Show that Multiplication of negative numbers is always positive eg. (-1)*(-1)=1
Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed?
How to allow page break inside a float environment? I'm using new float enviroment to show some examples in my document. This  enviroment contains tcolorbox which provides a gray background. The main text in the enviroment is a mixture of plain text and code blocks (in lstlisting env). Each float should have it's own caption and label.  In style file:  \usepackage{newfloat} \DeclareFloatingEnvironment[name={Example}]{suppfigure}   In the main document:  \begin{suppfigure}[!h] \caption{Some code example}\label{suppfigure1} \begin{tcolorbox}[arc=0mm, left=1pt, right = 1pt, boxrule=0mm,colback = {shadow-gray}]   Text  \begin{lstlisting}  int i = 0;  \end{lstlisting}  Some more text  \begin{lstlisting}  String.Format("{0,10:0.0}", -123.4567);   \end{lstlisting}  \end{tcolorbox}  \end{suppfigure}   How can I allow page break inside the suppfigure to extend my example to multiple pages? Where can I find header files in C on unix In Delphi, is TDataSet thread safe? I'd like to be able to open a TDataSet asynchronously in its own thread so that the main VCL thread can continue until that's done, and then have the main VCL thread read from that TDataSet afterwards. I've done some experimenting and have gotten into some very weird situations, so I'm wondering if anyone has done this before.  I've seen some sample apps where a TDataSet is created in a separate thread, it's opened and then data is read from it, but that's all done in the separate thread. I'm wondering if it's safe to read from the TDataSet from the main VCL thread after the other thread opens the data source.  I'm doing Win32 programming in Delphi 7, using TmySQLQuery from  as my TDataSet descendant. Undefined control sequence error on \fancy@reset, \f@ncyolh and \f@ncyorh when using newlfm class Is $\mathbb{Z}[x]/(x^2+1)$  isomorphic to  $\mathbb{Z}[i]$?   My attempt is that try to define a mapping $g$ from $\mathbb{Z}[x]$ to $\mathbb{Z}[i]$ by $g(f(x))= f(i)$, for $f(x)\in\mathbb{Z}[x]$. If it is possible then $\ker g$ is $(x^2+1)$? Am I on the right track? Please Help. How to determine whether the direction of rotating a 2D vector to another one is counter-clockwise or clockwise using the cross product? Vec2 a, b; // some random vector Vec2 cross = a.Cross(b); RelativeDirection(a,b,cross); // return true if the direction of                               // rotating a to b is counter-clockwise   How do I implement the function RelativeDirection()?  I got some suedo code from the textbook:  if a.Cross(b) &gt; 0 then ... else ...   but the cross product is a vector right? how do you compare it with an integer? Radius of convergence of a sum of power series I have  two series  $\displaystyle\sum_{n=1}^{\infty} a_n x^{n}$   $\displaystyle\sum_{n=1}^{\infty} b_n x^{n}$  with radius of convergence $2$ and $3$ respectively.  How can I find the radius of convergence for the following?     $\displaystyle\sum_{n=1}^{\infty} (a_n +b_n)x^{n}$   Can anyone help me? If 20 independent Bernoulli trials are carried out each with a different probability of success and therefore failure. What is the probability that exactly n of the 20 trials was successful?  Is there a better way of calculating these probabilities rather than simply summing together the combinations of success and failure probabilities? I tried to install WPS Office on my laptop (ubuntu 16.10 Gnome, 64 bit) with  Gdebi installer but it said   Dependency is not satisfiable: libpng12-0   Then I tried to open this library as follows  sudo apt-get install libpng12-0   But I encountered an error:  Reading package lists... Done Building dependency tree        Reading state information... Done Package libpng12-0 is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'libpng12-0' has no installation candidate   Can anyone help? Thanks a lot Git is automatically merging changes from a different branch to master I saw that it has been already reported that also they have earned courious badge but the badge has not been given. but their answers say that they must wait a day and the badge will be awarded.  In my case its 8/5 and it was 5/5 long time ago and I have never received a message telling that I have earned it. its just telling me that the next badge to be awarded is Curious badge.  