Prove that there exists $n_0 \in \mathbb{Z}$ such that $I = \{kn_0 | k \in \mathbb{Z}\}$ given that $I$ satisfies the following properties
Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$
The use of nominative "whom" (as in “persons whom it is foreseeable are likely to...”) From  page 48 of Law: A Very Short Introduction, by Raymond Wacks:     In other words, you owe a duty to persons whom it is foreseeable are likely to be harmed by your conduct.   To try to parse this, temporarily overlook 'it is foreseeable'. Then persons is the subject of the verb phrase are likely to be harmed. So , should the relative pronoun for persons be who instead? Is my parse wrong?   Are the (deleted) comments right: that this style predominates in law texts? If this is correct grammar, what is the general rule? Is it commonly used in other contexts? The Raspberry Pi B+ models have a protection circuit between the USB connector and the 5V net on the board.  They recommend putting a similar protection circuity on a Pi HAT before 'backpowering' the pi through its GPIO header along with a polyfuse.  I understand why this is the recommendation, but I'd like to understand more about how this circuit works.  I did some searching before posting this question and found information on using a MOSFET as a low voltage drop diode, but they all had the gate wired directly to ground without the pair of PNP's and the resistors.  What are they doing for this circuit?  Also, is this primarily using the body diode?  In which case, what's the relevant info on the datasheet that qualifies DMG2305UX for this application?  In the other circuits I found, it appeared low Rdson and Vgsth compatible with the circuit seemed like the relevant characteristics. Automatically building a database of forward and backward citations My goal is to produce a graph showing the linkages (and lack there of) between several fields that share a common subproblem by showing who cites whom.  There are many databases that show citations between papers: , , and Google Scholar.  However none allows me to download even part of their database.  Is there some database I have overlooked?  Has someone written a scrapper for one of these websites?    For reference, the fields I'm considering are Mechanics, Magnetic Resonance Imaging, Signal Processing, Geophysics, and several others. How do you send a HEAD HTTP request in Python 2? What I'm trying to do here is get the headers of a given URL so I can determine the MIME type. I want to be able to see if http://somedomain/foo/ will return an HTML document or a JPEG image for example. Thus, I need to figure out how to send a HEAD request so that I can read the MIME type without having to download the content. Does anyone know of an easy way of doing this? Using the Fitch system, how do I prove $((p \implies q) \implies p)\implies p$? Create an object with dynamic property names Help with Spivak Calculus Ch3 Problem 6a Yet again I find myself stuck on a Spivak question. This time it is simply the question that isn't clear to me. It states:  If $x_1, ..., x_n$ are distinct numbers, find a polynomial function $f_i$, of degree $n-1$ which is 1 at $x_i$ and 0 at $x_j$ for $j\neq i$. Hint: the product of all $(x-x_j)$ for $j\neq i$, is 0 at $x_j$ if $j\neq i$. This product is usually denoted by: $$\prod_{\begin{smallmatrix}{j=1}\\ {j\neq i} \end{smallmatrix}}^n(x-x_j)$$  To be honest I don't know where to begin. I understand how to create a polynomial equation for a given set of results but this is quite strange. I'm not sure exactly what the author is expecting.  The answer book shows:  $$f_i(x)={\prod_{\begin{smallmatrix}{j=1}\\ {j\neq i} \end{smallmatrix}}^n(x-x_j)}/{\prod_{\begin{smallmatrix}{j=1}\\ {j\neq i} \end{smallmatrix}}^n(x_i-x_j)}$$  Which I have expanded out just fine but don't know what to do with. test class with Batch and Callout I have a problem with FeedItems in my Test class, it's very weird behaviour Does someone explain to me why this happens?  Callout Class  public with sharing class Callout {  public static void runProccess(Case[] records) {     Database.executeBatch(new CalloutBatch(records), 10); }  public static void send(Case[] records) {     Case[] toUpdate = new Case[]{};       for (case record : records) {         HttpRequest req = new HttpRequest();         req.setEndpoint('callout:test/SOME_URL');         req.setMethod('POST');         req.setHeader('Content-Type', 'application/json');         req.setBody('');         Http http = new Http();         HTTPResponse res = http.send(req);          Map&lt;String, Object&gt; response = (Map&lt;String, Object&gt;) JSON.deserializeUntyped(res.getBody());          toUpdate.add(new Case(Id = (Id) response.get('Id'), Status = (String) response.get('Status')));     }      update toUpdate;  } }   CalloutBatch Class  global with sharing class CalloutBatch implements Database.Batchable&lt;Case&gt;, Database.AllowsCallouts{  private final Case[] records;  global CalloutBatch (Case[] records) {     this.records = records; }  global Iterable&lt;Case&gt; start(Database.BatchableContext BC) {     return records; }  global void execute(Database.BatchableContext BC, Case[] scope) {     Callout.send(scope); }  global void finish(Database.BatchableContext BC) { } }   Test Class  @IsTest  private class CalloutTest { @IsTest static void shouldReaciveData() {     String