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"Simulations" sums it up. Just the word "simulations" is enough to suck up any conceivable amount of processing power they will ever have.

Folding@Home runs about a petaflop these days, so they're planning to build the equivalent of about 1,000 Folding@Homes. But Folding@Home is barely making a dent in the number of proteins scientists want to fold. Just this one existing simulation project could probably saturate the proposed exaflop computer.

What if they want to simulate battle field conditions? Surely the number of possible "moves" between millions of troops, with many possible sets of actions and equipment and hundreds of thousands of vehicles on any terrain on the planet represents a number of possibilities significantly greater than a game of chess, with it's mere 32 game pieces that can only act in turns in very strictly defined, non-linear ways on a 64-space game board. But depending on which mathematician you listen to, just the number of possible moves in chess might be in the vicinity of 10^10^50, or the much lesser 10^120. How much processing power would it take to "solve" chess? Well, by these estimates, if every atom in the entire universe were harnessed into a quantum computer, and it worked on the problem for the entire life of the universe, it wouldn't have time to finish solving the game.

An exaflop is nothing when it comes to simulation. Any competent group of college science students could probably propose simulation software that makes useful work for a yottaflop machine. And if they wanted to simulate something like evolution, they'd still have to use really high abstraction levels to simplify things enough to get any results within a human's lifespan. I wonder if a yottaflop can even simulate one insect at the molecular level in real time, much lessa mouse, or several organisms interacting with each other and their environment.

Back to the military: just accurately simulating the possible variations they might want to try to improve a single firearm might keep a computer like the one they propose busy for a long time. It's not a question of what they'd ever come up with to do with all that power, its a question of how to prioritize the countless project for which each of which could occupy all its time. Or to simplify, optimize, and abstract the problems they're working on enough to get results even with an exaflop to play with.

I think you're maybe misunderstanding a little of how 'orbit' works. In order to go 'up' or 'down' in orbit, you really need to go faster or slower. That is to say, if you want to get into a higher orbit, you need to accelerate, and start moving faster around the Earth. You don't just push up perpendicular (normal) to the Earth's surface, that doesn't work. (Well, it will work temporarily, but it won't get you into a higher orbit. You'll just fall back down, because you're not escaping gravity. Remember, orbit is all about falling towards the Earth but moving fast enough to miss it, continuously.)

The satellite that was shot down yesterday was very, very close to the Earth's atmosphere. It was only one rotation, maybe less, away from starting to graze it (which means that it would slow down and begin to reenter and burn up). If we assume that when it was destroyed, pieces flew in all directions, some of them would have ended up with a greater net orbital velocity at the end. These pieces aren't the ones that exploded *up* (normal to the surface of the Earth), though, they're the ones that exploded *forward* (in the direction of the satellite's motion). They picked up some velocity and would end up in a slightly higher orbit as a result. I suspect it's not much of a higher orbit, though -- if anything, it probably just means they'll take a little longer to hit the atmosphere than other parts. It's tough to say without doing any calculations, but I doubt you have enough Delta-V to push the pieces into a long-term stable orbit. (Unless maybe the rocket fuel detonated.) The difference in velocities between high, long-term stable orbits and low atmosphere-grazing orbits is pretty substantial.

The pieces that flew off in other directions aren't really a huge concern, because they all end up in the same or lower orbits. Plus because you've blasted the satellite into little pieces and thus increased its surface area tremendously, it'll start slowing down on hitting the atmosphere much more quickly, and the pieces will burn up more completely on their way down.

My understanding is that what the Chinese did was quite different. The satellite they shot at was way out in a stable orbit, and thus the pieces it was reduced to stayed there as well. So now instead of a dead satellite floating around in orbit that's relatively easy to track and avoid, you have a vast cloud of small debris. Not an improvement at all.

How far do you take it? XP can run on an 8-MHz Pentium with 20MB RAM (for sufficiently low values of "run"). Can my pretty triangle (proven Turing-equivalent to a Core 2 Duo!) be said to run Vista?

ummm, a hexagon does not have a diameter

to clarify: what's called the travelling salesman problem is not the travelling salesman's problem. math should call the former the visiters' problem (or something), though this is your problem as well (with errands). there's no efficient algorithm for it.

real travelling salesman have a solved problem: walking along each road once, selling to each house along the way.

