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Has anyone thought of the idea that the universe is really shaped like a Klein Bottle? The interesting idea (if you believe this) is that the Universe would have zero volume and infinite surface area, since its boundary would wrap around like a mobius strip.

If someone speculates that Universe shapes like a horn, it could very well be a piece of the Klein Bottle.

I've ranted here before about the shoddy reporting that the New Scientist does. It's very curious to me that the only matches on Google for "Picard topology" are from this article. Can anyone shed some light on this situation? Picard groups are certainly well-known enough. If nothing else, it's something to be skeptical about. Is this really so new that nobody has ever mentioned in on the web, or is it just poor terminology? (Note that one of the scientists is quoted as using that term, but it's phrased in a way that makes it sound like the reporter put words in his mouth.)

Basically it's a cryptographic method that allows the same or nearly the same level of security as a regular public-key encryption scheme(based on factoring large numbers) but makes it computationally cheaper to encrypt the data.

Mostly right. ECC is based on the Discrete Log Problem, not factoring. The Discrete Log Problem is basically: given x, y find g such that g^x = y. That's easy for real numbers - you just take a log. The problem becomes rather more difficult in the case where you are working with integers mod some prime - that is, find an integer g, such that g^x mod p = y. That gives you Diffie-Hellman and El-Gamal. ECC is the same problem, but over the group of points of an elliptic curve over a finite field. You can show that this class of groups effectively maximises the difficulty of the Discrete Log Problem, and that's why the key sizes and computational efficiency is so much better.

Jedidiah

Basically it's a cryptographic method that allows the same or nearly the same level of security as a regular public-key encryption scheme(based on factoring large numbers) but makes it computationally cheaper to encrypt the data.

Mostly right. ECC is based on the Discrete Log Problem, not factoring. The Discrete Log Problem is basically: given x, y find g such that g^x = y. That's easy for real numbers - you just take a log. The problem becomes rather more difficult in the case where you are working with integers mod some prime - that is, find an integer g, such that g^x mod p = y. That gives you Diffie-Hellman and El-Gamal. ECC is the same problem, but over the group of points of an elliptic curve over a finite field. You can show that this class of groups effectively maximises the difficulty of the Discrete Log Problem, and that's why the key sizes and computational efficiency is so much better.

Jedidiah

Basically it's a cryptographic method that allows the same or nearly the same level of security as a regular public-key encryption scheme(based on factoring large numbers) but makes it computationally cheaper to encrypt the data.

Mostly right. ECC is based on the Discrete Log Problem, not factoring. The Discrete Log Problem is basically: given x, y find g such that g^x = y. That's easy for real numbers - you just take a log. The problem becomes rather more difficult in the case where you are working with integers mod some prime - that is, find an integer g, such that g^x mod p = y. That gives you Diffie-Hellman and El-Gamal. ECC is the same problem, but over the group of points of an elliptic curve over a finite field. You can show that this class of groups effectively maximises the difficulty of the Discrete Log Problem, and that's why the key sizes and computational efficiency is so much better.

Jedidiah

He's right, y'know... Solvable, in the sense of having a solution, is not the same as being computable.

Just because there is no algorithmic method of showing the termination of a program does not neccessarily mean that a proof of termination or non-termination doesn't exist (or that the program neither terminates or does not).

Cheers,
Craig

He's right, y'know... Solvable, in the sense of having a solution, is not the same as being computable.

Just because there is no algorithmic method of showing the termination of a program does not neccessarily mean that a proof of termination or non-termination doesn't exist (or that the program neither terminates or does not).

Cheers,
Craig

He's right, y'know... Solvable, in the sense of having a solution, is not the same as being computable.

Just because there is no algorithmic method of showing the termination of a program does not neccessarily mean that a proof of termination or non-termination doesn't exist (or that the program neither terminates or does not).

Cheers,
Craig

For those who don't understand why mathematicians are always striving for better than "probably true", the Mertens Conjecture [Mathworld] is the commonly given example. It was thought to be true. As the Mathworld article notes, as late as 1985, arguments were made that there was no counter-example for numbers less than 10^20 or even 10^30. However, in 1987, someone proved a counter-example DOES exist. It is somewhere between 10^12 and 10^65. In short, you could process billions of integers and find that the conjecture held, leading you to believe it was "probably true". But it is actually false.

The "diameter" you are measuring doesn't pass through the center of mass. If it did it would be a circle - as 5 seconds of thinking would show you.

