timo

I think that might depend on the constraints. I am not sure how to tackle your examples - I did not fully understand them. But imagine a pipe with some black powder in the middle and two bullets of equal mass on either side of the explosive. When the explosive explodes, there is some energy 2E released that (due to equal mass) spreads evenly over the two bullets - note that the energy E each bullet gets is independent of the mass of the bullets. Both bullets have opposite momentum of equal magnitude (so that the total of zero remains conserved). Since momentum and energy are related by [math] p = \sqrt{ 2mE} [/math] ( <- very simple rearrangement of classical physics formulas) you see that for the same explosion, i.e. the same energy released, using heavier bullets leads to the bullets having more momentum. That is not really the scenario you sketched but hopefully a start for you to a) show you that you can actually calculate the answer to such questions and b) show how it's roughly done. For say real guns you could make one of the masses significantly larger (that mass being the rifle) and still assume a constant energy released (that seems to make sense to me). The energy does not spread equally over bullet and rifle but of course conservation of energy (Erifle + Ebullet = Eexplosion) and momentum (pBullet = -pRifle) still hold and should be sufficient to get pBullet as a function of the bullet's mass.

Possibly, but I tend to doubt it. Exposure to public should -in average- lead to the conclusion that they are also just normal people. I had a related initial though, which I didn't post because it seemed slightly off-topic. But since you brought it up: My initial thought was that lawyers and physicians are considered prestigious professions because it's those professions that appear on tv most (alongside policemen and performers) - in a positive tone (soldiers, terrorists and other murderers appear a lot, too). Our impression of a lawyer is formed by Matlock (or insert any other tv series here; I don't really watch either), not by the reality of a lawyer sending the same "Dear, <insert name here>, we give you a deadline till <insert date here> to pay the money you own to our client." letter five times a day and taking 50 bucks for the fact that he knows to date to insert so that everything is formally correct.

I imagine you'd find it hard to do something in a masters course. I'd expect a good engineer can at least equal an average physicist when it comes to math skills. But I wonder: What physics knowledge do you expect a mechanical engineer to have? I expect classical mechanics, Lagrange mechanics and Thermodynamics at the chemists' level. That's stuff from the first two semesters in physics lacking relativity, statistics, quantum mechanics and field theory approaches. I might be wrong about my estimate of the physics a mechanical engineer learns (and also wrong about the physics taught in a US bachelor course) - it's not too unrealistic that they also learn electrodynamics, for example. I would imagine the lack of basic physics skills to be rather tremendous and am not sure if e.g. there is any physics research that a mechanical engineer -provided the skill I expected above- could work on out of the box - except maybe biophysics . Fixing on elite universities will probably not make it easier. In case you are a mechanical engineer: I might underestimate the physics they learn but do not overestimate it, either. There is a strong tendency among students to completely overestimate the stuff they learn and the level they learn it at: Chemists regularly think they learn as much physics as physicists plus their chemistry. Teaching students (this possibly exists in Germany only) for subjects A and B claim they learn as much A as the A students plus as much B as the B students plus pedagogics. Physicists and computer scientists fantasize that they were learning as much mathematics as mathematicians, ... . Quite obviously, that's all bullshit.

Not to forget that nature cares pretty little about what Prof. Einstein said or did not say. Just because he's the most famous physicist does not need to mean that everything he said makes sense or even is correct.

There is no point x such that for the pair a=2, b=-1 af(x)+bg(x) = 2x - 2x = 0 was [math] \neq 0[/math]. I meant to write "for all pairs (a,b)" in case that really wasn't obvious (like from the sentence before that one). EDIT: But perhaps it's best for you to just look up a definition from some webpage (Mathworld or Wikipedia), realize that "0" in af(x)+bg(x)=0 means the function O(x)=0 (and not a real number) and actually try to answer your question with that. The big difference is that O(x) is zero for every x, not just for one special choice of x.

