It’s not really new that mathematical notation is made there to be abused so as to be most efficient for the current context. But I just met an extreme case:

In the OpenMath3 and MathML3-content efforts, we are polishing the description of symbols’ so that they can be common. Discussion about using linear syntax appears, of course, and an extreme case about the need for context was just posted by James Davenport:

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.

This task describes the introduction of an angle symbol along with its input (QMath) notation as well as its presentation.

This task describes the identification of an appropriate symbol and declaring its notations for input in QMath.

Rendering Content Elements

Rendering Agent at Work

A rendering agent is a processor that, on behalf of a user, for example through an HTTP request, converts content-mathematical-expressions to presentation. We call this conversion, the delivery of the rendering agent. Typically, the delivery is happening within a user-interaction which lives within a context: information about this context can be multiple, it includes the preferred languages of the user, the preferred notational styles of the user or of the surrounding content. A rendering agent should be able to process arbitrary content

I remember having studied a book using what I remember as the british category school. This notation stipulates that function application is written left to right, ie that:

f∘g(x) should be written xfg

As far as I can tell, all the tools I have seen around that handle mathematical notations, both at input and output, do save brackets around binary operators by the use of the precedence of an operator: it basically says that an operator has a given precedence x and, if a term must be presented as an operand, a bracket is output if the child-binary operator has a lower precedence. (think a*(b+c))

I have just found that the precedence of convolution should not be a (linearly ordered) number…

OMDoc documents are made of mathematical items which can contain mathematical formulas; the latter may be presented to the screen or print media. The ActiveMath presentation system uses XSLT and Velocity to convert the OMDoc fragments to rendered HTML, XHTML, or PDF. For the mathematical formula to be presented well, an amount of notations are defined which produce XSLT templates.

This handbook page explains by examples how notations are conceived, authored, and used in ActiveMath.