Thursday, October 13, 2005
Today's colloquium was given by Eric Adelberger from University of Washington on their experiments for small scale gravity. They are using an extremely sensitive torsional pendulum apparatus coupled gravitationally by two rotating discs. All of them have holes so that 1/r^2 effect of two discs are cancelled and there is only non standard effects left at the first order. There are many checks of course. For example they discovered micron level non-smoothness of the inner surfaces of the holes by gravitation before the microscope!!! I think this is enough reason to trust them. Their conclusion so far is that there is nothing new down to 0.1 mm. But things get interesting at that scale because that is the length scale corresponding to measured dark energy. And as rumored by Lubos some time ago there seems to be something interesting around there. Data they have so far is not clean enough but seems to eye that the observed force is a little bit less than the theory. Since it is at the limits of their instrument, you should not get excited right now. He thinks that they can have enough tests in a few months to figure out whether it is some systematics or real signal. It seems that they are well away from the theoretical limits of the instrument design and currently limited by how fine they can manufacture their parts. We can expect much more accuracy in the following few years. Fingers crossed and waiting.
National Geographic Wildcam
There is a live webcam in National Geographic's website. You can watch a small pond in the middle of nowhere in Botswana, Africa. Many wild animals come there to drink water, wash themselves, have some rest or cool down. I sometimes leave it on my second monitor. It not only is a nonintrusive calming scene and an interesting watch but also reminds you how "unnatural" is our "modern" life. Everybody I showed it told me that being an animal there must be so peaceful. Well, at least when the lions around are full. Give it a try, it might be a better break than reading this blog.
Monday, October 03, 2005
OpenLaws
Today I am starting a cyber-social experiment. In history, as the societies get bigger we had to move to representative democracy. Now with the modern ways of communication and tools of the cyber world we have new collaborative creation devices. I think the most unbelievable one is the wikis. I want to try if we can use to write laws for the society with a wiki open to everybody. I call it OpenLaws. Do I believe that it is going to work? Absolutely not! But I would say the same thing for Wikipedia. So let's try. Go ahead and write laws of your dream world. I won't say much, hope it will build itself beyond my imagination. Comments are welcome.
Friday, September 30, 2005
DEVONthink for Scientists: Part II
In a recent post I gave an introduction to DT. Now let me give some examples of how to use it.
Here is one of the last things I did on my personal database. I put a new quotation from Sean Carroll. It was a small paragraph including the sentence "Things are not intrinsically interesting, they are found to be interesting by people". DT immediately suggested a small clipping from David Deutsch including the sentence "In the long run, the distinction between what is interesting and what is boring is not a matter of subjective taste but an objective fact". This is nothing but immensely thought provoking and I would never remember it. On the scientific side, it helps to locate standard references. For example, when I start writing something about renormalization, it reminds me to look at Peskin Chapter 12 which has a summary inside the database. I hand picked my notes and archive and organized them into folders very well. Hence it may not be very surprising.
Let's look at some bulk databases I built. Joanna Karczmarek of Rutgers kindly supplied me the abstracts of all the hep-th papers until July 2005, that makes 37586 items, 110,720 unique and 4,749,379 total words. This is not only a huge database but also very disorganized. Abstracts are collected into folders according to submission months. This is probably the best thing you can do to confuse an contextual search engine. But still DT did quite a nice job. Here is an example:
I chose the paper hep-th/9907001 of Parikh and Wilczek called "Hawking Radiation as Tunneling". First thing one can notice is that suggestions are dominated by the papers of the authors. This makes sense because authors' surnames are the most characteristic words for each document. But even this is not done blindly. DT offers bunch of Parikh papers, who wrote about the same subject extensively but no other Wilczek papers. Figuring out the authors' related other papers are not very interesting or useful, because I can do it myself. So here are the top five papers from other authors: hep-th/0505266 (Hawking Radiation as Tunneling through the Quantum Horizon), hep-th/0110289 (Radiation via Tunneling in the Charged BTZ Black Hole), hep-th/0504188 (Tunneling through the quantum horizon), hep-th/0503081 (Hawking Radiation as Tunneling for Extremal and Rotating Black Holes), hep-th/0207247 (Radiation via Tunneling from a de Sitter Cosmological Horizon). It is nice to see that some of the above papers are actually citing Parikh and Wilczek. Also the last one is interesting, in a sense that it is related but I wouldn't expect a computer to recognize that.
