Schrödinger's hat

Now you're just playing games with definitions. Some hands typed on a keyboard and a mouse clicked a few times. The cells exist. They work together to do things. They give the impression that they are a unified being a lot of the time. This doesn't prove that I posess a soul or a mind or whatever else philosophers want to call something that is separate from the laws of physics acting on the component parts. I have a strong impression that I have some kind of mind-like thing (although I can find no coherent definition of the term), but I also have a strong impression that the sky is turning sometimes, or that velocities add linearly -- both of these are wrong. Other than this, I have seen no evidence. I have seen plenty of evidence that I am some kind of physical/chemical phenomenon, on the other hand. The mechanisms behind many of the things my brain does are known, and if I imbibe certain chemicals I am inclined to do things that I would not otherwise. I would not call this conclusive, but until I see some evidence other than a vague impression, I see no reason to include a metaphysical mind/soul in my model of reality.

You forgot the rest of the sentence: 'or I am being decieved.' Descartes went on to use some circular logic to throw that latter part away, but I digress. Why is is objectionable for one to be an occurrance? I see plenty of evidence for me being an occurrance, or some kind of machine. My decisions and thought process are easily altered by the presence of chemicals in my blood, certain kinds of audio or visual input, or even the weather. How do I know this 'me' ness I feel isn't merely illusory? That it isn't some kind of facade which helps all the parts work together more smoothly. There are plenty of other things that various impressions and feelings I have tell me that are explicitly false.

FIrstly, I didn't think TFT monitors had anything resembling an electron gun in them, perhaps you meant CRT? If that is the case, the electronics involved are quite high voltage and could be a bit dangerous. Perhaps you would be satisfied by doing the experiment with a laser?

The most common situation for u substitution is to use the chain rule in reverse to turn something you can't integrate into something you can. What we want to do is manipulate our equation into something that looks like this: [math] \int f(u(x)) \frac{du}{dx} dx [/math] So a function of a second function, with the derivative of the second function multiplying the whole lot. Then we can use the chain rule to replace it with: [math] \int f(u) du [/math] A common example is: [math] \int x e^{x^2} dx [/math] First we observe that the derivative of [math] x^2[/math] is [math]2x[/math] So [math]u=x^2[/math] We can easily manipulate things to get a constant by multiplying by 1. [math] \int x e^{x^2} dx = \frac{2}{2} \int x e^{x^2} dx = \frac{1}{2} \int 2x e^{x^2} dx [/math] Which is: [math] \frac{1}{2}\int e^{u} du = \frac{1}{2} e^{u} + C = \frac{1}{2} e^{x^2} + C [/math] Hopefully this is enough that you can apply the same logic to your function. (If not, ask again and I'll elaborate further)

I'll just leave these here: [math] \Gamma(z)=\int_0^\infty t^{z-1}e^{-t} dt [/math] [math] \mathcal{L}[t^{z-1}](s) = \int_0^\infty t^{z-1}e^{-st} dt [/math] Where z is a complex number. (It may also help to find that eigenfunction I was talking about first. Then see how this applies to finding more eigenfunctions, Then you might be able to find a fixed point).

There are displays around that don't require glasses, such as the 3DS. They are still stereoscopic, but use prisims or similar to make you see a different image from different angles. They have the drawback of a rather limited range of viewing positions, but this is improving -- wider angle at least, I don't know if anything can be done about the limited distance range other than head tracking (which only works for one person).

Yup. stupid.

I've heard Brown and Churchill - Complex Variables and Applications recommended. I haven't encountered it personally.

This depends very much on the context. If it were on a number that is part of a larger expression including things like square root signs etc. I would almost always read it as some indication of a pattern that repeats. If it were on a number that is part of a computerised table (like a resizable spreadsheet), or on some kind of interactive diagram, I would almost always read it as a truncated or rounded decimal. I was thinking of this primarily in relation to graphics/engineering/computer programming/computer game type software I've encountered rather than mathematical notation.. For other cases (like a printed table, or static diagram, or a number on its own) it would be somewhat ambiguous, I'd lean towards the former if I were doing maths, the latter if it were something related to application. I agree with imatfaal that the tilde to the left is probably the best. It is certainly the most in keeping with (mathematical notation) conventions.

Another common notation for displaying some digits of a number is to use an elipsis, 1.414… (ascii character 133 for a single width version in most monospace fonts) This is the convention when you are storing more digits, but only displaying a few. It more frequently implies truncation (1.55555 would be truncated to 1.55…), but I have seen it for both. One thing to be careful of is that it is also used to imply a repeating digit/series of digits, or an infinite series (ie. 1 + 2 + 3 + ... would be the sum of all numbers from 1 to infinity) Interesting software. Have you heard of/looked at LaTeX (or lyx)? Latex is very much the traditional software for thist type of thing (This forum accepts latex code between [ math ] and [ / math ] tags without the spaces) if you want to try it out.) although it is more focused on good presentation than speed. I didn't download it, but I imagine it will be a lot more popular if you make it accept latex commands if you haven't already.

