Let $R=\mathbb{Z}[i]=\{a+bi\mid a,b\in\mathbb{Z}\}$, where $i^2 =−1$, and let $I=(3+i)$. Find the exact number of elements in $R/I$.
Order of some quotient ring of Gaussian integers
What is difference between the options "autoclean", "autoremove" and "clean"? How to search the internet for terms with special characters How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$ How   $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$   i have tried to do that by the following procedure:  $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2$  =$\frac{1}{n}(\sum_{i=1}^n X_i^2 - n\bar X^2)$  =$\frac{1}{n}(\sum_{i=1}^n X_i^2 - \sum_{i=1}^n\bar X^2)$  =$\frac{1}{n} \sum_{i=1}^n (X_i^2 - \bar X^2)$  Then i have stumbled. What counts as "observation" in Schrödinger's Cat, and why are superpositions possible? So if I understood correctly, Schrödinger's Cat is a thought experiment that puts a cat inside a box, and there's a mechanism that kills the cat with 50% probability based on a quantum process. The argument is that the cat now must be in a superposition of dead and alive.  This makes sense at first, but the state of the cat inside the box will affect the outside world in an observable way, right? For example if the cat dies, it might meow loudly which would be audible. If it didn't meow, it would produce a thud on the ground when it dies. And even if the ground was very solid, the redistribution of mass inside the box will affects its gravity field which means the whole universe theoretically immediately observe's the cat's death.  So extending this argument to all superpositions, the different states would cause different effects on the rest of the universe, usually a slight change in the gravity field is the minimum. This gravity perturbation would propagate throughout the universe, and even all the experimenters go to sleep with thick, thick earplugs, somebody or something in the universe is going to inadvertently observe the event and the superposition immediately collapses. Thus superpositions cannot exist beyond an extremely short amount of time.  What's wrong with my reasoning? Since I can't find any alternative to Linux'  elevation command, I have the following question:  How to define a PowerShell function which requires elevation? I mean UAC prompt.  Say, such function follows:  function system-check {     SFC /ScanNow }   System:  Windows 8.1 Pro 64-bit  PowerShell:   Major  Minor  Build  Revision -----  -----  -----  -------- 5      0      10586  117    EDIT1:  To be 100% understandable let me rephrase:   I run PowerShell as user I run the aforementioned function system-check I want the function to elevate in order to be able to execute the command; note, that I want the UAC prompt to appear I use time manager for a polygon layers with dates and it works. I would like to see the cummulative addition of my polygons. So I have 4 periods I want the poylgons from 1900-1950 to be added to the new ones in 1950-2000 etc in the next time step. How can I perform that or does another open source tool exist for that? How to get horizontal black line when coloring cells? Is the gradient a surface normal vector or does it point in the direction of maximum increase of f I'm having some trouble trying to visualize and physically understand what's happening with the gradient.  I understand that the following is true:   The gradient of f (grad(f)) points in the direction of maximum increase of f   However, later on, we are told that a gradient of a surface f (grad(f)) gives us the surface normal vector (i.e pointing away). How can that be? From the first statement, I thought that the gradient must be pointing in the direction of maximum increase of f - surely the direction of maximum increase of f should be a vector pointing in some direction on f itself. How can it be pointing outward as a surface normal vector? After removing any part the rest can be split evenly. Consequences? Why do electrons move towards anode in the Photoelectric effect experiment? Consider a Photoelectric effect experiment apparatus as shown in Figure 11.1    The variation of photocurrent with the voltage applied across $\mathrm{A}$ and $\mathrm{C}$ is as shown in Figure 11.3.    My question –     Why is the current non-zero even if the voltage is zero?    Consider a situation in which the voltage across the plates is zero. Photons of sufficient frequency strike the outermost surface of the cathode, emitter plate $\mathrm{C}$. Electrons become free from the attraction of nucleus as their net energy becomes positive.  Why I think that there should be no net current?  As soon as photons strike the outermost surface of the cathode(assumed to be $1$ to $10$ atomic diameters thick), electrons become free. Now, as the electrons have kinetic energy, they'll come out of that outermost surface. Now, the electrons(with kinetic energy) should have an equal probability of going in every direction of the surface. So, around half of the electrons should go in the right direction towards the anode and strike it and half should go to the left, trying to penetrate the inner surfaces of the cathode plate.   Now, the electrons which will go to the right will face some field due to the space charge present in the tube but nevertheless, some electrons will definitely reach the anode plate. After they strike with the anode plate, they'll face some resistance but still, some electrons will reach the low resistance copper wires connected to the anode plate and milliammeter.  Now, consider the electrons which go towards left. They'll definitely face resistance from cathode plate but some will still reach the copper wire.  