## May 15, 2010

### This Week’s Finds in Mathematical Physics (Week 298)

#### Posted by John Baez

In "week298" of This Week’s Finds, learn about finite subgroups of the unit quaternions, like the binary icosahedral group:

Then meet the finite subloops of the unit octonions. Get a tiny taste of how division algebras can be used to build Lie n-superalgebras that govern superstring and supermembrane theories. And meet Duff and Ferrara’s ideas connecting exceptional groups to Cayley’s hyperdeterminants and entanglement in quantum information theory.

Posted at May 15, 2010 6:06 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2216

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Yes, well, have fun at the Comlab conference, and say hi to Louis and Bob and Jamie and them from me. (I’ll just be sitting in my icy cold room, as usual, eating my beans and noodles and wondering about how I am going to get my next waitressing job.)

Posted by: Kea on May 16, 2010 4:15 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

In Supermembranes And The Signature Of Space-Time. Blencowe and Duff himself tell:

We conjecture (together with C. Hull and K. Stelle) that the (2,2) extended object moving in (10,2) spacetime may (if it exists) be related by simultaneus dimensional reduction to the (1,1) type II …

This is also reviewed in M theory (The Theory formerly known as strings).

On other hand, I have sometimes hear folklore about F-theory, telling that it really does not need the peculiar signature, but only to consider some infinitesimal extra dimension.

Posted by: Alejandro Rivero on May 17, 2010 1:35 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Regarding your Oxford slides, your first two slides are identical, and you have a bad line break at slide 12.

On slide 16, should you say something about normalizing rows to produce probability? I know we’ve discussed this before. Along with the idea of relative probabilities, you had the idea of matrix mechanics over Durov’s generalized rings.

Hmm, perhaps I should return to all that Progic material.

Posted by: David Corfield on May 17, 2010 2:27 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Thanks for catching those mistakes, David. Fixed!

The double first page was caused by the fact that this talk contains a mixture of Postscript figures (most of the talk) and a jpeg figure (the first page). Right now I don’t know how to get LaTeX to handle both those formats in the same file! I can use jpegs in pdflatex and Postscript in ordinary latex, which then can be turned into PDF. So I did this and then used Adobe Acrobat to combine two separate PDFs… and screwed up. I mention this boring stuff only in case some smart person knows a better solution.

I guess I’ll verbally mention the idea of ‘relative probabilities’.

I think there’s a lot of stuff left to be done with these ideas, and your ‘progic’ ideas.

Posted by: John Baez on May 17, 2010 7:26 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

I can use jpegs in pdflatex and Postscript in ordinary latex, which then can be turned into PDF.

Can you first turn your external postscript files into PDFs and then include them with pdflatex?

Posted by: Mike Shulman on May 17, 2010 9:54 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Scott wrote:

If you are using graphicx via

\usepackage{graphicx}

in the preamble, you *should* be able to specify file type via suffix:

Hmm. I’ve read things like this:

PDFLaTeX, when used with graphics or graphicx packages, can compile correctly PNG and JPG files into DVI or PDF, but is not able to handle EPS files. Conversely, the process of compiling with LaTeX to DVI and converting to PS and eventually PDF does support EPS, but does not support PNG and JPG.

… and that corresponds to my experience.

Hmm. I see that the document I just cited suggests a few workarounds, all mildly irksome, but probably better than what I actually did. One of them is a version of what Mike suggested: converting eps files to pdf using ‘epstopdf’ — a program I hadn’t heard of.

Posted by: John Baez on May 18, 2010 2:34 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

If you are using graphicx via
\usepackage{graphicx}
in the preamble, you *should* be able to specify file type via suffix:

\begin{figure}[htb]
\begin{center}
\includegraphics[width=5in]{filename.pdf}
\end{center}
\caption{pdf file type}
\label{abst}
\end{figure}

and

\begin{figure}[htb]
\begin{center}
\includegraphics[width=5in]{notherfile.jpg}
\end{center}
\caption{jpg file type}
\label{abst}
\end{figure}

I seem to remember having done this once. Of course, you can leave out the \begin{figure} … \end{figure} if you don’t want a bunch of bodies floating around.

For a hack, import the jpg into xfig as a picture, and then save the xfig file as an eps, and process using TeX and ghostscript (Texshop option), or else eps2pdf the file. That is how I got photos into this talk.

