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Mathematica's programming language is like that too... you can code a problem in many ways, and you can code it concisely and tersely or verbosely with lots of LongFunctionNames.

quotes

page from the tour

It's an awesome language but hardly anybody uses it to the full potential (of course since it only comes with $hundreds software package that might have something to do with it....)

There's no such thing as a 16-sided die, you moron. Although if it were up to me, I think I'd rather roll a Schmitt-Conway Biprism than a regular hexakaidecahedron.

So what happens when someone is programming in some language/dev-system that won't permit non-correct program ta be wrotted? Eh?

Ironically, it's provable that no Turing Complete language can be limited to create only correct programs for any non-trivial definition of correctness. Computer science is full of such fun things.

Proof: In your supposed language, there is a set of "correct" programs for some criterion, and a set of incorrect programs.

(Sub-proof: The program "return 1" is only correct for the problem consisting of "Always return true" and isomorphic problems, and "return 0" is only correct for the problem consisting of "Always return false" and isomorphic problems. Thus, for any given correct behavior there exists at least one incorrect program in your language, OR your language contains only a trivial number of programs and as such is hardly deserving to be called a "language".)

If there does not exist a Turing Machine (read: 'program') that can decide in finite time whether a given program meets a correctness criterion, then we can never decide whether a program is correct, because Turing Machines are the best computational tools we have. Therefore, suppose some Turing Machine M exists that accepts as input some suitable correctness criterion, and a program, and outputs whether that program meets the correctness criterion (in finite time).

By Rice's theorem, no such TM machine can exist. (Indeed, it's fairly trivial to show that any non-trivial definition of "correctness" must certainly include asserting that the program executes in finite time, and therefore a program that could check such a correctness criterion must be able to solve the Halting Problem, if such a machine existed.)

Since any language that only allowed one to express correct concepts is isomorphic to a Turing Machine that correctly validates the existance in that language, and no such TM can exist, no such language can exist, unless it is trivial and the correctness criterion is trivial. In which case as I alluded to earlier, it's hardly worth calling a "language" in the computer science sense.

In fact this sort of proof is one of the reasons I seriously wonder why so many people still seem to be pursuing this. (Fortunately, I see many signs that the field is waking up to this fact and good, dare I say useful research has been starting to get done in the past few years; perhaps the empirical successes of unit testing in software engineering has helped prompt this.)

BTW, no offense but I've been on Slashdot for a while, and I direct this at any replier; if you don't even know what Rice's Theorem is, please don't try to "debunk" this post. Many exceedingly smart people have dedicated their lives to computer science. I only wish I were smart enough to craft these tools, rather then simply use them. Rice's theorum has withstood their attack since 1953 . The odds of a Slashdot yahoo "debunking" this theorem are exceedingly low. (On the other hand, if you do understand what I was saying I of course welcome correction and comment; still, this is a fairly straightforward application of Rice's theorem and I can't see what could possibly go wrong. It's a pretty simple theorem to apply, it's just the proof that's kinda hairy.)

Also, your determination that grown biological computing is unlikely to ever achieve its ultimate goal, yeah, we prove that, don't we...

I do not believe biological computing is impossible; we are an obvious counterexample. What I don't think is going to happen is that at any given point, the most powerful (man-manufactured) computing device that exists is biological, because biological systems by definition require a life support system to support them, which must consume space, power, and resources better spent directly on the actual computation by a non-livi

I used to work for Wolfram Research, where Theo Gray works, several years ago. We actually made this ice cream for one of our Tech Support deparment Christmas parties. My boss's boss, Dave Withoff (a great guy!) set up the whole thing. His friend at the local university got the liquid nitrogen for us, and we made two batches - vanilla and strawberry. Very cool.

Mathworld is an indispensible resource for anyone who relies on mathematics! The site's of exceeding quality and is extremely complete.

For movie reviews, rottentomatoes.com is pretty good.

Mathworld is great for maths- related information. A website from the makers of Mathematica-- one reason why ideologically I prefer Mathematica over Maple, even though my uni uses the latter. Wolfram is simply more involved in education.

has everything you want to know about math. Scienceworld has some cursory scientific information as well.

has everything you want to know about math. Scienceworld has some cursory scientific information as well.

