A limit ordinal $\gamma$ is indecomposable if and only if $\alpha + \gamma = \gamma$ if and only if $\gamma = \omega^{\alpha}$ for some $\alpha$.
Indecomposable limit ordinals
How can I find out the size of a block device, such as /dev/sda? Running ls -l gives no useful information. My professor has stated the following lemma:  Let $U$ and $V$ be two vector spaces and $B$ a basis for $U$. If $f:B\rightarrow V$ is a map, then there exists a unique linear map $A: U \rightarrow V$ such that $A(x)=f(x)\ \forall\ x\in B$  How can I prove this lemma? (This is not a homework problem.) Continuity of $f(x)$ on $[a,b]$ implies continuity of $f(t) = \displaystyle \min_{a \leq x \leq t}f(x)$ at [a,b]. Why does the symbol for the multiplication operation change shape? messenger()->addMessage not working after a $form_state->setResponse In the submitForm of a custom form, I have this:    switch ($form_state-&gt;getValue('suscribe')) {   case 0:     $iStatus = 1;     break;   case 1:     $iStatus = 3;     $sFileName = 'sites/default/files/_private/bulletins/' . $form_state-&gt;getValue('adherent')[1] . ".pdf";     $response = new BinaryFileResponse($sFileName);     $response-&gt;setContentDisposition(       ResponseHeaderBag::DISPOSITION_ATTACHMENT,       'bulletin.pdf'     );     $form_state-&gt;setResponse($response);     break; }  $storage = \Drupal::entityTypeManager()-&gt;getStorage('member'); $id = $form_state-&gt;getValue('adherent')[0]; $entity = $storage-&gt;load($id); $entity-&gt;status = $iStatus; $entity-&gt;save();  \Drupal::messenger()-&gt;addMessage($this-&gt;t('Your wish has been recorded.'));   suscribe is a Yes/No radio button. $form_state-&gt;setResponse($response); triggers the automatic download of a pdf document.    The message (Your wish...) is correctly displayed in case 0. In case 1, the message shows only if I refresh the page (or go to another page). How can I have the message displayed without doing another action after submitting the form in case 1?    By the way, database is correctly updated in both cases. I'm using Drupal 8.6.10. WOL was working great on Bionic until yesterday's kernel update. How to diagnose/recover it?  Edit: if set to boot from 4.15.0-23 kernel it works. If set to boot from 4.15.0-24 it does not.  Edit2: WoL can be enabled with   sudo ethtool -s [card] wol g   but it isn't persistent. Searching for how to set it every boot.  Edit3: Per a comment by Kai-Heng Feng on this  this was an advertent change! (and therefore not a bug)  This is because the fix of LP: #1752772 has one commit [1] that disables WoL by default. Set WoL explicitly should solve your issue.  [1] https://github.com/torvalds/linux/commit/7edf6d314cd061e1d0a1b7bc0b511d64322c3f72 What would you consider to be the best exception type to throw when an extension method is called on a null instance (where the extension method does not allow it)? Since extension methods are nothing but static methods you could argue that it should be ArgumentNullException, but on the other hand they're used like instance methods so it might be more natural to use the NullReferenceException. Let's take the following example:  public static string ToInvariantString(this IFormattable value, string format) {     return value.ToString(format, CultureInfo.InvariantCulture); }   This way a NullReferenceException will be thrown if the value parameter is null.   The other example would be:  public static string ToInvariantString(this IFormattable value, string format) {     if (value == null) throw new ArgumentNullException("value");     return value.ToString(format, CultureInfo.InvariantCulture); }   EDIT: In some of the answers you have pointed out that an extension methods can be called like a static method and in those cases a null reference exception would be wrong, which is a great point, and actually one of my concerns, not sure why I forgot to mention that in the question in the first place.   Someone also pointed out that it's wrong to throw a NullReferenceException, and yes, it is. That's why I don't throw it, I just let it happen (let the CLR throw it) by not guarding the method.  