status = 'Closed';     Case caseRecord = new Case(Type = 'Web', Status = 'New');     insert caseRecord;      //insert new FeedItem(ParentId = caseRecord.Id, Visibility='AllUsers', Type = 'TextPost', Body = '');      System.assertEquals(0, [SELECT Count() FROM Case WHERE Id = :caseRecord.Id AND Status = :status]);      Test.startTest();     Test.setMock(HttpCalloutMock.class, new CalloutTestMock(caseRecord.Id, status));     Callout.runProccess(new Case[]{caseRecord});     Test.stopTest();      System.assertEquals(1, [SELECT Count() FROM Case WHERE Id = :caseRecord.Id AND Status = :status]); } }   Mock Class  public with sharing class CalloutTestMock implements HttpCalloutMock {  private Id recordId; private String status;  public CalloutTestMock(Id recordId, String status) {     this.recordId = recordId;     this.status = status; }  public HTTPResponse respond(HTTPRequest req) {     HttpResponse response = new HttpResponse();      response.setStatusCode(200);     response.setBody('{"Id": "' + recordId + '", "Status": "' + status + '"}');      return response; } }   This code works, but when I uncomment insert FeedItem in Test Class I receive an error message:     System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out   Class.Callout.send: line 23, column 1   Class.CalloutBatch.execute: line 20, column 1   FeedItem doesn't have an impact on this functionality, (I don't have any Process Builders, triggers etc. on FeedItem object) How to suppress Terminated message after killing in bash? Update: Failed to lock /var/cache/apt/archives/lock What does protection from [quality] do? Example 1) Let's say I have some random creature with protection from white, and my opponent plays , a white card with the effect "Destroy all creatures." Can my creature with protection from white be destroyed by a white card that isn't actually targeting it specifically?  Example 2) I was playing a match with a friend where I had  on the field against his 3 zombies. Is he able to declare his zombies as blockers even though Elite Inquisitor has "protection" from zombies?  If he can declare them as blockers, does protection mean that no damage is dealt by the zombies to  Elite Inquisitor? Prove that $2xy\mid x^2+y^2-x$ implies $x$ is a perfect square. Prove that $2xy\mid x^2+y^2-x$ implies $x$ is a perfect square. My work: $2xy\mid x^2+y^2-x \implies x^2+y^2-x=2xy\cdot k$ So,$x^2+y^2+2xy-x=(x+y)^2-x=2xy \cdot (k+1)$ And,$x^2+y^2-2xy-x=(x-y)^2-x=2xy \cdot (k-1)$ I found that for $x,y$ both odd, no solution exists. For $x$ even, and $y$ odd,no solution exists. Solution exists only for $x$ odd, $y$ even and $x$ even and $y$ even solution exists. Cannot do anything more. Please help! How should we understand the teletransportation thought experiment? True or false: if $f'(c)<0$, then $f$ is concave down at $x=c$? How can I determine the following statement is true or false?  If $f'(c)&lt;0$, then $f$ is concave down at $x=c$ ? How precisely do I need to post close to 7:00 (somewhere) to get hero of time? Timezones... are complicated. We have full hour offsets, half hour offsets...  And time keeping isn't always accurate. So does the Hero Of Time hat assume full or half hour offsets (or literally any timezone) and how close to 7:00 at somewhere do I have to post for my hat? Move files from multiple folders all into parent directory with command prompt I have multiple .rar files in multiple folders like this:    C:\Docs\Folder1\rarfile1-1.rar C:\Docs\Folder1\rarfile1-2.rar C:\Docs\Folder1\rarfile1-3.rar    C:\Docs\Folder2\rarfile2-1.rar C:\Docs\Folder2\rarfile2-2.rar C:\Docs\Folder2\rarfile2-3.rar    C:\Docs\Folder3\rarfile3-1.rar C:\Docs\Folder3\rarfile3-2.rar C:\Docs\Folder3\rarfile3-3.rar     I want to move all of the .rar files to the parent directory 'C:\Docs'. I have a lot more than 3 folders, so I was thinking of making a batch file or something. What would be the commands to do this? Thanks How often did Gandalf return to Hobbiton after the events of The Hobbit? It has been a while since I read the Lord of the Rings novels (don't have a copy at the moment) but I seem to recall he went to Hobbiton at least once, as the young hobbits were always excited to see him.   Did he travel there often? This is the code that's giving me trouble.   f = Frame(root, width=1000, bg="blue") f.pack(fill=X, expand=True)  l = Label(f, text="hi", width=10, bg="red", fg="white") l.pack()   If I comment out the lines with the Label, the Frame displays with the right width.  However, adding the Label seems to shrink the Frame down to the Label's size.  Is there a way to prevent that from happening? I got my dad a Logitech LS21 system for Christmas. It's a 2.1 system, but it's recognized in Windows 8 as a 2.0 system (it says there is no subwoofer connected).    How would I go about getting the sub recognized?  It plays some sound now, but it's so quiet you can't hear it without putting your ear right next to it. Insert only half a pdf page I'd like to include papers of mine which can easily be done with the pdfpages packages:  \usepackage{pdfpages} \includepdf[pages=-]{mypaper.pdf}   However, my thesis has an A5 format. Which means that each page is "half" an A4 page. Basically, I'd like two include each PDF page on two pages, like illustrated below:    While I found how to rotate the PDF using the angle parameter, I did not find how to split it...