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(Incidentally, if you're in the SF Bay Area you can play with triangular rollers at the Exploratorium. Amazingly they give a nice smooth ride.)

IAAP (I am a physicist). I do not work with lasers, but have taken a graduate level course in non-linear optics that primarily focused on lasers.

It is quite possible to damage a pilot's eyes at a range of a few miles, using only commercial laser systems. If done by competent individuals, it would probably involve a pulsed infrared laser (harder to detect, and the eye is more susceptible to near IR than to visible). A Nd:YAG laser (1064 nm) would be ideal.

Since a pulsed laser is used, there's no need for tracking the plane. A single 10-nanosecond pulse would be sufficient. At 10 - 20 pulses per second, you could just scan the sky in the area of the plane.

After reading the story, I did some rough calculations. For the above-mentioned laser, the laser beam would do damage (although likely not sufficient to totally blind the pilot) at ranges of up to two miles, and the beam would have a spot size several meters in diameter at that range. Obviously, with additional optics, range and spot size could be changed.

It seems to me that the laser could simply be mounted to a scope on a tripod (after some careful alignment), and that targetting by hand would work at least some of the time.

All this aside, I don't think the recent cases are anything to be worried about. More likely it's just a nutbar with a relatively weak visible laser (I assume the laser was in the visible range because the pilots reported it, and I doubt commercial planes are equipped to detect IR lasers). If it was someone serious, they'd be using IR lasers, and we wouldn't know until pilots started getting eye damage.

That said, the overall risk of plane crashes from this form of attack is low. If the airport and immediate area are kept secure (and they should be if only to guard against Stinger-style missile attacks), it's very unlikely someone with a commercial laser could get close enough to completely blind a pilot. Military or custom-built research lasers could blind from greater distances, but such systems are very finicky, and I can't see terrorists pulling that off.

Finally, I'd like to address a few points other people have brought up. If the polarization and angle of the beam are chosen correctly, virtually none will be reflected off the plane's window, and all will be transmitted (see Brewster's Angle). For modest laser powers, the damage to the retina will be localized to where the laser beam is imaged, leaving much of the pilot's vision intact. Bad for the pilot, but he could probably still land. For more intense beams, other damage mechanisms come into play (apparently for severe cases there is an actual popping sound perceived by the victim as the laser pulse creates a small shock wave inside the eye), and more of the victim's vision could be damaged.

Protective goggles aren't really an option, as they only protect against one wavelength. Attackers could then switch to a different type of laser (Ti:saph?). Combining goggles leads to virtually no light getting through.

References
Journal of Biomedical Optics 4(3), 337-344 (July 1999).
Big Sky Laser CFR-800 spec sheet

I should have said: "Quantum mechanical effects become significant in the realm of the extremely super small." Likewise: "Einstein's relativity becomes significant in the realm of extremely large values of velocity."

Close but no cigar, BECs (Bose Einstein Condensates), Nonlinear optical systems, and the Stern-Gerlach experiment (which gave evidence for electrons having spin) all occur on scales as large or larger than cell biology. Special relativity depends on velocity but General relativity does not, in fact velocity is a difficult idea to describe in General Relativity and not useful anyway. In full scale General Relativity all you have is spacetime and curvature.

What about: "Look around, and everyone will see that quantum mechanics is not tangible."

I guess the 8 foot tall, 2 ton NMR spectrometer I work on isn't tangible. How about this NMR magnet is it tangible? Of course I guess NMR isn't a quantum phenomenon.

I have not heard about the quantum economic model yet, or social Q.E.D.

Probably because quantum theory refers to a very specific set of axioms which cannot be applied willy nilly. While Darwin's Theory of Evolution likewise refers to a very specific logical framework, the general ideas of competition and selection can be simulated in a variety of other systems hence the proliferation of "Darwinian X" ideas floating around. On the the other hand, quantum statistical phenomena apply only in very specific cases and other than those systems studied by quantum physicists don't appear in economic systems or social interaction networks or other areas in which analogies to Darwinian Evolution can be made. On the other hand, there is a lot of useful work that can be done in applying quantum mechanical ideas to computation.

Google called me a wimp for not answering the non-mathematical questions. At MathWorld News,you can see how Eric and I answered all the other questions.

In physics and mathematics, I assure you it is e.

