$H = \left\{ id, (12)(34), (13)(24), (14)(23)\right\}$ is a subgroup of $A_4$.  A normal subgroup is a subgroup where the right and left cosets are the same.  To show that $H$ is a normal subgroup would I have to compute every single coset and show that the left and right cosets are the same?  Is there a clever way to do this problem?
Let H={e,(12)(34)} and K ={e, (12)(34), (13)(24), (14)(23)} be subgroups of $ S_4$ where e is identity element. Then which of following is true   H and K are normal subgroups of $S_4$ H is normal in K and K is normal in$ A_4$ H is normal in $A_4$, but not normal in $ S_4$ K is normal in $S_4$ but H is not   How should this question be approached, checking left and right cosets of these subgroup is very time consuming. Checking gh$ g^{-1}$ $\in $ H for every g in $ S_4$ is also time consuming.
How do I backup data (SMS/contacts) from a device with a broken screen? I just broke the screen of my (unrooted) Galaxy S.  How do I get back my SMS and contacts?   PS: I tried Kies but it won't detect my phone for some reason. (Kies works fine with another Galaxy I possess but not this one) Conditionally use 32/64 bit reference when building in Visual Studio Animation Cycles rendering remains incredible slow Inverse of Heine–Cantor theorem How to get ObjectContext to compile? How should I install HPLIP binary/proprietary plugin/driver for my HP printer or printer+scanner? Let's assume that one has some modern HP printer and/or scanner which is  supported by HPLIP package from official repository. Or supported by newer version of HPLIP package (see  about installation).  From official documentation () it is known that device needs proprietary/binary driver or plugin.  How should one install such plugin? Proofs using theorems instead of axioms I'm not sure how to prove these basic theorems in propositional calculus. Instead of using the standard axioms, we're supposed to use:   Deduction Theorem (if $\Phi, \alpha \vdash \beta$ then $\Phi \vdash \alpha \to \beta$),  Reductio (if $\Phi, \alpha \vdash \, $, then $\Phi \vdash \lnot \alpha$), Cut Rule (if $\Phi \vdash \alpha$ and $\Psi, \alpha \vdash \beta$ then $\Phi \cup \Psi \vdash \beta$),  Inconsistency Effect (if $\Phi \vdash \, $, then $\Phi \vdash \beta$ for every formula $\beta$), and  the Principle of Indirect Proof (if $\Phi, \lnot \alpha \vdash \, $, then $\Phi \vdash \alpha$),    as all the axioms can be deduced using these theorems.  I don't really know how to start the proofs without using the axioms:  i) prove that $\lnot(\alpha \to \beta) ⊢ \alpha$  ii) prove that $\lnot\alpha \vdash \alpha \to \beta$  Any suggestions on how to start these proofs or any insight at all would be greatly appreciated!  Thanks! What techniques are available for collapsing (or pooling) many categories to a few, for the purpose of using them as an input (predictor) in a statistical model?    Consider a variable like  (discipline chosen by an undergraduate student). It is unordered and categorical, but it can potentially have dozens of distinct levels. Let's say I want to use major as a predictor in a regression model.  Using these levels as-is for modeling leads to all sorts of issues because there are just so many. A lot of statistical precision would be thrown away to use them, and the results are hard to interpret. We're rarely interested in specific majors -- we're much more likely to be interested in broad categories (subgroups) of majors. But it isn't always clear how to divide up the levels into such higher-level categories, or even how many higher-level categories to use.  For typical data I would be happy to use factor analysis, matrix factorization, or a discrete latent modeling technique. But majors are mutually exclusive categories, so I'm hesitant to exploit their covariance for anything.  Furthermore I don't care about the major categories on their own. I care about producing higher-level categories that are coherent with respect to my regression outcome. In the binary outcome case, that suggests to me something like linear discriminant analysis (LDA) to generate higher-level categories that maximize discriminative performance. But LDA is a limited technique and that feels like dirty data dredging to me. Moreover any continuous solution will be hard to interpret.  Meanwhile something based on covariances, like multiple correspondence analysis (MCA), seems suspect to me in this case because of the inherent dependence among mutually exclusive dummy variables -- they're better suited for studying multiple categorical variables, rather than multiple categories of the same variable.  