I use dictionary.com a lot. It's not only useful for translating words (I'm not a native English speaker) and looking up synonyms, but also has a decent scope of information about technical terms.

Then there's of course Eric Weisstein's World of Science for everything related to physics, chemistry and mathematics.

"...mathematicians designate any theorem as 'trivial' once a proof has been obtained--no matter how difficult the theorem was to prove in the first place. There are therefore exactly two types of true mathematical propositions: trivial ones, and those which have not yet been proven."

P.S. Cantor didn't contradict that there was no largest integer; rather, he supposed that Z, N, R, etc. were infinite and then proceeded to show that |R| has to be greater than |Z| e.g. some infinities are larger than others.

I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

Q: What is often used by Canadians to help solve certain differential equations?
A: the Lacrosse transform.

The is a technique that makes certain differential equations a lot easier to solve - essentially you take a complicated D.E., substitute certain things in place of any derivatives you see by looking them up in a table, then solve the resulting equation using normal algebra, and finally transform it back also by looking up things in a table.

Q: Who knows everything there is to be known about vector analysis?
A: The Oracle of del phi!

Hmmmm, I don't get this one. Sorry. Anyone?

So, they just had to rely on the method of steepest descents.

A way to find the nearest local minimum of a function - works whenever the function is smooth near that minimum.

I know this is incredibly nerdy, but it sounds like some people would appreciate it if the jokes were explained to them...

Q: What did the constipated mathematician do?
A: He worked it out with a pencil!

OK, not going to try to explain this one.

Q: What's purple and commutes?
A: An Abelian grape.

A group is a set of things (think "numbers", but they could be sides of a cube, or colors, or anything you want) along with an operation defined on them (like addition or multiplication, but it doesn't have to work like those). When the operation on the group happens to be commutative (like 2+4 = 4+2), the group is called Abelian

Q: Why do you never hear the number 288 on television?
A: It's two gross.

A "gross" is a dozen dozen, or 144. Not a very mathematical joke.

Q: What do you get when you cross a mosquito with a rock climber?
A: Nothing. You can't cross a vector and a scalar.

The joke is referring to a Cross Product, an operation defined on two vectors. You can't take the cross-product of a vector and a scalar.

Q. How many mathematicians does it take to change a lightbulb?
A. 1, he gives the lightbulb to 3 engineers, thus reducing the problem to a previously solved joke.

When a mathematician needs to prove that A implies B, they may instead prove that A implies C where "C implies B" was already proved by someone else, or in a previous theorem.

Q: What's big, grey, and proves the uncountability of the reals?
A: Cantor's diagonal elephant.

The joke is referring to the Cantor Diagonal Argument, a proof technique that Cantor originally used to prove that even if you tried to associate one real number with every integer, there'd still be real numbers left over. (Amazingly, you can "count" the rational numbers - i.e. all of the possible fractional numbers. As a math major to show you sometime, it's a neat trick.)

Q: What's yellow and equivalent to the Axiom of Choice?
A: Zorn's Lemon.

Zorn's Lemma is a mathematical statement which turns out to be true if the Axiom of Choice is assumed to be true, or false if the Axiom of Choice is assumed to be false.

Q: What's yellow, normed, and complete?
A: A Bananach space.

A is space (a set of numbers with a lot of useful operations defined on them) that has a normalization operator defined, and is "complete", which means that the limits of all sequences you can define using numbers in the space are also in the space.

Q: What is very old, used by farmers, and obeys the fundamental theorem of arithmetic?
A: An antique tractorisation domain.

Q: What is hallucinogenic and exists for every group with order divisible by p^k?
A: A psilocybin p-subgroup.

A Sylow p-Subgroup is a certain type of subgroup (see the definition of a group above).

Q: What is often used by Canadians to help solve certain differential equations?
A: the Lacrosse transform.

The is a technique that makes certain differential equations a lot easier to solve - essentially you take a complicated D.E., substitute certain things in place of any derivatives you see by looking them up in a table, then solve the resulting equation using normal algebra, and finally transform it back also by looking up things in a table.

Q: What is clear and used by trendy sophisticated engineers to solve other differential equations?
A: The Perrier transform.

The Fourier Transform is also used in signal processing, including sound analysis and sound compression algorithms like MP3 and Ogg Vorbis.

