## Bi-Lipschitz equivalence and fixed points

Two metric spaces ${X,Y}$ are bi-Lipschitz equivalent if there is a bijection ${f:X\rightarrow Y}$ and a constant ${L}$ such that
${L^{-1}d_X(a,b) \le d_Y(f(a),f(b)) \le L\,d_X(a,b)}$ for all ${a,b\in X}$. In other words, ${f}$ must be Lipschitz with a Lipschitz inverse; this is asking more than just a homeomorphism (continuous with continuous inverse).

For example, the graph ${y=\sqrt{|x|}}$ is not bi-Lipschitz equivalent to a line. The cusp is an obstruction:

Indeed, the points ${A,B = (\pm \epsilon,\sqrt{\epsilon})}$ are separated by ${C=(0,0)}$ and lie at distance ${\sim \sqrt{\epsilon}}$ from it. So, the images ${f(A), f(B)}$ would lie on opposite sides of ${f(C)}$, at distance ${\sim \sqrt{\epsilon}}$ from it. But then the distance between ${f(A)}$ and ${f(B)}$ would be of order ${\sqrt{\epsilon}}$, contradicting ${d(A,B)\sim \epsilon}$.

There are other pairs of spaces which are homeomorphic but not bi-Lipschitz equivalent, like a circle and Koch snowflake.

What is the simplest example of two spaces that are not known to be bi-Lipschitz equivalent? Probably: the unit ball ${B}$ and the unit sphere ${S}$ in an infinite-dimensional Hilbert space.

In finite dimensions these are not even homeomorphic: e.g., the sphere has a self-homeomorphism without fixed points, namely ${R(x) = -x }$, while the ball has no such thing due to Brouwer’s fixed point theorem. But in the infinite-dimensional case ${B}$ and ${S}$ are homeomorphic. Moreover, there exists a Lipschitz map ${F\colon B\rightarrow B}$ such that the displacement function ${\|F(x)-x\|}$ is bounded below by a positive constant: no hope for anything like Brouwer’s fixed point theorem.

It’s hard to see what an obstruction to bi-Lipschitz equivalence could be: there are no cusps, nothing is fractal-like, dimensions sort of agree (both infinite), topologically they are the same… Here is a possible direction of attack (from Geometric Nonlinear Functional Analysis by Benyamini and Lindenstrauss). If a bi-Lipschitz map ${f\colon B\rightarrow S}$ exists, then the composition ${f^{-1}\circ R\circ f}$ is a Lipschitz involution of ${B}$ with displacement bounded from below. So, if such a map could be ruled out, the problem would be answered in the negative. As far as I know, it remains open.

## Irrational sunflowers

A neat way to visualize a real number ${\alpha}$ is to make a sunflower out of it. This is an arrangement of points with polar angles ${2 \pi \alpha k}$ and polar radii ${\sqrt{k}}$ (so that the concentric disks around the origin get the number of points proportional to their area). The prototypical sunflower has ${\alpha=(\sqrt{5}+1)/2}$, the golden ratio. This is about the most uniform arrangement of points within a disk that one can get.

But nothing stops us from using other numbers. The square root of 5 is not nearly as uniform, forming distinct spirals.

The number ${e}$ begins with spirals, but quickly turns into something more uniform.

The number ${\pi}$ has stronger spirals: seven of them, due to ${\pi\approx 22/7}$ approximation.

Of course, if ${\pi}$ was actually ${22/7}$, the arrangement would have rays instead of spirals:

What if we used more points? The previous pictures have 500 points; here is ${\pi}$ with ${3000}$. The new pattern has 113 rays: ${\pi\approx 355/113}$.

Apéry’s constant, after beginning with five spirals, refuses to form rays or spirals again even with 3000 points.

The images were made with Scilab as follows, with an offset by 1/2 in the polar radius to prevent the center from sticking out too much.

n = 500
alpha = (sqrt(5)+1)/2
r = sqrt([1:n]-1/2)
theta = 2*%pi*alpha*[1:n]
plot(r.*cos(theta), r.*sin(theta), '*');
set(gca(), "isoview", "on")

## 2015 syllabus

Reviewing Calculus VII material: one post from each month of 2015.

