No this isn’t about some kind of science fiction, this is about five Unicode characters that are useful for mathematics, but are generally invisible or should be. The characters are the zero-width space (U+200B), function apply (U+2061), invisible times (U+2062), invisible comma (U+2063), and the new invisible plus (U+2064). This post discusses each one in the context of mathematical text.

The zero-width space is a handy character that has no glyph “ink” and hence no ascent (height above the base line), no descent (depth below the baseline) and no width. In Word 2007 math zones you can insert it (type 200B <Alt+x>) into an empty argument if you don’t want a dotted box character to appear. RichEdit uses it for optional empty arguments to suppress the dotted box except when the insertion point resides inside an empty argument.

The function-apply character (U+2061) is used in the linear format as a binary operator that builds into a math function object. For example in a math zone, if you type sin2061<Alt+x> x and click on “Professional”, you get the math function object sin x. Naturally it’s easier just to type sin<space>x and have formula autobuildup do this for you, but underneath it’s the function apply character that’s controlling the build up process.

The invisible times (U+2062) is a bona fide binary operator and you can break on it and align to it. Unfortunately we didn’t have enough time to develop the uses for invisible times, so it’s not currently very useful. Unlike in Word 2007, it shouldn’t display a glyph, except for a thin space if at the end of a math zone. With it you could then effectively break an equation before any character, not just on binary, relational and some other operators. It would be nice to be able to have it display a multiplication times symbol × if it ends up being the best point for an automatic break. Word 2007 displays the invisible times as a dotted box surrounding a times sign, which is the glyph for it in the Cambria Math font.

The invisible comma (or separator) is supposed to convey the semantic of separating two variables or indices. For example the indices ij on a matrix element aij could be separated by the invisible comma to emphasize that ij isn’t the product of i and j. Word 2007 displays the invisible comma as a dotted box surrounding a comma, which is the glyph for it in the Cambria Math font.

The invisible plus (U+2064) is new with Unicode 5.1 and is supposed to carry the semantic of connecting a whole number like 3 with a fraction like ½ to give a quantity 3½ that has the value 3.5, not 1.5 (3/2). The invisible plus is well intended, but it’s also tricky to use. For one thing in ordinary arithmetic, addition is considered to have lower precedence than multiplication. So the value of the expression 4×3 + 1/2 is 12.5, not 14 (4×3.5). But 4×3<invisible plus>1/2 has the value 14. In this usage, the invisible plus has a higher precedence than multiplication.

Some more discussion of the invisible operators is given in Section 2.14 of Unicode Technical Report #25.

Over 70 SCORM 2.0 White Papers have so far been submitted as part of the SCORM 2.0 process. The Program Committee will continue to accept submissions until the end of August.

This open dialogue about the ideas and concerns brought up in the White Papers, and about the future of SCORM, is underway at www.letsi.org/scorm2. You can read papers in your area of expertise or look over the informal comments and blog posts accumulated on the wiki. And, of course, if you've got something to say, post a comment. (Or you can just send an email to scorm2@letsi.org and they will post it for you.)

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Dear Developers and Users of Mathematical Editors,

At the Doctoral Programm of the CICM Conference in Birmingham last week, one of the senior researchers challenged the developers of mathematical editors to sit together, compare their implementations, and find synergies.

That day, some of us sat together and thought that this might be a good point and should not only involve the developers but also users that might be able to point to valuable requirements and, in particular, take over the testing ;-)

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The Inter2Geo project has released final versions of deliverables on an educational ontology for geometric competencies and on the metadata format it will adopt in the future. Quotes from the abstracts:

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Two very interesting developments are happening that will improve Word 2007’s MathML support. The first is key for helping in getting Word 2007 math text into the scientific and technical publisher workflows and the second may help in this regard too. Specifically new transforms are now available in beta versions enabling Word to read and write MathML. These XSLT files are responsible for converting between Word’s native math format OMML and MathML 2.0. If you’d like to try out the new files (omml2mml.xsl and mml2omml.xsl), you can download them from the Microsoft Connect site using the invitation code: 0707-84P4-DPWT. Once you’ve downloaded the files, copy them to C:\Program Files\Microsoft Office\Office12 subdirectory, or wherever winword.exe is. Before doing so, you might want to change the current omml2mml.xsl and mml2omml.xsl files to omml2mml.xsl.bak and mml2omml.xsl.bak, respectively, in case you want to back out the update at a later date. But I doubt you will. The new ones are significantly better.

<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /> 

The second development is that Word 2007 will have a service pack release that enables it to read and write the ISO standard odf files as well as the native ISO standard OOXML files. In the odf standard, math zones are represented by MathML 2.0. So when Word converts to and from odf, it will use MathML 2.0 for all math zones. And it will use the files above to do the translations.

I recently had a very simple request… soooo simple: our user just wishes to copy the formula from Mathematica (which can copy it in MathML) and paste it on something that does web.

I just went around and tried… SeaMonkey should support that in editor and reader: copy a piece of HTML with MathML and paste it, didn’t even work… my 1/x became a place full of nbsps in three lines!

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Subscript and Superscript Bases

For proper math typography, it’s important to know the base of a subscript or superscript expression. For example, in Einstein’s equation E = mc2, the superscript expression c2 appears and c is the base, not mc. Knowing what the base is allows proper kerning of the base relative to the script (superscript or subscript) as well as providing more accurate semantics in interoperating with mathematical calculation engines.

