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== '''[[Hausdorff dimension]]''' ==
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''by  [[User:Melchior Grutzmann|Melchior Grutzmann]] (and [[User:Brandon Piercy|Brandon Piercy]] and [[User:Hendra I. Nurdin|Hendra I. Nurdin]])
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==Footnotes==
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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.
 
=== Informal definition ===
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.
 
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.
 
 
=== 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>) 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)]]''
 
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Latest revision as of 10:19, 11 September 2020

1901 photograph of a stentor (announcer) at the Budapest Telefon Hirmondó.

Telephone newspaper is a general term for the telephone-based news and entertainment services which were introduced beginning in the 1890s, and primarily located in large European cities. These systems were the first example of electronic broadcasting, and offered a wide variety of programming, however, only a relative few were ever established. Although these systems predated the invention of radio, they were supplanted by radio broadcasting stations beginning in the 1920s, primarily because radio signals were able to cover much wider areas with higher quality audio.

History

After the electric telephone was introduced in the mid-1870s, it was mainly used for personal communication. But the idea of distributing entertainment and news appeared soon thereafter, and many early demonstrations included the transmission of musical concerts. In one particularly advanced example, Clément Ader, at the 1881 Paris Electrical Exhibition, prepared a listening room where participants could hear, in stereo, performances from the Paris Grand Opera. Also, in 1888, Edward Bellamy's influential novel Looking Backward: 2000-1887 foresaw the establishment of entertainment transmitted by telephone lines to individual homes.

The scattered demonstrations were eventually followed by the establishment of more organized services, which were generally called Telephone Newspapers, although all of these systems also included entertainment programming. However, the technical capabilities of the time meant that there were limited means for amplifying and transmitting telephone signals over long distances, so listeners had to wear headphones to receive the programs, and service areas were generally limited to a single city. While some of the systems, including the Telefon Hirmondó, built their own one-way transmission lines, others, including the Electrophone, used standard commercial telephone lines, which allowed subscribers to talk to operators in order to select programming. The Telephone Newspapers drew upon a mixture of outside sources for their programs, including local live theaters and church services, whose programs were picked up by special telephone lines, and then retransmitted to the subscribers. Other programs were transmitted directly from the system's own studios. In later years, retransmitted radio programs were added.

During this era telephones were expensive luxury items, so the subscribers tended to be the wealthy elite of society. Financing was normally done by charging fees, including monthly subscriptions for home users, and, in locations such as hotel lobbies, through the use of coin-operated receivers, which provided short periods of listening for a set payment. Some systems also accepted paid advertising.

Footnotes

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