the tree

Well clearly not, because there clearly are - it doesn't take an particular knowledge to find them. That is (sort of) what I meant, when I said that a,b,c could be rational and there would still be no solutions. To be clear on that, a rational number x is one that can be described as x=a/b where a and b are integers and b is not 0. Every rational number has either a finite or an eventually periodic decimal expansion, and every number with a finite or periodic decimal expansion is rational. Not quite. (6x1000000)3+(8x1000000)3=10000003x(63 + 83), so your hypothetical example (as we've covered, is impossible anyway) is out by a factor of 10000002.

Well, if you relax the restraint a tiny bit and allow for rational a,b,c then there are still no solutions (if (a/b)3 + (c/d)3 = (e/f)3 then adf3 + cbf3 = ebd3) - I think there might be some room for relaxing the condition on n - perhaps even a way to get countably many solutions.

There'd be some who'd argue that statistics isn't a subset of mathematics in the same way that computer science isn't. Either way, mathematics is a much, much bigger area of study than statistics. Algebra, calculus, set theory, group theory, geometry, topology, operational research - are all mathematics but not statistics.

Was this with the standard yeast you get in wine making kits? See, everyone knows that nothing can survive in whisky - and that Ribena has an expiration date of more than a couple of years. But as I had this misfortune of discovering, when stored in the same bottle, the top can be covered in some kind fungi in the space of the week - I'm still not sure how that happened.

Only if by "you're in the Matrix" he meant "the Earth revolves around the sun".

Presumably, even if every school in a state decided to adopt a uniform policy, that still wouldn't be a reason for a state to make it the law? To the best of my knowledge the British government don't encourage school uniforms - it's just that nearly all the school they fund just happen to like them.

Oh really, are we using 'slut' as an objectively bad term still? Really, promiscuity and a bit of skin are frowned upon some religions but unless you're trying to present a religious argument against school uniforms this really isn't relevant. Facts, and questions of ethics don't really go together. Facts are largely about what is, and this is a question of what someone may or may not think should be. I'd like to see some formal studies on that, I really would. Was anyone, anywhere, ever claiming that they would?

I think it'd be fair to agree that as far as this thread is concerned, Marat is just trolling and doesn't really listen to music or know much about it. So we can move on. For the past week I've been living in a communal setting - because needs must, and this has resulted in a requirement for the non offensive approach when it comes to musical selection. Which has become very interesting although not all that much fun. I've come to realise that I like my music a lot louder and livelier than a lot of people which I wouldn't have thought since I tend to listen in a fairly passive fashion - never having been one to dance. Of course, the way that you listen to music is a whole other kettle of fish possibly not really related to musical preference.

Supposing we knew the answer to the question posed in the topic, why would we tell you?

Marat, I'm sorry to be repetitive but I'd really like you to address me directly here. I'm going to have to call you on this false dichotomy again. The sets of 'classical' and 'popular' music are neither exclusive nor exhaustive of the realm of music. You are honestly giving the impression that you have just simply never bothered to listen to any music outside of your comfort zone. For instance you gave: Which was released half a century ago and is no way representative of any genre to speak of. There's nothing wrong with age, of course, but I don't think anyone has gone out of their way to listen to that since. Further to that you ignored my point about John Cage's 4'33" incredible simplicity despite it being one of the greatest triumphs of contemporary classical music - it seems that when you say 'classical music' you are only referring to the archetypical symphonic orchestra which anyone who has really tried to branch out their cultural experience would know is far from representative of classical music. The fact is that there are wonderfully (and by that I mean genuinely full of wonder) classical pieces that revel in simplicity such as - I don't get the impression that you've ever made the effort to listen to that. And then there are immensely complex pieces in many other genres - do you really believe that an average musician from any background would find Pink Floyd's Dark Side Of The Moon in any way simple? In fact, as far as I can tell - you haven't even given an example of a specific composer or piece of music that you do like. You talk about classical music as if it were this closed set that stopped being contributed to some time in the past and somehow deserves a kind of reverence for this supposed ancient quality, but at the same time it feels like you've read about the experience of listening to music without ever having done so: anyone can copy the definition of numinous from a dictionary but to not give a single explicit example of when it has ever been experienced seems like an incredibly shallow description. Please answer at the very least these questions: Which composers/artists/musicians do you like in particular? Is there a piece of theirs that is your favourite? Is there any music at all that holds a particular personal significance to you? I'll at least then believe that you actually do listen to music. Because right now, I am not convinced. If it should ever occur to you to push your horizons a little then I can wholly recommend software such as last.fm to find new things that you might be interested in.

lulwut.

