Re: decorated characters

On 10/18/21 9:34 AM, David Farmer wrote:
> A previous thread cast expressions like  x'  as multi-character
> variables.
> I think that was the wrong way to think about it.  There can
> be multi-character variables, but a separate and possibly more
> common situation is "decorated" characters.


> Examples, and the way I would pronounce them, are:
> x prime
> x double prime
> x hat
> x bar
> x tilde
> x check
> x dot
> x double dot
> x star
> Note that some of those have the decoration above the character,
> and others have the decoration above and to the right (northeast)
> of the character.
> I would be happy to hear of examples where one pronounces the
> decoration first and the variable second.
> Suppose  J_0  is the 0th order Bessel function of the first kind.
> Is the subscript a decoration?  It looks like one to me.  And I
> pronounce it as  "J zero" not "J sub zero".

You could think of J_0 as a decorated symbol, but since there are
J_1, J_2,... one might better think of an "array" of Bessel functions.
Until one realizes that they've been generalized to complex numbers as
well, so that Bessel J is actually a function of *two* arguments \nu and z
in J_\nu(z).  In special functions, one often hears the \nu referred to
as a "parameter", and the z referred to as an "argument" (although
that terminology may be confusing to computer scientists).

> An intent like
>      intent=decoration($1, $2)
> or
>      intent=decoration($2, $1)
> indicating on the order of pronunciation, could tell AT how to
> say the decorated character.  That intent also conveys the idea
> that the decorated character is one mathematical object.
> I say "decorated character" and not "decorated variable" because
> the decoration might be on a function and not on a variable.
> The derivative f', and Fourier transform \hat{f}, are common
> examples of functions with decorations.

Indeed, although I'd be inclined to "decorated symbol",
and they can be operators as well as variables, functions,
and in fact any mathematical whatsit.

If you slide back into "semantics", you might want to be careful
about distinguishing the Fourier transform operator from the
Fourier transform *of* f (which will of course be another function).

> A flaw is that we have not conveyed the meaning, only the
> pronunciation.  What if there were an (optional?) 3rd argument
> to the decoration intent?  For example:
> decoration(f, hat, fourier-transform)
> decoration(f, prime, derivative)
> decoration(x, dot, time-derivative)
> decoration(x, bar, mean)
> decoration(J, 0, bessel)
> decoration(a, n, index)
> The last one would appear in
>     sum a_n x^n
> because the "n" is an index (of summation), and I would pronounce
> the "sub" in that case.
> If x' is just a new variable and the prime has no meaning,
> that could be a case to omit the 3rd argument to decoration.
> All this assumes AT actually needs help pronouncing decorated characters
> correctly.
> Regards,
> David


Received on Monday, 18 October 2021 14:07:23 UTC