Another example is the database I built from a snapshot of the open mathematics encyclopedia PlanetMath with 4570 entries, 31,007 unique and 3,053,620 total words. This is a nice one because titles are organized into subject folders. When you look at entry for the Einstein field equations, here are the top five related entries: Ricci tensor, Einstein summation convention, Schrodinger's wave equation, Hartman-Grobman theorem, linear time invariant systems. Relation of the forth one is not apparent to me, may be this is something I should learn. Here is a different one: If I take Hamilton equations, it gives symplectic manifold, Hamiltonian vector field, cotangent bundle, Examples of symplectic manifolds, Poincare 1-form. (Yeah, I am too lazy to put a link for each word.) I leave it to you to decide whether these are useful enough. May be not, but the main purpose of the program is to help your memory, and it does this very well.
Some people argue that the best thing to do is to throw everything inside a single database. Besides the performance issues, if you put all of the above inside your personal database, they will naturally dominate the suggestion results. So my advice is to go for the Pro version and experiment with many databases till you are satisfied with the combinations.
Here is my bottom line; DEVONthink may not be perfect (when compared to a human who does not forget) but there is nothing better or even something to compare out there.
I know that there some services which try to do the same with the whole internet. I tried blinkx and it gives awful results, nothing useful and I haven't tried Watson yet, it is Windows-only. Although, I think this is a beautiful idea and we will use and enjoy such tools soon. That will be the next Google.
Here is one of the last things I did on my personal database. I put a new quotation from Sean Carroll. It was a small paragraph including the sentence "Things are not intrinsically interesting, they are found to be interesting by people". DT immediately suggested a small clipping from David Deutsch including the sentence "In the long run, the distinction between what is interesting and what is boring is not a matter of subjective taste but an objective fact". This is nothing but immensely thought provoking and I would never remember it. On the scientific side, it helps to locate standard references. For example, when I start writing something about renormalization, it reminds me to look at Peskin Chapter 12 which has a summary inside the database. I hand picked my notes and archive and organized them into folders very well. Hence it may not be very surprising.
Let's look at some bulk databases I built. Joanna Karczmarek of Rutgers kindly supplied me the abstracts of all the hep-th papers until July 2005, that makes 37586 items, 110,720 unique and 4,749,379 total words. This is not only a huge database but also very disorganized. Abstracts are collected into folders according to submission months. This is probably the best thing you can do to confuse an contextual search engine. But still DT did quite a nice job. Here is an example:
I chose the paper hep-th/9907001 of Parikh and Wilczek called "Hawking Radiation as Tunneling". First thing one can notice is that suggestions are dominated by the papers of the authors. This makes sense because authors' surnames are the most characteristic words for each document. But even this is not done blindly. DT offers bunch of Parikh papers, who wrote about the same subject extensively but no other Wilczek papers. Figuring out the authors' related other papers are not very interesting or useful, because I can do it myself. So here are the top five papers from other authors: hep-th/0505266 (Hawking Radiation as Tunneling through the Quantum Horizon), hep-th/0110289 (Radiation via Tunneling in the Charged BTZ Black Hole), hep-th/0504188 (Tunneling through the quantum horizon), hep-th/0503081 (Hawking Radiation as Tunneling for Extremal and Rotating Black Holes), hep-th/0207247 (Radiation via Tunneling from a de Sitter Cosmological Horizon). It is nice to see that some of the above papers are actually citing Parikh and Wilczek. Also the last one is interesting, in a sense that it is related but I wouldn't expect a computer to recognize that.
Another example is the database I built from a snapshot of the open mathematics encyclopedia PlanetMath with 4570 entries, 31,007 unique and 3,053,620 total words. This is a nice one because titles are organized into subject folders. When you look at entry for the Einstein field equations, here are the top five related entries: Ricci tensor, Einstein summation convention, Schrodinger's wave equation, Hartman-Grobman theorem, linear time invariant systems. Relation of the forth one is not apparent to me, may be this is something I should learn. Here is a different one: If I take Hamilton equations, it gives symplectic manifold, Hamiltonian vector field, cotangent bundle, Examples of symplectic manifolds, Poincare 1-form. (Yeah, I am too lazy to put a link for each word.) I leave it to you to decide whether these are useful enough. May be not, but the main purpose of the program is to help your memory, and it does this very well.
