This Week's Finds in Mathematical Physics (Week 294)
John Baez
March 21, 2010
This Week's Finds in Mathematical Physics (Week 294)
John Baez
Sorry, I've been busy writing papers for the last couple of
months. But I'm not done with my story of electrical circuits! It
will take a few more episodes for me to get to the really cool part:
the symplectic geometry, the complex analysis, and how they fit
together using the theory of loop groups. I plan to talk about this
in Dennis Sullivan's seminar at the City University of New York later
this spring. I haven't written anything about it yet. So, I need to
prepare by discussing it here.
You'll understand why I need to prepare if you've heard about
Sullivan's seminar. It's a "Russian style" seminar, meaning that it's
modeled after Gelfand's famous seminar in Moscow. And what does that
mean? Well, Gelfand was famous for asking lots of questions. He
wanted to understand all that was said - and he wasn't willing to put
up with any nonsense. So, his seminar went on for hours and hours,
leaving the speaker exhausted.
Here's a nice description of it:
1) Simon Gindikin, Essay on the Moscow Gelfand Seminar, in Advances in
Soviet Mathematics, 16, eds. Sergei Gelfand and Simon Gindikin, 1993.
Available at http://www.math.rutgers.edu/home/gelfand/
Let me quote a bit:
The Gelfand seminar was always an important event in the very vivid
mathematical life in Moscow, and, doubtless, one of its leading
centers. A considerable number of the best Moscow mathematicians
participated in it at one time or another. Mathematicians from
other cities used all possible pretexts to visit it. I recall how
a group of Leningrad students agreed to take turns to come to
Moscow on Mondays (the day of the seminar, to which other events
were linked), and then would retell their friends what they had
heard there. There were several excellent and very popular
seminars in Moscow, but nevertheless the Gelfand seminar was always
an event.
I would like to point out that, on the other hand, the seminar was
very important in Gelfand's own personal mathematical life. Many
of us witnessed how strongly his activities were focused on the
seminar. When, in the early fifties, at the peak of antisemitism,
Gelfand was chased out of Moscow University, he applied all his
efforts to seminar. The absence of Gelfand at the seminar, even
because of illness, was always something out of the ordinary.
One cannot avoid mentioning that the general attitude to the
seminar was far from unanimous. Criticism mainly concerned its
style, which was rather unusual for a scientific seminar. It was a
kind of a theater with a unique stage director playing the leading
role in the performance and organizing the supporting cast, most of
whom had the highest qualifications. I use this metaphor with the
utmost seriousness, without any intention to mean that the seminar
was some sort of a spectacle. Gelfand had chosen the hardest and
most dangerous genre: to demonstrate in public how he understood
mathematics. It was an open lesson in the grasping of mathematics
by one of the most amazing mathematicians of our time. This role
could be only be played under the most favorable conditions: the
genre dictates the rules of the fame, which are not always very
convenient for the listerners. This means, for example, that the
leader follows only his own intuition in the final choice of the
topics of the talks, interrupts them with comments and questions
(a privilege not granted to other participants) [....] All this is
done with extraordinay generosity, a true passion for mathematics.
Let me recall some of the stage director's strategems. An important
feature were improvisations of various kinds. The course of the
seminar could change dramatically at any moment. Another important
mise en scene involved the "trial listener" game, in which one of
the participants (this could be a student as well as a professor)
was instructed to keep informing the seminar of his understanding
of the talk, and whenever that information was negative, that part
of the report would be repeated. A well-qualified trial listener
could usually feel when the head of the seminar wanted an occasion
for such a repetition. Also, Gelfand himself had the faculty of
being "unable to understand" in situations when everyone around
was sure that everything is clear. What extraordinary vistas were
opened to the listeners, and sometimes even to the mathematician
giving the talk, by this ability not to understand. Gelfand liked
that old story of the professor complaining about his students:
"Fantastically stupid students - five times I repeat proof,
already I understand it myself, and still they don't get it."
