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For those with maths, Messers Wolfram tell all. I like this one.

This experiment shows off wave/partical duality (it even has cool terminology). The cool bit about physics (yeah, it has cool bits) is the things it takes your head a while to get around.

OK, background: waves spread round corners. Think of a wave at a harbour mouth. The closer the gap is to the wavelength of the wave, the better it spreads (look up diffraction) (troll me, I know this is a gross over-simplification) - ever think about how you can hear but not see round corners? Light == really short wavelengths (nanometres), not like door width lengths (m) (doesn't bend well round the corner), sound == long wavelengths, kinda door-width like (m/cm ish) (bends very well round the corner).

So you get two bits of card with a light behind them, and a screen to shine light through them onto. The first card has one slit, so it shines a little line of light onto the second.

The second has two parallel slits in it, within range of the spread of light, and the light that gets through the first card onto a slit in the second card makes it to the screen.

Now the cool bit.

You get a ripple of light on the screen. Not a black screen. Not two lines showing up the second card shape. Ripples.

Now, modern physics can explain this. It's the wavefront from the first slit (think ripple hitting a harbour mouth) that spreads out in a circle and hits the next two slots, starting another ripple on the other side of both.

At the far wall, you get points where the peak of a wave from one slit hits the peak of a wave from the other, and you get a really tall peak. Or a trough and a trough, and get a really low trough.

For those with maths, Messers Wolfram tell all. I like this one.

This experiment shows off wave/partical duality (it even has cool terminology). The cool bit about physics (yeah, it has cool bits) is the things it takes your head a while to get around.

OK, background: waves spread round corners. Think of a wave at a harbour mouth. The closer the gap is to the wavelength of the wave, the better it spreads (look up diffraction) (troll me, I know this is a gross over-simplification) - ever think about how you can hear but not see round corners? Light == really short wavelengths (nanometres), not like door width lengths (m) (doesn't bend well round the corner), sound == long wavelengths, kinda door-width like (m/cm ish) (bends very well round the corner).

So you get two bits of card with a light behind them, and a screen to shine light through them onto. The first card has one slit, so it shines a little line of light onto the second.

The second has two parallel slits in it, within range of the spread of light, and the light that gets through the first card onto a slit in the second card makes it to the screen.

Now the cool bit.

You get a ripple of light on the screen. Not a black screen. Not two lines showing up the second card shape. Ripples.

Now, modern physics can explain this. It's the wavefront from the first slit (think ripple hitting a harbour mouth) that spreads out in a circle and hits the next two slots, starting another ripple on the other side of both.

At the far wall, you get points where the peak of a wave from one slit hits the peak of a wave from the other, and you get a really tall peak. Or a trough and a trough, and get a really low trough.

To hell with Gigabytes and Terabytes! What about gibibytes and tebibytes?

To hell with Gigabytes and Terabytes! What about gibibytes and tebibytes?

And a problem with electrical generators or something. And a chalk mark. Anyhow, here's a link about one of my heroes:
Steinmetz
-- ac at work

I'm really surprised that only two Fields Medals were awarded this time around- at least three have been awarded every four years since 1974. Is Preda Mihailescu's proof of Catalan's Conjecture considered too recent for consideration? I'd think that sort of thing, combined with his work on noted hot topic primality would make him an attractive candidate.
Of course, I'm sure they are many others who were also very deserving as well. No, I am not Dr. Mihailescu, and have never met him in fact; it's just when I saw that the Fields Medals were awarded, my first thought was, "I wonder if they gave one to that guy who proved Catalan's Conjecture?" As recent as the proof was (considering the slow, careful peer review that accompanies important purported mathematical proofs), I wasn't shocked to not see his named- I was far more surprised that the committee chose to not award the remaining two prizes to anyone.

I guess the article and most posts in this Slashdot topic are referring more or less to calculus (which is closer to analysis than algebra) and linear and second-order equations solving (a mostly insignificant part of algebra IMHO).

In France, you cannot enter state approved (the so-called technical elite) engineer schools without first passing school-specific competitive tests for which you prepare for two or three years after graduation.

Their you get mostly math and physics high level courses where I got taught what was called algebra: group theory, rings, fields, vector space, etc... (More info available at Eric Weisstein's world of mathematics)

In the business field, I don't have to use much maths whether theoretical or applied. I find what's more important is how you express your ideas and how you communicate with your co-workers, your boss, the customers.