It's completely different from the bicycle since the bicycle uses straight sides on the wheel and a curved surface - the exact opposite of the coin.

It's the seven side version of a reuleaux triangle, the center of mass oscillates as it rolls. Mark the center of mass and roll one if you can't prove it to yourself with less than 5 seconds of thought.

And it does not have constant "diameter" is has constant "width" - the two terms mean different things (and are the same for a circle and only a circle).

See http://mathworld.wolfram.com/CurveofConstantWidth. html .

Most 'random' number generators are actually pseudo-random numbers. And actually have a pattern.

Chaotic systems are actually quite controlloable in a very interesting way. The key property that makes a chaotic system so unpredictable is divergence -- if two copies of the system differ by delta, that delta will grow exponentially in time (doubling according to a coefficient call the Lyapunov coefficient). Yet, the divergence is never arbitrary. Instead, the divergence in chaotic systems happen within a space called the strange attractor - the diverging trajectories stay within in the attractor zone even as the split from each other.

If you map the strange attractor and nudge the system are the right point of the cycle, you can push the system into what ever mode of behaviro you want. Although you cannot predict the longterm behavior of the chaotic system, you can perturb it periodicaly to stabiize it or rapidlly shift its behavior. Scientists are looking at how to use this chaotic control theory to control unstable systems such as ultrahigh power lasers, manuerable jet aircraft, and heart tissue.

The key controlling a chaotic system is to understand how the chaotic system diverges (the shape of the strange attractor) and use that knowledge to deftly inject perturbations at just the right moment.

If you can evacuate the tunnel, and run maglev, then if you dig your tunnel correctly you would need no extra power to make it run: with a gentle slope going down and then back up you could let the train run under the influence of gravity, with zero departure speed and zero arrival speed. I wonder how long the trip would take.

Well, if we ignore friction then you don't actually have to worry about how you dig your tunnel, so long as the gravitational potential (roughly: height) of your destination matches that of your departure.

Ignoring the rotation of the earth and nonuniformity of its gravitational field, your trip would be fastest if the path were cycloidal, as this solves the brachistochrone problem. In this simple view, the problem is basically just finding a satellite orbit connecting the two points.

Accounting for the rotation of the earth is probably just a matter of setting up your path so that the earth rotates "just right" underneath it. However, the gravitational field will change quite a bit on your ideal path, since (roughly speaking) only those parts of the earth closer to the center than you are have any gravitational effect. So you've got to solve the brachistochrone in a varying gravitational field.

Maybe I'll solve that one sometime.

See article

I believe the number of primes less than n tends asymptotically toward n/log(n) as n increases. Given how big 2048 bit numbers are, I would suggest we can take n/log(n) as a pretty accurate count for ballpark work.

So, working sketchily, we're talking roughly 2^2040 primes in that range (note, that's pure ballpark, but it does give a rough order of magnitude - we're talking LOTS of primes, not just a few).

Jedidiah.

To first order, the speed of sound does not depend upon the pressure at all; rather it depends primarily upon the mean mass density and the temperature.

The reduction of sound speed at altitude is due to the reduction of temperature. The temperature rises again in the upper stratosphere (ozone heating) and then drops down to its coldest temperature at the mesopause (around 120 K, at 85 km). However, the temperature increases rapidly above that, getting back to room temperature by 110 km, and heading for 1000k and beyond by the time you get to LEO.

At high altitudes the mass density is decreasing as you get more and more atomic species (e.g. O rather than O2) as well as larger fractions of light constituents (e.g. H2, H), so the speed of sound is quite high at LEO. At altitudes above the "turbopause" (somewhere around 105 km) the components of the atmosphere are no longer well-mixed, thus the different component gases stand at their own scale heights.

see scale height and speed of sound

To first order, the speed of sound does not depend upon the pressure at all; rather it depends primarily upon the mean mass density and the temperature.

The reduction of sound speed at altitude is due to the reduction of temperature. The temperature rises again in the upper stratosphere (ozone heating) and then drops down to its coldest temperature at the mesopause (around 120 K, at 85 km). However, the temperature increases rapidly above that, getting back to room temperature by 110 km, and heading for 1000k and beyond by the time you get to LEO.