It has remained beyond my understanding how Gelfand could manage
all that physically for so many hours. Formally the seminar was
supposed to begin at 6 pm, but usually started with an hour's
delays. I am convinced that the free conversations before the
actual beginning of the seminar were part of the scenario. The
seminar would continue without any break until 10 or 10:30 (I have
heard that before my time it was even later). The end of the
seminar was in constant conflict with the rules and regulations of
Moscow State University. Usually at 10 pm the cleaning woman
would make her appearance, wishing to close the proceedings to
do her job. After the seminar, people wishing to talk to Gelfand
would hang around. The elevator would be turned off, and one
would have to find the right staircase, so as not to find oneself
stuck in front of a locked door, which meant walking back up to
the 14th (where else but in Russia is the locking of doors so
popular!). The next riddle was to find the only open exit from
the building. Then the last problem (of different levels of
difficulty for different participants) - how to get home on
public transportation, at that time in the process of closing up.
Seeing Gelfand home, the last mathematical conversations would
conclude the seminar's ritual. Moscow at night was still safe
and life seemed so unbelievably beautiful!
This is a great example of how taking things really seriously,
and pursuing them intelligently, with persistent passion, can infuse
them with the kind of intensity that leaves echoes resonating
decades later.
Sullivan's seminar is also intense, though it plays to a smaller
audience. Like Gelfand's, it's set in a tall building: in fact, a
30-story skyscraper called the Graybar Building, right next to Grand
Central Station. The first time I was asked to speak there, my talk
was supposed to start at 3 pm. But before that, there was an informal
"pre-talk" where people discussed math and sat around eating lunch.
Someone went down to get sandwiches, and I was asked what kind I
wanted. I said I wasn't hungry, but someone who knew better got me
one anyway.
My talk started at 3... and it went on until 9! I loved it: here was
someone who really wanted to understand my work. None of the usual
routine where everyone starts eyeing the clock impatiently as the
allotted hour nears its end. It was clear: this seminar would last as
long as it took to get the job done. And when we were done, we all
went out to dinner... and talked about math.
So, I should get back to my tale of electrical circuits. I'm really
just using these as a nice example of physical systems made of
components. Part of my goal is to get you interested in "open
systems" - systems that interact with their environment. My physics
classes emphasized "closed systems", where we assume that we've
modelled all the relevant aspects of what's going on, so the
interaction with the outside environment is negligible. Why?
It lets us use the marvelous techniques of symplectic mechanics -
Hamilton's equations, Noether's theorem giving conserved quantities
from symmetries, and all that. These techniques don't work for
open systems - at least, not until we generalize them. But almost
every device we design is an open system, in a crucial way: we do
things to it, and it does things for us. So engineers need to think
about open systems.
And mathematical physicists should too - because life gets more
interesting when you treat every system as having an "interface"
through which it interacts with its environment. For starters,
this lets you build bigger systems from components by attaching
them along their interfaces. We can also formalize the problem
of taking a system and decomposing it into smaller subsystems.
In engineering this is called "tearing". For example, we can
take this electrical circuit:
| |
| |
----- |
| | |
----- |
/ \ |
/ \ |
------------ |
| | |
------------ |
| | | |
| | \_______/
| |
| |
and tear it in two like this:
| |
| |
----- |
| | |
----- |
/ \ |
/ \ |
------------ |
| | |
------------ |
| | | |
| | \_______/
| |
| |
Giampiero Campa pointed out an article that's full of wisdom about
open systems, the history of control theory, and the cultural
differences between mathematics and engineering:
2) Jan C. Willems, In control, almost from the beginning until the
day after tomorrow, European Journal of Control 13 (2007), 71-81.
Also available as
http://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/2007.2.pdf
You don't need to know anything about control theory to enjoy this!
Well, it helps to know that "control theory" is the art of getting
open systems to do what you want. But it's always fun to begin
learning a subject by hearing about its history - especially from
somebody who was there.
Here's a passage that connects to the point I was just trying
to make:
One can, one should, ask the question if closed systems, as
flows on manifolds and
dx/dt = f(x)
form a good mathematical vantage point from which to embark on the
study of dynamics. In my opinion they do not, for a number of
reasons. First, in a good theory the state x should be derived from
a less structured model. A more serious objection is that closed
systems are not good concepts to deal with modeling. A model
usually consists of a number of interacting subphenomena that need
to be modeled one-by-one. In these sub-models, the influence of the
other subsystems needs par force to be viewed as external, and in
principle free. Tearing leads to models that are open.
