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small samples

Here are some more samples which involve embellishment of symbols.  To
clarify how semantical information might be attached, I've asked a
couple of questions in the notes.

The aim is to have a set of examples for the case of `strip' notation
(symbols on a baseline with slight deviation for embellishments
and fractions).  This leaves out matrices and commutative diagrams
and aligned equations, but should cover a lot of notation.  Again,
the approach is `notational' because I have the feeling that most
members of the ERB would rather approach things this way than through
semantics.  (We're talking about devising notation which can be
rendered to various forms with semantical cues, rather than devising
semantical structures and specifying how to render those.)

-Ron



---------------------------------------------------------
Item 9 / fonts

TeX:			{\bf R}, \mathbf R

Wolfram:		<b>R</b>

MINSE:			
Display-List (S):	
Display-List (MS):	
ISO 12083:		<bold>R</bold>



Notes:
1. The first style shown for TeX accords with standard text font
   changing mechanisms, the second is the method used with AMSTeX
   (actually "\bold" is used rather than "\mathbf") and AMSLaTeX.
   "\mathbf" takes a single argument, and "\mathbf R^n" would embolden
   only the "R".

2. I believe Bruce spoke of using the standard HTML (text) font change
   syntax for the Wolfram Proposal, but I know he mentioned that the
   specification within display list format was yet to be put forward.

3. I think Robert and Neil's display list format would use
   "<mattribute>", but I'm not certain about this.
   
4. I do find the AMSTeX, unary prefix operator, style of notation
   most natural since it accords with other operator-style methods of
   embellishing objects. Can this be done in the Wolfram approach?
   
   In the case of changing fonts on operators, Neil has said he thinks
   that allowing prefix operators on operators and also asking our
   parsers to interpret the result automatically as another operator will
   unduly complicate our parsers. This would imply that authors would
   have to enter something like "<mo ...>&bold;+</mo>" if we had a
   prefix-style font-changing mechanism. The same argument must also
   apply to other prefix "embellishments" of operators.

5. Suppose for the moment we have a paper in which the author uses
   emboldening to indicate vectors. Suppose also that the style for
   indicating the reals and complexes is to embolden the "R" and "C".
   Through the course of the paper we have objects mentioned in a variety
   of spaces, with some algebra and subscripting involved in giving the
   dimensions of the spaces (e.g., in TeX, "\mathbf R^{k^2_i+k_i-1}").
   How do we envision type-theoretic information being attached to
   all the various kinds of vectors? (And we should assume, to be
   difficult, that, say, "v" is used to denote vectors from spaces of
   different dimension as the article progresses; i.e. "v" is a bound
   variable used for different purposes within the article.) Do we
   envision that our type-theoretic information will contain notation
   itself?
   


---------------------------------------------------------
Item 10 / primes

TeX:			x', x^\prime, x^{\prime\prime}

Wolfram:		
MINSE:			
Display-List (S):	
Display-List (MS):	
ISO 12083:		


Notes:
1. TeX has two forms of priming, but Spivak eschewed the "x'" when
   he felt he could not get it to work in all situations which might be
   natural (so for AMSTeX only the forms using "\prime" are allowed.

2. I don't know how other notations handle the prime or plan to
   handle it. A superscripted "*" might also present a similar `odd
   case' if we opt for the "x'" style (i.e. treating the prime as a
   character already raised and in a smaller size). I would vote myself
   for treating a prime as a superscript and entering it as such
   (granting that people might change the `look' themselves by defining a
   macro).

3. For concreteness, how might some person or program associate
   differentiation to priming within a paper?
   

---------------------------------------------------------
Item 11 / overlines

TeX:			\overline s, \overline{s+t}

Wolfram:		
MINSE:			
Display-List (S):	
Display-List (MS):	
ISO 12083:		<overline>s</overline>,
                        <overline>s+t</overline>


Notes:
1. The `line' of the ISO overline can come in many styles (single,
   double, triple, dash, dot, bold, etc.). TeX has to stand on its head
   to get many of these. "\overline" and "\bar" are two related
   embellishment forms, the first `stretchy', the second not.

2. I'm uncertain as to how the Wolfram Proposal will handle these
   sorts of things. And how will something like an overline be related
   to other operators which are stretchy or not?


---------------------------------------------------------
Item 12 / overbar with subscript

TeX:			\bar x_1

Wolfram:		x^^&bar;_1

MINSE:			
Display-List (S):	<mscript>
                          <moverscript>x<mc>&bar;</moverscript>
                          <mc><mc>1
                        </mscript>

Display-List (MS):	
ISO 12083:		x<top>&bar;</top><inf>1</inf>



Notes:
1. This is a simple example, but it brings up a couple of points.
   If the overbar signifies conjugation, it's probably most likely that
   the proper "expression" (and, in any case, a possible expression) is
   one signifying the conjugate of the object "x_1". In this case,
   the Wolfram expression `should' be "x_1^^&bar;" (because the nesting
   of the display schemata is then correct). I assume the visual display
   will then be hard to get right (which only means `to look as it would
   look with the TeX coding', not that the standard visual display is
   logically correct).

2. And generally, it appears that the nesting of the various display
   schemata in the WP requires that *some* linear interpretation be
   given for the entire set of embellishments added to a character. (This
   is analogous to the standard problem of being required to *write* a
   matrix in row- or column-major form, but wanting to interpret it
   in both ways.)
   
   Bruce had a discussion in the WP about how `scripts' would `appear'
   from display list format (he spoke of following down a nested sequence
   of script operators, etc.). I took this to mean a literal sense of
   appearance, but perhaps he meant it in such a way as to imply
   symmetrical `meaning' (so that neither the bar nor the subscripting
   operator in the example above truly assume precedence within the
   display list). I need clarification in this.

3. And as with prime notation, how would a person or software specify
   that barring is to be read as conjugation? Does it come to individual
   interpretational annotations (semantical markup embedded within the
   mathematical text) or a declaration outside the body of a paper?
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