timo

I'm pretty sure they do. Just not necessarily with the first electron the come close to. If I remember correctly, the probability of an electron and a positron annihilating with each other when they meet (whatever "meeting" may mean exactly for elementary particles) decreases with their relative velocity (or equivalently: relative momentum). So very fast positrons may (and apparently do) survive some distance before they react with electrons and vanish.

As a function of velocity the kinetic energy is [math]E_{\rm kin} = \left( \frac{1}{\sqrt{ 1 - v^2/c^2}} -1 \right) mc^2[/math]. But there's little information to be gained from this relation other than that a) for 0<=v<c any possible positive energy can be achieved, and b) the reverse statement: For any finite kinetic energy, the corresponding speed of the object is always smaller than that of the speed of light (which in turn is a popular argument for why nothing can exceed the speed of light). It merely has a higher kinetic energy than you would expect from non-relativistic physics. The reason is that non-relativistic physics is wrong for high speeds.

That's probably true from a perspective of Newtonian gravity, yes. Nope. In Newtonian gravity, the (negative) potential energy E of an object of mass m that is a distance d away from a point-like gravitational source of mass M is given by [math] E = G Mm / d [/math], where G is the gravitational constant. This implies that an object with a kinetic energy larger than this E can get an arbitrary distance away from the gravitational source - get free from the attracting source. If you evaluate the velocity that would result in this kinetic energy you get the escape velocity. The equation for the potential energy I gave directly implies that for every pairs of masses m, M and for every non-zero distance between them, there is some (possibly high) kinetic energy that would allow the particle to break free of the attraction. Black holes are characterized by the fact that there are regions for which an escape from the attracting source is impossible irrespective of the kinetic energy - a feature that qualitatively does not exist in Newtonian Gravity (as I just explained). The boundary separating those regions from the normal regions is called the event horizon. In standard coordinates is forms a sphere with a radius that is called Schwarzschild radius. For people who never heard about relativity it seems suggestive to define a classical Schwarzschild radius at the distance at which the escape velocity calculated via E=mv^2/2 equals c, since no particle's velocity can ever exceed c. However, in relativity E=mv^2/2 is not the correct expression to calculate the kinetic energy as a function of velocity. In relativity, a particle of mass m can reach an arbitrarily high kinetic energy even for velocities <c. General relativity foots on a different concept of space and time than other physics (or chemistry, biology, engineering, ...). As a result, I don't think there is a standard interpretation of what acceleration is supposed to be in the context of your question. It's intriguing to answer your question with "infinite". Sounds cool, after all. But "not obvious what's meant with it" really is the better answer - despite sounding like a weak excuse. Btw.: Please use proper capitalization - and possibly also punctation.

Covariances for uncorrelated statistical variables vanish. Concerning a question about statistics that already is a stronger background than most of the people on this forum have. Many (most?) people using statistics actually have no proper mathematical training in it (like me, for example ). Btw.: you'll soon realize that in mathematics it is very common to define the letters/objects being used, and not simply assume that everyone will correctly guess what sigma-x-one is (sry, could not resist this comment)

It would certainly help if you could have been bothered to mention the context you have seen the equation in, and what the meaning of the symbols is, and what your scientific background is, ... . The keyword you are most likely looking for is "uncorrelated", followed by "small error".

In that case, the acceleration is neither c, not more or less than c. 13 m/s cannot be compared to 8 m/s^2 in a similar sense as five elephants not being comparable to 25 Kelvin. An escape velocity is conceptually not the same as an acceleration - and despite popular belief, thinking about black holes, photons, curved spacetime, and extra dimensions is no help for understanding concepts of school physics (but apparently more interesting than the school physics). Is "of course gravity obeys a 1/r^2 law" a claim or a question? The gravitational force of a point source (and sources well-approximated as a point) obeys a 1/r^2 law in Newtonian gravity. But considering that black holes are an effect of General relativity that is not present in Newtonian gravity, it's quite an extrapolation to claim that this property of Newtonian gravity necessarily holds true. I think you are underestimating how different General relativity is from school physics.

Exactly my though, especially when it came to the "research grants" part. I'm not completely convinced that ultimately mobilizing many people on the Internet (or via friends) via blunt "vote for me" slogans should be an important factor in the evaluation of a research project (except for social sciences, perhaps). Btw.: I actually do consider your post a kind of spam (in a milder form, though). But judging your thread in that category is for the forum moderators to decide, not for me. So don't worry about my negative attitude too much - and please don't consider it a personal attack on you, either.

It's been "already stated" around here: and here: iow: It has been explicitly mentioned in 100% of the replies you got until you re-posted here - and in almost half of all of the sentences therein. I think you should read replies you get a bit more carefully - to say the least.

