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Actually, infinity minus infinity is not well-defined. In fact, for any extended real number you choose (which includes +/- infinity), I can show you a sum/difference of inifinities equal to that number.

And you are correct: there can be bounded infinities, depending on your point of view. For example, see the one-point compactification of the real line. http://mathworld.wolfram.com/Compactification.html . In some sense, it all depends on your definition of distance.

For the visually-oriented minds:

http://map.gsfc.nasa.gov/m_mm/ob_techorbit1.html
http://www.physics.montana.edu/faculty/cornish/lag range.html

For the mathametically inclined:

http://scienceworld.wolfram.com/physics/LagrangePo ints.html

Further, if I realize that people are sneaking in, I may have to charge 80$ from everyone else to cover the sneakers loss of sales.

No you don't. You don't have to charge an extra penny to cover the lost sales.

You may decide to increase revenues because you aren't covering costs, or because you want a higher profit. You can increase revenues by increasing the unit revenue ($50-$80), or by increasing the number of sales. You can increase the number of sales by increasing demand (better product or marketing), or by convincing the "sneakers" to pay instead (such as by locking the back door and hoping they'll pay $50 to get in).

Now, the idea of "lost sales" assumes that the "sneakers" would pay if they can't get in; that's well discussed in other posts so I'll skip it. The idea of having to "cover" that, however, is a misdirection: you have to cover your costs, or your desired profits, but just losing sales (if you are losing sales) doesn't mean you have to cover that.

As I said, it's a misdirection, like the Missing Dollar Paradox. You don't have to cover the "sneakers"; you have to cover your costs. You don't have to cover more costs because somebody sneaks in, any more than you have to cover more costs because I jump up and down in my living room shouting "Bwibbity-bwippity-bwippity-blech!"

well, he proved that rule 110 is universal. He also discovered the smallest set of NAND/NOR axioms for boolean algebra.

well, he proved that rule 110 is universal. He also discovered the smallest set of NAND/NOR axioms for boolean algebra.

well, he proved that rule 110 is universal. He also discovered the smallest set of NAND/NOR axioms for boolean algebra.

well, he proved that rule 110 is universal. He also discovered the smallest set of NAND/NOR axioms for boolean algebra.

Have you seen wireworld? It is another cellular automata which has some very interesting properties.

Spelling is man-made, and is only defined by people who care enough to research it. I think my spelling is good enough to convey my point.

Math is something that is a part of nature. Look here:
http://britton.disted.camosun.bc.ca/fibslide/jbfib slide.htm/

and here:
http://mathworld.wolfram.com/FibonacciNumber.html/

Just think about your experience with math. There is usually one branch of it that is "hard to get". Once someone says that words that link your current understanding of things, to the way they are expressed as a group in math form, then suddenly BING!, you get it. And it's usually something like, "Oh man, so THAT'S it. Oh okay, now I understand." After that, it's easy. Have you never had this experience? I had this experience with geometry.

In fact, this is how you calculate momentum... Not for photons. This is how you calculate momentum for photons: p = h / lambda, where lambda is wavelength. Alternatively: p = hf / c, where E is energy, and f is frequency. More info here: http://scienceworld.wolfram.com/physics/Photon.htm l [wolfram.com] And here: http://scienceworld.wolfram.com/physics/Energy.htm l [wolfram.com] You can "back-calculate" a supposed mass for a photon, once you know its momentum, by using the p = mv equation. But this often called a "fictional" mass, because it is purely relativistic. If you took away a photon's speed, it would have neither mass nor momentum, and would essentially cease to exist. Mass as an fundamental physical quantity exists even in the absence of velocity. This cannot happen with a photon... Unless you subscribe to the view that photons do not always travel at c in vacuum. But I will not argue that here. Not enough space, and I don't want to be in a flamewar. -- Copyright 2005, all rights reserved. Duplication of this post or sig, without express permission, will be prosecuted. Oh, oops, did I duplicate that? Sorry bout that.

In fact, this is how you calculate momentum... Not for photons. This is how you calculate momentum for photons: p = h / lambda, where lambda is wavelength. Alternatively: p = hf / c, where E is energy, and f is frequency. More info here: http://scienceworld.wolfram.com/physics/Photon.htm l [wolfram.com] And here: http://scienceworld.wolfram.com/physics/Energy.htm l [wolfram.com] You can "back-calculate" a supposed mass for a photon, once you know its momentum, by using the p = mv equation. But this often called a "fictional" mass, because it is purely relativistic. If you took away a photon's speed, it would have neither mass nor momentum, and would essentially cease to exist. Mass as an fundamental physical quantity exists even in the absence of velocity. This cannot happen with a photon... Unless you subscribe to the view that photons do not always travel at c in vacuum. But I will not argue that here. Not enough space, and I don't want to be in a flamewar. -- Copyright 2005, all rights reserved. Duplication of this post or sig, without express permission, will be prosecuted. Oh, oops, did I duplicate that? Sorry bout that.