Posted by: Scott Carter on May 18, 2010 12:50 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Oxford slide no. 40: There is a classical no-cloning theorem?

(Or is it a QM no-cloning theorem that’s become a classic? :-)

Posted by: Tim van Beek on May 17, 2010 2:54 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

The last page of my slides was supposed to summarize only the ideas that will surprise and thrill the audience, so I get a standing ovation. The Wooters–Zurek no-cloning theorem for quantum mechanics will be very well known to them, since Bob Coecke and Samson Abramsky (both at Oxford) have shown how this theorem is related to the fact that the monoidal category of Hilbert spaces with its usual tensor product is non-cartesian. But for some reason few people study classical mechanics using the complete repertoire of fancy ideas that they use for quantum mechanics. So I wanted to mention that Aaron Fenyes also has a no-cloning theorem for classical mechanics.

This is related to my earlier point that classical mechanics can be formulated using ideas akin to Heisenberg’s ‘matrix mechanics’, but with a different rig replacing the complex numbers.

Aaron Fenyes is a grad student at the Perimeter Institute. A while back he sent me an email starting like this:

At the end of the notes from your 2008 classical mechanics course, it says, “I believe the non-Cartesian nature of this product means there’s no classical machine that can ‘duplicate’ states of a classical system. But, strangely, this issue has been studied less than in the quantum case!” After mentioning this to a couple people, I got tired of not being able to give a specific result, so I wrote up a no-cloning theorem for symplectic mechanics, and I figured you might find it useful to have lying around.

I’m trying to get his okay to make it available on the webpage for this talk.

There are in fact a number of monoidal categories relevant to classical mechanics, and none of the ones I know are cartesian:

• the category of symplectic manifolds and symplectomorphisms
• the category of Poisson manifolds and Poisson maps
• the category Mat(RMIN) described here
• the category of symplectic vector spaces and linear Lagrangian correspondences described in Alan Weinstein’s paper Symplectic categories

… and I would like to add, ‘the category of symplectic manifolds and Lagrangian correspondences’… but unfortunately we can’t compose Lagrangian correspondences unless a transversality condition holds! This problem is the real topic of Weinstein’s paper.

Posted by: John Baez on May 17, 2010 7:38 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Aha.

Is there another hunch of yours of the same caliber somewhere in lecture notes of an undergraduate class, so that I could write down a solution after getting tired of not getting any definite answers from those Perimeter guys?

Posted by: Tim van Beek on May 17, 2010 9:52 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Heh. This Week’s Finds and my seminar notes are packed with hunches of varying caliber. Sometimes you need to read between the lines a bit to see them — for example, if I say something ‘should’ be true, it means I believe it’s true but haven’t proved it. And sometimes, I’ve said something is true even though I haven’t proved it. By now I realize this is a bad habit… thanks to the following story.

Once I went to a talk where somebody said that for any ring $R$ there’s a one-object tricategory $Alg(R)$ consisting of $R$-algebras, bimodules and bimodule morphisms. I said “Really? Do you know if anyone has ever written that up?” And the speaker said “Sure! It’s in This Week’s Finds!” Which galled me, because while I knew it was true, I’d never seen a proof written up — and I realized then that by claiming it was true in This Week’s Finds, I’d reduced the chances of ever seeing a proof.

Posted by: John Baez on May 18, 2010 2:48 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

I think you know this, but lest any other reader think that that particular fact still remains to be proven, let me point out that it follows easily from Theorem 21 of this paper and from the main theorem of this one (the latter in fact shows that $Alg(R)$ is a symmetric monoidal bicategory).

Posted by: Mike Shulman on May 18, 2010 4:14 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

I think you know this, but lest any other reader think that that particular fact still remains to be proven, let me point out that it follows easily from Theorem 21 of this paper and from the main theorem of this one (the latter in fact shows that $Alg(R)$ is a symmetric monoidal bicategory).

And for eternity:

$n$Lab: tricategory – Examples

$n$Lab: monoidal bicategory – Examples

Posted by: Urs Schreiber on May 18, 2010 9:52 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

But seriously, isn’t the no-cloning theorem supposed to be of an inherently quantum mechanical nature? How does a classical version fit in?

(Well, I hope you get Aaron Fenyes’ permission to publish his idea/proof…and I’m curious about what the members of your audience in Oxford will say…).