It certainly brings a whole new meaning to "the halting problem"!

"Anyone attempting to produce random numbers by purely arithmetic means is, of course, in a state of sin." - saying basically that since the universe contains true randomness, it cannot be the product of a calculation.

How do we know it's true randomness and not just pseudorandomness that just looks like the real thing? Something like Rule 30?

A complete metric is a metric in which every Cauchy sequence is convergent. A topological space with a complete metric is called a complete metric space.http://mathworld.wolfram.com/CompleteMetric.html

Sorry, some corrections; http://reference.wolfram.com/mathematica/tutorial/TheSoftwareEngineeringOfMathematica.html

Most of the kernel and core is a variant of C, though not vanilla C so I wouldn't say that it necessarily has the same bug-rates. There is also use of Compile[] etc. The front-end is a separate application. The packages are entirely readable code.

Professors get paid to author College/High School math textbooks, ergo, good.

Software is written by Mathmaticans working for Private Firms, professors not needed, ergo BAD.

Much of the guts of Mathematica -- the high level functions -- are available for perusal at http://functions.wolfram.com/ Much of the reason why Mathematica works so well is due to careful consideration of branch cuts.

For example, what is Log[a b]? If you look in almost any math book, you'll find Log[a b] = Log[a] + Log[b]. That works fine most of the time, but tends to fail in devastating ways when combined with other functions. The open source programs tend to follow this "good enough" approach.

Log[a*b] == Log[a] + Log[b] + 2*I*Pi*Floor[(Pi-Arg[a]-Arg[b])/(2*Pi)]
is the very careful way to handle the log function.

http://functions.wolfram.com/ElementaryFunctions/Log/16/04/0005/

Much of the guts of Mathematica -- the high level functions -- are available for perusal at http://functions.wolfram.com/ Much of the reason why Mathematica works so well is due to careful consideration of branch cuts.

For example, what is Log[a b]? If you look in almost any math book, you'll find Log[a b] = Log[a] + Log[b]. That works fine most of the time, but tends to fail in devastating ways when combined with other functions. The open source programs tend to follow this "good enough" approach.

Log[a*b] == Log[a] + Log[b] + 2*I*Pi*Floor[(Pi-Arg[a]-Arg[b])/(2*Pi)]
is the very careful way to handle the log function.

http://functions.wolfram.com/ElementaryFunctions/Log/16/04/0005/

Please ask any question desired. I know more.

Is the Euler-Mascheroni Constant irrational?

I suppose it has something to do with the "openness" of the fonts, or something like that, but haven't complete (or nearly so) scientific font sets been around for a long time? Other posters have mentioned the TeX collections, and there's also Mathematica's fonts: http://support.wolfram.com/mathematica/systems/windows/general/latestfonts.html.
      Basically: what's new about the Stix font set?

Mathworld says affinity is another word for affine transform, which of course doesn't make sense here.

There's a relevant article in this month's Notices:

http://www.ams.org/notices/200710/tx071001279p.pdf

Specifically, the quote from the horse's mouth:

http://reference.wolfram.com/mathematica/tutorial/WhyYouDoNotUsuallyNeedToKnowAboutInternals.html

which contradicts the principles of mathematics, and of scientific method. I can answer *any* question you ask!

From the article:

Here it is. Just two states and three colors. And able to do any computation that can be done.
[...]
There were some simpler universal Turing machines constructed in the mid-1900s--the record being a 7-state, 4-color machine from 1962.

So basically there's a machine that has two states, each of which can be three colors, and that machine can perform ANY computation that an x86 cpu can perform. The code to add two 32 bit numbers in an x86 processor might be just a few bytes and the code to do the same thing with this 2,3 Turing machine might be thousands of bytes, but it can do it. It will be horribly inefficient and slow, but it can be done. They've proved that a 2,2 machine is impossible so a 2,3 machine was the simplest possible theoretical Turing machine. This paper proved that one exists. It doesn't have a practical application right now, but the article mentions possible molecular computers that can use this simple machines to do calculations on strands of molecules like DNA.

I find this site fairly useful:

http://mathworld.wolfram.com/

http://mathworld.wolfram.com/ is pretty decent, maybe a little formal but hey its maths

I was someone who was once considered to be exceptional in math. Unfortunately, I made the mistake of stopping at calculus.

