Could someone please verify whether my solution is okay?      Let $G$ be a finite cyclic group with $|G|=n$ and generator $x$. If $y=x^{k}$ and $gcd(k,n)=1$, then show that $y$ is a generator of $G$.   Let $y=x^{k}$ with $gcd(k,n)=1$. Then $|\langle y\rangle|=|y|=\frac{n}{gcd(k,n)}=\frac{n}{1}=n$. Then $G=\langle y\rangle$.
A cyclic group is a group that is generated by a single element.  That means that there exists an element $g$, say, such that every other element of the group can be written as a power of $g$.  This element $g$ is the generator of the group.  Is that a correct explanation for what a cyclic group and a generator are? How can we find the generator of a cyclic group and how can we say how many generators should there be?
Is there a way in photoshop to convert k values to alpha values? Basically, I would like to make it so that the lighter a pixel is (the more white it has) the more transparent it becomes. Black pixels would be opaque, white pixels would be transparent.  I currently use the slider in blending options on the first tab to get rid of the white and make it transparent but find it works slightly better if I use the magic wand tool with delete.  I figure that there is some blending option out there but there are so many and I'm not sure which to use to achieve my goal.  (Using Photoshop CS6) Was Thanos really committed or just a hypocrite? The title is extremely vague, because this question contains HEAVY spoilers for Avengers: Infinity War, specifically the very last scenes of the movie. I can't reasonably put the entire question in spoiler tags, so consider this your warning.  At the end of the movie, Thanos pretty much wins. He gets all the Infinity Stones, Gamora and Vision have been brutally murdered, and the rest of the cast completely beaten. Nobody could stop him. So he executes his plan.  As far as I know (unless I missed something), his plan is to eradicate exactly half of all mortals in the universe, and from his own words, in a fair lottery (random chance), without taking into account a person's wealth, social status, or anything else. Pure chance decides whether you are sacrified for the greater good or live through this greater good (according to him).  During the movie, Thanos and one of his underling (Ebony Maw) say themselves that being sacrified is an honor and should be appreciated. As you would expect, most people do not appreciate being sacrified.  At the end of the movie, Thanos uses all 6 stones to do a finger snap, deleting half the universe.  The question is: Did he exclude himself from the targets? Was there a 50% chance that Thanos himself would disappear right there?  As far as I know, this is not explicitly answered in the movie itself. I am looking for similar situations in the comics (or in the movie if I happened to miss something) where you could know, or have an educated guess, as to Thanos being the kind of person who follows through to the end, even if it means dying, or the kind of person who gets scared of death when it comes down to it, and comes off as an hypocrite.  Some people also mentioned his "deal" with Strange would make him spare Tony from the snap too. I'm also wondering about that, if there is a link between the two, but it's not the focus of the question.  The reason I'm wondering is that Thanos seems to me like a very extreme utilitarist. Even though he might be wrong about the value of his plan (Do the remaining half really become happy?), in-universe it seemed successful at least once (Gamora's home planet), so his reasoning is not bad, even if the premise might be horribly flawed. Excluding himself from the snap would make it into a straight villain, while considering himself equal to the rest would make him into a wannabe hero with very very evil methods. This makes a huge difference in the interpretation of the movie and his character. Let $d=GCD(n^2+5,n^3-5n^2+6n)$ show that $d|630$ How do you redirect to a page using the POST verb? When you call RedirectToAction within a controller, it automatically redirects using an HTTP GET.  How do I explicitly tell it to use an HTTP POST?  I have an action that accepts both GET and POST requests, and I want to be able to RedirectToAction using  POST and send it some values.  Like this:  this.RedirectToAction(     "actionname",     new RouteValueDictionary(new { someValue = 2, anotherValue = "text" }) );   I want the someValue and anotherValue values to be sent using an HTTP POST instead of a GET. Does anyone know how to do this? If $x_1,...,x_n$ are positive real numbers and if $y_k=1/x_k$, prove that $$\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2.$$   I've been learning induction, and I've come across this problem that I really can't even break down and begin to think about. I've been told it has something to do with Cauchy-Schwarz, but I cannot figure out how to apply it. I would appreciate help figuring out how to go about and formulate this proof. Thanks! Can I use a present-tense verb when writing about a past event? My sentence is     In Charles Dickens' book Oliver Twist, the stingy orphanage director doles out just one small bit of soup to the hungry orphans.   