Posted by: Tim van Beek on May 18, 2010 9:17 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

To discriminate quantum from classical, one does not rely on no-cloning but on no-broadcasting [H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B.~Schumacher (1996) Noncommuting mixed states cannot be broadcast. Physical Review Letters 76, 2818–2821. arXiv:quant-ph/9511010 . This is for example used in Environment and classical channels in categorical quantum mechanics and Classical and quantum structuralism. The reason is, that indeed, while even classical probability theory has no-cloning, it doesn’t have no-broadcasting. Broadcastable data is the same as decoherent data. So if you want to use no-cloning to discriminate anything, you have to be very precise about what you are talking, as is clear from the previous entries. On the other hand, no-cloning does play an important role in quantum computing. But it is a fact, that if one starts to list things people say are weird about quantum, that many can be found in classical settings. Pinpointing in conceptual terms “what is quantum” is not at all obvious.

Posted by: bob on May 19, 2010 9:16 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

You can now see Aaron’s paper on the website for my talk, or here:

Posted by: John Baez on May 19, 2010 2:24 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Thanks to Bob and John, I will take a look. I guess the first thing I need to understand is “broadcasting” in the Barnum-Caves-Fuchs-Jozsa-Schumacher-paper and broadcastable $\cong$ decoherent.

But it is a fact, that if one starts to list things people say are weird about quantum, that many can be found in classical settings. Pinpointing in conceptual terms “what is quantum” is not at all obvious.

This is a real surprise! (For someone who stopped thinking about these matters after passing the QM exam, that is).
Posted by: Tim van Beek on May 19, 2010 2:45 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

I find the distribution collapse heuristic of measurement is the most unclassical aspect of quantum whatnot. Otherwise it’s “just” a differential equation; and those are fairly classical.

Posted by: some guy on the street on May 20, 2010 6:23 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Tim van Beek wrote:

But seriously, isn’t the no-cloning theorem supposed to be of an inherently quantum mechanical nature? How does a classical version fit in?

There are certain cases where people have thought about the quantum version of a problem a lot more carefully than the classical version, and this is one.

When people think about duplicating quantum states they are fairly careful to describe the candidate ‘duplication process’ — which they intend to prove does not exist — as linear operator between Hilbert spaces, or perhaps a unitary operator. But when they think about duplicating classical states they often sloppily treat it as a mere function between sets, or perhaps a one-to-one and onto function. I too have been guilty of this.

However, processes in classical mechanics are not just arbitrary functions between sets! The set of states of a classical system has a lot of extra structure: it’s typically a symplectic manifold, or more generally, a Poisson manifold. And processes aren’t arbitrary functions: they need to respect this extra structure. So, they need to be symplectomorphisms, or Poisson maps, or something like that. When you take this into account, you see it gets a lot harder to duplicate classical states. In fact Aaron shows its impossible in the symplectic case.

If that sounds too fancy and mathematical, you may prefer this puzzle. Try to use the laws of classical mechanics to design a machine like this:

The machine has a hole in the top where you can insert in a little box with a ball in it. The ball can move around in the box, so it has some arbitrary position and momentum. The machine measures the position and momentum of the ball at a certain moment — let’s say when you push a big red button. And then, out of a slot at the bottom, it spits out two boxes with balls in them! A bell rings when this happens, and when the bell rings, the state of each ball in each box is guaranteed to be equal to the state of the original ball in the original box at the moment you pushed the big red button.

Can you do it?

You get to use gears, levers, pulleys, all sorts of stuff like that — anything whose behavior can be described using classical mechanics.

Posted by: John Baez on May 20, 2010 12:31 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

In case you’re wondering how the machine gets to measure the position and momentum of the ball in the box: I was imagining the machine can remove the top of the box, and then insert sensors of various sorts.

The impossibility of building this machine should resemble the impossibility of building a working perpetual motion machine or Maxwell’s Daemon. No matter how clever you are, you should fail.

Posted by: John Baez on May 20, 2010 12:52 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

You get to use gears, levers, pulleys, all sorts of stuff like that — anything whose behavior can be described using classical mechanics. … In case you’re wondering how the machine gets to measure the position and momentum of the ball in the box: I was imagining the machine can remove the top of the box, and then insert sensors of various sorts.