At high altitudes the mass density is decreasing as you get more and more atomic species (e.g. O rather than O2) as well as larger fractions of light constituents (e.g. H2, H), so the speed of sound is quite high at LEO. At altitudes above the "turbopause" (somewhere around 105 km) the components of the atmosphere are no longer well-mixed, thus the different component gases stand at their own scale heights.

see scale height and speed of sound

IMHO, the discovery of a real-world application of the idempotent law that was Boole's greatest accomplishment. One could argue that Lebnitz and Boole had independently discovered this. This is not unlike Hamilton's discovery of an application for non-commutative algebra.

Boole's contribution to logic was profound. First, a real world model for any mathematical property ensures the consistency of that model. Boole's work provided an abstraction for elementary set theory. The key to this abstraction is idempotency. The aggregate of set A and itself is the set A (i.e. A+A=A). Thus, Boolean algebra formalizes the basic set theoretic operations of union and intersection, which in turn is almost trivially isomorphic to a Boolean ring. I could create all kinds of stupid rules [insert your favorite slam on mathematics here] that have no meaning in the real world. Most importantly, Boole seemed to be the first to attempt to bridge the gap between abstract thought and mathematics. Admittedly there was some previous work in attempting to formalize|classify all syllogistic reasoning. It was the first step towards a unified theory of logic and ultimately what is hope to be a universal theory of symbolism (see Chomsky's mathematical linguistics).

The irony about mathematics is that often the best ideas are childishly simple. It's not the proof of deep theorems (although that has it's place) that often has the greatest impact. It's the fresh applications of mathematical rigour to some real world scenario. Thus, mathematics is often at it's weakest when done in isolation. Incidentally, Knuth's work in algorithm analysis was revolutionary. In a world described by (K-Complexity (AIT)|cellular automata|simple computer programs) algorithm analysis and ultimately a proof of P not= NP may be to hold the key to the fundamental laws of nature (i.e. physics, biology, and chemistry).

Incidentally, the Martin Davis' The Universal Computer is a great popular science book on this topic. A free copy of the introduction is here. This book manages to introduce the ideas of Turing (Turing-Post?) Machines and the Diagonal Method to the lay reader. The author is a respected logician and computer scientist who studied under Church and Post.

IMHO, the discovery of a real-world application of the idempotent law that was Boole's greatest accomplishment. One could argue that Lebnitz and Boole had independently discovered this. This is not unlike Hamilton's discovery of an application for non-commutative algebra.

Boole's contribution to logic was profound. First, a real world model for any mathematical property ensures the consistency of that model. Boole's work provided an abstraction for elementary set theory. The key to this abstraction is idempotency. The aggregate of set A and itself is the set A (i.e. A+A=A). Thus, Boolean algebra formalizes the basic set theoretic operations of union and intersection, which in turn is almost trivially isomorphic to a Boolean ring. I could create all kinds of stupid rules [insert your favorite slam on mathematics here] that have no meaning in the real world. Most importantly, Boole seemed to be the first to attempt to bridge the gap between abstract thought and mathematics. Admittedly there was some previous work in attempting to formalize|classify all syllogistic reasoning. It was the first step towards a unified theory of logic and ultimately what is hope to be a universal theory of symbolism (see Chomsky's mathematical linguistics).

The irony about mathematics is that often the best ideas are childishly simple. It's not the proof of deep theorems (although that has it's place) that often has the greatest impact. It's the fresh applications of mathematical rigour to some real world scenario. Thus, mathematics is often at it's weakest when done in isolation. Incidentally, Knuth's work in algorithm analysis was revolutionary. In a world described by (K-Complexity (AIT)|cellular automata|simple computer programs) algorithm analysis and ultimately a proof of P not= NP may be to hold the key to the fundamental laws of nature (i.e. physics, biology, and chemistry).

Incidentally, the Martin Davis' The Universal Computer is a great popular science book on this topic. A free copy of the introduction is here. This book manages to introduce the ideas of Turing (Turing-Post?) Machines and the Diagonal Method to the lay reader. The author is a respected logician and computer scientist who studied under Church and Post.

IMHO, the discovery of a real-world application of the idempotent law that was Boole's greatest accomplishment. One could argue that Lebnitz and Boole had independently discovered this. This is not unlike Hamilton's discovery of an application for non-commutative algebra.

