Why is it that a real scalar field is charge neutral whereas a complex scalar field (or a spinor field like Dirac theory) describe charged fields?
If we have a complex scalar $\phi$ we know that the gauge-invariant interaction with $A$ is given by $A^\mu J_\mu$, where $J$ is the Noether current of the $U(1)$ symmetry of the Lagrangian $$ J_\mu\sim \phi^\dagger\partial_\mu\phi-\phi\partial_\mu \phi^\dagger \tag{1} $$  If we have a real scalar instead, $J=0$ and the field doesn't couple to $A$: real particles aren't charged.  My question is: what if we took $$ J_\mu\sim \phi\partial_\mu\phi \tag{2} $$ to couple $\phi$ to $A$? The interaction is Lorentz invariant and renormalisable, but it is not gauge-invariant, which is probably a bad thing. At what point would this theory break down?  This is probably very naïve, but I think that the Feynman rules for this theory are straightforward. So I guess the theory makes sense at least perturbatively. The theory is probably flawed at a more fundamental level, but I cant't seem to find where (some kind of gauge anomaly perhaps?)
Prove that $∂A$ is closed given $∂A = \text{Cl}(A) − \text{Int}(A)$ Similar questions have been asked, but none with the given information. My textbook doesn't give me the fact that  $∂A = \text{Cl}(A) − \text{Int}(A)$. If that were the case, I could just state that definition and note that it's the intersection of closed sets.   I have very little knowledge of set theory and proofs, so I'm not sure how else to prove this. As always, I appreciate any help.  The question comes from “Introduction to Topology: Pure and Applied” by Colin Adams and Robert Franzosa.   "Let $A$ be a subset of a topological space $X$. Prove that $∂A$ is closed given  $∂A = \text{Cl}(A) − \text{Int}(A)$" After reading of:  I wonder if the  also applies to flagging of old questions.  I have the subjective feeling the longer the time between the question and the flagging is, the longer it takes to become reviewed.   Is flagging of old posts a waste of moderator time? Should I avoid that? I've got this one file, favicon.ico. It's actually a large group of nested files, but purely for example, just this one.  If I do cp favicon.ico favicon2.ico, Apache can read the second one just fine.  If I do mv favicon.ico favicon3.ico, Apache throws a 403 on favicon3.ico.  If I su apache, I can read the file as apache.  stat says both files are identical, except for the inode number.  Is there some hidden flag somewhere or something that I'm missing? I've never seen anything like this.  error_log says (13)Permission denied: access to /favicon3.ico denied Markov chains, cellular automata, and raster algebra I need to build a probabilistic cellular automaton for simulating land-use changes and am wondering how to implement this in a GIS. I understand that Markov chains are a way of doing this, and IDRISI has some sort of module for Markov chains, but I have no access to IDRISI.  that, with raster data, this can, in theory, be done with any GIS that can do raster algebra. Presumably this means that it can even be done in ArcGIS, and I think I've figured out how to build cellular automata in ModelBuilder (once I know with certainty how to make iteration work there).  Still, I can find next to no practical advice on this. Once I have two datasets, one showing the land use at a certain point in time (time A) and the other at a later time (time B), I imagine doing the following (correct me if I'm wrong):   If there are (for example) 3 types of land use, then there are 9 possible things that can happen to a cell from time A to time B; thus, reclassify each dataset such that each land use is represented by a different number (for example, 9, 11, and 14 at time A and 3, 4, and 5 at time B); Divide one raster by the other, obtaining a raster with 9 possible cell values. Determine the percentage of cells in the new raster that has each of the 9 values (I forgot, how does one do this in ArcGIS?); those will be the probabilities of each type of change (or lack thereof) in land use.   What I cannot figure out, though, is how to apply these probabilities during the next iteration. Finally, I wonder if there is a way of making an external factor (say, proximity to roads) influence the spatial distribution (not the overall likelihood) of change in my model.  Can anyone recommend where to find information on how to do this in ArcGIS? How to lazy load images in ListView in Android I am using a ListView to display some images and captions associated with those images. I am getting the images from the Internet. Is there a way to lazy load images so while the text displays, the UI is not blocked and images are displayed as they are downloaded?   The total number of images is not fixed. Is wildcard search possible in Gmail? $\{a_n\}$ be a sequence such that $ a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in... Let $\{a_n\}$ be a sequence of positive real numbers such that   $a_1 =1,\ \  a_{n+1}^2-2a_na_{n+1}-a_n=0, \ \ \forall n\geq 1$.   Then the sum of the series $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in...  (A) $(1,2]$, (B) $(2,3]$, (C) $(3,4]$, (D)$(4,5]$.   Solution attempt:   Firstly, we figure out what $\frac{a_{n+1}}{a_n}$ is going to look like. We get, from the recursive formula, $\frac{a_{n+1}}{a_n}=1+\sqrt{1+\frac{1}{a_n^2}}$ (remembering the fact that $a_n&gt;0$, the other root is rejected).   We know that, if $\lim_{n \to \infty}\frac{a_{n+1}}{a_n}&gt;1$, then $\lim a_n \to \infty$.   Further, $(a_{n+1}-a_n)= \sqrt{a_n(a_n+1)}&gt;0$. (Again, the other root is rejected due to the same reason).   Hence, $(a_n)$ increases monotonically.   Therefore, the largest value of $\frac{a_{n+1}}{a_n}$ is approximately $1+\sqrt{1+\frac{1}{1}} \approx 2.15$  Now, the sum can be approximated as $\displaystyle\frac{\frac{1}{3}}{1-\frac{2.15}{3}} \approx 1.3$ (In actuality, $\mathbb{sum}&lt; 1.3$).  So, option $(A)$ is the correct answer.   