Review reopen post queue suddenly increase
Lots of questions in the reopen queue
Is the double-slit experiment typically carried out in a vacuum? It is my understanding of the double-slit experiment that when a measuring device is activated, to measure which of the two slits a particle is travelling through, this measuring is responsible for a different outcome than when the device is left off. When the device is left off particles leave an interference pattern, suggesting a single particle (or its wave) interferes with itself; when the device is activated no interference pattern is left, and particles are observed to travel through a single slit.  Now, does all this imply that this experiment is typically carried out in a vacuum? If not, how is it that interaction with other particles in the environment of the experiment's set-up are not already influencing the outcome, regardless of whether the measuring device is on or left off?  This question may very well be a very naive one, due to my very limited understanding of particle interaction, vacuums and/or physics in general. If so, I would appreciate a brief, layman's, overview of where my understanding of the different subjects at play here is likely to be lacking. Why do we use the object instead of the subject pronoun in constructions like "stupid me"? I'm trying to find out how come we say lucky me and stupid us rather than lucky I and stupid we. My understanding is that this is not a recent invention, but a relic from the distant past where it was a dative construction. However, I don't see the reason for using any case other than the nominative in this construction to begin with.        He went surfing. → Lucky him went surfing.   I got pulled over. → Stupid me got pulled over.   They bought a house. → Thoughtless them bought a house.      This is rather intriguing. After thinking about it for a while, I couldn't come up with any other language, Germanic or otherwise, where simply introducing an adjective would dictate a change of grammatical case. This appears to be an English peculiarity. How far can it be traced back? Can anyone comment on its background? Let $f_n = \mathbb 1_{[-n,n]}$ for all $n \in \mathbb{N}$   Compute explicitly $f_n \star f_1$ for all $n \in \mathbb{N}$.  Show that $f_n \star f_1$ is the Fourier transform of $g_n = \frac{  \sin{(2\pi x)} \sin{(2 \pi nx)}}{\pi ^2 x^2}$  Show that $\| g_n \|_1 \to \infty$  Deduce that the Fourier transform is not surjective from $\mathcal{L}^1(\mathbb{R})$ to $C_0( \mathbb{R})$    Here, $C_0(\mathbb{R})$ is the space of continuous functions $f : \mathbb{R} \to \mathbb{R}$ that tends to $0$ at $-\infty$ and $+\infty$. I managed to compute 1) and proved 2). But I don't know how to prove 3). Moreover, I don't see how can 1), 2) and 3) prove 4). Can anyone help? help with some Hilbert style proofs in a propositional logic axiom system. I'm very new to classical propositional logic, and my lecturer is using a system with the following axioms:  A1. X→(Y→X)  A2. (X→(Y→Z))→((X→Y)→(X→Z))  A3. (¬Y→¬X)→(X→Y)  Use uniform substitution and Modes Ponens (MP) as rule of inference. Deduction Theorem also allowed.  I'd like to deduce the following:  (a) X, Y ⊢ ¬(X→(¬Y)) (conjunction, but expressed in terms of NOT and IMPLIES)  (b) X→((X→Y)→Y)  (c) (X→Y)→(((¬Y)→X)→Y)  (d) (((¬Y)→X)→Y)→(X→Y)  Remark: The last two are to show that the axiom (A3) can be replaced with a different axiom:  B3. (¬Y→¬X)→(((¬Y)→X)→Y).  Solution: Bram28 proves (b) and (c) below. I demonstrate (a) and (d).  (d) is alright; first, I claim that X, ((¬Y→X)→Y) ⊢ Y.  Axiom (A1) gives ⊢X→(¬Y→X) so that ((¬Y→X)→Y)⊢X→Y by Hypothetical Syllogism (HS), which Bram28 shows to be valid.  Now that we have ((¬Y→X)→Y)⊢X→Y just apply the deduction theorem to get what we want: ⊢((¬Y→X)→Y)→(X→Y).  (a) is slightly trickier. Axiom (A1) gives ⊢ Y→(X→Y) so that X, Y ⊢ (X→Y).  Next, by part (b), we have as a theorem that ⊢X→((X→¬Y)→¬Y).  Therefore X, Y ⊢(X→¬Y)→¬Y.  But ⊢(¬¬(X→¬Y))→(X→¬Y) by DN Elim, which Bram28 shows to be valid.  By HS, X, Y ⊢(¬¬(X→¬Y))→¬Y.  By Axiom (A3) and MP, X, Y ⊢(Y→¬(X→¬Y)).  Also, rather trivially X, Y ⊢Y.  Finally by MP we conclude X, Y ⊢¬(X→¬Y) as required. Prove $[A,B^n] = nB^{n-1}[A,B]$ I am trying to show that $[A,B^n] = nB^{n-1}[A,B]$ where A and B are two Hermitian operators that commute with their commutator. However, I am running into a little problem and would like a hint of how to proceed.  If A and B commute then $[A,B] = ABA^{-1}B^{-1} = e$ where e is the identity element of the group.  $$\therefore AB=BA$$  $$n=1; [A,B^1] = (1)B^0[A,B] = e$$ This statement is certainly true. however moving on to $n = 2$ I find...  $$[A,B^2] = AB^2A^{-1}B^{-2} = ABBA^{-1}B^{-1}B^{-1} = BBAA^{-1}B^{-1}B^{-1}$$  Where in the last step I have used the fact that A and B commute to rearange the terms. However, it is plain to see that this last term simply reduces to the identity as well and for the n = 2 case we have:  $$[A,B^2] = e \ne 2B[A,B] = 2Be = 2B$$  Clearly I have assumed something I shouldn't have. The fact that there is a multiplicative factor of n implies I should be adding things, but I thought if I kept it as general as possible, the answer should just fall out naturally. I don't want an answer please, only guidance. In a Cartesian system, I've got the slope, start point and distance of a line segment.  What's the formula to find the endpoint? Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of them, $2N-1$ are initial positions and velocity. Any function of these initial condition is a constant of motion, by definition. Also, there should be exactly $2N-1$ algebraically independent constants of motion.  On the other hand, Noether's procedure gives us integrals of motion as a result of variational symmetries of the action. These integrals of motion are also conserved but they are not always $2N-1$ in number. In consequence, we classify the system by their integrability.  