Now, as there is no potential difference applied between $\mathrm{A}$ and $\mathrm{C}$, there is no need of commutator and voltmeter. So, the ammeter is connected in series with the anode plates and cathode plates with copper wires.  Now, if there is steady state current in the circuit, electrons emitted to any one of the sides should dominate over the other. But the number of electrons emitted on both the sides is same, and all the electrons should face the same resistance as the current(electrons) should flow in a closed loop. But if the number and velocity of electrons coming from both the sides are same, there should be no net current. But there is. Why?  Edit –  From the answers of anna v and mmainville, it can be seen that the emission of electrons is dependent on the angle of emission of light.  But still, the question remains. If the electrons emitted towards anode have enough energy to complete the circuit and again reach the holes in cathode despite the resistance in between, the electrons emitted to the left should have enough energy to come to anode plate via copper wires and again get ejected through anode plate by the virtue of their kinetic energy and reach the cathode plate. In this case too, the current should be zero which is not true. Why? $\phi \times \theta$ is faithful iff $(|Z(H)| ,|Z(K)|)=1$ for faithful characters $\phi \in Irr(H)$ and $\theta \in Irr(K)$ . Number of triangles formed by $m$ lines Brie Cheese and expiration date I have a Brie Cheese that has not been opened. It shows an expiration date that is 6 weeks ago. Is it still safe to eat? Linked server returned message "Query timeout expired." We have two separate SQL Servers. On one server we have a data warehouse (DWH), on the other we have sales information database.  Now on the DWH server there is an ETL job that collects the information from the sales server. The job runs daily after midnight. The DWH collects the information via linked server from the sales database.  Now, the most time the ETL job runs without any problems. But sometimes it fails because of query timeout. We have found out, that there is a specific pattern: The failure happens every 11th day. So on the 11th day the ETL job fails to collect information.   The following error occurs:     SQLNCLI11 for linked server “my linked server” returned message “Query timeout expired.”   Note: The job fails usually 10 minutes after start.  We have searched everything and could not found out what this issue causes. Also we know that the amount of data is every time, almost the same. Also there is no any scheduled job that runs all 11 days or something.  The remote query timeout on the linked server is set to 0.  Our next step will be, to turn off the antivirus programm on the sales server, to check if this causes the problem.  Does anyone have any clue or idea, where i can search further to find the problem? Why can't Chromium suddenly access any partition except for /home? Would Beetee's electrocution plan work and kill the tributes in the lake? In , the second episode of the Hunger Games trilogy, one of the characters (Beetee) proposes a plan to kill some of the remaining tributes. He proposes wrapping a special wire that he has designed to withstand the vast energy of lightning around a tall tree and running it down to a salt water lake. Then when lightening strikes the tree (which it does predictably every twelve hours) anyone in the lake will be killed. (N.B. This is not Beetee's real plan, but it is the one he wants his allies to believe.)  I don't think his plan would work at all. Assuming that his wire can carry the enormous current delivered by a lightening strike (this is fiction after all) surely the current would short to ground through the salt water rather than taking a path through anyone standing or swimming in the water. The analogy I have in mind is that of birds standing untroubled on live electricity wires. They are unharmed because they do not provide a path to earth. But another analogy might be dropping a mains electric heater into a bath when you are in it. I think that kills you (though I'm not sure why.) Question :   Determine all natural numbers n such that 3   divides $n\cdot2^{n}+1$  Actually I don't have any ideas to approach but  my efforts :   I see $n=1,2,7,8,13,14$ so I think :   $n=6k+1$ and $n=6k+2$ $k\in \Bbb{N} $  If I'm not wrong but I don't know how ? I prove it ? Difference between Ubuntu and Linux Mint Can someone please tell me the differences between Ubuntu and Linux Mint ?   I thought they were almost the same thing until I downloaded Ubuntu 13.10 versions of softwares for my Linux mint 16 system, and half of them didn't work. (I did that as those are the links I got when I searched for Linux mint versions) Is voting to balance in the spirit of the site? The other day I saw an answer which was ok. Not great but not bad. So no reason to upvote or downvote. However the answer already had a single downvote. I felt this was unfair so upvoted the answer to keep it at what I felt was a fair score. It wasn't until afterward that it occurred to me that I should have perhaps solely voted on the content of the answer.  Is this a reasonable reason for an upvote or should I have left the answer alone? How do I remove objects after building them, for example a fence that I placed without taking into account its orientation? I have right clicked on it like you would a structure but nothing happened. I have also tried Shift+Alt+Click and nothing has happened. I couldn't think of any other solutions as this is my first game.