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It is not that de Branges is unqualified: he settled Bieberbach's Conjecture. Interestingly, much of the validation of de Branges work on Bieberbach's Conjecture was done by a team at the Steklov Institute, referred to in the MathWorld link in the article.

how much mass do we need to send out to stop the earths rotation totally

The angular momentum of the earth is approximately 7e27 kg m^2/s.

One kilogram on the earth's surface at the equator has angular momentum of about 3e9 kg m^2 /s. One kilogram in geosynchronous orbit has angular momentum 1.3e11 kg m^2/s. So, the earth can spin up approximately 5.4e16 kg to geosynchronous orbit, which is perhaps one hundred millionth of its mass. It's also about a million times the total amount of freight shipped in the US yearly.

Note that bringing any mass down from the elevator will spin the earth back up.

This helpful link will make it all clear.

I thought I was the only one! I'll get stuck on Wikipedia, Mathworld and Everything2 for hours on end, using Google as a sidebar for non-linked terms. By the time I'm done, I'll have like 20-25 different browser tabs open, detailing my trek from A to B.

Be careful: NP \ P != NP-complete.

A problem is only NP-complete if it is both in NP and in NP-hard. Many problems exist which we know are not P, but have not been able to show to be NP-complete.

In fact, according to this article, although we know that integer factorization is in NP, "[i]t is suspected to be outside of all three of the complexity classes P, NP-Complete, and co-NP-Complete".

Another interesting problem (not necessarily related to prime factoring) is graph isomorphism, which "seems to fall in a crack between P and NP-complete".

Too tired to think about this really, but I think you're thinking of region coloring, whereas the post you responded to was talking about vertex coloring. It's easy to make graphs that require more than four colors to color neighboring vertices differently, see here. Sounds like you're aware of this, just got switched around when you read it. Hope that helps. (I can tell I'm tired because it took me three tries to get "vertices" right...)

What do you make of this then?
The continuum hypothesis, first proposed by Georg Cantor, Eric Weisstein's World of Biography holds that the cardinal number of the continuum is the same as that of Aleph-1. The surprising truth is that this proposition is undecidable, since neither it nor its converse contradicts the tenets of set theory.

Isn't all information potential file data? Is Microsoft really doing something different than has been done before?

The article also states "WinFS uses a direct acyclic graph of items (DAG)."

The math goes back to the 1970's, as referenced by MathWorld Old math can be used in new ways. Is his a new way when it's used in the FS that Microsoft is attempting?

The articles also says: " the WinFS data model provides the following concepts to describe data structures and organizations: * Types and subtypes. * Properties and fields. * Relationships. * Constraints. * Extensibility. "

Does the new Reiser4 file system support any of these concepts? -- Is WinFS really as new and exciting as the marketing and media says it is?

Thanks.

Tonight on Fox: when being a smartass goes wrong.

I know, it's really George Lucas' fault.

The stuff shown by quantum mechanics is entirely physical, and you can see its effects quite easily. You're using the behavior of an electron encountering a step potential right now to read this post. If that doesn't satisfy, then just take a look at any image from an STM taken around atomic resolution.
Corral See all the waves?

The call for physical proof is significant:
mathematics can describe things which don't/can't exist in this universe. Just because mathematics says that I can take any solid and rearrange it into any other solid using rigid motions, even if they don't have the same volume (Banach-Tarski Paradox) doesn't mean that I should expect to actually accomplish this on any physical object.

Do these people even have electricity? Maybe we should be examining our priorities here... Clean drinking water for everyone, or email? I'd don't know about you guys, but I'd take food and water over 32 messages about increasing the size of my pen1s!

Perhaps this will clarify:

Physicist Paul Dirac

Repeating Decimal on MathWorld. 1/2 = 0.5 = 0.49999... - I'm not sure what's philosophical about it, it's perhabs a bit counter-intuitive, but many things in Maths can be so, and as far as I know the equality is a proven fact and not really open for philosophical discussion...?

Pulsars are rapidly rotating neutron stars with periods less than ~3.75s. When they were first discovered at the radio telescope at Jodrell Bank, England, their origin was unknown and they were thought to possible be signals from extraterrestrials. As a result, the first pulsar was named LGM-1, with LGM standing for "Little Green Men."

