Definition:Multiindex

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Definition

Definition 1

Let $\displaystyle m = \prod_{j \mathop \in J} X_j^{k_j}$ be a mononomial in the indexed set $\left\{{X_j: j \in J}\right\}$.

Such a mononomial can be expressed implicitly and more compactly by referring only to the sequence of indices:

$k = \left \langle {k_j}\right \rangle_{j \in J}$


and write $m = \mathbf X^k$ without explicit reference to the indexing set.


Such an expression is called a multiindex (or multi-index).


Definition 2

Let $J$ be a set.


A $J$-multiindex is a sequence of natural numbers indexed by $J$:

$\displaystyle k = \left \langle {k_j}\right \rangle_{j \in J}$

with only finitely many of the $k_j$ non-zero.


Definition 3

A multiindex is an element of $\Z^J$, the free $\Z$-module on $J$, an abelian group of rank over $\Z$ equal to the cardinality of $J$.


Modulus

Let $k = \left \langle {k_j}\right \rangle_{j \mathop \in J}$ be a multiindex.


The modulus of such a multiindex $k$ is defined by:

$\displaystyle \left|{k}\right| = \sum_{j \mathop \in J} k_j$


Also known as

Some sources hyphenate for clarity: multi-index.


Also see


Linguistic note

The plural is multiindices (or multi-indices).