edit: to be clear, this is about collapsing categories (not selecting them), and the categories are predictors or independent variables. In hindsight, this problem seems like an appropriate time to "regularize 'em all and let God sort 'em out". Glad to see this question is interesting to so many people! Possible Duplicates:             Voting on questions played a big role in the site when we didn't have the option to comment on, close, migrate, flag, or edit questions which did not fit the site or required modification to be useful and usable.  As it is, question votes are not much more than a popularity contest, and a lot of bandwidth is taken up on Meta regarding question votes.  Notably:        In most cases a downvote is being given where a good edit should be used instead, or a comment indicating that the question user needs to provide more information.  People are also downvoting in the case of duplicates (or suspected duplicates) when they don't want to do the work of finding the duplicate.  There is no need to downvote to 'alert' other users that a question isn't worth spending time on - if it's bad enough that it can't be fixed, and so bad that you want to be sure others avoid it then it's not worth keeping open.  These are all situations where the person downvoting should be taking a different action.  , and several questions have thrown ideas at how to increase question upvoting artificially, and a  was even implemented to encourage this behavior.  Given all this, I am now convinced that voting on questions is an unnecessary distraction.  Voting no longer provides any useful information about a question, and there are more than adequate means of fixing or removing questions that need work.  Voting makes more sense for sites like Meta where questions aren't really questions, but suggestions and discussions.  For purely Q/A sites such as SO, SF, SU, the voting is superfluous.  The only two possible positive things voting now provides are:   A way for users at lower rep levels to have a voice when they are unable to perform many of the other functions that are needed to fix a question (I contend this is useless, but perhaps they need some sort of outlet) A 'hook' to get new users into the reputation game (which fails frequently because questions aren't getting upvotes, especially questions from new users)   If that is truly the case, then we should consider removing question voting altogether, or removing the effect of question votes on reputation.   Am I wrong? What obvious and explicitly important role does question voting play in our community? Are you allowed to stack Divine Smite and Eldritch Smite on the same attack? I recently came across an IE7 only bug that I thought I'd share so when I come to this site 6 months from now to figure out the same thing, I'll have it on hand.   I believe the easiest way to recreate this bug would be the following html in a page with a declared doctype (it works correctly in "quirks mode" / no-doctype):  &lt;div style="overflow: auto; height: 150px;"&gt;     &lt;div style="position: relative;"&gt;[...]&lt;/div&gt; &lt;/div&gt;   In IE7, the outer div is a fixed size and the inner div is relatively positioned and contains more content (assuming the inner div causes an overflow). In all other browsers, this seems to work as expected.  Screenshot: Counter example for a result of intersection of subspaces I am struggling with this question from Halmos's text, please ignore the imperative language.  "Suppose that $L, M$  and $N$ are subspaces of a vector space. Show that the equation    $$L \cap (M + N) = (L \cap M) + (L \cap N)$$   is not necessarily true.  Since each of these subspaces has origin in them, clearly there intersections could not be empty. I wasn't able to formulate an example where this result did n't hold. Any help would be highly appreciated. Is there a simple method of parsing XML files in C#? If so, what? Where can I find the location of downloaded .deb files?  I want to copy this folder before I reinstall Ubuntu so I don't have to redownload the packages.  I am running Ubuntu 12.10 32-bit.  