If $a_n\to0$, there exists $\pm$ such that $\sum\limits_n\pm a_n$ converges
Forcing series convergence
Value of momentum of a particle in 1D box prepared in a particular state? What is the value of momentum of particle in 1D box in state $\sin(10\pi x/a)$? My understanding Standing waves representing particle in 1D box is not an momentum eigenstate so if we measure the momentum of particle in this state we may get any value of momentum which gives a distribution whose expectation value tends to zero. Answer key In the answer key it's given that the value of momentum is 5h/a and -5h/a.If we measure the momentum for this state we may get any of these two values.But is this a wrong reasoning because if these was the case,these should mean that 5h/a and -5h/a are the eigenvalue of the following wavefunction which is clearly not. What is the correct answer? What is the proper way to explain the twin paradox? The paradox in the  is that the situation appears symmetrical so each twin should think the other has aged less, which is of course impossible.  There are a thousand explanations out there for why this doesn't happen, but they all end up saying something vague like it's because one twin is accelerating or you need general relativity to understand it.  Please will someone give a simple and definitive explanation for why both twins agree on which twin is younger when they meet for the second time? When would you call java's thread.run() instead of thread.start()? How to remove \n from a list element? I know classical physics, quantum mechanics, special relativity, and basic nuclear physics. I would like to get into some particle physics.   Where can I find a good introduction? It might be useful to segment books by whether they require quantum field theory or not. Canonical vs. non-canonical terminal input I am studying for an exam and I am confused as to how canonical vs. non-canonical input/output works in Unix (e.g., curses).  I understand that there is a buffer to which "line disciplines" are applied for canonical input.  Does this mean that the buffer is bypassed for non-canonical input, or does it simply mean that no line disciplines are applied?  How does this process differ for input and output operations?  In the curses programs I have worked with that demonstrate canonical input, the input typed by a user is automatically entered either after a certain number of characters have been typed or a certain amount of time has passed.  Are either of these things considered "line disciplines" or is this something else entirely? Which browsers are officially supported, and what else do I need? $(\frac 3p)$ is the Legendre symbol here, not a fraction, if that wasn't clear.  This is what I have so far:  We know by Gauss's Lemma that $(\frac qp) = (-1)^v$ where $$v = \#\left\{1 \le a \le \frac{p-1}{2} \mid aq \equiv k_a \pmod{12}),\space -p/2 &lt; k_a &lt; 0\right\}$$  And we also know that multiples of $(\dfrac{p-1}{2}) \times 3$ must be between $0$ and $p/2$, between $p/2$ and $p$, or between $p$ and $3p/2$.   Representing $p$ as $12k + r$ seems like the next step, but I'm not sure where to go from there. I (as a middle-school student) was wondering how virtual particles even conserve energy of the entire system? I don't mean just the particle's energy, but conservation with respect to the surroundings?  Let me explain more, Einstein proved that $E = mc^2$ and equivalently, we can say $m = \frac{E}{c^2}$ . We however know mass causes curvature in space-time (gravity well). Now that statement being completely valid we can use Quantum Fluctuation, implying energy and time can be related by the relation:  $\Delta{E}\Delta{t}\approx\frac{\hbar}{2}$ and by that simple relation we can make the following points:   As the energy increases, the time allowed to be "borrowed" decreases, before it gives it back to the universe. As time increases, the energy allowed to be "borrowed" decreases.    These are completely satisfactory points I've made, I believe.  Now, I have read that the vacuum is filled with constantly bubbling virtual particles that get created from the energy "borrowed" by the above relation and is destroyed again, which satisfies the energy conservation laws.   This seems to imply that  in the short time the virtual particle exists, they must have a very small mass which is: $m_{1} = \frac{\Delta{E}}{c^2}$ and using basic classical mechanics rather than GR (for sake of simplicity) we calculate that the gravitational force mediated by this short-lived virtual particle;  $F_{g}= G\frac{m_1 * m_x}{r^2}$   where $m_x$ is an particle\mass in question. Nevertheless since we know the force set upon an object moves the object in a direction, we can say it would increase its kinetic energy by this factor: $F_g = \frac{1}{2}mv^2$ which is equal to $G\frac{m_1 * m_x}{r^2} = \frac{1}{2}mv^2$.  Surely, this implies that every time quantum fluctuations happen, energy of real-particles\masses increases and therefore violates energy conservation, but what prevents this violation from taking place?  If virtual particles are not really present and interact with objects like I thought, how can Hawking radiation (currently a large concept in black hole physics) then be accepted in physics community? Is there a way to retrieve the user id of @mention user from chatter post in Apex? We can write a before insert trigger on feeditem, but when we display the body (which has @mention user) it simply displays as text. Is there any workaround for this? Prove that given $A\subseteq\mathbb{R}$, if has a maximum $a_0$, then $a_0$ must be the supremum.   