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  1. This looks an awful lot like homework to me. You'll have to demonstrate that you've made an effort, before anyone else does.
  2. No. Comprehension and credulity have absolutely no bearing on truth or reality.
  3. Group actions can be performed on decks or hands of cards - so that could be considered an application of group theory, even though you wouldn't really think of it like that.
  4. It is considered in physics, like any other number. Things that carry no mass include the photon, gauge bosons and gluons. Nearly all type of energy create action 'at a distance'. Do you actually mean anything by this at all?
  5. I've taken a few philosophy classes at uni, none of the lecturers were theistic and I wasn't aware of any theistic classmates (even out of those who were studying theology). And even in high school - the entire religious studies department was staffed by atheists. On that note, I'd strongly recommend to all: This talk on the nature of good explanations by David Deutsch. Also maybe Chapter 13 of Peter Godfrey Smith's Theory and Reality. And (with a grain of salt) the paper Two Basic Types of Scientific Explanation by Carl Hempel.
  6. As I read it (google translator iyf): we start with an infinite set of apples - partitioned into subsets of 5. From each of those, two are taken and the set is repartitioned into sets of 7 and two apples are again removed from each subset. This is repeated indefinitely, with partitions into subsets of prime sizes. As I understood that: We start with apples {1,2,3,4,5,6,...} and in the first iteration we remove {5,6} U {10,11} U {15,16}... in the second iteration we remove {7,8} U {14,15} U {21,22}... in the third iteration we remove {11,12} U {22,23} U {33,44}... The question is simple enough to understand, after infinite iterations, what has been allowed through this sieve? The answer, I don't know - but some experimenting seems to show that a large amount of numbers do get through - so I'd be tempted to say that it's infinite. I'm sure this can be solved though - so the answer to the question poised in the topic - is yes.
  7. If we continue to put more satellites into Earth's orbit, then it would perhaps be easier to chuck them from the moon - with a lower escape velocity than from the Earth. Particularly if we were to continue space exploration then the moon would be a reasonable jumping off point. Though I'd still go for all of the above.
  8. Right, the vast majority of applications of electricity primary involve circuitry - for all intents and purposes, all of them. Lighting is the most simple circuit you're going to use - a power supply, a fuse, a switch and a light bulb. The type of thing you'd make with a kids science kit really is a scaled down version of exactly how that works. Even with added complexity such as two way switches or more complex components such as dimmer switches it's all incredibly simple and all the circuitry involved can be mapped out on half a post it note. A simple heater, or refrigerator for that matter, doesn't really vary in terms of circuit diagrams of a light and a dimmer switch - replacing the light with a heating or cooling component. The next step up in complexity is probably analogue timers, like those in a toaster. Once a (variable) capacitor is filled to the brim then a circuit breaker is activated, taking away current from the heating coils and the mechanism holding everything down, so that the toast can pop up. Still, very simple. Transistors are the key to further complexity, rather than cutting current on or off - it can be redirected - allowing for digital timers, digital clocks, transistor radios. Further components aren't strictly speaking required for a more or less unlimited increase in complexity: the silicon chip is essentially thousands of transistors, forming an immense circuit with loads of sub circuits that create logic gates necessary to run a computer, television, games console, digital radio, burglar alarm and thousands of other pieces of consumer electronics.
  9. Paper sizes aren't an international standard either. DVD cases are actually a lot more standard, although a CD/DVD itself would be a pretty good bet for standard sizes. Third world countries for that matter, are more likely to have a CD or two lying around than to still use Imperial measurements. edit: WolframAlpha suggests an AA battery as being 2 inches.
  10. That sort of testing testing is well, most of what statistical inference is, actually. Likelihood functions (amongst other things, but likelyhood is easy to work with), help form confidence intervals and hypothesis tests. So that for a given amount of data, you can say with some given percentage of confidence that the coin is fair, the octopus is working on random or otherwise. Things with a probability of less then 128, do happen - but somehow I doubt that the octopus wasn't trained in some way.
  11. I think, that the choice may be between using known results such as implicit differentiation - or by proving the limit from the [imath]\epsilon - \delta[/imath] definition of a limit. To do that, rather than considering [math]\lim_{n \to 0} \frac{a^n - 1}{n}[/math] it'd probably be easier to use [math]\lim_{n \to \infty} n (a^{\frac{1}{n}} - 1)[/math].
  12. Project Gutenberg.
  13. I wouldn't know exactly what sort of distribution you'd be looking at. The article presumes c(x)>0, c'(x)<0 and c''(x)>0 for c=1 being maximum strength, c=0 being minimum strength and 0<x<1 being the distance from the bottom, which allows for a whole host of functions. Of course, some empirical science should be able to provide information on that. Blotting paper perhaps?
  14. Machine? If you're using a machine that has a tank of coffee ready made then it'd have dripped in, and all been equally close to the coffee as it did so, so the issue wouldn't come up. But who uses drip machines anyway? Making coffee without a machine, stirring varies from making no difference at all (because you're not moving stuff vertically) or being quite dangerous (because you are).
  15. Based on the paper Recursive Binary Sequences of Differences by R. M. Richman, an article on the Guardian website tells us how to pour the perfect [two] cup of coffee. I think it was Alfréd Rényi who said "A mathematician is a device for turning coffee into theorems." The result is fairly intuitive: if there is a flavour gradient in the coffee pot then the average of the top and bottom should match the strength of the middle, when you're dividing the pot into two cups - assuming a well behaved distribution of flavour (I don't think you could apply the same principle to fairly sharing a bottle of beer, since there are no virtually dregs in the top or middle). The Guardian article implies that it's yet to be generalised to n cups, but I don't think that'd be too difficult - even if the method would be very computationally intensive.
  16. ! Moderator Note Oh, awesome! edit: oh :'(
  17. The foundations for both were laid in one paper in 1917. Einstein was perhaps the most famous person to have and well known for working on the Manhattan Project. Regardless, it's actual function is quantifiable description and prediction. Books wouldn't sell that much if they were made with the express intent of intimidating, nor if they were useless. You are free, of course, to speculate on the intentions of authors - but I would consider how making wild unsupported allegations reflects on you.
  18. I didn't mean particularly formal Riemann sums, just that it'd have been done by adding up values of 1/x. edit: oh and I'm pretty sure Styla is a spambot, which would explain the nonsense, I've reported the post.
  19. In that case you might want to use \mbox around plain text such as 'if', and use the \cases notation rather than trying to hack your way around a matrix. [math] f(x) = \begin{cases} \ln(1-x) & \mbox{if} \; x<0 \\ 1 & \mbox{if} \; 0\leq x < c \\ x^2 & \mbox{otherwise} \end{cases} [/math] See how that looks neater both in terms of code and output?
  20. Well, stimulated emission and the theories of relativity are kind of separate - even if they came from the same guy. Of course the Internet relies on general relativity as well (in fact, it'd kind of work without lasers, lasers just make it work well).Harnessing the power of the atom seems to be the most obvious product, if we're looking for examples of things we wouldn't have without his work.
  21. For a large part, I think measured by hand. Although the 'nice' values for the trig functions can be proven from the geometric axioms. Logarithms only came up in the past couple of centuries, they'd have been done by (and I'd imagine still are to an extent) by approximate Riemann sums.
  22. Matrices are something more specific than two dimensional arrays of numbers, though two dimensional arrays of numbers can be treated as matrices if you really want. Most simply, image data would be stored as each position vector (or set of Cartesian co-ordinates) being associated with one piece of colour data. (that what you'd call a bitmap). Transformations can then be performed on those position vectors (as per Arnold's cat) or on the colour data describing them.
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