If you view a closed system as an interconnection of two systems,
these two systems will be open. Systems that take into account
unmodeled external influences form therefore a much more logical
starting point. Third, many basic laws in physics address open
systems. For example, Newton's second law, Maxwell's equations, the
gas law, and the first and second laws of thermodynamics. A good
setting of dynamics should incorporate these important examples
from the beginning. Finally, closed systems put one in the absurd
situation that in order to model a system, one ends up having to
model also the environment.
These arguments seem obvious and compelling. Twenty five years
ago, it was my hope that system theory, with its emphasis on open
systems, would by now have been incorporated and accepted as the
new starting point for dynamical systems in mathematics. Better,
more general, more natural, more apt for modeling, offering
interesting new concepts as controllability, observability,
dissipativity, model reduction, and with a rich, well developed,
domain as linear system theory. It is disappointing that this
didn't happen. What seemed like an intellectual imperative did not
even begin to happen. Mathematicians and physicists invariably
identify dynamical systems with closed systems.
I think this will change. I think we just need to develop the right
framework for open systems. Luckily, a lot of this framework is
already available: concepts like operads, n-categories and the like
give very general ways of describing how to build big things by gluing
together little pieces. For example, a trained mathematician will
take one look at this:
| |
| |
----- |
| | |
----- |
/ \ |
/ \ |
------------ |
| | |
------------ |
| | | |
| | \_______/
| |
| |
and say "that's a morphism in a compact closed category". So, we just
need to focus these concepts on the problems of engineering, and
explain them in ways that engineers - as opposed to, say, topologists
or quantum field theorists - can enjoy.
For a deeper look at Willems' ideas on open systems, try this:
3) Jan C. Willems, The behavioral approach to open and interconnected
systems: modeling by tearing, zooming and linking, Control Systems
Magazine 27 (2007), 46-99. Also available at
http://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/2007.1.pdf
In particular, people who doubt that engineers could ever enjoy fancy
math like operads and n-categories should check out the box near the
end, on "polynomial modules and syzygies".
Now, I've been talking recently about "bond graphs". This is
a general framework for physical systems which treats variables
as coming in groups of four:
q - displacement
p - momentum
q' - flow
p' - effort
If we use the example of a massive object that can move back and
forth, q and p stand are its position and momentum, while q' and p'
are velocity and force. But if we use the example of an electrical
circuit, q is charge and p is something fairly obscure called "flux
linkage". Then their time derivatives are current, q', and voltage,
p'.
In both these examples the quantity q'p' has dimensions of power.
Bond graphers consider this very important: the idea is that when we
consider mixed systems, like an electrical motor pushing a massive
object around, it's *power* that flows from one part to another.
In "week289", I listed two examples of systems where q'p' does *not*
have dimensions of power: thermal systems and economic systems.
People do draw bond graphs of these, but they're considered
second-class citizens: they're called "pseudo bond graphs".
Jan Willems has some criticisms of the bond graph methodology,
including this obsession with power - and also its focus on q' and p'
at the expense of q and p. I've tried to give q and p more importance
in my discussion so far, since for people trained in classical
mechanics they're of utmost importance. But for people trained in
electrical circuits, it's q' and p' that seem important: they talk
about current and voltage all the time, and a bit less about the other
two.
Here's a summary of Jan Willems' criticisms of bond graphs, taken from
a little box in the above paper. I'll paraphrase a bit here and there:
The tearing, zooming, and linking methodology for modeling
interconnected systems advocated and developed in this article has
many things in common with bond graphs. Introduced by Paynter in
the 1960s, bond graphs are popular as a methodology for modeling
interconnected physical systems, especially in mechanical engineering.
For modeling physical systems, bond-graph modeling is a superior
alternative to signal-flow diagrams and input/output-based modeling
procedures.