I'd strongly suggest reading up on the prohibition in America, or for that matter just watching a few Mafia films. No no no, you're missing my point - if you get to declare that alcohol tastes bad an impose that opinion on the rest of us - then I should get to do the same with artificial sweeteners. And yes, they all do harm - they release insulin unnaturally when there is no sugar around for that insulin to deal with, this leads to all types of problems. I would argue that popping candy is unpleasant enough to warrant a poison and sailing and rock climbing is plenty dangerous. Not to mention how going out at night can lead to catching the flu. There are restrictions on buying cyanide, since it can be used as a weapon, but you can buy it. Seriously? Anecdotal evidence? What do you imagine the title of this website is? Because personal freedoms are important. (how many times with 'the poison is in the dose' have to be repeated?) This kind of statement is what gives the impression that you don't know what you're talking about: a single pint of beer wouldn't put a healthy person over the UK driving limit. Oh heavens there is an awful lot of stupid here.Alcohol is a general family of molecules, of which ethanol is a subset. Some people drink pure alcohol - it is a very bad idea but generally speaking not lethal and it makes for an okay bucket list item. Having laws doesn't stop a citizen being free, being unable to make choices about their individual lifestyle does. THE POISON IS IN THE BLOODY DOSE. It's pretty evident that you're not used to forming cogent arguments, there's no reason to point that out. Languages have conventions for a reason. Me and Captain Panic aren't the same person, I never mentioned alcohol helping people get along, please pay attention to whom you're addressing - it comes across as extremely rude otherwise. Please excuse whatever I missed in whateverthehell that was. Fixing that in order to produce a legible response was super exiting and not whatsoever tedious, thankyou.

This is true, for the most part. So presumably what you're suggesting is that if a certain freedom isn't strictly needed - then there'd be no harm in removing that freedom? Carrying that on, why not ban artificial sweeteners? Everyone knows they do more harm than good, no-one needs them and because I personally don't like the taste of them (I tried one once) that must mean that they objectively taste bad. Popping candy for that matter, sailing, music festivals, the right to leave the house at night, rock climbing, churches - why not ban all of those? The vast majority of people who drink never develop an addiction, that much is little bit obvious. You've never had a cup of coffee? You have any idea what is in that? Believe it or not, drinking does not have to lead to drunkenness. You seem to have some pretty large gaps in your understanding of what alcohol is (it is ethanol), what it does, how it's made, the concept of living in a free country, how arguments are formed and what the purpose of an ellipsis is - it would seem you need an awful lot more experience.

That doesn't seem to follow - there are some amazing examples of jazz violin playing and well, the fiddle is basically the foundation of all modern folk music. That definitely doesn't follow. I'm pretty sure everyone I know has at some point brought music just to piss off their parents, that's a rather essential phase in musical appreciation IMO. Really? Perhaps you're the only one amongst your friends that appreciates music to the extent that you do. This seems horribly biased to me. There is absolutely no way one man could listen to 'most' modern music, or for that matter think of all classical music as being at one level of intelligence. Everyone needs some more dubstep in their lives, the variation on repetition there is often incredibly subtle look up more or less anything by Netsky for an example of that. Personally, I've always felt that symphony was the weakest facet of classical music. Many composers were perfectly capable of an "overwhelming sense of intricate structure" on a single instrument (Beethoven's Für Elise being one of the most obvious examples) . And a lot of symphonic classical music it well, it reminds me very much of heavy metal being done wrong: lots of power chords and amps at 11 - but still with predictable dynamics and somewhat of an absence of poetry. I think you're drawing a bit of false dichotomy between 'classical' and 'popular', there are many artists who are certainly neither. Not to mention the slightly post and pre classical periods of music such as Baroque and Romantic. There is an awful lot of music out there. I really don't like the idea of there being a correlation between complexity and intelligence. Compare John Cage's 4'33" with Queen's Bohemian Rhapsody for instance - I'd say there is significantly more intelligence in the composition of the much simpler former. In terms of my listening habits, I do of course, have a last.fm account. And somewhere or other there is an SFN group. edit oh it's in my signature.

