## February 7, 2017

### Functional Equations I: Cauchy’s Equation

#### Posted by Tom Leinster

This semester, I’m teaching a seminar course on functional equations. Why? Among other reasons:

1. Because I’m interested in measures of biological diversity. Dozens (or even hundreds?) of diversity measures have been proposed, but it would be a big step forward to have theorems of the form: “If you want your measure to have this property, this property, and this property, then it must be that measure. No other will do.”

2. Because teaching a course on functional equations will force me to learn about functional equations.

3. Because it touches on lots of mathematically interesting topics, such as entropy of various kinds and the theory of large deviations.

Today was a warm-up, focusing on Cauchy’s functional equation: which functions $f: \mathbb{R} \to \mathbb{R}$ satisfy

$f(x + y) = f(x) + f(y) \,\,\,\, \forall x, y \in \mathbb{R}?$

(I wrote about this equation before when I discovered that one of the main references is in Esperanto.) Later classes will look at entropy, means, norms, diversity measures, and a newish probabilistic method for solving functional equations.

Read on for today’s notes and an outline of the whole course.

I don’t want to commit to TeXing up notes every week, as any such commitment would suck joy out of something I’m really doing for intellectual fulfilment (also known as “fun”). However, I seem to have done it this week. Here they are. For those who came to the class, the parts in black ink are pretty much exactly what I wrote on the board.

Here’s the overall plan. We’ll take it at whatever pace feels natural, so the section numbers below don’t correspond to weeks. The later sections are pretty tentative — plans might change!

1. Warm-up   Which functions $f$ satisfy $f(x + y) = f(x) + f(y)$? Which functions of two variables can be separated as a product of functions of one variable?

2. Shannon entropy   Basic ideas. Characterizations of entropy by Shannon, Faddeev, Rényi, etc. Relative entropy.

3. Deformed entropies   Rényi and “Tsallis” entropies. Characterizations of them. Relative Rényi entropy.

4. Probabilistic methods   Cramér’s large deviation theorem. Characterization of $p$-norms and power means.

5. Diversity of a single community   Background and introduction. Properties of diversity measures. Value. Towards a uniqueness theorem.

6. Diversity of a metacommunity   Background: diversity within and between subcommunities; beta-diversity in ecology. Link back to relative entropy. Properties.

Posted at February 7, 2017 11:25 PM UTC

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### Re: Functional Equations I: Cauchy’s Equation

Fun old Putnam problem: Cauchy + graph not dense $\implies$ $\mathbb{R}$-linear.

Posted by: Jesse C. McKeown on February 8, 2017 1:05 AM | Permalink | Reply to this

### Re: Functional Equations I: Cauchy’s Equation

Which Putnam exam is this from?

For those who don’t feel like bashing their heads over this, you can apply rot-13 to see a solution:

Abgvpr gung fhpu na nqqvgvir ubzbzbecuvfz zhfg or D-yvarne, juvpu vzcyvrf gung vgf tencu jvyy or n irpgbe fhofcnpr (bire gur svryq D) bs gur erny cynar. Rvgure vgf tencu vf pbagnvarq va n fgenvtug yvar (fb vf tvira ol zhygvcyvpngvba ol n erny fpnyne: gur E-yvarne pnfr), be ryfr vgf tencu unf gjb E-yvarneyl vaqrcraqrag ryrzragf k, l, jurapr pbagnvaf nyy engvbany yvarne pbzovangvbaf bs k, l, juvpu gura svyy bhg n qrafr fhofrg bs gur cynar.

Posted by: Todd Trimble on February 9, 2017 9:55 PM | Permalink | Reply to this

### Re: Functional Equations I: Cauchy’s Equation

Very nice solution!

Posted by: Tom Leinster on February 9, 2017 10:16 PM | Permalink | Reply to this

### Re: Functional Equations I: Cauchy’s Equation

I don’t exactly remember what year it was from (or how exactly the problem was worded), but I know it was before 2001, because it came up in the practise sessions that year… I was a first-year undergrad, and probably even more annoying than I am now.

Posted by: Jesse C. McKeown on February 10, 2017 1:22 AM | Permalink | Reply to this

### Re: Functional Equations I: Cauchy’s Equation

I guess everyone here knows, but solutions $f \colon \mathbb{R} \to \mathbb{R}$ of $f(x + y) = f(x) + f(y)$ that are not of the form $f(x) = a x$ are all nonmeasurable and thus can only be proved to exist using nonconstructive methods, e.g. with the axiom of choice. So, one can if so inclined argue that for all practical purposes they don’t really exist.

(For all I know their existence is equivalent to the axiom of choice… someone must know.)

Posted by: John Baez on February 8, 2017 6:40 AM | Permalink | Reply to this

### Re: Functional Equations I: Cauchy’s Equation

It’s not as known as you might think. Andreas Blass proved that the assumption “every vector space over every field has a basis” + ZF implies the axiom of choice. A nice encapsulated proof may be found here, section 3.

But as to whether “every vector space over a fixed ground field $k$ has a basis” implies AC: as far as I know little is known. For example, it seems to be open in the case $k = \mathbb{Q}$. This MO thread discusses this point.

My gut tells me that there’s no way that a specific vector space like $\mathbb{R}$ having a $\mathbb{Q}$-basis would be nearly enough to prove AC. It’s just too restricted in scope. Also, while such a basis for $\mathbb{R}$ is somewhat nasty from the point of view of descriptive set theory (it can’t be Borel for instance), it need not be too nasty: see this MO answer by Joel David Hamkins for some information. Curiously, this type of thing came up in Tom’s other recent Café thread, where I mentioned this old result of Sierpinski that Hamel bases can’t be analytic sets (i.e., continuous images of Borel sets) – but they can be projections of complements of projections of closed sets, according to Joel’s answer!

Posted by: Todd Trimble on February 8, 2017 2:17 PM | Permalink | Reply to this

### Re: Functional Equations I: Cauchy’s Equation

I guess everyone here knows […]

I only know because I read Tom’s notes :-)

Posted by: Simon Willerton on February 8, 2017 8:49 PM | Permalink | Reply to this

### Re: Functional Equations I: Cauchy’s Equation

In one of life’s odd moments of synchronicity, I was just reading about this very topic earlier today, in Jeremy Gray’s The Hilbert Challenge (Oxford UP, 2001). I started sampling this volume on Google Books because one of my research topics made unexpected contact with Hilbert’s 12th Problem, and I have only the most casual acquaintance with that whole area.

Posted by: Blake Stacey on February 9, 2017 7:49 PM | Permalink | Reply to this

### Re: Functional Equations I: Cauchy’s Equation

What should my mathematical background be to understand your notes? This topic is very, very appealing.

Posted by: David on March 10, 2017 8:34 PM | Permalink | Reply to this

### Re: Functional Equations I: Cauchy’s Equation

I’m glad you find it appealing!

I advertised it not only to staff and PhD students, but also to our 4th- and 5th-year undergraduates, telling them that they should be well-placed to understand it if they had a good grasp of our compulsory 3rd-year course Honours Analysis. Only a small portion of that course is actually needed for what we’re doing with functional equations, but I thought that was about the right level in terms of general mathematical experience.

There will be bits and pieces of more advanced stuff that we’ll need soon, but again it’s the kind of thing that most of our mathematics undergraduates pick up in their 3rd or 4th year. Of course, education systems vary enormously from country to country and even university to university, but I hope that gives a rough indication.

Posted by: Tom Leinster on March 10, 2017 9:39 PM | Permalink | Reply to this

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