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The biggest thing you gain is that you no longer are at the mercy of the "Rocket Equation." Your "Specific Impulse" which (since you have no knowledge of this aspect) you can think of as the "mass efficiency" of your orbital craft becomes essentially infinite -- your available reaction mass becomes the entire Earth, even though you don't have to carry it with you.

        http://scienceworld.wolfram.com/physics/RocketEqua tion.html

Also, you also missed schemes involving energy recovery from inbound cargoes. If you extract energy from cargoes going down through regenerative braking, you can use that energy to bring up the outbound cargoes. (Up = trading kinetic for potential energy. Down is the reverse. This discounts orbital velocity, but so long as the up & down traffic balances, it works out.)

--SCZ

d((x0,y0),(x1,y1)) = |x1-x0| + |y1-y0|

You can think of it as 1st in the series of metrics

d((x0,y0),(x1,y1)) = ( |x1-x0|^p + |y1-y0|^p )^(1/p)

where for p=1 you get this metric and for p=2 you get the traditional Euclidian one.

You have three complete cycles. That is more then enough for the Nyquist Limit

This cycle is known as the Pacific Decadal Oscillation. The theory is that this is caused by interference effect between the sunspot cylces and El Nino/La Nina.

And this would seem to affect fishery catches fishery catches

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

A good programmer always checks the type of his variables. ;-)

From your math classes, you might recall that "n" denotes a Natural Number.

Although the link above suggests that there is no general agreement on whether 0 is included in the set of Natural Numbers, I doubt anyone here has taken a math class where "n" was anything other than a positive integer.

You can get a mathematica package to do Soduku http://library.wolfram.com/infocenter/MathSource/5 690/

It's true what you say, but 'real' (sorry ;)) mathematicians say 0/0 is undefined.

Ok, here's a mind bender: you say that there are infinately many integers, and there are infinately many non-integers between each pair of integers, does this mean that there are more non-integers than there are integers? infinity * infinity is still infinity... My answer would be that, yes, it is still infinity, it's just an infinately larger infinity... You could even say that it's a significantly larger value of infinity... What's infinity divided by infinity? :)

These things have been defined quite well. The number of positive integers (N) is 'aleph null', or countably infinite. It's exactly equal to Z (including zero and negative integers) and Q (fractions), because there is a 1:1 mapping. It's less than the number of real numbers (R) because there is no 1:1 mapping (cantor's diagonal argument). As I recall it's an open question whether there is an infinity that is less than aleph 1 but less than aleph 0.

For more about this, see cardinal numbers.

My favorite math riddle?

The Condom Problem

The best I can offer is to use complex numbers to transform the equation, something like the Laplace Transform. I have totally forgotten how to do any of this stuff, but I'm fairly sure it's what you're looking for.

I first read about hypergame from Smullyan (I don't remember which book), but the mathworld reference is at http://mathworld.wolfram.com/Hypergame.html

There's a good write up of this on MathWorld.

And God said "In Any Right Triangle, The Area Of The Square On The Hypotenuse Is Equal To The Sum Of The Areas Of The Squares On The Other Two Sides, and This is Good !.

"He then Ordered his followers to read http://mathworld.wolfram.com/ and to register, for this was Good, Too"

"On the sixth fractal equation, God contemplated his finished creation in its vast splendor and saw that it was "Good", also "((The New Gen. 1:31))

On the next meeting, our Pastor, Hon. Prof of advanced Quantum Particles Physics Peter Hawkins, will explain us how God managed to create Free Will and reminded us about it by also creating Quantum Incertanity Principle as a practical example..

By - The First Church of the Flying Scientific Spaghetti Monster, reformed -

See, there is a theological answer to all your question ! 8p

Now Begone, Unbeliever !! 8)

One of the definitions of a parabola (the one I got when I was in high school anyways) is that it's the collection of points equidistant (the same distance) from a point, the focus, and a line, the directrix. One of the side notes of this is that the lines coming from the directrix are all at right angles to the directrix. So basically, if you have nicely collimated (made up of all parallel rays--a good assumption given that we're however many millions of miles away from the sun) and they're all coming in at right angles to the parabola's directrix (this is where the aiming comes into play) all the incoming rays of light will be aimed directly at the focus of the parabola. By widening or narrowing your parabola (making sure that it is still, in fact, a parabola) you can change your focal length (how far out you shoot).

