I'm wondering what the math would be to getting a grenade arc (like you're throwing it). In my game you would know the point in 3D that you'd be aiming at so the idea would be to arc it from it's starting point to exactly that point, but I'm not sure of the math on getting all the points between so I can smoothly move it without physics. Any advice or pointing in a direction would be great. Thank you!
If I have a point which I want to hit at the end or during a parabolic arc, how would I calculate the needed x and y velocity?
show that Cov(X+ Y, X-Y)= Var(X) - Var(Y) Given matrices $A$ and $B$, where that $AB = A + B$, prove $AB = BA$. I keep coming up with AB = AB. It seems like basic algebra, but for the life of me, I'm getting nowhere :/. Someone help please? Let us consider that $u\in C^2(\Omega)\cap C^0(\Omega)$ and satisfying the following equation .  $\Delta u=u^3-u , x\in\Omega$ and   $u=0 $   in  $\partial \Omega$  $\Omega \subset \mathbb R^n$ and bounded .   I need hints to find out what possible value $u$ can take ?  Thank you for ur hints in advance . Validate/verify PDF files integrity Is there any tool which I can run on the PDFs archives (all directories) and finally it lists/identifies the corrupt/invalid PDFs? I have hundreds of PDF files (related to documentation etc.) on my computer (Windows machine), and very often I receive/have to send dozens of PDFs via email. It is now a normal routine that the PDF I receive or send is corrupted. This creates serious trouble sometimes, when the source file (e.g. Word file or the Tex file) is missing/not available instantly. Checking these thousands of PDFs is not possible in finite time, so I have searched for a tool which I can run once and it would scan all PDFs (in directories and sub-directories), and finally I would get a list of files which I should re-create. Up to now, it seems there is no such tool. What is a terminal and how do I open and use it? How to cleanly launch a GUI app via the Terminal? It is well known that quantum mechanics and (general) relativity do not fit well. I am wondering whether it is possible to make a list of contradictions or problems between them?  E.g. relativity theory uses a space-time continuum, while quantum theory uses discrete states.  I am not merely looking for a solution or rebuttal of such opposites, more for a survey of the field out of interest. find the value of limit Find the value of $$ \lim_{n\to\infty} \sum_{k=n+1}^{2n} \frac{1}{k} $$ I think it become a series with each denominator approach infinity. so the value of the limit is 0. am i true ? Does $\sum_{n=1}^\infty \frac{\cos{(\sqrt{n})}}{n}$ converge? Show if a user is online, idle, or offline on their account page? I want to display the status of the user on the account page. I added the following code in the theme I am using. It works, but users are never displayed offline. Why? What is wrong in the code? user.html.twig &lt;div class=&quot;bs-field-status&quot;&gt;   {% if status == 'Online' %}     &lt;i class=&quot;user-online fa fa-circle fa-lg&quot;&gt;&lt;/i&gt; Online   {% elseif status == 'Absent' %}     &lt;i class=&quot;user-absent fa fa-circle fa-lg&quot;&gt;&lt;/i&gt; Absent   {% else %}     &lt;i class=&quot;user-offline fa fa-circle fa-lg&quot;&gt;&lt;/i&gt; Offline   {% endif %} &lt;/div&gt;  bootstrap_subtheme_front_office.theme &lt;?php /**  * @file  * Bootstrap sub-theme.  *  * Place your custom PHP code in this file.  */  use Drupal\Core\Database\Database;  /**  * Implements hook_entity_presave().  */ function bootstrap_subtheme_front_office_preprocess_user(&amp;$variables) {   // get user object   $user = $variables['elements']['#user'];   //- The user has logged in at least once   if ($user-&gt;getLastLoginTime()) {     if (account_is_logged_in_less_then_thirty_minutes($user-&gt;id())) {       $status = 'Online';     }     else {       $status = 'Absent';     }   }   else {     $status = 'Offline';   }   $variables['status'] = $status; }  /**  * @param $uid  *  * @return bool  */ function account_is_logged_in_less_then_thirty_minutes($uid) {   $connection = Database::getConnection();   $query = $connection-&gt;select('sessions', 'sessions')     -&gt;fields('sessions', ['sid', 'uid', 'timestamp'])     -&gt;condition('sessions.uid', $uid, '=')     //- chef if the user was online in 30 minutes (60 * 30)     -&gt;condition('sessions.timestamp', \Drupal::time()         -&gt;getRequestTime() - (60 * 30), '&gt;')     -&gt;execute();   //- Get result.   $results = $query-&gt;fetchAll(\PDO::FETCH_OBJ);   return (count($results) &gt; 0) ? TRUE : FALSE; } I have an Excel file that has a bunch of VBA and macro code in it.  When I open the file in Excel I can choose not to 'enable' them - so the values in the fields all stay as they were during the last save.  I need to manipulate the values as they were last saved - so I don't want the macros (which look at the current date and update values accordingly) to run.  When I open it via our dot net code:   Dim oxlRep As Excel.Application Dim oWBRep As Excel.Workbook Dim oSheetRep As Excel.Worksheet Dim oRngRep As Excel.Range oxlRep.Open(path)   the vb code runs - throwing off the values.  I've been looking for a way to open it without macros, or in 'secure' mode where the macros aren't run.  If I simply double click the file and don't choose to enable macros the values are all there as I want them.    