timo

- Malaysia seems like a good choice to me. Probably an interesting choice for the guys who have to read your essay, too. - The question is not which Malaysian technology had a big impact on the rest of the world but how technology and engineering (wherever their research origins) affect Malaysia. - Obvious choices for technologies that certainly influenced Malaysia are computers and telecommunication. That obviousness makes them pretty boring choices, though.

No, I was not referring to any particular book. I was just seeing that none of the books sounds like having an introduction to Lagrangian and Hamiltonian Mechanics. I do not know if it's necessary to buy a book that explicitly covers the topic. The topic should be part of any (theoretical) mechanics book for physicists. You can check your introduction book, perhaps it's in there, already. I just thought that the gap "introductionary physics, <no analytical mechanics>, <no Quantum Mechanics>, Quantum Field Theory" looked quite big.

Two quick comments: - Science and especially physics textbooks are usually referred to by the name of the author, not by the title. - If you want to understand a textbook, you cannot expect to read it in the same manner and speed as a novel. Depending on how in-depth the books are, you might already have enough reading stuff for the next two years. - Bonus comment: I see no book on Quantum Mechanics and I am not sure if the physics intro book covers Hamiltonian Mechanics (check that). Also, I am not sure if a masters in Biochemistry offers a sufficient mathematical background to understand the more advanced physics topics.

Apart from nitpicker loopholes ("measure the inhomogenity"), I think the answer is no. Disclaimer: I did not read the post above prior to writing this, I just knew someone would come up with that. You'd have to define what "clocks ticking at the same rate" means. Due to different setups, you have kind-of different comparison scenarios for the canonic understanding of "clocks going slower". All halfways-sensible scenarios for checking whether the two clocks go at the same rate (via a 3rd observer) I can currently think of, I get to the answer: Most likely not. And no one told you that the conditions for a black hole are usually given in what comes closest to the frame of rest of the respective object? Suppose everyone was too busy with 61-dimensional quantum wormholes at the Planck scale (where space is quantized). In other words: Contrary to its name and popular belief, Relativity primarily deals with things that are not relative to a frame of reference. Following that spirit, the statement of whether something is a black hole or not should also be independent of the frame, meaning an object that is no black hole at rest also is no black hole in any other frame. I'm not exactly a BH-expert, though. I'm not sure what you'd plot. Even in the non-relativistic case, the acceleration due to gravity is a) a vector, not a scalar. b) dependent on the frame of reference. Solve r=2Gm/c² with c being the speed of light, G being the gravitational constant and r being the radius of the sphere for the total mass (including the 100 initial kgs) m. Express the speed of an object via its relativistic mass (which is just the laymen term for energy, btw), plug that into the relation for time dilatation and start explaining . It's not that we didn't have an explanation (in the sense of "we define spacetime as... and eigentime as... and therefore time dilatation), though. I'm quite happy with the standard explanation . The coffee on my table is probably in the gravitational field of earth. Luckily, it's acceleration is not observable. You somehow have to define what a gravitational field is, though. As I previously said, you'd ideally find a description of the gravitational field that is not relative to an observer. Since at least the understanding of acceleration that you most likely have in mind (the F=ma one) does not hold to that definition, you cannot equate that acceleration and the gravitational field. Unlikely. Except for gauge theories with massive interaction fields, everything usually works fine by just assuming that a 1 kg rock has a mass of 1 kg. You are more likely looking for a good book on General Relativity and/or Differential Geometry. The dictionary particle physics <-> your notation is mass <-> rest mass energy <-> relativistic mass. The Higgs Field is responsible for the rest mass of an elementary particle. Non-elementary particles can (and do) get their masses through other mechanisms. You get your points across quite well, I think.

