Prove that  $\sin(\theta) + \sin(\theta+2\pi/3) + \sin(\theta+4\pi/3) = 0 $   I was able to solve it using trigonometric identities but I want a method which uses complex numbers.  Any help would be appreciated.
Prove that $\sin(\theta) + \sin(\theta+2\pi/3) + \sin(\theta+4\pi/3) = 0 $ for any angle $\theta$.   This came up in the context of electricity:  It is common in electrical power engineering to use three-phase circuits with sinusoidal currents out of phase with each other by 120 degrees. The benefit of this is that the currents sum to zero at the neutral point meaning less copper is required for the system.    However, while I know that the three phasors always sum to zero, and can see visually, as in the figure that this is true at every point, how can this be shown more formally, or algebraically?
"Contact us"... and then what? Display command in xterm titlebar My Bash Prompt is currently setting the xterm titlebar using the following sequence:  PS1='\033]0;\u@\h&gt;\w\007'   Is there an easy way to display the current command in the titlebar. For example, if I am tailing a file using tail -f foo.log, I want my titlebar to say tail -f foo.log. Prove that if $A$ is a set with $m$ elements, $B$ is a set with $n$ elements, and $A \cap B = \emptyset$, then $A \cup B$ has $m+n$ elements. How best can I discover what is up with my electrical bill? This might be subjective, I don't know. However, it's a serious problem for me. My electrical bill is outrageous. It's three times as large as my next-door neighbor, and four or five times the size of my neighbor's across the street. It's worse in the summer (I live in central TX), but I've had the air conditioner inspected, and while the house is right at the maximum capacity for our unit, the unit should still be able to take care of the load. That also doesn't explain why our winter bills are larger than the comparables.  What I would like to do is check the amount of current drawn by each running appliance and calculate where all my money is going. I just don't know how to go about doing this. Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer?  The square roots need to be irrational. Is there a command for large middle delimiters consistent with \bigl and \bigr? After browsing through related threads, I am now of the understanding (please correct me if I am wrong) that the rule of thumb is to use the \bigl,\bigr pair with brackets, parentheses, etc. for operators like sums, products, and integrals, and \left,\right in all other cases.  I only recently found out about \middle , which has solved what had been a vexing problem for me: delimiters for function arguments (slashes, bars) not scaling properly. Leafing through my cheatsheet has not been of much help, so I wish to ask: is there an equivalent of \middle for \bigl and \bigr, or would the use of \middle along with that pair not be gauche? skype 4.3 cannot connect to account on ubuntu 14.04 LTS I have being using Skype for a long time on my Ubuntu 14.04 LTS. From this morning on, Skype simple cannot connect to my account anymore.  That is my Skype version:     Skype 4.3.0.37   © 2014 Skype and/or Microsoft   My Ubuntu system:  LSB Version:    core-2.0-amd64:core-2.0-noarch:core-3.0-amd64:core-3.0-noarch:core-3.1-amd64:core-3.1-noarch:core-3.2-amd64:core-3.2-noarch:core-4.0-amd64:core-4.0-noarch:core-4.1-amd64:core-4.1-noarch:cxx-3.0-amd64:cxx-3.0-noarch:cxx-3.1-amd64:cxx-3.1-noarch:cxx-3.2-amd64:cxx-3.2-noarch:cxx-4.0-amd64:cxx-4.0-noarch:cxx-4.1-amd64:cxx-4.1-noarch:desktop-3.1-amd64:desktop-3.1-noarch:desktop-3.2-amd64:desktop-3.2-noarch:desktop-4.0-amd64:desktop-4.0-noarch:desktop-4.1-amd64:desktop-4.1-noarch:graphics-2.0-amd64:graphics-2.0-noarch:graphics-3.0-amd64:graphics-3.0-noarch:graphics-3.1-amd64:graphics-3.1-noarch:graphics-3.2-amd64:graphics-3.2-noarch:graphics-4.0-amd64:graphics-4.0-noarch:graphics-4.1-amd64:graphics-4.1-noarch:languages-3.2-amd64:languages-3.2-noarch:languages-4.0-amd64:languages-4.0-noarch:languages-4.1-amd64:languages-4.1-noarch:multimedia-3.2-amd64:multimedia-3.2-noarch:multimedia-4.0-amd64:multimedia-4.0-noarch:multimedia-4.1-amd64:multimedia-4.1-noarch:printing-3.2-amd64:printing-3.2-noarch:printing-4.0-amd64:printing-4.0-noarch:printing-4.1-amd64:printing-4.1-noarch:qt4-3.1-amd64:qt4-3.1-noarch:security-4.0-amd64:security-4.0-noarch:security-4.1-amd64:security-4.1-noarch Distributor ID: Ubuntu Description:    Ubuntu 14.04.3 LTS Release:    14.04 Codename:   trusty   Any suggestion why or how to fix it?  