I don't even know where to start here. Also, if it is a group homomorphism, I have to determine whether they are injective or surjective. Any help would be great, thanks!
When is mapping $g$ to $g^{-1}$ a group homomorphism? It just means that the map maps the identity to the identity and inverses to inverses. So does that mean it's only a homomorphism if the inverse is itself?
I'm looking for some insight into how Stack Overflow releases are deployed and managed.   What software/tooling is used? What time are releases generally performed? How many and what steps are involved?   I'm assuming it's not too stringent given the light team working on the codebase. But considering the high rate of change, I could see some kind of tooling and automation becoming increasingly relevant, so I'm curious to learn what kind of decisions are being made to deal with the changes in scale and feature set over time.  Is this information available somewhere? Stack Overflow has left! Create any number of paragraph fields in a multiple value field on node create form I have a paragraphs field that takes a term reference.  I need to count how many terms there are for a specific vocabulary and then create that many instance of the paragraphs multiple value field for when creating a node of a certain type. Can this be done with Hook_form_alter ?  It is important that these fields are loaded when the node creation form is loaded rather than the user having to click add more  Any help would be greatly appreciated  I have tried this but when saving the node it doesn't save the new fields  function hook_form_alter(&amp;$form, $form_state, $form_id) {    if ($form_id == 'my_form') {    $item = $form['field_features']['widget']['0'];    // get taxonomy terms   $query = \Drupal::entityQuery('taxonomy_term');   $query-&gt;condition('vid', "my_voc");   $tids = $query-&gt;execute();   $terms = \Drupal\taxonomy\Entity\Term::loadMultiple($tids);    $count = count($terms);    $i = '1';   foreach($terms as $term){     $form['field_features_para']['widget'][$i] = $item;     $form['field_features_para']['widget'][$i]['#delta'] = $i;     $form['field_features_para']['widget'][$i]['#weight'] = $i;     $i ++;   }  } } Change Default Scrolling Behavior of UITableView Section Header Approximating commuting matrices by commuting diagonalizable matrices Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A_n$ and $B_n$ of matrices such that   $A_n \rightarrow A$, $B_n \rightarrow B$.  Each $A_n$ is diagonalizable  and the same for each $B_n$.  For every $n$, $A_n$ commutes with $B_n$.    Moreover, it would be nice if the following property was additionally satisfied: if $A,B$ are real, then $A_n,B_n$ can be chosen to be real as well. Problem with Refsection Between vs. Among - "Difference [between | among] one thing and other things." Consider the following two versions of a question:1   Version 1: What is the difference in meaning between "[to be] not invited" and similar negation forms?  Version 2: What is the difference in meaning among "[to be] not invited" and similar negated forms?    Personally, I think Version 1 might subtly suggest a Star Topology (see below) with the target sentence "[to be] not invited"at the center.  And Version 2 might subtly suggest a more free-form comparison like the Mesh or Fully Connected, with the target sentence simply "one among many":   Between approx equal "Star" and among approx equal "Fully Connected"?  Don't get me wrong: I do think this is quite subtle, not always applicable, and I don't think anyone would intuit such a "network topology" meaning without further explanation.   So the main question is, are both grammatically correct?  If yes, is there a semantic difference to choose one over another, perhaps as I suggested?  Or does the plurality aspect of between and among trump other semantic and stylistic issues?    1. In my question , another user provided a very helpful edit. However, the change of the word between to among prompted this question. How to plot and visualize a single linear vector in 3D? I am creating table in LaTeX and I need to get the text to perform line break automaticaly when it's too long. And if possible, I'd like to get the text alignment to the left. Is it possible to handle this situation? Shut down computer from keyboard Is there a keyboard shortcut available to shutdown the computer?  I know I can use the Power button, but I would like to be able to it from the keyboard.  If such a shortcut does not exist already, how do I set one? Do gravitons interact with each other? I always thought the non-linearity of Einsteins field equations implies that there should be direct graviton-graviton interactions. But I stumbled upon  which argues:     If gravitons exist, then, like photons and unlike gluons, gravitons do   not interact with other particles of their kind. That is, gravitons   carry the force of gravitation but are not affected by it. This is   apparent due to gravity being the only thing which escapes from black   holes, in addition to having an infinite range and traveling in   straight lines, similarly to electromagnetism.   Is Wikipedia correct? If not, why not? And what then are the arguments that there must be graviton-graviton interactions?    (As of this question being asked, the above paragraph has been removed from Wikipedia.) node.js conflicts: /sbin/node vs /usr/bin/node I have two Ubuntu 12.10 machines: machine A is a VMWare VM and machine B is and old Acer laptop. On both machines, I installed node using apt-get. However machine A installs the main binary as /usr/bin/node and machine B as /usr/bin/nodejs. Here are some behaviors I observed:   Both machine returns /usr/bin/node in response to the which node command. However, there is no such file on machine B. Issuing the command node works for machine A, not B. On B, I have to use nodejs instead. On B, there is a binary called /sbin/node, A does not have it.   These behaviors causes inconsistency problem when developing on both machines. For now, I worked around by renaming the binaries on B as such:  sudo mv /sbin/node /sbin/node-sbin sudo ln -s /usr/bin/nodejs /usr/bin/node   Is there a way to really fix this, so that node will be installed as /usr/bin/node instead of /usr/bin/nodejs? Addition is to Integration as Multiplication is to ______ Addition is to Integration as Multiplication is to ______ ? Everyone knows that definite integration  is "a way to sum continuum-many terms" in a rough sense. Can we "multiply continuum-many factors"  in a similar sense? Regular Expression: Numeric range I have a Dell XPS 12 running Ubuntu 13.10 and it doesn't appear to find it's wireless.  I have tried the suggestions  but they made no difference  peter@xps:/lib/firmware$ ls -l /lib/firmware/iwlwifi-7260-7.ucode  -rw-r--r-- 1 root root 682892 Aug  4 12:50 /lib/firmware/iwlwifi-7260-7.ucode peter@xps:/lib/firmware$ sudo lshw -c network   *-network UNCLAIMED             description: Network controller        product: Wireless 7260        vendor: Intel Corporation        physical id: 0        bus info: pci@0000:06:00.0        version: 63        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress cap_list        configuration: latency=0        resources: memory:f7c00000-f7c01fff  peter@xps:~/backports-3.10-2$ grep IWL7260_UCODE_API_MAX drivers/net/wireless/iwlwifi/iwl-7000.c  #define IWL7260_UCODE_API_MAX   7     .ucode_api_max = IWL7260_UCODE_API_MAX,         \ peter@xps:~/backports-3.10-2$ dmesg | grep iwl peter@xps:~/backports-3.10-2$ iwconfig lo        no wireless extensions.  peter@xps:~$ find / -name *7260-6* peter@xps:~$ find / -name *7260-7* /lib/firmware/iwlwifi-7260-7.ucode   peter@xps:~$ dmesg | grep -i -A 20 intel [ 1318.013278] Intel(R) Wireless WiFi driver for Linux, in-tree:d [ 1318.013281] Copyright(c) 2003-2013 Intel Corporation [ 1318.015154] cfg80211: World regulatory domain updated: [ 1318.015157] cfg80211:   (start_freq - end_freq @ bandwidth), (max_antenna_gain, max_eirp) [ 1318.015159] cfg80211:   (2402000 KHz - 2472000 KHz @ 40000 KHz), (300 mBi, 2000 mBm) [ 1318.015161] cfg80211:   (2457000 KHz - 2482000 KHz @ 40000 KHz), (300 mBi, 2000 mBm) [ 1318.015162] cfg80211:   (2474000 KHz - 2494000 KHz @ 20000 KHz), (300 mBi, 2000 mBm) [ 1318.015164] cfg80211:   (5170000 KHz - 5250000 KHz @ 40000 KHz), (300 mBi, 2000 mBm) [ 1318.015165] cfg80211:   (5735000 KHz - 5835000 KHz @ 40000 KHz), (300 mBi, 2000 mBm)     What I have found is that it appears to be loading an old version of the firmware but I don't know how or why  peter@xps:/lib/firmware$ modinfo iwlwifi | grep 7260 firmware:       iwlwifi-7260-6.ucode peter@xps:/lib/firmware$ ls *7260-6* ls: cannot access *7260-6*: No such file or directory   Any suggestions of how to get this setup very welcome. How to use \graphicspath? I am using MikTex 2.9 and TeXstudio for my TeX preparation. I would like to set the path to my figures folder, which contains all the figures, plots etc. required for my work.   I used \graphicspath to set the directory, but it gives me an error message:  \documentclass[12pt]{article}  \usepackage{graphicx}  \graphicspath{D:/LATEX/Reports@IIT/figures}  \begin{document} \includegraphics[width=4.5cm,height=4cm,angle=0]{logo.jpg}  \end{document}      Error: Package pdftex.def Error: File `logo.jpg' not found   However, when I prepend the filenames of the images with the same path in the \includegraphics command, it works just fine:  \documentclass[12pt]{article}  \usepackage{graphicx}  \begin{document}  \includegraphics[width=4.5cm,height=4cm,angle=0]{D:/LATEX/Reports@IIT/figures/logo.jpg}  \end{document}   Why is \graphicspath not setting the path? How to launch Ubuntu in Windows Subsystem for Linux in Windows 10 S? When asking a new question and typing in the title we get a list of similar questions named     Questions that may already have your answer     That suggestions contains posts across all tags. Like the example in the screenshot it returns similar questions for Python, jQuery, SQL, C#, Java, C++, ...  This makes the list less useless unlikely people find a question that already answers theirs. If people actually find a similar question in that list many duplicates could be avoided.  So can tags I entered below the question be considered to update the list? And can the tag textbox maybe put above the question title? Path bar missing in file dialog after running software updater for Ubuntu 17.04 After running software updater, the path bar in the file dialog (for example the one you get when choosing Open File in in Google Chrome or Firefox) is missing. This is frustrating since it's no longer possible to copy a path from the terminal (for example) and paste it into the file dialog. Here is a screenshot of the dialog:     How can I get back the path bar? $(\Bbb R,+)$ is a topological group. Is there any other group structure on $\Bbb R$ such that it is still a topological group and this group is not isomorphic to $(\Bbb R,+)$ ?  Refer to  for the general problem. In that problem I have conjectured that there is no such group however I couldn't prove it.