The main question is should it be doles or doled? Could it be either? Compute $\mathbb{P}\{ W_t < 0 \, \, \text{for all} \, \, 1 < t < 2\}$ for a Brownian motion $(W_t)_{t \geq 0}$ Let $(W_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion. Calculate $$\mathbb{P}\{ W_t &lt; 0 \, \, \text{for all} \, \, 1 &lt; t &lt; 2\}.$$ I can only think that this will be conditional on $W_1$. Please tell how to proceed? Does the Great Old One warlock's telepathy from the Awakened Mind feature take an action? Does the Great Old One warlock's telepathy from the Awakened Mind feature take an action?  It doesn't seem too clear to our group and we have basically ruled it as equivalent to talking. Prove that any two consecutive terms of the Fibonacci sequence are relatively prime Prove that any two consecutive terms of the Fibonacci sequence are relatively prime.   My attempt:    We have $f_1 = 1, f_2 = 1, f_3 = 2, \dots$, so obviously $\gcd(f_1, f_2) = 1$. Suppose that $\gcd(f_n, f_{n+1}) = 1$; we will show that $\gcd(f_{n+1}, f_{n+2}) = 1$. Consider $\gcd(f_{n+1}, f_{n+2}) = \gcd(f_{n+1}, f_{n+1} + f_n)$ because $f_{n+2} = f_{n+1} + f_n.$ Then $\gcd(f_{n+1}, f_{n+1} + f_n) = \gcd(f_{n+1}, f_{n}) = 1$ (gcd property).  Hence, $\gcd(f_n, f_{n+1}) = 1$ for all $n &gt; 0$.  Am I on the right track? Any feedback would be greatly appreciated.  Thanks, How to make a file (e.g. a .sh script) executable, so it can be run from a terminal Probably a really simple one this - I'm starting out with C# and need to add values to an array, for example:  int[] terms;  for(int runs = 0; runs &lt; 400; runs++) {     terms[] = runs; }   For those who have used PHP, here's what I'm trying to do in C#:  $arr = array(); for ($i = 0; $i &lt; 10; $i++) {     $arr[] = $i; } How can I delete an account on a specific site? My account has 20 (!) sites associated to it.  I only use it for 3-5 sites.  To reduce clutter, I would like to delete my profiles that I do not use much.  I have went through , but when I enter the profile URL as:  http://russian.stackexchange.com/users/5772/alex   it gives the error message:     oops! this isn't a link to a valid profile   What procedure do I go through to delete my profile only on certain sites? override at runtime __setattr__ Is there a way to transfer money and items from iOS to Android? I've been playing Jetpack Joyride on iOS, and I just got an Android device. Can I move my things, such as money and unlocks, from my iOS device to Android? iOS: How to change app language programmatically WITHOUT restarting the app? AuraClientService.postProcess: error in processing [RangeError: Maximum call stack size exceeded] - Client Side Action I have a lightning component that is using ui:inputSelect.   The component code:              &lt;div class="slds-form-element__control"&gt;                 &lt;ui:inputSelect class="slds-select" aura:id="quals" change="{!c.getPreCodes}"&gt;                     &lt;aura:iteration items="{!v.questions}" var="q"&gt;                          &lt;ui:inputSelectOption aura:id="question" text="{!q.Id}" label="{!q.Question_Text__c}"/&gt;                     &lt;/aura:iteration&gt;                 &lt;/ui:inputSelect&gt;             &lt;/div&gt;   The controller:      getPreCodes : function(cmp, event, helper){         var selected = cmp.find("quals").get("v.value");          var action = cmp.get("c.getPreCodes");         action.setParams({             "qualID" : selected         });         action.setCallback(this, function(response){             var state = response.getState();             if (state === "SUCCESS") {                  console.log('precodes: ', response.getReturnValue());              }         });          $A.enqueueAction(action);     }   The basic logic is that the uiSelect is dynamically populated on init (this works fine). When a user makes a selection the "change" parameter function is fired to retrieve a list from the database and pass it to another component.   However, the change function {!c.getPreCodes} seems to be called recursively causing a      "Something has gone wrong. AuraClientService.postProcess: error in   processing [RangeError: Maximum call stack size exceeded] . Please try   again."   error. If I comment out the enqueueAction the function only runs once (nothing is called recursively) so I suspect it has something to do with the controller call. Quarks in a hadron: where does the mass come from? What's better to use in PHP for appending an array member,  $array[] = $value;   or  array_push($array, $value);   ?  Though the manual says you're better off to avoid a function call, I've also read $array[] is much slower than array_push(). What are some clarifications or benchmarks? Is there any way to make a method with more than one instance? I've noticed this on occasion with a variety of applications. I've often thought it was because the output was cancelled early (ctrl+c, for example) or something similar, and zsh is filling in a new line character. But now curiosity has gotten the best of me, since it doesn't seem to do this in bash.   zsh    bash    The Sequence program is something I pulled from a book while reading on Java certifications and just wanted to see if it would compile and run. I did notice that it does not use the println() method from the System.out package/class. Instead it uses plain old print().  Is the lack of a new line character the reason I get this symbol?