I think I favor the ArgumentNullException (that's what I've use so far) but I still think there is at least room to argue for an against the NullReferenceException since it seems more natural in most places where the method is going to be used. Removing Ubuntu from UEFI Bios Menu Cardinality of Irrational Numbers I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem:   Show that $|\mathbb{I}|=|\mathbb{R}|$,  How can I do that?  Do I need to assume the Continuum Hypothesis in order to that statement to be true?  Thanks Front facing camera stopped working on the iPhone 4s First, a word about the format - I don't know whether this was a movie or a TV series, b/c I saw it aired in pieces on PBS back in the 1980s. I don't even recall if it was originally voiced in english, dubbed over, or subtitled. I'll just call it a movie for now.  The movie seemed to be set in Europe in what looked like the 1960's or 70's, where almost all the people were gone. The protagonist was probably a 20-something man, who routinely visited a psychiatrist. I think the protagonist also had a girlfriend, but most of the dialogs seemed to be between the man and the doctor.   It was kind of a mystery where all the people had gone, and I think the man was nominally seeing the doctor to cope with the stress of it all, or had amnesia and suspected he knew The Truth. However, it turns out that the doctor discovers that the man is somehow the key to the reality that the characters share, and during hypnosis sessions, discovers he can manipulate the man and thereby reality. To make it easier for himself, the doctor gives the man a code word to return the man to a hypnotic trance on command. The code word, I am positive, was 'Antwerp'.  I don't think I ever saw the end of this, but the 'truth' might have been that World War III was fought with neutron bombs, and that the man was actually just dreaming everything as he wandered around the city, dying. Or not. It would be nice to know how it turned out, but I don't even have the name of the show. Settings API enable default settings on theme install? I am still learning settings API so please forgive my noob question. I have three fields: a checkbox, colorpicker and text field (they are working), but I would like to do this: when the theme is installed add "checked" state to my checkbox but allow user to "unchecked" it ,and if empty add default color to my colorpicker... I have no idea how to do this so please help me!  My code looks like this now (thanks to @Chip Bennett! ) :  &lt;?php add_action('admin_menu','create_theme_options_page'); function create_theme_options_page(){ add_options_page('Theme Options','Theme Options','administrator',__FILE__,'build_options_page'); }    function build_options_page(){    ?&gt;    &lt;div id="theme-options-wrap"&gt;    &lt;div style="float:left;margin-top:19px;position:absolute;" class="icon32" id="icon-tools"&gt;&lt;br/&gt;&lt;/div&gt;    &lt;h2&gt;My options page&lt;/h2&gt;    &lt;form method="post" action="options.php"&gt;    &lt;?php settings_fields('plugin_options');?&gt;    &lt;?php do_settings_sections(__FILE__); ?&gt;    &lt;p class="submit"&gt;    &lt;input name="Submit" type="submit" class="button-primary" value="&lt;?php esc_attr_e('Save changes');?&gt;"/&gt;    &lt;/p&gt;    &lt;/form&gt;    &lt;/div&gt;    &lt;?php    }       add_action('admin_init','register_and_build_fields');      function register_and_build_fields(){      register_setting('plugin_options','plugin_options','validate_setting');      add_settings_section('main_section','&lt;br/&gt;&lt;br/&gt;Main Settings','section_cb',__FILE__);      add_settings_field('banner_heading','Banner Heading:','banner_heading_setting',__FILE__,'main_section');      add_settings_field('display_sidebar','Display Sidebar:','display_sidebar_setting',__FILE__,'main_section');      add_settings_field('display_colorpicker','Display Colorpicker:','display_colorpicker_setting',__FILE__,'main_section');      }        //Install Iris colorpicker        add_action( 'admin_enqueue_scripts', 'mw_enqueue_color_picker' );        function mw_enqueue_color_picker( $hook_suffix ) {        // first check that $hook_suffix is appropriate for your admin page        wp_enqueue_style( 'wp-color-picker' );        wp_enqueue_script( 'my-script-handle', get_template_directory_uri() . '/js/my-script.js', array( 'wp-color-picker' ), false, true );        }           //Colorpicker          //Now how to add a default color on theme install?          