How odd. Are you a physicist? Are you a mathematician? IANA physicist... Yet. I am in my third year of undergrad work for a degree in physics, and currently I work for the Collider Detector at Fermilab experimental collaboration. And yet, I have never heard of anyone refer to an order of magnitude as anything other than a factor of 10.

For the mathematics side, check out wolfram's mathworld site. In the parentheses, in the line of text after the first expression, we see that the exponent used in scientific notation (which has a base of 10) is called the "order of magnitude."

I believe that's actually XOR, thank you. Tyranny of OR would basicly be the tyranny of apathy, "Yeah, you could have either one, I don't care, as long as you have atleast one."

IANAM (I am not a mathematician), although I did pass Calc 3, Linear Algebra and AP Statistics, but a tesseract projection shouldn't be infinitely large. You can think about the projection into 3D by analogy: let's say you start with a square, and want to make a cube. All you have to extrude the square upwards. If you look from the top down, however, it looks almost like a square still. Then, if you look at it isometrically, it looks like you dragged a copy of the square at a 45 degree angle from the original and connected it point for point with the original. The 3D projection of a hypercube similarly looks like two cubes, seperated orthoginally depending on the hypercube's rotation in 4D, but connected point for point. At the correct orientation in 4D, it'll actually be a cube. Here's some sites that explain it better and include pretty pictures:
http://www.geom.uiuc.edu/docs/outreach/4-cube/
http://casa.colorado.edu/~ajsh/sr/hypercube.html
http://mathworld.wolfram.com/Hypercube.html

Perhaps, but probably not in our lifetimes. This is the holy grail of modern mathematics.

We are clearly capable of tracking things through space with very, very low margins of error.

To know where something is now doesn't mean you can predict where it will be in the future. Not within a "space is big" margin of error.

So what's the bottleneck here?

The very thing that makes it want to hit us: gravity. That is, the Three-Body Problem, an 3 is at the lower end of influences that come into play over the next 25 years.

I guess I'd just like to see the math on how they come up with these numbers.

Nobody's stopping you; it's not a secret. Go get it.

But get ready for some heavy lifting; as you dig into it you'll very quickly realize why they didn't try to put any in a popular news article.

I'm not too up on it myself but you can start with phase spaces, I think, though that hardly touches the real fun, which is the probabilistic aspect of determining the path of an object through all of the influences of the solar system... while I'm not up on the details I do know they don't use naive formulations of that problem, they've got some powerful and brain-bending tricks to prevent the estimate from diffusing too quickly.

No, It's more like comparing Windows Paint with Pokémon Project Studio.

If I wanted to bring packages like Corel, Xara or Photoshop into the picture, then we're talking Mathematica here.

They are totally similar in that both don't do scalar calculations and can graph. The one is limited to 2D graphs, and the other to 4D.
-shrug-

Both are mere toys.

Ah...what metric are you using to determine the fractal dimension?

If you're talking about box count, this is not as accurate a metric as you are claiming. If you're talking about exact Capacity Dimension, then you do have a point: it can be used. It is also very difficult to determine precisely in almost all cases. It is difficult to prove that something actually has such a dimension other than by empirically showing that the box count is consistent enough to imply a fractal nature.

Box count is a fairly useful metric, actually. However, a better one that seems to show the same information while having little to do with fractals is edge density. It's also faster to compute.

A particular example of the maths is here.

No! Bad monkey!

They have these things called "math books" filled with problems. The advantage there is that you can work with pencil and paper and actually struggle through problems.

Need help? A better place - one with actual information about how things work, rather than just how to solve certain kinds of problems - is Mathworld. This one will stick with you, too.

I still use it, and I'm getting my Masters in Engineering.

While people have suggested maple and mathematica they are both fairly expensive (even for students). magma is another paid alternative.

mupad is a decent alternative, that, while not open source, is free to download.

To comment on the many pencil+paper posts, I would like to add that a whiteboard is an essential mathematical tool. Besides ease of erasure, they allow for much more collaboration than a pencil and paper.

Mathematica is actually available for Linux, as well.