Start with khan academy It'll get you from whatever level you're at up to a decent introduction to complex numbers. Then I'd probably head to a library (if you have acess to a university library), or texbook store and get one of the more popular texbooks on the subject (hopefully someone else here can make some suggestions as to which one, I always forget the names of such things). Come over to the chat for (slightly) quicker and more interactive answers to questions you might have as you go. Bear in mind it can take a while for someone with the relevant knowledge to start paying attention (many people are quite idle), so hang around for at least 20 minutes or so.

Not a bad idea. A ~ to the left would be another good way of doing that. [math] \begin{array}{r} 3.2 \\ {\sim}4.1\\ 1.4 \end{array} [/math] Or a small one somewhere in the box

The amplitude is more closely related to number, if anything. The energy is directly proportional to the wave number/frequency, or inversely proportional to the wavelength/period. In systems where the particles do not interact (think photons in an EM field) the amplitude is related to the total energy. At any given frequency it determines the number of particles (each with an amount of energy that depends on the frequency). For interacting systems -- and when you include particles with rest mass -- things get somewhat more complex, but the general theme of energy inversely proportional to wavelength remains.

Hmm, the only thing I can think of is that writing things in scientific notation generally implies they have been rounded. There are a number of conventions for writing numbers with a margin of error, perhaps one of these might be useful: You can parenthesise the values that might vary according to the uncertainty or error: [math]1.41(4)[/math] This would imply that it was somewhere between 1.4135 and 1.4144999... and for some reason something (such as rounding) has introduced an error which effects the last digit. Similarly 1.4(14) would imply that there is an error that effects the last two digits (uncommon to use two digits except in high precision work). You can write the error explicitly: [math]1.414\pm 0.0002[/math] Or if the number had been rounded before you encountered it and you did not know whether it was 1.4142, 1.4135 or 1.4144 (or anything in between) you would write: [math]1.414\pm 0.0005[/math] You can write an interval: something like [1.4135,1.4145) If it's part of an equation or derivation you can use approximately equals: [math] a = \sqrt{2} \approx 1.414[/math] If you are indicating this in ascii it's common to use ~ or ~= as in sqrt(2) ~= 1.414 By convention, when you write something in scientific notation, it is implied that there is only a certain amount of information, ie. 1.414 could imply 1.41400000..... but [math]1.414\times 10^0[/math] Implies nothing about the later digits (you might have rounded them, or they may not have been there). This is not as strong a statement as the others. It doesn't say you have any further information, rather than saying you don't have any further information. Your idea about a paler colour would work quite well for the case of a single piece of software (I wouldn't use dashes or dots as they are often used to imply a repeating digit), but does not make for a very good convention (hard to do on a blackboard/in pencil/in plain text etc). More conventional is to write numbers that have infinite precision as rationals, radicals, or integers ie. write [math]\frac{1}{8}[/math] or [math]\pi[/math] Thus it is implied that anything you write as a decimal has limited precision.

All these do is concentrate solar energy. There's really not much to the design; anything roughly parabolic and shiny will do the trick. It's also not that hard to get close to 100% efficiency. The problem comes in using the sunlight you have concentrated to do something. If you just want to heat something nearby, it's a good plan. You can also use the heat to power some kind of generator (and get the losses inherent in a heat engine). There are designs around for solar collectors that work this way (and I think some commercial plants), but the main focus seems to be on photovoltaics.

You can use [math] and [/math] tags and LaTeX to produce symbols. Ie. [math]\int_{x=0}^a\int_{y=0}^{\sqrt{a^2 - x^2}}x\;dy\,dx[/math] Will produce: [math]\int_{x=0}^a\int_{y=0}^{\sqrt{a^2 - x^2}}x\;dy\,dx[/math] Did you do anything work so far? Exactly where are you stuck? It's good form to post your attempt along with the question here. That way we can address the points where you are stuck without just handing you the answer. At any rate, here's a few things which might get you started if you're completely stuck: What does [math]y = \sqrt{a^2 - x^2}[/math] remind you of? Perhapsit will be more apparent if it's written [math]x^2 + y^2 = a^2[/math] What would that mean for the limits of r? If you draw a circle, which angles result in a positive x value? Perhaps you could draw this region in cartesean coordinates,