Bond graphs view each system interconnection in terms
of power and energy. The variables associated with terminals are
assumed to consist of an effort and a flow, where the (inner)
product of effort and flow is power. Connections are formalized by
junctions. Using a combination of junctions and component
subsystems, complex physical systems can be modeled in a systematic
way. The power interpretation automatically takes care of
conservation of energy. The philosophy underlying bond graphs is,
as stated by P.J. Gawthrop and G.P. Bevan,
Power is the universal currency of physical systems.
The idea that terminal variables come in pairs, an effort and a
flow, with efforts preserved at each interconnection and the sum of
flows equated to zero at each interconnection, is appealing and
deep. But, in addition to weak mathematical underpinnings and
unconventional graph notation with half arrows, bond graphs have
some shortcomings as a modeling philosophy, as explained in the
section "Bond-Graph Modeling". The main points discussed in that
section are the following:
1. The requirement that the product of effort and flow must be
power is sometimes not natural, for example, in thermal
interconnections.
2. In connecting terminals of mechanical systems, bondgraph modeling
equates velocities, and sets the sum of the forces equal to zero. In
reality one ought to equate positions, not velocities. Equating
velocities instead of positions leads to incomplete models.
3. Interconnections are made by means of terminals, while energy is
transferred through ports. Ports involve many terminals
simultaneously. The interconnection of two electrical wires
involves equating two terminal potentials and putting the sum of
two terminal currents to zero. The product of effort, namely, the
electrical potential, and flow, namely, the electrical current, for
an electrical connection has the dimension of power, but it is not
power. Power involves potential differences, while the
interconnection constraints involves the terminal potentials
themselves. It is not possible to interpret these interconnection
constraints as equating the power on both sides of the interconnection
point.
4. In many interconnections, it is unnecessary to have to worry
about conservation of energy.
Willems has his own methodology, which he explains. I'll need to learn
about it!
I'll get into the deeper aspects of electrical circuits next Week.
There are just a few leftovers I want to mention now. I told you
about five basic 1-ports in "week290": resistors, capacitors,
inductors, voltage sources and current sources. Each was defined by a
single equation involving q, p, q' and p, and perhaps the time
variable t. These five are the most important 1-ports. But there are
some weirder ones worth thinking about. Here they are:
1. A "short circuit". A linear resistor has p' = R q'. If
the resistance R equals zero, you get a "short circuit". Now
the relation between voltage and current becomes:
p' = 0
So, there's always zero voltage across this circuit element -
it's a perfect conductor. Or in the language of bond graphs:
there's always zero effort across this 1-port.
2. An "open circuit". If you take a linear resistor and say
its resistance is *infinite*, you get an "open circuit". Now the
relation between voltage and current becomes:
q' = 0
So, there's always zero current through this circuit element - it's a
perfect insulator. Or in the language of bond graphs: there's always
zero flow through this 1-port. By the way, the word "open" here
has nothing to do with "open system".
The point of these examples is that most linear resistors let us treat
current as a function of voltage *or* voltage as a function of
current, since R is neither zero nor infinite. But in the these two
limiting cases - the short circuit and the open circuit - that's not
true. To fit these cases neatly in a unified framework, we shouldn't
think of the relation between current and voltage as defining a
function. It's just a relation!
In the world of algebraic geometry, a relation defined by polynomial
equations is called a "correspondence". One way to get a correspondence
is by taking the graph of a function. But it's important to go beyond
functions to correspondences. And my claim is that this is important
in electrical circuits, too.
But here are some even weirder one ports:
3. A "nullator". Here we bend the rules for 1-ports and impose
*two* equations:
p' = 0
q' = 0
I don't think you can actually build this thing! The Wikipedia
article sounds downright Zen: "In electronics, a nullator is a
theoretical linear, time-invariant one-port defined as having
zero current and voltage across its terminals. Nullators are
strange in the sense that they simultaneously have properties
of both a short (zero voltage) and an open circuit (zero current).
They are neither current nor voltage sources, yet both at the
same time."
4. A "norator". Here we bend the rules in the opposite direction
and impose *no* equations:
Yes, that's a picture of no equations. Truly Zen: what is the sound
of no equations clapping? I don't think you can build this thing
either! At least, not by itself....
