Is there a canonical metric for a Lie group manifold?
Constructing a metric on a Lie Group
How can I capture KeyDown event on a WPF Page or UserControl object? I have a Page with a UserControl on it.  If the user presses Esc while anywhere on Page I want to handle.  I thought this would be as easy as hooking up the PreviewKeyDown event, testing for the Esc key, and then handling it.  However, when I placed I breakpoint in the event handler I found it was never getting called.  I thought perhaps the UserControl might be getting hit, so I tried PreviewKeyDown there... same result.  Does anyone know the proper place to test for a KeyDown or PreviewKeyDown on a Page object? Let be $V$ a vector space over the complex numbers with inner product. Let be $T$ and $S$ linear operators in $V$ with adjoint operators $T^*$, and $S^*$ respectively. I know that it satisfies this: $\left&lt; T(x),y \right&gt;=\left&lt; x,T^{*}(y) \right&gt;$ $\left&lt; S(x),y \right&gt;=\left&lt; x,S^{*}(y) \right&gt;$    $\forall x,y \in V$ But, How can I prove that the adjoint operator of $ST$ is $(ST)^{*}$? And how can I  prove that $(ST)^{*}=T^{*}S^{*}$? How to render a background image in 2.8? I'm trying to "add" an 3D object onto a photo. I have added the photo to the scene in three ways so far, but none of them render (eevee):   Camera > Background Images Empty a. Add > Empty > Image (Empty gets added to current Collection) b. Add > Image > Backround (Empty gets added to Scene Collection)   These all show up in the 3D View and I was able to align the camera and the object so its perspective matches the scene in the photograph. But when I Render the scene, I only see the object, not the background image.  For the latter, it's simple: Empties are not rendered at all. There are few questions about this, since 2.8 has changed how it handles background images. The most popular seems this one (it includes references to others):      But I seem not to be able to find a method to have the background image render.  Another method might be to use the Compositor (Nodes) to Mix or Alpha Over an Image with the Render Layers. But this does not give the same alignment as achieved with the background image (also this requires that the World background does not get rendered (Alpha channel); for Eevee, this can be achieved by checking Film>Transparent).  What other options are there?  Docs: Busy Device on Umount I often experience a problem to umount a directory:  umount /mnt/dir umount: /mnt/dir: device is busy   There are many reasons why the device is busy. Sometimes there are processes running which have open locks on it, sometimes there are other directories mounted on top of /mnt/dir.  My question:  What are the steps to check why a directory couldn't be unmounted.  I know there are many reasons, but it's ok if you explain a specific solution.  [EDIT]  [X]  running processes on mounted volumes.   [X] another volume is mounted on top of a volume we want to unmount   [_] NFS locks the volume we want to unmount Difference between [NSMutableArray array] vs [[NSMutableArray alloc] init] form in a .tpl.php file This is the code in my template.php file.  function studiga_theme_theme($existing, $type, $theme, $path) {         $items['add_property_form'] = array(         'render element' =&gt; 'form',         'template' =&gt; 'node--14',         'path' =&gt; drupal_get_path('theme', 'studiga_theme') . '/templates/node',     );      return $items; }    function custom_add_property_form($form, &amp;$form_state) {     $form['first_name'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('First name')),     );     $form['last_name'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Last name')),     );     $form['address'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Address')),     );     $form['city'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('City')),     );     $form['state'] = array(         '#type' =&gt; 'select',         '#options' =&gt; array(             'default' =&gt; 'State',             '...' =&gt; '...',         ),     );     $form['zip'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Zip')),     );     $form['email'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Email')),     );     $form['phone'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Phone')),     );     $form['submit'] = array(         '#type' =&gt; 'submit',         '#value' =&gt; 'Submit',     );      return $form; }   This is the code in node--14.tpl.php.                                          Contact Information                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  &lt;div class="row"&gt;     &lt;div class="small-12 medium-12 large-8 large-offset-2 columns"&gt;         &lt;?php print render($form['submit']); ?&gt;     &lt;/div&gt; &lt;/div&gt;     But no fields are appearing - what am I doing wrong? You can't have a straight forward  in a tpl.php file can you? All of my Firefox extensions were spontaneously and simultaneously disabled (converted to Legacy status) a few moments ago.  