Q:

I know this is incredibly nerdy, but it sounds like some people would appreciate it if the jokes were explained to them...

Q: What did the constipated mathematician do?
A: He worked it out with a pencil!

OK, not going to try to explain this one.

Q: What's purple and commutes?
A: An Abelian grape.

A group is a set of things (think "numbers", but they could be sides of a cube, or colors, or anything you want) along with an operation defined on them (like addition or multiplication, but it doesn't have to work like those). When the operation on the group happens to be commutative (like 2+4 = 4+2), the group is called Abelian

Q: Why do you never hear the number 288 on television?
A: It's two gross.

A "gross" is a dozen dozen, or 144. Not a very mathematical joke.

Q: What do you get when you cross a mosquito with a rock climber?
A: Nothing. You can't cross a vector and a scalar.

The joke is referring to a Cross Product, an operation defined on two vectors. You can't take the cross-product of a vector and a scalar.

Q. How many mathematicians does it take to change a lightbulb?
A. 1, he gives the lightbulb to 3 engineers, thus reducing the problem to a previously solved joke.

When a mathematician needs to prove that A implies B, they may instead prove that A implies C where "C implies B" was already proved by someone else, or in a previous theorem.

Q: What's big, grey, and proves the uncountability of the reals?
A: Cantor's diagonal elephant.

The joke is referring to the Cantor Diagonal Argument, a proof technique that Cantor originally used to prove that even if you tried to associate one real number with every integer, there'd still be real numbers left over. (Amazingly, you can "count" the rational numbers - i.e. all of the possible fractional numbers. As a math major to show you sometime, it's a neat trick.)

Q: What's yellow and equivalent to the Axiom of Choice?
A: Zorn's Lemon.

Zorn's Lemma is a mathematical statement which turns out to be true if the Axiom of Choice is assumed to be true, or false if the Axiom of Choice is assumed to be false.

Q: What's yellow, normed, and complete?
A: A Bananach space.

A is space (a set of numbers with a lot of useful operations defined on them) that has a normalization operator defined, and is "complete", which means that the limits of all sequences you can define using numbers in the space are also in the space.

Q: What is very old, used by farmers, and obeys the fundamental theorem of arithmetic?
A: An antique tractorisation domain.

Q: What is hallucinogenic and exists for every group with order divisible by p^k?
A: A psilocybin p-subgroup.

A Sylow p-Subgroup is a certain type of subgroup (see the definition of a group above).

Q: What is often used by Canadians to help solve certain differential equations?
A: the Lacrosse transform.

The is a technique that makes certain differential equations a lot easier to solve - essentially you take a complicated D.E., substitute certain things in place of any derivatives you see by looking them up in a table, then solve the resulting equation using normal algebra, and finally transform it back also by looking up things in a table.

Q: What is clear and used by trendy sophisticated engineers to solve other differential equations?
A: The Perrier transform.

The Fourier Transform is also used in signal processing, including sound analysis and sound compression algorithms like MP3 and Ogg Vorbis.

Q:

I know this is incredibly nerdy, but it sounds like some people would appreciate it if the jokes were explained to them...

Q: What did the constipated mathematician do?
A: He worked it out with a pencil!

OK, not going to try to explain this one.

Q: What's purple and commutes?
A: An Abelian grape.

A group is a set of things (think "numbers", but they could be sides of a cube, or colors, or anything you want) along with an operation defined on them (like addition or multiplication, but it doesn't have to work like those). When the operation on the group happens to be commutative (like 2+4 = 4+2), the group is called Abelian

Q: Why do you never hear the number 288 on television?
A: It's two gross.

A "gross" is a dozen dozen, or 144. Not a very mathematical joke.

Q: What do you get when you cross a mosquito with a rock climber?
A: Nothing. You can't cross a vector and a scalar.

The joke is referring to a Cross Product, an operation defined on two vectors. You can't take the cross-product of a vector and a scalar.

Q. How many mathematicians does it take to change a lightbulb?
A. 1, he gives the lightbulb to 3 engineers, thus reducing the problem to a previously solved joke.

When a mathematician needs to prove that A implies B, they may instead prove that A implies C where "C implies B" was already proved by someone else, or in a previous theorem.

