I am relatively new to LateX, and am a bit confused with the \begin{equation} command. I use $$...$$ to enter the math mode in a new environment. However, I find that when I combine $$..$$ with \begin{equation}, the equation is not centered. However, for some reason, \begin{equation} seems to work with the ..\ method of writing equations better. Is there any reason for this? Thanks!
I once learnt, that it is uncool to use $$, but why is that? Why does \[ not act as a abbreviation to \begin{align}? I noticed, that there is a difference, since one cannot use &amp; and \\ inside a block started with \[.
Can't find Corruption Is it appropriate to put a comma before "which"? On Server Fault, we were discussing  in chat. There are non-native speakers in the chat and somehow we got on the topic of grammar and proper use of commas.   The sentence of interest:     I have just got a new Ubuntu LAMP VPS server, which I am trying to configure.   Some people thought that the comma was superfluous. Others did not.   When is it appropriate for a comma to preceed the word which, and is the example sentence above correct?    Also, before you vote to close this question as a dupe of , please consider the differences between the two questions and ignore the similarity of the titles. The OP's example sentence was restructured in that question instead of addressing the proper use of commas. Diagnostics for logistic regression? iOS 9.2 double touch fingerprint sensor I was messing around this morning and found that if I quickly double touch (NOT double-click) the fingerprint sensor on my iPhone 6S+ (iOS 9.2), the current app (or the home screen) slides down to the bottom half-screen, showing a blurred home screen background on the top half.  It doesn't matter if I use a finger that the sensor recognizes.  It goes back to normal after about 8 seconds or if you double-touch again, but I can't figure out what this mode is for??  Is this a bug or some hidden feature I haven't heard about yet? Which stats / gear can get the best our of wasteland exploration? Trying to get the best out of wasteland exploration here! I get it that you'd obviously need a big gun for survival out there, but apart from that, I'm wondering which stats / gear can get the best out of exploration.    How would you choose who to send out exploring from your dwellers. Which stat is the most important to choose explorers? Strength or endurance maybe? Does it make much of a difference equipping combat outfits? If so, whats the most important stat to boost through outfits? I found  where a technician with a lot of experience seems to clean his PCBs using water, and then he just puts it in the sun so it can dry. I was just amazed to see the circuit keeps working as intended, he says he keeps doing this all the time and several times, however I doubt this is normal.  Which leave me some questions:   What are the recommended methods to clean huge PCBs like the one in the video? What happens to charged capacitors when water shorts them out? Solder joints do not get any kind of corrosion? Is the antisolder mask waterproof? Are omniscience and omnipotence mutually inconsistent? I see this in The God Delusion by Richard Dawkins:       If God is omniscient, he must already know how he is going to intervene to change the course of history using his omnipotence. But   that means he can't change his mind about his intervention, which   means he is not omnipotent.   Is this argument sound? What is a NumberFormatException and how can I fix it? How do HttpOnly cookies work with AJAX requests? JavaScript needs access to cookies if AJAX is used on a site with access restrictions based on cookies. Will HttpOnly cookies work on an AJAX site?   Edit: Microsoft created a way to prevent XSS attacks by disallowing JavaScript access to cookies if HttpOnly is specified. FireFox later adopted this. So my question is: If you are using AJAX on a site, like StackOverflow, are Http-Only cookies an option?  Edit 2: Question 2. If the purpose of HttpOnly is to prevent JavaScript access to cookies, and you can still retrieve the cookies via JavaScript through the XmlHttpRequest Object, what is the point of HttpOnly?  Edit 3: Here is a quote from Wikipedia:     When the browser receives such a cookie, it is supposed to use it as usual in the following HTTP exchanges, but not to make it visible to client-side scripts.[32] The HttpOnly flag is not part of any standard, and is not implemented in all browsers. Note that there is currently no prevention of reading or writing the session cookie via a XMLHTTPRequest. [33].   