Obviously, I had to go through high level science and tech courses for which math is a life-saving requisite. However, my day-to-day job is not, unfortunately, to find new undiscovered tools and theories for my field, but just to use abacae and tools of the trade. And yet I've co-authored some patents.

Math is quite an interesting subject. It's a shame that it's taught entirely bottom-up though (eh, I was taught base conversion before multiplication, bare with me :)). Anyone tried the reverse? You could learn a major part of the definitions and theories first, and then apply them to solve equations. That would need quite some maturity, which you can get through learning english (or french in my case) and some philosophy first.

English is the #1 required skill everywhere anyhow.

In a nutshell, you start with a composite number, write down the prime factors in decimal in ascending order of size and so get a new number, which may either be prime or composite. If prime, then this is the Home Prime of the number you started with. If composite, repeat.

All the numbers below 100 reach their Home Primes rather quickly - except for 49 (and thus, 77). This one is expanded to 95 steps by now and has grown into a 194 digit number which is becoming increasingly difficult to factor. The last 15 steps of this sequence were done by Paul Leyland of Microsoft Research, Cambridge, and myself. The 95th step is factored down to a 153 digit composite which we are struggling with right now. See Patrick De Geest's page for the current status of this problem.

A difference between Home Primes and the 196 problem is that Home Primes can be shown to exist for every number - the probability that no Home Prime is found in as the number of expansion steps goes to infinity converges to zero. It is, however, quite possible that the number gets too large to factor with today's resources before the Home Prime is found. (That seems to be happening in the 49 case right now.) AFAIK no guarantee of a terminating sequence exists for the Palindrome problem, it is possible (and likely) that the 196 sequence spins off into infinity without ever becoming palindromic.

As far as I can tell, there is no practical use for either Palindromic numbers of Home Primes. It's just recreational - a way to spend spare time no better or worse than board games or sports on the TV. Except it involves massive amounts of cpu time and pretty advanced algorithms (i.e. ECM and GNFS) - the study of which I find extremely intriguing. It's probably one of the geekiest ways to spent your time (not that I were proud of that, I merely can't help it).

Alex

(Note: phi(n) is the number of primes less than n (Euler's totient function, I believe). phi(p) = p - 1 for prime p, and phi(pq) = phi(p)phi(q) for p relatively prime to q (note, this step breaks if p or q aren't prime))

A slight error: phi(n) is the number of positive integers less than n, which are relatively prime to n (ie. gcd(n,x)=1). Therefore, if p is a prime, it is also relatively prime to all smaller integers, so phi(p)=p-1.

The function that tells the number of primes smaller than n is pi(n), the prime counting function.

Refs: Totient Function Prime Counting Function (MathWorld's luckily back online!)

(Note: phi(n) is the number of primes less than n (Euler's totient function, I believe). phi(p) = p - 1 for prime p, and phi(pq) = phi(p)phi(q) for p relatively prime to q (note, this step breaks if p or q aren't prime))

A slight error: phi(n) is the number of positive integers less than n, which are relatively prime to n (ie. gcd(n,x)=1). Therefore, if p is a prime, it is also relatively prime to all smaller integers, so phi(p)=p-1.

The function that tells the number of primes smaller than n is pi(n), the prime counting function.

Refs: Totient Function Prime Counting Function (MathWorld's luckily back online!)

http://mathworld.wolfram.com/SquareRoot.html
Read and learn.

The correct definition of a hyperplane is here.

No.

That's not a question in general relativity. The curvature of a spacetime can be measured without considering it as being embedded in manifold of higher dimension. How? Here is the common demonstration of the idea without using the language of math which makes the idea harder to convey. If you want a more rigourous explanation see here. Note that the curvature we are concerned with here is the Gaussian curvature which is intrinsic, ie it can be measured without considering directions outside of the dimension of interest.

Consider the surface of sphere, any ball is a reasonable approximation. Now consider the following path. Starting at the equator while facing west (these are all well defined directions if you use the right hand rule and call north the direction of your thumb then east follows the curvature of your fingers and west is opposite east). Now go 1/4 of the circumference of the circle west, turn to face north. This is a 90 degree turn. Go to the north pole. Now turn 90 degrees again (again this is a well defined operation, when facing any direction a 90 degree turn is accomplished by orienting yourself such that the direction previously over your right shoulder is now the direction you are facing). Now continue to the equator. You should be at the original starting point, and another 90 degree turn will leave you facing west, your original direction of departure.