Q: What's big, grey, and proves the uncountability of the reals?
A: Cantor's diagonal elephant.

The joke is referring to the Cantor Diagonal Argument, a proof technique that Cantor originally used to prove that even if you tried to associate one real number with every integer, there'd still be real numbers left over. (Amazingly, you can "count" the rational numbers - i.e. all of the possible fractional numbers. As a math major to show you sometime, it's a neat trick.)

Q: What's yellow and equivalent to the Axiom of Choice?
A: Zorn's Lemon.

Zorn's Lemma is a mathematical statement which turns out to be true if the Axiom of Choice is assumed to be true, or false if the Axiom of Choice is assumed to be false.

Q: What's yellow, normed, and complete?
A: A Bananach space.

A is space (a set of numbers with a lot of useful operations defined on them) that has a normalization operator defined, and is "complete", which means that the limits of all sequences you can define using numbers in the space are also in the space.

Q: What is very old, used by farmers, and obeys the fundamental theorem of arithmetic?
A: An antique tractorisation domain.

Q: What is hallucinogenic and exists for every group with order divisible by p^k?
A: A psilocybin p-subgroup.

A Sylow p-Subgroup is a certain type of subgroup (see the definition of a group above).

Q: What is often used by Canadians to help solve certain differential equations?
A: the Lacrosse transform.

The is a technique that makes certain differential equations a lot easier to solve - essentially you take a complicated D.E., substitute certain things in place of any derivatives you see by looking them up in a table, then solve the resulting equation using normal algebra, and finally transform it back also by looking up things in a table.

Q: What is clear and used by trendy sophisticated engineers to solve other differential equations?
A: The Perrier transform.

The Fourier Transform is also used in signal processing, including sound analysis and sound compression algorithms like MP3 and Ogg Vorbis.

Q:

I know this is incredibly nerdy, but it sounds like some people would appreciate it if the jokes were explained to them...

Q: What did the constipated mathematician do?
A: He worked it out with a pencil!

OK, not going to try to explain this one.

Q: What's purple and commutes?
A: An Abelian grape.

A group is a set of things (think "numbers", but they could be sides of a cube, or colors, or anything you want) along with an operation defined on them (like addition or multiplication, but it doesn't have to work like those). When the operation on the group happens to be commutative (like 2+4 = 4+2), the group is called Abelian

Q: Why do you never hear the number 288 on television?
A: It's two gross.

A "gross" is a dozen dozen, or 144. Not a very mathematical joke.

Q: What do you get when you cross a mosquito with a rock climber?
A: Nothing. You can't cross a vector and a scalar.

The joke is referring to a Cross Product, an operation defined on two vectors. You can't take the cross-product of a vector and a scalar.

Q. How many mathematicians does it take to change a lightbulb?
A. 1, he gives the lightbulb to 3 engineers, thus reducing the problem to a previously solved joke.

When a mathematician needs to prove that A implies B, they may instead prove that A implies C where "C implies B" was already proved by someone else, or in a previous theorem.

Q: What's big, grey, and proves the uncountability of the reals?
A: Cantor's diagonal elephant.

The joke is referring to the Cantor Diagonal Argument, a proof technique that Cantor originally used to prove that even if you tried to associate one real number with every integer, there'd still be real numbers left over. (Amazingly, you can "count" the rational numbers - i.e. all of the possible fractional numbers. As a math major to show you sometime, it's a neat trick.)

Q: What's yellow and equivalent to the Axiom of Choice?
A: Zorn's Lemon.

Zorn's Lemma is a mathematical statement which turns out to be true if the Axiom of Choice is assumed to be true, or false if the Axiom of Choice is assumed to be false.

Q: What's yellow, normed, and complete?
A: A Bananach space.

A is space (a set of numbers with a lot of useful operations defined on them) that has a normalization operator defined, and is "complete", which means that the limits of all sequences you can define using numbers in the space are also in the space.

Q: What is very old, used by farmers, and obeys the fundamental theorem of arithmetic?
A: An antique tractorisation domain.

Q: What is hallucinogenic and exists for every group with order divisible by p^k?
A: A psilocybin p-subgroup.

A Sylow p-Subgroup is a certain type of subgroup (see the definition of a group above).

