Schrödinger's hat

To elaborate slightly on what Banks said: We do slowly add elements to the periodic table. As we smash together larger and larger nuclei, sometimes we manage to get a large nucleus to stick around long enough to do some measurements on it. To give you an example, one of the most recent additions (the most recent addition?) to be officially recognised is Copernicium (number 112). These elements are generally far too short lived to be found in nature (with half-lives between milliseconds and seconds). There are some predictions that some of the heavier elements still to be discovered will be more stable (between seconds and days, still not stable enough to find in nature unless you were very carefully picking through the remnants of a very recent supernova). The only possible use I can think of for some of these ultra-heavy elements is if you want a portable source of very specific energies of radiation. I can think of no good reason as to why you'd want this. Maybe for some kind of medical imaging? They would also be inferior in many ways to most currently used (natural and synthetic) elements used for this purpose due to very short shelf life and difficulty of production. TL;DR version: Yes, we've run out of elements that can be found in nature (be it on earth, stars or other planets). With the possible exception of finding the tiniest trace amounts of some of the elements we've already made in the lab (but not found in nature). It'll look much the same, maybe with an extra line or two on the bottom which anyone except nuclear physicists will leave off/ignore due to them being too radioactive to ever be useful. No, other planets will be made of the same stuff as Earth.

No, non-polynomial means non-polynomial. Quite distinct from non-deterministic polynomial (something that would be polynomial given a non-deterministic turing machine). If it turns out that [math]P=NP[/math] non-polynomial would exclude NP. Pointing out that I didn't use the exact same jargon as someone else is barely relevant. Wrong again. Verification will return true if the given input is a solution to the problem. If verification were possible without a test input, then primality testing would be trivial.

The idea is that you have a number [math]N[/math] which measures the size of your problem. This could be the number of digits in a number, or the number of elements in an array you want to sort. If something is polynomial time, it means that it takes no more than [math]N^c[/math] where c is a constant operations to solve the problem. This is good, because polynomials grow slowly compared to exponentials (stuff like [math]c^N[/math]) or similar. So as N gets very large, the number of operations required is still managable for a polynomial time algorithm (although as c gets bigger, this is less and less true). A good example is factoring. There is no known (classical -- ask if you're really curious) factoring algorithm which is polynomial in the number of digits. As a result adding a digit (approximately) doubles the amount of work required. Checking the factors, however, consists of a multiplication which is polynomial. You'd have to (again, very approximately) double the number digits to double the amount of work required. Which brings me to [math]P \mbox{ vs. } NP[/math] Khaled, please check your facts before posting. [math]NP[/math] Stands for non-deterministic polynomial. This is a set of problems (factoring is one) where the answer can be checked in polynomial time. This includes problems in [math]P[/math] where the answer can be calculated (not just checked) in polynomial time as well as problems that have no known solution in polynomial time (but can still be checked that quickly). (ie. all problems in P are in NP, but it is not known if all problems in NP are in P As Khaled said, it is not known if these problems (the ones that can be checked but not solved quickly) have a fast solution that is undiscovered [math]P=NP[/math] or not [math]P\neq NP[/math]

Didn't read much further than this: But you missed an [math]x^2[/math] and had a couple of odd things on later lines that looked related. Try getting the given equation to look more like your expansion of [math](x-a)^2[/math] [math]x^2 + \frac{3q}{p^2}x + \frac{4}{p^2} = 0[/math] So: [math] \frac{3q}{2p^2} = a [/math] and [math] \frac{\pm 2}{p} = a[/math] [math]\implies \frac{3q}{2p^2} = \frac{\pm 2}{p}\; \implies\; 3q = \pm 4p[/math] The [math]\pm[/math] must be + or we don't have both p and q > 0.

Seeing as this was advertising and the link was removed. For anyone wondering how it's done: Adding the digits of a number results in a number with the same value modulo 9 So 7 + 8 + 3 + 4 + 1 is 5 (mod 9) Subtract the sum of digits, 23, which is also 5 (mod 9) This guarantees you have a number that is 0 (mod 9) So from the sum of digits having the same value mod 9 property, we know that if we add the digits we'll get a number that is 0 mod 9. Erasing one digit reduces that value by the value of that digit. So sum the digits that are read out, 9 - sum is the value of the removed digit.