So you've traced out a triangle: a closed path with three vertices, but you've made 3 90 degree turns so the sum of the interior angles is greater than 180 degrees in violation of euclidean geometry. Therefore you know your 2 dimensional world: which is the surface of the sphere, is curved. Note that is is not necessary for this surface to actually be curved "into" anything. If the sphere is the spacetime of a universe then there is by definition nothing outside of the surface of the sphere to consider, all of space and all of time are contained on the surface. The surface is still curved, but it doesn't "curve" into anything, that's just a property of the spacetime.

I'm not sure I can make it any clearer, but if you consider Occam's razor you'll see that it doesn't make sense to thing about curved spacetimes as being embedded in some higher dimension. Since it is possible to measure curvature without appealing to a higher dimension (remember we never left the surface of the sphere in the above example) then you don't need the higher dimension, all the information required is contained in your local spacetime.

No.

That's not a question in general relativity. The curvature of a spacetime can be measured without considering it as being embedded in manifold of higher dimension. How? Here is the common demonstration of the idea without using the language of math which makes the idea harder to convey. If you want a more rigourous explanation see here. Note that the curvature we are concerned with here is the Gaussian curvature which is intrinsic, ie it can be measured without considering directions outside of the dimension of interest.

Consider the surface of sphere, any ball is a reasonable approximation. Now consider the following path. Starting at the equator while facing west (these are all well defined directions if you use the right hand rule and call north the direction of your thumb then east follows the curvature of your fingers and west is opposite east). Now go 1/4 of the circumference of the circle west, turn to face north. This is a 90 degree turn. Go to the north pole. Now turn 90 degrees again (again this is a well defined operation, when facing any direction a 90 degree turn is accomplished by orienting yourself such that the direction previously over your right shoulder is now the direction you are facing). Now continue to the equator. You should be at the original starting point, and another 90 degree turn will leave you facing west, your original direction of departure.

So you've traced out a triangle: a closed path with three vertices, but you've made 3 90 degree turns so the sum of the interior angles is greater than 180 degrees in violation of euclidean geometry. Therefore you know your 2 dimensional world: which is the surface of the sphere, is curved. Note that is is not necessary for this surface to actually be curved "into" anything. If the sphere is the spacetime of a universe then there is by definition nothing outside of the surface of the sphere to consider, all of space and all of time are contained on the surface. The surface is still curved, but it doesn't "curve" into anything, that's just a property of the spacetime.

I'm not sure I can make it any clearer, but if you consider Occam's razor you'll see that it doesn't make sense to thing about curved spacetimes as being embedded in some higher dimension. Since it is possible to measure curvature without appealing to a higher dimension (remember we never left the surface of the sphere in the above example) then you don't need the higher dimension, all the information required is contained in your local spacetime.

If you're referring to a differential gravitational attraction similar to that involved in Roche Limit" deformation of orbiting bodies, then no. What you're suggesting implies that the gravitational pull on the equator is significantly stronger than that on the polar regions. Since gravitational attraction drops off exponentially as the distance between the two bodies increases, that kind of differential pull only occurs when the gravitational bodies are relatively-speaking quite close together.

Perhaps if the moon had suddenly increased in mass a thousand-fold, but not possible due to distant stars or planets.

I would strongly disagree with your comment.

Up until a few hundred years ago science and arts were one of the same. Looking back trough the course of history a hell of a lot of famous inventors, scientists and mathematicians were also artists.
Look at things like the works of Leonardo Davinci , the elements or any old biology book you care to mention.

Just because you have a high level of creativity and inspiration doesn't mean that you can't do the math or apply engineering first principles to a project.
Sure, some of the projects out there will be purely created artistically, and some may be enginered(very hard to do with software!) but a lot of projects and probably most of the best ones will be a mix of artistic inspiration and creativity, and engineering principles.

Personally when I start to code on the 'Unknown' I play around with a few creative ideas, then re work those creative ideas into an well designed piece of software.

Enter Mathematica.

Mathematica has an add-on available which lets you link to Excel, and link from Excel to Mathematica. You can send that equation over to Mathematica, have it evaluate it, and send it back.

I've never used this feature, but I hear that it's a real life-saver. Plus, Mathematica is one of the easiest math programs to start using, and one of the most powerful if you keep using it.

