Schrödinger's hat

Well assuming they move faster than light, and that relativity holds. Sending something faster than light in one frame is the same as sending a message back in time in another. Once you can send a message back in time you run into all sorts of things like the grandfather paradox. If this isn't enough I can try and explain exactly how it entails sending a message back in time,

Could you explain your reasoning a little more? I think that perhaps noone has answered because they're not sure what you did. If you give us a little more context we're more likely to be able to find your mistake. I'd approach this by changing to cylindrical coordinates and doing the whole integral from scratch, but that's largely because calc 1 was a while ago and I don't remember the shortcut for solids of revolution. I can explain this if you like, but chances are high that that's not the method that was intended for solving this problem. Use of the forum's latex support is appreciated to make reading it easier Something along the lines of [math]V=\int\limits_{\pi}^{e}\int\limits_a^b f_{\text{foo}}\times x^3\;dxdy[/math] To produce: [math]V=\int\limits_{\pi}^{e}\int\limits_a^b f_{\text{foo}}\times x^3\;dxdy[/math]

Well the potentials for the various atoms/molecules that make up the crystals are made of can be described as series of trigonometric functions. I wouldn't be surprised if you wound up with the same types of symmetries in your plots as are found in crystals.

Where did you get this from? I've never heard such an idea except in scifi.

Well, there's speed of light to consider. The star has to actually have time to form. This takes a fairly long while by our standards. On top of that, there's a factor called metallicity to consider. The large percentage of the heavier elements there are in a star, the less fusion occurs at a given temperature. The less fusion that occurs, the less radiation pressure there is holding the star up. The less pressure there is holding it up, the denser it gets. And once it gets to a certain level of density it'll overcome degeneracy pressure and form a neutron star or black hole, exploding in a supernova as it does so. The modern universe has a higher proportion of heavier elements, so smaller stars form now than they did in the earlier universe. I don't remember exact figures, but low-metallicity stars in the early universe were up to 100s of times bigger than our sun. I wouldn't think that anything 1000s of times as large would last very long (ie. after the fusion started, it would slow the collapse of the gas cloud, but never stop it, and the cloud would never form a stable star before exploding).

There's a few ways. You can look at the surface (the size, temp etc) and the mass of the star and extrapolate using thermodynamics. You can look at the spectra of the reactions happening in it. By seeing which elements are fusing and at what rates you can form a relation between temp and pressure.

That's true, but charges of one kind tend to attract charges of the other, so on large scales they tend to even out, and gravity dominates.

No body knows for sure what low gravity does to a body. The effects of microgravity are well known, and they are not good. They include loss of bone density, heart problems, digestive problems among others. We do not know exactly how much gravity is needed to mitigate/prevent this. On top of this, the animals instinctive behaviors could be harmful in a low gravity environment. Smaller animals would probably be effected less, as their weight is less significant in the first place.

You're going to have to describe what you're asking a little bit more clearly. Are the pressures gauge pressure or absolute? If you meant absolute pressure, are you sure you meant for the seawater to be 0psi and not the desalinated water? As you posted this in applied mathematics, some more description of the context or model you are using would be helpful.Then we can be more helpful.

Uhmm, the electromagnetic field does contribute to the stress-energy tensor. It will have a gravitational field different to a non-magnetic object, albeit very very very very slightly different. I don't quite understand what you mean by circular loop in this context. If the loop is conductive you get an induced current which will result in the magnet slowing. However, this is a purely electromagnetic phenomenon. Even non-conductors will react to the EM-field very mildly. I can confidantly say that at all but incredibly high energies the electromagnetic and gravitational effects will be independant. For any experiment you can conceive of, you can consider the EM field and its interactions with charges separately from the gravitational field/spacetime curvature and interactions between masses separately. Then add the two sets of effects to get the right result. For incredibly strong fields you might need to consider the energy-momentum they contain, but otherwise you could ignore them for the gravitational effects. I do not know (and I do not know whether anyone knows) if there are coupling terms between the EM field and the gravitational field at higher energies. Or even if that makes any sense in terms of general relativity.