Full disclosure: I work for Wolfram Research. But oh -- the irony! I am also a columnist for Math Games at maa.org, and I wrote an article about the Quantian Distribution. I didn't want a spammer to start using quantian.org just as the distro was getting popular, so I bought it, and provided a redirect to the main Quantian site. So now, I'm getting doubly Slashdotted. Huzzah. A student should definitely be getting Mathematica for Students -- but check with the college first. They might be on a Mathematica Campus, and can get it for free.

Full disclosure: I work for Wolfram Research. But oh -- the irony! I am also a columnist for Math Games at maa.org, and I wrote an article about the Quantian Distribution. I didn't want a spammer to start using quantian.org just as the distro was getting popular, so I bought it, and provided a redirect to the main Quantian site. So now, I'm getting doubly Slashdotted. Huzzah. A student should definitely be getting Mathematica for Students -- but check with the college first. They might be on a Mathematica Campus, and can get it for free.

Full disclosure: I work for Wolfram Research. But oh -- the irony! I am also a columnist for Math Games at maa.org, and I wrote an article about the Quantian Distribution. I didn't want a spammer to start using quantian.org just as the distro was getting popular, so I bought it, and provided a redirect to the main Quantian site. So now, I'm getting doubly Slashdotted. Huzzah. A student should definitely be getting Mathematica for Students -- but check with the college first. They might be on a Mathematica Campus, and can get it for free.

What on earth is the point of working out sine or cosine with a TABLE? How is that more useful or 'fundamental' than using a calculator?

Both options are really 'black box magic occurs here'.

Mathworld gives the infinite-term series you should be summing up instead if you really want to remove some black box magic.

The insanely great mathworld is a great place to start. Pick a subject and start reading. If you don't understand something just follow the link to its definition, and pop the stack when you're ready to move on.

So true... honestly when I first started my research I discovered that many previous researchers (who had given me access to their data) had unreliable data (due to one reason or another). The data I had initally started running tests on I had to totally scrap for new data which proved to be the most dificult task of all my research (I was infact working with bankrupt company data which is always hard to get clean hands on). I lucked out and found what many researchers would claim the nervana of data sources in my field through much work and promisses to tripple check or add to prominant researchers data. I think discovery of good trustworthy data was the longest process of all. And in the end was not getting results fast enough, and funding was saddly pulled despite publications and good results shown quickly after. I like the saying "Junk in Junk out" and never wanted to produce more junk.

** On an interesting note, there are tests to run to see if data has been made up. Such as Benford's Lawor a more User friendly review of Benford's law. Or Zipf's Law.

But hey, as the joke goes, 97% of statistics are made up on the spot anyway so guess the data doesn't really matter.

So true... honestly when I first started my research I discovered that many previous researchers (who had given me access to their data) had unreliable data (due to one reason or another). The data I had initally started running tests on I had to totally scrap for new data which proved to be the most dificult task of all my research (I was infact working with bankrupt company data which is always hard to get clean hands on). I lucked out and found what many researchers would claim the nervana of data sources in my field through much work and promisses to tripple check or add to prominant researchers data. I think discovery of good trustworthy data was the longest process of all. And in the end was not getting results fast enough, and funding was saddly pulled despite publications and good results shown quickly after. I like the saying "Junk in Junk out" and never wanted to produce more junk.

** On an interesting note, there are tests to run to see if data has been made up. Such as Benford's Lawor a more User friendly review of Benford's law. Or Zipf's Law.

But hey, as the joke goes, 97% of statistics are made up on the spot anyway so guess the data doesn't really matter.

What has space exploration to do with humanity?

and solve the mathematical mystery of the number 137

To join that 'elite' group you need to insert another 3 in the middle.

Or add ".03599976" to the end, although those last two or three digits may be subject to change.

Prove the Euler-Mascheroni Constant to be irrational... or not. Either will work.

However, there are also established facts that are observations, not theories. These observations may be explained by a number of theories. If those theories contradict or are inconsistent with other observations or they are insufficient to explain other observations, they are regarded as incorrect or incomplete. Most theories that "were never accepted to begin with" were not accepted because they didn't adequately explain all available data. Gathering more data will not provide better evidence for incorrect theories.

If I provide a "theory" that objects shaped like a torus will never fall to the surface of the Earth, regardless of their density, and observational evidence shows that the "theory" is wrong (You drop a doughnut and it falls to the floor), no amount of additional evidence will convince anyone (reasonable) that my "theory" was correct. Revisiting it is pointless--let's move on to other theories to show how they do or do not work with the available evidence, or let's gather more evidence to test well-established theories.

