From: baez@math.removethis.ucr.andthis.edu (John Baez)
Subject: This Week's Finds in Mathematical Physics (Week 297)
Organization: University of California, Riverside
Sender: baez@math.removethis.ucr.andthis.edu (John Baez)
Newsgroups: sci.physics.research,sci.physics,sci.math
Also available at http://math.ucr.edu/home/baez/week297.html
May 9, 2010
This Week's Finds in Mathematical Physics (Week 297)
John Baez
This week I'll talk about electrical circuits and Dirichlet forms.
But first: knot sculptures, special relativity in finance, lazulinos,
some peculiar infinite sums, and a marvelous fact about the number
12.
Here are some cool sculptures of knots by Karel Vreeburg:
1) Karel Vreeburg,
http://www.karelvreeburg.nl/site/kunstwerken/357933_Beelden.html
The polished forms emerge from rough stone much as mathematical
abstractions emerge from physical reality. And I'm reminded of what
Michelangelo said: "Every block of stone has a statue inside it, and
the task of the sculptor is to discover it."
Next - remember that big glitch in the stock market last Thursday,
when the Dow Jones dropped 9.2% in less than an hour, and then bounced
back? For a while, about a trillion dollars had evaporated!
The worst part is, nobody knows why. But apparently one part of the
problem was that some electronic communication systems were lagging
behind, seeing a delayed view of what was really going on. But guess
how long this lag was. Just 0.1 seconds!
That's only three quarters the time it takes light to circle the
Earth. But these days it's considered an unacceptably long time for
computer trading. So, we've reached the point where special
relativity is important in economics! The Newtonian concept of "the
same time at different places" is no longer adequate:
A 1-millisecond advantage in trading applications can be worth $100
million a year to a major brokerage firm, by one estimate. The
fastest systems, running from traders' desks to exchange data
centers, can execute transactions in a few milliseconds - so fast,
in fact, that the physical distance between two computers
processing a transaction can slow down how fast it happens. This
problem is called data latency - delays measured in split
seconds. To overcome it, many high-frequency algorithmic traders
are moving their systems as close to the Wall Street exchanges as
possible.
This quote is from:
2) Richard Martin, Wall Street's quest to process data at the
speed of light, Information Week, April 23, 2007. Also available
at http://www.informationweek.com/news/infrastructure/showArticle.jhtml?articleID=199200297
See also:
3) Kid Dynamite's World, Market Speed Bumps,
http://fridayinvegas.blogspot.com/2010/05/market-speed-bumps.html
where someone comments:
What I suspect happened (following on moments after KD's
explanation ends) is that some meaningful trigger point on stop
loss orders was exceeded. This could have been a small wave of
selling from Bloomberg running the video of the crowd getting
agitated in Greece (which was at about 2:40PM EST), but whatever
the case - a wave of selling started. That in turn brought the
price down, which triggered some stop loss orders, which in turn
fueled more stop loss orders, along with any humans and machines
that just sold on the steep drop.
However, given the heavy volume at the time, the HFT systems that
would normally jump in (albeit at much lower bids) didn't even get
to see accurate representations of the order books, because I was
seeing at least a 100ms delay in quotes from ARCA (the only ECN I
measured accurately).
So, at least with ARCA and probably the other exchanges as well,
everyone was running with at least a 100ms delayed snapshot of the
world. Given that I stopped calculating this delay when my own
software shutdown at 2:41PM (4 minutes before the peak of chaos),
this is probably understating matters somewhat.
If you can't see that the order book is missing bids because you
are operating 100ms behind the actual trades taking place, then
there is a meaningful window when the bids in the order book can
all be taken out before anyone even knows that they should be
placing bids!
Further, once you recognize that you are operating with stale
information (and 100ms is quite stale if you are seeing the markets
plunge the way they were), there is no way you are going to enter
orders, since you don't have any clue where to place them, and if
you do - you place them with much wider spreads than normal, which
in conjunction with market sell orders brings the trading price
down along with the bid/ask midpoint.
I guess it's just a matter of time before *general* relativity becomes
important in finance. I thank Mike Stay and Henry Baker for bringing
this issue to my attention.
I also enjoyed this blog post by Mike:
4) Mike Stay, Lazulinos,
http://reperiendi.wordpress.com/2010/04/27/lazulinos/
It's about a newly discovered quasiparticle with astounding
properties. If you want to really understand what's going on, read
the paper by Alexander Craigie - there's a link at the end of Mike's
post.