I've never had any experience with PVC+heat, but I know of a metal pipe-bending technique that is used to get around this problem. You cap one or both ends, fill the pipe with sand, sit it vertical, (this is for thinner steel pipe, so maybe something coarser might work?) and heat it with a blowtorch Then when you bend it, the sand prevents the pipe from collapsing. You still need a reasonably large radius for the bends, but it stops some of the distortion.

All of the strange effects of relativity come from the concept that there is a speed that is the same in all frames. It is not really important that it is the speed of light. It could be the speed of sandwich and it would have the same effect. Although, even if it were the speed of sandwich, you would still be unable to pass such a speed with finite momentum. It would do at least one of these things. Causality could stay intact if there was a detectable absolute reference frame (thus breaking lorentz symmetry). If this was the case, things would only be able to exceed the speed of light in some directions/frames of reference. In this absolute frame, the past would always effect the future. In other reference frames, there would be a seemingly arbitrary hard limit on the speed you could send a signal. If Lorentz symmetry were retained, then superluminal travel is equivalent to time travel, at least in principle.

Umm, there are plenty of chemicals (CO2 comes to mind) that sublime, or go directly from solid to gas. I'm sure some pure elements do, too. Unfortunately I can't remember which ones off of the top of my head. With regards to the light stuff. I don't really understand what you're talking about. Light isn't really gas-like, and it certainly never turns into a solid or a liquid. In some circumstances you can get light turning into matter though

Well, you can't really point to the number 2.7183... either. But it pops up all the time, too. Generally we formulate all of our equations around non-imaginary things being the ones we can measure. Whenever we have an equation that outputs something imaginary, we juggle it around a bit until it's real. The one exception I can think of is quaternions/clifford algebras. Directions in three dimensions can be represented as different roots of -1. In much the same way i can be used to represent a direction in 2d.

Come to think of it, mechanical is a much easier way of doing it. If you wanted them to move at will, just mount them so they are somewhat springy, with a collar which will cause them to contract then attach the ends of the petals to a central point you could then attach a bycicle break cable or something similar, and move the other end of it in your pocket or similar when you wanted them to move

Plenty of things. They're intimately linked to trigonometric equations. Quantum wavefunctions are, in general, imaginary. Although it's the real magnitude of the complex number that represents a probability.

As imatfaal said, Khan academy is excellent. I would recommend against using wolframalpha for your homework too much. If you do you will learn less. Otherwise it is very good.

Is there any advantage to embedding in a higher dimension, other than slightly easing the cognitive burden?

And that's without getting into the statistics aspect of things. I don't think I'll worry you too much with that. The basic idea is: we don't have to worry as much as we thought about error building up, because the errors in different directions cancel out most of the time. Using statistics we can figure out when we have a 99% chance or 60% or whatever of the errors not building up to a point that it matters. I'm always surprised by how complex/involved the answers to simple concepts can get. You can usually simplify it significantly with hand-waving, but there's usually at least a few 'why's left over. Reminds me of this. Also, I was going to make a few more diagrams to make it more digestable, but I got a bit impatient/lazy.

I would think a much better experiment would be to get a double-insulated (possibly vacuum-filled) chamber with a glass window, a microbalance, and a laser place the sample in the chamber, weight it, then hit it with the laser until it's red-hot. If you really care about this, and you can find no relevant experiments maybe you should run it yourself? Making the chamber out of a thermos should not be too hard; lasers are cheap. The most expensive part would be high quality optics which do not heat up when hit with the laser. Construct the apparatus, then ask nicely at the nearest insitution which owns a microbalance whether you, or a willing member of staff could place it on the balance and press the button. Results should be immediate and scientists will be more inclined to reproduce your experiment if you confirm your hypothesis. If you are indeed right, the Nobel prize money should more than off-set your costs, so what have you got to lose?

argh, this is extremely hard to read. use [math]\phi \Phi \int\limits_a^b \Delta \frac{numerator}{denominator}\sin{(x^2)}\nabla \partial \mathbf{D}\mathcal{R}\hat{a}[/math] And so on to do symbols. Producing this type of thing: [math]\phi \Phi \int\limits_a^b \Delta \frac{numerator}{denominator}\sin{(x^2)}\nabla \partial \mathbf{D} \mathcal{R}\hat{a}[/math] There's a full tutorial for using latex in this forum stickied in the maths forum in case you haven't used latex before.