## Richardson Extrapolation and Midpoint Rule

The Richardson extrapolation is a simple yet extremely useful idea. Suppose we compute some quantity ${Q}$ (such as an integral) where the error of computation depends on a positive parameter ${h}$ (such as step size). Let’s write ${Q=A(h)+E(h)}$ where ${A(h)}$ is the result of computation and ${E(h)}$ is the error. Often, one can observe either numerically or theoretically that ${E(h)}$ is proportional to some power of ${h}$, say ${h^k}$.

The quantity ${A(h)-2^k A(h/2)}$ is an approximation to ${(1-2^k)Q}$ in which the main source of error is cancelled out. Thus, writing

$\displaystyle Q\approx \frac{2^kA(h/2)-A(h)}{2^k-1}$

should give us a much better approximation than either ${A(h)}$ or ${A(h/2)}$ were.

It is not quite intuitive that one can improve the accuracy of ${A(h/2)}$ by mixing it with a less accurate approximation ${A(h)}$.

For a concrete example, let’s apply the trapezoidal rule to the integral ${\int_{-1}^1 e^x\,dx=2\sinh 1}$ with step size ${h=2}$. The rule gives

$\displaystyle A(2) = \frac{2}{2}(f(-1)+f(1)) = 2\cosh 1$

which is quite wrong: the error is ${2e^{-1}\approx 0.74}$. Little surprise, considering the geometry of this approximation:

With the smaller step we get

$\displaystyle A(1) = \frac{1}{2}(f(-1)+2f(0)+f(1)) = 1+\cosh 1$

with the error ${\approx 0.19}$. The rule being of second order, this error is about ${4}$ times smaller.

The Richardson extrapolation gives

$\displaystyle \frac{4A(1)-A(2)}{4-1} = \frac{f(-1)+4f(0)+f(1)}{3} = \frac{4+2\cosh 1}{3}$

reducing the error by the factor of ${16}$: it’s now only ${0.012}$. And this did not require any additional evaluations of the integrand.

The formula ${\frac{f(-1)+4f(0)+f(1)}{3}}$ may look familiar: it’s nothing but Simpson’s rule. But the above derivation is much easier than what is typically shown in a calculus textbook: fitting parabolas to the graph of ${f}$ and integrating those.

All this is nice. But what if we started with the Midpoint rule? Let’s find out, following the example above. With ${h=2}$,

$\displaystyle A(2) = 2f(0) = 2$

which is off by ${0.35}$ (the Midpoint rule is generally twice as accurate as Trapezoidal). Then

$\displaystyle A(1) = f(-1/2)+f(1/2) = 2\cosh(1/2)$

with error ${\approx 0.1}$. Extrapolate:

$\displaystyle \frac{4A(1)-A(2)}{4-1} = \frac{4f(-1/2)-2f(0)+4f(1/2)}{3} = \frac{8\cosh(1/2) - 2}{3}$

and the error drops to ${0.010}$, slightly less than for Simpson’s rule.

So, the extrapolated-midpoint (EM) rule appears to be slighly more accurate than Simpson’s, with an equal number of evaluations. Why isn’t it in textbooks alongside Simpson’s rule, then?

A few factors make the EM rule impractical.

1. There is a negative coefficient, of ${f(0)}$. This means that the rule can give a negative result when integrating a positive function! For example, ${\int_{-1}^1 \exp(-9x^2)\,dx}$ evaluates to ${-0.39}$ with EM rule. Simpson’s rule, as pretty much any practical quadrature rule, preserves positivity.

2. The sample points for EM are less convenient than for Simpson’s rule.

3. The gain in accuracy isn’t that great. The reason that Midpoint outperforms Trapezoidal by the factor of ${2}$ has to do with how these rule approximate ${\int_{-1}^1 x^2\,dx=\frac23}$. Midpoint gives ${0}$, Trapezoidal gives ${2}$; the former is twice as accurate. To compare Simpson’s and EM rules, we should consider ${\int_{-1}^1 x^4\,dx}$ since both are of the ${4}$th order of accuracy: they evaluate cubic polynomials exactly. The results are ${1/6}$ for EM and ${2/3 }$ for Simpson’s: these are nearly equidistant from the correct value ${2/5}$. The difference in accuracy is less than ${15\%}$.

