Ito's process and martingale
martingale and stochastic Integral
fsck error on boot: /dev/sda6: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY I dual boot Ubuntu and Windows 7. I was browsing on FireFox in Ubuntu and all of a sudden I get some error message, so I rebooted.  Now when I try to boot into Ubuntu I get this message:  fsck from util-linux 2.26.2 /dev/sda6 contains a file system with errors, check forced. /dev/sda6: Inodes that were part of a corrupted orphan linked list found.  /dev/sda6: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY.         (i.e., without -a or -p options) fsck exited with status code 4 The root filesystem on /dev/sda6 requires a manual fsck  Busybox v1.22.1 (Ubuntu 1:1.22.0-15ubuntu1) built in shell (ash) Enter 'help' for a list of built-in commands.  (initramfs) _   How can I fix this? Find all functions $f:\mathbb{N_{0}} \rightarrow \mathbb{N_{0}}$ satisfying the equation $f(f(n))+f(n)=2n+3k,$ for all &amp; $n\in \mathbb{N_{0}}$, where $k$ is a fixed natural number.  A friend of mine suggested me to use recurrence relation. While it is unfortunate that I have problems understanding recurrence relation. Please refer information to understand difference equation, homogeneous equation &amp; auxillary equation.  Thanks in advance! Kindly don't down vote for not showing my attempt. I am seriously not able to understand the question and the process suggested to use. How do I render a Twig template and load a CSS file from a controller? Is this considered a good practice for rendering a Twig template and loading a CSS file from a controller?   DefaultController.php  namespace Drupal\acme\Controller;  use Drupal\Core\Controller\ControllerBase; use Drupal\Core\DependencyInjection\ContainerInjectionInterface; use Symfony\Component\DependencyInjection\ContainerInterface; use Drupal\Core\Template\TwigEnvironment;  class DefaultController extends ControllerBase implements ContainerInjectionInterface  {   /**   * @var Drupal\Core\Template\TwigEnvironment   */   protected $twig;    public function __construct(TwigEnvironment $twig)   {     $this-&gt;twig = $twig;   }    public static function create(ContainerInterface $container)   {     return new static(       $container-&gt;get('twig')     );   }    /**    * hello    * @param  string $name    * @return string    */   public function hello($name)    {      $template = $this-&gt;twig-&gt;loadTemplate(       drupal_get_path('module', 'acme') . '/templates/hello.html.twig'     );      $markup = [       '#markup' =&gt; $template-&gt;render([ 'name' =&gt; $name ]),       '#attached' =&gt; [ 'css' =&gt; [           drupal_get_path('module', 'acme') . '/assets/css/acme.css'         ]       ]     ];      return drupal_render($markup);   } }   acme.routing.yml  acme_hello:   path: '/acme/hello/{name}'   defaults:     _content: '\Drupal\acme\Controller\DefaultController::hello'     _title: 'acme Title'   requirements:     _permission: 'access content'   hello.html.twig  &lt;div class="acme-hello-text"&gt;   &lt;h1&gt;Hello {{ name }}!&lt;/h1&gt; &lt;/div&gt; On Windows 7 can one make UAC on an admin account prompt for the password, just like a limited account? I am one of the very few who likes the security of UAC and so I have the UAC set to "Always Notify" on my Admin account. But I would be even happier if the system would prompt me for a password, when notifying me.  I can get what I want by adding a second non-admin user and using that account.  But somehow it seems wasteful to have two accounts set up when I will always use the non-admin account day-to-day.  So I am wondering if there is any setting that would cause an Admin account to behave password-prompting-wise as a standard account? "Failed to download repository information" error According with Mumford the answer is yes, but there are some obscure points in the proof.  We know that there exists a very ample line bundle on $A$ since every abelian variety is projective. Hence, via some corollaries of the Theorem of the Cube, we get an ample line bundle whose restriction to the kernel of $n_A$ is both the trivial bundle and very ample. This should imply that the dimension of $\ker n_A$ is $0$.  Furthermore, Mumford says that the fact that the dimension of the kernel is $0$ implies the surjectivity of $n_A$.  Is anybody able to clarify these two points to me?  Thanks in advance! How to correctly install libpng12-0 on the Ubuntu 19.10? For  I need to install libpng12-0 package on the Ubuntu 19.10 system.  I know that it is available .  So I have decided to download this package manually and install it by hand.  On previous Ubuntu versions (such as 18.04 LTS) this trick ends with success, but here on 19.10 it fails in very strange way:   wget http://mirrors.kernel.org/ubuntu/pool/main/libp/libpng/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb sudo apt-get install ./libpng12-0_1.2.54-1ubuntu1.1_amd64.deb  0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/116 kB of archives. After this operation, 285 kB of additional disk space will be used. Get:1 /home/user/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb libpng12-0 amd64 1.2.54-1ubuntu1.1 [116 kB] Selecting previously unselected package libpng12-0:amd64. (Reading database ... 212864 files and directories currently installed.) Preparing to unpack .../libpng12-0_1.2.54-1ubuntu1.1_amd64.deb ... Unpacking libpng12-0:amd64 (1.2.54-1ubuntu1.1) ...  