Next, an observation from Robert Baillie. Take this series:
Pi/Sqrt(8) = 1 + 1/3 - 1/5 - 1/7 + 1/9 + 1/11 - ...
Square each term, add them up... and you get the square of
the previous sum:
Pi^2/8 = 1 + 1/3^2 + 1/5^2 + 1/7^2 + 1/9^2 + 1/11^2 + ...
Don't tell undergraduates about this - they are already confused
enough!
And finally, a comment from Nora Ganter. If you look at the
cohomology of the symmetric groups, you find an element of order 12 in
H^3(S_n,Q/Z) for n greater or equal to 4. But the third cohomology of
a group classifies ways of extending it to a 2-group. So whenever you
realize a finite group as a group of permutations of 4 or more things,
you automatically get a way of extending it to a 2-group!
I would like to understand this better. In particular, the
number 12 here should be related to the fact that
pi_{k+3}(S^k) = Z/24
for k >= 5. After all, stable homotopy groups of spheres are related
to the cohomology of symmetric groups, since the group completion of
the classifying space of the groupoid of finite sets is Omega^infinity
S^infinity - see "week199" if you don't know what I'm talking about
here. But I'm confused about the numbers 12 versus 24 here, and also
the role of Q/Z coefficients.
Does someone know a place where you can look up cohomology groups of
the symmetric groups?
Next: electrical circuits!
Last week I discussed electrical circuits made of (linear) resistors
and "grounds" - places where wires touch an object whose electrostatic
potential is zero. I want to fill in some missing pieces today.
Suppose we have such a circuit with n wires dangling out of it. I've
been calling these "inputs" and "outputs" - but today I don't care
which ones are inputs and which ones are outputs, so let's call them
all "terminals".
We saw last time that our circuit gives a function
Q: R^n -> R
This tells you how much power the circuit uses as a function of the
electrostatic potential at each terminal.
It's pretty easy to see that Q is a "quadratic form", meaning that
Q(phi) = sum_{i,j} Q_{ij} phi_i phi_j
for some matrix Q_{ij}, which we can assume is symmetric. And it's
easy to see that Q is "nonnegative", meaning
Q(phi) >= 0
I wildly guessed that every nonnegative quadratic form comes from a
circuit made of resistors and grounds. Since then I've learned a few
things, thanks to Ben Tilly and Tom Ellis.
For starters, which nonnegative quadratic forms do we get from circuits
built only from resistors? We certainly don't get all of them. For
example, if n = 2, every circuit built from just resistors has
Q(phi) = c (phi_1 - phi_2)^2
for some nonnegative number c. So, we'll never get this quadratic
form:
Q(phi) = (phi_1 + phi_2)^2
even though it's nonnegative. In general, for any n, we can get
a lot of quadratic forms just by connecting each terminal to each
other with a resistor. Such circuits give precisely these quadratic
forms:
Q(phi) = sum_{i,j} c_{ij} (phi_i - phi_j)^2
where the numbers c_{ij} are nonnegative. We can assume without loss
of generality that c_{ii} = 0. The numbers c_{ij} are *reciprocals*
of resistances, so we're allowing resistors with infinite resistance,
but not with zero resistance.
It turns out that quadratic forms of the above type are famous:
they're called "Dirichlet forms". People have characterized them in
lots of ways. Here's one: they're the nonnegative quadratic forms
that vanish when phi is constant:
phi_i = phi_j for all i,j implies Q(phi) = 0
and also satisfy the "Markov property":
Q(phi) >= Q(psi)
when psi_i is the minimum of phi_i and 1. This characterization
is Proposition 1.7 here:
5) Christophe Sabot, Existence and uniqueness of diffusions on finitely
ramified self-similar fractals, Section 1: Dirichlet forms on finite
sets and electrical networks, Annales Scientifiques de l'École Normale
Supérieure, Sér. 4, 30 (1997), 605-673. Available at
http://www.numdam.org/numdam-bin/item?id=ASENS_1997_4_30_5_605_0
Sabot doesn't prove this result, which he considers "well known".
