[seqfan] Re: detective work related to "Floretions"

[Joerg Arndt]

* Robert Munafo <mrob27 at gmail.com> [Nov 21. 2009 14:11]:
> A term like "nonassociative algebra" might be suitable for some readers, but
> it confuses me. I still think "algebra" is how I solved word problems in 7th
> grade (well not quite, but close :-).

http://en.wikipedia.org/wiki/Algebra_over_a_field

> 
> The purpose of my investigations is to make this stuff useable by people who
> need to be shown simple and concrete examples. I understand that
> "commutative" refers to the idea that "3 x 7 = 7 x 3" and that's about it. I
> don't know what "sub-algebra" means, or how "an algebra" can "contain"
> quaternions?

A subset of the elements that still is an algebra,
prototypical: R is a subalgebra of C,
Q is a subalgebra of the quaternions.

http://en.wikipedia.org/wiki/Algebra_over_a_field#Subalgebras_and_ideal

(apparently it is "subalgebras", not "sub-algebra" as I put it).

About "concrete":  the operations as given now are
concrete as can be, but bloody obscure!
This is due to the epic notation fail.
Improving notation does not render things less concrete,
but will make things more parsable.


> If a "projection operator" is something that takes one type of
> number for input and produces another type as output, let's just say that,
> and avoid the jargon. I don't need to have taken college level maths courses
> to to calculate these sequences.

A map f is a projection iff f(f(x))=f(x) \forall x

cf. under http://en.wikipedia.org/wiki/Projection_(mathematics)

"
In linear algebra, a linear transformation that remains unchanged if
applied twice (p(u) = p(p(u))), in other words, an idempotent
operator. For example, the mapping that takes a point (x, y, z) in
three dimensions to the point (x, y, 0) in the plane is a projection.
...
"

Note the "linear"s may be dropped.
Example: sorting is a projection.


> 
> Also, I really do prefer the separate letters a,b,c,d,e,f,...,o,p rather
> than a subscripted coefficient. In my opinion, subscripted coefficients are
> for cases where the number of components is variable, unlimited, and/or
> exceeds the number of letters in your alphabet(s). None of those applies
> here: we have exactly 16 components all the time. It gets worse when there
> are two or more dimensions worth of subscripts, like when defining matrix
> multiplication.

Letters are OK up to about five symbols
(e.g. fine and custom for quaternions).
But _not_ for 16.  Trust.  Me.

> 
> As for the special-named functions like "ibase()", I just need to understand
> and document them, because that's they appear in the existing OEIS database
> entries. Think of my work as a Rosetta stone for Creighton's work (-: There
> were three languages on the Rosetta stone. Here, we have Creighton's
> language, formal algebra language, and "I just want to calculate the d**n
> thing" language.

If a Rosetta stone shall serve as a translation device
then I suggest to include standard math notation.
With proper math lang the "calculate" version
will be trivial.  And _way_ less obscure than it is now.

Does the python script linked under
"Symbolic Calculator in Python" on the page
  http://fumba.eu/sitelayout/Floretion.html
not rate as "code from hell" for you?
Compared to this IOCCC appears to be
a competition in clarity.


> [...]

cheers,   jj