This post describes the subscript/superscript base rules used by Word 2007 and RichEdit 6 in building up math text from the linear format. The rules are good, but not infallible, and users can overrule them either directly in the linear format or after they are built up into the Professional format.

Unicode math alphabetics: Ordinarily when a user types an ASCII letter or a Greek lower case letter α..ω (along with some variants), the letter is automatically converted to the corresponding Unicode math italic letter. These special mathematical letters, along with the basic set of Latin letters in Fraktur, script, and open-face math styles, are reserved for mathematical variables . Accordingly if a subscript or superscript follows such a letter, that letter is considered to be the base. In linear format if you type E=mc^2<space>, you get E = mc2, where the letters are given by math italic characters (not used here in this blog post). In particular, c would be given by the math italic c, U+1D450, rather than by the ASCII c, U+0063. This single math italic c is the base of the superscript expression c2. For more information on the math alphabetics, please see Section 2.1 of the Unicode Technical Report #25.

Numbers: A consecutive string of ASCII digits is treated as a base. So in  the expression 1002, the 100 is the base of the superscript expression and has the mathematical meaning of “one hundred squared”. This quantity is typed in as 100^2.

ASCII letter strings: Since mathematical variables are almost always represented by math alphabetics, a consecutive string of ASCII letters is treated as a base. So in the superscript expression sin-1, the base is “sin”. Actually this case is usually handled by the function name mechanism described next. You can enter an ASCII letter string by turning off the italic button before you type or by selecting the corresponding math italic letters and then turning off the italic button. Be sure to turn the italic button back on if you want to enter math italic variables.

Function names: when a consecutive string of English alphabetics is typed followed by a space or bracket of some kind, the resulting math italic string is “folded” down to the corresponding ASCII letter string and compared to entries in a mathematical function dictionary. If found, the folded version of the string is used followed by the function-apply operator U+2061. The dictionary includes trigonometric functions like sin, cos, tan, etc., along with many other famous math function names. Users can modify this dictionary. If the function-apply operator is then followed by a subscript or superscript, that script is transferred to the function name, and the function name becomes the base of the script expression. This is handy for typing in expressions like sin-1x.

Embellished operators: If an operator character precedes a subscript or superscript, the operator is the base. For example, in the expression +­2, the + is the base.

Built-up math objects: If a built-up math object such as a stacked fraction precedes a subscript or superscript, that object is the base.

Superscript a subscript object: Exceptions to the rule above occur for superscripting a subscript object and subscripting a superscript object. In both of these cases, the combination is turned into a subsup object, which has special typography, typically placing the superscript over the subscript.

Opaque strings: Opaque strings are whatever is inside a \begin \end expression. Such strings are bases if followed by a subscript or superscript. This is the catch-all method of letting most any mathematical text be a subscript/superscript base. The user is cautioned to use reasonable choices so that the result is understandable to readers.

Complex script characters: In Indic scripts like Devanagari, a number of Unicode characters may be combined to form a character “cluster”. If such a cluster is followed by a subscript or superscript, the cluster becomes the base. However, this doesn’t occur for Arabic ligatures, for which only the last character is treated as the base. One can force the whole ligature to be the base by putting it inside a \begin \end expression, i.e., by making it an opaque string.

Ordinary text: Expressions resulting from the linear format “rate” are called ordinary text and are useful as variables when you want to spell out the variables’ names. Such ordinary text strings are treated as bases.

Meet the Project Office and many partners in Monterrey, Mexico. You can find us on the campus of UANL and at the WIRIS and JEM Thematic Network booths at the exhibition fair.

Ulrich Kortenkamp (PHSG)
Axel M. Blessing (PHSG)
Christian Dohrmann (PHSG)
Yves Kreis (ULUX)
Colette Laborde (CABRI)
Jean-Marie Laborde (CABRI)
Tomas Recio (UCAN)
Ramon Eixarch (M4M)

ICME11 Details...

Tim Bray indicates that almost all the legal impediments before an open-source java are cleared. Now this really means there might be, one day, a clean java, something that does cause headaches to distributions!

I much love the GNU Classpath license… really… as simple as that: do whatever you want to “link to it” but there rest is the GPL-full-land.

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I'd like to invite you to participate in a little survey on mathematical knowledge items, possible issues with them, and how to solve issues. It will take around 10 minutes to answer the questions.

The results will be used in order to improve the mathematical argumentation ontology that is used in the semantic wiki SWiM and thus the assistance in tracking and solving issues there.

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The local user meetings have a central role in gathering the community of practice. They intend to help providing a complete European coverage. The objectives of the Local User Meetings are:

• Present Intergeo

Those meetings need to present Intergeo to the users: The need of a common file format for interoperability, the need of a web platform to share ressources, the need of the ontology and the curriculum mapping to share resources across all European countries.

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There’s a wind for more content construction in the ActiveMath group, with at least two projects at the University focussed on creating content (and a adapt platform and…). And MathML starts to play an important role there.

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More and more, I cannot understand why the default subversion toolset exports logs in a format different than RSS.

And actually I have found one from codingmonkeys.de which I slightly polished.

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