Definition:Multiindex
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Definition
Definition 1
Let $\ds m = \prod_{j \mathop \in J} X_j^{k_j}$ be a monomial in the indexed set $\set {X_j: j \mathop \in J}$.
Such a monomial can be expressed implicitly and more compactly by referring only to the sequence of indices:
- $k = \sequence {k_j}_{j \mathop \in J}$
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 $J$:
- $\ds k = \sequence {k_j}_{j \mathop \in J}$
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 $k = \sequence {k_j}_{j \mathop \in J}$ be a multiindex.
The modulus of such a multiindex $k$ is defined by:
- $\ds \size k = \sum_{j \mathop \in J} k_j$
Also known as
Some sources hyphenate for clarity: multi-index.
Also see
Linguistic Note
The plural of multiindex (or multi-index) is multiindices (or multi-indices).