Since a picture is worth a 1000 words, check out the second image at Mathworld's article.

I hear ya.

>>> "they could not exceed the speed of light in a vacuum. Nothing can."

I think you'll find that the theories show nothing can traverse the s.o.l boundary, that still allows for superluminal particles but makes them hard/impossible to detect.

http://scienceworld.wolfram.com/physics/Superlumin al.html

Tachyons aren't just a plot tool for StarTrek script writers!

Dr. Stephen Wolfram has been doing a similar type of research using his "A New Kind of Science".

Wolfram Tones

WolframTones works by taking simple programs from Wolfram's computational universe, and using music theory and Mathematica algorithms to render them as music. Each program in effect defines a virtual world, with its own special story--and WolframTones captures it as a musical composition.

Surprisingly even just looking for a peak over 165 doesn't change the conclusions too much.

We can use the Poisson distribution to model random arrivals (like text message requests). For example, if the average number of messages sent per second is 138, but they arrive randomly over time, then there is approximately a 1% chance of getting over 165 messages in any particular second we choose to examine.

Or, if the average TM's per second is only 119, then the probability of going over 165 in any single second is 1/37,000. Since there are 28,800 seconds in the 8-hour working day, if the average is below 119 (which it surely is), it would be unlikely even to see a single disruption in an 8-hour period. I admit I was surprised the numbers came out this way, but they did. It ain't gonna happen.

Mankind's future in computing must build on immoveable foundations of theory and logic if it is to progress into a realm where machines of IQs in the millions work at our behest. Advocating some sort of ad hoc "practical" computing barbarism is very short-sighted, dangerous, and regressive.

Too bad there are no such foundation

(p1 + p2)[)[initial]=(p1 + p2)[final]

Uh, we know the first two p(momentum, vector) values(and that's if we know the mass of the asteroid, which isn't necessarily true), but not the second two. In my math classes, we learned that was 1 equation (vector valued) and two unknown vectors. I don't think anyone can solve that, and no, conservation of kinetic energy won't work because the internal energy changes big time in most non-particle scale collisions. In Mechanics, many of our college educated comrades learned of a way to resolve this textbook documented issue with the simple aide of a constant e, which details the elasticity of the interaction. Unfortunately, e is not easy to determine through theory, and is also just a model (and a bad one at that), and therefore an experiment is usually called for (and usually a lot of them). 'Nuff said.

http://scienceworld.wolfram.com/physics/Collision. html

The classic example of "crowd wisdom" is the jellybeans-in-the-jar experiment, often used in introductory MBA classes to convince people that open markets value securities (basically) fairly. The experiment goes like this: the professor brings a jar of jellybeans and asks everyone to guess how many there are in the jar. The individual estimates may vary quite a lot, but the average of the estimates in the class is usually close, in fact often closer than the closest estimate of any of the students depending upon the size of the class. That is in a situation where the students have very little information about the jar and perhaps no experience with such estimates. If there were greater experience and/or they were allowed more information then presumably the individual and average estimates would be even closer. Basically, this can be described as the "Central Limit Theorem" in action- that the standard deviation of averages is smaller than the standard deviation of the individuals by a factor of the square root of the sample size, as illustrated in this applet or in this Mathworld description. The CLT actually says more- that as the sample size increases, the distibution of averages approaches that of the normal ("bell curve") distribution, so the distribution of avergaes is roughly normal, and then techniques designed to analyze the normal distribution can be applied with greater certainty.

http://scienceworld.wolfram.com/astronomy/LeapYear .html

Oh geez I am dumb. It must be late. Coulda always made it a sub and checked to see if it was >= 1900 ...

why

Someone or another asks something like this everytime anything related to black holes comes up on Slashdot.