This is in principle, if not practise, very similar to the idea of trusted computing. There are security benefits, but many consider the potential price (manufacturers/microsoft being able to dictate which OS/software you are allowed to run) too much. As I said, it depends on how the preview is generated. I believe google render an image of the webpage on their servers, if this is the case it is perfectly safe. (I have also read something about them writing a simple html renderer entirely in canvas, but I don't know if they use it there). Someone less security minded might just put it in an iframe/div or wrap the webpage in some piece of javascript that somehow changes its size. Not really sure how it's done, but if it were done in a way that resulted in the webpage running any scripts, it could be dangerous.

The actual act of putting a font on a system doesn't lead to executing arbitrary code that I'm aware of (although this doesn't rule out vulnerabilities in the font rendering), but this is a common vector for malware regardless. The fonts will come wrapped in an installer that also contains malware, it is this installer (an executable) rather than the font itself that contains the virus/malware. I seem to recall there was some issue about truetype fonts being able to carry viruses (windows 98?), but I imagine this has been fixed by now. Definitely, these will run with the same priveliges as the account launching them, and so can do anything you could do. As xitten said, this would be the easiest way of getting a virus listed. To elaborate, running any untrusted executable (.exe in windows) file (whether it's OMGINSTALLTEHSMILEYS.exe or thisisMSWordItstotallylegit.exe) is a terrible idea and is almost equivalent to handing the author your system on a plate. Recent versions of windows are slightly better (UAC will ask you if you want to give the file priveliges to do many things), but doing this is still doing 90% of the work for whoever is trying to comprimise your system. It depends. This is probably at least as safe as visiting the website. Depending on how the preview works it could result in running untrusted javascript. This would require some vulnerability in your browser for any malware/virus to be installed, even if whoever wrote the website doing the previewing wasn't very security conscious. It's probably the safest of the things you listed. This /shouldn't/ be a problem, but microsoft have this insane obsession with autorun. They appear to have gotten over it recently, but many versions of windows will automatically run executables on an inserted drive (if they are named/arranged correctly). They even went so far as to automatically turn autorun back on with patches even if the user had disabled it several times. This is an extremely common vector for viruses/worms/malware. There have also been many reports of USB sticks coming infected from the factory. For this reason you should always disable autorun or first format your USB sticks in a computer that does not have this stupidity. I'm not really up with windows 7 on this, but I've heard many accounts of autorun viruses in both XP and Vista

There's a nice tree structure you can access via the exercises. If you pick an area that interests you, you can see what's required before you learn it, and what it leads to.

a. You assume this mapping is one-to-one. If the unprovable statements in this symbolic system do not correspond to physical phenomena, then 3 does not follow. b. You assume the symbolic system required to describe the universe is of at least the same level of complexity as one to which Godel's incompleteness theorem applies. c. This does not prove that an alternate system cannot be constructed to explain those phenomena that do not fit within the first (and then a third and a fourth and so on).

We could just as easily have another scenario, planet in an unstable orbit, maybe. I'm ignoring the life already present issue for now, as well. I'll assume there is none for the time being. So now we have the question of whether it's ethical to colonize a planet that will become uninhabitable in the future. This is true of all planets, even earth, it's just a matter of when. So let's try a bisection: Is it unethical (on the basis of how long the planet will be around) to colonise a planet that will be destroyed: Immediately after colonisation? After one or two generations? After centuries? Millenia? Megayears? Gigayears? I think I'd answer yes to the first one, no to the last, and a big fat it depends on all the others. Next question is: Depends on what?

Any simplification (including the one I gave) leads to another point where you can point and say, 'how do you know that?'. You could say something along the lines of 'by looking at other stars we know that....', but that doesn't say anything about why we think the sun will act in a similar way to other stars.

No. What're you supposed to do, come with someone and only eat from their plate?

Ever so slightly more refined than that. Start with your most constrained employee, and work towards one that can take the most shifts. That way you're less likely to have to backtrack. Bear in mind that the work required for this kind of approach increases extremely quickly with number of employees/shifts, so your client may turn around and wonder why their system hangs for 3 hours every morning if they get a few new staff and want you to fix it.

