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Difference between revisions of "Multiple sequence"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m0654002.png" />-tuple sequence, of elements of a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m0654003.png" />''
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''$k$-tuple sequence, of elements of a given set $X$''
  
A mapping of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m0654004.png" />-th power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m0654005.png" /> of the set of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m0654006.png" /> into the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m0654007.png" />. An element (or term) of a multiple sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m0654008.png" /> is an ordered set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m0654009.png" /> elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m06540010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m06540011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m06540012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m06540013.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m06540014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m06540015.png" />). The element is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065400/m06540016.png" />.
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A mapping of the $k$-th power $\mathbb{N}^k$ of the set of natural numbers $\mathbb{N}$ into the set $X$. An element (or term) of a multiple sequence $f : \mathbb{N}^k \rightarrow X$ is an ordered set of $k+1$ elements $(n_1,\ldots,n_k,x)$, where $x = f(n_1,\ldots,n_k)$, $x \in X$, $(n_1,\ldots,n_k) \in \mathbb{N}^k$, i.e. $n_j \in \mathbb{N}$ ($j = 1,\ldots,k$). The element is also denoted by $x_{n_1 \ldots n_k}$.

Latest revision as of 18:52, 12 October 2014

$k$-tuple sequence, of elements of a given set $X$

A mapping of the $k$-th power $\mathbb{N}^k$ of the set of natural numbers $\mathbb{N}$ into the set $X$. An element (or term) of a multiple sequence $f : \mathbb{N}^k \rightarrow X$ is an ordered set of $k+1$ elements $(n_1,\ldots,n_k,x)$, where $x = f(n_1,\ldots,n_k)$, $x \in X$, $(n_1,\ldots,n_k) \in \mathbb{N}^k$, i.e. $n_j \in \mathbb{N}$ ($j = 1,\ldots,k$). The element is also denoted by $x_{n_1 \ldots n_k}$.

How to Cite This Entry:
Multiple sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiple_sequence&oldid=33584
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article