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f(a + b) = f(a) + f(b)

LordMorpheus said:
superposition

It's also a big chunk of what makes linear systems so nice to work with.
Click to expand...

i was tempted to go with linearity but i thought there was another name given to this property before I knew what linearity and superposition were. maybe not.

now if anyone could tell me whether or not the inverse error function is linear, that'd be nice. right now, it seems like it's not but i might have messed up something :| be linear damnit.
 
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Limit.jpg
 
There are no wrong answers up there. They all mean more or less the same thing (at least as far as an engineer cares about it).

A mathematician might understand more about the differences than I do, but really you could say additive, superposition, or distributive and most people would understand what you're talking about.
 
Linearity is really the most general property. Superposition (from physics) and the distributive property (from... elementary school? lol) are consequences of linearity. Linearity is usually given as something like
1) f(x+y) = f(x)+f(y) (also called additivity or superposition as mentioned in this thread)
2) f(a*x) = a*f(x) (also called homogenity)
Although 2) is really an obvious consequence of 1) if you're working with real numbers.

Also, OP: the inverse error function is *not* linear. A simple way to see this is to graph it. Or you can hit up wiki to find some series expansions that are complex looking & clearly nonlinear.

This fact should be obvious b/c the inverse of a linear function is *always* linear. Error function can't be linear b/c it's the integral of a non-constant expression, exp(-x^2).

Note: f(x) = constant is *not* linear according to the above definition. So it isn't a counter-example for why inverse of linear function is linear. In fact, f(x) = a*x+b isn't linear either in the previous definition unless b=0.
 
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Leros said:
Looks like "additive function" is the base mathematical concept behind the general principle of superposition.

http://en.wikipedia.org/wiki/Superposition_principle
Click to expand...
Any definition I can find, including the linked wiki page, invariably ties the term to linear systems.
From your wiki link:
In physics and systems theory, the superposition principle [1], also known as superposition property, states that, for all linear systems,
The superposition principle holds because, by definition, a linear system must be additive.

Or from http://www.wisegeek.com/what-is-the-principle-of-superposition.htm
In physics and engineering, the principle of superposition is the additive property of any linear function or system.

So additive property is more general than superposition, and as stated in the OP without additional conditions, I think additivity is the correct term here.


For those who think the answer is distributivity - from the definition, it requires two binary operators. If f is a normal function, then it's a unary operator. Arguing that we can look at this as an operator whose first argument is a function and the second a number is also wrong, because binary operators require the same domain. Besides, right-distributivity doesn't even make sense here.
 
eLiu said:
Linearity is really the most general property. Superposition (from physics) and the distributive property (from... elementary school? lol) are consequences of linearity. Linearity is usually given as something like
1) f(x+y) = f(x)+f(y) (also called additivity or superposition as mentioned in this thread)
2) f(a*x) = a*f(x) (also called homogenity)
Although 2) is really an obvious consequence of 1).
Click to expand...
2) follows from 1) for real numbers, but in general it doesn't have to. See e.g. antilinear map that's quite common in physics.
 
"Distributive property" - have you people made it even to an Algebra 2 class in high school yet?! Wow. No, it's not distributive property.
 
The error function is not linear, nor is its inverse. It can be approximated as linear over portions of its range (i.e. near 0 and at large values of the argument), depending on the accuracy you need, but it's definitely not linear.
 
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