I actually was thinking of "what city would I like to live in". :-) You are correct that not all "capita"s are equally relevant and probably a total of grad students plus professors is a better denominator. Less refined census data than that is easier to come by, though. I do think that bigger cities support more schools and/or bigger graduate departments, other things being equal. So, in a vague statistical correlation sense just bulk census data gets you part of the way there.

If the real question is "what university would I like to be near" then a city is also the wrong aggregation unit, so not only the normalization but also the aggregation should change. I believe per university/per student or per professor/group output is what most academics would like to know for bragging rights or even funding priority reasons, but they usually make such evaluations themselves on a per department basis.

If you read the paper or click on the maps you will actually see that they DO NOT CORRECT for local population density. So, the metric in question is absolute rather than "per capita" productivity. This doesn't entirely invalidate it, but it calls into question how you would verbalize or interpret the results.

I mean, if 8 of the top 10 cities for science *by any metric* are also 8 of the top 10 cities by population you have said something less interesting. These cities are already top cities for "being" at all. :-)

It would be far more interesting to normalize in a per capita sense. There are clearly some major outliers in that sense scrolling around on the map. Vancouver lept out at me, but I'm sure others could find them as well. Now, wouldn't it be nice if the fancy visualization researchers helped us along in that task? :-)

I don't believe you understand the situation or my argument. It's not nearly as strong as P=NP or not. There could be non-hypergeometric family simplifications that do better than the Dai linear sums, and there can be other numerical methods that also do better (and there are for some kinds of options). This new paper just shows that one possible approach to simplify a formula won't work - a formula-to-be-improved already only compelling because Dai et al compare it to naive, strawmen alternatives they found in a textbook, not actual competitive methods available in the options pricing literature. So, it's not a paper to be rejected, but it really doesn't change the world much. It says, "don't look *here* for a way to simplify that other non-optimal approach".

The naive CRR (Cox, Ross, Rubinstein) method for pricing options is O(n^2) where n is the number of levels in a recombinant binomial pricing lattice. That is, a lattice like a binary tree, but where you have cross links connecting nodes. The naive approach requires visiting each one of these nodes and hence O(n^2) and the error of the produced option goes down only proportional to the node spacing. For at least 15 years this problem has been converted to "linear time" (really the important relation is between the price error and the CPU time) by means of a variety of extrapolation methods (this began with Richardson extrapolation) using evaluation with two trees to get a much smaller error. There are in fact numerical methods that for special options can do slightly better than this. Broadie 1996 is one reference. While pretty fast and very easy to understand, there are yet faster methods using adaptive mesh crank-nicolson PDE solvers that do a bit better. Just a couple of years ago, Dai, et al. published a paper showing how to get linear time an entirely different approach involving combinatorial sums. This may have improved performance bounds for some exotic options, but did NOT do much for improving real-world implemented algorithmic performance of pricing the European and American options that are so commonly traded on exchanges, in the US and worldwide. So, at least for the most important class of options Dai et al was kind of a snoozer. The paper referenced in the summary above is entirely a follow-up paper to Dai, et al 2008. This new paper merely shows that there is no "short cut" in evaluating the relevant sums with hypergeometric functions, a kind of special function common in mathematical physics. So, in short, all this says is that the already "non fastest method" cannot be made faster by one numerical methods approach. It is certainly deserving of publication and dissemination, but changes the world not at all.

For some critical discussion of the "productivity", this recent thread might also be of interest. In the article in question Bjarne claims credit (dubiously IMO) for saving 'years of development time' on any complex project [ Google, DNA matching, etc. ] where people happened to use C++ instead of some alternative. http://www.reddit.com/r/programming/comments/cdncx/linus_about_c_productivity_again/?utm_source=web&utm_medium=twitter

Yes, if Hawking's idea about black hole radiation is true, all gravity fields should radiate. Without even walking across the room you create thermal radiation at a fantastically small temperature no matter how small your ass is, just by virtue of your tiny gravitational field. There is no "sufficiently large gravity well" to generate the radiation, only sufficiently large to generate *measurable* radiation.

In the case of black holes, the radiation of stellar or galactic mass singularities is absolutely miniscule. Evaporation is only a "noticeable" process for very tiny black holes with the mass of an asteriod packed into the space of a proton.

As for what you can or cannot picture, that is your issue. I am just letting you know the basic phenomenon is much more broad and actually much more fundamental than a black hole event horizon membrane. The membrane and virtual pairs may begin (but not end) arguments and derivations or motivate theoretical preferences for resolving various issues, but it is misleading to call that imagined scenario the essence of the process. Physics teaching often suffers from "historical bias". Because some physicist first imagined things a particular way or convinced his peers a particular way, this is often the path used to motivate things to a popular audience. The truth is that after some thought and generalization it may be much less sensitive to the original motivating visual picture.