Verticle applications will be a different story, but there's very little that can run on LINUX that can't run on Solaris already. Never mind what gets written in Java.

Other neat tricks on Google:

1. Do a search on "1" and then "2" ... "9" and for each number plot the number of hits returned versus the number, and you end up with something like Benford's Law.
2. On Babelfish, enter a phrase in one language, and keep translating it through the set of languages until you hit a limit cycle.

We see a full moon every 28 days.

We see a full moon every 29.5 days on average. See this page for the computation and exact value of the synodic period.

when you buy a new 1000baseT ethernet card it can potentialy hit 1,000,000,000 bits per second. The storage and networking worlds use the true meanings of Kilo = thousand, Mega = million, Giga = billion, Tera = trillion, Peta = Quadrillion ...

You don't see peaple bitching that they paid to travel a Kilometer in a taxi but they only went 1000 meters...

The problem is actually that WAY back in conputing history the term kilo was misappropriated because there was no verbal shorthand for 1024. Since then we have come up with the alternative terms of KiBi(KiloBinary) = 1024(2^10); MeBi = 1,048,576(2^20); GeBi = 1,073,741,824(2^30); TeBi = 1,099,511,627,776(2^40)...

Check out http://mathworld.wolfram.com/Byte.html
http://www.bipm.org/en/si/prefixes.html
http://physics.nist.gov/cuu/Units/binary.html
for more info.

I must admit that they sound a bit funny though...

If, however, the phrase "exists one and only one straight line which passes" is replaced by "exists no line which passes," or "exist at least two lines which pass," the postulate describes equally valid (though less intuitive) types of geometries known as elliptic and hyperbolic geometries, respectively.

There was a furor that surrounded the nuetrino when it was first thought up and they did think that it was so weakly interacting that they'd never find it. (...we found it)

The Higgs boson is another case in point; (...). Do we write off the Higgs boson because we don't have a detector for it?

The history of physics is full of theorized particles or fields that never panned out. We just never read about them in textbooks, because who wants to learn things that are wrong. We tell glamorous annecdotes about the hard to find particles that were discovered (e.g. Neutrino) and ignore the theories that were failures (e.g. Ether). So, our perceptions are colored, and we over-value theory.

Please don't fall into that trap. Look at things objectively and critically. No matter how good the theory looks, experimental confirmation is necessary. Physicists believed in the existence of Ether right up until the 1887 Michelson-Morley experiment.

PS. I wanted to provide links to more examples than just Ether, but for the exact reasons I outlined above I had diffculty locating web pages on other failures.

I mean really.... because there's never been a movie/tv show ever that tried to portray a car as having emotions....

Let's get this clear: Yes, car's have been portrayed as having emotions lots of times.

That is not original.

What is original is proposing this as a real technical solution to an actual problem, as opposed to simple anthropomorphy.

Since human faces have been drawn forever, I suppose the idea of Chernoff Faces would be obvious too. It is not.

Free fall = orbit = Effect of gravity is negated by the centripital force. ( things do not orbit around something else, they both orbit around a common mass point ) For the Gravity Vector and Centripital vector, If equal, then object stays in sky, due to orbit. Zero gravity = stuff flies off into space. Escape velocity = sustained centripital force nessesary to overcome force of gravity. Free fall in vomit_Comet is a type of orbit. ( kind of like the orbit of a comet crashing into the earth ). http://www.astronomynotes.com/gravappl/s3.html ( as long as the distance is finite, there is some gravitational force ) Henry Cavendish measured the gravitational constant with freaking CANDLES! http://scienceworld.wolfram.com/biography/Cavendis h.html http://www.geocities.com/neveyaakov/electro_scienc e/cavendish.html

Does this mean we can expect the whole dating-and-mating process to be reduced to an algorythm?

Nah, if there is an algorithm for that at all, you can bet your a** it's going to be of exponential complexity. Not only that, but if it is ever solved, the universe will implode or something equally nasty. There are some things man is not meant to tinker with. Stick with trying to reduce NP-complete problems to polynomial time, it's much safer and easier. Or you could try solving the good old Collatz Problem.