Instead, he points us to this book, which is not only fun to read,
but also free:
6) P. G. Doyle and J. L. Snell, Random Walks and Electrical Circuits,
Mathematical Association of America, 1984. Also available at
http://www.math.dartmouth.edu/~doyle/
You may wonder what random walks and diffusions on fractals have to do
with electrical circuits! The idea is that we can take a limit of
electrical circuits that get more and more complicated and get a
*fractal*. The electrical conductivity of this fractal can be
reinterpreted as heat conductivity, using the analogies described back
in "week289". And then we can study the heat equation on this fractal.
This equation says how heat diffuses with the passage of time.
But there's nothing special about *heat*. We can use the heat
equation to describe the diffusion of just about anything. We could
even use it to describe the diffusion of tiny drunken men who stumble
around aimlessly on our fractal! And that's where "random walks" come
in.
It turns out that in situations like this, the heat equation is
completely determined by a quadratic form called a "Dirichlet form".
But it's not a quadratic form on R^n anymore: it's a quadratic form
on a space of real-valued functions on our fractal.
In fact Dirichlet forms were first studied, not for finite sets or
fractals, but for nice regions in Euclidean space - the sort of
regions you'd normally consider when studying the heat equation. In
this case the Dirichlet form arises from the Laplacian:
Q(phi) = - integral phi Laplacian(phi)
where phi is a function on our region. The moral is that we should
think of any Dirichlet form as a generalized Laplacian!
There's a huge literature on Dirichlet forms. Most of it focuses on
analytical subleties that don't matter for our pathetically simple
examples. For a little taste, try this review of two books on
Dirichlet forms:
7) Review by Daniel Stroock, Bull. Amer. Math. Soc. 33 (1996) 87-92.
Also available at
http://www.ams.org/journals/bull/1996-33-01/S0273-0979-96-00617-9/
Among other things, he mentions a simpler characterization of Dirichlet
forms. We're only considering quadratic forms
Q: R^n -> R
and it turns out such a form is Dirichlet iff
Q(phi) >= Q(psi)
whenever
|phi_i - phi_j| >= |psi_i - psi_j|
for all i,j. It's a fun exercise to see that this is equivalent to
our previous characterization. And there's a simple physical idea
behind this one: a circuit made of resistors will use more power when
the potentials at different terminals differ by bigger amounts!
Okay... I'm digressing a bit. Let's get back on track.
We've seen that the quadratic form of a circuit made from resistors is
Dirichlet whenever the circuit is of a special form: namely, when it
has one resistor connecting each pair of terminals.
But what about other circuits made from resistors, like this?
x x
| |
o-----------o
/ \ |
/ \ |
/ o--------o
| / \ |
| / \ |
o---o-----o-----o
| | |
x x x
Here the x's are the terminals, but there are also other vertices,
which I'll call "internal vertices". Also, not every vertex is
connected to every other vertex. Do we get a larger class of
quadratic forms if we allow more general circuits like this?
No! All we get are Dirichlet forms!
For starters, it doesn't matter that not every vertex is connected
to every other vertex. We can connect them with wires that have
infinite resistance, and nothing changes. (Remember, we're allowing
infinite resistance.)
So, the only interesting thing is the presence of "internal vertices".
Why are the quadratic forms of circuits with internal vertices still
Dirichlet forms?
This follows from Sabot's Proposition 1.8. Let me explain the idea.
Suppose, for example, that we have a nonnegative quadratic form in 3
variables
Q: R^3 -> R
Then we can get a quadratic form in 2 variables by taking the
minimum of Q as the third variable ranges freely:
P(phi_1, phi_2) = min_{phi_3} Q(phi_1, phi_2, phi_3)
Physically this corresponds to taking a circuit with 3 terminals,
like this:
x x
\ /
\ /
\ /
\ /
x
and treating it as a circuit with 2 terminals by regarding
the third terminal as an internal vertex:
x x
\ /
\ /
\ /
\ /
o
This means we let the potential at this vertex vary freely; by the
principle of minimum power, it will do whatever it takes to minimize
the power. So, we get a new circuit whose quadratic form is
P(phi_1, phi_2) = min_{phi_3} Q(phi_1, phi_2, phi_3)
More generally, we can take a nonnegative quadratic form in n
variables, and take any subset of these variables, and get a new
quadratic form by this minimization trick. And Sabot claims that if
the original form was Dirichlet, so is the new one. He doesn't prove
this, but I think it's easy - try it!