One way of determining this (not necessarily the only way) is by using empirical measurements and observing other stars. If you look at enough of them, you can generate something called a Hertzsprung Russell diagram http://en.wikipedia.org/wiki/Hertzsprung%E2%80%93Russell_diagram There's a lot of info not on one of these that you can get from the spectrum of a star, but suffice to say for now that we know that our sun is on the main sequence. We then go and look at various star clusters. There's a range of ways of finding the age of a star cluster (we know from some simple energy equations, and measuring luminosity, that big stars burn fast, for example). You can measure the ratios of large to small stars, count the number of supernovae (and remnants thereof) and stellar formation events, and so on. Once you do that, you start to notice that stars follow a pattern based on their age. Depending on their mass (and I don't feel like going into the mass calculations right now) and metallicity (basically how much of the star is made of heavier elements) they follow a fairly standard pattern as they age. Stars above a certain mass get brighter and hotter as they run out of fuel, and then finally start burning Helium and turn off of the main sequence. You can also predict where on the main sequence a star of given mass and metallicity will be at a given age from your HR diagram. (more here) http://en.wikipedia.org/wiki/Main_sequence#Evolutionary_tracks There are plenty of things I have skimmed over, or ignored here feel free to ask more on any point I've glossed over. This is a subject somewhat bigger than one forum post can contain. As such most documentaries tend to leave it at 'because that guy said so'. Adding a little bit more would be 'because of blah, because that guy said so' so there's not really much point. The only alternative is to assume your watcher has a large amount of patience and can follow a more technical explanation while you explain the whole lot (and probably take a lot longer to do so). Sadly documentary writers seem to be expecting less and less of their audiences recently.

Oblig http://xkcd.com/221/

Yes, this is certainly possible. Thinking about it briefly, the vague form of a completely naive algorithm (basically just brute force it) that would do it in roughly [math]O(n^{\sqrt{n}})[/math] time comes to mind. This would be between somewhat and completely impracticle for more than 50 people. There are probably many improvements that anyone competent would think of along the way, but in general it would still scale horribly. I don't know the techniques used to do it quickly, though I know they exist. A good starting point may be to look into integer or linear programming and the associated mathematics. If this is a problem that falls in this domain (and it smells rather similar, I have seen linear programming stuff on a crossword solver) then this would likely be the best approach. Expect a fair deal of linear and other abstract algebras. So it may be fairly heavy going. Other approaches/fields that come to mind would be pathing or chess playing algorithms.

This is tantalisingly familiar, and brings connotations of surveying and naviagtion to my mind. A word similar to menstruation (but not exactly), cannot think of exactly when/where I heard it. Aha, this seems to back up my vague recollections, although I'm still no closer to knowing where I originally heard it. http://www.ayton.id.au/gary/Science/Ma_mens.htm

Heh, whenever I see an exponential of something other than a number I think of a chinese lecturer I had who had a lot of trouble pronouncing 'exponential map'. As a result I often call it the exploitation map, which is somewhat fitting as it is so useful.

As capn said, not all file formats support metadata or other information that won't affect the output. Even in some that don't, it is possible to hide a message. You can put your message in the least significant bits of a picture or audio file, they will not change the color/sound perceptably, but would allow you to write something someone else could later read (perhaps in a text editor if you were very clever, or maybe a hex editor or some custom encoding). This process is called stenography. I've also been wracking my brains trying to remember the source, but I recall reading about some early programmers putting messages in areas of the machine/source code that they prevented the machine from ever getting to. Something like: if (1== 0) { int The_code_below_this_block_does_something_complicated_and_hard_to_understand } Although I'm certain the language was not a c-like one. You could probably even do this with raw machine code (so that it was a message when encoded in -- say -- ascii) if you prevented the computer from executing that section. Similar tricks could/probably have been employed in many circumstances. This being said, there are formats that don't have metadata and in which every bit of the file makes a significant difference to the output (raw text comes to mind, although there is probably room in there somewhere to hide a message -- access times, permissions and such; although it wouldn't all be preserved when you copied the file/accessed it etc).

Does anyone else think of this when they tell someone to label their axes? (Yes, I realise I didn't bother with units, scale, or a title.) Or do you have any other weird associations with common phrases?