I understand that document.cookie is blocked when you use HttpOnly. But it seems that you can still read cookie values in the XMLHttpRequest object, allowing for XSS. How does HttpOnly make you any safer than? By making cookies essentially read only?   In your example, I cannot write to your document.cookie, but I can still steal your cookie and post it to my domain using the XMLHttpRequest object.  &lt;script type="text/javascript"&gt;     var req = null;     try { req = new XMLHttpRequest(); } catch(e) {}     if (!req) try { req = new ActiveXObject("Msxml2.XMLHTTP"); } catch(e) {}     if (!req) try { req = new ActiveXObject("Microsoft.XMLHTTP"); } catch(e) {}     req.open('GET', 'http://stackoverflow.com/', false);     req.send(null);     alert(req.getAllResponseHeaders()); &lt;/script&gt;   Edit 4: Sorry, I meant that you could send the XMLHttpRequest to the StackOverflow domain, and then save the result of getAllResponseHeaders() to a string, regex out the cookie, and then post that to an external domain. It appears that Wikipedia and ha.ckers concur with me on this one, but I would love be re-educated...  Final Edit: Ahh, apparently both sites are wrong, this is actually a . IE6 &amp; 7 are actually the only browsers that currently fully support HttpOnly.  To reiterate everything I've learned:   HttpOnly restricts all access to document.cookie in IE7 &amp; and FireFox (not sure about other browsers) HttpOnly removes cookie information from the response headers in XMLHttpObject.getAllResponseHeaders() in IE7. XMLHttpObjects may only be submitted to the domain they originated from, so there is no cross-domain posting of the cookies.   edit: This information is likely no longer up to date. How do I run .sh scripts? Whenever I open a .sh file, it opens it in gedit instead of the terminal. I can't find any option similar to Right Click → Open With → Other Application... → Terminal.  How do I open this file in the terminal? In Section 9 of Chapter 6, Lang constructrs for the first time a non-abelian extension as a splitting field $K$ of $X^n - a$ when the ground field $k$ does not contain primitive $n$-th roots of unity $\zeta_n$. He constructs an injective homomorphism of the Galois group $G(K/k)$ into the matrices of type   $$ M=\begin{pmatrix} 1 &amp; 0 \\ b &amp; d \end{pmatrix} \quad \hbox{with}\ b\in \mathbb{Z}/n\mathbb{Z} \quad \hbox{and} \quad d\in (\mathbb{Z}/n\mathbb{Z})^*,$$  where $(\mathbb{Z}/n\mathbb{Z})^*$ the multiplicative group of integers modulo n.  Afterwards, he states in Theorem 9.4 that if $n$ is an odd positive integer prime to the characterstic and $[k(\zeta_n):k]=\varphi(n)$ (i.e, the primitive roots of unity are not in the ground field), then the above homomorphism is, in fact, an isomorphism. Besides, the commutator group is $G(K/k(\zeta_n))$, so $k(\zeta_n)$ is the maximal abelian subextension of $K$.  Lang proves the theorem inductively over the number of prime factors in the decomposition of $n$ (i.e. if $n=9= 3\cdot 3$, then the proof "would take two steps" to complete). The induction start with $n$ prime is written clearly and I agree with it.   Now come the questions:   I do not understand Lang's argument to prove that the commutator subgroup is the subgroup $\begin{pmatrix}1 &amp; 0 \\ b &amp; 1\end{pmatrix}$, $b\in \mathbb{Z}/n\mathbb{Z}$. What is this projection on the diagonal and how does one see that the factor group of the commutator group is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^*$?   By the way, direct computation of $x\, y\, x^{-1}\, y^{-1}$ for two arbitrary matrices $x=\begin{pmatrix} 1 &amp; 0 \\ b &amp; d \end{pmatrix}$, $y=\begin{pmatrix} 1 &amp; 0 \\ c &amp; e \end{pmatrix}$ yields a matrix $\begin{pmatrix} 1 &amp; 0 \\ b(1-e)+c(d-1) &amp; 1 \end{pmatrix}$. As $n$ is odd, $2 \in (\mathbb{Z}/n\mathbb{Z})^*$, so taking $e=1$ and $d=2$, we indeed reconstruct the above-mentioned group.   The second (inductive) part of the proof is a complete mystery to me with virtually every sentence being a Theorem which needs an extra proof. Can someone give me some hints?   2.a. Lang writes $n=p\,m$ and takes $\alpha$ to be the root of $X^n-a$. It is clear that $\beta=\alpha^p$ is a root of $X^m -a$. Now comes the argument "By induction we can apply the theorem to $X^m-a$". I agree that we can apply it here but the result is that $K=k(\beta,\zeta_m)$ is the splitting field of $X^m-a$. In the diagram below, there appears $k(\beta,\zeta_n)$ out of nowhere.   2.b. Now Lang proceeds to apply the theorem on $X^p-\beta$ over $k(\beta)$. Bergman's "Companion" reminds to check that $[k(\beta,\zeta_p):k(\beta)]=\varphi(p)$ before applying the theorem and gives a hint "$[k(\zeta_p):k]$ and $[k(\beta):k]$ are relatively prime". I disagree with this hint. Take $n=21= m p$ where $m=3$ and $p=7$. Then $[k(\zeta_p):k]=\varphi(p)=p-1=6$ but $[k(\beta):k]=m=3$ and both are not co-prime. Can we apply the theorem here at all? Or if so what is the argument? If Bergman's argument is valid, where is the fault in my logic?  