Sabot calls this trick for getting new Dirichlet forms from old ones
the "trace map". He also describes another trick, the "gluing map".
This lets us take the Dirichlet form of a circuit and get a new
Dirichlet form by gluing together some terminals. For example, we
could start with this circuit:
x x
\ /
\ /
\ /
\ /
x
and glue the top two terminals together, getting this circuit:
x
/ \
/ \
\ /
\ /
x
Both the trace map and the gluing map have interesting category-
theoretic interpretations. For example, the gluing map lets us
*compose* electrical circuits - or more precisely, their Dirichlet
forms - by gluing the outputs of one onto the inputs of another.
Finally, suppose we allow grounds as well as resistors. Sabot
considers circuits of this sort in the following beautiful paper:
8) Christophe Sabot, Electrical networks, symplectic reductions, and
application to the renormalization map of self-similar lattices,
Proc. Sympos. Pure Math. 72 (2004), 155-205. Also available as
arXiv:math-ph/0304015.
He only considers circuits of a special form. They have no internal
vertices, just terminals. As before, each pair of terminals is
connected with a resistor. But now, each terminal is also connected
to the ground via a resistor! Such circuits give exactly these
quadratic forms:
Q(phi) = sum_{i,j} c_{ij} (phi_i - phi_j)^2 + sum_i c_i phi_i^2
where c_{ij} and c_i are nonnegative numbers.
Let's call these "generalized Dirichlet forms". I believe these
generalized Dirichlet forms are characterized by the Markov property:
Q(phi) >= Q(psi)
when psi_i is the minimum of phi_i and 1.
These generalized Dirichlet forms don't include *all* the nonnegative
quadratic forms. Why? Because, as Ben Tilly pointed out, they don't
include quadratic forms where the cross-terms phi_i phi_j have
positive coefficients. So, for example, we don't get this:
Q(phi_1, phi_2) = (phi_1 + phi_2)^2
Sabot claims that generalized Dirichlet forms are closed under the
trace map and gluing. Given this, the same argument I already
sketched shows that *every* electrical circuit built from resistors
and grounds has a quadratic form that's a generalized Dirichlet form!
So, it's all been worked out...
Even better, Sabot explains how quadratic forms on a vector space V
give Lagrangian subspaces of T*V. This is the trick I used last week
to introduce wires of zero resistance.
A wire with zero resistance would use an infinite amount of power if
you put a different electrostatic potential at each end. KABANG! -
the ultimate "short circuit"! So, wires with zero resistance are not
physical realistic, but they're useful idealizations: they serve as
identity morphisms in the category-theoretic description of electrical
circuits. Circuits containing these wires can still be described
using Lagrangian subspaces. These subspaces *don't* come from
quadratic forms. But they are limits of subspaces that do.
Now we can make this more precise. There's a manifold consisting of
all Lagrangian subspaces of T*V - the "Lagrangian Grassmannian".
Sitting in here is the set of generalized Dirichlet forms on V. We
can take the closure of that set and get a space C(V). Points in C(V)
correspond to circuits built from resistors, grounds, and wires of
zero resistance. Sabot says this space is discussed here:
9) Y. Colin de Verdiere, Reseaux electriques planaires I,
Comment. Math. Helv. 69 (1994), 351-374. Also available at
http://www-fourier.ujf-grenoble.fr/~ycolver/All-Articles
So, Sabot, Verdiere and the rest of the Dirichlet form crowd have done
almost everything I want... *except* phrase their results in the
language of category theory! And that, of course, is my real goal: to
develop category theory as a language for physics and engineering.
Last week I gave a preliminary try at describing a category whose
morphisms are electrical circuits built from resistors and grounds.
I said:
Claim: there is a dagger-compact category where:
An object is a finite-dimensional real vector space.
A morphism S: V -> W is a Lagrangian subspace of T*V x T*W.
We compose morphisms using composition of relations.
The tensor product is given by direct sum.
The symmetry is the obvious thing.
The dagger of a subspace of T*V x T*W is the corresponding
subspace of T*W x T*V.
The problem was that this category has too many morphisms. If we only
want physically realistic circuits - or *almost* realistic ones, since
we're allowing wires of zero resistance - we should work not with all
Lagrangian subspaces of T*(R^m) x T*(R^n), but only those lying in the
subset C(R^m x R^n). So, let's try:
Claim: there is a dagger-compact category where:
An object is a natural number n.