Yes, I'm single ... why do you ask?

The biggest problem with cannibalism, is that you spread certain diseases more easily. Things like Mad Cow Disease, is an example, bus just similar things applied to humans. If there is this Extinction Level Event (ELE) then we need to ensure the gene-pool survive, I mean make use of it, to get rid of AIDS. But then we are back at the level where the NAZI's where and that is a loss in humanity no matter how you look at it.

Arguably the strongest will survive, but the sum-of-the-parts will be lower then the whole. The people we need to survive is not necessarily the "strong". Nerds and geeks is what we need, basically a lot of engineers (yup, I am an Engineer), and then physical labor to build things quick or sacrifice themselves for the greater good, like the firemen of Chernobyl. Have you ever considered what will happen once there are no people to look after Nuclear Power plants, I would imagine in a countries like the USA these plants are geographically so densely distributed that if one fails the fall-out will trigger evacuations and thus failures of the others in close proximity, ultimately leaving you with a cascading domino-effect.

The people that you will need to solve this type of problem is not really the "strong". I would imagine, the "red-necks" will probably have the highest change of survival but then humanity will be gone, and we will be back in the dark ages.

I would imagine either things turnout or at least start to happen as it did in the movies, Deep Impact (FEMA) and possibly a messed up society as in Twelve Monkeys or even Mad Max.

As an engineer my biggest problem, would be to effectively restart our technology level in preferably less then 10 years. Else we might loose it altogether. Knowledge transfer would be important, else we might have a society like The Postman or the TV series, Jerimaia. The problem with most of today's technology is that the components are extremely specialized, you need high-tech to replicate them, and most parts are incremental improvements on a previous version. For example, you need a C compiler, to compile the new-improved C compiler. Some technologies are extremely recursive in this manner. So to the point that if a crucial part is lost, it might be lost forever. Basically how many technologies, can you start from ground principles ?

Things like a surface mount sintered Tantalum Cap, or a microprocessor, or even a DVD reader, to learn new generations about reading, math, etc. Can you restore your backups ? Maybe without the magnetic mantle we might have a huge EM pulse that messes up all magnetically stored data. If the step-function input is too big, we might be back in the dark ages.

Ok one part of me feel like, yeah !!! But it is the transients before we get there that scares me.

Once the GPS system fails it will have repercussions on everything that depends on it. Hmmm, NTP for one. Some utility companies even use it to monitor electric load on the powergrid, the mass movement of charge, etc. Most complex control systems are useless without accurate inputs. So how big was the "margin-of-error" people designed in that lowest-bidder control systems for that Nuclear Power Plants ? If you look at what happened at Chernobyl with un-self-sustained Nuclear Powerplants you have to start worrying.

If that is not enough to worry about, what will the effects be on the worlds international food supply ? I think we have all started to notice the "Weird" weather. Zetatalk (of Planet X fame) has nice pictures correlating the changes in the magnetic field with changes in temperature. I mean evolution happens over millennia can a significant part of the Earth's food supply handle a severe step-function input, and what will the transient response be like ?

Also have a look at these:
http://www-ssg.sr.unh.edu/406/Review/rev6.html
http://ds9.ssl.berkeley.edu/LWS_GEMS/movies/6magne t.mov
http://ds9.ssl.berkeley.edu/LWS_GEMS/movies/6sec6. mov

See Sampling Theorem.

If I don't make much sense to you (frankly, I don't make much sense to me), head over to Mathworld where I'm sure they do a much better job of explaining this.

However I was surprised to learn that the top human checkers players can easily trounce the computer. I would guess that checkers would be orders-of-magnitude a "simpler" problem than chess. Maybe it's that chess gets all the buzz, since it's considered to be the ultimate thinking-man's game.

Checkers has all but been solved. See this Mathworld article for more info. Basically, there's an estimated 10^12 to 10^18 different positions in a game, with a possibility for only having to solve 10^9 of them. With sufficient memory (Beowulf cluster, anyone?) checkers can be completely solved such that you can guarantee either a win or at worst a draw for the first person to move.

