The Unapologetic Mathematician

Mathematics for the interested outsider

Two Mean Value Theorems

We’ve got two different analogues of the integral mean value theorem for the Riemann-Stieltjes integral.

The first one says that if \alpha is increasing on \left[a,b\right] and f is integrable with respect to \alpha, with supremum M and infimum m in the interval, then there is some “average value” c between m and M. This satisfies

\displaystyle\int\limits_{\left[a,b\right]}fd\alpha=c\int\limits_{\left[a,b\right]}d\alpha=c\left(\alpha(b)-\alpha(a)\right)

In particular, we should note that if f is continuous then the intermediate value theorem tells us that there is some x_0 with f(x_0)=c. That is, there is some x_0 such that

\displaystyle f(x_0)=\frac{1}{\alpha(b)-\alpha(a)}\int\limits_{\left[a,b\right]}fd\alpha

When \alpha(x)=x this gives us the old integral mean value theorem back again.

So why does this work? Well, if \alpha(a)=\alpha(b) then both sides are zero and the theorem is trivially true. Now, the lowest lower sum is L_{\alpha,\{a,b\}}(f)=m\left(\alpha(b)-\alpha(a)\right), while the highest upper sum is U_{\alpha,\{a,b\}}(f)=M\left(\alpha(b)-\alpha(a)\right). The integral itself, which we’re assuming to exist, lies between these bounds:

\displaystyle m\left(\alpha(b)-\alpha(a)\right)\leq\int\limits_{\left[a,b\right]}fd\alpha\leq M\left(\alpha(b)-\alpha(a)\right)

So we can divide through by \int_{\left[a,b\right]}d\alpha=\alpha(b)-\alpha(a) to get the result we seek.

We can get a similar result which focuses on the integrator by using integration by parts. Let’s assume \alpha is continuous and f is increasing on \left[a,b\right]. Our sufficient conditions tell us that the integral of f with respect to \alpha exists, and the integration by parts formula says

\displaystyle\int\limits_{\left[a,b\right]}fd\alpha=f(b)\alpha(b)-f(a)\alpha(a)-\int\limits_{\left[a,b\right]}\alpha df

But the first integral mean value theorem tells us that the integral on the right is equal to \alpha(x_0)\left(f(b)-f(a)\right) for some x_0. Then we can rearrange the above formula to read

\displaystyle\int\limits_{\left[a,b\right]}fd\alpha=f(b)\alpha(b)-f(a)\alpha(a)-\alpha(x_0)\left(f(b)-f(a)\right)
\displaystyle=f(a)\left(\alpha(x_0)-\alpha(a)\right)+f(b)\left(\alpha(b)-\alpha(x_0)\right)
\displaystyle=f(a)\int_{\left[a,x_0\right]}d\alpha+f(b)\int_{\left[x_0,b\right]}d\alpha

So there is some point x_0 so that the integral of f is the same as the integral of the step function taking the value f(a) until x_0 and the value f(b) after it.

March 28, 2008 Posted by John Armstrong | Analysis, Calculus | | 1 Comment

On a Lighter Note

Someone got ahold of one of my students’ homeworks from last semester.

March 28, 2008 Posted by John Armstrong | Uncategorized | | No Comments

Hoax!

Isabel at God Plays Dice cites (uncharacteristically credulously) this story of a 13th-century monk who supposedly discovered the Mandelbrot set. How surprising, since we didn’t see it until we had computer graphics. But there’s a lot more anachronism just beneath the surface. There are so many topics that feed into the standard representation of the Mandelbrot set that we have so completely internalized we hardly know how to think without them.

First, this story asks us to believe that Brother Udo worked with complex numbers 300 years before Tartaglia and Cardano hesitantly advanced them. They were motivated by problems that could not be solved without them, while here they’re just being used for multiplications that can be perfectly well defined over the reals. Why would he have even thought to use complex numbers? Sure, maybe he was using a geometric description that happened to exactly correspond, but he wouldn’t have thought of it as complex multiplication. Still, more on that possibility later.

Next the story asks us to believe that Brother Udo graphed algebraic relations 400 years before Descartes introduced analytic geometry. So he somehow had the idea that algebra and geometry were interlinked so far in advance of everybody else and didn’t tell anyone? Again, maybe the construction was wholly geometrical.

Next the story asks us to believe that Brother Udo was not only graphing any old algebraic relations, but using the graphical representation of complex numbers 500 years before Argand described that interpretation. Again, this is something we internalize in high school (or earlier) and so we forget how late in the game it actually came along.

Okay, so could there have been a totally geometrical construction that these historians are simplifying into the function z\mapsto z^2+c? I doubt it. Mostly because you run it 70 times and then.. what? In rendering of the Mandelbrot set you know whether to keep or toss a point by whether you eventually go outside the circle of radius 2. So now we need to posit an unspoken proof that once a point leaves that circle it can never return. No reference is made to that result either.

Of course, the page cites sources. But the only source directly on point for the main content of the story is this one:

[5] Schipke, R.J. and Eberhardt, A. “The forgotten genius of Udo von Aachen”, Harvard Journal of Historical Mathematics, 32, 3 (March 1999), pp 34-77.

but this “Harvard Journal of Historical Mathematics” doesn’t seem to exist either. In fact, a Google search on that journal’s name gives… a bunch of references back to this very story! Let me be clear about this:

Just because someone includes a bibliography doesn’t mean they’re not forging the bibliography too!

And, of course, the page itself says it was published April 1, 1999. It’s not even a new hoax. But as I show above, you don’t need to see that cue to realize it’s a joke. You just have to stop, think, and really understand what someone is trying to sell you. You have to be skeptical about any assertion of fact. And it seems a lot of people aren’t.

This story got included in course materials for a course on fractal geometry taught at Yale by my teaching mentor Michael Frame, with help by Benoît Mandelbrot himself. John Allen Paulos wrote it up for ABC news. Sure, maybe they knew it was a hoax and passed it along as such. Paulos wrote his article on April 1, 2001, after all. But then it gets swept up by woo like “Life Technology”, “The All-Oneness Hadron Materia”, and badly-researched psychology texts. And nobody tags it as the hoax it is.

So here I am. This story is false. Unquestionably false, like faking the moon landing or a face on Mars. False like psychic powers and divine cameos on corn chips. False like that girl who got so high on LSD she microwaved the baby she was taking care of. Those of you who love linking to big-name skepti-bloggers might want to point them here so we can squelch this thing once and for all.

March 28, 2008 Posted by John Armstrong | rants | | 12 Comments