A morphism S: m -> n is a point in C(R^m x R^n).
We compose morphisms using composition of relations.
The tensor product is given by direct sum.
The symmetry is the obvious thing.
The dagger of a subspace of C(R^m x R^n) is the corresponding
subspace of C(R^n x R^m)
There are a few things to check here. I haven't checked them all.
By the way: in case you actually want to study this stuff, I should
point out that Sabot's second paper uses "Dirichlet form" to mean what
I'm calling a generalized Dirichlet form, and uses "conservative
Dirichlet form" to mean what I'm calling a Dirichlet form. So, be
careful.
Also, here's another worthwhile reference:
10) Jun Kigami, Analysis on Fractals, Cambridge U. Press. First
60 pages available at
http://www-an.acs.i.kyoto-u.ac.jp/~kigami/AOF.pdf
It's full of information on Dirichlet forms and electrical circuits.
And it gives yet another characterization of Dirichlet forms! I don't
love it - but I might as well tell you about it.
A Dirichlet form on R^n is a nonnegative quadratic form that
vanishes when phi is constant:
phi_i = phi_j for all i,j implies Q(phi) = 0
and satisfies
Q(phi) >= Q(psi)
whenever
psi_i = phi_i if 0 < phi_i < 1
1 if phi_i > 1
0 if phi_i < 0
This is yet another way to say that power decreases when the
potentials at the terminals are closer together.
Kigami also explains the relation between Dirichlet forms and Markov
processes. His Theorem B.3.4. says that for a measure space X, there
is a one-to-one correspondence between Dirichlet forms on L^2(X) and
strongly continuous semigroups on L^2(X) that map functions in L^1(X)
to functions of the same sort, and map nonnegative functions whose
integral is 1 to functions of the same sort. Such semigroups are
called "Markov". The classic example is provided by the heat
equation! But in our electrical circuit example, we're considering
the pathetically simple case where X is a finite set.
One simple thing that deserves to be emphasized is that a Dirichlet
form is not a kind of quadratic form on an abstract vector space.
It's a kind of quadratic form on a space of functions! In particular,
in my discussion above, R^n really means the algebra of functions on
an n-element set - and in the second dagger-compact category mentioned
above, the objects should really be finite sets. I was just working
with a skeletal subcategory, to make things less intimidating.
Okay, I'll stop here for now. Later I want to bring inductors and
capacitors into the game... and loop groups!
-----------------------------------------------------------------------
Quote of the Week:
"Discussions about theoretical engineering research often feels like
visiting a graveyard in the company of Nietzsche. From the beginning
of my career until now, I have always been hearing that 'the field is
dead', 'circuit theory is dead', 'information theory is dead', 'coding
theory is dead', 'control theory is dead', 'system theory is dead',
'linear system theory is dead', 'H_infinity is dead'. Good science,
however, is always alive. The community may not appreciate the
vibrancy of good ideas, but it is there. The absence of this impatience
is one of the things that makes working in a mathematics department
simply more pleasant." - Jan C. Willems
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Addendum: My friend Bruce Smith wrote:
I can't tell for sure, from what you wrote about grounds in week297
(and the last few Weeks), whether you are aware of this way to think
about them: there is a 1-1 correspondence between circuits that can
include grounds, and circuits that can't. To implement it, starting
with a circuit that can include grounds, just add an extra
terminal, call it "G" for "ground", and replace every internal
ground with a 0-resistance connection to that terminal G. Also, in
your thinking about potentials at terminals, replace "the potential
at T_i" with "the potential difference between T_i and G" (or
equivalently but differently, require that the potential at G is
always 0).
(I'm pretty sure you must be aware of this, but somehow it didn't
show up as a simplifier in your explanation as much as, or as
explicitly as, I thought it ought to.)
If 0 resistance bothers you, note that it can be reduced away (by
eliminating internal terminals in your resulting circuit) unless
you had a ground directly connected to a terminal; if you were
allowing that, then in your new circuit you'd better be allowing
direct connections between two terminals, but I presume that
whatever difficulties this causes in either case are essentially
the same.
For more discussion, visit the n-Category Cafe at:
http://golem.ph.utexas.edu/category/2010/05/this_weeks_finds_in_mathematic_58.html
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