Uhmm, we can't really help you with this without knowing the different directions. Also don't double-post/post a thread which doesn't have meaningful content. You can edit your posts to add more information, or reply to the thread you already made.

Shape memory alloy actuators or muscle wire sound like the way to go. You can get it from robotics/rc supply places. I don't know of a specific supplier to recommend, but google returns a lot if you search for either of those terms. It bends when a current is put through it. You could glue it to the back of the flower and cause it to open/close. Control shouldn't be too bad. Just lead some wire to your pocket with a battery pack and a switch. It will probably drain power whilst active, so glue it together in the shape you want the flowers to be in most of the time.

Continuing from where I left off. With [math]f=xy[/math], our partial derivatives are no longer constant. If we change [math]x[/math] by a little bit, the amount [math]f[/math] changes depends on y. You can see this easily enough, even without calculus. If we take [math]f(x+\Delta x,y)=(x+\Delta x)y = xy + \Delta x y[/math]. So the amount f changes when we change x by a small amount depends on the value of y, or [math] \left.\frac{\partial f}{\partial x}\right|_{y=y_0} = y_0 [/math] Same goes for [math]y[/math] at [math]x_0[/math]. So our error formula for multiplication winds up being: [math]\Delta f = y\Delta x + x\Delta y[/math] Now if we have a certain number of significant digits for [math]x[/math], call it [math]n[/math], and a certain number of significant digistfor [math]y[/math], call it [math]m[/math], then we know that the maximum values for the errors in each term are less than: [math]\Delta x = \frac{x}{10^{n-1}}[/math] and [math] \Delta y = \frac{y}{10^{m-1}}[/math] Looks complicated, but this is just a way of saying 'the last significant digit might vary'. An example might help: Say we have: 10.2 Then [math]\frac{10.2}{10^{3-1}}=0.102[/math] In truth this is an overestimate, because we might have 9.3, divide by 10 to get 0.9 No matter how bad our rounding error, we know the original number was in the range of 9.25 to 9.3499... so 0.9 is an overestimate, but an upper bound is all we need to illustrate why it's the number of digits that matters. Using this information, we go back to the equation [math]\Delta f = x\Delta y + y\Delta x[/math] and plug in our values to get: [math]\Delta f = \frac{xy}{10^{n-1}} + \frac{xy}{10^{m-1}}[/math] Or a number n digits smaller than xy plus a number m digits smaller than xy If m and n are different, one of these will be ten, a hundred, or thousands of times as big as the other, so we only need to calculate our error from the number with the least significant digits. So our answer will have meaningful digits up to the mth or nth digit from the left, whichever is smaller. Using some real numbers with more exact rounding error. [math]345.92 \times 0.164[/math] The maximum errors that we might have accumulated if we'd produced these by rounding another number are 0.005 and 0.0005. Our non-rounded result is 56.731 Our error will be [math]345.92\times 0.164 \times 0.05 + 345.92\times 0.164 \times 0.005[/math] or [math]0.28 + 0.028[/math] We can throw away the 0.028 because really only the 0.28 matters. This will change everything up to the third digit, so we round to 56.7 Although our error is big enough to change the 7, we still write it down because it carries some information. On top of this rounding would change the 6. Hopefully this makes it clear why we use number of digits for multiplication.

Could you post a list of links to all these sources? 15 pages of posts is a lot to trawl through to find them. The only paper I recall seeing a link to was the one on the redefinition of 'synchronised'. It took me a while to work out the consequences of such a definition, one of them is a non-constant speed of light. Light would wind up moving faster towards than away from you or vice versa. I'll address this in more depth when I finally get around to making those simulations. You often refer to others but don't post journal names/dates etc.