On the other hand, I seem to remember that massless objects (like photons) are *always* travelling at c in some direction. This leads to some oddities, but none that can't be resolved in some way.
Massless objects which don't travel at c would have zero momentum-energy, and would not have many ways of interacting with massive stuff in an observable fashion. They may "exist" in some philosophical sense, but there's no reason for physicists to care.
Any one searching for some of Voevodsky's work should look for his name in the UIUC K-theory preprint archive. This paper is a good introduction to his homotopy theory, and if you have access to a research library, you may find a book he recently wrote with Suslin and Friedlander, "Cycles, Transfers, and Motivic Homology."
I don't see why the donut-coffee cup description of topology upsets people so much. It's just an old joke. The Berkeley math dept. sells (or used to sell) mugs that had a yellow sign reading, "Caution! Not a donut." Perhaps the sensitive mathematicians would be appeased if we used other functionally equivalent analogies, like, "Topology tells you that you can take off your shirt without removing your jacket," or, "you don't really need to put on your socks before your shoes."
I hope you're aware that when scientists refer to the Earth's poles changing, they mean the north-south orientation changes (that is, compasses start pointing south, then go back to pointing north) and not that the actual Earth did flip-flops.
This is not necessarily true. Since the Earth is not a rigid body, it is possible for the mass distribution to shift over geological time periods. Substantial evidence for historical changes in the Earth's axis of rotation has been accumulated, although as a non-geologist, I cannot say whether it is widely accepted. You may want to search for True Polar Wander for more detailed information. This phenomenon is quite different from the magnetic field shifts to which you seem to be referring.
Okay, I see why your puzzle is related to the discrete log problem. You're looking for the order of 2 in (Z/(n+1)Z)* with the shuffle you described, or (Z/(n-1)Z)* with the shuffle you meant to describe. It doesn't seem to work too well with odd n.
the discrete log problem is specifically, given integers y, g, p, find a (preferably minimal)solution x to the problem
y = g^x mod p, 0 = y p
g really ought to be a generator of (Z/pZ)* if you want the discrete log to be well-defined.
if you shuffle a 52 card deck perfectly 7 times (divide the deck exactly in half, always have the top half drop the first card, drop exactly one card after another) then you end up with the original order of the deck.
Wrong. The shuffle you describe produces a 52-cycle in S_52. In particular, you have to repeat it 52 times to get the original order. However, if the bottom half drops the first card, you get 2 fixed points, a transposition, and 6 8-cycles, meaning 8 shuffles are required.
I don't see many patterns that are immediately obvious for n cards, except for the 2m-periodicity when n=2^m. Is there a general solution?
Even the management and bio majors could kick your ass in diferential equations, so no "stupid drunk frat boys" comments.
Those stupid drunk frat boys...
Ow, my ass.
On-topic bit: It is interesting that you back up your criticism of the statistical methods used in the article by citing anecdotal evidence of Windows users' cluelessness. I think it unlikely that the members of your fraternity made up a non-trivial sample space. Despite its flaws, the article made the point that Linux is increasingly perceived as a viable target, and that as its acceptance as a server platform increases, the likelihood that a given installation is vulnerable seems to rise.
In fact one volcanic eruption in south america in the early 90's spewed about 10 times as much chlorine into the upper atmosphere as all the industrial chemicals user in human history!
I believe you are referring to Mount Pinatubo, in the Philippines. See this link for a discussion of volcanic chlorine emissions, in particular, why CFC's may have a stronger effect on stratospheric ozone. I'd like to see a source for your figure of 2 orders of magnitude.
Certain economists had a running joke about classifying the Philippines as a Latin American country during the Asian economic boom of the 80's and early 90's, and as an Asian country during the Latin American boom (and the corresponding Asian crisis) a few years ago. Given the recent financial situation in Argentina, your geographic mistake is understandable.
Li(x) is defined as the integral from 0 to x of (1/t) dt. And apparently the number of primes below x (usually denoted pi(x)) is pretty well approximated by Li(x).
Not quite: note the obvious initial logarithmic divergence. Informally, you can just change the integrand from what you had to dt/(log t), but you really ought to work around the singularity at 1. Some people change the bounds of integration to start at 2 to avoid this. It simply shifts the function by a small constant (about 1.05)
I'm a bit surprised no one here has mentioned Pierre Deligne's 1974 proof of the Weil conjectures, in particular the analogue of RH for smooth projective varieties over finite fields (for which he was awarded a Fields medal in 1978). This is perhaps the strongest "evidence" for the original hypothesis (unless you find the brute force calculation convincing), and it has other interesting consequences, for example the resolution of Ramanujan's tau conjecture (ref: Hartshorne's Algebraic Geometry).
There is a nice discussion of potential avenues of attack on the Riemann Hypothesis at the end of chapter 5 in Patterson's text on the Zeta Function (Cambridge Studies 14), including some vague ideas on why a purely analytic strategy is not likely to be successful.
