Invisible Operators

MathML uses an "invisible multiplication" character to explicitly denote when two variables or expressions are being multiplied. There is also an "invisible apply function" character that differentiates between multiplication and functions. For example, ab might mean "a times b", or the two-letter variable "ab". Similarly a(b+c) might mean "a times b + c" or it might mean, "apply function a to the argument b+c". By inserting the proper invisible character, either multiplication or apply function, the semantics become clear.

MathFlow will automatically recognize common functions and will handle "invisible multiplication" and "apply function" accordingly. You can apply these commands manually, by choosing the appropriate command from the Invisible characters palette:

You can also manually insert invisible multiplication by pressing the spacebar following an identifier. By doing so, and continuing with our previous example, the "a" and the "b" will each go into identifier tokens of their own, with the InvisibleTimes operator token in between. The "a" and the "b" will remain in the customary italic typeface, giving you a visual cue that you have encoded a multiplication.

If you hit the space bar after anything other than an identifier (following an operator symbol, for example), the MathFlow Editor will insert a space character as usual.