... such as the BBP formula, which is amazingly simple (and easy to memorize!), converges at more than one hex digit per iteration, and can be used to extract an arbitrary hex digit.

that's interesting, I wish I had modpoints

From Mathworld at Wolfram Apparently Pi shows up in the bible twice. Weird.

Also, wikipedia has a rather complete coverage of the topic. http://en.wikipedia.org/wiki/Pi

pi is really irrational...

Obviously, you can approximate any irrational number as p/q, and your approximation gets better as q gets larger. So, if x is the number that you're trying to approximate, then take the absolute value of x - p/q. Then, find the smallest number u (the infimum) for which abs(x - p/q) is less than 1 / (q^u). This value u is the irrationality measure of x. For u=1, x is rational, (mathworld isn't quite clear about u between 1 and 2...) for u greater than 2, x is transcendental.

For pi, u ~= 8.0161. So pi is quite irrational.

We learned about this when studying for the Putnam my freshman year.

Not really...

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

No. They reiterate a formula in their head.

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

Game theory may be more accurately described as a branch of mathematics, which has applications in many areas, including economics... and military theory.

Proof (if you still don't get it):
Starting premises:
1. {Integers} is a subset of {Real numbers}.
2. The intersection of {real numbers} and {imaginary numbers} is the empty set {}: no real numbers are imaginary.

An imaginary number is a complex number that has zero real part. An imaginary number can therefore be written as a real number multiplied by the "imaginary unit" i (equal to the square root <picture of square root of -1>).

As all integers are real numbers, no integers are imaginary numbers.

Then again, if you don't know what an exponential is, never mind.

Actually, 6 is the perfect number -- not 7.

In fact, this is how you calculate momentum...
Not for photons.

This is how you calculate momentum for photons:

p = h / lambda, where lambda is wavelength.

Alternatively:

p = hf / c, where E is energy, and f is frequency.

More info here:
http://scienceworld.wolfram.com/physics/Photon.htm l

And here:
http://scienceworld.wolfram.com/physics/Energy.htm l

You can "back-calculate" a supposed mass for a photon, once you know its momentum, by using the p = mv equation. But this often called a "fictional" mass, because it is purely relativistic. If you took away a photon's speed, it would have neither mass nor momentum, and would essentially cease to exist. Mass as an fundamental physical quantity exists even in the absence of velocity. This cannot happen with a photon...

Unless you subscribe to the view that photons do not always travel at c in vacuum. But I will not argue that here. Not enough space, and I don't want to be in a flamewar.

In fact, this is how you calculate momentum...
Not for photons.

This is how you calculate momentum for photons:

p = h / lambda, where lambda is wavelength.

Alternatively:

p = hf / c, where E is energy, and f is frequency.

More info here:
http://scienceworld.wolfram.com/physics/Photon.htm l

And here:
http://scienceworld.wolfram.com/physics/Energy.htm l

You can "back-calculate" a supposed mass for a photon, once you know its momentum, by using the p = mv equation. But this often called a "fictional" mass, because it is purely relativistic. If you took away a photon's speed, it would have neither mass nor momentum, and would essentially cease to exist. Mass as an fundamental physical quantity exists even in the absence of velocity. This cannot happen with a photon...

Unless you subscribe to the view that photons do not always travel at c in vacuum. But I will not argue that here. Not enough space, and I don't want to be in a flamewar.

For those of us that are joining late, remember that as you move faster through space the universe around you seems to speed up AND space itself seems to contract--from your frame of reference distances are shorter, and you thus do not need to travel as far.

Space doesn't contract for the traveler. The traveler seems to contract when viewed by an outside observer, since the speed of light is constant in all reference frames. It's called Lorentz Contraction.

What don't you try Mathematica on OS X ? According to this link

http://www.wolfram.com/news/dbak.html

the database access functionality is built in to 5.1 and above. I wasn't sure from what you said whether what you want to do is possible with this approach, but it might be worth considering ?

math-atlas tends to have better material on advanced(i.e. undergraduate and graduate level) mathematical topics. Here's an example: mathworld page on motives, and math-atlas page on motivic. Another example: math-atlas page found by looking for ``Postnikov towers", and a page from mathworld on Postnikov systems. math-atlas doesn't have a history of disappearing for long stretches of time either. At any rate, all of the big math sites(Planetmath, wikipedia, mathworld, math-atlas) have good material and are worth checking out.

math-atlas tends to have better material on advanced(i.e. undergraduate and graduate level) mathematical topics. Here's an example: mathworld page on motives, and math-atlas page on motivic. Another example: math-atlas page found by looking for ``Postnikov towers", and a page from mathworld on Postnikov systems. math-atlas doesn't have a history of disappearing for long stretches of time either. At any rate, all of the big math sites(Planetmath, wikipedia, mathworld, math-atlas) have good material and are worth checking out.

http://mathworld.wolfram.com/ -- great resource, not necessarily only about Mathematica (TM), but math in general.