Some people argue that the best thing to do is to throw everything inside a single database. Besides the performance issues, if you put all of the above inside your personal database, they will naturally dominate the suggestion results. So my advice is to go for the Pro version and experiment with many databases till you are satisfied with the combinations.
Here is my bottom line; DEVONthink may not be perfect (when compared to a human who does not forget) but there is nothing better or even something to compare out there.
I know that there some services which try to do the same with the whole internet. I tried blinkx and it gives awful results, nothing useful and I haven't tried Watson yet, it is Windows-only. Although, I think this is a beautiful idea and we will use and enjoy such tools soon. That will be the next Google.
The Efficient Academic
There is a new online discussion group for productivity in scholarly work. It is called The Efficient Academic. I don't know why, but my impression is that people in the academia is much more disorganized than the "business" people. We try to cover it as a source for creativity, but is it really a legitimate excuse? Is there anything we should learn from the business self-help books (like the excellent GTD)? It is nice to hear others' experiences, solutions and tools.
Sunday, September 25, 2005
Twenty-five Questions for String Theorists
Gordon Kane and his collaborators have just published a review paper on the phenomenological aspects of string theory: hep-th/0509157. If you are an outsider to string theory (like me), you may wonder what are these people are really working on. String theory is advertised as a very beautiful solution to the unification of physical laws (or the question to the answer 42). But if they established such a theory what are those thousands of strings theorists trying to figure out? One answer is to find the M-theory, the non-perturbative completion of the string theory. The other answer is to connect proposed Planck scale physics to our experimental capabilities. This new paper is a compilation of 25 open questions in this second category and also it is a good resource for outsiders to understand the cutting-edge issues and where the problems may lie. I just wonder what percent of string theory is along these lines.
Saturday, September 24, 2005
Extracting numbers from AdS/QCD
This Wednesday's seminar at CTP was given by Emanuel Katz of Boston University (see him here). He talked about his recent work on AdS/QCD correspondence, published in hep-ph/0501128. The main program here is to find a 5D string theory such that QCD is its induced theory on the 4D boundary of this bulk. So far nobody could came up with such a theory. QCD has a very rich structure with asymptotic freedom, chiral symmetry breaking and many particle fields. Unless you came up with the right matter content and exact SU(3) symmetry there is no hope that you can have reasonable calculations for low energy. (Low energy is interesting because high energy QCD is easy to due to asymptotic freedom.)
Katz and his collaborators took a simpler and different approach. Instead of finding a full fundamental theory, they tried to find a few relevant terms in the bulk, fix their coefficients by QCD data and compute other data and see if there is anything matches (which is not very likely a priori). They took an AdS space with a boundary where we live. Fifth dimension in the bulk, corresponds to the energy scale of the interactions on the boundary. As you move away from the boundary energy scale gets smaller (length scale gets larger). So if you want to study confinement you need to study the region near the boundary. In their model this is done by putting another brane parallel to the boundary as an infrared cut-off and study the region in between. Physics is imposed by the boundary conditions on the 'infrared brane'. There are four free parameters in the Lagrangian which are set by number of colors and three experimental values like the rho meson mass, the pion mass, and the pion decay constant. Surprising thing is after building this simple but very unrealistic model they can calculate many other experimental parameters very accurately (within %10 in most of the cases). He showed additional results with strange quark in the talk and even they are still very good (within %30). Actually this is better than the first versions of computationally intensive lattice-QCD.
I always thought that even if they find a dual theory for QCD, it would be much more complicated and will not be practical for anything beyond intellectual joy. But this work made me step back and take a second look. May be there is something here.
Even more important question is whether it has a fundamental meaning or not. Spring-mass system and an LC circuit have dual theories (actually same theories with different coordinate names). But it does not mean that one is made out of the other. Are we really living on a brane and is standard model just a projection of a (5+n)D theory? Probably I will never learn the answer.
Update: I just saw that Jacques Distler has a slightly more technical review of this paper.