Virtual particles are really terms in a perturbation expansion that in some respects are terms similar to real particles and in other respects are not. For example, traveling backward in time is something they get to do, having negative energy, and so on. What they can and cannot do and why is context dependent and relies upon the actual formal derivations and properties of what is going on. So, as a "reasoning device" they fail most laymen, and in my opinion are not very intuition building.

So, to answer (1), yes -- just an analogy. (2) would be correct if the answer to (1) were "no", but it isn't. :-)

As I've referred to above, "capture" and "escape" of "virtual" particles is all a bunch of highly specific visualization related to a black hole or event horizon, but the actual result pertains to all accelerating reference frames and all spacetime curvature. Though Hawking himself might disagree with me, I find it pedagogically misleading to "explain" the possibility of this thermal radition in terms of processes only happening at a literal even horizon.

This is actually an interesting case of the strong principle of equivalence -- that gravity is locally indistinguishable from an accelerating frame of reference for all physical processes. (The weak principle of equiv is only about graviational forces, but the quantum vacuum is broader physics than that.) Specifically, you can derive Unruh radiation from quantum vacuum transformations *or* you get the same numerical temperature as starting from the idea that an accerelating reference frame "event horizon" is the same as a gravitational event horizon. I derived that latter in high school in the mid 80s, actually, to prove to myself that strong P of E held in this case. It's a relatively easy exercise in hyperbolic functions and basic calculus to compute the asymptotic trajectory of uniformly accelerating frame and back out the effective accerelation event horizon. Plug that in to Hawking's formula for a black hole and you get Unruh's result for acceleration. (They really call it Fulling-Davies-Unruh since it was done three times independently after the Hawking-Bekenstein results.)

I would agree with another responder here that not mentioning the thermal character of the radiation and words suggesting its monochromicity makes this particular result a little dubious, but I have not read the arXive article.

The process need not actually be distributed over space -- the escaping particle travels, yes, but the actual energy conversion happens when and where the escaping is first created.

Now, its creation is a quantum state transition which has a "magical" quality in the same way that, say, a photon escaping an atom's electron shell does. There is no extended energy transport process at all. The electron makes a quantum jump simultaneously with the photon field of the world gaining a new photon traveling away. Indeed, with visible light, the wavelength of the photon -- hundreds of nanometers -- can easily exceed the spatial scale of the atoms electron shell, usually a few nm. So, the photon kind of just "appears".

The responder has it right. You are missing that the virtual pair has no net energy initially and one escapes. So, the outside world is getting heavier and the black hole lighter - to conserve total system energy. You are thinking of the "virtual" counterpart as having mass, but it does not. It's "virtual".

As I mentioned above, one does not need a black hole for this -- all curved space should release thermal energy, though the rate is usually immeasurably small. Google Unruh effect and read about it in relation to the Sokolov-Ternov effect which has been observed since the 1970s. There is not perfect interpretational consensus about all this, though.

Popular visualizations and even the notion of "virtual particles" do not allow very accurate reasoning with regards to Hawking radiation. In particular, the "promotion process" from "virtual" to "real" is just a crutch for proving something to all orders in perturbation theory. Shortly after Hawking-Bekenstein, Bill Unruh proved that simply being in a uniformly accelerated reference frame creates a perception of thermal background radiation coming from the background -- at a temperature equivalent to the pseudo-event horizon of the acceleration for the duration of the acceleration. You see, while if you move at a constant rate any photon will catch you just as quickly as if you were standing still (basic special relativity) if instead you accelerate forever, you asymptotically approach the speed of light, but there are photons far enough behind you that will never catch you. How far behind they need to be depends on how fast you are accelerating. So, every acceleration corresponds to a pseudo event horizon. As soon as one stops accelerating the photons can catch up to you. Unruh's result does *not* depend on the permanence of the horizon, but works for temporary accelerations. So, the horizon does not need to be "permanent" for the "promotion" to occur. A better way to think about Hawking radiation is any gravitation field (any curved space, that is) decaying via thermal radiation, or space itself providing some "resistance to acceleration" or intrinsic acceleration-only viscosity where the energy taken away from the acceleration is converted to thermal radiation. The image of a virtual pair around an event horizon is not, ultimately, how the result holds or is proven or even what the process is "about". It's more like an "inspiration to a derivation" than something to be taken so literally.