Non-theories (like those proposed by creationists) are also a waste of time in the scientific communtiy because they can never be tested. I could propose that there is a tribe of invisible gnomes living in my garden. How would you test that? Well, say we look for their footprints? Oh, I forgot to mention that they don't leave footprints. OK, what about other kinds of evidence: maybe they're visible in the IR? No, they're completely transparent at all wavelengths. I'll just keep adding on addition superpowers to these gnomes until you give up in frustration, but you never were able to provide any evidence to prove my theory wrong, so it must be right and the truth. . .

About there being a correct theory: I didn't really state that well. Yes, you are correct, there can be a correct theory, which is the truth. However we can never know that the theory is 100% correct. We will never have collected all available evidence to test any theory, so all we can do is agree to call a well-tested theory a law. The scientific method has no (meaningful) aspirations toward finding absolute truth (*) because we will not know it when we have it. The scientific method is simply a way for us to make the universe meaningful in ways that are consistent from one observer to another.

* There are statements (let's call them axioms or postulates), like Einstein's Relativity Principal: "Every observer in the Universe must experience the same natural laws", that seem like absolutes, and are, in fact absolutes, but they are not the truth, they are the building blocks of our sciences. If we can't use those, then there is nothing that we can trust about our sciences, as everything is just magical because the laws don't apply to everyone equally. Notice that this axiom says nothing about knowing what those natural laws are, just that everyone follows them.

They're like the field axioms of algebra http://mathworld.wolfram.com/FieldAxioms.html.

Or, maybe Euclidian space is a better example. In 2-D Euclidian space, given any straight line and a point not on it, there "exists one and only one straight line which passes" through that point and never intersects the first line, no matter how far they are extended.

This is intuitively correct, except that it's impossible to prove for all cases. That's because it's an axiom for given, limited set of geometries (Euclidian geometries). There are other, self-consistent geometries which are non-Euclidian

"indigo", the coolest

Planck's distribution indicates that higher frequency (shorter wavelength) indigo and violet are WARMER than red.

also.. if you can SEE the green in the plants wouldn't that mean they're NOT absorbing it?

See Wein's Displacement law.
http://scienceworld.wolfram.com/physics/WiensDispl acementLaw.html

IF you ever had to code a linear system solver using Guassian ELimination for educational purposes you have a love hate relationship with pivoting:

Damn You Engineering Fundamentals!

Excel is for chemistry students and management types. Go Go gadget MATLAB.

A little Googling shows this to be something called a Hadamard Transform.
In the Early to Mid '90s, fast computers had to churn away to make fuzzy cubes and other simple objects.

With better computers and better printers the rendering should be faster and the reduction phase not as extreme. Also with larger Holographic plates the results should be less fuzzy.

Does anyone know the state of Computer generated Holograms? Real geeks wouldn't make holograms with old fashion photographic plates, but in the guts of their over-clocked AMD boxen.

There are a lot of them.

As a matter of fact 2048 bits can represent numbers up to 2^2048 ~= 3.23x10^616. By the Prime Number Theorem there is about 2.27x10^613 primes in that space. If each prime required 2048 bits to store, then storing all the 2048 bit primes requires more petabytes then you and I are likely to ever see.

(Jeez I hope I didn't do the math wrong. The most convenient calc I have available to me as I write this in the Windows calc)

Perhaps, but the statement that every positive integer can be written as a unique product of prime numbers is known as the Fundamental Theorem of Arithmetic -- so I'd still consider it worth knowing. ;-)

I do not 'get' what is interesting or useful about prime numbers.

Prime numbers are useful in encyption, but those are very large primes that are difficult to factor. To a mathematician, prime numbers are fascinating. The distribution of primes, which is specifially random but generally predictable (check out the MathWorld article for more details) is of particular interest.

For example, the Riemann Zeta Function of n, which is the infinite sum of all terms k^(-n)[k varies from one to infinity], is expressable as the infinite product (1-p1^(-n))*(1-p2^(-n))*(1-p3^(-n))... [where pn is the nth prime number]. This is a mind-boggling connection. For example Zeta(2)= (pi^2)/6 = (1 + 1/2 + 1/4 +1/8 ...)= ((1-2^(-2))*(1-3^(-2))*(1-5^(-2))...) There is therefore a connection between pi and the distribution of the primes. Math is crazy stuff.