Now, you may wonder why electrical engineers bother talking about
things that don't actually exist. That's normally the prerogative
of mathematicians. But sometimes if you combine two things that
don't exist, you get one that does! This is often how we introduce
new kinds of things. For example, i x i = -1 lets us introduce the
"imaginary" number i in terms of the "real" number -1.
As far as 1-ports go: if I have one equation too many, and you have
one too few, together we're just right. So, there's a 2-port called
the "nullor", which is built - theoretically speaking - from a
nullator and a norator. Remmber, a 2-port is specified by by two
equations involving q_1,q'_1,p_1,p'_1, q_2,q'_2,p_2,p'_2, and perhaps
the time variable t. Here are the equations for the nullor:
p'_2 = 0
q'_2 = 0
So, the first wire acts like a norator while the second acts like
a nullator. To see why engineers like this gizmo, try this:
4) Wikipedia, Nullor, http://en.wikipedia.org/wiki/Nullor
For more, try these:
5) Herbert J. Carlin, Singular network elements, IEEE Trans. Circuit
Theory, March 1965, vol. CT-11, pp. 67-72.
6) P. Kumar and R. Senani, Bibliography on nullors and their
applications in circuit analysis, synthesis and design, Analog
Integrated Circuits and Signal Processing 33 (2002), 65-76.
Here's the last 1-port I want to mention:
5. The "memristor". This is a 1-port where the momentum p is a
function of the displacement q:
p = f(q)
The function f is usually called the "memristance". It was
invented and given this name by Leon Chua in 1971. The idea
was that it completes a collection of four closely related 1-ports.
In "week290" I listed the other three, namely the resistor:
p' = f(q')
the capacitor:
q = f(p')
and the inductor:
p = f(q')
The memristor came later because it's inherently nonlinear. Why?
A *linear* memristor is just a linear resistor, since we can
differentiate the linear relationship p = Mq and get p' = Mq'.
But if p = f(q) for a nonlinear function f we get something new:
p' = f'(q) q'
So, we see that in general, a memristor acts like a resistor whose
resistance is some function of q. But q is the time integral of
the current q'. So a nonlinear memristor is like a resistor
whose resistance depends on the time integral of the current that
has flowed through it! Its resistance depends on its history.
So, it has a "memory" - hence the name "memristance".
Memristors have recently been built in a number of ways, which are
nicely listed here:
7) Wikipedia, Memristor, http://en.wikipedia.org/wiki/Memristor
Electrical engineering journals are notoriously resistant to the
of open access, and I don't think there's a successful equivalent
of the "arXiv" in this field. Shame on them! So, you have to nose
around to find a freely accessible copy of Chua's original paper
on memristors:
8) Leon Chua, Memristor, the missing circuit element, IEEE
Transactions on Circuit Theory 18 (1971), 507-519. Also available at
http://www.lane.ufpa.br/rodrigo/chua/Memristor_chua_article.pdf
To see what the mechnical or chemical analogue of a memristor is like,
try this:
9) G. F. Oster and D. M. Auslander, The memristor: a new bond graph
element, Trans. ASME, J. Dynamic Systems, Measurement and Control
94 (1972), 249-252.
Also available as http://nature.berkeley.edu/~goster/pdfs/Memristor.pdf
Memristors supposedly have a bunch of interesting applications, but
I don't understand them yet. I also don't understand "memcapacitors"
and "meminductors". The above PDF file also contains a New Scientist
article on the wonders of these.
To wrap up the loose ends, I want to tell you about Tellegen's
theorem. Last week I started talking about electrical circuits and
chain complexes. I considered circuits built from linear resistors.
But now let's talk about completely general electrical circuits.
Last time I said an electrical circuit has "vertices", "edges" and
"faces":
o---------o---------o
|/////////|/////////|
|/////////|/////////|
|//FACE///|///FACE//|
|/////////|/////////|
|/////////|/////////|
o---------o---------o
The faces come in handy: electrical engineers call them "meshes".