The radiation emitted from black hole related events, such as quasars, gamma ray bursts, and Hawking radiation, for that matter, comes from processes near-sometimes very near, but still OUTSIDE, the event horizon. As long as you're outside the horizon, there are trajectories that escape.

As for,

Also, if a black hole was created at explosion, was this even more massive then we can see, yet the black hole swallowed up a majority of the explosion and what we see, is just a small glimpse of it?

According to the literature on very massive stars, there as mass ranges that results in the star collapsing completely into a black hole such that no significant amount of matter or radiation gets away at all.

Check out How Massive Single Stars End their Life. Figure 1 is particularly enlightening. It's a pretty math-free article, so I think anyone who's generally interested in this stuff can follow it, maybe with a bit of help from Wikipedia and Science World.

http://tones.wolfram.com/about/faqs/howitworks.htm l

What are the historical antecedents of WolframTones?

Ideas of "generative music" or "algorithmic composition" go back a long way. Mozart, for example, was said to have a scheme for composing minuets based on throwing dice. In the early 20th century, composers like Schoenberg considered formal matrix-like methods, and especially in connection with early synthesizers there was interest in deriving music from electronic and other physical processes. In the late 20th century, many experiments were done using 1/f noise, fractals, L systems, and even cellular automata. Most often, explicit randomness was taken as the foundation, and extensive layers of post processing were done. The publication of A New Kind of Science led to a new approach, and much purer ways to derive music from the computational universe--culminating in WolframTones.

At first I thought it was strange how classically Baroque the sample on http://tones.wolfram.com/about/how.html sounds. Not wanting to sound snobby, but the sounds being generated wouldn't sound so genuinely 'new' to someone who has listened to a number of dense notational compositions of Baroque Fugues. Then I remembered Bach was a mathematician, and some of the Fugues Baroque composers wrote can be considered mathematical excercises as much as anything.

It's thought many passages of these Fugues were written without the aid of a piano, merely working from clusters of notes and their variations. As a result they go like automata, sounding like machines set in motion that develop variations and themes, and for a lot of people are quite unlistenable. I've played a Bach Fugue and the basic technique I was taught was 'be the machine!'

Nice try, but the Terms of Use already assert copyright (use of content must be licensed and cannot be reproduced, rebroadcast, etc.) and prohibit mirroring, and automated/scripted access to the site.

Here's mine >8)

http://tones.wolfram.com/id/Ge9y3BYq2uasToZXknwS8u 3VTbkcva7cWG5qRtysSVaPVJ1n

Hmm..interesting, you don't get to modify if you use that link.

Evryone else post theirs too

http://tones.wolfram.com/id/GrAMpRZwE6hkGUO9WuKXM2 l27hZDSO5btPgMHlEfHAx9g2G

1. Grab the URL from the bottom of the generate page
   
   • http://tones.wolfram.com/id/Ge0VOcDtDGMSHE1qTfMi 30N7BgRQF8HB4rsF1vv3MUZQOob
2. Take the ID from the end
   • Ge0VOcDtDGMSHE1qTfMi30N7BgRQF8HB4rsF1vv3MUZQOob
3. Append this to http://tones.wolfram.com/SMSMathematica/NKM/sound. jsp?id=
   • http://tones.wolfram.com/SMSMathematica/NKM/sound. jsp?id=Ge0VOcDtDGMSHE1qTfMi30N7BgRQF8HB4rsF1vv3MUZ QOob
4. Open it in your browser.
5. No shit (profit!)