Not exactly sure what you mean by "at rest". Here's some thoughts: Ground state of a particle in a box [0,L]: [math]\psi(x) = \alpha \sin (x\pi/L) \Theta(x) \Theta(L-x) [/math], with alpha some non-zero normalisation constant. => 1: [math]\left< p \right> = \left<\psi | \hat p \psi \right> = | \alpha |^2 \int_0^L dx \ \sin(x\pi/L) i\hbar \partial_x \sin(x\pi/L)[/math]=[math] i \hbar | \alpha |^2 \pi/L \int_0^L dx \sin(x\pi/L) \cos(x\pi/L) = 0[/math], with the zeroness of the integral following from the anti-symmetry of the integrand around L/2. 2:[math]\left< E_{\text{kin}} \right> = \left< p^2/2m \right>= \left< \psi| \hat p^2 \psi \right>/2m = |\alpha|^2 / 2m \int_0^L dx \ \sin(x\pi/L) (-\hbar^2 \partial_x^2) \sin(x\pi/L)[/math][math] =|\alpha|^2 \hbar^2 \pi^2/(2mL^2) \underbrace{\int_0^L dx \ \sin^2(x\pi/L)}_{=L/2} =|\alpha|^2 \hbar^2 \pi^2/(4mL) > 0[/math]. Comment for a large box: [math]|\alpha |^2 = 2/L \Rightarrow \lim_{L \to \infty} \left<E_{\text{kin}} \right> = 0[/math], in that case. Time-dependent case: It's easy to see that for the ground state above, [math]\left< x \right>(t) = L/2 \Rightarrow v(t) \approx \frac{d}{dt}\left< x \right>(t) = 0[/math] because the additional term due to time, [math] \exp (-i E t/\hbar) [/math], cancels out for all t. Note that E = Ekin+Epot, where Epot depends on how you set up the value of the potential of the box (and that all of above statements hold true independently of Epot).

Why would you want to calculate something for something when the something isn't defined for the other something? You could just define it as some thing but something tells me you are probably looking for something like the Gamma function.

Things already experience different time passings in normal gravitational fields. You could get an upper limit for the expansion difference by guessing how much Neil Armstrong has grown during his trip to the moon. But you should be aware that sizes of everyday-objects are independent of what cosmologists refer to when they say space expands. They are determined by the strengths of the interactions. E.g. the geometry of a molecule is strongly determined by the electromagnetic interactions between the atoms' nuclei and the electrons. If you put an H2 molecule into an expanding space, then the size of the molecule will remain the same, just as if you put a pencil on an expanding rubber sheet and the expand the sheet. Effectively, that unequal expansion is what allows to see the expansion - if you'd put a 2nd pencil on the rubber sheet, you'd see it moving away from the 1st pencil.

I think if we were 10 times bigger than yesterday but have no way to find out (i.e. there is no way in which this expansion causes any effect at all), then we can as well ignore that expansion.

Dunno when you contacted the admin, but keep in mind that the admins are not robots but human people who sometimes sleep or simply do other things than browsing the internet. From my experience: Posting links to other websites generally is not a problem (even when they are selling stuff). Problems arise when the mentioning the website was the sole reason for posting, not a useful addend to the post.

- Time being relative to mass and energy does not sound true. The statement doesn't even make much sense to me. What should that mean? - Different objects can have a different understanding of the time-direction similarly as the forward-direction of two cars differ when one travels to the north and the other one travels to the north-east. Seen from the perspective of the first car, there is a point where the 2nd car will have travelled 1 km into the forward direction (of the 1st car), i.e. 1 km north. At this point, however, the 2nd car will have travelled [math]\sqrt{2}[/math] km in its forward direction ([math]\sqrt{2}[/math] km north-east = 1 km north + 1 km east). In that sense, objects can disagree on a distance travelled in a certain direction. In the same sense (except for some different signs), objects can disagree on the distance travelled in the time direction. - Consider two objects: Yourself as the observer in the laboratory and some gas molecule bouncing around. As said, you and the molecule can have a different understanding and disagree on the amount of time passed. Neglecting gravity and assuming the molecule to have a constant speed relative to you, the factor by which the two "distances" (=experienced time) differ is [math]\gamma^{-1}= \sqrt{1-v^2/c^2}[/math], where gamma is called the Gamov Factor, v is the velocity and c is the speed of light. You might (but shouldn't) say that time stops for the molecule, if [math]\gamma^{-1}[/math] approaches zero. - Now, let's take a classical gas and assume the velocity of the molecules did approach 0 m/s as the temperature approaches 0 K. In that case, the factor by which their time measurements differ from yours approaches one. In other words: Their time-direction and their time measurements approach yours. That's pretty much the opposite of what you probably meant by time stopping. - I see no good reason why someone would want to include gravity into that question (I see some bad reasons, though ). In short: No, you can't freeze time at 0 K.