Paul B.

F.P.???

I'm not sure if it was a photoshop plugin or a standalone filter, but the filter was able to derive sharp pictures from the bokeh of photographs. ...and it's really not all that breakthrough-ish. Nearly anyone who's taken a signal processing class will have done this. The simplest version is an unsharp mask.

Here's the basic idea: you assume some "spreading" of the data happened, and you assume its shape. Then you try to undo what happened - perform the inverse.

There are two problems with this. First, the original convolution you assumed (that "spreading") is destructive to information. There exists no unique inverse mapping. You have to pick one, and hope that what it yields looks right.

Second, without making some major assumptions (that signal processing people aren't usually keen to make) there is no way to differentiate between true signal and noise. The noise, along with the blurry edges, also get sharpened. You can mitigate this somewhat with your choice of inverse mapping. Again, you pick something that looks right.

They do have some prior information going into this - they know the equipment that took the pictures - but pretty much nothing they do will exactly restore the information that was lost. Math isn't magical enough to do that.

For the hardcore:

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

and follow the links from there. :)

Not in set theory.

The == operator in PHP has nothing to do with set theory. It's called the "equal" operator, so either it should behave in a manner consistent with equality or it should be named and written differently.

You're the one trying to convert "spam" into an integer.

Really. Tell me, where in

$a == $b

Everyone STARTS somewhere different. Then, through hard WORK, they END UP somewhere better, and where they END UP is limited only by their DRIVE and BELIEF that they can get there.

It's equation fifteen on this page.

Happy to help!

--grendel drago

Incomplete, and thus inaccurate. Nanotechnology is likely to make petrochemical synthesis from alternative raw materials possible, and alternate materials may partially replace plastics. (Pla However, the key difficulty in replacing oil is that it is a pre-existing compact form chemical energy storage. Nanotechnology might concievably produce a replacement energy storage technology-- although the high energy profit ratio and reasonable energy density of gasoline make this somewhat questionable. However, if you are using nanotechnology to synthesize a form of energy storage, you still need the energy as an input in one form or another. Nanotechnology is primarily materials science; it does not provide new energy sources.

A space elevator is highly unlikely to be developed before it is rendered unnecessary by other nanotech-based technology.

Um? A space elevator allows for access to space via a Conservative system. Unless losses to entropy are far higher than projections for fullerene conductors indicate, there is NO more long-term economical way for bulk materials space transport; the costs are prinicpally capital set-up costs, of which the bulk are projected for the initial semi-orbital space deployment... of just the sort that a beanstalk can ameliorate.

Furthermore, large scale space solar is the ONLY kind of energy source capable of sustaining humanity to a Singularity scenario, barring a suprise in a GUT breakthough well beyond what is expected at this point from string theory models. Nanotech is primarily an improvement in materials science, not physics. It has the potential to improve energy storage and transportation, but very limited possibilites for improving energy production.

Besides which, I said Transhumans will go off-planet - not every monkey-ass primate who can wipe his nose. If the latter die by the billions, it's NOT a disaster.

Unfortunately, until said Transhumans are living in a self-contained biosphere, they are probably reliant for production of their food supply on said monkey-ass primates. Even then, the transhumans will need a self-sustaining technological infrastructure. Haven't you read Marooned in Realtime? Merely because something is stupid does not mean it is safe to remove from your local economy nor ecology. Additionally, diminished biodiversity within a species-- even (trans)humans-- is generally a bad thing; too much risk of single point of failure from epidemics and the like.

The Fields Medal is the rough equivalent to the Nobel Prize, for mathematics. It is worth less, monetarily, however.

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

Not that its a bad thing. Remember the Google Labs Aptitude Test? Or the billboard with the digits of e on it? This is more of the same. Its a way to screen out potential employees by motivation and skill in a real world environment. Think of it as a summer-long job interview at minimum-wage or less.

It has side benefits, like helping out the OSS community (that is, if the students don't do negative work, drawing more of the mentors' time than the usefulness they contribute). But, first and foremost, its about screening potential employees.

|up, down> + |down, up> is not an eigenstate of the spin flip operator. Correct.

However, that doesn't mean you can't apply the spin-flip operator to it. I don't know how to explain it in more detail, other than to try and give an experimental realization, which I don't think is what you're looking for.