But they're really just mathematical fictions. When you look at
a circuit you don't see faces, just vertices and edges:
o---------o---------o
| | |
| | |
| | |
| | |
| | |
o---------o---------o
So, just for fun, let's leave out the faces today. Let's start with a
graph, and orient its edges:
o---->----o---->----o
| | |
| | |
V V V
| | |
| | |
o----<----o---->----o
This gives a vector space C_0 consisting of "0-chains": formal linear
combinations of vertices. We also get a space C_1 of "1-chains":
formal linear combinations of edges, and a linear map
delta
C_0 <-------- C_1
defined as follows: for any edge
e
x --------> y
we have delta(e) = y - x.
This gadget
delta
C_0 <-------- C_1
is a pathetically puny example of a chain complex: we call it a
"2-term chain complex".
If we take the duals of the vector spaces involved, our 2-term
chain complex turns around and becomes a 2-term "cochain complex":
d
C^0 --------> C^1
Here d is defined to be the adjoint of delta:
(df)(e) = f(delta e)
for any 0-cochain f and any 1-chain e.
What can we do with such pathetically puny mathematical structures?
First, in any electrical circuit, the current I is a 1-chain.
Moreover, Kirchoff's current law says:
delta I = 0
meaning the total current flowing into any vertex equals the total
current flowing out. Last week I stated this law for closed circuits
made of resistors, but it's true for any closed circuit as long as the
current isn't changing too rapidly with time. Indeed, we can take it
as a mathematical definition of what it means for a circuit to be
"closed". By "closed" here, I mean that no current is flowing in
from outside.
Second, in any electrical circuit, the voltage V is a 1-cochain.
Moreover, Kirchoff's voltage law says:
V = d phi
meaning that we can define a "potential" phi(x) for each vertex x,
with the property that for any edge
e
x --------> y
the voltage V(e) is the difference phi(y) - phi(x). This law is
true for all circuits, as long as the current isn't changing too
rapidly with time.
Third, the power dissipated by the circuit equals
V(I)
Here we are pairing a 1-cochain and a 1-chain to get a number.
Again, we talked about this last week, but it's true in general.
But now comes something new!
Let's compute the power V(I) using Kirchoff's voltage law and
Kirchoff's current law:
V(I) = (d phi)(I) = phi(delta I) = 0
Hey - it's zero!
At first this might seem strange. The power is always zero???
But maybe it isn't so strange if you think about it: it's a version of
conservation of energy. In particular, it fails when we consider
circuits with current flowing in from outside: then delta I doesn't
need to be zero. We don't expect energy conservation in its naive
form to hold in that case. Instead, we expect a "power balance
equation", as explained in "week290".
But maybe it *is* strange. After all, if you have a circuit built
from resistors, why should it conserve energy? Didn't I say resistors
were dissipative?
I still don't understand this as well as I'd like. The math seems
completely trivial to me, but its meaning for circuits still doesn't
seem obvious. Can someone explain it in plain English?
Anyway, this result is called "Tellegen's theorem". Clearly you have
to be in the right place at the right time to get your name on a
theorem! It doesn't have to be hard. It just has to be new and
important. If I'd been there when they first discovered numbers,
2+2=4 would be called "Baez's theorem".
Still, you might be surprised to discover there's a whole book
on Tellegen's theorem:
10) Paul Penfield, Jr., Robert Spence and Simon Duinker, Tellegen's
Theorem and Electrical Networks, The MIT Press, Cambridge, MA, 1970.
Part of why this result is interesting is that depends on such minimal
assumptions. Typically in circuit theory we need to know the voltages
V as a function of the currents I, or vice versa, before we can do
much. For example, for circuits built from linear resistors, we have
a linear map
R: C_1 -> C^1
such that
V = RI
This is Ohm's law. But Tellegen's theorem doesn't depend on this, or
on any relationship between voltages and currents! Indeed, we can take
two *different* circuits with the same underlying graph, and let V be
the voltage of one circuit at one time, and let I be the current of
the other circuit at some other time. We still get
V(I) = (d phi)(I) = phi(delta I) = 0
so long as Kirchoff's voltage and current laws hold for each circuit!