In order to prove this, I have written the following:  Let $A\subseteq \mathbb{R}$, nonempty set because $a_{0}$ $\in A$ for hypotesis. Then, since $a_{0}$ $\in A$ is the largest element of the set $A$   (i.e. for each element $A$ of A we have that $ a\leq a_{0}$) $A$ is bounded above. Then, A has a supremum. Let $s$ be the least upper bound of $A,$ so we have that for each element $a$ of $A$, $a\leq s$, for being $s$ an upper bound. In particular, $a_0\leq s$, which implies that $s=a_0$.   Then, we have that $\sup(A)=\max(A)=a_0$, which is exactly what we wanted to prove.  Observation: this proof is true if $s \in A$   I'm not sure if my proof is correct. I will be thankful if someone tells me. Possibility of recovering files from a dd zero-filled hard disk I have "zero filled" (complete wiped) an external hard disk using dd, and from what I have heard: people said you should at least "zero fill" 3 times to be sure that the data are really wiped and no one can recover anything.  So I decided to scan the disk once again after I've zero filled the disk. I was expecting the disk to still have some random binary left. It turned out that it has only a few sequential bytes in the very beginning. This is probably the file structure type and other headers stuff. Other than that, it's all zeros and nothing else.  So if we have to recover any file from a zero filled disk, ...how? From what I've heard, even you zero fill the disk, you should still have some data left. ...or could dd really completely annihilate all data? Sometimes when attempting to capture a Pokémon, the mon will be wiggling in the ball and the app will disconnect from servers (for mobile connection reasons or whatever), and I'll be left staring at a ball containing Schrodinger's Pokémon. I'll sigh at the loss (or in the case of a 500CP Pinsir, scream in agony), and reload the game.  I thought maybe encounters in general couldn't handle disconnecting well, but sometimes immediately after a Pokémon broke free of a ball, the little icon in the upper left would appear, the game would stall for a bit, but it would recover.   Is there anything I can do to avoid this? Should I be waiting it out, instead of reloading?  (and before anyone says it, yes I have filed a bug report) Yesterday, my laptop caught a virus which caused massive damage. Since them, I have been trying to recover important files before reformatting my computer, a task the virus has not made easy.   Restoration points predating the attack have been deleted.  Most of my folders show empty.  My Start menu is essentially empty, with the exception of Trillian and Mirror's Edge. The same goes for my Desktop, which only has programs which were installed after the attack.  Searching for files through my computer is pretty much useless, as it only rarely brings up anything. I suspect most of my files have not been deleted. While my folders show empty, uTorrent still does display them and I can open them from here. Unfortunately, when I select Open Containing Folder, the folder still shows as completely empty even if I'm currently watching a video from that very folder. Further adding evidence to the not-deleted-just-missing theory, the data recovery software I'm using (Restoration) cannot find only find an handful of the missing files.   If they were deleted, I could do a forensic recovery to get them back but since they're probably still somewhere on my computer, just out out of my reach, I can't find them.   Under those circumstances, is there a way I can recover those files? Filling cells with sequential numbers in an Excel (2003) macro I need to fill an excel column with a sequential series, in this case from -500 to 1000.  I've got a macro to do it, but it takes a lot of lines for something that seems like it should be a single function [something like FillRange(A2:A1502, -500, 1000, 1)].  But if that function exists, I can't find it.  Is the following as simple and elegant as it gets?  'Draw X axis scale Cells(1, 1).Value = "mV" Cells(2, 1).Value = -500 Cells(3, 1).Value = -499 Cells(4, 1).Value = -498  Dim selection1 As Range, selection2 As Range  Set selection1 = Sheet1.Range("A2:A4") Set selection2 = Sheet1.Range("A2:A1502")  selection1.AutoFill Destination:=selection2 Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus Given a string of JSON data, how can I safely turn that string into a JavaScript object?  Obviously I can do this unsafely with something like:  var obj = eval("(" + json + ')');   but that leaves me vulnerable to the JSON string containing other code, which it seems very dangerous to simply eval. Any Video formats that render with Alpha Channel? A Variation of the even-town odd-town problem How to find the determinant of this $3 \times 3$ Hankel matrix? Today, at my linear algebra exam, there was this question that I couldn't solve.   Prove that $$\det \begin{bmatrix}  n^{2} &amp; (n+1)^{2} &amp;(n+2)^{2} \\  (n+1)^{2} &amp;(n+2)^{2}  &amp; (n+3)^{2}\\  (n+2)^{2} &amp; (n+3)^{2} &amp; (n+4)^{2} \end{bmatrix} = -8$$   Clearly, calculating the determinant, with the matrix as it is, wasn't the right way. The calculations went on and on. But I couldn't think of any other way to solve it. Is there any way to simplify $A$, so as to calculate the determinant?