What determines the height at which a superconductor levitates? Can you just place it randomly in the magnetic field? What are the limits of superconductors?
Quantum levitation, flux pinning -- basically, when a superconductor floats above a permanent magnet, is really fascinating. But does the strength of the magnetic field influence the superconductor's height? And what relationship/formula suggests this?  For example, if I had a coil wrapped around a metal object and then ran a small current through it, a small magnetic field would be generated and a superconductor exposed to it would float about it at some height that I put it on.  But what if I increased the current, so that the magnetic field also increased? What happens to the superconductor? Does its height change? Or does it just get harder to "pull out" of its levitating state?
For a submerged object, buoyant force ($F_b$) is defined as:  $$F_b = V_{\text{submerged}} \times \rho \text{ (density)} \times g \text{ (gravitational constant)}$$   Conceptually, the buoyant force equation says that buoyant force exerted is equal to the weight of the volume of water a given object displaced. Why? I went online to   and found the following explanation, but it seems like non-sequitur to me:      Explanation: When an object is removed, the volume that the object occupied will fill with fluid. This volume of fluid must be supported by the pressure of the surrounding liquid since a fluid can not support itself. When no object is present, the net upward force on this volume of fluid must equal to its weight, i.e. the weight of the fluid displaced. When the object is present, this same upward force will act on the object. "Do you know where's Linda?" vs "Do you know where Linda is?" If a user is the type of person to consecutively get downvoted to potential negative reputation, so be it. What's the reasoning to maintain a minimum of one?  One reason that may justify this is of trying to not deter new users from the site who haven't read the FAQ and unknowingly and wrongly ask or answer a question. But most of us acknowledge this and not downvote in such a scenario.  So what's the reasoning behind this lower bound of 1? function a () {     return "foo"; }  a.b = function () {     return "bar"; }  function c () { }; c.prototype = a;  var d = new c(); d.b(); // returns "bar" d(); // throws exception, d is not a function   Is there some way for d to be a function, and yet still inherit properties from a? Are data members allocated in the same memory space as their objects in C++? I see questions/information online about how to get rid of corpses but I'm not sure what the consequences to having them lying around are.  I know that I can use a morgue to store them and that they will eventually be taken away in a hearse. However, I've always just left them lying around. This doesn't seem to have any obvious consequences.  Do they upset prisoners or staff (e.g. increasing the danger level beyond what death normally does)? Make them sick? Slow down pathing? Is this something that's not implemented yet or is leaving dead bodies all over the place meant to be completely fine?  The wiki's pages for  and  don't shed any light on this. All I'm finding is the how of getting rid of corpses/glitches related to it but not why it matters in the first place. All I've seen that's related is that apparently prisoners can loot guard corpses for keys, but there's no obvious consequences for prisoner corpses which are what I tend to end up with.  So why not just leave them in the corridors? What are the best ways to improve a site's position in the Google search results? Statement: $[a, b] \cap \mathbb{Q}$ in $\mathbb{Q}$ is not compact. Thus the interior of all compact subsets of $\mathbb{Q}$ is $\emptyset$.   I am trying to understand the first sentence. I read that a closed subspace of a compact space is compact, so for example, consider the unit interval $[0, 1]$ which is a compact space. Take a closed subspace $[0, 1] \cap \mathbb{Q}$ of $[0, 1]$. This set is closed since it just consists of all the rational numbers in between $0$ and $1$, including $0$ and $1$. So it is a closed subspace of a compact space. But why isn't this compact? Does there exist a function such that $\lim_{x \to a} f(x) = L$ for all $a \in \mathbb R$ but $f(x)$ is never $L$? Does there exist a function $f:\mathbb R \to \mathbb R$ such that $\lim_{x \to a} f(x) = L$ for   all $a \in \mathbb R$ but $f(x) \neq L$ for all $x$?   