I need help to solve initial-value quasilinear problem: $xu_y - yu_x = u$, $u(x,0)=h(x)$  Here what I did: The initial curve $\Gamma: &lt;x=s, y-0, z=h(s)&gt; $  The characteristic equation $\frac{dx}{dt}=-y$, $\frac{dy}{dt}=x$, $\frac{dz}{dt}=z$.. So $x=-yt + s$, $y=xt$ and $z=h(s)e^t$. Solve for $s$ I get $s=\frac{x^2 + y^2}{x}$ then solution for this problem is  $u=h(\frac{x^2 + y^2}{x})e^{y/x}$. Is it right? I am not sure because the answer from text book, PDE 4th Edition by Fritz John, is $u=h(\sqrt{x^2+y^2})e^{arctan(y/x)}$  Please help.  Thanks
I am trying to find the equation of characteristic curves and solution of the partial differential equation $$x\frac{\partial u}{\partial y}-y\frac{\partial u}{\partial x}=u$$ $u(x,0)=\sin(\frac{\pi}{4}x)$ .   What I did: $$\frac{dx}{-y}=\frac{dy}{x}=\frac{du}{u}$$ with  From first two equalities $$\frac{dx}{-y}=\frac{dy}{x}$$ by solving we get $x^2+y^2=c_1$ for some constant $c_1$. Now I am stuck with the third and rest of the equalities to come up with a suitable manipulation to solve the problem. How can I do this? Any help would be great. Thanks.
Using "recently" with past simple Is it fine to use recently with the past simple tense in the following sentence?     Did you read any books recently?   Is it grammatically correct?  Or do I have to use:     Have you read any books recently? So i was reading about  on Wiki and I found a interesting thing written there .    It says that the S. radius of the universe is as big as the size of the universe?   How is this possible?  Since most the universe is empty space shouldn't the S. radius of our universe be significantly smaller then 13.7 light years? I can use ls -ld */ to list all the directory entries in the current directory. Is there a similarly easy way to just list all the regular files in the current directory? I know I can use find  find . -maxdepth 1 -type f   or stat  stat -c "%F %n" * | grep "regular file" | cut -d' ' -f 3-   but these do not strike me as being overly elegant. Is there a nice short way to list only the regular files (I don't care about devices, pipes, etc.) but not the sub-directories of the current directory? Listing symbolic links as well would be a plus, but is not a necessity. We have a org wide email addressed which has been "Verified" into our sandbox. I understant org wide email addressed is unsupported metadata, so this was achieved as pre deployment step.   Into our repo, we have a workflow that send email alert to that org wide email address. "  Workflow   ... &lt;senderAddress&gt;noreply@mycompany.com.au&lt;/senderAddress&gt; &lt;senderType&gt;OrgWideEmailAddress&lt;/senderType&gt;    The problem we are facing is that we are still getting the following error even if this org wide email address is verified:      You must specify a verified Org Wide Email Address for this sender   type.   Is that a bug in the metadata api or expected behaviour ? Ideally we would like to not have to do that as a post deployment step.   PS: We are running DX under api version 43.0 Super+D key shortcut does not show desktop I recently installed Saucy Salamander on my computer. Previously in Ringtail, Super+D used to show the desktop. However for this release, the Super+D does not seem to work properly... Is it a bug? Get width of a given text as length Probability that an integer number having Poisson distribution is even The probability $P(X=n)$ that an event X takes place $n$ times in a fixed period of time follows the Poisson distribution with parameter $\lambda$ i.e.  $$ P(X = n) = e^{-\lambda} \frac{\lambda ^ n}{n!}$$  I have to evaluate the probability that the event $X$ takes place an even number of times. I know that:   $$ P(X \text{ is even} ) = e^{-\lambda} \cdot \sum_{k=0}^{+\infty} \frac{\lambda^{2k}}{(2k)!}$$  but I can't solve the series.  I guess that I have to use the fact that $e^\lambda = \sum_{n = 0}^{+\infty} \lambda^n/n!$, but I got stuck.  How can I evaluate $P(X \text{ is even})$ (alternative solutions appreciated). What if the answers of a historical locked question become obsolete? My understanding is that a question is historically locked because the information in it is too valuable for the question to be deleted.   But if the "valuable information" becomes obsolete, what happens then?  We cannot modify the question content, or the answers for a historically locked question, right? Irreducible polynomial in field extension I have this problem      Let $L/K$ be a finite field extension and $|L:K|=m$. Let $f(x)$ be an irreducible polynomial of degree $n$ in $K[x]$ and $\gcd(m,n)=1$. Prove that $f$ is irreducible over $L$.   I'm just a beginner so I'm really confused with the $\gcd(m,n)=1$, I don't know how to use it. So help me with this problem. Thank you In Cocoa KVO, why doesn't a change on a NSMutableArray proxy notify observers? download file on form submission I have created a form which has only a submit button. In the submit function I am generating csv file and saving it to the files folder using fputcsv function.  But Now I need extra functionality of submit. After saving the file it should download the file also.  How to do this? Any one have some idea about this? Electrons have spin 1/2, and as they are charged, they also have an associated magnetic moment, which can be measured by an electron beam splitting up in an inhomogeneous magnetic field or through the interaction of the electrons's magnetic moment with an external magnetic field in spectroscopic measurements.   