I'm a bit fascinated by this paper, which you can get online:
11) G.F. Oster and C.A. Desoer, Tellegen's theorem and thermodynamic
inequalities, J. Theor. Biol 32 (1971), 219-241. Also available at
http://nature.berkeley.edu/~goster/pdfs/Tellagen.pdf
They give a decent description of Tellegen's theorem, and they use
it to derive something they call "Prigogine's theorem", which is
supposed to be in here:
12) Ilya Prigogine, Thermodynamics of Irreversible Processes, 3rd
edition, Wiley, New York, 1968.
I don't understand it well enough to give a beautiful lucid
explanation of it. But it's not complicated. It's an inequality
that applies to closed circuits built from resistors and capacitors, or
analogous systems in chemistry or other subjects.
According to Robert Kotiuga, the chain complex approach to electrical
circuits goes back to Weyl:
13) Hermann Weyl, Repartici?on de corriente en una red conductora,
Rev. Mat. Hisp. Amer. 5 (1923), 153-164.
He also recommend these references:
14) P. Slepian, Mathematical foundations of network analysis,
Springer, Berlin, 1968.
15) H. Flanders, Differential Forms with Applications to the Physical
Sciences, Dover, New York, 1989, pp. 79-81.
16) S. Smale, On the mathematical foundations of electrical network
theory, J. Diff. Geom. 7 (1972), 193-210.
17) G. E. Ching, Topological concepts in networks; an application of
homology theory to network analysis, Proc. 11th. Midwest Conference on
Circuit Theory, University of Notre Dame, 1968, pp. 165-175.
18) J. P. Roth, Existence and uniqueness of solutions to electrical
network problems via homology sequences, Mathematical Aspects of
Electrical Network Theory, SIAM-AMS Proceedings III, 1971, pp. 113-118.
By the way: if you've been paying careful attention and reading
between the lines, you'll note that I've been advocating the study
of the category where an object is a bunch of points:
x x x
and a morphisms from one bunch of dots to another is
graphs with loose ends at the top and bottom:
x x
| |
| |
| |
o |
/ \ |
/ \ |
/ o |
| / \ |
| / \_____/
| |
| |
x x
Here the circles are vertices of the graph, while the the x's are the
"loose ends". We compose these morphisms in the visually evident way,
by gluing the loose ends at the top of one to the loose ends at the
bottom of the other.
I would like to know all possible slick ways of understanding
this category, because it underlies fancier categories where
the morphisms are electrical circuits, or Feynman diagrams, or
other things.
For one thing, this category is "compact closed". In other words,
it's a symmetric monoidal category where every object has a dual.
Duality lets us take an input and turn it into an output, like this:
x
|
|
|
o
/ \
/ \
/ o
| / \
| / \
| | |
| | |
x x x
or vice versa.
And in fact, this category is the free compact closed category on one
self-dual object, namely x, and one morphism from the unit object to
each tensor power of x. The unit object is drawn as the empty set,
while the nth tensor power of x is drawn as a list of n x's. So, for
example, when n = 3, we have a morphism that looks like a "trivalent
vertex":
o
/|\
/ | \
/ | \
x x x
Using duality we get other trivalent vertices, like this:
x
|
|
o
/ \
/ \
/ \
x x
and the upside-down versions of the two I've shown so far.
In this category, a morphism from the unit object to itself is just a
finite undirected graph. Or, strictly speaking, it's an isomorphism
class of finite undirected graphs!
For electrical circuits it's also nice to equip the edges with
orientations, so we can tell whether the current flowing through the
edge is positive or negative. At least it *might be* nicer - everyone
seems to do it, but maybe it's bit artificial. Anyway, if we want to
do this, we should find a category where the morphisms from the unit
object to itself are finite *directed* graphs.
I think this is the free compact category on one object +, the
"positively oriented point" and one morphism from the unit object to
any tensor product built by tensoring a bunch of copies of this object
+ and then a bunch of copies of its dual, -. So, among the generating
morphisms in this compact closed category, we'll have four trivalent
vertices like this:
/|\
/ | \
| | |
V V ^
| | |
+ + +
/|\
/ | \
| | |
V V ^
| | |
+ + -
/|\
/ | \
| | |
V V ^
| | |
+ - -
/|\
/ | \
| | |
V V ^
| | |
- - -
We then get other trivalent vertices by permuting the outputs or
turning outputs into inputs.