I found such a function in $\mathbb Q \to \mathbb Q$, where $f(\frac{p}{q})= \text{first p+q digits of }\pi$ satisfies the above condition. However, I have not been able to extend it to reals. So, is such a function possible, and if it is, is there any explicit example preferably related to the above function in rationals? Could someone please explain? I couldn't find anything on the internet, everything talks about how to go about it in some way, but nothing says exactly what it is.   Also, what is a fully trusted assembly and how do they differ from one another?  I have a MS certification exam and this is the only topic that I just don't understand.  EDIT: Thanks guys. Now I have a better understanding of security in .NET. I was able to pass my certification exam. Automatically adding user and password to openvpn script I have a working script, which allows me to do a vpn connection. It looks like this:  openvpn --config VPNBook.com-OpenVPN-Euro1/vpnbook-euro1-tcp443.ovpn   All is good, but I have to input the publicly available vpn and password every time I run this script. (Don't worry, I'm not going to do banking or anything serious over a free VPN, this is more for learning).  This is the username and the password available publicly on their website:  Username: vpnbook  Password: he2qv5h   After firing up, the script prompts me for username and password which now I need to copy and paste.   Tue Aug 15 11:54:51 2017 library versions: OpenSSL 1.0.2l  25 May 2017, LZO 2.08 Enter Auth Username: vpnbook Enter Auth Password:    Is there a way of doing it so the script inputs the values on its own?  I tried echoing it, but it isn't that easy.  Or do I have to edit the command itself, somewhere in the files? If yes, can someone point me in the right direction for doing it? Show $a^p \equiv a \mod 2p$ I've stucked at the following problem:  Let $p$ be an odd prime number. Show that for all $a\in\mathbb{Z}$  $$a^p \equiv a\ (\mathrm{mod}\ 2p)$$  How can I prove it? I know within Fermat's theorem that $a^p \equiv a\ \mathrm{mod}\ p$. How do I just restart the nautilus after adding a nautilus script without having to log out. Prove Uncountable set minus a countable set is uncountable Let B be a countable set, and let A be an uncountable set. Prove that A-B is uncountable Add layer to a QGIS group using Python What does it mean for a QFT to not be well-defined? It is usually said that QED, for instance, is not a well-defined QFT. It has to be embedded or completed in order to make it consistent. Most of these arguments amount to using the renormalization group to extrapolate the coupling up to a huge scale, where it formally becomes infinite.  First of all, it seems bizarre to me to use the renormalization group to increase resolution (go to higher energies). The smearing in the RG procedure is irreversible, and the RG doesn't know about the non-universal parts of the theory which should be important at higher energies.   I have read in a note by Weinberg that Kallen was able to show that QED was sick using the spectral representation of the propagator. Essentially, he evaluated a small part of the spectral density and showed that it violated the inequalities for the field strength $Z$ that are imposed by unitarity + Lorentz invariance, but I do not have the reference.  So what I would like to know is :   What does it take to actually establish that a QFT is not well-defined?   In other words, if I write down some arbitrary Lagrangian, what test should I do to determine if the theory actually exists. Are asymptotically free theories the only well-defined ones?   My initial thought was: We would like to obtain QED (or some other ill-defined theory) using the RG on a deformation of some UV CFT. Perhaps there is no CFT with the correct operator content so that we obtain QED from the RG flow. However, I don't really know how to make that precise.   Any help is appreciated. Showing that the product of all $n$th Roots of Unity is $(-1)^{n+1}$ I need to show that the product of all the $n$th roots of unity is $(−1)^{n+1}$.   Is there a way to do this by induction? If there is, I can't seem to figure it out. Are there other, perhaps more efficient, methods of proof? Suppose the proper length of a train is longer than that of a bridge, and the bridge can't bear the total weight of the train but can bear it partially. As the train goes very fast, it becomes shorter than the bridge and it can be entirely on the bridge. As a result, the bridge will collapse. However, in the train's frame, the train is even longer than the bridge, and it is only partially on the bridge; so the bridge will not collapse. What is the solution to this paradox? Probability of winning the game "1-2-3" Dividing polynomial $f(x)$ by $x-3$ and $x+6$ leaves respective remainders $7$ and $22$. What's the remainder upon dividing by $(x-3)(x+6)$?