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Can Mathematica construct a set of equations?

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  Can Mathematica construct a set of equations? Ralph Dratman <ralph.dratman@gmail.com> - 2011-05-30 10:35 +0000
    Re: Can Mathematica construct a set of equations? "Dr. Wolfgang Hintze" <weh@snafu.de> - 2011-05-31 11:45 +0000

#2794 — Can Mathematica construct a set of equations?

Given a set of N points Pn in the real plane, all within a distance d
of each other,

In vector notation,

   || Pj - Pk || <= d,   1 <= j,k <= N

or written out, say for N=3,

|| P1 - P2 || <= d,
|| P2 - P3 || <= d,
|| P3 - P1 || <= d.

That is fine for 3 points, but suppose I have 10. Then the long
version is Choose[10,2] = 45 equations, and I don't particularly want
to write them out by hand. Can Mathematica do that for me, and give me
the equations in a notebook?

I'm not even sure how to represent the position vectors so I can refer
to xj or yk later on. How do I set up vector-sub-j and its components
x-sub-j and y-sub-j ?   Would that be a list of N lists of length 2?
Or is there a more specific vector notation?

Sorry to be so clueless. Thank you.

Ralph

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#2833

Yes, it can.

Let us first define the distance between vectors as
dif[x_?VectorQ, y_?VectorQ] := Sqrt[Dot[(x - y),(x - y)]]

The equations (inequalities) can then be easily created at once thus

eqs[n_] :=
And @@ (Flatten[
Table[eq[i, k] = dif[p[i], p[k]] <= d, {i, 1, n - 1}, {k, i + 1, n}]])

Finally let the variables, i.e. the components of the vectors, be 
defined as
(this is the case of two dimensions x and y)

p[1]={px[1],py[1]}, p[2]={px[2],py[2]},  ...

We can make the assigment using

Table[p[i] = {px[i], py[i]}, {i, 1, n}];

Now choosing n (for instance = 3) we can "solve" the equations for the 
variables

n=3
sol = Solve[eqs[n], Flatten[Table[p[i], {i, 1, n}]]]

If the equations contain inequalities we should use Reduce instead of 
Solve.

PS: the generalization to more than two dimensions, say k, can be made 
using a numbering scheme replacing and extending x and y:

p[i] = {p[i,1],p[i,2],...,p[i,k]}

PPS: I guess it needs a very fast computer to actually carry out the 
solving procedure for n in the range of 10.

Hope this helps,
Wolfgang

"Ralph Dratman" <ralph.dratman@gmail.com> schrieb im Newsbeitrag 
news:irvrtb$8jo$1@smc.vnet.net...
> Given a set of N points Pn in the real plane, all within a distance d
> of each other,
>
> In vector notation,
>
>   || Pj - Pk || <= d,   1 <= j,k <= N
>
> or written out, say for N=3,
>
> || P1 - P2 || <= d,
> || P2 - P3 || <= d,
> || P3 - P1 || <= d.
>
> That is fine for 3 points, but suppose I have 10. Then the long
> version is Choose[10,2] = 45 equations, and I don't particularly want
> to write them out by hand. Can Mathematica do that for me, and give 
> me
> the equations in a notebook?
>
> I'm not even sure how to represent the position vectors so I can 
> refer
> to xj or yk later on. How do I set up vector-sub-j and its components
> x-sub-j and y-sub-j ?   Would that be a list of N lists of length 2?
> Or is there a more specific vector notation?
>
> Sorry to be so clueless. Thank you.
>
> Ralph
> 

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