Lukac’s Characteristic Functions

I was curious to see the proof for this theorem, so I’ve started reading Lukac’s Characteristic Functions. It’s a good book, but as written expresses absolutely no interest in the subject of probability. While about a subject that is interesting because of its connection with probability — a characteristic function is the Fourier transform of a probability distribution function — the discussion is entirely in terms of characteristic functions. For example, studying the characteristic functions for infinitely divisible distributions is motivated by the question of which characteristic functions have the property that every n-th root is also a characteristic function.

Applications of Fractional Derivatives

In ordinary calculus, you define “integral derivatives” — the first derivative, second derivative, etcetera. If you think of differentiation as an operator D that takes functions to functions, then the higher-order derivatives are just Di for natural numbers i. As far back as Liouville, mathematicians have defined fractional derivatives, extensions of this definition to real numbers, i. There is more than one possible definition, Wikipedia page gives the usual definition, which is in terms of the Laplace transform.

I’ve never known if fractional derivatives were good for anything, or were just a historical curiosity. (They are a special case of singular integral operators, which are useful in PDEs.) This very brief paper discusses an application of fractional derivatives to models of particles in a liquid. This sounds like it should be related to Brownian motion, and it is, but the processes that arise are related to more general Lévy processes than just Brownian motion.

Magistra’s Mathematical Ghost

I ran across this interesting post by a historian who was an undergraduate mathematics major. She found her old linear algebra notes, and was surprised to find how little of it she still understood:

It’s not just that I can’t answer this question now, it’s that I can barely comprehend even what it means. The terminology bounces through my brain, stirring vague imprecise echoes… It’s disconcerting enough to come across the ghost of your own self. It would be even more disconcerting to know for sure that part of your intellect is now forever closed off to you.