I guess we may assume that the machine can use classical electrodynamics, too, and that the sensors act noninvasive, that is they have no influence on the system that they monitor.

Can you do it?

I have no idea, but my educated guess is that it would be wiser to try to prove that it cannot be done.
Posted by: Tim van Beek on May 20, 2010 3:41 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Tim wrote:

I guess we may assume that the machine can use classical electrodynamics, too …

Well, Aaron proved his no-cloning theorem for systems with finitely many degrees of freedom — that is, where the phase space is a finite-dimensional symplectic manifold. Field theories, like electromagnetism, have infinitely many degrees of freedom. So, his theorem doesn’t really apply. But I bet it could be generalized.

This could be a fun project for someone (hint, hint). But still, I think it’s fun to see what we can and can’t do using just classical mechanics, not field theory.

… and that the sensors act noninvasively, that is they have no influence on the system that they monitor.

Well, that’s a tricky question. I just want you to assume that the formalism of Hamiltonian mechanics applies. So the question is whether measurements that have no influence on the system being measured are allowed by Hamiltonian mechanics. Or at least very little influence. We all believe that in classical mechanics you can measure the position of a big boulder by bouncing a tiny pebble off it without affecting the boulder’s momentum very much — you can make the effect as small as you like. But still, this is the sort of thing you need to check, not just take for granted.

For example, if we have two particles with position and momentum $(q_1, p_1)$ and $(q_2, p_2)$ respectively, can we make $q_1$ affect $q_2$ without any effect the other way? We have to see if the resulting time evolution is symplectic.

For example, consider this linear transformation:

$(q_1, p_1, q_2 , p_2) \mapsto (q_1, p_1, q_2, p_2 + q_1)$

where the position of the first particle affects the momentum of the second particle, but there’s no effect the other way. Can this happen? No, because the symplectic 2-form

$d p_1 \wedge d q_1 + d p_2 \wedge d q_2$

gets sent to

$d p_1 \wedge d q_1 + d p_2 \wedge d q_2 + d q_1 \wedge d q_2$

So we say this transformation is not symplectic, and we know that no process in classical mechanics can make it occur.

But this is just one example; one has to think about this stuff in general and figure it out!

Posted by: John Baez on May 21, 2010 1:18 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

“ but unfortunately we can’t compose Lagrangian correspondences unless a transversality condition holds! ”

Reminds me of a similar problem for Chas-Sullivan string topology for which the hoped for way out of transversality is to look for an A_infty structure

anyone worked on that?

Posted by: jim stasheff on May 18, 2010 12:45 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Jim wrote:

Reminds me of a similar problem for Chas-Sullivan string topology for which the hoped for way out of transversality is to look for an $A_\infty$ structure.

Anyone worked on that?

I don’t understand this stuff well enough to give you a really illuminating answer, but try this:

Posted by: John Baez on May 19, 2010 2:28 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Thanks for this very classical TWF episode! It had starring almost all my favorite characters (octonions, E8, higher gauge theory, etc.) and had as always an excellent storytelling. All that makes it very hard to accept that TWF will be off the air very soon!

There will be a movie, eh a book one day, right?

Posted by: Christian on May 18, 2010 9:38 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

I’m glad you enjoyed this epsiode! TWF is not going off the air: TWFMP is becoming TWF and the range of themes will grow, but if I learn new cool stuff about time-honored themes I doubt I can resist talking about it.

You can also look forward to the movie version of The Octonions — I’m busy rounding up 8 stars. Unfortunately Snow White and the Seven Dwarves are locked into a lifetime contract with Disney.

Posted by: John Baez on May 19, 2010 2:32 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Has anyone heard what became of the kid’s movie “Mono, the adjoint functors and the diagram c(h)ase”?

Posted by: Tim van Beek on May 19, 2010 2:49 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Whew! I had a bit of a scare when reading the above comment about TWFMP being taken off the air. Thanks for the clarification John. Awesome episode as always. Cheers!

Posted by: Michael on May 23, 2010 9:36 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

I noticed that at Woit’s some wrestling has been going on concerning how classification of tripartite entangled states (under SLOCC) may be connected to Duff et all string theory stuff. Maybe of interest here at the cafe is that Aleks Kissinger and I found that this SLOCC-classification is actually directly connected to the classification of commutative Frobenius algebras on 2d Hilbert space in Hilb; see col 1 in arXiv:1002.2540 which states that there are only two kinds of commutative Frobenius algebras, special ones and “anti-special ones” (yes, we have been told many times that this is a horrible name), and this follows directly from the entanglement classification result.