However, that only holds true if you define m as m := E/c^2

I'm not sure to what extent it applies to your case, but maybe this helps: 1) The top method to find publications nowadays is Google Scholar. 2) Many papers that are available for purchase are also available for free (via arXiv, some university server, ...). Scholar tends to identify such free sources quite well (at least for all of my publications), for this either 2a) click of the "PDF from xxx" button right to the search result 2b) click on "all xx versions", browse through them, and see if there is a version that is available to you. Btw.: The idea of citing papers is that you read them during your thesis and found them helpful and relevant for the topic at hand. The idea is not that after you finished your work you Google 100 publications with an appropriate-sounding title because everyone else also has 100 papers in their bibliography.

a seems to be a vector by definition. Hence, it is not clear to me what [math]a_{ij}[/math] and A are supposed to be. It would probably become clear by "reverse-engineering" what you wrote. But I have the feeling that not having defined your mathematical objects may hint to where your problem lies. Hint: [math] \vec a = \sum_i x^i \vec u_i = \sum_{ij} x^i {\bf 1}_i{}^j \vec u_j = \sum_{ijk}x^i \underbrace{A^{-1}{}_i{}^j A_j{}^k}_{={\bf 1}_i{}^k} \vec u_k = \sum_{ijk}\underbrace{x^i A^{-1}{}_i{}^j}_{=y^j} \underbrace{A_j{}^k \vec u_k}_{=\vec v_j} =y^j \vec v_j = \vec a [/math], where A is a suitable transformation matrix, 1 is the identity matrix, and x, y are the coefficients of vector a in the basis of the u and v, respectively. If you're unfamiliar with the index notation, a notation closer to the one you seem to use would be [math]{\bf a} = \vec x \vec {\bf u} = \vec x A^{-1} A \vec {\bf u} = \vec y \vec{\bf v}[/math] (where now bold stands for a vector and the arrow merely indicates a tuple). The important point is only that you get the idea, anyways.

You already mentioned the way to solve your problem yourself: I = O-I (edit: where O and I in this case stand for the areas of the circles O and I, respectively). Start with that, rewrite it in terms of the radii of the two circles and solve for the radius of the outer circle. Since you already have 277 posts here I assume you know that there is a dedicated "homework help" section, and that this is not homework.

I tend to get headaches and a strange (dry) feeling in my throat in my flat in winter time due to the air being too dry. The rather obvious solution is to put a small bowl of water on the heater (anywhere in the room should actually do the trick, too). I don't really consider this a waste of energy. And even though it's not a very scientific statement, my gut feeling tells me that the bowl of water can well compete with expensive electricity-driven devices.

The statement as you put it is not true and should therefore resist being proven. Take for example a=-3 and b=-2. You'd get -3 < -3 + -2/2 = -4, which is obviously not true. Did you possibly meanprove that if a < b are real numbers then a < (a + b)/2 < b ?

No need to apologize for the variable names, it was merely a suggestion of mine. Yes, I do know what your professor means. But I don't want to give too much help for your homework assignment. Try out this code: #include<stdio.h> int main() { int i, tens, remainder; for (i=0; i<20; i+=1) { tens = i / 10; // division by 10, throw away the remainder remainder = i % 10; // the "%" is the modulo operation giving the remainder of above division. printf("%i = %i * 10 + %i \n", i, tens, remainder); } getch(); } The connection to your time conversion problem hopefully becomes clear once you understand what this code snippet does. Btw.: I only use c++ myself, so above code is untested. If it does not compile, I hope that necessary corrections are obvious.

I don't think that your problem is 12hrs time vs. 24hrs time. The problem is that time 01:23 is not 123 minutes into the day, but rather 60*1 + 23 minutes. C assumes that the digit codes 0123 and 0224 stand for the decimal numbers 123 and 224, and correctly (within this assumption) returns 224-123 = 101. I recommend converting the given time into "minutes since midnight". Unrelated hint: I strongly recommend giving your variables sensible names, e.g. "duration" rather than "ave".

As a matter of fact we recently had quite a fight about the question what matter is in the German Wikipedia. The bottom line (for me) indeed is that there is no unique field-spanning definition.

Learning advanced topics does not necessarily have to happen in school. Particularly, it is not necessary to skip topic X before you can start learning topic Y.

I don't see how your school curriculum prevents you from learning more advanced topics. I would recommend paper-based textbooks over online resources.

That is an interesting statement for that a) it sounds very reasonable, and b) I have never considered that aspect myself, before.

Consider a set of carbon atoms. Why do you call subsets of this set "molecules" in the first place?

0) The standard method to put an idea on paper is using a pen or a pencil. I'll assume you were asking about a scientific publication. 1) Scientific papers are usually written about results, not about ideas. 2) The two most common programs to write papers are latex or Microsoft Word. I also use latex for posters (with the "beamerposter" package), but I imagine standard presentation software would also work well. 3) For structuring of the content, just look at similar publications by others.

Why? You didn't even bother to specify why you are asking this question and what your background is. I see little point in working out a proof based on group theory if you don't even know what algebra (also known as "abstract algebra" in the US ) is. And in the case that you are currently learning algebra, then showing you a proof would probably violate the "homework rules" of this forums. And if you knew algebra, you wouldn't ask such a question. And if you weren't looking for an algebra-based answer, then you seemingly forgot to add what kind of answer you are looking for.

Newton became a fellow in Cambridge at age 25, which I believe implies being paid for doing science. According to (German) Wikipedia his scientific career effectively ended when he was transfered to another non-scientist position. Einstein was a professional physicist, and I am quite sure that he expected his employer to pay his salary. Some of his works may stem from a time where he was not paid for conducting research, though. Franklin is well-known only in the US, so I don't know about him. I don't believe Newton and Einstein were expecting money for their successes. Also, I doubt that any of them was into science only for the money (just as today, there are other fields that you go in if you are only interested in money). But I would imagine that just as any other professional scientist they expected to be paid for their work - at least the part they did on a paid position.

I have the feeling that the amount of people who locked your thread with a question is smaller than "all" - possibly even as low as one.