Sorry, I think I didn't read your post carefully enough. The definition you gave starts to give me headaches but I think putting the "for all x" behind the equation might fix it. Anyways, I was thinking about the following definition (like said, it seems I didn't read your post carefully enough - sorry again) that seems to make a bit more sense to me: - Two functions f(x), g(x) are linearly dependent if there is a pair (a,b) with [math]a \neq 0[/math] or [math] b \neq 0 [/math] such that af(x)+bg(x)=0, where 0 is the zero function, i.e. a function that is zero for all x. This implies that (af+bg)(x) must be zero for all x. equivalently: - The negation would be that two functions f(x),g(x) are linearly independent if for any pair (a,b) with [math]a \neq 0[/math] or [math] b \neq 0 [/math] there exists at least one point x such that [math]af(x) + bg(x) \neq 0[/math]. Sidenote: By "for any x" I meant "there exists one x such that ...". You hopefully see that this is different from "for all x it is true that ... "

I'd say if there is no guarantee that a+b exists (are you sure that it is not given in your system?) then it makes no sense to substitute it with y or at least not to substitute it with y and assume that y exists (in the system). I don't know what system you are talking about but note that e.g. mathematical groups do have closure under the operation as an axiom .

Your own definition (which you should have been given in the first post, btw) sais "for all x", not "for any x". Try a=1, b=-1 and x=1.

Well, I think I figured that after I posted: I think the idea was that for each pair of dots there is a direct connection line/path. In that case to draw it every dot except the starting and the end dot would need an even number of connections (one by which you approach the dot and one that you leave it with). That conflicts the demand that every dot has three connections. I think the problem was mentioned in "Fermat's Last Theorem"; at least I am fairly sure I read it in a non-math book somewhere.

So did your son.

Behold my amazing math skills:

Closure of a group under the operation(s)? E.g. if a is a real and b is a real then a+b is just another real which you can call it y if you like.

You rearrange the equation for T1 (i.e. in the form T1=....), then type the right-hand side into the calculator of your choice. Note that XY + XZ = X(Y+Z), in case your problem is rearranging.

What exactly do you mean by proof? The proof that the series converges? Iirc, the series does not converge under all possible criteria for convergence. You should find something in any good calculus book (math for mathematicians at university level). In the unlikely case that you speak German you could use O. Forster: "Analysis 1". EDIT: Or, seeing Klaynos' post, http://en.wikipedia.org/wiki/Convergence_of_Fourier_series

Typo correction: Should read [math] V_f = V_e \ln \left[ \frac{ M_r + M_f } { M_r } \right] = V_e \ln \left[ 1+\frac{ M_f } { M_r } \right] [/math] where the rightmost term added by me is a trivial step from Janus version but shows the critical value, the mass-to-fuel ratio, more prominently. Note: Mr is the mass of the rocket without the fuel. Not sure if that was a typo or if I missed something: I think I disagree with the "need to throw out causality": When velocities are not restricted then the full set of events with times >t lie in the future of any event at t. Order of cause and effect seem to be conserved in nonrelativistic mechanics. That pretty much meets my requirements for causality, as far as I see right now.

- The speed depends on the distance of two objects. The relative speed that the distance between two points increases is (according to the all-mighty ... Wikipedia) 75 km/s per megaparsec. From that you can calc out yourself how far two objects must be apart for their distance to increase faster than the distance a beam of light would locally cross in the same small time interval. - This increase in distance is not a velocity. It is conceptually a bit different. The typical picture is along the lines of an ant sitting on the surface of the balloon next to a dot marked on the surface of the balloon. There's two different ways that the distance between the ant and the dot can increase: By the ant walking away from it on the balloon surface or by the ant sitting still and the balloon increasing in size (or combinations, of course). The former is restricted by the maximum speed/velocity of the ant. The increase in balloon size (-> expanding of the universe) is not due to the ant or the dot moving and not restricted by the ant's maximum speed. Btw, you can easily verify that an expanding balloon also satisfies Hubble's Law. The reason why I said that all: Being something different than an object moving around in space, the increase in distance between two points in space (dots) due to cosmic expansion is not restricted by light-speed so there's nothing special to be expected for two points whose increase in distance due to expansion exceeds light speed. - The influence of velocity on time that you seem to have in mind is an influence due to velocities, not due to expansion. So your last two sentences probably don't apply. - While I've typed a few more letters than you have in your question, the answer is still very brief. So don't worry if not everything made sense. I think the key point you are missing is that expansion is somewhat different from movement. The balloon is a very nice and prominent example because it is a non-static background (like spacetime in cosmology is) but still familiar to most (unlike spacetime in cosmology).