I can't help but hope there's a slicker desciption of this category.
Anybody know one?
>From directed graphs we can get chain complexes, and we've seen how
this is important in electrical circuit theory. Can we do something
similar to all the morphisms in our category?
Well, we can think of a directed graph as a functor
X: G -> Set
where G is category with two objects, "vertex" and "edge",
and two morphisms:
source: edge -> vertex
target: edge -> vertex
together with identity morphisms. We can think of G as the
"Platonic idea" of a graph, and actual graphs as embodiments of
this idea in the world of sets.
Taking this viewpoint, we can compose a directed graph
X: G -> Set
with the "free vector space on a set" functor
F: Set -> Vect
and get a gizmo that's like a graph, but with a vector space of
vertices and a vector space of edges. A category theorist might
call this a "graph object in Vect".
This may sound scary, but it's not. When we perform this process,
we're just letting ourselves take formal linear combinations of
vertices, and formal linear combinations of vertices. So all we
really get is a 2-term chain complex.
This sheds some light on how graphs are related to chain complexes.
In fact, we can turn this insight into a little theorem: the category
of graph objects in Vect is equivalent to the category of 2-term chain
complexes. There's a bit to check here!
In short, waving the magic wand of linearity over the concept of
"directed graph", we get the concept of "chain complex". So, there
should be some way to take compact closed category I just described
and wave the magic wand of linearity over that, too. And the result
should be a category important in the theory of electrical circuits.
There's a closely related result that's also interesting.
Suppose we have a directed graph:
x---->----x---->----x
| | |
| | |
V V V
| | |
| | |
x----<----x---->----x
This looks a bit like a category! In fact we can take the free
category on a directed graph: this is called a "quiver". And if we
wave the magic wand of linearity over a category (in the correct way,
since there are different ways), we get a category object in Vect.
But the category of category objects in Vect is *also* equivalent to
the category of 2-term chain complexes! Alissa Crans and I called a
category object in Vect a "2-vector space", since we can also think of
it as a kind of categorified vector space. See Section 3 here:
19) John Baez and Alissa Crans, Higher-dimensional algebra VI: Lie
2-algebras, Theory and Applications of Categories 12 (2004), 492-528.
Available at http://www.tac.mta.ca/tac/volumes/12/15/12-15abs.html
and also as arXiv:math.QA/0307263.
This idea was known to Grothendieck quite a while ago - read the paper
for the history. But anyway, I think it's neat that we can take the
bare bones of an electrical circuit:
o---->----o---->----o
| | |
| | |
V V V
| | |
| | |
o---->----o---->----o
and think of it either as a graph, or a category, or a graph or
category object in Vect, namely a chain complex - but moreover, we can
also think of it as an endomorphism of the unit object in a certain
compact closed category!
If you made it this far, you deserve a treat:
20) Astronomy Picture of the Day, Cassini spacecraft crosses
Saturn's ring plane, http://apod.nasa.gov/apod/ap100215.html
Saturn's rings edge-on, and a couple of moons, photographed by the
Cassini probe! Shadows of the rings are visible on the northern
hemisphere.
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Quote of the Week:
"... I have almost always felt fortunate to have been able to do
research in a mathematics environment. The average competence level
is high, there is a rich history, the subject is stable. All these
factors are conducive for science. At the same time, I was never able
to feel unequivocally part of the mathematics culture, where, it seems
to me, too much value is put on *difficulty* as a virtue in itself. My
appreciation for mathematics has more to do with its clarity of
thought, its potential of sharply articulating ideas, its virtues as
an unambiguous language. I am more inclined to treasure the beauty
and importance of Shannon's ideas on errorless communication, algorithms
such as the Kalman filter or the FFT, constructs such as wavelets and
public key cryptography, than the heroics and virtuosity surrounding
the four-color problem, Fermat's last theorem, or the Poincar? and
Riemann conjectures." - Jan C. Willems
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Addendum: For more discussion, visit the n-Category Cafe at:
http://golem.ph.utexas.edu/category/2010/03/this_weeks_finds_in_mathematic_55.html
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