To regain my mastery of mathematics, I decided to take a single math problem very seriously. I figured that I would try to
understand the solution by grounding all ideas down to postulates.

I figured that this was a great way to learn mathematics anew and really get advanced. I soon learned that there were wonderful
math resources on the web. Wikipedia is really great. There's also MathWorld.com.,
PlanetMath, MathForum.org, and
Cut-The-Knot.org.

Being pretty ambitious, I chose Fermat's Last Theorem and Andrew Wiles's solution as my jump off point. I started this adventure
in 2004. Since then, because the problem is so tough, I started blogging through the different threads of the problem and I find
myself recreating the history of mathematics from the perspective of number theory.

I am not sure that this approach would work for everyone but if you are a solid problem solver, it can really make advanced
mathematics more fun. If you are interested to see what I came up with, you can check out my blog a My math blog.
I also started a general math blog.

Best of luck in learning mathematics.

-Larry

Wolfram called; they want their concept back.

He was just wanted to prove he knew the Intermediate Value Theorem. Give him a few weeks, and he'll explain it in terms of L'Hospital's Rule.

He was just wanted to prove he knew the Intermediate Value Theorem. Give him a few weeks, and he'll explain it in terms of L'Hospital's Rule.

It seems that your whole argument is that ODF should have implemented Excel bugs in the standard just in case a user wants to use ceiling in openoffice, then save as Excel, open in Excel and modify the formula so it's not ceiling any more, then import that back into openoffice and save as an Excel file.

And this makes you so angry that you hope ODF will fail?

The function is called ceiling. The mathematical definition is here: http://mathworld.wolfram.com/CeilingFunction.html
ODF is correct, OOXML is wrong. Nuff said.

"Also, this may be a stupid question, but I wonder how one measures the 'randomness' of a generator?" There are lots of places that talk about this. A simplistic explanation of what it means to be a good PRNG is simply to provide a sequence of numbers with no correlations that matches the desired distribution. (http://mathworld.wolfram.com/RandomNumber.html). Books on modeling and simulation often have good explanations of this. This page (http://csrc.nist.gov/rng/) has a good overview, including simple descriptions of 16 statistical tests of interest.

Mersenne Twister is not a random number generator, it's a pseudo-random number generator.

Randomness is measured as entropy. See here for details: http://mathworld.wolfram.com/Entropy.html

Congratulations: you've got some of the potentially most interesting classes to use technology in - but that potential will be wasted if you just use the tablet and projector to show Powerpoint slides.

When you're designing your class, think: what can the tablet do that would be useful that could not have been done without it. Powerpoint fails this test miserably - an overhead projector would do just as well.

Here are some possible uses that do pass the test:

• Use symbolic math software to help students visualize the math, and to explore interesting problems that cannot be handled without it. Mathematica is everybody's pet favorite, of course - but I would argue that it's grotesquely overpowered and complex for most of what you'll need. Instead, take a look at something like Ron Avitzur's Graphing Calculator - the name doesn't do justice to what is a particularly elegant little program.
• For Physics, use the tablet to analyze physical data. One of the best uses here is to film objects in motion, then transfer the video to the tablet (or get a cheap webcam and record directly on the tablet), and analyze the results frame-by-frame - your students will come out with a much better understanding of motion. A free package for video analysis is Physmo.
• For more sophisticated experiments, check out what the folks at PASCO have to offer - their sensors are reasonably inexpensive.
• If you do a Google search, you'll find a wealth of Java applets that simulate concepts in Physics - when contextualized by discussion, physical experiments, and "what if" explorations, these can be tremendously useful. Without this framework, though, they are no better than the film loops of old.

One last suggestion: don't hog the tablet - let your students use it too. You can set up a problem, and invite students to come up and work through it individually or in groups, showing their thought process to the rest of the class. The students will learn much more, and everybody - including you - will have a lot more fun.

Good luck!

Wouldn't work. See http://mathworld.wolfram.com/UtilityGraph.html What you really need is to allow interconnections to go over or under each other.

I was under the impression that this latency issue was caused by the fact that there is no positive solution to the utility problem. Essentially, each core is connected directly to the other two, in a planar graph. There's no way to connect each of 4 cores to the other three without the connections intersecting, at least if the connections are made on anything topologicically the same as a convex subset of the plane (that is, no planar graph exists).