2.c. Even assuming that we can apply the theorem in 2.b., then we can say something about $k(\alpha, \zeta_p)$ and not $k(\alpha,\zeta_n)$. How do we proceed to checking everything Lang states in the last section of the proof (basically, everything after the diagram)?  EDIT: I have found the essentially same question Allow linking (suggested) edits to one comment As the feature-request  elaborates on, some suggested edits seem too radical without the context of comments they are based on.  I fully endorse this request and would like to take it one step further:  Add the option to "suggest* edit based on this comment" next to comments, which would specifically highlight that comment below the suggested edit, such that edits that otherwise would be deemed "too radical" at first sight can be put into a better context. On approval, said comment would also become linked-to by the revision comment.  Since it makes sense to upvote such a comment due to it providing information justifying an edit, that link could replace the flagging-flag (which is anyway hidden after an upvote) with the edit-pen that indicate edited comments, like so:        (mockup based on a  , coming to cinemas never)  In fact, for the editor's convenience, the clicked-on base-comment might also be added to the bottom of the post and preselected (e.g. for one-key deletion or conversion to a quote)    * The "suggest" would not be shown for users with the required edit privilege (&lt;2k rep or OP), yet this option would allow to link an edit to a helpful comment and include aforementioned preselect-insertion of the comment at the post's bottom for convenience Intuition for groups This is quite a non-standard question, certainly for mathematics, though I believe it is no less important (for me and my peers, i.e. grads).  The course I am reading so far introduced us to Groups, rings, fields, etc. in the first year, progressing to characters, reps etc in the latter years.  However, even with representation theory I still don't feel like I have a good intuition.  How do you gain intuition in such an area. I know there will be a response of doing examples, but this doesn't really help - I've done lots already.  Am I doomed. A question on limsup Double vs. BigDecimal? Accelerating particles to speeds infinitesimally close to the speed of light? How can I efficiently drive an LED? I understand that I can not connect an LED directly to a battery because it will draw too much current. Thus, there must be something else in the circuit to limit the current. What options are there? Are some methods more efficient than others? I'm having troubles with rendering my datatable correctly. There are actions added but it seems the datatable is not high enough to view it correctly. Is it something I'm doing wrong? Or is it a bug?    This is the code for the datatable  &lt;div class="slds-text-heading_medium"&gt;Search results&lt;/div&gt; &lt;lightning:datatable data="{! v.persondata }"                      columns="{! v.personcolumns }"                      keyField="id"                     onrowselection="{! c.getSelectedPerson }"                     hideCheckboxColumn="true"  /&gt;   Here is the code for initializing the columns  //init datatable component.set('v.personcolumns', [             {label: 'Last Name', fieldName: 'LastName', type: 'text'},             {label: 'First Name', fieldName: 'FirstName', type: 'text'},             {label: 'Mailing Zip/ Postal Code', fieldName: 'PersonMailingPostalCode', type: 'text'},             {label: 'Mailing City', fieldName: 'PersonMailingCity', type: 'text'},             {label: 'Billing Street', fieldName: 'PersonMailingStreet', type: 'text'},             {label: 'Email', fieldName: 'PersonEmail', type: 'text'},             {label: 'Primary Phone', fieldName: 'PersonHomePhone', type: 'text'},             {label: 'Secondary Phone', fieldName: 'PersonOtherPhone', type: 'text'},             {type: 'action', typeAttributes: { rowActions: actions}}         ]);   And the code for adding data (the response is a list of Accounts)  component.set('v.persondata',response.getReturnValue()); Woman planning to visit me from Ukraine to the US applied for a visa and such, claims to have paid for the visa but needs 500 dollars more for the airfare from Ukraine to the US. She says strict rules apply that the airfare cannot be purchased by US person, and must be purchased in Ukraine by person applying for the visa because of terrorist issues, etc..   Is there truth to such? How do I install minecraft 1.9 on mac?