H and L for high and low will be your off and on (I don't know if H is always on). The funny squiggle is a rising edge. (A signal that has just turned on). Qn seems to be some kind of variable value, I'm not 100% sure what it means in this context, but I would conjecture that it means they will retain their previous value, or will be set depending on the value of the corresponding Dn (ie. Q1 will be set to D1 on the bottom two rows). This fits with MR resetting them to L and the fact that they will only change if the corresponding D (input) is high or low. So Q3 would maintain the value that Q3 last had, unless MR was high (in which case it would go low) or both of the input enable values were low. In the second case (both IE values are low) then Q3 will be set to whatever D3 is when the clock pulse goes from 0 to 1 (the chip detects a rising edge on CP). Also, as long as the clock signal remains low, Q3 will be kept at its current value.

Missed this somehow before my other post. I gather from seeing your other posts around, that you are reasonably mathematically literate. So you may be able to derive the Lorentz transforms from some simple thought experiments. Here's one to get you started (it should yield a time dilation formula). Consider a pair of paralell plates or mirrors with a photon or laser beam (or other speed of light object) bouncing between them. Simulation of this situation here: http://www.refsmmat....tml#light-clock The object/photon/whatever (let's call it a ball to avoid confusion with wave stuff) is bouncing between the plates which are (for convenience) 1 light second apart. The ball takes 1 second per bounce, (and the local clocks will tick once for every time it bounces) Now consider a frame in which the whole contraption is moving up. The ball bouncing off of a plate is an event, the fact that it happens cannot change. But in a different frame, the time and place of an event might change. The ball has moved a different distance (to meet the moving plate). Some assumptions which should help: 1. The ball is moving at the same speed in both frames (that is the magnitude of its velocity is constant). 2. The dimensions of space are independant. Change in velocity upwards will not cause any non-classical or otherwise unexpected change in sideways distances or positions. 3. Constant velocity. Something moving inertially (at constant speed) in one frame will be moving at constant speed in other frames. From this and other similar situations, you can derive special relativity logically with no need for further experimental results (other than the constancy of the speed of light). Once you have a working formula for time dilation, you can derive length contraction as well (a good thought experiment is to consider a car moving on a rail and the observations of the moving and stationary observer -- add beacons, laser beams (a pair of lasers fired from the center of the car simultaneously at either end of the car is useful) etc as you like to see what will happen). Other notes: Keep careful track of signs, probably the easiest thing to muck up. Velocity of [math]\frac{\sqrt{3}}{2}c[/math] leads to convenient numbers. You may see factors of [math] \sqrt{1 - \frac{v^2}{c^2}}[/math] pop up a lot, it helps to give them a name. Traditionally we set [math]\gamma = \frac{1}{ \sqrt{1 - \frac{v^2}{c^2}}}[/math] You don't really need to keep track of three directional dimensions, you can happily set z=0 v_z =0 and leave it out. Taking derivatives and keeping track of some calculus identities can sometimes sidestep a bit of algebra (but you only need algebra to do the calculations). Feel free to post your progress/requests for further help here if you decide to have a go at it (or if you want to try, but are still a bit lost)

The cause of all the weirdness is that there is a speed that is the same for all observers. If B sees the light travelling at c (relative to him because he's at rest in his frame), and A sees the light travelling at c relative to him, and speed is distance/time, then something has to change with your definitions of distance and time. B's seconds and A's seconds are not the same. B's metres and A's metres are not the same.

I don't quite follow. You're saying there could be mathematical objects for which there would be no real world representation? Depending on your definition of representation, we already have those.

'applications in computers' is an awfully wide field to dismiss out of hand, it includes a lot of computer science (including mathematics) as well as the simple grunt-work of sitting down to program something in C or assembly. It has a lot of practical interest if you're trying to operate a debugger. It also has some pedagogical interest. By understanding that radix is an arbitrary choice, you come to better understand decimal arithmetic. It can also help to drive home the distinction between repeating decimals and irrational numbers. Many students lump these together as 'number that keeps going'. If you're not thinking in terms of using different bases you'll also have a devil of a time programming anything using large numbers that works efficiently (or even understanding the explanations of said things). There are tricks from number theory using exotic bases, such as a primorial or other variable radix, or base e. Sometimes they can result in tighter data packing or faster algorithms. Granted, looking at a sum and inferring that it is in another base is an exercise which has little mathematical value, but then so is finding the derivatives of a bunch of polynomials along with almost everything else encountered in a highschool or college exam. Sadly 'get the students to do a bunch of related, but worthless calculations and hope they think about the concepts involved' seems to be the main way of teaching maths these days.