Re: [Axiom-developer] [Axiom-mail] manipulating series not as streams but as sumations

[Raymond Rogers]

Ralf,  Attached is an example of the analysis I was thinking about.
In theory it adds nothing to the information content; but in practice:
The human race has been developing, arguing, and changing the 
mathematical notation for a long time.  The present incarnation is not 
perfect but I think as a vechile of communication it is preferable to 
any particular human language with its deliberate ambiguities.  I say 
deliberate because it must accommodate poetry and background 
associations.  For instance "closed set"  is supposed to bring to mind a 
certain type of set but does not contain the true definition.  
Especially the definition that is used in point-set topology.

Although I haven't found it yet; if Axiom is to be a mathematical 
language it must have a definition along the lines of BNF or my favorite 
the railroad tracks/yard of Pascal.  This diagram/definition can be read 
"automatically" and a parser generated for use by any old program that 
has is compatible with it.  I would presume there would be c programs 
that read the output of yacc and produce target sources and  
executables.  Personally I would aim for grep or awk as the final 
program they are runtime ASCII programs.  But I have a penchant for 
simplicity.

Ray

On 10/21/2014 11:27 AM, Ralf Hemmecke wrote:
> On 10/21/2014 04:39 PM, Raymond Rogers wrote:
> > Let's think about this.
> > I think the question can be interpreted as a formating/display problem
> > for the underlying code.
> If you believe so... Let me state it a bit differently what
> datastructure comes out of the series command. Or rather what is stored
> inside the data structure.
>
> Here an example:
>
> (1) -> Z==>Integer; Q==>Fraction Z;
>                                                                     Type:
> Void
> (2) -> f(q:Q):Q == truncate(abs(q*cos(q::Float))+1)*(q/2+1)
>
>     Function declaration f : Fraction(Integer) -> Fraction(Integer) has
>        been added to workspace.
>                                                                     Type:
> Void
> (3) -> s: Stream Q := stream(f, 0)
>     Compiling function f with type Fraction(Integer) -> Fraction(Integer
>        )
>
>               3 7 15 31 63 127 255 511
>     (3)  [0,1,-,-,--,--,--,---,---,---,...]
>               2 4  8 16 32  64 128 256
>                                                Type:
> Stream(Fraction(Integer))
> (4) -> p:=series(s)$UnivariateFormalPowerSeries(Q)
>
>     (4)
>           3  2   7  3   15  4   31  5   63  6   127  7   255  8   511  9
>       x + - x  + - x  + -- x  + -- x  + -- x  + --- x  + --- x  + --- x
>           2      4       8      16      32       64      128      256
>     +
>       1023  10      11
>       ---- x   + O(x  )
>        512
>                           Type:
> UnivariateFormalPowerSeries(Fraction(Integer))
>
> The only thing that s (the Stream) knows is a function object f (if you
> like you can use f=random). It cannot look inside f except to ask for
> more values. Remember AXIOM code is compiled.
>
> If from that you would somehow be able to extract a pattern, great, but
> I doubt.
>
> > Of course I am not that familiar with the Axiom internals.
> Well, you might come from another CAS. AXIOM *is* different. It is in
> general *not* dealing with expression trees. So what you have in mind
> (adding presentation) does not work for Stream of Series.
>
> That is not saying that it is impossible, but rather that the Stream and
> PowerSeries domains are not appropriate for what you have in mind. (And
> at the moment I don't think there is anything like what you want in Axiom.)
>
> Ralf
>
> PS: BTW, why didn't you continue on the mailing list?

--
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George Santanyana

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