I do not 'get' what is interesting or useful about prime numbers.

Prime numbers are useful in encyption, but those are very large primes that are difficult to factor. To a mathematician, prime numbers are fascinating. The distribution of primes, which is specifially random but generally predictable (check out the MathWorld article for more details) is of particular interest.

For example, the Riemann Zeta Function of n, which is the infinite sum of all terms k^(-n)[k varies from one to infinity], is expressable as the infinite product (1-p1^(-n))*(1-p2^(-n))*(1-p3^(-n))... [where pn is the nth prime number]. This is a mind-boggling connection. For example Zeta(2)= (pi^2)/6 = (1 + 1/2 + 1/4 +1/8 ...)= ((1-2^(-2))*(1-3^(-2))*(1-5^(-2))...) There is therefore a connection between pi and the distribution of the primes. Math is crazy stuff.

Brief overview. There is also the Prime Spiral.

Brief overview. There is also the Prime Spiral.

Mathematica is available on Linux.
There is also Octave for the Maple fans.

Being a power-of-2 times a product of distinct Fermat Primes, this means that a regular 7710-gon is constructible with ruler and compass.

Being a power-of-2 times a product of distinct Fermat Primes, this means that a regular 7710-gon is constructible with ruler and compass.

A couple years ago, there was a proposed proof to the Poincare conjecture- not the Perelman proof which AFAIK still holds together, but another attempt which was soon found to have an insurmountable problem. When the proof was first announced, the Mathworld news item ran, Poincaré Conjecture Purportedly Proved, and when the hole in the proof (essentially, an unproven step used in the proof) came to light, the headline was Poincaré Conjecture Purported Proof Perforated.

A couple years ago, there was a proposed proof to the Poincare conjecture- not the Perelman proof which AFAIK still holds together, but another attempt which was soon found to have an insurmountable problem. When the proof was first announced, the Mathworld news item ran, Poincaré Conjecture Purportedly Proved, and when the hole in the proof (essentially, an unproven step used in the proof) came to light, the headline was Poincaré Conjecture Purported Proof Perforated.

"The Music of The Primes" by Marcus du Sautoy, Harper Collings 2004.
http://www.amazon.com/exec/obidos/tg/detail/-/0060 935588/qid=1099578709/sr=8-3/ref=sr_8_xs_ap_i3_xgl 14/103-5861881-0888602

Although the title sounds exactly like what you are looking for, it doesn't really talk about music made by primes. Having studied some number theory, I found the math in the book to be fairly basic - but it does give an interesting account of the history of the people involved.

If you are interested in getting a high-level grasp of the questions involved in the distribution of prime numbers (hint: zeros of the Riemann Zeta function, perhaps http://mathworld.wolfram.com/RiemannZetaFunctionZe ros.html), then this book is worth the read - even if you have no math background.

The Travelling Salesman Problem remains NP-complete, and I don't think Linux or any other open source software is going to change that. So, in other words, your options may come down to an inefficient solution like you have now, or an efficient solution that takes longer to calculate than the inefficient route takes to travel. :)

I mean you take the "PING" program and run ping, you see 90% packet loss with a 6 seconds average latency, I believe I still have the files laying around from that data ping xxx.com > pinglog

Anyhow when you start to study Error corrector/erasure code you quickly realize that at best you can only reduce errors, IE a (DB increase if you will in S/N) By increasing used bandwidth you increase S/N. Shannon's law type of stuff.

Anyhow If can be proven mathematically that no matter how much you add you can not get 100% perfect data cross. Unless you "close the loop" using feedback as to what was lost, hence TCP.

Now I found you can and combine Error Correction and retransmission to get 100% of your signal/data across. And by using the right type of error correction you didn't need to increase latency to do your retransmissions. (The how, would be a very long discussion to get into that I'd want to take offline)

Now if we take this further I can deliberately strip my codes across a connection to intentionally cause a predictable and manageable amount of packet loss.
For example I know I have a T3 (45Mbs) on one side and an dual channel bonded ISDN on the other 128Kbps (this shows how far back I had worked on this) I know that the network between my T3 and the ISDN is probably better then 10Mbps at all point up until the "last mile".

