## The day is coming

November 19, 2013

## Riemann on Riemann sums

November 16, 2013

Though Riemann sums had been considered earlier, at least in particular cases (for example, by Cauchy), the general version we consider today was introduced by Riemann, when studying problems related to trigonometric series, in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. This was his Habilitationsschrift, from 1854, published posthumously in 1868.

Riemann’s papers (in German) have been made available by the Electronic Library of Mathematics, see here. The text in question appears in section 4, Ueber den Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit. The translation below is as in

• A source book in classical analysis. Edited by Garrett Birkhoff. With the assistance of Uta Merzbach. Harvard University Press, Cambridge, Mass., 1973. MR0469612 (57 #9395).

Also zuerst: Was hat man unter $\displaystyle \int_a^b f(x) \, dx$ zu verstehen?

Um dieses festzusetzen, nehmen wir zwischen $a$ und $b$ der Grösse nach auf einander folgend, eine Reihe von Werthen $x_1, x_2,\ldots, x_{n-1}$ an und bezeichnen der Kürze wegen $x_1 - a$ durch $\delta_1$, $x_2 - x_1$ durch $\delta_2,\ldots,$ $b - x_{n-1}$ durch $\delta_n$ und durch $\varepsilon$ einen positiven ächten Bruch.  Es wird alsdann der Werth der Summe

$\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots$ $\displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)$

von der Wahl der Intervalle $\delta$ und der Grössen $\varepsilon$ abhängen.  Hat sie nun die Eigenschaft, wie auch $\delta$ und $\varepsilon$ gewählt werden mögen, sich einer festen Grenze $A$ unendlich zu nähern, sobald sämmtliche $\delta$ unendlich klein werden, so heisst dieser Werth $\displaystyle \int_a^b f(x) \, dx$.

In Birkhoff’s book:

First of all: What is to be understood by $\displaystyle \int_a^b f(x)\,dx$?

In order to establish this, we take the sequence of values $x_1,x_2,\ldots, x_{n-1}$ lying between $a$ and $b$ and ordered by size, and, for brevity, denote $x_1 - a$ by $\delta_1$, $x_2 - x_1$ by $\delta_2,\ldots,$ $b - x_{n-1}$ by $\delta_n$, and proper positive fractions by $\varepsilon_i$. Then the value of the sum

$\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots$ $\displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)$

will depend on the choice of the intervals $\delta_i$ and the quantities $\varepsilon_i$. If it has the property that, however the $\delta_i$ and the $\varepsilon_i$ may be chosen, it tends to a fixed limit $A$ as soon as all the $\delta_i$ become infinitely small, then this value is called $\displaystyle \int_a^b f(x) \, dx$.

(Of  course, in modern presentations, we use $\Delta_i$ instead of $\delta_i$, and say that the $\delta_i$ approach $0$ rather than become infinitely small. In fact, we tend to call the collection of data $x_1,\dots,x_{n-1}$, $\varepsilon_1,\dots,\varepsilon_n$ a tagged partition of ${}[a,b]$, and call the maximum of the $x_{i+1}-x_i$ the mesh or norm of the partition.)

## Analysis – HW 5 – Newton’s method

November 16, 2013

This set is due Friday, December 6, at the beginning of lecture.

Newton’s method was introduced by Newton on De analysi in 1669. It was originally restricted to polynomials; his example in Methodus fluxionum was the cubic equation

$x^3-2x-5=0.$

Raphson simplified its description in 1690. The modern presentation, in full generality, is due to Simpson in 1740. Here, we are mostly interested in the dynamics of Newton’s method on polynomials.

## AlgoRythmics

November 15, 2013

This link should take you to the YouTube channel of Algo-rythmics, or see their website.

Different sorting algorithms (bubble sort, insertion sort, quicksort, selection sort, shell sort) illustrated through folk dance.

## Shirt

November 8, 2013

Isn’t this cute?

All the kids in Francisco’s class drew themselves. And then they got a class shirt with all the drawings. I think the idea is to wear it every Friday.

I got one for myself as well.

(Bemmy is the one farthest to the right, in the front row.)

## Weierstrass function

November 7, 2013

Weierstrass function from 1872 is the function $f=f_{a,b}$ defined by

$\displaystyle f(x)=\sum_{n=0}^\infty a^n\cos(b^n\pi x)$.

Weierstrass showed that if

• $0,
• $b$ is an odd positive integer, and
• $\displaystyle ab>1+\frac32\pi$,

then $f$ is a continuous nowhere differentiable function. Hardy proved in 1916 that one can relax the conditions on $a,b$ to

• $0,
• $b>1$, and
• $ab\ge 1$.

Here, I just want to show some graphs, hopefully providing some intuition to help understand why we expect $f$ to be non-differentiable. The idea is that the cosine terms ensure that the partial sums  $\displaystyle f(m,x)=\sum_{n=0}^m a^n\cos(b^n\pi x)$, though smooth, have more and more “turns” on each interval as $m$ increases, so that in the limit, $f$ has “peaks” everywhere. Below is an animation (produced using Sage) comparing the graphs of $f(m,x)$ for $0\le m<20$ (and $-10\le x\le 10$), for $a=1/2$ and $b=11$, showing how the bends accumulate. (If the animations are not running, clicking on them solves the problem. As far as I can see, they do not work on mobiles.)

Below the fold, we show the same animation, zoomed in around $0$ by factors of $100$, $10^4$, and $10^6$, respectively, illustrating the fractal nature of $f$.

## Continuous nowhere differentiable functions

November 7, 2013

Following a theme from two years ago, we will have a final project for this course, due Wednesday, December 18, by noon, but feel free (and encouraged) to turn it in earlier. (As discussed in lecture, the project is voluntary for some of you. Contact me if you are not sure whether it is required or voluntary for you.)

There are many excellent sources on the topic of continuous nowhere differentiable functions. Johan Thim’s Master thesis, written under the supervision of Lech Maligranda, is available online, here, but feel free to use any other sources you find relevant.

Please choose an example of a continuous nowhere differentiable function, either from Thim’s thesis or elsewhere, and write (better yet, type) a note on who it is due to and what the function is, together with complete proofs of continuity and nowhere differentiability. Though not required, feel free (and encouraged) to add additional information you consider relevant for context.

(For an example of what I mean by relevant additional information: Weierstrass function is $\displaystyle f(x)=\sum_{n=0}^\infty a^n\cos(b^n\pi x)$ where $0, $b$ is an odd positive integer, and $\displaystyle ab>1+\frac32\pi$. It may be interesting to add a discussion of precisely what conditions are needed from $a,b$ to ensure (continuity and) nowhere differentiability; Weierstrass original requirements are more restrictive than necessary. For another example, Schoenberg functions, discussed in Thim’s thesis, give a natural example of a space filling curve, so consider including a proof of this fact.)

Please take this project very seriously (in particular, do not copy details from books or papers, I want to see your own version of the details as you work through the arguments). Feel free to ask for feedback as you work on it; in fact, asking for feedback is a good idea. Do not wait until the last minute. At the end, it would be nice to make at least some of the notes available online, please let me know when you turn it in whether you grant me permission to host your note on this blog.

Here is a list of the projects I posted on the blog, from last time:

Contact me (by email) as soon as you have chosen the example you will work on, to avoid repetitions; I will add your name and the chosen example to the list below as I hear from you.

List of projects:

• Joe Busick: Katsuura function.
• Paul Carnig: Darboux function.
• Joshua Meier: A variant of Koch’s snowflake.
• Paul Plummer: Lynch function.
• Veronica Schmidt: McCarthy function.