this is my profile: Send data from Lightning Web Component (LWC) to a Visualforce page I am adding an LWC component in visualforce and need to add a handler for an event fired in the LWC component on the visualforce page. When googling all I can find is examples of how to do this with aura component using application events. LWC controller: handleSearchClick() {              const searchinfo = {          shipper: this.shipper,         system: this.system,         product: this.product,         status: this.template.querySelector('c-lwc-multi-select').value(),         period: this.period       };     this.dispatchEvent(new CustomEvent(         'dosearch',          {             detail: { data:  searchinfo},             bubbles: true,             composed: true,         }     ));     console.log({ 'lwcNominationSearch searchinfo': searchinfo }); }  AURA app: &lt;aura:application extends=&quot;ltng:outApp&quot; access=&quot;GLOBAL&quot;&gt; &lt;aura:dependency resource=&quot;c:lwcNominationSearch&quot; /&gt; &lt;aura:dependency resource=&quot;markup://force:*&quot; type=&quot;EVENT&quot;/&gt; &lt;/aura:application&gt;  Visualforce: &lt;div id=&quot;rcontainer&quot;&gt; &lt;/div&gt;  &lt;script&gt;     $Lightning.use(&quot;c:NotificationSearch_ltngoutApp&quot;, function()     {         $Lightning.createComponent(         &quot;c:lwcNominationSearch&quot;,         {},         &quot;rcontainer&quot;,         function(cmp)         {             console.log('component created');              //$A.eventService.addHandler({ &quot;event&quot;: &quot;c:myEvent&quot;, &quot;handler&quot; : visualForceFunction})               $A.eventService.addHandler({                 event: 'dosearch',                 handler: function (event) {                          console.log(&quot;vf event success&quot;);                 }             });          });     }); &lt;/script&gt; Storing the original document class page layout with geometry How to change Facebook username? Do I have to re-register a new Facebook account to change my facebook.com/USERNAME?  Or are there any ways to change it? Java String array: is there a size of method? Reprojecting MODIS Swath data to WGS84 using GDAL Recursive time series forecasting model In a recursive forecasting model, let's say you are trying to predict sales of Target for the next month and you will append that prediction to your input and predict the month after. Basically, your target is Sales quantity, which you lagged. But let's say you have other continuous numerical features like Price which you will lag as well. As your model is making predictions on Sales_QTY, it will feed it back to sales_qty as input so you will never ran out of values for that one. So how do you deal with your other features (other than the target) in order to generate them as next months inputs in a recursive forecasting model (because eventually you will run out of them)? DO you create a sub-model and try to predict them as well? Distance between footnote and the line above it This is rather difficult to describe with fancy pictures with red circles but it's easily reproducible.  Whenever I click Leave Closed in the re-open queue on MSO the "nominate this question for re-opening" pop-up appears. I normally press escape out of shock and a leave closed vote is recorded.  This is the latest review it's happened to me on:   You can also test it on , I clicked reopen this time and a leave closed vote was recorded but I also got a red popup stating that an error had occurred.  This is also occurring when the Edit &amp; Reopen button is clicked, the edit box opens for a second before being overridden by the popup. I'm experiencing a bug where when the machine comes back from standby a monitor connected via displayport (MST) does not come back on, however the windows are still in the area it cover if it did come on.  The only way to recover from this short of a restart is to disable and reenable the display now I can do that via the command line just fine but when the display is disabled all the windows get rearranged onto the remaining screen.  Is there a way to persist the window positions and restore them so my script could do this sequence:-   enumerate windows, store positions display off (messing up positions) display on (making the display work again, yay!)  restore position of the windows   I can do step 2 and 3 just fine but I'm lost for step 1 and 4  I know I can force a position per app but that's not what I want as I could be doing different things and have the same apps in different positions, I also have multiple terminals open, I need the current positions storing somehow and restoring.  Any help would be appreciated!