Even using a calculator as a crutch can be beneficial. Using it to get the answer so you can be more confident of your understanding is, more often than not, good. Using it to get the answer and leaving it at that is a good way to lose any progress you've already made. <mandatory car analogy> Calculators are tools, just like cars. And in the same vein they are very useful. I'll happily use a CAS to do tedious algebra that I can learn nothing new from, or use a computer when I want to add more than a few numbers. Just like I'll use a car when travelling more than 2km and want to get there quickly. The problem comes when the calculator is the first thing you reach for. If you use the car for every tiny thing like going to the letterbox to collect your mail, you'll become increasingly unfit. When you get to the base of a mountain which the car can't climb or get stuck in somemud, you'll be completely helpless. </mandatory car analogy>

Part of the issue (and I think this is what Dr. Rocket may have been getting at) is that this should have been done before the student hits university. It should be just as much (if not more) the responsibility of the highschool teacher to do this as the university lecturer. Spending most of first year concentrating on fixing incorrect models and teaching critical reasoning is one way of routing around a faulty primary/secondary school system, but it is symptomatic of a wider issue. This being said I have seen enough lecturers who are bad at teaching and TAs who don't know calculus to see Capn's point. While this should not be enough to stop a student learning the correct concepts at this level (if the have an adequate background in critical reasoning skills and have already been absolved of their incorrect models), it most certainly does not help the process along. Part of the issue here is that staff are often required to teach as part of their duties without requiring/being given much in the way of training. While they are invariably intelligent (and capable of learning on their own), the fact that they are obliged to be there means that some of them spend far less time thinking about/researching teaching methods than needed.

Humm. Choose your poison it has applications in everything involving continuous variables (and many things that don't). Everything from mechanics to quantum physics, signal analysis, population modelling, chemistry, economics. It'd almost be quicker to name things that don't involve calculus. Name something you find interesting that involves non-discrete variables and someone here should be able to point you to an application of calculus nearby. Understanding it may be a bit of a stretch without doing the basic stuff first, but it might at least help motivate you when slogging through it.

In my experience a large part of the problem is that a lot of education is so compartmentalised. They get 'today we're doing trig identities' in maths class, and that's all they do. Nothing else in their education is required for that unit, and it isn't applied anywhere else. When I tutor highschool and first year uni (14 - 17 y.o) students I find they can solve the equations, and multiply constants by fractions, but it requires a significant amount of goading and encouragement. The usual conversation goes something like: "Now, do you remember from maths last year/in year 10/etc how to solve an equation like this for t?" "But we're doing physics, not maths." After the exam on any given unit they (perhaps rightly as they are rarely contradicted in this) come to the conclusion that they will never need that piece of information again and stop paying any attention to it. Another thing that comes up a lot is 'are we allowed to do that?' They get so used to only using (and only being allowed to use, even if they see another way around) a single method on a single type of problem that they think it must be cheating to use knowledge external to the unit they are doing. The third major issue I've seen (mostly in physics and maths) is that very little attention is paid to what they are actually doing. Barring a few contrived problems at the end of the chapter there isn't really any context or reason behind what they are learning. On top of this, no attention is paid to understanding the methods that are being applied. The focus is instead on performing a series of steps memorized by rote. I'm not convinced the calculators are at fault here. When deprived of a calculator, many students will fall back on some algorithm or method that they learned by rote to do whatever it is they are doing. This is often just as much of a black box as a calculator (they put the numbers in, crank the handle, even if the handle is in their head, and the answer pops out). Much of their mathematical knowledge seems to be stored as a big list of rules (ie. when adding fractions you cross multiply) with no reason or rationale behind them other than 'the book said so'. This becomes quite apparent when you look at the mistakes being made (ie. cross multiplying fractions whenever they are encountered even if they already have a common denominator or are being divided etc). The calculator merely allows people to limp along for a few more years (basically by acting as external storage for their list of rules) where otherwise they would have failed years ago.