Katz and his collaborators took a simpler and different approach. Instead of finding a full fundamental theory, they tried to find a few relevant terms in the bulk, fix their coefficients by QCD data and compute other data and see if there is anything matches (which is not very likely a priori). They took an AdS space with a boundary where we live. Fifth dimension in the bulk, corresponds to the energy scale of the interactions on the boundary. As you move away from the boundary energy scale gets smaller (length scale gets larger). So if you want to study confinement you need to study the region near the boundary. In their model this is done by putting another brane parallel to the boundary as an infrared cut-off and study the region in between. Physics is imposed by the boundary conditions on the 'infrared brane'. There are four free parameters in the Lagrangian which are set by number of colors and three experimental values like the rho meson mass, the pion mass, and the pion decay constant. Surprising thing is after building this simple but very unrealistic model they can calculate many other experimental parameters very accurately (within %10 in most of the cases). He showed additional results with strange quark in the talk and even they are still very good (within %30). Actually this is better than the first versions of computationally intensive lattice-QCD.
I always thought that even if they find a dual theory for QCD, it would be much more complicated and will not be practical for anything beyond intellectual joy. But this work made me step back and take a second look. May be there is something here.
Even more important question is whether it has a fundamental meaning or not. Spring-mass system and an LC circuit have dual theories (actually same theories with different coordinate names). But it does not mean that one is made out of the other. Are we really living on a brane and is standard model just a projection of a (5+n)D theory? Probably I will never learn the answer.
Update: I just saw that Jacques Distler has a slightly more technical review of this paper.
Friday, September 23, 2005
Notes on making science
Read this and replace every "art" with "science". It surprisingly makes sense.
Update: Another short piece I would recommend along these lines is some "Frank" advice on making right choices in life. Does anybody have a proof that the given algorithm is the best?
Update: Another short piece I would recommend along these lines is some "Frank" advice on making right choices in life. Does anybody have a proof that the given algorithm is the best?
Tuesday, September 20, 2005
DEVONthink for Scientists: Part I
Managing information is a very important part of a modern scientists life. Lab notebooks, ideas scribbled on papers, proceedings, papers to be written, papers written 10 years ago, papers to be read, papers to be refereed and of course physics blogs. Many people have many different ways to cope with it. Some are very tidy with classified hanging folders and highly maintained list of bibliographies and some are very messy. I personally hate bunch of papers flying around filling desk and shelf space. I can never find things there. What can I do, I was using Yahoo before reading my first paper! So for the last few years I was in a search for effective collecting techniques for information. My dear idea notebook served me quite a while but it was just better than nothing. I couldn't easily cross reference and it didn't integrate into paper folders. But finally I feel like the issue is settled for quite a long while. Today I will talk about my final decision: DEVONthink.
DT is a (Mac-only) database management software for every kind of electronic document. It lately became well-known (read fashionable) thanks to Steven Johnson's NYTimes article and following blogs (here and here). Probably journalists and writers are the primary market for this software but I believe it is an invaluable tool also for scientists. So I'll try to give review (or a praise) of it for you.
Structure of the database for your documents in DT, looks like the good old file structure of your hard drive with folders and subfolders. You put in any kind of document (like txt, rtf, pdf, html, tex as well as image and quicktime files and yes doc!) inside and classify in a dedicated place where you can search inside the documents, edit and give links (with wiki capabilities) to each other. Up to this point you have many alternatives that you can do the same. But DT shines now on with its unmatched, intelligent text indexing abilities.
DT looks (at least) two things in your database. Which words go with which words inside the documents you chose and which files go which files inside the folders you classified. Using these data it decides which are the words that is characteristic to a file or a folder. It uses this information for the two magic buttons. First one is "Classify". When you put a new document "Classify" suggests you where to put it in your database. I "usually" know where I want to put it but just for fun I always click and see with joy that my folder is among the top suggestions and it really helps when I am undecided. Second one is "See Also". This button is like your personal librarian. It suggests documents related to the current one. After a while it becomes a common thing that DT reminds you little notes written down from a book or an abstract from an ArXiv read months ago and already forgotten. I hadn't recognized how much I forget, until using this and I am 24.
Both functions start to give reasonable results after tens of documents placed in a few folders and get better and better as your structure get bigger and better organized into a few levels of subfolders. I am very satisfied with the speed. It works seamlessly on my 1.5GHz, 512MB Powerbook with my primary database which has about 1.5K documents with 75K unique and 1.7M total words. (There are so much unique words because I have Turkish documents as well in the database. I am planning to move them into a separate database later. And I should say that it works also perfect in Turkish. It (statistically significant) sometimes guesses the (folder) name of columnists from just the text!)