So, what is the difference between the  and ? Why do non- have less integrals of motion when they should always have $2N-1$ constants of motion? Virtual box is not working on windows 10 What is the best way to explain how to choose between "its" and "it's"? Probably one of the most frequent grammar mistakes in the English language is:     The dog sat on it's mat.   Since spelling checkers don't catch it, and it is even logical, since you would correctly write:     The dog sat on Fluffy's mat.   What is the best way to explain to a learner of English how to choose between it's and its? What is the safest way to clean up /boot partition? I have 200 MB assigned for the /boot partition.  Whenever I try to update the kernel, I receive an error message that basically states /boot is full.  What can I do to cleanup /boot and remove/backup the older kernels? Drop or change automatic migration-rejection based on non-existing tags I believe this feature creates more issues than it solves and thus would like to see it gone. (This is an out-growth of , which I repost here as it concerns several/all sites.)  The main reason why I think so is that there can be numerous reasons other than the subject being off-topic why some tag does not exist on some site, for one thing it could be a spelling, typesetting, or other formalistic reason. For example, for the two sites I know best MathOverflow and Mathematics hardly any of the most popular tags have a strictly matching tag on Mathematics but only because on MO there is the convention that certain tags are formatted in a specific way, which is not in place on Mathematics (for example nt.number-theory vs number-theory).  On top of that a user choosing the wrong site in the first place is likely not too well aware of the tags of the site they are posting too so that the question might be poorly tagged in the first place.         There is an additional detail I do not understand (it is obsolete if the actual request should be followed but in case it is not):   Even if on the source site a tag has a synonym and that synonym exists as a tag on the target site the migration is rejected. I do not understand the logic behind this at all. Possibly this is just an oversight.   I could also think of some other ways to improve the process while keeping the idea, but since I believe that it is best dropped I do not (yet) elaborate on this. Can a homomorphic encryption scheme be made CCA2 Secure? Is it possible to modify a homomorphic encryption scheme so that it can be CCA2 secure?  From the definition of a homomorphic scheme, it seems that it is malleable, which would result in lack of CCA2 security. So, if we use the regular method to convert a CPA secure scheme to CCA2 secure scheme (using a hash of the message and a random value), will it still preserve its homomorphic property? Please correct me if my understanding is wrong. I'm building an object detector using HOG features and linear SVM. Some of the regions of the object are more "distinctive" so I would like to give more weight to the features extracted from those regions. (e.g. imagine we want to detect between mugs and glasses, then the most "distinctive" part is the mug's handle)  How could I do this? For sure there should be a more intelligent way than just replicating the features of those "distinctive" regions. Why does a relay have a minimum applicable load? I am trying to find relays for my application and I read  which looks fine but specifies     Minimum applicable load: 10mV 10µA   In my circuit, it is expected that the relay closes but no voltage and current is applied. You can think of it as 2 relays in series where one is open and one is closed, so there is no current. This sounds like something I'd do since I was at school.  Why would a relay require a minimum voltage and current on the load side? Is it allowed to operate that relay under my conditions or not? What could possibly break in a relay if I don't respect this requirement? What does "minimum applicable load" mean? When and how do I need to consider this value? Is "do something by date X" inclusive? Possible Duplicate:          For example, if John says:      Return it to me by March 24th.    Does it mean that I need to return it to him before 23:59 p.m. March 24th, or before 23:59&nbsp;p.m. March&nbsp;23rd? I have bought return tickets from London to Lille Europe on Eurostar. The trains on my outward and inbound journeys both stop at Calais Frethun.  Will Eurostar let me get on in Calais on the inbound? What about getting off at Calais on the outbound leg? I suspect leaving the train early is easier than getting on the train 'late'. Story identification - "bless the beasts that bless us" I'm trying to trace what was probably a short story rather than a full novel from my childhood.  I think it starts in the present day (so the 80s or thereabouts) and there was some kind of alternate reality bleed which meant there were (i think) ghost boars or pigs, they were becoming more physical with time, and so problematic. The family escaped when the family dog led them to and through a portal which they couldn't perceive to a safer place.  In the end, they ended up in some kind of rural community who had a religion / custom that required them to say "Bless the beasts that bless us" but some folk shortened it to BTBTBU which irritated the girl who was the principle character.  I used to read a lot of kids short story sci-fi and fantasy including Nicolas Fisk but I don't know whether this was from a compilation of several authors or a single author's collection. Can an abstract class have a constructor?  If so, how can it be used and for what purposes? How to correctly add a path to PATH? I'm wondering where a new path has to be added to the PATH environment variable. I know this can be accomplished by editing .bashrc (for example), but it's not clear how to do this.  This way:  export PATH=~/opt/bin:$PATH   or this?    export PATH=$PATH:~/opt/bin Wilson and Fermat's Theorem