How can I do this? How do I withdraw a paper from a journal after it was accepted but before it was fully published? On the advice of a senior colleague, I decided that I do not want to publish the paper anymore and with that journal. I wrote the editor several times; he has received the emails, but refuses to reply. I know he gets the emails because if its on other matters he has replied but not on the paper. I am ready to forfeit the processing fee I gave them but they refuse to reply and have already published the abstract of the paper to their site but not the full paper. Is legal action an option and do I have the right to sue? What are the guidelines in matters like these? Centos7 can't boot with shim-x64-15-7.el7_9.x86_64 The floowing is serial console output: BdsDxe: starting Boot0003 &quot;CentOS Linux&quot; from HD(1,GPT,A51EC18C-6831-4D47-B93E-B20DBC3F30BF,0x800,0x64000)/\EFI\centos\shimx64.efi  UEFI: Attempting to start image. Description: CentOS Linux FilePath: HD(1,GPT,A51EC18C-6831-4D47-B93E-B20DBC3F30BF,0x800,0x64000)/\EFI\centos\shimx64.efi OptionNumber: 3.  !!!! X64 Exception Type - 0D(#GP - General Protection)  CPU Apic ID - 00000000 !!!! ExceptionData - 0000000000000000 RIP  - 00000000BF2E5D5C, CS  - 0000000000000038, RFLAGS - 0000000000010006 RAX  - 00000000BFF5FE50, RCX - 000000000000001F, RDX - 00000000BFF31F28 RBX  - 00000000BF352FB0, RSP - 00000000BFF31E50, RBP - 00000000000023C0 RSI  - 00000000BFF31F28, RDI - 0000000000000068 R8   - 00000000BDB8C018, R9  - 000000000000211E, R10 - 00000000BDC6C010 R11  - 00000000BFF31E9A, R12 - 00000000BE791068, R13 - 00000000BF339018 R14  - 00000000BEE21018, R15 - 0000000000000068 DS   - 0000000000000030, ES  - 0000000000000030, FS  - 0000000000000030 GS   - 0000000000000030, SS  - 0000000000000030 CR0  - 0000000080010033, CR2 - 0000000000000000, CR3 - 00000000BF401000 CR4  - 0000000000000668, CR8 - 0000000000000000 DR0  - 0000000000000000, DR1 - 0000000000000000, DR2 - 0000000000000000 DR3  - 0000000000000000, DR6 - 00000000FFFF0FF0, DR7 - 0000000000000400 GDTR - 00000000BF3EEA98 0000000000000047, LDTR - 0000000000000000 IDTR - 00000000BEE1F018 0000000000000FFF,   TR - 0000000000000000 FXSAVE_STATE - 00000000BFF31AB0 !!!! Find image based on IP(0xBF2E5D5C) /build/work/af60adde42b1d1ad5be2a01e4924bb905248/google3/blaze-out/k8-opt/genfiles/third_party/edk2/ovmf_x64_csm_debug_workspace_dir/ovmf_x64_csm_debug_edk2_files_dir/Build/OvmfX64/DEBUG_CLANG38/X64/OvmfPkg/8254TimerDxe/8254Timer/DEBUG/Timer.dll (ImageBase=00000000BF2E4000, EntryPoint=00000000BF2E5AB5) !!!! How to add D-link dwm-156 to usb switch mode for automatic eject of driver CD / switch to modem mode? I am running Ubuntu 13.10, 32-bit.  I had every time need to go terminal to "eject" the windows driver CD,nor gone to disk &amp; eject windows driver CD to make the d-link dwm-156 gsm modem functioning through network manager.  It seems that usb mode switch is not working.  But on my friend's computer with ubuntu 14.04 ,usb mode switch is working and i can connect d-link dwm-156 to mobile broadband through network manager as soon as i get in.  Is there any way to make usb mode switch to eject the windows driver CD of d-link dwm-156 in my ubuntu 13.10,32-bit at start up?   The lsusb output is Bus 002 Device 013: ID 2001:a706 D-Link Corp before ejecting the CD (modem as storage device of windows driver).  The lsusb output is Bus 002 Device 005: ID 2001:7d01 D-Link Corp after ejecting the CD (as modem).   I don't want to use the driver in the CD,as i need to send sms &amp; find the data usage. Is a   select *  from myView   faster than the query itself to create the view (in order to have the same resultSet):  select * from ([query to create same resultSet as myView])   ?  It's not totally clear to me if the view uses some sort of caching making it faster compared to a simple query. Bounded in Probability and smaller order in probability I wanted to prove that if $X_n$ is bounded in probability and $Y_n = o_p(X_n)$, then $Y_n \rightarrow 0$ in probability   I  know the following definitions that is $X_n$ is bounded in probability meaning that   $P(|X_n|&lt;M)&gt;1-\epsilon$  and I know that $Y_n = o_p(X_n)$ implies that $Y_n/X_n \rightarrow0$ in probability My question is about one particular usage of static keyword. It is possible to use static keyword to cover a code block within a class which does not belong to any function. For example following code compiles:  public class Test {     private static final int a;         static {         a = 5;         doSomething(a);     }     private static int doSomething(int x) {         return (x+5);     } }   If you remove the static keyword it complains because the variable a is final. However it is possible to remove both final and static keywords and make it compile.   It is confusing for me in both ways. How am I supposed to have a code section that does not belong to any method? How is it possible to invoke it? In general, what is the purpose of this usage? Or better, where can I find documentation about this?