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Topic: Is there a name for this notation?
Replies: 8   Last Post: Oct 9, 2011 9:17 PM

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Kaba

Posts: 289
Registered: 5/23/11
Re: Is there a name for this notation?
Posted: Oct 9, 2011 11:37 AM
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Kaba wrote:
> Kaba wrote:
> > In the case of "base 2", this can be seen as a sparse form of the
> > traditional base-2 representation. Let I subset NN. Then the value of I
> > is given by
> >
> > V(I) = sum_{i in I} 2^i,

>
> A straightforward generalization to represent all positive real numbers
> is to allow I subset ZZ, with I countable.


Let b in NN, b >= 2. A _number_ in sparse-base-b representation is a set
A subset [ZZ times (0, b) subset NN], such that

(i, v_1) in A and (i, v_2) in A ==> v_1 = v_2.

The value of A is given by

V(A) = sum_{(i, v) in A) v 2^i.

Examples follow.

In sparse-base-2:

0 = {}
1 = {(0, 1)}
2 = {(1, 1)}
3 = {(0, 1), (1, 1)}
4 = {(2, 1)}
5 = {(0, 1), (2, 1)}
6 = {(1, 1), (2, 1)}
7 = {(0, 1), (1, 1), (2, 1)}

In sparse-base-3:

0 = {}
1 = {(0, 1)}
2 = {(0, 2)}
3 = {(1, 1)}
4 = {(0, 1), (1, 1)}
5 = {(0, 2), (1, 1)}
6 = {(1, 2)}
7 = {(0, 1), (1, 2)}

In sparse base-4:

0 = {}
1 = {(0, 1)}
2 = {(0, 2)}
3 = {(0, 3)}
4 = {(1, 1)}
5 = {(0, 1), (1, 1)}
6 = {(0, 2), (1, 1)}
7 = {(0, 3), (1, 1)}

--
http://kaba.hilvi.org



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