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On the other hand, I seem to remember that massless objects (like photons) are *always* travelling at c in some direction. This leads to some oddities, but none that can't be resolved in some way.
Massless objects which don't travel at c would have zero momentum-energy, and would not have many ways of interacting with massive stuff in an observable fashion. They may "exist" in some philosophical sense, but there's no reason for physicists to care.
Any one searching for some of Voevodsky's work should look for his name in the UIUC K-theory preprint archive. This paper is a good introduction to his homotopy theory, and if you have access to a research library, you may find a book he recently wrote with Suslin and Friedlander, "Cycles, Transfers, and Motivic Homology."
I don't see why the donut-coffee cup description of topology upsets people so much. It's just an old joke. The Berkeley math dept. sells (or used to sell) mugs that had a yellow sign reading, "Caution! Not a donut." Perhaps the sensitive mathematicians would be appeased if we used other functionally equivalent analogies, like, "Topology tells you that you can take off your shirt without removing your jacket," or, "you don't really need to put on your socks before your shoes."
I hope you're aware that when scientists refer to the Earth's poles changing, they mean the north-south orientation changes (that is, compasses start pointing south, then go back to pointing north) and not that the actual Earth did flip-flops.
This is not necessarily true. Since the Earth is not a rigid body, it is possible for the mass distribution to shift over geological time periods. Substantial evidence for historical changes in the Earth's axis of rotation has been accumulated, although as a non-geologist, I cannot say whether it is widely accepted. You may want to search for True Polar Wander for more detailed information. This phenomenon is quite different from the magnetic field shifts to which you seem to be referring.
Okay, I see why your puzzle is related to the discrete log problem. You're looking for the order of 2 in (Z/(n+1)Z)* with the shuffle you described, or (Z/(n-1)Z)* with the shuffle you meant to describe. It doesn't seem to work too well with odd n.
the discrete log problem is specifically, given integers y, g, p, find a (preferably minimal)solution x to the problem
y = g^x mod p, 0 = y p
g really ought to be a generator of (Z/pZ)* if you want the discrete log to be well-defined.
if you shuffle a 52 card deck perfectly 7 times (divide the deck exactly in half, always have the top half drop the first card, drop exactly one card after another) then you end up with the original order of the deck.
Wrong. The shuffle you describe produces a 52-cycle in S_52. In particular, you have to repeat it 52 times to get the original order. However, if the bottom half drops the first card, you get 2 fixed points, a transposition, and 6 8-cycles, meaning 8 shuffles are required.
I don't see many patterns that are immediately obvious for n cards, except for the 2m-periodicity when n=2^m. Is there a general solution?
Even the management and bio majors could kick your ass in diferential equations, so no "stupid drunk frat boys" comments.
Those stupid drunk frat boys...
Ow, my ass.
On-topic bit: It is interesting that you back up your criticism of the statistical methods used in the article by citing anecdotal evidence of Windows users' cluelessness. I think it unlikely that the members of your fraternity made up a non-trivial sample space. Despite its flaws, the article made the point that Linux is increasingly perceived as a viable target, and that as its acceptance as a server platform increases, the likelihood that a given installation is vulnerable seems to rise.
In fact one volcanic eruption in south america in the early 90's spewed about 10 times as much chlorine into the upper atmosphere as all the industrial chemicals user in human history!
I believe you are referring to Mount Pinatubo, in the Philippines. See this link for a discussion of volcanic chlorine emissions, in particular, why CFC's may have a stronger effect on stratospheric ozone. I'd like to see a source for your figure of 2 orders of magnitude.
Certain economists had a running joke about classifying the Philippines as a Latin American country during the Asian economic boom of the 80's and early 90's, and as an Asian country during the Latin American boom (and the corresponding Asian crisis) a few years ago. Given the recent financial situation in Argentina, your geographic mistake is understandable.
Li(x) is defined as the integral from 0 to x of (1/t) dt. And apparently the number of primes below x (usually denoted pi(x)) is pretty well approximated by Li(x).
Not quite: note the obvious initial logarithmic divergence. Informally, you can just change the integrand from what you had to dt/(log t), but you really ought to work around the singularity at 1. Some people change the bounds of integration to start at 2 to avoid this. It simply shifts the function by a small constant (about 1.05)
I'm a bit surprised no one here has mentioned Pierre Deligne's 1974 proof of the Weil conjectures, in particular the analogue of RH for smooth projective varieties over finite fields (for which he was awarded a Fields medal in 1978). This is perhaps the strongest "evidence" for the original hypothesis (unless you find the brute force calculation convincing), and it has other interesting consequences, for example the resolution of Ramanujan's tau conjecture (ref: Hartshorne's Algebraic Geometry).
There is a nice discussion of potential avenues of attack on the Riemann Hypothesis at the end of chapter 5 in Patterson's text on the Zeta Function (Cambridge Studies 14), including some vague ideas on why a purely analytic strategy is not likely to be successful.