Now let say that in 1996 MAE West was experiencing 20% packet loss (I have logs of it) almost every afternoon, but I still needed to get my data to some ISDN line on the other side of this gigabit router that is over saturated with bazillions of TCP connections.

Is this situation what if I use an error correction code set to recover 25% loss but with retransmissions? If I send 128K I'd have 102Kbps arrive. Then there is the overhead of the code itself to deal with. But knowing where my loss if occurring at I can deliberately send 25% more data to that connection. So about 170 Kbps gets sent. 20% lost in the backbone and I am guaranteed that the remaining 5% will be lost at the ISP's ISDN terminal server before being sent down the ISDN line. So I will have a known state of 25% lost data, but 128Kbps going over my connection.

Are you following me? What ever the lost across the backbone is become irrelevant since the lost at the connection will force me to at 25% loss.

Now what If I were to use a perfect code (turn out I found a number of perfect erasure codes)
Perfect codes

So start with 128K create 170K, I lost X% across the net lost 25% -X at the ISP and receive 128K. Using a perfect code for 25% loss I recover 100% of my data that needed 128Kbps to get across.

But this only works using retransmissions that are placed back into the error corrected stream! -- YES, Make sure everyone know John Sokol wrote this first since this was the big revelation that made this work!

By doing this it allows you to increase the hamming distance for the next error recovery so you can tolerate future lost data better. While recovering 100% of the data sent first pass.

Lets take the Hamming (7,4) Code it's a perfect code. It has a hamming distance of three. This allow you to correct 1 error and detect 2.

http://www.dcs.st-and.ac.uk/~sal/school/CS3010/L ec tures/forhtml/node3.html

For error you can only correct for 0.5 - 1/2 the hamming distance, but for Erasures you can correct for 1 - the hamming distance.

So I can correct for 2 lost packed out of 7 I send, but only 4 are data with 3 added overhead.

But these are block codes, and are not very efficient or effective. So used brute force and later Genetic algorithms to search for long perfect cyclical codes for Erasures.
The closest thing to the codes I found is called "hagelbarger codes"

The main fear is that of a catastrophe occuring. That is a catastrophe in the mathematical sense. A very good way to familiarise yourself with the concept is by going through the first chapter of James Murray's Mathematical Biology, an Introduction.
In this, Murray discusses mainly the equilibria in population dynamics, i.e. what concentrations of foxes and rabbits are likely to be able to coexist in a regular pattern.
When a catastrophe occurs, the equilibrium has been so perturbed that the system is unpredictable until it finds a new one. Once this new equilibrium has been reached, it is not possible to "go back". This is called a hysteresis effect.
Population dynamics is very similar to weather in that sense.
If you're wondering why global warming is bad, suppose we produce a sufficient ripple in the system (e.g. through rapid increase in carbon monoxyde concentrations) to initiate a catastrophe, which brings to say five degrees centigrade hotter or lower on average. This would have such monstruous consequences for the planet it's very hard to imagine.
For some extreme predictions, see Hubert Reeves' three scenarii (unfortunately I was unable to find this information in english, apparently Reeves' book Mal de Terre (Earthake) has not been translated into English)

The Bernoulii equation relates pressure and energy on a streamline, the Euler equations express the conservation of momentum for an inviscid fluid.

See http://scienceworld.wolfram.com/physics/EulersEqua tionofInviscidMotion.html

Other favorite is the pseudoscience behind marketing industry equations like
opportunities to see (OTS). I can never remember these offhand because they are so meaningless, but it is fascinating to think of the millions of dollars spent on ie outdoor billboards, where the advert is sold on the basis of these "opportunities to see." See also A Test of the Direct/Indirect BBD and Other Exposure Distribution Models.

I don't have the charset to write it up here, but those of you that are familiar with it no that there is no bound (no pun intended) to it's utility. Here's a link to the equations on MathWorld.The extensions of this theorem, the Laplace integral and Z-transform are equally important. These equations enable us to project functions of time onto periodic functions which we can then use to analyze the frequency content of the original function. They also enable us to easily solve differential equations and discrete difference equations.

Mike

That would be the Euler-Lagrange equation.

Not exactly, see here for the expression of K (eq. 12).