This can be solved directly by creating chips with multiple planes on which connections can be made, or indirectly by running messages through other cores, at the cost of latency. Then again, I have no idea if multi-layer chips are in production.

Feel free to read about base if you care, but I doubt you do.

A real number can be represented using any integer number as a base (sometimes also called a radix or scale). The choice of a base yields to a representation of numbers known as a number system. In base , the digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ... are generally used as symbols representing the decimal numbers 10, 11, 12, ...).

Feel free to read about base if you care, but I doubt you do.

A real number can be represented using any integer number as a base (sometimes also called a radix or scale). The choice of a base yields to a representation of numbers known as a number system. In base , the digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ... are generally used as symbols representing the decimal numbers 10, 11, 12, ...).

Feel free to read about base if you care, but I doubt you do.

(what's with the random bolding of words?)

Of course Gimp doesn't count. You want a spreadsheet program for that, or at least a calculator. There's plenty of calculators for Linux. If you're really really into counting and arithmetic, there's that program from Wolfram that will run on Linux.

...they're off visiting all the other jillions of interesting sentients throughout the universe?

It seems to me in order for the "Fermi paradox" to be a problem, you've got to assume that the development of intelligent, spacefaring sentients is really, really, common.

Suppose, for example, we assume that we get found by someone detecting our radio broadcasts. According to this, the first commercial radio broadcast was in 1920. The wave-front from that broadcast is now a sphere ~43 light years in radius. According to this, the Milky Way galaxy has a diameter of 100,000 light years.

Using a 2D (because I don't have the math or the data for a 3D) model: a disc of radius 43LY has area 43*43*pi = 5.8E3 LY^2. For the galaxy, A=50000*50000*pi = 7.8E9. So our broadcast sphere has covered 0.00007% of our own galaxy.

So even if there is another sentient spacefaring species out their zipping around in their FTL ships, they'd have to be looking really hard just to get down to the granularity necessary to look in our little corner of the galaxy.

And what if you assume the development of sentient life is unlikely? What if the nearest one is in, say, the LMC? What if FTL travel is impossible, or just really hard? We might never meet one.

It seems this is a real life application of the "Sultan's Dowry problem"

http://mathworld.wolfram.com/SultansDowryProblem.h tml

The daughters are the beautiful women willing to marry you. The dowry is "satisfaction with your marriage". Then you have to estimate 'n' by the number of beautiful women willing to marry you who have presented so far; this is the most difficult part of the task, particularly if it's been only one so far.

Note that thinking about the problem this way is a pretty good bar to sex OR marriage, particularly if you try to explain it to your potential mates.

No, one of them actually succeeds.

(This post brought to you courtesy of the Well-Ordering Principle.)

x = [R+scos(1/2t)]cost

y = [R+scos(1/2t)]sint

z = ssin(1/2t)

for s in [-w,w] and t in [0,2pi).

With interactive pictures!

http://mathworld.wolfram.com/MoebiusStrip.html

The other guy posting is correct: refraction is only a property of the material and doesn't depend on the shape of the optic. Ultimately where the light rays go and their dependence on the shape of the optic enters into it via Snell's law.

If you take a slab of material with a constant index of refraction, you can change the path of light rays going through it by changing the shape of the surface of a material, just like you describe. Another way to do it is to take a flat slab of material, like looking into one end of a cylinder, and change the light ray paths by changing the index of refraction as you go through the material. An example of the second case is a GRIN lens. There are plenty more examples of the variable index situation, such as acoustic waves being channeled or reflected through thermal layers in the ocean or through the atmosphere.

FRAUD ALERT -- FRAUD ALERT -- FRAUD ALERT

Slashdot editors apparently don't read the comments on the stories they post. Also, Slashdot editors apparently didn't listen in Physics class. This is the fifth time in 3 years that they have fallen for the same fraud, if I count correctly. Some of my other comments:

Max Planck would be very sad about this.

Distinguish between real science and junk science.

Planck's constant is so small that interactions between electromagnetic waves and molecules cannot be chemically specific. The 2,000 MHz radiation from WiFi is felt as heat, a very, very small amount of heat, almost certainly not measurable.

