[Axiom-developer] "empty list" results, inverse trig substitutions, "Floats" in "solve"

[James]

I'm new to axiom, using it for calculations.  There seems to be odd behaviour
when using "solve" on a list of equations.  This is with the most recent axiom
binary package, Version Axiom (May 2009), on Ubuntu 8.10 "intrepid".  For
instance:

 (8) -> solve([a=3+x,b=1-x,x=2],[a,b])
 ...
 (8)  []
                         Type: List List Equation Fraction Polynomial Integer

So, the answer is "the empty list"?!  That's not very useful, and seems not
correct either.  What am I missing here?

Then instead:

 (9) -> solve([a=3+x,b=1-x,x=2],[a,b,x])
 (9) ->
   (9)  [[a= 5,b= - 1,x= 2]]
                         Type: List List Equation Fraction Polynomial Integer

That works.  Why?

Similarly, axiom seems confused about finding substitutions.  For instance:

 (12) -> solve([ tan(bt)=(a*r)/(s*(r-d)), x=100*cos(bt), y=d*sin(bt)],[x,y,bt])
 (12) ->
    (12)  [[]]
                                   Type: List List Equation Expression Integer

Here, "the empty list" again - why?

If, instead, I "spell it out" for axiom, by taking the arctan instead, then

 (13) -> solve([bt=atan((a*r)/(s*(r-d))),x=100*cos(bt),y=d*sin(bt)],[x,y,bt])
 (13) ->
    (13)
    [
                 (100r - 100d)s                        a d r
      [x= ----------------------------, y= ----------------------------,
           +-------------------------+      +-------------------------+
           |  2           2  2    2 2       |  2           2  2    2 2
          \|(r  - 2d r + d )s  + a r       \|(r  - 2d r + d )s  + a r
                   a r
       bt= atan(--------)]
                (r - d)s
      ]
                                   Type: List List Equation Expression Integer

Axiom doesn't know about substituting inverse trig functions by itself?

But then, again,

 (14) -> solve([bt=atan((a*r)/(s*(r-d))),x=100*cos(bt),y=d*sin(bt)],[x,y])
 (14) ->
    (14)  [[]]
                                   Type: List List Equation Expression Integer

Arrrrgh!  Ok, why is that again, returning "the empty list" when using "solve"
with the truncated list of variables?


On anther topic, "Floats" in "solve", where this works, mixing integers and
floats:

 (16) -> solve([a=3+x,b=1-x,x=2.0],[a,b,x])
    ...
    (16)  [[a= 5.0,b= - 1.0,x= 2.0]]
                            Type: List List Equation Fraction Polynomial Float

and this works:

 (17) -> solve([a=3+x,b=1-x,x=2],0.001)
   ...
   (17)  [[x= 2.0,a= 5.0,b= - 1.0]]
                                    Type: List List Equation Polynomial Float


doing this, using "x=2.0" instead of "x=2":

 (18) -> solve([a=3+x,b=1-x,x=2.0],0.001)
   ...
   There are 20 exposed and 3 unexposed library operations named solve
      having 2 argument(s) but none was determined to be applicable.
      ...
   Cannot find a definition or applicable library operation named solve
      with argument type(s)
                       List Equation Polynomial Float
                                    Float


Arrrrgh! - the dreaded "none was determined to be applicable"!
Does that make sense, that?


Thanks in advance for any clues!  Are these bugs?


James