Is the procedure correct?   I have been noticing a handful of this type of questions (based on approximations) lately, and the goal is to find out where the sum / the limit of the sequence might lie.   Is there any "definitive" approach that exploits the recursive formula and gives us the value, or does the approach varies from problem to problem? Why isn't The Flash unstoppable? I've recently gotten into The Flash TV show, and so I've been reading about his powers. What I don't understand is why his typical rogues pose a threat - it seems like there should be very little limit to his powers.  He can go back in time, so if he loses a battle or the bad guys do something, he can go back and preempt it. In the new 52, he can all but predict the immediate future, so he can see bad things that are going to happen and stop them. He can phase matter through matter, or let matter pass through him. This means he is all but immune to weapons fire.  I haven't read enough of the old comics so I don't the know challenges he's had. For a guy who can't be attacked, can time-travel and can race around the globe several times in a minute, I can't imagine there is much that could pose a threat. Auto comment "possible duplicate of ..." missing on voting to close as duplicate I have a profile on AskUbuntu.com which is recognized on my StackEchange profile (). I have already asked and answered questions there. But even though I'm logged in stack exchange, Askubuntu does not see me as logged in. I have a long history of problems with that and though I have been able to access it up to end 2018, and I don't understand why it can't do it anymore. I have seen multiple questions about similar problems, but they all are older than 2018 and do not really helped me solve it. Amongst other, they proposed to log in with the same user and password, but using the same user and password on two different log in systems is a security hole. either your credential cookies are shared, or one should use different logging info. EDIT: it has been suggested to deactivate third party cookies blocking, but I do not want to ruin my browser's tracking protection. Using same modulus for RSA I know that there exist some attack when using same modulus.      But with a little modification, $m$ is the plain-text $N$ is the RSA modulus $r_1, r_2$ is the random padding $e, s$ is the public exponent $C_e, C_s$ is the cipher-text  encrypt as follow $$С_e = (m + r_1)^e   \bmod N$$ $$С_s = (m + r_2)^s   \bmod N$$  If attacker only knows $C_e, C_s, r_1, r_2, e, s, N$. Is it possible to know $m$?  It seems the  does not work? jQuery animate() What conservation law corresponds to Lorentz boosts? Noether's Theorem is used to relate the invariance of the action under certain continuous transformations to conserved currents. A common example is that translations in spacetime correspond to the conservation of four-momentum.  In the case of angular momentum, the tensor (in special relativity) has 3 independent components for the classical angular momentum, but 3 more independent components that, as far as I know, represent Lorentz boosts. So, what conservation law corresponds to invariance under Lorentz boosts? I know that we don't have automatic login for the Data Explorer, but what's with the automatic timeout/logout?  There have been times that I am logged into the site, maybe I walk away for a little bit, and I come back to being logged out.    Is there a reason for logging us out of Data Explorer?  If so, is it possible to either remove the timeout or extend the period of time before timeout? If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$ If a group $G$ is isomorphic to $H$, prove that ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$.   Can someone provide a step by step solution? Explaining along the way our strategy in proving this. How many individual lights can an arduino control I want to use a regular Arduino board to control LEDs - what is the most lights I can control individually from the board?  Is there a tutorial on how to hook it up? How can I double the newlines in an output stream Ubuntu GNOME 17.04: wi-fi not working -- mac address keeps changing? My Panda USB wi-fi adapter works just fine on 16.10, but when I try to connect to my wi-fi router in 17.04, GNOME network manager reports "Connection failed." I did some tinkering, and noticed that my MAC address for my wifi adapter, according to GNOME, is DIFFERENT every time I make it forget my wifi settings and try to reconnect. Any leads on a possible fix or work-around?   I'm running Ubuntu GNOME 17.04, kernel 4.10.0-19-generic, GNOME 3.24.0. I have two given names and one surname. Is it a problem if I omit the surname from my ticket? I have two names as my given name in the passport and I typically just use this as my first and last name. However, my passport has a surname too which I almost never use. Due to this, I mistakenly provided my given name as first and last name.   As per the passport,  Surname : AAAA Given name(s) : BBBB CCCC   In the ticket, First name : BBBB Last name : CCCC  The credit card I've used to make this booking has my name as BBBB CCCC.  Will this create a problem at the travel time?  Update : the airline says that they can't update the name in the ticket. They are saying that since they have made a remark of the passport details boarding will not be a problem but I might face problems at immigration. Do they check name details in itenerary to exactly match passport at the time of immigration? Is that true in your experience?  Update : Travelled without any problem, neither at boarding gate, immigration nor after arrival at border police control. Problem. Use Tutte's 1-factor theorem to prove that every 3-regular graph with no bridges has a 1-factor.  I honestly have no idea how to approach this problem. I have drawn a couple of graphs to gain some insight to no avail. Any hints or tips will be greatly appreciated.