Boole's contribution to logic was profound. First, a real world model for any mathematical property ensures the consistency of that model. Boole's work provided an abstraction for elementary set theory. The key to this abstraction is idempotency. The aggregate of set A and itself is the set A (i.e. A+A=A). Thus, Boolean algebra formalizes the basic set theoretic operations of union and intersection, which in turn is almost trivially isomorphic to a Boolean ring. I could create all kinds of stupid rules [insert your favorite slam on mathematics here] that have no meaning in the real world. Most importantly, Boole seemed to be the first to attempt to bridge the gap between abstract thought and mathematics. Admittedly there was some previous work in attempting to formalize|classify all syllogistic reasoning. It was the first step towards a unified theory of logic and ultimately what is hope to be a universal theory of symbolism (see Chomsky's mathematical linguistics).

The irony about mathematics is that often the best ideas are childishly simple. It's not the proof of deep theorems (although that has it's place) that often has the greatest impact. It's the fresh applications of mathematical rigour to some real world scenario. Thus, mathematics is often at it's weakest when done in isolation. Incidentally, Knuth's work in algorithm analysis was revolutionary. In a world described by (K-Complexity (AIT)|cellular automata|simple computer programs) algorithm analysis and ultimately a proof of P not= NP may be to hold the key to the fundamental laws of nature (i.e. physics, biology, and chemistry).

Incidentally, the Martin Davis' The Universal Computer is a great popular science book on this topic. A free copy of the introduction is here. This book manages to introduce the ideas of Turing (Turing-Post?) Machines and the Diagonal Method to the lay reader. The author is a respected logician and computer scientist who studied under Church and Post.

IMHO, the discovery of a real-world application of the idempotent law that was Boole's greatest accomplishment. One could argue that Lebnitz and Boole had independently discovered this. This is not unlike Hamilton's discovery of an application for non-commutative algebra.

Boole's contribution to logic was profound. First, a real world model for any mathematical property ensures the consistency of that model. Boole's work provided an abstraction for elementary set theory. The key to this abstraction is idempotency. The aggregate of set A and itself is the set A (i.e. A+A=A). Thus, Boolean algebra formalizes the basic set theoretic operations of union and intersection, which in turn is almost trivially isomorphic to a Boolean ring. I could create all kinds of stupid rules [insert your favorite slam on mathematics here] that have no meaning in the real world. Most importantly, Boole seemed to be the first to attempt to bridge the gap between abstract thought and mathematics. Admittedly there was some previous work in attempting to formalize|classify all syllogistic reasoning. It was the first step towards a unified theory of logic and ultimately what is hope to be a universal theory of symbolism (see Chomsky's mathematical linguistics).

The irony about mathematics is that often the best ideas are childishly simple. It's not the proof of deep theorems (although that has it's place) that often has the greatest impact. It's the fresh applications of mathematical rigour to some real world scenario. Thus, mathematics is often at it's weakest when done in isolation. Incidentally, Knuth's work in algorithm analysis was revolutionary. In a world described by (K-Complexity (AIT)|cellular automata|simple computer programs) algorithm analysis and ultimately a proof of P not= NP may be to hold the key to the fundamental laws of nature (i.e. physics, biology, and chemistry).

Incidentally, the Martin Davis' The Universal Computer is a great popular science book on this topic. A free copy of the introduction is here. This book manages to introduce the ideas of Turing (Turing-Post?) Machines and the Diagonal Method to the lay reader. The author is a respected logician and computer scientist who studied under Church and Post.

Here is some more information.

Congratulations, you're much more mathematically talented than average. Most folks reading this theorem see a small speck flying in the stratosphere, then hear a faint *whooosh* several seconds later.

I think I can summarize the collective "wha" by saying, I do really appreciate postings on abstract mathematics, but I don't have a clue what your talking about. In fact, I could have a PhD in mathematics and be a respectable researcher and only have a foggy notion.

With that said, I included a couple of links below:

Wikipedia's explanation on the problem

an insanely terse definition with a bibliography of the originally sited papers

Now, armed with all those definitions of all the unfamiliar terms in that paragraph, complete with links to the terms used in the definitions (which are themselves complete with links to all the terms used in the definitions of the definitions, ad nauseum), you've got all you need to understand those two paragraphs! Isn't the Internet great?

Now, armed with all those definitions of all the unfamiliar terms in that paragraph, complete with links to the terms used in the definitions (which are themselves complete with links to all the terms used in the definitions of the definitions, ad nauseum), you've got all you need to understand those two paragraphs! Isn't the Internet great?

