I'm supposed to solve $u_{xx}-3u_{xt}-4u_{tt}=0$ with initial conditions $u(x,0)=x^2$ and $u_t(x,0)=e^x$.   So I factored the problem into $(u_x-4u_t)(u_x + u_t)$ and set each equal to 0 and found the 2 solutions to be $(x,t)=f(x+t/4)$ and $u(x,t)=g(x-t)$  Then I get $u(x,t)=1/2[\phi(x+t/4) + \phi(x-4)]$ + 1/2c $\int_{x-t}^{x+t/4} e^{s} ds$  =1/2[$(x+t/4)^2 + (x-t)^2$] + 1/2c[$e^{x+t/4}-e^{x-t}$]  I multiply out the first part then and get a solution of:  1/2[$5/2xt-15/16t^2$]+1/2c[$e^{x+t/4}-e^{x-t}$]  But the book has a solution of $x^2 + t^2/4 + 4/5[e^{x+t/4}-e^{x-t}]$  I don't understand how they get 4/5 in front of the exponentials and how $1/2[\phi(x+t/4) + \phi(x-4)]$ turns into $x^2 + t^2/4$ .  Can anyone see where I went wrong?
Solve $$u_{xx}-3u_{xt}-4u_{tt}=0$$ where $u(x,0)=x^2$ and $u_t(x,0)=e^x$.  My workings so far: I have factored the differential equation in the following way: $$(\delta_x-4\delta_t)(\delta_x+\delta_t)=0$$ where $\delta_x=\frac{\delta}{\delta x}$ etc. Now if we let $v$ be the solution to $(\delta_x+\delta_t)u$ then we have the following two equations: \begin{eqnarray*} (\delta_x+\delta_t)u=u_x+u_t=v\\ (\delta_x-4\delta_t)v=v_x-4v_t=0 \end{eqnarray*} Now for $v$ we simply find $$v=h(t+4x)$$ where $h$ is an arbitrary function of one variable. Now what remains is find $u$ such that $$u_x+u_t=h(t+4x)$$ I am stuck here, I thought about making a change of variables $\zeta=x+t$ and $\eta=x-t$ and thus using the product rule to show that $u_x=u_{\zeta}+u_{\eta}$ and $u_t=u_{\zeta}-u_{\eta}$ and thus $u_x+u_t=2u_{\zeta}$ and we need to solve $$u_{\zeta}=h(t+4x)$$ (I left out the factor 2 because $h$ is an arbitrary function). Do I simply integrate now and conclude $$u=f(\zeta)h(t+4x)+g(\eta)=f(x+t)h(t+4x)+g(x-t)$$ This seems wrong to me...  Some help would be greatly appreciated!
Array index out of bound behavior Why does C/C++ differentiates in case of array index out of bound #include &lt;stdio.h&gt; int main() {     int a[10];     a[3]=4;     a[11]=3;//does not give segmentation fault     a[25]=4;//does not give segmentation fault     a[20000]=3; //gives segmentation fault     return 0; }  I understand that it's trying to access memory allocated to process or thread in case of a[11] or a[25] and it's going out of stack bounds in case of a[20000]. Why doesn't compiler or linker give an error, aren't they aware of the array size? If not then how does sizeof(a) work correctly? When to use "click" and "click on" Can apostrophe 's' be used with 'etc.'? Suppose that $y_1(x)$ and $y_2(x)$ are solutions of the differential equation $y''+py'+qy=0$ on $I$. How can I show that if $y_1$ and $y_2$ vanish at the same point then they are not linearly independent?   Here is my attempt to prove the problem. Since $y_1(x)$ and $y_2(x)$ are solutions of the differential equation $y''+py'+qy=0$ on $I$, it is enough for me to show that the Wronskian of $y_1$ and $y_2$ denoted by $W(y_1,y_2)$ is zero. Now since $y_1$ and $y_2$ vanish at the same point say $p$ we have $y_1(p)=0$ and also $y_2(p)=0$.   Solving for $W(y_1,y_2)(p)$ we have: $y_1(p)y_2'(p)-y_2(p)y_1'(p)=0$ since $y_1(p)=0$ and also $y_2(p)=0$.   Am I correct? Thanks No Drupal Site Found when clearing cache I installed drush using composer following the instructions here:   When trying to run drush cc all I get this error:     No Drupal site found, only 'drush' cache was cleared.[warning] 'all'   cache was cleared. [success]   I added this line:  export PATH="/Applications/MAMP/bin/php/php5.6.2/bin:$PATH"   to my .bash_profile and restarted bash and MAMP. Still no luck. Any ideas?   This is my drush status:   Drupal version         :  7.36                                                               Site URI               :  http://default                                                     Database driver        :  mysql                                                              Database hostname      :  localhost                                                          Database port          :                                                                     Database username      :  root                                                               Database name          :  srod                                                     PHP executable         :  /usr/bin/php                                                       PHP configuration      :                                                                     PHP OS                 :  Darwin                                                             Drush script           :  /Users/srod/.composer/vendor/drush/drush/drush.php   Drush version          :  7.0-dev                                                            Drush temp directory   :  /tmp                                                               Drush configuration    :                                                                     Drush alias files      :                                                                     Drupal root            :  /Applications/MAMP/htdocs/srodsite                             Site path              :  sites/default      I'm running it from Drupal project folder. Ignore Pulses Less than a Given Width in a Pulse Train Let's say I have a 5V pulse train that varies between 0 and 75000 pulses per second. Pulse width, let's say, is 5 microseconds. How could I calculate or determine the best way to filter out all pulses less than 4 microseconds in width? How to describe a guy who is popular with girls? Perhaps I should make it clear:  - He naturally attracts girls.  - He doesn't chase girls and have no intention for any relationship.  - You just see him often together with girls. A doubt from Milnor's "Topology from a Differentiable Viewpoint". This is a doubt from Milnor's "Topology from a Differentiable Viewpoint".     For a smooth $f:M\to N$, with $M$ compact, and a regular value $y\in N$, we define $n(f^{-1}(y))$ to be the number of points in $f^{-1}(y)$. It is easy to see that $n(f^{-1}(y))$ is finite. It can also be seen that $n(f^{-1}(y))$ is locally constant; i.e. there is a neighbourhood of $y$ such that for every point $y'$ in that neighbourhood, $n(f^{-1}(y'))=n(f^{-1}(y))$.   The proof for the statement in bold is the following:     Let $x_1,x_2,\dots,x_n$ be the points of $f^{-1}(y)$, and choose disjoint neighbourhoods $U_1,U_2,\dots,U_n$ of these points, which are mapped diffeomorphically into neighbourhoods $V_1,V_2,\dots,V_n$ of $y$. The required neighbourhood is $(V_1\cap V_2\cap\dots V_n)-f(M-U_1-U_2-\dots U_n)$.   I don't understand how $(V_1\cap V_2\cap\dots V_n)-f(M-U_1-U_2-\dots U_n)$ is open. I know that $(V_1\cap V_2\cap\dots V_n)$ is open. However, how can one prove that $f(M-U_1-U_2-\dots U_n)$ is closed? I can see this working if $f$ is surjective, but not otherwise. `sudo echo "bla" >> /etc/sysctl.conf` permission denied Disclaimer: I'm pretty novice at sysadmin stuff.  I'm trying to set up port forwarding in an AWS EC2 instance, this has to be done in the command-line because I don't want to go in and edit anything, it has to be automatic (it's part of a build process).   sudo echo "net.ipv4.ip_forward = 1" &gt;&gt; /etc/sysctl.conf      Permission denied   The weird thing is I've been (successfully) using sudo for pretty much every command that required su privileges. If I do sudo su before the command (trying it out by hand in an ssh session), then it works.  Reasons behind this? Possible solutions that don't involve sudo su or manual edits? If tenured staff are virtually unsackable, why is the drive to find funding so strong? Is there a way to search the latest version of every file in TFS for a specific string or regex? This is probably the only thing I miss from Visual Source Safe...    Currently I perform a Get Latest on the entire codebase and use Windows Search, but this gets quite painful with over 1GB of code in 75,000 files.    EDIT: Tried the powertools mentioned, but the "Wildcard Search" option appears to only search filenames and not contents.  UPDATE: We have implemented a customised search option in an existing MOSS (Search Server) installation. Does a Warlock receive the benefit of their familiar's Magic Resistance trait? The Warlock has access to special forms for their find familiar spell through the Pact of the Chain feature. I'm wondering if these special forms, the Quasit, Imp, and Pseudodragon in particular, allow the player character to share the familiar's Magic Resistance feature. In the Monster Manual each of these creatures has a sidebar that states that the familiar shares its Magic Resistance feature with the companion they are bonded to, but the PHB doesn't mention this in any of the creatures' stat blocks.   For example, in the Variant: Pseudodragon Familiar sidebar on page 254 of the MM it says:     "While the pseudodragon is within 10 feet of its companion,        the companion shares the pseudodragon's Magic Resistance trait."   This feature seems clear in the MM but I'm led to believe that it wasn't intended for Player Characters since mention of it is absent from the stat blocks in Appendix D of the PHB (pages 307-309).  Would a 3rd level warlock who chose one of these familiar forms through the Pact of the Chain feature benefit from advantage from saving throws from spells and other magical effects due to these special forms, or was this feature only meant for powerful NPCs and enemy spellcasters who had made this link with these familiars themselves? How do I enable hibernation on Wubi? I installed Ubuntu on my Windows 7 Sager laptop using Wubi. Hibernate (i.e. suspend to disc) is not an option from the power icon, only suspend, shutdown, etc. Hibernate is also not an option from my battery/lid close options. I understand that hibernation is disabled by default in Ubuntu 12.04. I tried running pm-hibernate but I get the following message:  Looking for splash system... none  s2disk: Snapshotting system   and then the computer just hangs with a black screen. According to the documentation  if this fails then I can't enable hibernate but it offers no help in making pm-hibernate succeed.  