1. 2012

    More on scale invariant functions

    Back in February I did a post on scale invariant functions of one variable. These are functions that satisfy the condition

    $$f(\lambda x) = C(\lambda) f(x)$$

    Depending on which source you look at, you might find a more specific definition, but I think this is the most general condition that you can sensibly use to call a function scale invariant. Under this definition, I showed that all scale invariant functions of interest to physicists are power laws, of the form

    $$f(x) = ax^k$$

    Homogeneous functions

    There is a related concept called a homogeneous function (thanks to that site’s creator, Ondřej Čertík, for pointing it out), defined as those functions \(h(x)\) which satisfy

    $$h(\lambda x) = \lambda^k h(x)$$

    for some \(k\), called the degree of homogeneity. While the definition is similar to the one I’m using for scale invariant functions, it’s less restrictive. A homogeneous function has to satisfy the condition for only one particular choice of \(C(\lambda)\), not any arbitrary choice, so you can have homogeneous functions which are not scale invariant.

    Multivariate scale invariant functions

    The definitions of homogeneous functions apply to multivariate functions too, so let’s see …

  2. 2012

    Scale invariance and the power law

    With all the fuss about SOPA and PIPA, plus having actual work to do, I haven’t been able to write anything in a while. So I figured it’s time for some good old fashioned physics. Today I would like to introduce the concept of a scale invariant (or dilatation invariant) function. This special class of functions is defined by the property that when you scale (or dilate) the argument of the function by some factor, it’s equivalent to scaling the value of the function by some related factor.

    $$f(\lambda x) = C(\lambda) f(x)$$

    \(C(\lambda)\) is something that depends on \(\lambda\), but not on \(x\).

    To understand why physicists find scale invariant functions so fascinating, we have to go way back to the definition of an analytic function, the power series expansion. Pretty much every mathematical function used in physics can be expressed as some power series, like this:

    $$f(x) = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots$$

    Even if the function isn’t normally written like this, it’s enough that it can be. (The sine and cosine, for example.)

    Now, what happens when you plug some value with units into …