Q: What is often used by Canadians to help solve certain differential equations?
A: the Lacrosse transform.

The is a technique that makes certain differential equations a lot easier to solve - essentially you take a complicated D.E., substitute certain things in place of any derivatives you see by looking them up in a table, then solve the resulting equation using normal algebra, and finally transform it back also by looking up things in a table.

Q: What is clear and used by trendy sophisticated engineers to solve other differential equations?
A: The Perrier transform.

The Fourier Transform is also used in signal processing, including sound analysis and sound compression algorithms like MP3 and Ogg Vorbis.

Q:

I know this is incredibly nerdy, but it sounds like some people would appreciate it if the jokes were explained to them...

Q: What did the constipated mathematician do?
A: He worked it out with a pencil!

OK, not going to try to explain this one.

Q: What's purple and commutes?
A: An Abelian grape.

A group is a set of things (think "numbers", but they could be sides of a cube, or colors, or anything you want) along with an operation defined on them (like addition or multiplication, but it doesn't have to work like those). When the operation on the group happens to be commutative (like 2+4 = 4+2), the group is called Abelian

Q: Why do you never hear the number 288 on television?
A: It's two gross.

A "gross" is a dozen dozen, or 144. Not a very mathematical joke.

Q: What do you get when you cross a mosquito with a rock climber?
A: Nothing. You can't cross a vector and a scalar.

The joke is referring to a Cross Product, an operation defined on two vectors. You can't take the cross-product of a vector and a scalar.

Q. How many mathematicians does it take to change a lightbulb?
A. 1, he gives the lightbulb to 3 engineers, thus reducing the problem to a previously solved joke.

When a mathematician needs to prove that A implies B, they may instead prove that A implies C where "C implies B" was already proved by someone else, or in a previous theorem.

Q: What's big, grey, and proves the uncountability of the reals?
A: Cantor's diagonal elephant.

The joke is referring to the Cantor Diagonal Argument, a proof technique that Cantor originally used to prove that even if you tried to associate one real number with every integer, there'd still be real numbers left over. (Amazingly, you can "count" the rational numbers - i.e. all of the possible fractional numbers. As a math major to show you sometime, it's a neat trick.)

Q: What's yellow and equivalent to the Axiom of Choice?
A: Zorn's Lemon.

Zorn's Lemma is a mathematical statement which turns out to be true if the Axiom of Choice is assumed to be true, or false if the Axiom of Choice is assumed to be false.

Q: What's yellow, normed, and complete?
A: A Bananach space.

A is space (a set of numbers with a lot of useful operations defined on them) that has a normalization operator defined, and is "complete", which means that the limits of all sequences you can define using numbers in the space are also in the space.

Q: What is very old, used by farmers, and obeys the fundamental theorem of arithmetic?
A: An antique tractorisation domain.

Q: What is hallucinogenic and exists for every group with order divisible by p^k?
A: A psilocybin p-subgroup.

A Sylow p-Subgroup is a certain type of subgroup (see the definition of a group above).

Q: What is often used by Canadians to help solve certain differential equations?
A: the Lacrosse transform.

The is a technique that makes certain differential equations a lot easier to solve - essentially you take a complicated D.E., substitute certain things in place of any derivatives you see by looking them up in a table, then solve the resulting equation using normal algebra, and finally transform it back also by looking up things in a table.

Q: What is clear and used by trendy sophisticated engineers to solve other differential equations?
A: The Perrier transform.

The Fourier Transform is also used in signal processing, including sound analysis and sound compression algorithms like MP3 and Ogg Vorbis.

Q:

I do know that the holy grail is the search for larger primes.

Actually, finding large primes is pretty easy. Taking a large number and finding its prime factors is not. This conjecture/proof doesn't seem to have any immediate bearing on cryptography.

TTFN

I don't think it really has anything to do with what the language "needs". It has more to do with what the programmers want. I mean, if we didn't want these features that made programming easier, we'd all be coding in rule 110.

I was about to ask the same thing, but you beat me to it.

Avogadro's number is a defined constant, so far as I can tell.

And since a molecule of C-12 is defined to be 12 amu, and since 1 mole of x-amu molecules masses x grams... isn't this already settled?

