How can we obtain $\sum_{l=1}^{m}(-1)^{l+1} \binom{m}{l} \equiv 1\pmod{p}$
Proving $\sum_{k=0}^n(-1)^k\binom nk=0$
I am a third year (starting fourth year in the fall) PhD student in mathematics. I've passed all qualifying exams and am currently doing research. As far as I can tell, I am not doing poorly. I have the good fortune of having a great advisor, being in a very supportive department, and having friends and family who genuinely care about my success.   The fact is research is hard. It appears to consist primarily of staring at a problem for days and days and days without getting anywhere. Sometimes, rarely, I do figure something out and that feels wonderful, but the overwhelming majority of my time appears to be spent banging my head against a mostly figurative wall. I am not complaining about the material being hard, and I am not averse to putting in hard work, but I get frequently discouraged when I realize the vast volumes of mathematics that I yet know nothing about (and probably never will). It's very hard to quantify progress - in particular, there are too few tangible returns after too many hours worked. I find myself thinking along the lines of "Oh, if only someone actually smart were thinking about this problem they would have solved it in moments" and so on.   I've talked about this to some number of people; here is some advice I have received:   Take a day off. Putting in hours upon hours of trying things doesn't magically lead to a solution, particularly if the brain is tired and just wants to sleep. Have a hobby. Since math research doesn't exactly provide instant gratification, a hobby might provide some instead.  Talk to other graduate students. Realize that many graduate students go through this.   I'm interested to know how other folks have dealt with being discouraged as a graduate student. Does it get better with time and experience? Is this a sign that research is not for me and that I should seriously consider a life outside academia? Dynamically adding a form to a Django formset I want to dynamically add new forms to a Django formset, so that when the user clicks an &quot;add&quot; button it runs JavaScript that adds a new form (which is part of the formset) to the page. A diagram which is not the torus At page 17 of Munkres' Elements of Algebraic topology it says, referring to fig. 3.6, that "the diagram does not determine [the torus]. It does more than paste opposite edges together":   Is it so? I tried to glue them together, but end up with a normal torus. Typing the ZIP code doesn't yield the company page I need to create an edit component for an SObject from which I do not know the objectApiName at design time. I only know the Id of the SObject. objectApiName could be any valid SObject-name.  Now I want to use lightning:inputField   This needs to be wrapped by lightning:recordEditForm  In case I hardcode it to "Order", provide a vaild Order Id and let the controller load the order and assign it to the Parent-attribute, this works flawlessly:  &lt;aura:attribute name="recordId" type="String" default="" /&gt; &lt;aura:attribute name="Parent" type="SObject" default="{}" /&gt; &lt;aura:attribute name="ParentType" type="String" default="" /&gt; &lt;b&gt;{!v.ParentType}({!v.recordId})&lt;/b&gt; &lt;br/&gt; &lt;lightning:recordEditForm recordId="{!v.recordId}"  objectApiName="Order"&gt;     &lt;lightning:messages /&gt;     &lt;lightning:inputField fieldName="Name" /&gt;     &lt;lightning:button class="slds-m-top_small" variant="brand" type="submit" name="update" label="Update" /&gt; &lt;/lightning:recordEditForm&gt;   It looks like below and even the Update-button works as expected and saves the record:    But what I really want is  &lt;lightning:recordEditForm recordId="{!v.recordId}"  objectApiName="{!v.ParentType}"&gt;   Note the {!v.ParentType} instead of the hardcoded "Order"  As an great unfortunate, this only collapses to that error-popup    In the screenshot you see, that the ParentType is correctly bound to v.ParentType (right above the button) which a I also verify in the console  Suspicion  I set the ParentType-attribute as a result of APEX call (asynchronously). This seems to be too late for the lightning:recordEditForm. It tries to render with objectApiName at a time where it is still blank, hence it crashes during initialization...  Question  Is there a clean way to avoid this and get something to fly with an dynamic objectApiName?  Trying to use $A.createComponents()  As @sfdcfox suggested I jump on the $A.createComponents() which turned indeed out to be tricky, exactly as sfdcfox has predicted. Im following this .  Added a button to the markup    Added a handler to the js-controller:  makeEdit : function(cmp, evt, hlp) {      console.log('ParentId :: ', cmp.get("v.ParentId") );      console.log('ParentType :: ', cmp.get("v.ParentType") );      $A.createComponents([             ["lightning:recordEditForm",{                 "recordId"          : cmp.get("v.ParentId"),                 "objectApiName"     : cmp.get("v.ParentType"),             }],             ["lightning:inputField",{                 "fieldName" : "Name"             }]         ]         ,function(components, status, errorMessage){             if (status === "SUCCESS") {                 var wrapper         = components[0];                 var inner           = components[1];                 wrapper.set("v.body", inner);             }             else if (status === "INCOMPLETE") {                 console.log("No response from server or client is offline.")                 // Show offline error             }             else if (status === "ERROR") {                 console.log("Error: " + errorMessage);                 // Show error message             }         }     ); },   Now I can press the button - but nothing - I repeat: n o t h i n g - gets rendered at all. What is my fault here? Note that the console.log get output and the attribute are holding good values. Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]\in\mathbb{R}$ Let $[z_1,z_2,z_3,z_4]$ denote the  of the complex numbers $z_1,z_2,z_3,z_4\in \mathbb{C}$. Show that the distinct points $z_1,z_2,z_3,z_4\in\widehat{\mathbb{C}}$ lie on a generalized circle if and only if $[z_1,z_2,z_3,z_4]\in\mathbb{R}$.    I saw this statement claimed in , and wondered how one would go about showing it. I know of some facts we could work with:   First, the definition: A  in $\mathbb{C}$ is either a circle or a line in $\mathbb{C}$. I know that a Mobius transformation maps generalized circles to generalized circles. I know that if $f$ is a Mobius transformation and $z_j'=f(z_j)$, $j=1,2,3,4$, then $[z_1',z_2',z_3',z_4']=[z_1,z_2,z_3,z_4]$.   I have posted a potential proof as an answer below. Please let me know if you think it's complete. Thank you! Ownership of content in Community Wiki posts Several ways to prove that $\sum\limits^\infty_{n=1}\left(1-\frac1{\sqrt{n}}\right)^n$ converges Does iCloud Drive automatically download all files? I am using OS X 10.10 with iCloud Drive. I am thinking of upgrading to a bigger storage plan (at the moment: 5GB).   I am running this Version on a MacBook Air with 128GB.      If I buy the 500GB package, does it mean that I fill up my local hard   drive automatically when I upload things to the iCloud Drive?   iTunes has the Option of stream the music i didn't download yet.   How does iCloud Drive behave? Is there any Online Document? How do I safely investigate a USB stick found in the parking lot at work? I work at an embedded software company. This morning I found a USB stick in the parking lot in front of the building. With all the stories of "dropped USB stick attacks" in mind, I'm obviously not going to just plug it into my laptop. OTOH, I'm curious to know whether this was actually an attempt to compromise our systems, or it's really just an innocent case of somebody accidentally losing a USB stick. How do I safely inspect the USB stick without risking exposure?  I'm worried not just about malware and crafted file system images; there's also stuff like power surge attacks:.  EDIT: Many of the answers seem to assume I want to keep the drive and use it afterwards. I have no interest in that at all, I know USB sticks are cheap, and that it wouldn't be mine to keep anyway. I only want to know whether this was indeed a semi-targeted attack, partly out of curiosity whether this actually happens in real life and not just in security papers, but also so that I could warn my coworkers.  I want to know how I would figure out whether the stick contains malware. And that's not just a matter of looking at the drive contents and seeing a suspicious autorun.inf or a carefully crafted corrupt file system - I very much also want a way to inspect the firmware. I sort-of expected that there were tools for extracting that and comparing to known-good or known-bad binaries. Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen Suppose I have a set of 100 integers. I randomly choose 10 of those, make a note of which ones I selected, and repeat the process. What is the expected number of times this process must be repeated before I see all 100 integers?  I'm also greatly interested in how this is calculated as I'm trying to increase the expected number of times this process is repeated by changing the set sizes. How can I make a plane's edges taper inwards? How can I deform one edge or multiple edges of a plane or cube so they are tapered in and rounded equally? Layover at Narita, Japan. Do I need a transit visa? I will be transferring between two international flights at Tokyo Narita airport. I'm scheduled to depart the same day I arrive. I am planning to stay within the airport during the layover. Does anyone know if I will require a transit visa at Narita? When do "well" and "good" mean the same? Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic. Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more explicit way or is this explicit enough?    Problem Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.  Solution Recall that isomorphism preserves order of elements and hence if there exists an isomorphism from $\phi: \mathbb{C}-\{0\} \mapsto \mathbb{R}-\{0\}$, then $x \in \mathbb{C} - \{0\}$, and $\phi(x) \in \mathbb{R} - \{0\}$, then $\vert x \vert = \vert \phi(x) \vert$. Now note that the element $i \in \mathbb{C} - \{0\}$ has order $4$. However, no element in $\mathbb{R}-\{0\}$ has order $4$. Hence, no isomorphism can exist. Hence, the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.    Thanks What makes matter travel along geodesics? The relativistic explanation of gravity is geometric, the motion of a body in a field of space-time distortion can be described as being at rest and travelling along a geodesic of that field, but why does it travel along that geodesic? What causes it to move? Some icons are correct, and some are whack. Turns out it's not just the hot questions list, but the whole icon set. I am trying to use the Directory.GetFiles() method to retrieve a list of files of multiple types, such as mp3's and jpg's.  I have tried both of the following with no luck:  Directory.GetFiles("C:\\path", "*.mp3|*.jpg", SearchOption.AllDirectories); Directory.GetFiles("C:\\path", "*.mp3;*.jpg", SearchOption.AllDirectories);   Is there a way to do this in one call? Can't understand how to draw it. Repairing corrupt MP4 I took a video from my Android phone (Sony Xperia P, if you need to know), and before I could stop recording it, the phone memory got full and the phone got hanged too. I rebooted the phone, the video was there, but I couldn't play it from any media player on Android. It's in mp4 format.    So I tried to open it on PC and Mac with different video player, like VLC, quick Time, Windows Media Player etc., but none of them can play it. I know that the video is corrupted, but I believe it's repairable, since it got more than a GB in size. Is there anything that I can try  to repair the video file? The video is really important.