Enter Mathematica.

Mathematica has an add-on available which lets you link to Excel, and link from Excel to Mathematica. You can send that equation over to Mathematica, have it evaluate it, and send it back.

I've never used this feature, but I hear that it's a real life-saver. Plus, Mathematica is one of the easiest math programs to start using, and one of the most powerful if you keep using it.

About 450 disks? We find about 100 disks ever semester. One time I built a Depth 2 Menger Sponge out of the floppy disks. It's a type of 3D fractal. Unfortunately, the tape didn't hold and the whole thing fell apart. It was quite impressive while it lasted though.

In fact, the shortest day of the year in the Northern hemisphere oscillates regularly between December 21 and 22. This page has a table of the soltices up to 2009.

I would like to understand the math better, specifically to see if it might have applications to software. I'd also like to plot the superhighway, or understand how they are doing it. But only have a year of college math. Where is a good and free place to learn about it online? Been to Mathematica.

I would like to understand the math better, specifically to see if it might have applications to software. I'd also like to plot the superhighway, or understand how they are doing it. But only have a year of college math. Where is a good and free place to learn about it online? Been to Mathematica.

I would like to understand the math better, specifically to see if it might have applications to software. I'd also like to plot the superhighway, or understand how they are doing it. But only have a year of college math. Where is a good and free place to learn about it online? Been to Mathematica.

I would like to understand the math better, specifically to see if it might have applications to software. I'd also like to plot the superhighway, or understand how they are doing it. But only have a year of college math. Where is a good and free place to learn about it online? Been to Mathematica.

I would like to understand the math better, specifically to see if it might have applications to software. I'd also like to plot the superhighway, or understand how they are doing it. But only have a year of college math. Where is a good and free place to learn about it online? Been to Mathematica.

I would like to understand the math better, specifically to see if it might have applications to software. I'd also like to plot the superhighway, or understand how they are doing it. But only have a year of college math. Where is a good and free place to learn about it online? Been to Mathematica.

Well, you can find out what it is here. I suppose it comes up in crypto work, but I'm not familiar with the details. The website has some graphs to go with the equations and description, plus hyperlinks to related stuff.

octave:1> [2 1; 1 1]**8
ans =
1597 987
987 610

I believe I ran the spec95 benchmarks. The difference between a uniprocessor and a dual processor were only about 1-2% if I recall, so the overhead is small, but it exists. If you are really interested send me a note (mukasa@jeol.com) and I will run it again and send you the results.

I wouldn't call myself a mathematician, I just use Mathematica a lot. As I said I am running Mathematica on a dual G4 box and as far as I know none of the standard distribution is SMP aware. I suppose this is due to the difficulty of parallelizing the language since most machine specific behaviour is abstracted away. Otherwise it runs very well on the Mac. I mostly use it for symbolic calculations and numerical algorithm development so I don't make a lot of plots, but from the Mathematic on OS X presentation from Wolfram it seems there are some advantages to using Quartz for graphics.

Then again, I do believe there was a version of Mathematica for the Connection Machine that was SMP aware. Maybe they can dust it off and reincorporate it into the standard version.

Actually, it's more complicated than that; the group velocity of a pulse can exceed the speed of light, but the leading edge of the pulse cannot move faster than c (thus respecting relativity).

For more details check this out.

100 degrees in 2pi radians

400 actually, and the units are called gradians.

http://mathworld.wolfram.com/topics/MinimalSurface s.html

Nice job indeed!

That's easy. Take a 4-cube. Pass it through 3-space, and what you'll see is an infinitely small cube at the point in the center of the 4-cube which will then grow to the cross-sectional size of the 4-cube, and shrink back down. Not too hard to wrap your head around. Tesseracts are a totally different matter...

I disagree. That nicely describes what happens with a sphere, but not so much with a cube.

I guess it depends on how it's passing through 3-space. The best way to imagine this stuff is to imagine a 3-D object passing through 2-space. If you pass the cube through corner first, you'll get something like what you describe, except the cross sections will be triangular most of the time. Edge on, you'll get rectangles.

Face-on, your 2-space will see nothing until the 3-cube hits it, then the 2-space will see a square just sitting there until the 3-cube is all the way through.

What's the difference between a tesseract and a 4-cube? According to Eric Weisstein, it seems they're the same thing.