--grendel drago

Come on, the timecube guy is obviously a master at modern UI deign and html layout. :-)

Seriously though, here are some sites whose design I like:

Sweetcode

Mathworld

openrbl.org

perldoc

Paul Borke's website

the Joel On Software forums

the Tech Report (a debatable choice, but the best of its type)

Dmitry's Design Lab

The problem with creating objects that exactly obey the Golden Ratio, however, is that the ratio is given exactly by (1+SQRT(5))/2, or approximately 1.6180. It's an irrational number, not the easiest thing to work with in terms of defining dimensions of physical objects. Thus, reasonable approximations like 5:3 have long been used. Frankly, I'm not exactly sure why then the standard for widescreen televisions is 16:9, when 15:9 might have more natural aesthetic appeal. My best guess is that a 16:9, or 1.777...:1 ratio is simply slightly closer to the aspect ratio of big-screen movies than 5:3 would be, but not as wide, as presumably you might also want to use your television for television, and frankly, there's no reason to have the local evening news in 1.85:1 anamorphic widescreen. Most films shown in the theater are 1.85:1. These can generally be cropped to 1.78:1 with basically no loss of information. Sometimes DVDs are released in 1.85:1 format; these require even 16.9 televisions to provide a very small amount of letterboxing (almost unnoticeable).

And then there are those in 2.35:1 format: these films would be nearly unwatchable on standard 4:3 or even 5:3- black bars would cover well over half of the screen. Why would a film be shot in such a large ratio? Because a 2.35:1 film, projected onto a suitable screen, occupies an incredibly large chunk of your field of vision, making watching such a film an immersive experience- exactly the sort of thing that would put your butt in a seat rather than waiting to watch it in the comfort of your own home.

The Poincare Conjecture was proven last month. (Maybe.)
If the proof turns out to be correct, all your Origami is mathematically equivalent to a ball (3-sphere).
Conclusion: Nerds (who play with Origami) are now mathematically equivalent to professional sports players (who play games involving a ball). Amazing, isn't it?

(Don't try to explain this to a sports player.)

The Poincare Conjecture was proven last month. (Maybe.)
If the proof turns out to be correct, all your Origami is mathematically equivalent to a ball (3-sphere).
Conclusion: Nerds (who play with Origami) are now mathematically equivalent to professional sports players (who play games involving a ball). Amazing, isn't it?

(Don't try to explain this to a sports player.)

Well, that depends on how long you want to hold the gases. At any given temperature, the molecules of a gas at thermal equilibrium (or practically speaking, anywhere close to it) will have some distribution of speeds. Some molecules will travel faster, some slower. The mathematical expression characterizing this range of speeds is the Maxwell distribution. Here's a mathematical treatment of the Maxwell distribution; this page presents a nifty Java applet showing how this equilibrium takes place.

Note that a plot of population vs. speed, the Maxwell distribution tails off at higher velocities, but never actually goes to zero. In an atmosphere, this means that a small number of molecules will periodically get kicked up to above escape velocity through collisions with other molecules in the gas. If they happen to be heading the right direction, then they will escape into space.

Each molecule in a gas (on average) has roughly the same amount of kinetic energy. Earth's atmosphere contains very little hydrogen and helium because these light elements travel faster for a given amount of kinetic energy and escape more readily. A good part of the velocity distribution for these species is above escape velocity. Oxygen and nitrogen (not to mention water vapour and carbon dioxide) are significantly heavier, and bleed off at a much lower rate.

Moving to Mars. The surface gravity is only about forty percent that on Earth, if I remember correctly. It's a much shallower gravity well, and escape velocity is much lower (5 km/s on Mars vs. 11 km/s for Earth). Since kinetic energy is a function of the square of velocity, it takes a significantly smaller push to move a molecule out of Mars' hold. Nevertheless, there actually is still only a very small tail of the Maxwell distribution that sits above Mars' escape velocity.

I should also mention that there are sputtering processes that remove gas from the Martian atmosphere. Lacking a strong magnetic field to deflect the solar wind, a significant amount of gas is lost to sputtering, as well.