But thank you for playing.

(Check Mathworld for the details.)

I hold that the chaotic nature driven by these un-modelable nonlinearities which extend beyond the molecular level will mean we will never develop accurate long-term forecasts, although our statistical predictions should hold out fine.

yes, stats are clearly limited as a modeling tool, I prefer the 'let us simulate realiity' or modeling approach (ie: cellular automata), it gives a better understanding of the problem at hand. Stats are just as good as data set are, no more, but models reflect our own (lack of?) knowledge of the modeled thing.

Sure. Acceptance is sigmoidally[1] related to effectiveness.

You'll never get 100% effectiveness of a policy, but below a certain acceptance level you'll get almost no effectiveness, and above a slightly higher acceptance level you'll get almost total effectiveness.

Working to decrease acceptance is the opposite of political effectiveness; and the people working the system now are working against themselves, and getting us killed in the process. So they try to prevent us from knowing how to kill, as though that will prevent others from killing us.

If a model rocket was a Glock, this wouldn't just be a story on a corner of /.

[1] Like this sigmoid with translation and scaling to fit in the box from (0,0) to (1,1).

True white light (like what you get from the sun) consists of an equal spread of energy across all frequencies.

Actually, the Sun's spectrum is close to a blackbody spectrum, which is not "white" in the exact sense you mention. (Hence why you'll hear the Sun referred to as a "yellow star," and other stars referred to as "blue".) It also has a series of absorption lines in it.

It's an interesting proposition. Anybody with a math background learning Go for the first time has probably thought of similar things; the board is so abstracted (there are no "home rows" as in Chess, for example.)

But the same thing could have been accomplished much more quickly by simply displaying a 2-d board and defining how the edges are connected (see mathworld for the notation.) The usual Go board configuration is the "disc", but you could play Go on the Klein bottle if you wished.

In the end, I think topological Go would be easiest played in 2 dimensions with the players keeping in mind the edge identifications (hey, they're Go players, they're smart), but it would be very cool to do game replays with the full 3-d effect. It would probably be rather intimidating for the punters but, hey, they're Go players...

(Would some people be better at various Go topologies? That would be interesting. Perhaps a 1d disc player would be a 3kyu Klein bottle player. I doubt it, but if it did happen, it would be quite interesting. Are there different skills that come into play?)

It's an interesting proposition. Anybody with a math background learning Go for the first time has probably thought of similar things; the board is so abstracted (there are no "home rows" as in Chess, for example.)

But the same thing could have been accomplished much more quickly by simply displaying a 2-d board and defining how the edges are connected (see mathworld for the notation.) The usual Go board configuration is the "disc", but you could play Go on the Klein bottle if you wished.

In the end, I think topological Go would be easiest played in 2 dimensions with the players keeping in mind the edge identifications (hey, they're Go players, they're smart), but it would be very cool to do game replays with the full 3-d effect. It would probably be rather intimidating for the punters but, hey, they're Go players...

(Would some people be better at various Go topologies? That would be interesting. Perhaps a 1d disc player would be a 3kyu Klein bottle player. I doubt it, but if it did happen, it would be quite interesting. Are there different skills that come into play?)

And besides, once you are in space, without having to worry about air resistance, it's trivially easy to build up that extra velocity.

The lack of dynamic pressure in a vacuum is one of the reasons rockets build velocity so fast at the end of flight. But much more important is that the rocket is relatively light. The momentum of the rocket exhaust that barely moved the rocket at take off now makes the lighter rocket scream.

Check out the physics here

Generally somewhere between 250-300 km (where air drag starts to become important) and 1000 km (where the inner van allen radiation belt starts to get serious). Low earth orbit usually implies a modest inclination to the equator, (i.e., the lowest achievable from the launch site). The Space Shuttle flies in low Earth orbit.

For more information see this article from ScienceWorld

Since all we know is that the die has six sides we can simplify it to say that the odds are 1 in 6. And since we believe that there are two possible outcomes for this star, we assume that the odds are 1 in 2.