Posted by: bob on May 21, 2010 7:35 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

I’m pleased that hyperdeterminants are slowly gethering more interest. They are hard to get into but rewarding when you do.

The work on connections between M-theory and quantum information entanglement is intriguing. I hope it leads somewhere. It is curious that this work links the 2x2x2 hyperdeterminant and the octonions which were both worked on by Cayley. I wonder how long Cayley thought about possible connections just because they are both algebraic structures defined on 8 component vectors.

There is another curious historical coincidence for the 2x2x2x2 hyperdeterminant that was constructed by Schläfli. It is a degree 24 polynomial. Schläfli also discovered the 24-cell and in both cases the appearance of the number 24 can be linked to the importance of the number 24 in bosonic string theory.

Posted by: PhilG on May 23, 2010 10:07 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

PG:

There is another curious historical coincidence for the 2x2x2x2 hyperdeterminant that was constructed by Schläfli. It is a degree 24 polynomial. Schläfli also discovered the 24-cell and in both cases the appearance of the number 24 can be linked to the importance of the number 24 in bosonic string theory.

Why don’t you back up the last statement by appropriate references?

Posted by: Arnold Neumaier on May 26, 2010 6:31 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

I was being lazy but I thought these things have been mentioned here before or can be found in google.

The connection between the 24-cell and string theory has been mentioned by JB e.g. here or here.

The connection between the 2x2x2x2 determinant and elliptic curves which therefore links it to string theory is described here.

I suppose it’s also interesting that both the hyperdeterminant and the 24-cell are four dimensional so you might try to find a more direct connection using that.

Posted by: PhilG on May 27, 2010 8:44 AM | Permalink | Reply to this

### OT: amusing crop-circle link

Not remotely connected with anything in the TWF, but this link gives an attempt to decode a crop circle recently found in England as a statement of Euler’s trigonometric identity. (This story appears other places, so I’m pretty sure that the physical crop circle does exist – rather than being photoshopped – but beyond that…). On the assumption that it was done by human beings, there’s still niggling questions (surely the “corrected-version” brackets would, by the principle of only putting things in when they’re needed, imply the pi isn’t in the exponent?). Anyway, it’s mildly amusing to see how people will spend their time.

Posted by: bane on May 24, 2010 9:38 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

Mike Duff’s students William Rubens and Leron Borsten are here at the school on Foundational Structures in Quantum Computation and Information. Over coffee William just told me about this new paper:

It gives an actual map relating a certain class supersymmetric black holes to patterns of entanglement of 4 qubits. I don’t understand it, but one key piece seems to be the Kostant-Sekiguchi correspondence between nilpotent orbits of $SO_0(4,4)$ in $so(4,4)$ and nilpotent orbits of $SL(2,\mathbb{C})^4$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$.

Here I am just parroting the paper. But what does the second appearance of ‘nilpotent’ in the above sentence mean? I know what a nilpotent element of a Lie algebra is, but not what a nilpotent element of a vector space is. My best guess is that we should use the vector space isomorphism

$so(4,4) \cong (sl(2,\mathbb{C})^4 \oplus \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$

to interpret elements of $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ as elements of $so(4,4)$. Then we can say whether or not they are nilpotent.

The entanglement classes of 4 qubits are all the nonzero orbits of $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ under the action of $(sl(2,\mathbb{C})^4$. I don’t know what we leave out if only consider nilpotent orbits.

Leron Borsten told me something like this. The allowed charges for a certain class of STU black holes correspond to points in a certain 8-dimensional lattice… perhaps the integral split octonions. We can think of this lattice as sitting inside $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$. Then the entropy of the black hole is the square root of the absolute value of the $2 \times 2 \times 2 \times 2$ hyperdeterminant.

He said this was nicely explained here:

Posted by: John Baez on May 27, 2010 12:08 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 298)

I’ve added an addendum to week298 — a letter from Péter Lévay explaining more about quantum entanglement, black holes and exceptional algebraic structures.

Posted by: John Baez on August 15, 2010 7:40 AM | Permalink | Reply to this

Post a New Comment