Now, armed with all those definitions of all the unfamiliar terms in that paragraph, complete with links to the terms used in the definitions (which are themselves complete with links to all the terms used in the definitions of the definitions, ad nauseum), you've got all you need to understand those two paragraphs! Isn't the Internet great?

Now, armed with all those definitions of all the unfamiliar terms in that paragraph, complete with links to the terms used in the definitions (which are themselves complete with links to all the terms used in the definitions of the definitions, ad nauseum), you've got all you need to understand those two paragraphs! Isn't the Internet great?

A theorem which states that the analytic and topological "indices" are equal for any elliptic differential operator on an n-dimensional compact differentiable C^infinitiy boundaryless manifold.

And this is the least technical definition I have come across so far.

Trawling thru the USENET I found:
The Atiyah-Singer expression is:

{ ch(V|X^g)(g) * U(N^g) * Td(X^g) / det (1-g | (N^g)*) } [X^g]

Apparently the INVARIANCE THEORY, THE HEAT EQUATION, AND THE ATIYAH-SINGER INDEX THEOREM is a good source too.

And This book:
"The Atiyah-Singer index theorem and Elementary number theory" F. Hirzebruch and D. Zagier (Publish or Perish)

Moderate this comment
Negative: Offtopic Flamebait Troll Redundant
Positive: Insightful Interesting Informative Funny

"The null results returned by these experiments have two possible interpretations: (1) There is no ether; (2) the ether local to the Earth is entrained in its orbit around the Sun. (1), of course, is the orthodox line. The constancy of the speed of light for all observers is a _postulate_ that follows from accepting this interpretation..."

Heh. The reason (1) is the "orthodox line" is because (2) is ruled out by stellar aberration. Michelson-Morley says that if the ether exists, it must be dragged along with the earth. Stellar aberration says that if the ether exists, it must be stationary.

I imagine that this is the cause of NASA's perplexing refusal to take up Mr. Hogan's proposal.

And thus I see that you have not studied this :]

I don't wish to pretend to understand all the issues, but I have suffered through enough physics that I believe myself to be capable of giving a simplified and reasonably correct overview (bearing in mind that I haven't had a physics class for a while now) --

Yes, we all like Relativity... on a large scale.

Just as we all like the Standard Model... at the very small scales.

The problem is that the two models are inconsistant. In fact, they outright contradict each other in what they say about the universe--space cannot be both smooth and discrete at the same time!

Thus, we need to harmonize the two, and this is why we have not one but several theories of quantum gravity. The problem has been that we simply do not have the observations to tell which theory is correct.

So some of the real issues here concern very fundamental issues of what matter is composed of and how it behaves and things of that nature. That's a bit more than, say, giving a better model of the spring by adding one more polynomial term to our approximation of Hooke's law.

Quicksort was invented in 1961, while my favorite algorithm, the Cooley-Tukey FFT, was invented in 1965.

Anyone have a more recent favorite?

#!/usr/bin/perl -w

my ( $count, $low, $high ) = @ARGV;

defined $count and defined $low and defined $high or
die( "USAGE: rand.pl count low (inclusive) high (exclusive)\n" );

my $x = 0.5;
my $r = 3.6;

for ( $i = 0; $i < 256; $i++ )
{
$x = $r * $x * ( 1 - $x );
}

for ( $i = 0; $i < $count; $i++ )
{
$x = $r * $x * ( 1 - $x );
my $bits = ( $x * 1000 ) - ( int ( $x * 1000 ) );
print int( ( $bits * ( $high - $low ) + $low ) ) . "\n";
}

#!/usr/bin/perl -w

my ( $count, $low, $high ) = @ARGV;

defined $count and defined $low and defined $high or
die( "USAGE: rand.pl count low (inclusive) high (exclusive)\n" );

my $x = 0.5;
my $r = 3.6;

for ( $i = 0; $i < 256; $i++ )
{
$x = $r * $x * ( 1 - $x );
}

for ( $i = 0; $i < $count; $i++ )
{
$x = $r * $x * ( 1 - $x );
my $bits = ( $x * 1000 ) - ( int ( $x * 1000 ) );
print int( ( $bits * ( $high - $low ) + $low ) ) . "\n";
}

As opposed to what? British Units?

As opposed to binary units like gibibyte (GiB).