That reminds me of a large set of advanced math information published by Wolfram Research: see mathworld.wolfram.com. Individual pages have many hyperlinked terms which are very handy when exploring a certain subject. I don't know if I agree that it's a better way to teach the material though; I think books can work just as well, as long as the quality is there. After some time spent getting familiar with a good textbook, I find that I start building a sort of a mental map, and I can quickly flip pages to look stuff up. This is similar to how links work.

Someone, I think his name was Dedekind, might disagree... check it out.

Dedekind was a contemporary of Cantor, and proposed a clever definition of real numbers which conceives of each as a pair of sets. All members of S1 are less than any member of S2, and furthermore, S1 has no greatest member. This is a perfectly consistent (and interesting) formulation of the reals by an eminent 19th century mathematician; surely it can't be too silly to refer to real numbers as a set.

Actually, that's a *terrible* way to generate random numbers. Even discounted psychological factors (ask someone to pick a number between one and ten, and there's a significantly higher probability than 1/10 that they'll pick 7), it's pretty flawed: the chances of a person in the street picking an integer are pretty high, even though there are uncountably more real numbers than integers. The chance of them picking a number less than, say, Graham's number are pretty high, but almost all numbers are larger than g64. Moreover the Kolmogorov complexity of any number they can name in a finite time will by definition be finite - whereas the vast majority of numbers are of infinite Kolmogorov complexity and hence inexpressible. This last thing is what I believe the author is getting at when he tries to define randomness as a quantity.

Actually, that's a *terrible* way to generate random numbers. Even discounted psychological factors (ask someone to pick a number between one and ten, and there's a significantly higher probability than 1/10 that they'll pick 7), it's pretty flawed: the chances of a person in the street picking an integer are pretty high, even though there are uncountably more real numbers than integers. The chance of them picking a number less than, say, Graham's number are pretty high, but almost all numbers are larger than g64. Moreover the Kolmogorov complexity of any number they can name in a finite time will by definition be finite - whereas the vast majority of numbers are of infinite Kolmogorov complexity and hence inexpressible. This last thing is what I believe the author is getting at when he tries to define randomness as a quantity.

Well, it depends on what you mean by practical. On a computer, the only representable numbers are integers and rational numbers, both of which belong to countable sets. (Recall that 3.14159 is the rational number 314159/100000, so all the floating point numbers in a computer are rationals.)

The distinction between countable and uncountable sets (integers vs. reals, for example) is very important in mathematics for establishing properties about functions (e.g. calculus), measuring the size of sets (e.g. probability), and determining whether two sets share similar properties (e.g. topology). Outside the domains of pure and applied mathematics, I'm not sure that the abstract concepts of countability and uncountability are all that practical.

As a historical aside, the mathematician Georg Cantor, who discovered that there are different sizes of infinity, spent his last days in an insane asylum.
These links may be interesting.

Given an infinite supply would it matter if you counted by ones or by tens? How would this differ?

Your question could be interpreted this way: What is the difference between the sets A = {1, 2, 3, 4, 5, ...} and B = {10, 20, 30, 40, 50, ...}? Well, clearly set A contains set B as a proper subset, so A has elements that B does not. However, both sets are countable, since the function f(x) = 10x establishes a one-to-one correspondence between A and B.

Well, it depends on what you mean by practical. On a computer, the only representable numbers are integers and rational numbers, both of which belong to countable sets. (Recall that 3.14159 is the rational number 314159/100000, so all the floating point numbers in a computer are rationals.)

The distinction between countable and uncountable sets (integers vs. reals, for example) is very important in mathematics for establishing properties about functions (e.g. calculus), measuring the size of sets (e.g. probability), and determining whether two sets share similar properties (e.g. topology). Outside the domains of pure and applied mathematics, I'm not sure that the abstract concepts of countability and uncountability are all that practical.

As a historical aside, the mathematician Georg Cantor, who discovered that there are different sizes of infinity, spent his last days in an insane asylum.
These links may be interesting.

Given an infinite supply would it matter if you counted by ones or by tens? How would this differ?

