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Difference between revisions of "User:Richard Pinch/sandbox-CZ"

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(Start article: Zipf distribution)
(Start article: Weierstrass preparation theorem)
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=Weierstrass preparation theorem=
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In [[algebra]], the '''Weierstrass preparation theorem''' describes a canonical form for [[formal power series]] over a [[complete local ring]].
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Let ''O'' be a complete local ring and ''f'' a formal power series in ''O''[[''X'']].  Then ''f'' can be written uniquely in the form
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:<math>f = (X^n + b_{n-1}X^{n-1} + \cdots + b_0) u(x) , \,</math>
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where the ''b''<sub>''i''</sub> are in the maximal ideal ''m'' of ''O'' and ''u'' is a unit of ''O''[[''X'']].
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The integer ''n'' defined by the theorem is the '''Weierstrass degree''' of ''f''.
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==References==
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* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=208-209 }}
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=Zipf distribution=
 
=Zipf distribution=
 
In [[probability theory]] and [[statistics]], the '''Zipf distribution''' and '''zeta distribution''' refer to a class of [[discrete probability distribution]]s.  They have been used to model the distribution of words in words in a text , of text strings and keys in databases, and of the sizes of businesses and towns.
 
In [[probability theory]] and [[statistics]], the '''Zipf distribution''' and '''zeta distribution''' refer to a class of [[discrete probability distribution]]s.  They have been used to model the distribution of words in words in a text , of text strings and keys in databases, and of the sizes of businesses and towns.

Revision as of 06:42, 8 September 2013

Weierstrass preparation theorem

In algebra, the Weierstrass preparation theorem describes a canonical form for formal power series over a complete local ring.

Let O be a complete local ring and f a formal power series in O''X''. Then f can be written uniquely in the form

\[f = (X^n + b_{n-1}X^{n-1} + \cdots + b_0) u(x) , \,\]

where the bi are in the maximal ideal m of O and u is a unit of O''X''.

The integer n defined by the theorem is the Weierstrass degree of f.

References


Zipf distribution

In probability theory and statistics, the Zipf distribution and zeta distribution refer to a class of discrete probability distributions. They have been used to model the distribution of words in words in a text , of text strings and keys in databases, and of the sizes of businesses and towns.

The Zipf distribution with parameter n assigns probability proportional to 1/r to an integer rn and zero otherwise, with normalization factor Hn, the n-th harmonic number.

A Zipf-like distribution with parameters n and s assigns probability proportional to 1/rs to an integer rn and zero otherwise, with normalization factor \(\sum_{r=1}^n 1/r^s\).

The zeta distribution with parameter s assigns probability proportional to 1/rs to all integers r with normalization factor given by the Riemann zeta function 1/ζ(s).

References

How to Cite This Entry:
Richard Pinch/sandbox-CZ. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-CZ&oldid=30416