Eric Weisstein's World of Mathematics

The best way to learn math is in a classroom with an instructor who lectures a little and expects student participation a lot. But since you can't be a full-time student, you'll have to make do without it for the most part. I strongly reccommend against striking out on your own until you've taken the full calculus sequence, though. Whether you need to start at the beginning or not is at your discretion, but take actual courses with an actual teacher up through multivariable calc, which will be the third semester or fourth/fifth quarter.

Whether or not it's useful to put undergraduates through a course in proof-writing before jumping into advanced courses is subject to a lot of debate (my position is that it is useful but not essential). But in your case, since you won't have feedback from a professor when you're trying to learn the advanced stuff, it's an absolute necessity. If possible, take a course like that from a local college or university. If you can't, I reccommend a textbook called Chapter Zero by Carol Schumacher. Carol was my academic adviser at Kenyon College, and I took a class from her based on that book. You won't learn a lot of math from it, but you'll learn what math is, and you'll learn how to learn math.

After that, go to mathworld.wolfram.com and look around. It's a great resource for learning stuff, and even better for finding topics you want to explore further. Find something that sounds interesting and that you can basically understand the description of. A lot of colleges and universities have course catalogs available online--find some schools teaching an undergraduate class on the topic you picked, then look for the most common textbook. Buy it. Read it, working through problem sets, until you get tired of it. Go back to mathworld and repeat.

http://mathworld.wolfram.com It's free and it has everything I've ever needed to look up. Beats digging out the old Trig Books :-)

http://mathworld.wolfram.com/

This isn't completely what you want, but it is a very good reference site for mathematics, from the fine people who brought us Mathematica. And it's free, and as we all know, free is good.

I think that
(1) It's an "on the average" relationship, not an exact relationship - that is, if it says there are 100 primes in a long interval it may be off by a few percent, but if it says there's 1 prime in a short interval it may be off by +/- 1. So it's no good for the bisection search.

(2) The Zeta function is not that easy to compute anyhow.

Here's a brief explaination of the Zeta function given by mathworld...

... on the Riemann Hypothesis:

Riemann Hypothesis

Apparently there's a distributed computing project called ZetaGrid which has calculated the first 50 billion zeros out ... if you're bored of SETI@Home, this might be a nice change of pace.

Riemann Hypothesis
Riemann Zeta Function
Also, there's some rather technical details on the subject, from Stephen Wolfram's (A New Kind of Science) pet site.

Apparently there's a distributed computing project called ZetaGrid which has calculated the first 50 billion zeros out ... if you're bored of SETI@Home, this might be a nice change of pace.

Riemann Hypothesis
Riemann Zeta Function
Also, there's some rather technical details on the subject, from Stephen Wolfram's (A New Kind of Science) pet site.

However, for more details, you would have to look it up in a book on number theory or something like that.

However, for more details, you would have to look it up in a book on number theory or something like that.

However, for more details, you would have to look it up in a book on number theory or something like that.

That would be Russell's Antinomy, not Cantor's Paradox. Russell's is just the Greek Barber problem in the language of set theory. Cantor's paradox deals with the carninality of the universal set.

That would be Russell's Antinomy, not Cantor's Paradox. Russell's is just the Greek Barber problem in the language of set theory. Cantor's paradox deals with the carninality of the universal set.

Actually, we've done something like that as a technology demonstration for wireless access to webMathematica.

I'd always wondered how long it would be before the companies that produce software like Mathematica and Maple would port their software to PDAs. When I went to college at Rose-Hulman IT we were all issued notebooks which ran Maple and CAD software. We used Maple in all of our Calc classes and were able to use it on tests once we proved our ability to do that particular type of problem by hand first. The CAD software could have easily been on higher power workstations. If Maple had been on our PDAs it would have lowered the cost of going to the college by a few thousand dollars (high end notebooks were really expensive back in '95, and sometimes still are)

The main problem is that PDAs were nearly non-existant at that time, but today I can see PDAs like the iPaq doing a grand job of running some of this higher end math software.

Of course cheating would run pretty rampant with wireless transmitting of email and text, not to mention the ability to store files with crib sheets on them. I'm still not sure how our profs back in the day thought they were ensuring that we didn't cheat on our calc exams back then. I think it was more of a matter of honor than anything.

Like, why not just go straight cellular and connect to the internet or your home beowulf cluster?

Why stop there? Put a webMathematica server up, and access it though your PDA.