function display_colorpicker_setting(){          $options = get_option('plugin_options');          echo"&lt;input type='text' id='color1' name='plugin_options[display_colorpicker]' value='{$options['display_colorpicker']}'/&gt;";          }            //Sidebar checkbox           //Now how to add a default state "checked" on theme install?           function display_sidebar_setting(){           $options = get_option('plugin_options');           echo "&lt;input type='checkbox' name='plugin_options[display_sidebar]' {$options['display_sidebar']} id='display_sidebar' /&gt;";           }              //Banner Heading             function banner_heading_setting(){             $options = get_option('plugin_options');             echo "&lt;input name='plugin_options[banner_heading]' type='text' value='{$options['banner_heading']}'/&gt;";             }               //Validate our settings here               function validate_setting($plugin_options){                  return $plugin_options;               }                 function section_cb(){                } ?&gt; I am new to Ubuntu, but I believe I have Ubuntu 11.10 installed correctly.  I am trying to install Acroread using AdbeRdr9.4.6-1_i386linux_enu.deb that I downloaded from the Adobe web site.  How do I do this? In Monopoly, Is it OK for a third party to make a trade with a player who is about to lose? I play with some serious board gamers, but we have neglected Monopoly for a number of years, so our knowledge is mostly based on high school &amp; house rules.  We played last week and enjoyed it, but there was a piece of contention. There are times where we decide that we don't like one of our friends for some reason and don't want them to win (i.e. we landed on their property a bunch, or they refused to make a deal, etc.).  So when player 1 has some property and player 2 lands on it and cannot pay the rent their assets (houses/hotels get liquidated), their property gets transferred to the other player. (Please correct me if I'm wrong.) In this case, though, player 3 "traded" the amount of money owing for a piece of property (way overpaid). In this case they did it so that player 1 would not get the properties. Player 2 had no chance of winning the game (all properties mortgaged, no cash to speak of and many of the other properties with 3+ houses/hotels on them).  To summarize: Is it OK for a third player to purchase/trade property or cash to/from a player who is about to lose? Using conjugates to find a limit with a cubic root: $\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$ I currently have $$\lim_{h\to 0}\frac{\sqrt[3]{h+1}-1}{h}$$  Now, I know that when you have square root instead of a cubic root it's easy. You just multiply by the conjugate and simplify afterwards. If it were a sqrt I know that the limit is 1/2. I know that the result here is 1/3 but I can't seem to get there. I always end up with 1/4 due to getting something like: $$\frac{h+1-1}{h(\sqrt[3]{h+1}+1)(\sqrt[3]{h+1}+1)}$$  After simplifying and evaluating the limit you end up with $$\frac{1}{(\sqrt[3]{0+1}+1)(\sqrt[3]{0+1}+1)}$$  which turns out to be $1/4$.  What am I doing wrong? Galois extension is (not) transitive Let $ K/ L /F $ be fields. If $K / L$ is Galois and $ L / F $ is Galois, then $ K / F$ is Galois. We mentioned this very quickly in today's class without justifying. But I have trouble seeing this. Any help is appreciated.  Our definition for $ K / F$ being Galois is the $F$-automorphism group $G = G( K / F) : = \{ \sigma \in aut (G) : \sigma_F = id_F \}$ fixes only $F$, i.e. $ \{ x \in K : \sigma (x) = x \text { for all } \sigma \in G  \} = F$.  But we also showed that $ K /F$ is Galois if and only if $ K/F$ is normal and separable. Remove pseudo nodes in QGIS How to know whether Lagrange multipliers gives maximum or minimum? Residue theorem with $\frac{1}{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) d\theta$. The following integral should be doable using the residue theorum:     $$\frac1{2\pi}\int_{0}^{2\pi}e^{\cos\theta}\cos(n\theta) \,d\theta$$ What do brokers do with bad stock?