Anyone may have theories. Someone could say, for example, that pigs have started flying and they have been eating the bees. (The bees are dying because of bad management; the organic beekeepers aren't having problems.) The only real science, however, is based on what is already known through experimentation. That requires an understanding of what is known.

I know you didn't mean that seriously, but I think you might be surprised how thin the foil has to be for light to pass through it. The quantity of interest is "skin depth". You can calculate it with the formula here, which uses several constants that are pretty easy to find:

frequency of visible light: 600 THz (source)
conductivity of aluminum: 3.8 x 10^7 siemens per meter (source)
permeability of free space: 1.3 x 10^(-7) weber per ampere meter (source)

I calculated that the skin depth of aluminum is 8 nanometers. This means that the thickness of aluminum needed to stop 99.9% of the light is one 400,000th of an inch. For comparison, this is 10,000 times thinner than the thinnest aluminum foil available from McMaster-Carr (it's a company that sells materials for scientific research, among other things). Since the atomic radius of aluminum is 125 pm, this foil would be only 250 atoms thick, and would still block 99.9% of the light.

By the way, if you've never used it, you should check out Google's calculator. It handles units for you, so it makes calculations like this really fast.

The problem is that most articles that are made accessible to lay-people lose their usefulness for people who actually need to use that information. It is fine for a wikipedia article to contain a long but understandable description of a topic. But the concise technical description must remain in a prominent position. Wikipedia is first and foremost a reference source. The precise technical information is more important than the "How Stuff Works" version.

My opinion, as a mathematician, is that the first paragraph of an article should contain a concise, precise description of a topic, with all the technical terms linking to separate articles. This really doesn't take up much space, and is the format most useful for students of the subject. The average joe's description is going to be several times longer and contain less precise information, so it should go towards the beginning of the body test.

Look at http://mathworld.wolfram.com/ articles to see good examples. An article like Lie Group contains the entire definition in the first sentence, making that page a very good reference for people who already know some abstract algebra. Compare with the wikipedia page, Lie Group, which has a less concise and less clear definition. Wikipedia does have a lot more in the way of historical information, and a less dense explanation of the topic, but it still isn't very accessible. It also lacks the more or less obligatory link to the wolfram page. So what audience can make use of that article?

The problem is that most articles that are made accessible to lay-people lose their usefulness for people who actually need to use that information. It is fine for a wikipedia article to contain a long but understandable description of a topic. But the concise technical description must remain in a prominent position. Wikipedia is first and foremost a reference source. The precise technical information is more important than the "How Stuff Works" version.

My opinion, as a mathematician, is that the first paragraph of an article should contain a concise, precise description of a topic, with all the technical terms linking to separate articles. This really doesn't take up much space, and is the format most useful for students of the subject. The average joe's description is going to be several times longer and contain less precise information, so it should go towards the beginning of the body test.

Look at http://mathworld.wolfram.com/ articles to see good examples. An article like Lie Group contains the entire definition in the first sentence, making that page a very good reference for people who already know some abstract algebra. Compare with the wikipedia page, Lie Group, which has a less concise and less clear definition. Wikipedia does have a lot more in the way of historical information, and a less dense explanation of the topic, but it still isn't very accessible. It also lacks the more or less obligatory link to the wolfram page. So what audience can make use of that article?

I claim this

http://reference.wolfram.com/mathematica/guide/Sum maryOfNewFeaturesIn60.html
"Full-featured source-level debugger, including breakpoints, watchpoints and stepping."

If you haven't used Mathematica, you have no idea how badly the debugging sucks (prior to the new version.)

George Hart's Wolfram Mathematica notebook for Canonicalizing Polyhedra is at the following link:

http://library.wolfram.com/infocenter/Articles/201 2/

Wolfram and Hart have been working together for at least a decade.

Some of the demos are pretty funny. http://demonstrations.wolfram.com/AddingWholeNumbe rs/ I never imagined computers could do such things.

I clicked on the Demonstration Project link, then browsed through the list of demos and decided to try the Monty Hall Problem demo.

It brings me to a flash application which lets me experiment with the problem by clicking on doors and then seeing where the prize is. Actually, it doesn't. It gives me two options: I can download a "live version", or I can watch a demo of someone else clicking on doors and seeing where the prize is. Hello? This is flash, it's already interactive! Gah..