Could swap be my problem? It looks like I have very small swap:  user@ubuntu:~$ cat /proc/swaps Filename                Type        Size    Used    Priority /host/ubuntu/disks/swap.disk            file        262140  0   -1   The advice on  is only for the author's set up (e.g. I don't have an Ubuntu install disk since I used Wubi) and he says that 'INFO: This will not work for 12.04, resume from hibernate work differently in 12.04.'  Any advice? I really need to get hibernate working to use my laptop as a, er, laptop.  Thanks I was so good with windows, now I’m feeling like a complete idiot.  I have an SSD for ubuntu and want to install the majority of my programs to a separate HDD but I have no clue how. All searches I have tried only turn up with dual-boot installations.   (RESOLVED -> ) I am also having trouble locating "/etc/". Specifically, to modify Conky. Searching for this term only turns up instances where people are telling you to access this file, not how to get to it. Web applications that want to force a resource to be downloaded rather than directly rendered in a Web browser issue a Content-Disposition header in the HTTP response of the form:  Content-Disposition: attachment; filename=FILENAME  The filename parameter can be used to suggest a name for the file into which the resource is downloaded by the browser.  (Content-Disposition), however, states in  (The Filename Parameter) that the file name can only use US-ASCII characters:     Current [RFC 2045] grammar restricts   parameter values (and hence   Content-Disposition filenames) to   US-ASCII.  We recognize the great   desirability of allowing arbitrary   character sets in filenames, but it is   beyond the scope of this document to   define the necessary mechanisms.   There is empirical evidence, nevertheless, that most popular Web browsers today seem to permit non-US-ASCII characters yet (for the lack of a standard) disagree on the encoding scheme and character set specification of the file name. Question is then, what are the various schemes and encodings employed by the popular browsers if the file name “naïvefile” (without quotes and where the third letter is U+00EF) needed to be encoded into the Content-Disposition header?  For the purpose of this question, popular browsers being:   Firefox  Internet Explorer  Safari Google Chrome  Opera He is loved.   This is something that I've always kind of wondered.  In a sentence like this, is loved a verb or an adjective? Can it be considered either? Use Complete Induction of set theory to prove $\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + ... + \frac{1}{\sqrt{n}} > \sqrt{n}$. Proove by the Complete Induction for every $n\in \mathbb{N}, n \geq 1$ $$\frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + ... + \frac{1}{\sqrt{n}} &gt; \sqrt{n}$$.  I know only the basis - $\frac{1}{\sqrt{1}} \geq \sqrt{1}$.  But, I dont know how to do the change from the regular induction to the induction of Set theroy(Complete Induction). public static IQueryable&lt;TResult&gt; ApplySortFilter&lt;T, TResult&gt;(this IQueryable&lt;T&gt; query, string columnName)   where T : EntityObject {   var param = Expression.Parameter(typeof(T), "o");   var body = Expression.PropertyOrField(param,columnName);    var sortExpression = Expression.Lambda(body, param);   return query.OrderBy(sortExpression); }   Because the type for OrderBy is not inferred from sortExpression I need to specify it something like this at run time:  var sortExpression = Expression.Lambda&lt;T, TSortColumn&gt;(body, param);   Or  return query.OrderBy&lt;T, TSortColumn&gt;(sortExpression);   I don't think this is possible however as TSortColumn can only be determined during runtime.  Is there a way around this? If every compact set is closed, then is the space Hausdorff? Express symmetric polynomial $\prod_{i < j} (X_i+X_j)$ in terms of elementary symmetric functions Exercise: Define a polynomial $\Sigma(X_1,\ldots,X_n)$ as \begin{align*} \Sigma(X_1,\ldots,X_n) = \prod_{i &lt; j} (X_i+X_j) \end{align*} This is a symmetric polynomial, quite clearly. I want to express $\Sigma$ as a polynomial in the elementary symmetric functions, for $n=2,3,4$. Case $n=2$ is trivial: $\Sigma(X_1,X_2)= \sigma_1 = X_1+X_2$. Case $n=3$ is easy. I write $\Sigma$ as follows: \begin{align*} \Sigma(X_1,X_2,X_3) = (\sigma_1 - X_1)(\sigma_1 - X_2)(\sigma_1 - X_3), \end{align*} then apply  formally, obtaining \begin{align*} \Sigma(X_1,X_2,X_3) = \sigma_1^3 - \sigma_1 \sigma_1^2 + \sigma_2 \sigma_1 - \sigma_3 = \sigma_2 \sigma_1 - \sigma_3. \end{align*} The case $n=4$ seems harder to me, and I'm unable to work it out. I hope there is some trick to avoid unnecessary computations. My attempt is to write down $\Sigma$ completely: \begin{align*} \Sigma = (X_1+X_2)(X_3+X_4)(X_1+X_4)(X_2+X_3)(X_2+X_4)(X_1+X_3) \end{align*} and then multiply factors in groups of two, obtaining: \begin{align*} \Sigma = (\sigma_2 - X_1X_2-X_3X_4)(\sigma_2 - X_1X_4 - X_2X_3)(\sigma_2 - X_2X_4-X_1X_3), \end{align*} but it doesn't seem to help much. How can I solve the exercise?