On the other hand, a photon is neutral - how can one measure its spin if there's no magnetic moment? How do we know it has spin 1? Full Name Requirement of Commit Phase The Commitment form on Area 51 requires your full name to be provided. Is this publicly viewable? Why not just use the existing usernames? Troubleshooting advice for: -bash: cannot execute binary file: Exec format error I have a mining rig and am trying to run a program ./garlicoind  but I get this error:  -bash: ./garicoind: cannot execute binary file: Exec format error   Is this a common problem? What does it indicate? I want to make a bootable USB thumb drive to install Windows on my computer. I don't want to use my old DVD to create an image and then upgrade to SP1, so I decided to download an untouched .iso file that already has SP1 in it. My question is this, is there a difference between these two files files?     en_windows_7_home_premium_with_sp1_x64_dvd_620851.iso   vs     en_windows_7_home_premium_with_sp1_x64_dvd_u_676549.iso   What does dvd_u mean, is it somehow different from plain dvd? Also will these work with my retail Home Premium key or do these work only for thoe with TechNet/MSDN accounts? How to render a pixelated, occluded wireframe with freestyle? I'd like to combine these features:    The pixelated look of a wire material with anti-aliasing disabled. Proper occlusion of hidden lines.   A wireframe material produces the first feature, freestyle produces the second. Can a line set be created that produces neither double pixels nor gaps? Line thickness alone does not seem to be the answer.  There is the trick with two materials and z-offset, but I have trouble getting a really clean solution at the outline of the object. So a freestyle solution would be best.    The main problem is that freestyle (to the right) produces double pixels: This problem occurred to me when I came across a similar problem where the radii were taken over only the primes. That question was unanswered, but it seems to me infinitely many circles of radius $1/2, 1/3, 1/4...$ can fit into a unit disk. The area of all those circles would be $\pi \sum_2^\infty 1/n^2 = \pi^3/6 -\pi$, which is less than the area of the unit disk $\pi$. But can the circles actually be packed with no overlaps? How do I add a Pokestop to an area that has none? It is an area where there are tons of businesses but unless I drive about a mile from my work area there are no Pokestops.   I want to add one since we have 1000's of people in the work area, a lot of them play Pokemon Go, and we all can't leave work to drive a mile away to get to a Pokestop area.   Plus it would be nice to be able to capture Pokemon on our breaks. Why Does The Background Image Not Show in Orthographic View? I suspected that installing xfce4 cause some issues in Ubuntu so I removed them using apt-get remove xfce4 xfce4-goodeis.  But when I ran 'dpkg -l' I still can see some libraries and packages for xfce4:  $ dpkg -l | grep -i xfce rc  libexo-1-0:amd64                              0.10.2-2                                 amd64        Library with extensions for Xfce rc  libgarcon-1-0                                 0.2.1-1                                  amd64        freedesktop.org compliant menu implementation for Xfce rc  libxfce4ui-1-0                                4.11.0-0ubuntu1~ppa0.13.10.1             amd64        widget library for Xfce - Gtk+2 variant rc  libxfce4ui-common                             4.11.0-0ubuntu1~ppa0.13.10.1             all          common files for libxfce4ui rc  libxfce4util6                                 4.10.1-1                                 amd64        Utility functions library for Xfce4 rc  libxfcegui4-4                                 4.10.0-2                                 amd64        Basic GUI C functions for Xfce4 rc  libxfconf-0-2                                 4.10.0-2                                 amd64        Client library for Xfce4 configure interface rc  mousepad                                      0.3.0-2                                  amd64        simple Xfce oriented text editor rc  thunar                                        1.6.3-1ubuntu1                           amd64        File Manager for Xfce rc  xfce4-appfinder                               4.10.1-1                                 amd64        Application finder for the Xfce4 Desktop Environment rc  xfce4-clipman                                 2:1.2.3-2ubuntu1                         amd64        clipboard history utility rc  xfce4-mixer                                   1:4.10.0-1ubuntu2                        amd64        Xfce mixer application rc  xfce4-notes                                   1.7.7-3ubuntu2                           amd64        Notes application for the Xfce4 desktop rc  xfce4-panel                                   4.10.1-1ubuntu1                          amd64        panel for Xfce4 desktop environment rc  xfce4-power-manager                           1.2.0-2ubuntu1                           amd64        power manager for Xfce desktop rc  xfce4-session                                 4.10.1-1ubuntu1                          amd64        Xfce4 Session Manager rc  xfce4-settings                                4.11.1-0ubuntu1~ppa0.13.10.1             amd64        graphical application for managing Xfce settings rc  xfce4-terminal                                0.6.2-3ubuntu1.1                         amd64        Xfce terminal emulator rc  xfce4-volumed                                 0.2.0-0ubuntu1                           amd64        volume keys daemon rc  xfdesktop4                                    4.11.2-0ubuntu1~ppa0.13.10.1             amd64        xfce desktop background, icons and root menu manager rc  xfwm4                                         4.11.1-0ubuntu1~ppa0.13.10.1             amd64        window manager of the Xfce project   I copy-pasted all these packages\libraries to form an apt-get remove command to remove all of these dependencies, and the command ran normally, but running dpkg -l | grep -i xfce showed these dependencies again !  Why are these dependencies still installed ?