So in my opinion there are 3 good websites on the internet:
E-Print Archives
Mathworld and Scienceworld
Federation of American Scientists

Of the three, 2 are distinctly not for profit, but rather so that scientists can get some work done again and who know's why wolfram put mathworld and scienceworld online. As far as more liberal arts stuff, the only online thing I know of is jstor.org and I think that might require paying for, but my university pays for it if it does. I found all those sites very useful and suggest that you check them out if you haven't already done so.

-Scott

So in my opinion there are 3 good websites on the internet:
E-Print Archives
Mathworld and Scienceworld
Federation of American Scientists

Of the three, 2 are distinctly not for profit, but rather so that scientists can get some work done again and who know's why wolfram put mathworld and scienceworld online. As far as more liberal arts stuff, the only online thing I know of is jstor.org and I think that might require paying for, but my university pays for it if it does. I found all those sites very useful and suggest that you check them out if you haven't already done so.

-Scott

I take some issue with the mathematics of the cited reviewer as well though. He says:

Not only is this optimism controversial and expressed without adequate justification, it is also inconsistent with the equally controversial and equally unjustified pessimism expressed on the final page of the main text, where we are told that a problem with which Cantor wrestled throughout his life - whether any set is intermediate in size between the set of positive integers and the set of sets of positive integers - is 'for ever undecidable'.

The problem Cantor wrestled with was the Continuum Hypothesis, the question of whether there is a cardinality of sets between the countable sets (like the integers) and the uncountable sets like the real numbers (and the set of subsets of the integers). In fact, this _is_ undecidable as it has been shown that one can assume either that the hypothesis is true, or that it is false, and still derive no contradiction with the standard axioms of set theory. From Mathworld:

Godel showed that no contradiction would arise if the continuum hypothesis were added to conventional Zermelo-Fraenkel set theory. However, using a technique called forcing, Paul Cohen (1963, 1964) proved that no contradiction would arise if the negation of the continuum hypothesis was added to set theory. Together, Godel's and Cohen's results established that the validity of the continuum hypothesis depends on the version of set theory being used, and is therefore undecidable (assuming the Zermelo-Fraenkel axioms together with the Axiom of choice).

IANA logician, but I think this is solid ground for claiming that the hypothesis is "forever undeciable".

When GE told Charles Proteus Steinmetz that he couldn't smoke at work anymore he said "If the cigar goes, Steinmetz goes!"


Since the guy practically invented AC, they kept both him and his cigars.

If dying stories are dying, then is that statement dying as well? Argh - Cantor paradox!

I'm sorry for taking awhile to respond to your questions.

I really don't mean to sound like I'm bashing PlanetMath. I really do appreciate the site, and hope to see it grow and expand. I believe it will. I think it's a great resource.

And I didn't recommend it just because it's an open alternative to Mathworld in the sense of being a clone. It does have statistics content that Mathworld lacks. It's different enough to be treated differently.

I am aware of the fact that it is organized by subject and alphabetically. I don't mean to imply that PlanetMath is completely disorganized or incomplete to the point of being unusable, just that there are features of it that reduce its usability or usefulness.

Just as an example, compare the Mathworld entry and the PlanetMath entry for the normal distribution. I use this just an example, but it is particularly apt for statistics.

The first thing you will note is that the Mathworld entry is much longer and covers content in much more depth. Not only is more content covered, but more is explained. The PlanetMath entry is much like a glossary entry, the Mathworld entry is more explanatory.

The other feature to note is that the Mathworld entry contains a number of figures, which are extremely helpful in understanding distributions. The PlanetMath entry lacks any figures.

There are also citations given the Mathworld entry, and none in the PlanetMath entry.

There are also oddities before you even get to the entry on the normal distribution in PlanetMath. For example, note that in browsing the N section in order to find the normal distribution, there is an entry comprising a proof that the normal distribution is a distribution. It seems to me that this should be integrated into the entry on the normal distribution, not exist as a separate entry.

These are just examples, but I believe they are fairly representative of issues I have come across when using PlanetMath.

I really don't want PlanetMath to become a Mathworld clone (I already feel a little uncomfortable about the name of the site), but I do think PlanetMath could use some improvements in certain ways.

Again, though--I think it's a great site, and I generally never search Mathworld without searching PlanetMath also, because there are many times Planetmath has information Mathworld lacks.

Why haven't I contributed? I guess one reason is just that I'm extremely busy. Another, I suppose, is that I'm intimidated by the "ownership" process. And there is a part in the back of my mind that is concerned about whether or not if I contribute, the material will be lost, either due to financial failures, loss of interest, or lawsuits by Mathworld over the similarity of the site.