dpkg: error processing archive /home/user/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb (--unpack):  unable to install new version of '/lib/x86_64-linux-gnu/libpng12.so.0': No such file or directory No apport report written because the error message indicates an issue on the local system                                                                                          Errors were encountered while processing:  /home/user/libpng12-0_1.2.54-1ubuntu1.1_amd64.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   What is wrong here? How can I get libpng12-0 installed on Ubuntu 19.10 system?    Note: the same happens on 19.04, but it is near EOL. Footnotes in tables? I would like to have footnotes in tables, but the \footnote command does not work inside a table environment.  Googling one finds that this is , at least in the UK.  That link contains a list of solutions to the problem, but since the author of the FAQ does not believe that this is good style, no comparison is made between the different methods.  I am sure that there are good typographical reasons why in most situations it is undesirable to have footnotes in tables (although I have never come across them), however I have recently felt the need to do just this in a paper I am currently writing.  Therefore I'd like to ask two questions:   What are the arguments against footnotes in tables? If one really must go against the advice in 1., which method (whether considered in the  or not) would you recommend?   Thanks in advance.  Added  In response to ShreevatsaR's comment: I do not much care for where the text of the footnote appears.  I do however insist on the footnote marks being attached to the relevant entries in the table, so that I can avoid having to reference "the second entry in the eighth line of the table", etc... I've read a great article about the . But there are still some points that are unclear.  Why do we say...   school project but not school's project? car service but not car's service or even cars' service (plural form)? apostrophe usage but not apostrophe's usage or apostrophes' usage? Alignment issue in 'Your Sites' section in the Android app There is some unwanted extra spaces in between each site in the Your Sites section.  Steps to reproduce:   Touch the Nav Icon in the left top corner Touch the EDIT link in YOUR SITES Scroll the screen; there is unwanted spaces in between the each site   For reference I recorded the same:    Configurations:   App Version: 1.0.85 Device Manufacturer: Motorola Device Model: XT1033 OS Version: 5.1 (2) Which package can be used to write BNF grammars? My kids are playing Minecraft on 2 PCs, creative mode.  They are connected by the way of one of them starts a single player game on the other one than joins that game over the LAN.  Can I, using Minecraft PE on my iPad, join their game as well? Add new JDBC Driver to connect Mathematica to SQL database Locking an iPhone if button is broken? My friend's iPhone's sleep/wake button no longer works. Is there any way to turn off the screen besides setting up auto-lock and waiting a few minutes? A book on quantum mechanics supported by the high-level mathematics I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators etc, certainly the modern mathematics). If there isn't something similar please give me a reference to the book that is strictly supported by mathematics (given a set of mathematically descripted axioms author develops the theory using mathematics as a main tool). The scroll bar is hidden on MacBook if the user doesn't move the mouse for some time.  This pattern is also used in other devices too, for example Android devices.  The scroll bar does not take much place but it informs a user of where on page he is.  Is it a good approach hiding the scroll bar when the user is presumed viewing the content? In my object record type i have created formula field. Now i want my formula field to work as below.  If Basement__c = NA and Property__r.Basement__c = NA OR  Basement__c = Crawl and Property__r.Basement__c = Crawl OR  Basement__c = Slab and Property__r.Basement__c = Slab Then it will return Type__c(formula field) = "MAtch"  And (Another rule) Basement__c DOES NOT contain any of the following "N/A,Crawl,Slab,None and Property__r.Basement__c Does not contain any of this then formula field will return "HAs basement"  How can i achieve this? Please help. Custom 500 Error page: Is there a way to theme a fatal error? In /core/includes/errors.inc, when there is a fatal error it outputs a response with the message "The website encountered an unexpected error. Please try again later." and a back-trace, if error_level is set to verbose, and without the back-trace otherwise.  It seems like at this point, there's no way, with an event subscriber to capture the response and alter it.  Is that correct?  Or is there a way to wrap the output from line 268 of errors.inc in Drupal 8.4 with markup?    Note, there's another , but that doesn't offer a complete solution.  Also, I looked into .htaccess and Apache ErrorDocument directive, but that . Once Apache passes off to PHP, it can no longer use that directive.  That's for real server errors. How to prove reflexivity, symmetry and transitivity for the following relation? To env or not to env