Your question could be interpreted this way: What is the difference between the sets A = {1, 2, 3, 4, 5, ...} and B = {10, 20, 30, 40, 50, ...}? Well, clearly set A contains set B as a proper subset, so A has elements that B does not. However, both sets are countable, since the function f(x) = 10x establishes a one-to-one correspondence between A and B.

There are only so many properties of real numbers that we can actually describe in words. (Same number as the cardinality of N.) Most of these properties are either almost always true (i.e. the set of numbers for which it's not true is not dense in R) or almost always false. A random number is one that is on the "almost always" side for each of these properties; i.e. transcendental, normal, non-computable, non-compressible, etc.

There are only so many properties of real numbers that we can actually describe in words. (Same number as the cardinality of N.) Most of these properties are either almost always true (i.e. the set of numbers for which it's not true is not dense in R) or almost always false. A random number is one that is on the "almost always" side for each of these properties; i.e. transcendental, normal, non-computable, non-compressible, etc.

The sorts of people who reject the Axiom of Choice (disclaimer: I'm still undecided on the matter) insist on a "constructive" set theory--meaning you can't pull examples of sets that "ought to exist" out of thin air, you have to build them out of the Zermelo-Fraenkel Axioms (minus the Axiom of Choice, of course).

They have a distinction between truth and provability. A statement is true if no counterexample exists (can be constructed), and a statement is provable if there exists a proof of it using the ZF axioms. Using the words "truth" and "provability" in that way, it's clear that the unprovability of the continuum hypothesis is itself proof of its truth. If a counterexample could be constructed (a set with cardinality greater than that of the integers and less than that of the reals), the hypotheis would be provably false. But since it's known to be unprovable, it must be impossible to construct such a set. And the nonexistence of a counterexample is the definition of truth.

It may not actually be inconsistent to use a version of set theory that includes the negation of the continuum hypothesis as an axiom (I'll call it the NCH axiom for Negation-Continuum-Hypothesis), but very few mathematicians (even those who accept the axiom of choice) would accept such a system. Informally, axioms are supposed to be self-evident truths. Even the Axiom of Choice merely extends a statement that is provably true in the finite case to the infinite case, but the NCH axiom asserts, for no self-evident reason, the existence of an exotic set with properties that aren't even trivial to define. The Continuum Hypothesis is technically unprovable, but unless you're actually doing formal mathematics you can safely think of it as true.

I thought "omega" was already taken in number theoretical circles--the surreal number consisting of up-up-up-up-... ("up-hat")? Hell, Cantor broke out the Hebrew numbers to express his weird idea. That link uses omega in its own way. This guy really should have tried a little harder.

defining what a random number is supposed to look like (without circularly using the word 'random') is impossible. If you can define exactly what it should look like, then you can use that definition to create (or compress (see below)) a random number. It would not, then, be random

defining what a random number is supposed to look like (without circularly using the word 'random') is impossible. If you can define exactly what it should look like, then you can use that definition to create (or compress (see below)) a random number. It would not, then, be random

FYI, the continuum hypothesis is neither true nor false (or BOTH true and false, depending on how you think about it :).

It is independent of the rest of set theory... much like Euclid's parallel postulate is to geometry. You can assume it's true, or assume it's false, and you get different versions of set theory in the end. Similar to the existence of both euclidean and non-euclidean geometries.

Many people don't realize that there are multiple versions of something as fundamental to mathematics as set theory! Check out the Axiom of Choice for another example of something that's neither true nor false in set theory.

My favorite proof involving cardinality and set theory is the proof that there are the same number of integers as fractions... so simple that a school kid can understand every step, yet so profound a conclusion!

FYI, the continuum hypothesis is neither true nor false (or BOTH true and false, depending on how you think about it :).

It is independent of the rest of set theory... much like Euclid's parallel postulate is to geometry. You can assume it's true, or assume it's false, and you get different versions of set theory in the end. Similar to the existence of both euclidean and non-euclidean geometries.

Many people don't realize that there are multiple versions of something as fundamental to mathematics as set theory! Check out the Axiom of Choice for another example of something that's neither true nor false in set theory.

My favorite proof involving cardinality and set theory is the proof that there are the same number of integers as fractions... so simple that a school kid can understand every step, yet so profound a conclusion!