Perhaps this will make more sense... let's work in "vector notation"...

define |up, up> = (1, 0, 0, 0)
|up, down> = (0, 1, 0, 0)
|down, up> = (0, 0, 1, 0)
|down, down> = (0, 0, 0, 1)

The spin flip operator for the first electron can now be written in matrix form as

0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0

Since this matrix is unitary (see http://mathworld.wolfram.com/UnitaryMatrix.html), then this is a valid time-evolution operator in quantum mechanics.

I guess I really haven't said anything new here, so I doubt that actually clarifies anything. But I'm not sure what it is you don't understand.

When read, /dev/urandom device will return as many bytes as are requested. As a result, if
there is not sufficient entropy in the entropy pool, the returned values are theoretically
vulnerable to a cryptographic attack on the algorithms used by the driver. Knowledge of how
to do this is not available in the current non-classified literature, but it is theoretically
possible that such an attack may exist. If this is a concern in your application, use
/dev/random instead.

Quote:

Incidentally, the Simplex method -- unlike differential calculus-based methods for more general problems like the Kuhn-Tucker method -- is quite programmable on a computer, and quite efficient.

In theory, the simplex method is a non-polynomial-time (NP) algorithm, and actually of the worst kind. It is an extremely clever algorithm: it jumps from point to point in order to check whether it is optimal or not, and it can do so exhaustively and in a coherent way. However, it has to check many many points, so many that in theory is the worst thing you can apply to an LP problem.

However, it works quite efficient in practice, and for reasonable problem sizes it spits the solution very fast so that someone can put a simplex solver in a loop and bootstrap it with no remorse.

Of course, nowadays, Interior Point Methods dominate the picture. They are polynomial-time (P) algorithms, they work extremely fast, and they are also very very reliable.

And here I thought it was a mark of divine sanction for the Axiom of Choice. After all, doesn't it just scream Banach-Tarski Paradox to you?

I am waaaay out of my depth, but it seems that natural lasers are pretty common.
And apparently there is an observed nebula colder than the microwave background.
And I mean that nebula is really really big, so maybe just the distibution of temperatures within it would allow for the possibility of really cold regions.
Basically my argument is against your putting "naturally" in quotes. My thesis here is that there is nothing we can do which isnt reproduced by purely unintelligent processes. All we can do is organize things and tidy them up a bit to get useful work out of them.

You can try http://integrals.wolfram.com/ instead; keep in mind that with integrals, anything complex doesn't come out "correctly". (not in a very "human" form, a lot of e^ and ln)

According the standard SI naming convention, gigabyte *is* 1,000,000,000 bytes.

If you're looking for the correct, though rarely used prefix that floats in the the computer world, it's "gibi".

kilo - 10^3
kibi - 2^10 = 1024
mega - 10^6
mebi - 2^20 = 1024^2 = 1,048,576
giga - 10^9
gibi - 2^30 = 1024^3 = 1,073,741,824

and so on.

Link here.

This is the same guy who calculated the 1 Quadrillionth hexadigit of Pi (no, not digit. It is in base 16).

This might not be as impressive as it sounds. There are algorithms that allow you to find base-16 digits in pi without calculating any of the preceding ones.

The BBP formula is one he mentions, and Bellard's formula that was actually used is listed at MathWorld under "Pi Formulas."

If he'd actually come up with one of these, I would be very impressed. As it is, it sounds like he just put the brute force behind it.

As a counter example, how about the four color theorem?

Still, if we make that: "Any idea that can't be convincingly explained without using several different colours and animation has a serious flaw in it somewhere.", I think we'll be in perfect agreement. Especially since that sort of idea usually involves dancing hamsters or so.

Is N Composite or Prime has been traditionally considered easier than actual factoring.

The page you link says "While this had long been believed possible", which means your "as many thought" is incorrect; see also the AKS news. They also say "this algorithm is still impractical".

Is N Composite or Prime has been traditionally considered easier than actual factoring.

The page you link says "While this had long been believed possible", which means your "as many thought" is incorrect; see also the AKS news. They also say "this algorithm is still impractical".

Prime Factorization = The factorization of a number into its constituent primes, also called prime decomposition.

Factoring ~= Factorization

Can you really do prime factoring in a heartbeat then?

Prime Factorization = The factorization of a number into its constituent primes, also called prime decomposition.

Factoring ~= Factorization

Can you really do prime factoring in a heartbeat then?

Prime Factorization = The factorization of a number into its constituent primes, also called prime decomposition.

Factoring ~= Factorization

Can you really do prime factoring in a heartbeat then?

In August 2002, Agrawal et al. discovered a deterministic algorithm (AKS ) for determining if a number is prime that runs in polynomial time (large polynomial though).

This means that prime testing is not NP-Hard or NP-Complete as many thought (not proved though!) it was. The same can happen to factorization. There is no guarantee that one day some smart guy will come up with a revolutionary idea that would lead many cryptosystems into oblivion. Who knows?