1. Grab the URL from the bottom of the generate page
   
   • http://tones.wolfram.com/id/Ge0VOcDtDGMSHE1qTfMi 30N7BgRQF8HB4rsF1vv3MUZQOob
2. Take the ID from the end
   • Ge0VOcDtDGMSHE1qTfMi30N7BgRQF8HB4rsF1vv3MUZQOob
3. Append this to http://tones.wolfram.com/SMSMathematica/NKM/sound. jsp?id=
   • http://tones.wolfram.com/SMSMathematica/NKM/sound. jsp?id=Ge0VOcDtDGMSHE1qTfMi30N7BgRQF8HB4rsF1vv3MUZ QOob
4. Open it in your browser.
5. No shit (profit!)

without paying for it to be sent to your phone, obviously.

i came up with this:
http://tones.wolfram.com/id/GNkwrIzW4skvtw8KRaBTon 7s9efTRKB2FSdvGwCCrIYcTmyf
and would like to take the midi and actually put real synths and drums on it.

http://tones.wolfram.com/id/G1aG90C9HT8B689djr5D7B VCYdxjun4riZW4

Quite frankly, RTFA. I got sick of his own horn tooting on that site. You'd think that he was the only person that had anything to do with chaos theory and emergent behaviour; although he doesn't use those terms it seems.

"It's a rather direct consequence of a core phenomenon of Stephen Wolfram's science: that programs with very simple underlying rules can generate great complexity of behavior."

http://tones.wolfram.com/about/faqs/howitworks.htm l

His science? Please.

fuck yeah: http://tones.wolfram.com/id/GQiOhCXNt66xea45dmBBem DjUR8sf62hPQzn4AkqAZijHtjNlW can i copyright this so i can sue you all for listening to it without paying me? or does this fella own all the material generated? but anyway now i can put it onto flash sites without getting in trouble.

see:
http://tones.wolfram.com/tsfaqs/windows/configureq uicktime.html

Hmm. God Bashers... Hmm.

It is so. Distressing to me how many people are so rational when it comes to most parts of life, but then, when it comes to God, are all. Hmm. Anti. Just Anti.

How often do we complain about lack of education for poor kids? Do we say "oh, you are poor, so you must be stupid?"

Since when does uneducated = stupid?

As to non-stupid people. Newton? Sir Issaac? He was stupid? I cant really vouce for the source but it doesnt stray from what I Have read about newton (http://scienceworld.wolfram.com/biography/Newton. html):

"Although his methodology was strictly logical, Newton still believed deeply in the necessity of a God. His theological views are characterized by his belief that the beauty and regularity of the natural world could only "proceed from the counsel and dominion of an intelligent and powerful Being." He felt that "the Supreme God exists necessarily, and by the same necessity he exists always and everywhere."

I mean, fine. You dont believe in God. All good. So you will burn in hell. No big deal. But, why, oh why be so anti somone who DOES believe in God (not the troll whose comments I havent bothered reading, he is just a flamebait mark him so with mod points. But all the other Theists out here)? And, sheesh, if you ARE going to be anti, at least be clever about it, instead of implying that the uneducated are neccessarily stupid.

And because neither of the above really explains anything, here's a better explanation of Bulk Modulus (With lots of neat graphs & charts on the left side of the page)

And because neither of the above really explains anything, here's a better explanation of Bulk Modulus (With lots of neat graphs & charts on the left side of the page)

He was wrong to say "almost guarantee." He would've been correct if he had used the word "average". The mean of a geometric distribution with 1/n chance of success on any trial is n-1.

Geometric Distribution

I forgot, I used Bernoulli Distribution to do the maths, you can check it out here.

Irrelevant.

A Bell Curve---better known as a normal distribution or Gaussian---is an extremely common pattern for large numbers of data points to fall into when measuring something natural. It's a curve that is key to any kind of statistical psychology.

That some stupid book happened to have the name of the curve in its title says nothing about whether a normal distribution is a sensible result to expect. It's like saying you won't eat dinner off china because you don't like China - it's complete nonsense.

1... is... not... a... prime...

For info on why, see the mathworld prime number entry.

Interestingly, it says that, at one time, 1 was considered prime and 2 was not. Pretty amazing, considering importance of the Fundamental Theorem of Arithmetic.

1... is... not... a... prime...

For info on why, see the mathworld prime number entry.

Interestingly, it says that, at one time, 1 was considered prime and 2 was not. Pretty amazing, considering importance of the Fundamental Theorem of Arithmetic.

