I have a problem with proving the below sequent in intuitionistic logic: $$ \neg \neg A \to \neg \neg B \vdash \neg \neg (A \to B) $$ I stuck here:  What should I put in $6th$ line? Is my approach even correct? I noticed that Generally, I have problems with proving sequents with double negation in intuitionistic logic. Is there any pattern for solving such examples?
Does the equivalence $$\neg\neg (P \rightarrow Q) \leftrightarrow (\neg\neg P \rightarrow \neg\neg Q)$$ hold in propositional intuitionistic logic? In propositional classical logic the equivalence holds obviously since $P \leftrightarrow \neg\neg P$. However, the similar-looking implication $\neg\neg (P \vee Q) \rightarrow (\neg\neg P \vee \neg\neg Q)$ is . The validity of the backward implication comes up in  as well.
How to modify a text file? What is the meaning of the density of a distribution at a point? I know how to use PDFs to calculate probabilities, but I don't think I understand them. For example, at $X=0$ the PDF of the standard normal distribution is $\approx 0.4$. Does this have any useful meaning? $ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $ $$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $$   (can this be duplicate? I think not) I tried it using many methods  $1.$ Solve this conventionally taking $1^\infty$ form in no luck  $2.$ Did this, expand $ {(1+x)^{\frac1x}}$ using binomial theorem got $\frac13$ then grouped coefficients of $x^0$ and it cancelled with $e$ then took coefficient of $x$ cancelled with $\frac{ex}{2}$ and so on   very messy right ? at last I got $\frac13$ but that's not the expected answer!  I must have went wrong somewhere can anyone help me with this. overset text over equal sign in in an Align environment I'm trying to make a longtable that is wider than the textwidth at a certain moment. I've tried different things always ending up with a different problem:  First possibility: If I use the adjustwidth environment the longtabu does not align correctly. The left margin stays fixed and the right margin gets out of the page. This in contrast, if I do the exact same thing to make a tabu (instead of longtabu), it works perfectly. I think using the adjustwidth is the best way to go, but I can't manage to get it working.  Second possibility: I've found a solution using the fullwidth package. That package does the trick on small longtables, but freezes when I'm trying to do it on my big longtable. The command \setlength\LTleft{-2cm} didn't work as well.  The final option I tried was to use geometry. But that changes the margin of the whole page thus making a page break which I'm not interested in. The code I used for this was  \newgeometry{left=1.2cm} \begin{longtable} long table stuff \end{longtable} \restoregeometry   Here's an example of what I want contra what I get  \documentclass[a4paper]{article} \usepackage{longtable} \usepackage{chngpage} \usepackage{lipsum} \usepackage{tabu}  \begin{document} \section{This is the original textwidth.} \lipsum[1]  \section{This is the width I need for my longtabu} \begin{adjustwidth}{-1in}{-1in} \begin{tabu} to \linewidth {*{13}{|X[l]} |} 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ \end{tabu} \end{adjustwidth}  \section{This is what happens} \begin{adjustwidth}{-1in}{-1in} \begin{longtabu} to \linewidth {*{13}{|X[l]} |} 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ 01 &amp; 02 &amp; 03 &amp; 04 &amp; 05 &amp; 06 &amp; 07 &amp; 08 &amp; 09 &amp; 10 &amp; 11 &amp; 12 &amp; 13 \\ \end{longtabu} \end{adjustwidth}  \end{document} Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$ Newline in a WPF-label? How can I add a newline in the text of a label in WPF such as the following?  &lt;Label&gt;Lorem    ipsum&lt;/Label&gt; Local html file AJAX Call and jQuery Woes The function auto.arima in the forecast package of R is a powerful tool to identify the best ARIMA(p,d,q) model for a given data series. In the   of the function, they report the following:  auto.arima(y, d=NA, max.p=5, max.q=5, max.order=5, max.d=2, start.p=2, start.q=2, ...)   According to the above function it seems that, by default, auto.arima tries to find the best model among all the combination having:   p equal or lower than 5 q equal or lower than 5 d equal or lower than 2 order equal or lower to 5, i.e. p + q + d  ≤ 5.   However, why does the default code stops at these conditions? Is there any particular reason why p, q and d should be equal or lower than 5, 5 and 2 and their sum lower than 5? Uniform convergence for sequences of functions 8.4 - Use entity browser in a module block from in same way as I would in a custom block I am having trouble adding an image picker to a block form defined in a custom module.  If I create a custom block type I can define an image field as entity_reference and in the block's form view tell it to use entity_browser and media_entity_browser.  This provides me with an picker with which I can select my image:    I am trying to recreate this in a block provided by my custom module, without much success  // /src/Plugin/Block/MyCustomBlock.php  function blockForm($form, FormStateInterface $form_state)  {     ...      $form['image'] = [       '#type' =&gt; 'entity_browser',       '#entity_browser' =&gt; 'media_entity_browser',       '#title' =&gt; $this-&gt;t('Image'),       '#default_value' =&gt; $this-&gt;configuration['image'] ?? null,       '#weight' =&gt; '0',     ];      ... }   This gives me the following.  Note the lack of the bounding box, title and option to edit remove items:     Exporting the configuration files for the working form view gets me this:  // core.entity_form_display.block_content.lead_block.default.yml      ...  field_image:     weight: 1     settings:       entity_browser: media_entity_browser       field_widget_display: rendered_entity       field_widget_edit: true       field_widget_remove: true       selection_mode: selection_append       field_widget_display_settings:         view_mode: default       open: false     third_party_settings: {  }     type: entity_browser_entity_reference     region: content   ...   But if I try and use 'type' =&gt; 'entity_browser_entity_reference' in my blockForm function the field disappears altogether.  Looking into various core files I can see that entity_browser is a @FormElement and entity_browser_entity_reference is a @FieldWidget but I don't really understand the difference.  As you might guess I am fairly new to Drupal and I feel like I am missing something fundamental.  I (think I) understand that custom block types are actually nodes internally and that they are subject to different rules to blocks which I define in my custom modules, but I don't fully grasp what the differences here are.  Is is it possible to use the entity browser in my custom module block form in the same way as it is used in the form which I build through the admin's custom block type builder?  Thank you. Expected value in an urn problem In an urn there are $n$ red balls and $m$ blue balls. I extract them without replacement. Let $X$=time of first blue. What is $E(X)$?  I found PMF of $X$ and it is, if $k &gt; n+1$  $$P(X=k) =\frac{n(n-1) \dots (n-k+1)m}{(m+n)(m+n-1) \dots (n+m-k+1)}$$  How could I evaluate $E(X)$?  Edit: I'm looking for an explicit form of $E(X)$ How to call drupal_bootstrap from custom file in subfolder? I'm trying to call drupal_bootstrap from 'temp.php' in a subfolder called 'custom' my code as follow:  chdir('../'); define('DRUPAL_ROOT', getcwd()); require_once DRUPAL_ROOT . '/includes/bootstrap.inc'; drupal_bootstrap(DRUPAL_BOOTSTRAP_FULL);   Once I call the temp.php I get Drupal homepage.   I also tried an explicit physical path but with no use, code as follow:  define('DRUPAL_ROOT', '/var/htdocs/drupal7'); require_once '../includes/bootstrap.inc'; drupal_bootstrap(DRUPAL_BOOTSTRAP_FULL);   temp.php located under the following folder structure:  - var -- htdocs --- drupal7 ---- custom  ----- temp.php How can I make a rounded indentation on a cylinder? I'm trying to do a cosmetic bottle in blender but i don't how to do this thing in the image. How to do this round edge on the cap. Can anyone help me? How do I reboot a Ubuntu Server from the command line? sudo reboot now never seems to work for me. Because its a server, I don't have a window manager installed, so I can't do it from a GUI (which always seems to work).    I've had the problem for years on different versions of the operating system. It happens in virtual machines and on real metal. The same problem exists with sudo shutdown now, too.  I've lived with it in the past. The difference now is I don't want to just hit the power button because I might lose data. How does the wording "for each" work? Some cards in magic has something happen for each {something}. For instance:  makes you gain 2 life for each creature you control. Tapping  adds one green mana to your mana pool, and you gain 1 life for each non-land card revealed.  How is this life gained: all at once or "one bit" at a time? Say I tap Selvala and 1 land, and 2 non-lands are revealed: do I gain 2 life, or do I gain 1 life two times? Would  have one or two +1/+1 counters placed on it if I had it in play at that point? Passing data between two controllers using storyboards $ KC $ spaces imply $ US $ spaces , but vise versa is false. In the $ US $ space , each convergent sequence has unique limit.      In the $ KC $ space , every compact subset is closed.      It easy to show that $ KC $ spaces imply $ US $ spaces.   The example below show that $ US $ spaces do not imply $ KC $ spaces.      Let $ X = [ 0, 1 ] $ with the the usual topology and attach  a new point z whose neighborhood are open dense subsets of $ [ 0,1 ]$. Observe $ [ 0,1 ]$ is a compact non-closed  of $X$ and then $X$ is not a $KC$ space. However no sequence in $ [ 0,1 ]$ converge to $z$ and in particular  all convergent in $X$ have unique limits.   can you give me more details?   Why $X$ is not closed? Why no sequence in $ [ 0,1 ]$ converges to $z$ and in particular all convergent sequences in $X$ have unique limits ? Excel: identifying if all values in a row are not equal Cumulative Sum without Script