Nevertheless, even the most pessimistic estimates suggest that an atmosphere similar to Earth's would last tens of thousands on years on Mars. A short lifespan in terms of planetary evolution--a long time for human beings. Even the Moon would take from one to ten thousand years (depending upon who you ask) to bleed off an Earth-like atmosphere. Recall that Mars has surface features strongly suggestive of flowing surface water. (Liquid water requires an appreciable atmosphere, otherwise it just boils off.) That sort of erosion takes a long time to happen, which further supports the notion that Mars can hold on to an atmosphere, at least for a few million years at a time.

Absurd! Computers are a clone of the 66 year old Turing machine. See here.

>People are individuals, not averages.

If you know anything about Statistics, then Markov
and Chebychev tells us people don't stray too far from averages.

>People are individuals, not averages.

If you know anything about Statistics, then Markov
and Chebychev tells us people don't stray too far from averages.

Well I must have one lame internet connection, because I can't get those Canadian web sites to work.

I much perfer Eric's definitions of Light Year and Parsec.

Well I must have one lame internet connection, because I can't get those Canadian web sites to work.

I much perfer Eric's definitions of Light Year and Parsec.

Well I must have one lame internet connection, because I can't get those Canadian web sites to work.

I much perfer Eric's definitions of Light Year and Parsec.

Big American Physics Lesson Party Time!!!

There's plenty of us that need this, I'm sure.

1) Basic parabolic flight path
You throw a ball straight out, it curves toward the ground. The harder you throw, the farther it goes before it hits the ground.

2) The earth is round.
If you go in a perfect, absolute straight line perpendicular to the ground wherever you're standing, you'll end up going higher and higher as the earth curves away below you.

3) Perpendicular flight path on a scale where the curvature of the earth becomes a factor
If you throw something really freakin' hard, it'll go far enough for the earth to curve away before it hits the ground.
Gravity continues to pull towards the center of the earth, so the ball continues to fall in that direction. However, since the ball was fast enough to fly past the ground the first time, and since above the atmosphere there's not much friction to slow it down, the ball will keep orbiting the earth until something stops it.

You can reach whatever altitude you want maintaining 1 mph the whole time, however, you'll start falling the moment you turn off your rocket unless you get enough lateral velocity to get into an orbit.

Escape velocity is nyah an entirely different concept from orbits. Escape velocity tells you when you're going fast enough that you can turn off your rocket and keep drifting towards infinity, or how fast the muzzle velocity on your space cannon has to be to get your cannonball to Mars.

If this is incomprehensible it's because /. hates my asciiart visual aids. Taco, you square! Quit opressing my art!

more recreational math

some lego

In the first half of the last century, Hardy was very proud for being a "pure mathematician", whos results would never ever have practical implications. He worked on number theory.

1975, Rivest, Shamir and Adleman realized that the results of Euler, Fermat and Lagrange (see above) are not only true, but can be used as an asymmetric cipher. It's known as the "RSA algorithm". I think You can call that "a useful application".

It may take centuries until a mathematical truth finds it's useful application. If You only demand results with obvious practical use, you won't get far.

How Eric can keep from writing "Fuck CRC Press" on every page, I do not know.

(BTW, what a nice way to discover my ISP has a transparent proxy.)

Access Denied to IP Address 213.176.138.15

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You forgot to give your source.

It's so simple when you put it in plain english ...

It doesn't appear that the paper will survive the two years...

From the site:
It is unclear as of this writing if Dunwoody's proof will last even a fraction of that duration.

In fact, it appears that the purported proof has already been found lacking, judging by the facts that (1) the abstract begins, "We give a prospective [italics added] proof of the Poincaré Conjecture" and (2) the revised April 11 version of the preprint contains a small but significant change in title from "A Proof of the Poincaré Conjecture" to "A Proof of the Poincaré Conjecture?" In particular, a critical step in the paper appears to remain unproven, and Dunwoody himself does not see how to fill in the missing proof.

Dunwoody

It seems as if he missed a step and couldn't figure it out.

The link to mathworld.wolfram.com from the post says:

In April 2002, M. J. Dunwoody produced a five-page paper that purports to prove the conjecture. However, according to the rules of the Clay Institute, the paper must survive two years of academic scrutiny before the prize can be collected.

So, why the excitment about this later Perelman paper? Has the Dunwoody paper been debunked?