You can also ask for related documents for just the selected part of the text which is very useful. But unfortunately it can not do the reverse yet. I mean, it can not suggest a certain paragraph of a document. So putting into too long pieces (like books) probably won't be very useful (may be just for search). Steven johnson suggest that the ideal length is 50-500 words, but I think physics papers are OK in length. Getting a relevant paper is enough most of the time, you can figure out where to read (which is the conclusion).
What I put in? Notes for my ideas, notes from books and papers I read, daily musings, abstracts (sometimes whole papers), blogs, news, columns, chapter summaries for Sakurai and Peskin, This Week's Findings in Mathematical Physics archives, theorems from Wikipedia etc etc etc.
One more unique feature is the fuzzy search. When you turn it on it not only searches for the words you supplied but also "related" words and words that look like yours with some suffixes and misspellings.
What else I like about it?
* There is a full screen edit mode. It takes you away from all the distractions of internet. Just you and your thoughts. I write everything here now. Plain texts can be seen green on black background which is very easy on the eye.
*If you like adding equations (preferably LaTeX) to your notes (like any physicist). You can use universal RTF format and Equation Service to put inline LaTeX created mini PDF formulas. You can go back and forth LaTeX and PDF as many times as you like. Mac services menu makes this process just a single key combination. (I use cmd+.)
* Integrates perfectly with your Safari. Again thanks to services menu any selection goes to your database with single key combination (I use cmd+[ for txt and cmd+] for rtf). There is also a browser in DT good for html capture and site sucker for whole site downloads.
* You can read your RSS and Atom feeds and save items as separate files with a script. There is a great Apple script support if you know, if not there is big library of them.
* You can have replicates (synchronized mirrors) of files in different folders.
* Exporting files as you put them in. This is very important if you want to move somewhere else one day. For example you can't do it with OneNote.
* They have a fast tech-support and lively forums with lots of writers, journalists, students and academicians.
* And many other things I haven't yet used so far; like index, summarize, concordance, searchable file comments etc.
DT comes in two flavors Personal (single database, no site sucker, no RSS etc) for $40 and Pro (everything above) for $75. There is %15 discount for students (yes grads also). You can try it for 150 using hours for free.
I hope I have convinced that you have never seen a software like this before, highly recommended. I wished there were some competition.
In the second part of my review (praise) I will give additional examples for using it as a scientist. Please share your experience in comments if you are a DT user.
UPDATE: Here is the second part.
DT is a (Mac-only) database management software for every kind of electronic document. It lately became well-known (read fashionable) thanks to Steven Johnson's NYTimes article and following blogs (here and here). Probably journalists and writers are the primary market for this software but I believe it is an invaluable tool also for scientists. So I'll try to give review (or a praise) of it for you.
Structure of the database for your documents in DT, looks like the good old file structure of your hard drive with folders and subfolders. You put in any kind of document (like txt, rtf, pdf, html, tex as well as image and quicktime files and yes doc!) inside and classify in a dedicated place where you can search inside the documents, edit and give links (with wiki capabilities) to each other. Up to this point you have many alternatives that you can do the same. But DT shines now on with its unmatched, intelligent text indexing abilities.
DT looks (at least) two things in your database. Which words go with which words inside the documents you chose and which files go which files inside the folders you classified. Using these data it decides which are the words that is characteristic to a file or a folder. It uses this information for the two magic buttons. First one is "Classify". When you put a new document "Classify" suggests you where to put it in your database. I "usually" know where I want to put it but just for fun I always click and see with joy that my folder is among the top suggestions and it really helps when I am undecided. Second one is "See Also". This button is like your personal librarian. It suggests documents related to the current one. After a while it becomes a common thing that DT reminds you little notes written down from a book or an abstract from an ArXiv read months ago and already forgotten. I hadn't recognized how much I forget, until using this and I am 24.
Both functions start to give reasonable results after tens of documents placed in a few folders and get better and better as your structure get bigger and better organized into a few levels of subfolders. I am very satisfied with the speed. It works seamlessly on my 1.5GHz, 512MB Powerbook with my primary database which has about 1.5K documents with 75K unique and 1.7M total words. (There are so much unique words because I have Turkish documents as well in the database. I am planning to move them into a separate database later. And I should say that it works also perfect in Turkish. It (statistically significant) sometimes guesses the (folder) name of columnists from just the text!)