Numerical creation, evaluation, integration and interpretation of R^n -> R functions

The gradient effectively just is the collection of partial derivatives wrt. to the different parameters x_k. [math]\nabla = \left(\frac{\partial}{\partial x}, \frac{\partial}{\partial x}, \dots \right)[/math]. Or akin to the notation in the code above: grad = (dx,dy,...). Note that in above the variables gradNum and gradAna are the squared magnitudes of the gradient (taking the square of the difference vector rather than the difference of the squares would probably give a better error estimate, btw). So initially there is no reason to evaluate additional points. However, you might need to do that for the Hesse matrix, depending on your implementation.

Yep, it's just the next-better approximation No, but writing one is not too tedious, so: from scipy.xplt import * # comment out if not available class fR2toR: """An f: R²->R, with partial derivatives. """ def value(self,x,y): """f(x,y)""" return cos(x)*cos(y) def dx(self, x,y): """df(x,y)/dx""" return -sin(x)*cos(y) def dy(self, x,y): """df(x,y)/dy""" return -cos(x)*sin(y) class ForwardStepDifferentiation: """Simple forward step.""" def __init__(self,delta): """ init with stepsize delta.""" self.delta = delta def dx(self,f,x,y): """partial diff of f wrt. to x.""" return (f.value(x+self.delta,y)-f.value(x,y))/self.delta def dy(self,f,x,y): """partial diff of f wrt. to y.""" return (f.value(x,y+self.delta)-f.value(x,y))/self.delta class CentralDifferentiation: """1st order taking half a step in both directions.""" def __init__(self,delta): """ init with stepsize delta.""" self.delta = delta def dx(self,f,x,y): """partial diff of f wrt. to x.""" return (f.value(x+self.delta/2.,y)-f.value(x-self.delta/2.,y))/self.delta def dy(self,f,x,y): """partial diff of f wrt. to y.""" return (f.value(x,y+self.delta/2.)-f.value(x,y-self.delta/2.))/self.delta class ExtrapolatedDifferentiation: """ 'Extrapolated' differentiation.""" def __init__(self,delta): """ init with stepsize delta.""" self.c1 = CentralDifferentiation(delta/2.) self.c2 = CentralDifferentiation(delta) def dx(self,f,x,y): """partial diff of f wrt. to x.""" return (4*self.c1.dx(f,x,y)-self.c2.dx(f,x,y))/3. def dy(self,f,x,y): """partial diff of f wrt. to y.""" return (4*self.c1.dy(f,x,y)-self.c2.dy(f,x,y))/3. def getSqError(f,method): """ get summed squared error of the method over [0:9]x[0:9]. """ error = 0. for x in arange(100)*0.1: y = arange(100)*0.1 gradAna = f.dx(x,y)**2+f.dy(x,y)**2 gradNum = method.dx(f,x,y)**2 + method.dy(f,x,y)**2 for i in arange(30): error += abs(gradAna[i]-gradNum[i]) return error def plotError(f,method,windowNum): """ plot error over stepsize. """ window(windowNum) x = 1.*arange(20) y = 1.*arange(20) for i in arange(20)+2: x[i-2] = 1./i method.delta = x[i-2] y[i-2] = getSqError(f,method) plg(y,x) f = fR2toR() # test function m1 = ForwardStepDifferentiation(.1) m2 = CentralDifferentiation(.1) m3 = ExtrapolatedDifferentiation(.1) print "Some simple test, no plots" print "Error 1= ",getSqError(f,m1) print "Error 2= ",getSqError(f,m2) print "Error 3= ",getSqError(f,m3) print "Plots over stepsize; comment out if scipy.xplt not available" plotError(f,m1,1) plotError(f,m2,2) plotError(f,m3,3) raw_input("Press <enter> to ... errr ... leave.") winkill() The super-low error of the 3rd method is a bit strange, though.

...and she probably knows better than Phi's other wives.

[math] \frac{\partial}{\partial x} f(x,y) \approx \frac{f(x+\Delta x, y) - f(x,y)}{\Delta x} [/math], as a simplemost example. More sophisticated methods can be applied just as in the 1D case, the idea of keeping all but the variable you differentiate to constant should remain. The gradient should follow from the partial differentials just as in the analytical case.