Despite being impractical, the EM rule has an insight to offer. By moving two evaluation points from ${\pm 1}$ to ${\pm 1/2}$, we made the coefficient of ${f(0)}$ go from positive ${4/3}$ to negative ${-2/3}$. (The coefficients are determined by the requirement to evaluate ${\int_{-1}^1 x^2\,dx}$ exactly.) So, there must be an intermediate position at which the coefficient of ${f(0)}$ is zero, and we don’t need to compute ${f(0)}$ at all. This idea leads to the two-point Gaussian quadrature

$\displaystyle \int_{-1}^1 f(x)\,dx \approx f(-1/\sqrt{3})+f(1/\sqrt{3})$

which on the example ${\int_{-1}^1 e^x\,dx}$ beats both EM and Simpson’s rules in accuracy, while using fewer function evaluations.

After writing the above, I found that the Extrapolated Midpoint rule has a name: it is Milne’s rule, a member of the family of open Newton-Cotes formulas.

## Poor man’s discrete cosine transform

The discrete cosine transform can be understood as a trigonometric interpolation problem. Given some data, like ${y = (3,1,4,1,5,9)}$, I want to find the coefficients ${A_0,\dots,A_5}$ such that

$\displaystyle \sum_{k=0}^5 A_k\cos kx_j = y_j,\quad j= 0,\dots,5$

with ${x_j=\pi j/6}$. (This is different from the unitary scaling preferred in image processing.) Matlab’s signal processing toolbox, as well as “signal” package of Octave-Forge both have dcp command. But the former is expensive and the latter is not entirely painless to install. Let’s work with the standard tool fft, readily available in Matlab, Scilab, and Octave.

The full Fourier series reduces to cosines when the function is even. Therefore, we need to extend ${y}$ by an even reflection. One might imagine such an extension as

$\displaystyle (3,1,4,1,5,9,9,5,1,4,1,3)$

but this is incorrect. Thinking of the six ${y}$ values as the samples of a function at ${0,\pi/5,2\pi/5,\dots,\pi}$ helps: the even reflection across ${\pi}$ should not duplicate the boundary value at ${\pi}$. Also, ${0}$ gets reflected to ${2\pi}$, which is the same as ${0}$ due to periodicity, so this value also should not be repeated. The appropriate reflection is

$\displaystyle (3,1,4,1,5,9,5,1,4,1)$

In Matlab terms,

y = [3 1 4 1 5 9]
z = [y, y(end-1:-1:2)]
w = fft(y)

The output is real, as desired. It also has the same kind of symmetry as the input.

$\displaystyle (34, -10.618, 7.618, -8.382, 5.382, 8, 5.382, -8.382, 7.618, -10.618)$

The term ${34}$ is simply the sum of the ${z}$-values, but the constant term in Fourier expansion is the mean value. So, division by the data size (${10}$) is needed to get the coefficients. Now that we have them, it seems that the interpolating function should be

$\displaystyle 3.4 -1.0618\cos x+0.7618 \cos 2x -\dots -1.0618\cos 9x$

but this is incorrect. Although this function does pass through the given points,

it wiggles too much due to high frequency terms such as ${\cos 9x}$. The aliasing comes into play: at the multiples of ${\pi/5}$, the function ${\cos 9x}$ is the same as ${\cos x}$.

Similarly, ${\cos 8x}$ should be replaced by ${\cos 2x}$ in the interpolating function, etc. This results in a much smoother interpolant, which is a linear combination of cosines up to ${\cos 5x}$.

$\displaystyle y= 3.4 -2.1236\cos x + 1.5236\cos 2x -1.6764\cos 3x +1.0764\cos 4x + 8\cos 5x$

Here is the complete Matlab code for this process of reflection, transforming and subsequent “folding” of the coefficients.

y = [3 1 4 1 5 9]
z = [y, y(end-1:-1:2)]
w = real(fft(z))/length(z)
a = [w(1), 2*w(2:end/2), w(end/2+1)]

Taking real part of FFT makes sure that nothing imaginary slips in through the errors associated with numeric computation.

RESTYLEgc is a pretty old (2008-11) MIT-licensed project by Brian Gibson, which enables, among other things, replacing the default stylesheet of embedded Google Calendar with a custom one. One can see it in action on the front page of Stony Brook math department.