You can also ask for related documents for just the selected part of the text which is very useful. But unfortunately it can not do the reverse yet. I mean, it can not suggest a certain paragraph of a document. So putting into too long pieces (like books) probably won't be very useful (may be just for search). Steven johnson suggest that the ideal length is 50-500 words, but I think physics papers are OK in length. Getting a relevant paper is enough most of the time, you can figure out where to read (which is the conclusion).
What I put in? Notes for my ideas, notes from books and papers I read, daily musings, abstracts (sometimes whole papers), blogs, news, columns, chapter summaries for Sakurai and Peskin, This Week's Findings in Mathematical Physics archives, theorems from Wikipedia etc etc etc.
One more unique feature is the fuzzy search. When you turn it on it not only searches for the words you supplied but also "related" words and words that look like yours with some suffixes and misspellings.
What else I like about it?
* There is a full screen edit mode. It takes you away from all the distractions of internet. Just you and your thoughts. I write everything here now. Plain texts can be seen green on black background which is very easy on the eye.
*If you like adding equations (preferably LaTeX) to your notes (like any physicist). You can use universal RTF format and Equation Service to put inline LaTeX created mini PDF formulas. You can go back and forth LaTeX and PDF as many times as you like. Mac services menu makes this process just a single key combination. (I use cmd+.)
* Integrates perfectly with your Safari. Again thanks to services menu any selection goes to your database with single key combination (I use cmd+[ for txt and cmd+] for rtf). There is also a browser in DT good for html capture and site sucker for whole site downloads.
* You can read your RSS and Atom feeds and save items as separate files with a script. There is a great Apple script support if you know, if not there is big library of them.
* You can have replicates (synchronized mirrors) of files in different folders.
* Exporting files as you put them in. This is very important if you want to move somewhere else one day. For example you can't do it with OneNote.
* They have a fast tech-support and lively forums with lots of writers, journalists, students and academicians.
* And many other things I haven't yet used so far; like index, summarize, concordance, searchable file comments etc.
DT comes in two flavors Personal (single database, no site sucker, no RSS etc) for $40 and Pro (everything above) for $75. There is %15 discount for students (yes grads also). You can try it for 150 using hours for free.
I hope I have convinced that you have never seen a software like this before, highly recommended. I wished there were some competition.
In the second part of my review (praise) I will give additional examples for using it as a scientist. Please share your experience in comments if you are a DT user.
UPDATE: Here is the second part.
Friday, September 16, 2005
Three dimensions from strings, or may be seven...
First we thought that strings live in 26 dimensions. Then it turned out that it is 10. Now string theory says that we should either be in a 3 or a 7 dimensional brane. That's what Karch and Randall's analysis claim in hep-th/0506053. You can also read the recent AIP news article here.
The argument is that if the universe (of course the 10D one) is initially filled with branes of all dimensions and sizes, eventually only 3 and 7 dimensional ones will survive and dominate the universe. So the dimension of our world is not chosen by landscape or some fundamental principle, but by the cosmological dynamics. I can't go into the details here, but it is interesting that branes with dimensions have different reasons to dilute away.
I am certainly not an expert, but the arguments here seems to be very basic. If they survive it might be a major milestone in string theory. All we need is an anthropic principle to choose 3D ones ;) Actually these things are already in the market. Structures will not be stable in universes with 7 space and 1 time dimensions. But Karch and Randall suggest that if the expansion of the universe is dominated by something other than branes, there is quite a good chance that 3D ones dominate 7D ones anyway.
It looks interesting, I'll keep an eye on these.
The argument is that if the universe (of course the 10D one) is initially filled with branes of all dimensions and sizes, eventually only 3 and 7 dimensional ones will survive and dominate the universe. So the dimension of our world is not chosen by landscape or some fundamental principle, but by the cosmological dynamics. I can't go into the details here, but it is interesting that branes with dimensions have different reasons to dilute away.
I am certainly not an expert, but the arguments here seems to be very basic. If they survive it might be a major milestone in string theory. All we need is an anthropic principle to choose 3D ones ;) Actually these things are already in the market. Structures will not be stable in universes with 7 space and 1 time dimensions. But Karch and Randall suggest that if the expansion of the universe is dominated by something other than branes, there is quite a good chance that 3D ones dominate 7D ones anyway.
It looks interesting, I'll keep an eye on these.