I think you have an incorrect view of QM, at least your statement seems to indicate that. The next best correct statement would be "when the wave-function is non-zero in a region R, then the particle can be found in that region". Finding it in a region, however, is a measurement process, meaning you change the system. QM quick-guide how tunneling should be seen (several simplifications and non-stated assumptions implicit): - Particles are in a state |state>, an element of a Hilbert space. States can be represented by wave-functions f(x). - The Hilbert space is more or less (might depend on how you treat the normalization constraint) just a vector space. - A vector space has a basis. A nice base for the Hilbert space is a base of states which are eigenstates to the Hamiltonian (Energy-eigenstates). - If |n> is an EV to the Hamiltonian with energy En, then from the Schrödinger equation, the equation of motion of |n> is f(x,t) = exp(-i En t/hbar) f(x,t=0). => The probability density for finding a particle in some location x remains constant if it was in an energy eigenstate. - For the potential you sketched, there is no energy eigenstate for which the whole wavefunction is constrained inside the middle sink (that's probably what you meant with the exp(-x/a)). => A wave-function f(x,t=0) for an initial state (at t=0) that is constrained to being inside the middle sink is not equal to the WF of a single energy-eigenstate but can be expressed as a linear combination of energy eigenstates, e.g. |state> = c1 |1> + c2 |2>. The coefficients (c1 and c2 here) add up in such a way that the probability for finding the particle outside =0. - Due to |1> and |2> having different energies, the equation of motion allows that the probability of finding the particle outside does not need to remain 0 under time evolution. => Even though |state> initially had zero probability for finding the particle outside the nucleus, this does not have to hold true under time-evolution.

If the ideas are based on an existing physics model, then the appropriate place is the subsection of that physics model. If the ideas are based on physics but not on a concrete model, then probably general physics. If the ideas are based on your thoughts alone, then the Speculations forum is probably best-suited: http://www.scienceforums.net/forum/forumdisplay.php?f=59

And neither is there any object in the room that causes the pencil to fall down. Strange world.

That does not exactly answer my question, ecoli. By "the decision" I meant the allowing or dis-allowing of waterboarding for the CIA and the implications under US rights, not the charta of human rights.

How can the sun affect the motion of Jupiter when it can't even affect the trajectory of your pencil?

As a semi-related question: The CIA is the foreign intelligence service, right? That means it operates entirely outside the US? Who would be potentially affected by the decision? Everyone? Only people outside the US? Only non-US citizens? Only non-US citizens outside the US (Guantanamo not counted as US territory, then)?

Hence my non-understanding of Severian's post. You can either interpret the question in which case the issue depends on how you interpret(ed) the question. Or you can take is "as is", in which case I think the question is technically incorrect. I wouldn't be too surprised if ethical judgements can only be made on actions, not on objects (with object being meant sufficiently wide so that nuclear power fits in - it surely isn't an action) . I even think "judging actions" pretty much is the definition of ethics (but that's just a layman's guess - clarification from a professional would be highly welcome). Actually, I already had the idea that the question might be much smarter than it initially sounds. We have 12 replies to what -without interpretation- is not even a question. If you are interested in how humans react to improperly stated questions (either on the "positive side" of interpreting the missing parts or the "negative side" of blindly/zombiely reacting to emotionally-loaded keywords) that might be an interesting result. I'm not sure how you'd measure the people not having responded, though. Perhaps by counting the number of views (12:68 then, btw) or by somehow measuring the amount of discussion.

I don't really understand that. The problem where to store the reaction waste, safety problems and problems with military abuse (those three being the only possible problems with nuclear power I can think of at the moment) seem to have obvious solutions in the case of the sun.

The main difference comes from the two articles being written by different people . Further differences come from that you possibly didn't read the two articles carefully enough. Simply from skimming (hm, perhaps I shouldn't claim that others hadn't read the articles ... ), I see two things that are different/contrary to your post: 1) Article 2 doesn't claim that x0 sufficiently close to x* was necessary for convergence, but that (together with f being twice-differentiable) it was sufficient for convergence to x*. 2) Article 1 explicitly gives examples where the x_n do not converge (bottom of the article), so "no restrictions what so ever as to the choice of x_0 (the initial value)" is true in so far as that no restrictions are explicitly mentioned, but not in the sense that the article claimed there were none.

How do you quantify gravity? My guess would be that "twice of our planet" means that the acceleration due to gravity at the surface is twice that of earth. The 5-times mass can then be countered by a larger radius, i.e. you are comparing values at different distances from the respective centers.