Perhaps those aren't really good reasons not to contribute. I probably will contribute at some point. But I still do have minor (but notable) concerns about how the site operates.

Here's the other one:

http://mathworld.wolfram.com/topics/Probabilityand Statistics.html

I am a statistician of sorts (my training isn't in statistics per se, but that's what I do research on), and I'm sorry to say that I'm not aware of any good online statistics references.

There are some sites that come close.

Mathworld, for example, has some excellent reference material on statistics, but beyond some very basic or introductory material, it tends to become sparse quickly. It's typical of much of what's out there: lots of material on mathematics, but not statistics in particular. I also have ethical objections to Wolfram, and so feel uncomfortable supporting any site hosted by his company.

PlanetMath: is a good alternative to Mathworld, filling in some material that Mathworld lacks. It has the benefit of being open. However, PlanetMath suffers from the problem of being extremely disorganized. Many of the entries seem incomplete or lacking in depth. Finally, like Mathworld, it doesn't treat statistics as much as other branches of math.

HyperStat is a good online resource for introductory statistics. I've actually referred to it a couple of times in my research when I can't remember exactly what some formula is, and don't trust my memory of it. It covers introductory material in depth, but doesn't go into fundamentals or intermediate or advanced material. It's also sort of commercial, disorganized, and poorly designed.

Statsoft Electronic Textbook covers more advanced material, but doesn't seem to provide much explanation or background. It's really more a guide to doing analyses in STATISTICA than anything else.

Finally, I've noticed the Statistics Glossary more and more, but it really is a glossary more than an explanatory reference. It also doesn't get further than very introductory topics.

In short, there is a huge niche for a comprehensive, open, in depth statistics resource ala Mathworld or PlanetMath. Perhaps PlanetMath will become more organized and complete. I've thought about contributing to PlanetMath, but I don't feel completely comfortable with it.

I would have killed for a slashdot story like this 3 or 4 years ago when I was making my calc requirements. One of the best things about using the web for study is the diversity of material out there. You aren't just limited to the dead tree on your desk to help you understand the material.

BTW, Anyone studying math who hasn't been turned on to http://mathworld.wolfram.com should definitely check it out.

Faradaycage

I guess you where sleeping your way through the optics lectures: These lenses could definitely work. If you look at the picture
you see that there are two fluids: brown one on top and a blue one on the bottom. If you remember Snell's law (ray bends towards the normal in the denser medium), you can conclude from the picture that the 'brown' fluid has a higher refractive index than the 'blue' fluid. The left picture thus resembles a hollow/concave/negative lens and the right picture resembles a convex/positive lens. Of these the positive (on the right) can be used to form a real image (one you can capture on a CCD or a retina), whereas the negative only forms a virtual image.

A colleague of mine did his internship at the group that invented these and my boss still works part-time at Philips.

He should read up on what happened to Eric Weisstein's Mathworld website. In short Weisstein licensed a publisher to produce a printed "snapshot" of the website. After the book came out the publisher sued him and had his web site shut down for a year because it was infringing the book's copyright.

He should read up on what happened to Eric Weisstein's Mathworld website. In short Weisstein licensed a publisher to produce a printed "snapshot" of the website. After the book came out the publisher sued him and had his web site shut down for a year because it was infringing the book's copyright.

So we're flying a large, noisy, semi-empty garage in space, and it is so under-staffed (2 people instead of 2.5 required to maintain it) that we can't even use it for scientific experiments.
Actually, in a recent issue of Nature they spoke of planning experiments to search for Lorentz violation aboard the ISS. They include a Michelson-Morley type experiment and some involving atomic clocks (?!). The experiments are being pushed by Stephan Schiller and Peter Wolf (the men who brought you OPTIS) They state they hope for this to begin around 2005. Unfortunatly, a subscription is required to view the article and it was rather light on detail, yet it shows there is indeed hope for the ISS as a science platform.

No - it's not a scam nor perpetual motion machine. A company has already built submarines on this principle that are being used as autonomous research drones. Here's announcment about the Slocum Glider. Here's a couple of action shots of it being deployed. My advice would be to talk a couple of college physics courses to undertand how BUOYANCY works.
Granted it's more complicated in air (larger because air is so dilute when compared with water) however with advances in composite materials, it is certainly doable.