It's time consuming more than anything, not to mention a little bit pointless. If you want to calculate log10(5) by hand then recall that logb(x) =ln(x)/ln(b)where ln() is the natural logarithm, and you could always calculate that from the integral by the trapezium rule or something.

For one I think that should be [math]\begin{array}{rcl} x &=& 5 \cdot \log_2 (x) \\ 2^x &=& 2^{5} \cdot 2^{\log_2 (x)} \\ 2^x &=& 2^5 \cdot x \end{array}[/math] Yeah, at most two - [imath]y(x) = 5 \cdot \log_2 (x) - x[/imath] has just the one turning point.I severely doubt there's a closed form expression for either.

You are by no means the only one.

The university applications process also involves a personal statement in which you describe your interest and motivations as well as other hobbies and what you do outside of the classroom - lots of people take up new hobbies during their time in VI-form to make themselves sound more interesting and wind up having some real fun in the process. You'll find lots of opportunities to join in on sports, volunteering or maybe to take up a part time job. In terms of books to keep your interest, I'd whole heartedly recommend The Universe In A Nutshell by Stephen Hawking - it's essentially a light version of A Brief History Of Time so it makes for relatively (heh) easy reading while also giving you a strong impression of the breadth of modern physics.

Okay I'm presuming you're in the UK already and haven't yet pursued A-levels and whatnot? If so A-levels are definitely the way to go for a science degree, whether you do them at a VI-form or a college is up to you, some are better than others but that all depends on your area and again what's right for you. You'll want to have maths and physics under your belt, maybe further maths and another science to boot. Beyond that, read around the subject, learn independently and take great care in picking the right course when it comes to applying for university.

No. An internal combustion engine works on the basis that it is the stuff that is burning which pushes the pistons. That'd be disastrously inefficient, but not difficult to make. On what sort of scale? I'm sure you could just buy a stove kettle, a dynamo and a fan on more or less any high street - putting all that together might pose a challenge but that's all detail.

I suppose you could model user behavoir, clicking from one page to another with a rather large Markov chain, although I'm not sure how the data from that would help optimise the choice of adverts. Another approach that I'd propose would be: Assume that you only need to 'try' an advert once, that is that once a user has seen it and not clicked, they aren't going to click it that session. Find the probability of an advert being clicked at all (the amount of times the advert has been clicked divided by the amount of unique users exposed to it). Multiply that probability by the profit from a click, to get the expected profit per user for each advert. (this should be a very, very, small number). Order the adverts from highest expected profit to lowest. Either by some complicated click counting with cookies and whatnot, or by just by depth from the home page - expose your users first to the most potentially profitable adds, then once those haven't gained a click move on to trying the others.

The thing is, there's kind of a very wide of curves that could define the shape of the roof, it could be quadratic or a sine wave, my guess would be a catenary. I certain amount of trial and error is probably needed to get that to work how you want it. Edit: Right yes, just played with Maple for a while, this isn't too hard but the formulae are a bit ugly. Let's say we have parameters l,w,h for the desired length, width and height. You should be able to put together an equation u(x,y) for the height of the building at a given point. Too much help here would really jeopardise your work, so I wont give you further hints until you say what you've done. Mine was based around cosh but if you do something entirely different then please do tell. I'm assuming you've done the above so that u(x,0) is the interesting cross-section, and u(0,y) is a rectangle of size l*h. And that the centre of the floor is at (0,0). Then some quick sketching should show you that the volume of the cuboid with sides at x and -x can be given by l*u(x,0)*|x|. Plot, differentiate, find the stationary point etcetera. Play aimlessly with software of your choosing, make pretty pictures, have fun. I'm sure you can integrate to find the area under the curve, do that for different h. Repeat part 2 for each of those. Compare the ratios. Check your notes for an arc length formulae.

The general formula for a finite geometric series is [imath]\sum\limits_{i=0}^{n}r^i = \frac{r^{n+1}-1}{r-1}[/imath], it's pretty easy to subsitute in [imath]r=\frac{1}{c}[/imath]. I can't see why [imath]c \in \mathbb{N}[/imath] should be a requirement at all.

Honestly, if you can't work out to copy/paste a front-end by yourself, then you need to spend more time on basic computing skills before trying to build an interactive website. Well yeah, but a much smaller site could live off of free software such as phpbb, I'd say as a rule - hosting costs would be a bigger concern than software.

I think you should concentrate on your exam.