Kurt Pagani wrote:
>
> On Sunday, 12 October 2014 20:32:50 UTC+2, Waldek Hebisch wrote:
> >
> > Kurt Pagani wrote:
> > >
> > > It was indeed not very difficult. A simplex can be represented most
> > > favourably as a matrix (rows = points of F) and FreeAbelianGroup will
> do
> > > the rest.
> >
> > I admit that I got lost in this thread. However, what you do looks
> > strange so I wonder if you really want this. Namely, simplical
> > complexes are normally defined on vertices from some abstract
> > set. It does not matter what the nature of set is, just which
> > point are equal or not. Of course, you can take Expression(Integer)
> > as your set, but for computability one prefers finite sets.
> > In case of finite set using some initial interval of integers
> > is normal practice.
> >
> >
> Of course you're right from an abstract point of view, however, for
> practical purposes '*abstract simplicial complexes''* are not very useful.
AFAIK this is approach used when computing homology or cohomology.
That is first you need to fing simplicial complex homeomrphic
(or homotopy equivalent) to your space, this probably is done
by hand. Once you have abstract simplicial complex you can
compute boundary operators (as matrices over Z^n). Then,
you calculate Smith form of "Laplace" operator made from boundary
operators, this gives you homology.
> The example given - using Expression(Integer) - is rather
> misleading/unfortunate as Simplex(n,k,F)
> is meant for 'practical' fields F like Q,R,C. Since Expression(Integer)
> 'has Field' it also works for this setting.
>
>
> > In case of de Rham theory one has a manifold and triangulates
> > it. Triangulation means that we have mapping from some abstract
> > simplical complex into our manifold and this mapping is a
> > homeomorphizm. De Rham duality says that homology of the mainfold
> > (which is the same as homology of the complex) with real coefficients
> > is in duality with homology of complex of differential forms.
> > The duality is given by integration of a form over a chain.
> >
> That is the goal. Integration of forms over chains. To do this we need a
> representation which is usually a simplicial chain in a F^n (F=R,Q). We
> have the 'forms' now we need the chains.
>
> > Note, that in such setting coefficients of forms are functions
> > on the mainfold, while elemenst of chain are functions from
> > simplices to the manifold. There is extra condition, namely
> > if lower dimensional simplex S1 is a side of higher dimensional
> > simplex S2 function on S1 must agree with restriction of function
> > on S2.
> >
> Yes, but it depends on the context. There is a (comprehensive) book
> 'Geometric Integration Theory' by Hassler Whitney, Princeton University
> Press 1957, which covers the different aspects.
>
>
> > Now, I do not see how your domain Simplex implement simplices.
> > In abstract case simplex is given by list of vertices (list
> > because we want orientation). Two simplices are equal if
> > one is evan permutation of the other. Odd permutation
> > change sign.
> >
> >
> Well, again, you look at it from an abstract point of view and 'your'
> objects live on the manifold. The approach in DERHAM is to look at a chart
Actually, DERHAM does not imply charts. You can have mainfold M as
a subset of R^n and extend your forms form M to R^n. AFAICS DERHAM
will work fine for such calculation. There is also another point of
view: manifold is determined by its ring of smooth functions. So
you can indentify manifold with ring of functions + derivatives.
If such ring is represented as subring of Expression(Integer) then
DERHAM can be used.
> (diffeomorph to some Euclidean space) and the simplices/chains are part of
> that, in other words pre-images of the traingulations. A simplex is an
> 'oriented list' of points [P_0,..,P_k] and geometrically the convex hull of
> those. The points are e.g. in R^3: P0=[0,0,0], P1=[1,0,0], P2=[0,1,0],
> then [P0,P1,P2] is the standard 2-simplex in R^3 (a triangle). Therefore
> the representation as matrix i
>
> 0 0 0
> 1 0 0
> 0 1 0
>
> One may quickly check degeneracy or compute the volume of the simplex.
OK. But there are few problems:
- we need to identify simplices that differ only by even permutation
of vertices and make sure that odd permutations change sign.
In textbook one can divide free module spanned on simplices
by a submodule, but in practical implementations such quotient
of modules would cause serious troubles, so it is better
to identify simplices early
- you consider only affine simplices, but once you have more than
one chart curviliner simplices are unavoidable
> Yes, very unfortunate. I wanted just a 'flat' Union like Union([T(j) for
> j in 1..k ]) for some type T(i). Apparently there is no such method?
No such construct. There is Any which places no restriction on
members but Any should be avoided if there are alternatives.
>
> > I do not see why you want to take union way: what you
> > want is a module where coordinates come from different spaces.
> > So, if any recusive direct product would be more appropriate.
> > However, it looks simpler to define the whole direct product
> > from the start and if needed define components by restriction.
> >
>
> You are certainly right. One should probably merge Simplex(n,k,F) into a
> Simplex(n,F) simplices in F^n.
Yes.
--
Waldek Hebisch
[email protected]
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