Repetition-Operator in mathematical formulas?
J�rgen Will
is there a repetition-operator in mathematical formulas? I want to
abbreviate multiple sum signs, multiple product signs, multiple integral
signs and similar into one operator. Is the application of the product
operator (\Pi) onto other operators usual in mathematics?
Chip Eastham
The "dot-dot-dot" (ellipsis) is commonly used to
indicate omitted terms or signs of the kind you
ask about. Multiple (nested) sums, products, and
integrals can be expressed as a sum, product, and
integral respectively, with appropriate changes
in indexing or domain of integration.
If you are asking if the repetition can be formally
defined using the \Pi symbol, I've never seen it
done. It seems to me the crux of a usable symbol
is how clearly it defines the relationship between
scoping (indexing) from one operator to the next
repetition. Where the relationship is simple, the
ellipsis should usually suffice (be preferred). If
the relationship is not so simple, it is probably
worthwhile to develop the definition by induction/
recursion.
regards, chip
J�rgen Will
news:601c1c83-5906-4713...@p23g2000vbl.googlegroups.com...
On Sep 27, 10:30 am, "J�rgen Will" <jw...@onlinehome.de> wrote:
> The "dot-dot-dot" (ellipsis) is commonly used
"ellipsis" in this context.)
What I want, is to abbreviate the above notation.
Chip Eastham
> "Chip Eastham" <hardm...@gmail.com> schrieb im Newsbeitragnews:601c1c83-5906-4713...@p23g2000vbl.googlegroups.com...
> On Sep 27, 10:30 am, "Jürgen Will" <jw...@onlinehome.de> wrote:> The "dot-dot-dot" (ellipsis) is commonly used
>
> Do you mean "..." - Sum1 Sum2 ... Sum n? (I don't know what is the word
> "ellipsis" in this context.)
> What I want, is to abbreviate the above notation.
Yes, there is a special ellipsis character in Unicode,
but ... covers it well in ASCII.
Say you want to abbreviate a nested summation:
SUM1 SUM2 ... SUMn
A nested summation is itself a summation, so we can
abbreviate by yet another SUM symbol. The problem
is with clarifying what the summation is over!
In a simple case all the nested sums would have
limits on a single index that are independent.
The nested SUM's can be reordered arbitrarily
in that case, and the "abbreviated" SUM is
taken over a Cartesian product of the limits
for the nested SUM's.
In a complicated case, the limits of a nested
SUM depend on the indexes of the SUM's outer
to that one. For example:
N i j
SUM SUM SUM f(i,j,k)
i=1 j=1 k=1
In this particular case we could rewrite the
summation as:
SUM f(i,j,k)
(i,j,k) in A(N)
where A(N) = {(i,j,k)| 1 <= k <= j <= i <= N}.
For other summations it can be more difficult
to explain what the limit/domain of summation
is. But a unifying principle is that nested
sums make a sum, nested products make a
product, and nested integrals make an integral.
regards, chip
J�rgen Will
> For example:
> N i j
> SUM SUM SUM f(i,j,k)
> i=1 j=1 k=1
> In this particular case we could rewrite the
> summation as:
> SUM f(i,j,k)
>(i,j,k) in A(N)
> where A(N) = {(i,j,k)| 1 <= k <= j <= i <= N}.
Ok, I think that hepls. This notation (with index sets) is usual. I have to
see if this notation brings the same clarity as what I want with the
repetition operator.