Division by zero is completely meaningless. Yes there are cases where division by zero creates a removable singularity, and for continuity's sake you can define a new curve/sequence/function/whatever with the convenient value. But that doesn't make the division meaningful...

You're assuming that the familiar logical system of Calculus 101 is the only way of defining concepts like "zero" and "infinity". But that's not true. There are alternate approaches that I'm not qualified to get into (basically, some mathematicians are trying to resolve the ambiguities Newton tapdanced around when he invented Calculus).

And even if you don't get into that kind of quibble, division by zero is only undefined in the limited context of "standard" real numbers. It makes perfect sense to define division by zero in the context of complex numbers.

Ah you cry, but the time taken by a pendulum does not depend on the weight, well yes but the pennies slightly raise the center of gravity of the bob you see...

Slightly off-topic, but interesting none the less: a normal pendulum does not take the same time to reach the bottom no matter where it is released, contrary to popular belief. The correct curve is actually an inverted cycloid, and the finding of this curve was deemed the 'tautochrone problem.' Obligatory mathworld linkage: Tautochrone Problem.
Of course, this is an example of where reality and theory conflict: constructing a clock with a tautochronic (a made up word?) pendulum wouldn't matter enough due to friction etc.; the semi-circle is just fine for pendulum based clocks. =)

Prime numbers are defined to be both integers and positive.

At least, that's what Eric Weisstein thinks.
So do The Prime Pages

Gotta be positive, por favor.

here

Don't forget the Prime Spiral.

This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting. While drawing a grid of lines, he decided to number the intersections according to a spiral pattern, and then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam's slightly more formal prose, it "appears to exhibit a strongly nonrandom appearance"

More info.

I think the conventional 0.01s to 1000s figures are how long they last in gamma rays. Do the bursts last longer in other forms of light?

Also, does anyone know what the Wolf-Rayet star has to do with anything? Is it a possible burst candidate? I read the origional article and the gsfs.nasa.gov link and I didn't find any mention of this.

Does the "Wolf-Rayec" star classification refer to a massive star about to collapse?

All Astronomy Picture of the Day ( here) says about it is that Wolf-Rayec stars are around 40x the mass of the sun and provides a broken link :(

To confuse the issue more, Weisstein's World of Science ( here )says Wolf-Rayec stars are:
"A type of star whose spectra consist of very wide emission bands as well as absorption lines in the violet region. These lead astronomers to conclude that these stars are surrounded by rapidly expanding shells of gas. Wolf-Rayet stars are classified as irregular variable stars, and are sometimes also called W stars."

From this one might assume Wolf-Rayet stars might already have undergone an event which might have caused a GRB (gamma ray burst)?

Part of the problem of mathemtics is that there is only a finite symbol set available to us (at least with TEX), so we tend to use the same symbol to mean different things in different fields. I'd try to pick up a book that has an index of notation. (Most have them, you just have to remember to look.) Otherwise, start with an introductory advanced math text (Eggen, Smith, and St. Andre, A Transition to Advanced Mathematics comes to mind), and that should give you the foundations to move onto other books, as any good book will introduce any specialized notation. Another good resource is MathWorld. You can't exactly type in the symbols that you want, but you can search on terms that are appearing around the symbol to try to get a topic, and then things are well cross-referenced, so you can back up to a lower level of understanding if needed.

-Shylock0

But doesn't it [Ceres] have a satellite? -- and -- What would we qualifty that as, because a satellite must orbit a planet.

It doesn't appear that Ceres has any satellites. But, there are 31 asteroids that do! That doesn't make them planets though...they're just small asteroids with really small moons.

Can anyone remind me what that sequence of numbers is called that vaguely predicts the distances of planets from the Sun?

Yep, its the Titius-Bode Law. Ceres does fit into this. But the reason we don't have a planet in between Mars and Jupiter is because "many astronomers think the asteroid belt is where a planet tried to form, but was pulled apart before it could solidify, caught between the strong opposing tugs of Jupiter and the sun's gravity." Quote taken from here.

Why does a planet _have_ to be a shpere...How perfect a sphere?

Well.... Ceres's shape is too distorted. Its shape is not spherical enough to be like regular planets. And, to get really technical, no planet is really a sphere. Due to rotation, all planets have a slightly distorted shape.

Everyone who reads memepool would probably think the same. :)