I think I've now got the hang of this: more intuitively, the traffic network can instead be modelled by an AC electrical circuit. The fluid dynamical aspects of flow along a single road might perhaps be incorporated by combining both models into a hydraulic analogue computer. Next time you're stuck in a traffic jam, try figuring out whether it's a UTM. Or would be, when/if ever it got moving again ... WFL On 5/22/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:

The "standing wave" phenomenon in heavy traffic mentioned in the article is straightforward application of fluid dynamics, which I had ample opportunity to analyse during a particularly excruciating expedition along the London orbital motoway years ago.

I had to view the video in full-screen mode before I could follow it; now I understand the mechanical paradox, which still surprises me. But I don't see how the mechanical model maps to a traffic situation.

WFL

On 5/21/15, Mike Stay <metaweta@gmail.com> wrote:

...

This short link also works and has no tracking data. http://theconversation.com/the-maths-of-congestion-springs-strings-and-traff...

On Thu, May 21, 2015 at 3:19 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:

...

See embedded video. New to me, though I had previously heard rumours ...

http://theconversation.com/the-maths-of-congestion-springs-strings-and-traff...

[no line breaks in above link!]

WFL

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

My traffic computer has solved that already (via the anthropic principle) -- it's an oracle! WFL On 5/24/15, Veit Elser <ve10@cornell.edu> wrote:

...

On May 23, 2015, at 4:20 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:

Next time you're stuck in a traffic jam, try figuring out whether it's a UTM. Or would be, when/if ever it got moving again …

What, you would rather have the halting problem? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

http://www.huffingtonpost.com/2015/05/24/john-nash-dead-a-beautiful-mind_n_7...

I don't know much about the equilibrium theory he won a Nobel prize for. But when I was 12 (in 1959) someone gave me Martin Gardner's "Scientific American Book of Mathematical Puzzles and Diversions," which had a chapter on The Game of Hex, which was invented independently by Piet Hein in 1942 and by John Nash in 1946. I have greatly enjoyed Hex ever since. Later, I learned the beautiful definition of an (abstract) smooth n-dimensional manifold in a college course. It was nice to learn that every manifold was in fact a submanifold of some Euclidean space R^p, in fact for some p <= 2n. Soon after that, we were taught the definition of an (abstract) Riemannian metric on a manifold, which was extremely satisfying and felt like exactly the right definition for making a manifold into a metric space. It was natural to wonder if every Riemannian manifold was also a submanifold of some Euclidean space, carrying the inherited Riemannian metric. Then someone told me that Nash had proved this. Originally he showed that for a smooth (C^oo) isometric embedding of a compact n-manifold into R^p, this was always possible for any p >= n(3n+11)/2. It's now known that at least for a compact smooth Riemannian n-dimensional manifold, it has an isometric embedding into R^q for any q >= n(n+1)/2 + max(2n, n+5). This compares with the earlier result of p >= n(3n+11)/2 = n(n+1)/2 + n(n+5). Maybe a more surprising result is that Nash originally proved that for a smooth manifold M^n (of differentiability class C^oo again) then for any dimensional Euclidean space R^s, s >= n+1, into which M has a smooth "short" embedding (one in which no distance is magnified), then: M has a C^1 *isometric* embedding in R^s. (The original theorem used R^(s+1), but N. Kuiper immediately showed this extra dimension was not necessary.) The strangeness of this result is exemplified by the familiar square torus R^2 / Z^2. This can be mapped via a short embedding to a small torus of revolution in R^3, and by the Nash-Kuiper theorem can be isometrically embedded C^1 into R^3. You probably won't be able to think of a way to do this, but recently an explicit such embedding has been found: < http://www.emis.de/journals/em/images/pdf/em_24.pdf >. ——Dan

On May 24, 2015, at 8:06 AM, Erich Friedman <erichfriedman68@gmail.com> wrote:

http://www.huffingtonpost.com/2015/05/24/john-nash-dead-a-beautiful-mind_n_7...

<< Rayleigh's monotonicity law says that paradoxes like this can't occur in purely resistive networks. >> Nor it seems can they occur in simple AC bridge circuits; so presumably this monotonicity extends to general AC circuits. My facile notion looks to be sadly dead in the water. Traffic engineers seem interested in queueing (5 vowels?!) theory rather than more classical properties of networks. WFL On 5/26/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:

I hadn't really thought this through properly --- just noticed that the video compares two systems comprising masses and springs, which is formally identical to a circuit with inductance coils and capacitors.

However the paradox involves the height of the mass [which unexpectedly increases after one supporting string is cut]. It's unclear to what property of AC (or indeed of traffic flow) this might correspond!

So some direct bijection between traffic flow and (say) current would be required to make the analogy work. Traffic engineers must surely have already investigated the matter; I have yet to conduct a search.

WFL

On 5/24/15, James Propp <jamespropp@gmail.com> wrote:

...

It'd be interesting to see an AC circuit analogue of Braess' paradox, because of the contrast with DC circuits: Rayleigh's monotonicity law says that paradoxes like this can't occur in purely resistive networks.

Jim Propp

On Saturday, May 23, 2015, Fred Lunnon <fred.lunnon@gmail.com> wrote:

...

I think I've now got the hang of this: more intuitively, the traffic network can instead be modelled by an AC electrical circuit.

The fluid dynamical aspects of flow along a single road might perhaps be incorporated by combining both models into a hydraulic analogue computer.

Next time you're stuck in a traffic jam, try figuring out whether it's a UTM. Or would be, when/if ever it got moving again ...

WFL

On 5/22/15, Fred Lunnon <fred.lunnon@gmail.com <javascript:;>> wrote:

...

The "standing wave" phenomenon in heavy traffic mentioned in the article is straightforward application of fluid dynamics, which I had ample opportunity to analyse during a particularly excruciating expedition along the London orbital motoway years ago.

I had to view the video in full-screen mode before I could follow it; now I understand the mechanical paradox, which still surprises me. But I don't see how the mechanical model maps to a traffic situation.

WFL

On 5/21/15, Mike Stay <metaweta@gmail.com <javascript:;>> wrote:

...

This short link also works and has no tracking data.

http://theconversation.com/the-maths-of-congestion-springs-strings-and-traff...

...

On Thu, May 21, 2015 at 3:19 PM, Fred Lunnon <fred.lunnon@gmail.com

<javascript:;>>

...

wrote:

...

See embedded video. New to me, though I had previously heard rumours