How can I re-enable them? Firefox 66.0.3 (64-bit) Is there a specific purpose in terms of efficiency or functionality why the k-means algorithm does not use for example cosine (dis)similarity as a distance metric, but can only use the Euclidean norm? In general, will K-means method comply and be correct when other distances than Euclidean are considered or used?  [Addition by @ttnphns. The question is two-fold. "(Non)Euclidean distance" may concern distance between two data points or distance between a data point and a cluster centre. Both ways have been attempted to address in the answers so far.] Is there a unique solution for this quadratic matrix equation? I want to model the transfer function of a resistor mathematically, and am after some resources / tips on if it's possible to do so.  For example, if I put 1V into a 1 Ohm resistor, I want to know how long and how (shape) it will take to reach a steady state temperature, given 25 deg C ambient.  I.e.   $$ \frac{T(s)}{V(s)} = ??? $$  Where T(s) is the laplace transform of temperature T(t), measured in say Kelvin.  I understand there are a lot of factors which will affect this model (material, size, shape, ambient (?)), but I'm not sure where to even start. A  shows the heat rise at a certain load, but not really how it gets there. I can't find a time-domain function for the temperature rise either. What is wrong with this query? (Unknown error parsing query) How can be solved the equation $x^3+x-1=y^2$ in positive integers?    I know this equation defines an elliptic curve, but this seems to be a non-elementary way to solve this question. Is there a more elementary solution?   By the way I found three solutions: $(1,1), (2,3)$ and $(13,47)$. Is this related to the law group property of elliptic curves? Thanks for further answers. Let $T$ be a self adjoint operator on a finite dimensional inner product space. And let $4$ and $5$ be the only eigenvalues of $T$. Show $T^2-9T + 20I = 0$ .  proof:Let $T$ be a self adjoint. And let $4 , 5$ be the only eigenvalues of $T$. Then $Tv = 4v, Tv = 5v$ for some $v \neq 0$. Then $Tv = 4v, Tv = 5v$ if and only if $(T-4I)v = 0, (T-5I)v=0$.  Then $ (T-4I) (T-5I)v=0$ implies $(T^2 -5T -4T + 20I^2)v = 0$, implies   $(T^2 -9T + 20I)v = 0$, so we have  $T^2 -9T + 20I = 0  $. Can someone please check this? if it's not correct, could I please have some help on it? Thank you Area 51 is miscounting my proposal followers I know of one person who is following my proposal who doesn't show up on the proposal webpage.    My proposal is for quantitative finance: .  It shows 19 people following the proposal (on the side of the page), and the total listed at the top of the page is 19 followers.  Yet one user is not listed: .  This makes me wonder if other followers are also not being counted? What causes a force field to be "non-conservative?" Apex PMD: Problem: Validate CRUD permission before SOQL/DML operation Background   I am using the Apex PMD plugin for VS Code and it's giving me this problem:     Validate CRUD permission before SOQL/DML operation   For this line of code:  insert contentVersion;   Which is part of this method:  private void attachReport(Id recordId) {      ContentVersion contentVersion = new ContentVersion(         versionData = Blob.valueOf(buffer.toStr()),         title = 'Import Report',         pathOnClient =  StringUtils.format('/Import-Report-{0}.txt', DateTime.now().getTime()),         FirstPublishLocationId = recordId);      insert contentVersion; }   Questions    Why am I getting the problem? What should I do to not get the problem?     Reference What are pros and cons for MacPorts, Fink and Homebrew? I'm just migrating from Ubuntu Linux to Mac, and everything is new and I'm re-learning a lot of stuff.  On Linux I had the excellent apt-get to manage software packages. I googled for an alternative on Mac and found about MacPorts, Fink and Homebrew.  I will use this computer primarily to develop Ruby on Rails applications.  So, what are the differences between them? Which are the upsides and downsides? Which one is best maintained and has more packages? Bootstrap CSS Active Navigation We are developing a new Lightning Community to replace our Partner Community. However, when we use our visualforce pages with LightningStylesheets=true on the Lightning Community, the Lightning styling is not applied.  I understand that this is the case because Lightning Communities are not Lightning Experience.  Is there a way to force LightningStylesheets anyway? I am trying to stay away from having to rebuild/restyle all our legacy visualforce pages. Several badge descriptions contain raw HTML I was reading through some badge descriptions and noticed that Synonymizer's description needs to be updated. As it stands, there is raw HTML (an href) in the description:    EDIT: after reading comments by @ColeJohnson, I have come to the understanding that this post comes off as whiny. I want to make it very clear that I wasn't ever trying to complain. My first reaction was to fix the perceived error myself. When I realized I was unable to do so, I figured the best course of action would be to find someone who would be able to make the necessary edits. Hence the post here.