessler]</ref><ref>[http://www.nist.gov/pml/pubs/sp447/index.cfm Weights and Measures Standards of the United States: A brief history], from the NIST website.</ref>


 
''[[National Institute of Standards and Technology|.... (read more)]]''
=== Hausdorff measure and dimension ===
Let ''d'' be a non-negative real number and ''S'' ⊂ ''X'' a subset of a metric space (''X'',''ρ'').  The ''d''-dimesional Hausdorff measure of scale ''δ''>0 is
:<math> H^{d*}_\delta(S) := \inf \{\sum_{i=1}^\infty r_i^d : S\subset\bigcup_{i=1}^\infty B_{r_i}(x_i), r_i\le\delta \}</math>
where B<sub>''r''<sub>''i''</sub>(''x''<sub>''i''</sub>)</sub> is the open ball around ''x''<sub>''i''</sub> ∈ ''X'' of radius ''r''<sub>''i''</sub>.  The ''d''-dimensional Hausdorff measure is now the limit
:<math> H^{d*}(S) := \lim_{\delta\to0+} H^{d*}_\delta(S)</math>.
As in the Carathéodory construction a set  ''S'' ⊂ ''X'' is called ''d''-measurable iff
:<math> H^{d*}(T) = H^{d*}(S\cap T)+ H^{d*}(T\cap X\setminus S)</math> for all  ''T'' ⊂ ''X''.
A set ''S'' ⊂ ''X'' is called Hausdorff measurable if it is H<sup>''d''</sup>-measurable for all ''d''≥0.
 
''[[Hausdorff dimension|.... (read more)]]''


{| class="wikitable collapsible collapsed" style="width: 90%; float: center; margin: 0.5em 1em 0.8em 0px;"
{| class="wikitable collapsible collapsed" style="width: 90%; float: center; margin: 0.5em 1em 0.8em 0px;"
|-
|-
! style="text-align: center;" | &nbsp;[[Hausdorff dimension#Literature|notes]]
! style="text-align: center;" | &nbsp;[[National Institute of Standards and Technology#Literature|notes]]
|-
|-
|
|
{{reflist|2}}
{{reflist|2}}
|}
|}

Revision as of 22:31, 26 January 2012

National Institute of Standards and Technology

by Paul Wormer, Milton Beychok and John R. Brews

(PD) Logo: National Institute of Standards and Technology

The National Institute of Standards and Technology (NIST) is a United States federal agency within the U.S. Department of Commerce.[1] The institute was founded in 1901 with the aim to advance measurement science, standards, and technology. NIST was known between 1901–1988 as the National Bureau of Standards (NBS).

NIST has an operating budget of about $1.6 billion[2] and operates in two locations: Gaithersburg, Maryland, and Boulder, Colorado. NIST employs a staff of about 2,900 scientists, engineers, technicians, and support and administrative personnel. About 2,600 associates and facility users from academia, industry and other government agencies complement the staff.[3]

History

Article 1, Section 8 of the United States Constitution grants the U.S. Congress the power to "To coin money, regulate the value thereof, and of foreign coin, and fix the standard of weights and measures". In June 1836, almost fifty years after the U. S. Constitution was ratified, the U.S. Senate and the U.S. House of Representatives adopted a joint resolution establishing a U.S. Office of Weights and Measures within the U.S. Department of the Treasury. From that date until March 1901, the Office of Weights and Measures was administered mostly by the U.S. Coast Survey, later renamed as the U.S. Coast and Geodetic Survey (USC&GS), within the U.S. Department of the Treasury.[4] Ferdinand Rudolph Hassler, a professor of mathematics, served as the head of U.S. Coast Survey as well as the Office of Weights and Measures from 1836 to 1843.[5][6]

.... (read more)

Retrieved from "https://citizendium.org/wiki/index.php?title=CZ:Featured_article/Current&oldid=829691"