Thanks in advance. Characterizing all ring homomorphisms $C[0,1]\to\mathbb{R}$. This is something I've been trying to work out this evening.      Let $R$ be the ring of continuous real-valued functions on $[0,1]$ with pointwise addition and multiplication. For $t\in [0,1]$, the map $\phi_t\colon f\to f(t)$ is a ring homomorphism of $R$ to $\mathbb{R}$. I'm trying to show that every ring homomorphism of $R\to\mathbb{R}$ has this form.   Suppose otherwise, that there is some $\phi\neq\phi_t$, and thus there is some $f_t\in R$ such that $\phi(f_t)\neq \phi_t(f_t)=f_t(t)$. Define $g_t=f_t-\phi(f_t)1\in R$. Here $\phi(f_t)1$ is the constant function sending $[0,1]$ to $\phi(f_t)$. Then $g_t(t)\neq 0$. My first small question is why does $\phi(g_t)=0$? It seems only that $\phi(g_t)=\phi(f_t)-\phi(\phi(f_t)1)$.  I would like to conclude that there are only finitely many $t_i$ such that $g(x)=\sum g_{t_i}^2(x)\neq 0$ for all $x$. Then $g^{-1}=1/g(x)\in R$, but $\phi(g)=0$, contradicting the fact that homomorphisms map units to units. How can we be sure there are only finitely many $g_{t_i}$ such that the sum of their squares is never $0$? Thanks. FM Transmitter oscillation problem I've been trying to get this simple FM transmitter to work over the past few days and I could use some help in a few areas.      First iteration was on a breadboard with all parts included.  The exception being I didn't have a 0.1uH coil handy but I had another coil rated at 0.7uH (using my LCR meter).  I also didn't have a C4 value of 4.7pF but I had a 10pF value.  I figured I could just tune it later.  I hooked a scope up to it and got no oscillation.  I tried connecting the probe to the gate and the collector separately and together.  Still no oscillation.  Second iteration.  I removed the audio component of the circuit.  Essentially everything to the left of C2.  I had gotten that working no problem anyway so I figured I would concentrate on the oscillator.  This time I tried different values for the inductor ranging from 20uH to 1mH and different fixed values for the variable cap (10pf, 20pf, 30pf).  These were all ceramic caps. No love.  Third iteration. I tested the transistor to make sure that wasn't the issue.  Seems fine.  PN2222A variety. (Also tried 2N3904 and BC547). I then modeled this up on a piece of perf board.  I even found a SMD inductor of 0.1uH so now I thought I would be good.  Tested all the pieces on the meter before installation. Still nothing. I'm not seeing anything on the spectrum analyzer either.  Questions:   Do you think my benchtop power supply is causing a problem?  All the various versions of this type of FM transmitter circuit on YouTube and the Internet all seem to use a battery.  Any chance the 60Hz main power is messing with things?  I've put .1uF and higher caps on the power rails. I have a 200Mhz SIGLENT SDS1202X-E scope.  I've built and tested other oscillators up to a few MHz before with no issues.  Any thoughts on that being the issue?  I would expect to see something on the scope right? Any chance the probe is dampening the oscillations?     I know there are a lot better FM transmitter circuits out there I was just experimenting with this (and others like it) and couldn't get any to work. Can a git submodule be created retroactively? AC: Why differentiate between Ground and Neutral? Current flows through a conductor connecting points at different potentials.  Leaving aside multi-phase details, common/conventional AC systems use a 3-wire setup:   Wire-1: a line/live/hot/phase wire presenting a point that oscillates between 2 potentials. Wire-2: a neutral wire presenting a point of unknown/unspecified and varying potential, that nevertheless presents some fixed/specified potential difference to Wire-1 at least some of the time. Wire-3: a ground/earth wire presenting a point at 0V potential difference to its immediate physical surroundings.   Wire-1 and Wire-2, in addition to some device that is to be powered, are used to construct a closed electrical circuit. Wire-3, leaving aside EMI/shielding concerns, is used to ensure that current will flow through it, rather than the device's user, if there ever occur any faults and the device's user comes in contact with Wire-1 or Wire-2.  Additionally to this however, Wire-2 and Wire-3 are at some point connected. This is done to ensure that Wire-2's potential remains close to that of Wire-3 .. which seems to be important for some reason.  