I used computers and software that taught persistence, team work, problem solving and resiliance... It was called Mathematica, oh, and I got one heck of a lesson in Calculus while I was at it!

iterate, iterate, for god sakes newton was above iterating

He was? I guess he had nothing to do with the Newton-Raphson method then.

The key here is the energy required. Space travel is still dominated by propulsion. That is, the engines and the fuel they need, and the fuel needed to launch that fuel to orbit, etc., is where most of the cost is.

It is important to travel on a trajectory, called the transfer orbit, that requires the least energy. For a high thrust spacecraft, the minimum energy trajectory is called a Holman transfer. Simply, it is an orbit that just touches the orbits of both planets. The periapsis, the closest point to the sun, touches the orbit of the one planet and the apoapsis, the furtherest point, touches the other planet. For this to work, the destination planet needs to be half an orbit away when the spacecraft arrives. This is a lot easier to see in a picture.

For Earth to Mars, the spacecraft launches and then the thrusters fire to change the spacecraft's orbit of the sun from Earth's orbit to the transfer orbit. It then travels half of the transfer orbit and fires its thrusters to change its orbit to match Mars. This can be done by aerocapture, aerobraking or propulsion. The opportunity for a Holman transfer to Mars occurs every 26 years. It is based on the length of the orbit for the bodies being transferred between. The return trip also needs to be a Holman transfer to save fuel. The opportunity does not occur until many months after arrival. I forget the actual number. That is why Mars trips will have a long stay on Mars before returning.

Low thrust is different. Low thrust spacecraft thrust all or most of the time during the trip and the trajectory is more complicated. It is not usable for manned flight because it is to slow but is useful for unmanned spacecraft sometimes.

This is called Celestial Mechanics. When you add propulsion, it becomes Orbital Mechanics.

The best site I have found is NASA's Spacefligh Basics.

Also good is this site.

For explanation of gravity assists see this site.

Also see, Science World at Wolrram

Agreed that it's kind of silly. While reading the /. abstract, I said, "Hm... I wonder how long this computer (2GHz Centrino laptop) would take." It took 8.72 seconds to calculate Pi to a million decimal places, and display them (which always takes longer than anything else, it seems), using the Chudnovsky formula http://mathworld.wolfram.com/PiFormulas.html, in Mathematica 5.1.
    I did not RTFA (I'm not new here), but might, since I want to know what slow-ass implementation this guy used to calculate that.

Calculating pi is a series of mathematical operations where you can't do the next one without the prior because you need the remainders. Supercomputers are super due to a heck of a lot of CPUs all working on different parts of a problem that can be broken into chunks. How exactly do you break a series of operations that depend on the priors into chunks for a supercomputer to rip through?

Use the BBP Formula. Pifast is just a benchark, like all benchmarks it's rather silly. The record is for PCs, the top 500 supercomputers are benchmarked using another silly benchmark (LINPACK).

Calculating pi is a series of mathematical operations where you can't do the next one without the prior because you need the remainders.

That's not entirely true. There exists a digit-extraction algorithm for computing pi starting at the nth digit, without the need to compute any other digits. The only catch is that it only works on base 16.

Calculating pi is a series of mathematical operations where you can't do the next one without the prior because you need the remainders.

That's not entirely true. There exists a digit-extraction algorithm for computing pi starting at the nth digit, without the need to compute any other digits. The only catch is that it only works on base 16.

the code itself is structured as a proof

I just wrote some Haskell code to manipulate formal power series in Haskell. One thing that was cool was that I was able to take propositions that I could prove mathematically and simply rewrite them as code. It was pretty mind-blowing. Things that were traditionally hard to compute became one liners because of lazy evaluation. On the other hand, almost trivial changes to the code that still resulted in true mathematical propositions didn't result in working code. Essentially the problem was to do with what precisely depended on what. The wrong mathematical proposition and a1 depends on a2 which depends on a3 up to infinity so the code never terminated. In fact, it's very easy to write code that looks provably correct but doesn't terminate. I just came across a paper on this very subject - the fact that some things we take for granted in mathematics aren't true of guaranteed to terminate functional programs. Pity I can't find the link.