Due to various changes on Google’s side made since 2011, the version available from Google Code link above no longer works. So I put up a GitHub repo with slightly modified files; only the two PHP scripts and the CSS file are included, the optional resources are not.

### Keeping up with the changes in Google URL structure

Line 120 of restylegc.php changed from

$url = "https://www.google.com/calendar/embed?" .$_SERVER['QUERY_STRING'];

to

$url = "https://calendar.google.com/calendar/embed?" .$_SERVER['QUERY_STRING'];

Line 19 of restylegc-js.php changed from

$url = "https://www.google.com/calendar/" .$_SERVER['QUERY_STRING'];

to

$url = "https:" .$_SERVER['QUERY_STRING'];

Line 42 of restylegc-js.php changed from

$replacement = '"https://www.google.com/calendar'; to $replacement = '"https://calendar.google.com/calendar';

### Better wrapping of text in narrow calendars

Line 1181 of restylegc.css changed from

  white-space:nowrap;

to

  white-space:pre-wrap;

Line 1263 of restylegc.css changed from

  white-space:nowrap;

to

  white-space:normal;

white-space:normal;

to the rule

.ie6 .agenda .event-title

although in 2015, IE6 isn’t quite as big a deal anymore.

Tangentially related to better space use: changed font-weight:bold; to font-weight:normal; in line 1263, the rules for .agenda .event-summary-expanded

## Rarefaction colliding with two shocks

The Burgers equation ${v_t+vv_x =0}$ is a great illustration of shock formation and propagation, with ${v(x,t)}$ representing velocity at time ${t}$ at position ${x}$. Let’s consider it with piecewise constant initial condition

$\displaystyle u(x,0) = \begin{cases} 1,\quad & x<0 \\ 0,\quad & 02 \end{cases}$

The equation, rewritten as ${(v_t,v_x)\cdot (1,v)=0}$, states that the function ${v}$ stays constant on each line of the form ${x= x_0 + v(x_0,0)t}$. Since we know ${v(x_0,0)}$, we can go ahead and plot these lines (characteristics). The vertical axis is ${t}$, horizontal ${x}$.

Clearly, this picture isn’t complete. The gap near ${1}$ is due to the jump of velocity from ${0}$ to ${2}$: the particles in front move faster than those in back. This creates a rarefaction wave: the gap gets filled by characteristics emanating from ${(1,0)}$. They have different slopes, and so the velocity ${v}$ also varies within the rarefaction: it is ${v(x,t)=(x-1)/t}$, namely the velocity with which one has to travel from ${(1,0)}$ to ${(x,t)}$.

The intersecting families of characteristics indicate a shock wave. Its propagation speed is the average of two values of ${v}$ to the left and to the right. Let’s draw the shock waves in red.

Characteristics terminate when they run into shock. Truncating the constant-speed characteristics clears up the picture:

This is already accurate up to the time ${t=1}$. But after that we encounter a complication: the shock wave to the right separates velocity fields of varying intensity, due to rarefaction to the left of the shock. Its propagation is now described by the ODE

$\displaystyle \frac{dx}{dt} = \frac12 \left( \frac{x-1}{t} + 0 \right) = \frac{x-1}{2t}$

which can be solved easily: ${x(t) = 2\sqrt{t} +1}$.

Similarly, the shock on the left catches up with rarefaction at ${t=2}$, and its position after that is found from the ODE

$\displaystyle \frac{dx}{dt} = \frac12 \left( 1 + \frac{x-1}{t} \right) = \frac{x-1+t}{2t}$

Namely, ${x(t) = t-\sqrt{2t}+1}$ for ${t>2}$. Let’s plot:

The space-time trajectories of both shocks became curved. Although initially, the shock on the right moved twice as fast as the shock on the left, the rarefaction wave pulls them toward each other. What is this leading to? Yet another collision.

The final collision occurs at the time ${t=6+4\sqrt{2} \approx 11.5}$ when the two shock waves meet. At this moment, rarefaction ceases to exist. The single shock wave forms, separating two constant velocity fields (velocities ${1}$ and ${0}$). It propagates to the right at constant speed ${1/2}$. Here is the complete picture of the process:

I don’t know where the holes in the spacetime came from; they aren’t predicted by the Burgers equation.