Now the part I don't understand is why there needs to be a distinction between Wire-2 and Wire-3 at the power socket, if there is none a few meters further down the line.  I have tried to look this up, but all answers I could find so far seem incomplete. The answers depend on how the question is phrased:   If the question is phrased as "Why do we need Wire-3 in addition to Wire-2" the answer is because "Wire-2 may be at a substantial potential difference to its surroundings/user and thus endanger him/her if he/she ever comes in contact with it or Wire-1". If the question is phrased as "Why do we need Wire-2 in addition to Wire-3" the answer is because "Wire-2 is needed to form a closed electrical circuit" or phrased somewhat differently "Wire-2 is needed to create a potential difference to Wire-1 and thus for current to flow" .. with the argument further being that when taking practical considerations into account Wire-3 can't provide a reliable/stable potential difference to Wire-1 like Wire-2 can.   This doesn't really answer why there's a need to differentiate between Wire-2/Wire-3 though, considering how    Wire-3 remains Wire-3 and maintains 0V potential difference to its surroundings/user, regardless of whatever else happens around it .. since that is what it's supposed to do, or phrased differently, since that is the reason for Wire-3 being useful in the first place .. right?   and   Wire-2 is connected to Wire-3   What am I missing here?    Why is it safe to touch Wire-3 but not Wire-2, or why can Wire-3 provide a level of protection that Wire-2 can't?  Why differentiate between Wire-2 and Wire-3 at the power socket but then connect them further down the line? counting number of specific string in each row I have a SNP table looks like this  A N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N C C N N N N N N N N N N C C N N N C C N N N N N N N N N N N N N N N N N   I want to count total number of N in each row? any idea how to do that What does a "< <(...)" redirection mean? I've seen that rvm (ruby version manager) is installed using the following command:  bash &lt; &lt;(curl -s https://raw.github.com/wayneeseguin/rvm/master/binscripts/rvm-installer )   So as I understand we get the script content and pass it to the bash (I believe &lt; &lt; and &lt;&lt; is the same thing?) I am interested in the &lt; &lt; part, found following description on the net:     &lt;&lt; token    Means use the current input stream as STDIN for the program until token is seen.    This is somehow not clear for me, can someone make an example or explain it in more simple way? What would be an English equivalent for the Mexican Spanish word tocayo? In Mexican Spanish (not sure if other Spanish speaking countries use the word too) we call "tocayo" to those people that share the same name as us (but not necessarily the same last name i.e., Juan Gonzalez and Juan Carlos Castillo are tocayos).  I was wondering if an equivalent for this word exists at all in English or maybe even an expression? My C# application is returning 0xE0434352 to Windows Task Scheduler but it is not crashing How to set default magnification for all windows How to control RAM / swap usage Is there a way (application) to control RAM / swap usage policy at application level.  I have some applications that I use all the time and I want them to stay in RAM all the time and there are also application that I use quite seldom and they might be thrown to swap at demand. I know that default configured windows should give me best performance but I want to check whether I can get more :).  I use windows 7. I'm trying to find the birth certificate of my wife's great grandmother who was probably born in Durban, South Africa in 1884. I'm currently unable to find any online resources for this.  If such a thing does not exist online, is there any other method of retrieving a copy or at least just the details? Working in Cycles.  Say I have a "disco ball" with a glossy shader. I'm pointing a spot light to it, and I'm trying to have the reflections of that light illuminate the diffuse material on the walls with the patterns from the ball.    I get this:    but I'd like to get something like:    For the illustration I placed a point light inside the ball an removed faces. But what I'd like to do is move the light around the ball so that the light reflected from it changes like it would do in real life.   Is that possible at this point?  EDIT:  Just for the record here's the image rendered in LuxRender (yet another program to learn...) Uniform convergence of alternating series