Usually we run this code within the month that the files are created, so we haven't seen this problem in the 3 or 4 years that it has been working.  Now I need to go back to some of the old files and run some archival code...    Anyone have a suggestion? Surface tension: the paper clip experiment In the paper clip experiment, the surface tension of water prevents the clip from falling, thus we can assume it exerts a force of $mg$ (weight of clip) upwards.   However, if you try to pull the clip out of water, surface tension opposes this motion also. If for simplicity we assume that the clip is a rectangle of length $l$ and breadth $b$, and the surface tension is $T$, then the liquid exerts a downward force of: $$T×2(l+b)$$  What is going on here? Why is there an inconsistency in the direction along which the force acts? No space following macro without argument I am using texlive 2013, and just ran the following command to update the packages.  sudo ./tlmgr update  -all   After it finished, I tried to compile tex documents that uses tikz package as the following MWE shows:   \documentclass{article}  \usepackage{tikz}  \begin{document}       aaaa  \end{document}   Then pdflatex complains; it said:   (/usr/local/texlive/2013/texmf-dist/tex/latex/pgf/utilities/pgfrcs.sty  (/usr/local/texlive/2013/texmf-dist/tex/generic/pgf/utilities/pgfutil-common.tex  ! I can't find file `pgfutil-common-lists.tex'.  l.556 \input pgfutil-common-lists.tex   The problem is that, pgfutil-common-lists.tex is indeed in the directory:   /usr/local/texlive/2013/texmf-dist/tex/generic/pgf/utilities/   Why is that? pdflatex should be able to find it, right? I even download the  again, and replace the pgf directory in   /usr/local/texlive/2013/texmf-dist/tex/generic/   Nothing changed at all........... Any idea? Thank you for your critics.  %=====================================================================  Then followed the comments, I type the following command in the terminal:    sudo mktexlsr gives me command not found;  mktexlsr gives me:  mktexlsr: /usr/local/texlive/2013/../texmf-local: directory not writable.     Skipping...  mktexlsr: /usr/local/texlive/2013/texmf-config: directory not writable. Skipping...  mktexlsr: /usr/local/texlive/2013/texmf-dist: directory not writable. Skipping...  mktexlsr: /usr/local/texlive/2013/texmf-var: directory not writable. Skipping... mktexlsr: Done; Forkpsewhich pgfutil-common-lists.tex, nothing pops up. Why the CMB has not been dispersed so far? Imagine you have a box of black body radiation. What happens if you open the box for a long time? It becomes dispersed and no radiation remains in the box.   Now, apply this example to the Cosmic Microwave Background radiation. The CMB has been produced about 380,000 years after the Big Bang. Giving that the space is flat as many observations suggest, that radiation has been produced in a universe with no boundaries.   Now, my question is this: in these conditions, why that radiation has not been dispersed completely so far? It is true that the radiation has been produced everywhere in the space but giving that the space is infinite, why has not it been dispersed do far?   Calculations in the standard textbooks are done in such a way as if the CMB has been within a physical box, however expanding with the universal expansion. But, this is not the actual situation. The CMB has not been and is not enclosed by walls of a box, so it must have completely dispersed so far. What has prevented this to happen? A Good Book for Mathematical Probability Theory Show that $$\lim_{x \to \infty} \frac{\log(1 + e^x)}{x} = 1$$ How do I prove this? Or how do we get this result? Here $\log$ is the natural logarithm. A method for evaluating sums/discrete functions by assuming they can be made continuous and differentiable? Suppose I had a function that satisfied the property $f(x)=f(x-1)+g(x)$.  For any $x\in\mathbb N$, it is easy enough to see that this boils down to the statement  $$f(x)=f(0)+\sum_{k=1}^xg(k)$$  If we return to our functional equation and differentiate it $n$ times, we get  $$f^{(n)}(x)=f^{(n)}(x-1)+g^{(n)}(x)$$  Once again, for any $x\in\mathbb N$, we have  $$f^{(n)}(x)=f^{(n)}(0)+\sum_{k=1}^xg^{(n)}(k)$$  One could then integrate both sides.  For example, this asserts that  $$\sum_{k=1}^xg(k)=xf'(0)+\int_0^x\sum_{k=1}^tg'(k)dt$$  If the sum inside the intagral is generalized to any upper bound.  How would I justify this for discrete functions?    An example of this is letting $f_p(x)=\sum_{k=1}^xk^p$, as it seems to follow the above result.  We can see from the above that  $$f_p(x)=a_px+p\int_0^xf_{p-1}(t)dt$$  $$a_p=1-p\int_0^1f_{p-1}(t)dt$$  which appears to be a true recursive formula.    Under what conditions can I extend discrete functions $f(x)$ and $g(x)$ into continuous differentiable functions and apply this method? Proving $\arg(zw)=\arg(z)+\arg(w)$ Ubuntu 11.10 is installed, and I have updated my system via Ubuntu Software Centre, and after the update 3 updates remain in the list. But I am unable to select them, the tickboxes cannot be checked and Ubuntu Software Centre gives no error when I try to check.  Also, below it says "There is no update to be installed" and even though there seems to be updates Install Updates button is unclickable.  What should I do in order to install these updates?