My idea would have been to catch growing numbers soon enough, something like when a number is stored as [math]a= a1 * a2[/math] and you want to multiply it with a number b you'd schematically do while (a1 * b > something large) { a2 = a2 * 1.e20 a1 = a1 / 1.e20 } while (a1 * b < something small) { a2 = a2 / 1.e20 a1 = a1 * 1.e20 } a1 = a1*b; Another alternative would be to store the numbers as [math]a = a1 \cdot 10^{a2}[/math]. I recommend considering the idea with the logarithms, though. It's how I store probability distributions with a large range of entries and I am quite happy with it. And to agree with DH and what I said previously: The best solution is not to need a range exceeding the double range.

It's a bit little you are telling: How large and how small? What kind of simulation and more importantly what kind of data and calculations? The most obvious solution is to rearrange your calculation not to run into the problem. Or you can store the large number as two (or more) large factors which when multiplied would give the large number or you can store the numbers as their logarithms or try if Mathematica (or a similar program) can handle it or look for a custom real number implementation or ... . It's really hard to give a sensible reply when the problem is given as "I have a problem with large numbers, what can I do?". EDIT: Oh ... and are we talking about integer, natural or real numbers?

Since you did not say what Operating System you use it's probably some Windows: Try right-clicking the html file, select something like "open with" and then select your web browser (Internet Explorer and Firefox being the most common). Alternatively, from your webbrowser select something like "File->Open" and then chose to open the html file.

triclino: 1) Existence is one of your axioms. Axioms are usually (read: always) not proven. You might have meant that with your part A (I see no steps other than noticing it is an axiom) so that's perhaps just semantics. 2) To disprove x and y ("and" and "or" being meant as the boolean operations in this point) it is sufficient to disprove x or y ( not( x and y) = (not x) or (not y) ). It is not necessary to disprove x and to disprove y. According to mainstream logic, at least. 3) What are you complaining about? You were asking for proofs that you were given quite direct hints how to construct them (see my previous post, for example) and you finally even got the full proof presented. Do you complain about the proof not being a proof or about labeling it as "proof by contradiction"? Note: This is a real question, it is not really clear to me. In the former case: Try being more clear. In the latter case: I see how one could interpret it not being a proof by contradiction (directly showing that all zeros are the same) but I also see how one could be interpreted as such (showing that there cannot be two distinct zeros). I don't think the issue is worth bitching about. 4) I am not sure if I understood what you meant with your bus example (I am not even sure if it is a syntactically correct sentence) but perhaps this helps: Assuming we have agreed that if there cannot be a 2nd zero and we ask ourselves if there can be N>2 distinct zeros. You can then just pick two of them and by the same construction show that they are equal which again contradicts the assumption that the N zeros were distinct. Not sure if that's what you meant, though. 5) Please do not use write in allcaps and use proper punctation. I do not see a sensible pattern in your capitalization and as a non-native speaker/reader and that really decreases readability.

As a matter of fact, the definition on Wikipedia does not have this circular dependency:

I find it extremely hard to answer your question because you say that you are aware that a very comparable machine exists and is about to start running soon but yet do not mention what you expect from SSC.

I then possibly do not understand what you guys mean. I was talking about jumps between shells. In particular I only considered the subset of jumps from an excited state to the ground state. If you consider jumps form n+1 to n (n being the main quantum number) the issue becomes even worse because the wavelength of the individual jumps then also diverges as n increases. Perhaps I was missing explicitly saying that if a subset of possible emission wavelength already does not lead to a finite sum that then whole set does not, either?

Or towards infinity: 1) In classical QM there is an infinite number of energetically-distinct orbitals for the hydrogen atom => there is an infinite number transitions from excited orbitals to the ground state => there is an infinite number of possible transitions for the hydrogen atom alone. 2) All these transitions are bound to release an energy < 13.7 eV => All the corresponding wavelengths of the transitions are greater than some threshold wavelength. (1) + (2) => There is no finite result.

A constant charge density on the surface of a torus.