Gravity is neither destructive nor counter to destruction. We often refer to entropy as something that 'forces' things to become more disordered, but in this context it does not have the same meaning as 'a force'. The former just meaning a tendancy towards disorder, the latter is a precise physical meaning about how it changes momentum/potential energy. Entropy goes up if you consider a closed isolated system (which the solar system along with all the matter it ejected is a reasonable approximation of, the gas that left the sun is mostly just sitting around in interstellar space and even if this is not good enough you can consider the galaxy closed for almost all purposes). At any rate, even if we only consider the solar system as it is now (whether or not we exported enough entropy at some time in the past to be more ordered than the original gas cloud), it's still becoming more disordered (the sun and planets are slowly coming into equilibrium with space around us). The sun is becoming more disordered as fusion occurs, we can lower our local entropy because we have the benefit of sitting in an energy gradient (with the still hot sun on one side and cold space on the other). This still increased the overall entropy, even if we wound up with a local pocket of low entropy (I still don't know if this is the case and I'm not quite sure how to go about setting up the calculation). Yes, this is pretty much the concept of entropy. If you have one hot atom and a bunch of warm ones, the hot one tries to get rid of its energy, heats up a warm one and so on until they are all at the same temperature. Thermodynamics doesn't say that everything will cool down, just that it'll all come to the _same_ temperature (whatever that may be). Once you include some stuff from cosmology (including expansion) you figure out what that temperature is (if expansion is accelerating then it will tend to 0 given infinite time). As to why it hasn't smoothed out yet. I think the anthropic principle sums it up fairly nicely. If the universe were a lot hotter (/less spread out) life (like us at least) couldn't exist. If the universe were a lot colder (/more even temperature) life (like us at least) couldn't exist. Therefore the universe must be about this smoothed out (ie. cooled down somewhat but not too much) if we are to exist. Even if we aren't the only way life can exist in the universe, and life that did exist in the early universe would be very different and would have the same two conditions as true (and thus come to the same conclusion). Same goes for any life that may form later after all/most of the stars burn out.

Invite some athletes to eat there, then watch the news. (Or don't do this as it would be immoral)

Well you could try some broad classes of functions to see if they work. My thought process went something along these lines, edited slightly due to the benefit of hindsight. Just put in the most general forms of exponentials/polynomials/etc you can integrate and see what comes out. You'll find that basic exponentials and trig functions give you functions involving polynomials, so you can rule out those. Things like [math]e^{f(x)}[/math] get quite difficult to integrate so maybe leave them out. Polynomials in are going to give you polynomials out, so this looks promising. High integer powers of t give you highly negative integer powers of s. So this train of logic leads us to consider negative powers of t, fractional powers, or complex powers [math]t^q[/math] where q is not a positive integer. Looks like our most promising candidate. Try some fractional powers around 1 You can try putting in set numbers by trial and error, or try it for a general case (will probably involve special functions and all sorts of nasty stuff). Failing that, go back to [math]e^{f(x)}[/math] (already knowing the answer I can tell you it's the first case). There are some more advanced ways of approaching this problem, but I am barely aware of them (let alone knowing them well enough to teach someone else). A fourier transform is another transform a lot like a laplace, but double sided and involving imaginaries. Yes, unfortunately I don't know how to do that off hand, and I weakly suspect it may not be possible. An eigenfunction is a way of generalising the concept of an eigenvector. In linear algebra you get an operator (usually represented as a matrix). You find vectors which, when multiplied by this matrix will give you the same vector back, but scaled by the eigenvalue. You can extend this concept to apply to functions (rather than vectors) and other operators (not just matrices) You can think of your function (let's say the function [math]sin(t)[/math]) as a bit like an infinite dimensional vector which takes on a value for every index (value of t) In this context you can think of a derivative or a laplace transform as an operator much like the matrix. For example the class of functions [math]e^{ax}[/math] are the eigenfunctions of the derivative operator with eigenvalue [math]a[/math] [math]\frac{\partial}{\partial x} e^{ax} = a e^{ax}[/math] and [math]e^x[/math] has an eigenvalue [math]a=1[/math] so it is the fixed point. The function I found (which is lying waiting for you in most tables of laplace transforms) is an eigenfunction, but its eigenvalue is not 1, so it is not a fixed point. It's also unclear to me whether others exist or not (namely an eigenfunction with eigenvalue 1 which we're after). I suspect that they either don't, or are quite hideous.

Humm, bit of a tricky one. Basically they are asking if you can find some function f(t) which, when you put it through a laplace transform, comes out as the same function. I can get [math] \mathcal{L}(f(t)) = a F(s)[/math] (our old friend [math]\pi[/math] shows up, too). Getting this far is as simple as looking at a table of standard Laplace transforms and looking for something that is of the same form in both columns (which I'll leave as an exercise for you to complete as this is the homework help section). As a concrete example for the Fourier transform, you'd use the gaussian. [math]\mathcal{F}(e^{\frac{-x^2}{2}}) = e^{\frac{-s^2}{2}}[/math] (give or take a pi depending on your definition of fourier transforms) We've got a bit of wiggle room here because we can use the coefficient of [math]x^2[/math] to modify the coefficient in order to force it to be 1. Unfortunately the function that works for the laplace transform isn't an exponential so I don't know if it's possible to get exactly the same function out (including the coefficient being one) You may be able to get the answer by fiddling with shifting/scaling theorems and such, I'm not sure. Making further progress without trial and error (and understanding what's going on further) would be aided by understanding of eigenfunctions of operators and hilbert spaces