Since on Mathematics stackexchange I didn't get an answer, I'll try it here, since people here are more familiar with this topic (general relativity related).  I am reading a dissertation of Porfyriadis "Boundary Conditions, Effective Action, and Virasoro Algebra for $AdS_3$", and I am trying to solve a system of DE to get the appropriate diffeomorfism (page 31 onward).  I am trying to solve a system of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this:  $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$  which you get by taking a Lie derivative of background metric and setting it equal to certain $\mathcal{O}(r^n)$ terms.  $g_{ij}$ are given, from metric ofc. I need to assume that the solution (since I'm looking for components of $\xi^\mu$( which is a vector with components $\xi^t, \xi^r, \xi^\phi$) is given with power series of the form:  $\xi^\mu=\sum\limits_{n}\xi^\mu_n(t,\phi)r^n$,  and this is to be seen as expansion around 1/r (expansion around $r=\infty$).  Now when I plug this in the ODE I get this  $\frac{2}{l^2}\sum_n\xi^r_nr^{n+1}+2\sum_n\xi^t_{n,t} r^n+\frac{2}{l^2}\sum_n\xi^t_{n,t}r^{n+2}=\mathcal{O}(r)$, where  $\xi^\mu_{n,i}$  is the derivative with the respect to i-th component.  What troubles me is, how to expand this? Do I set n=0,-1,-2,... until my O(r) terms cancel each other out? Or?  I'm kinda stuck, at this seemingly easy point.  In the thesis he gets 6 equations with coefficients, first one should be:  $\xi^r_{n-1}+l^2\xi^t_{n,t}+\xi^t_{n-2,t}=0,\ n\ge 2$,  But I am not getting this. What am I doing wrong?    EDIT: For further clarity: The metric is that of $AdS_3$ given with line element:  $ds^2=-\left(1+\frac{r^2}{l^2}\right)dt^2+\left(1+\frac{r^2}{l^2}\right)^{-1}dr^2+r^2 d\phi^2$,  and the differential equations in question are given by solving $\mathcal{L}_\xi g_{\mu\nu}=\mathcal{O}(r^n)$, where $\mathcal{O}(r^n)$ are the fall off conditions. In the dissertation, he took the deviation of nonzero components of the metric to be subleading, that is:  $\mathcal{L}_\xi g_{tt}=\mathcal{O}(r)$ $\mathcal{L}_\xi g_{rr}=\mathcal{O}(r^{-3})$ $\mathcal{L}_\xi g_{\phi\phi}=\mathcal{O}(r)$, while others are $\mathcal{O}(1)$.  Solving Lie derivative gives me 6 equations, which I should solve by plugging in the above ansatz ($\xi^\mu=\sum\limits_{n}\xi^\mu_n(t,\phi)r^n$), but this is the part I get stuck.  ADDENDUM:  I was looking at other components, and have noticed that I have $g_{rr}$ factor with some of them. That term in metric is:  $g_{rr}=\left(1+\frac{r^2}{l^2}\right)^{-1}$  Now, is it legitimate thing to expand this around $r=\infty$ so that I can put $r$ terms inside the sums (assumed solution)?
I am solving a series of ODE, such that each DE is equal to some degree of term that I'm expanding to. For instance, one DE is this:  $\xi^r\partial_r g_{rr}+2g_{tt}\partial_t\xi^t=\mathcal{O}(r)$  $g_{ii}$ are given, from metric, but that's not important. I need to assume that the solution (since I'm looking for components of $\xi^\mu$, which is a vector with components $\xi^t,\xi^r,\xi^\phi$) is given with power series of the form:  $\xi^\mu=\sum\limits_{n}\xi^\mu_n(t,\phi)r^n$, and this is to be seen as expansion around $1/r$ (expansion around $r=\infty$).  Now when I plug this in the ODE I get this  $\frac{2}{l^2}\sum_n\xi^r_nr^{n+1}+2\sum_n\xi^t_{n,t} r^n+\frac{2}{l^2}\sum_n\xi^t_{n,t}r^{n+2}=\mathcal{O}(r)$, where   $\xi^\mu_{n,i}$   is the derivative with the respect to i-th component.   What troubles me is, how to expand this? Do I set n=0,-1,-2,... until my $\mathcal{O}(r)$ terms cancel each other out? Or?  I'm kinda stuck :\
Prove that $\binom{m+n}{m}=\sum\limits_{i=0}^m \binom{m}{i}\binom{n}{i}$ I need to proof this following equality :  $$\binom{m+n}{m}=\sum_{i=0}^m \left(\binom{m}{i}\binom{n}{i}\right)$$  This is what I did combinatoric proof:  Left : subset with $m$ members from $m+n$  Right : $\binom{m}{i} \rightarrow$ The number of subsets in set $m$ could also be written as $2^m$ AND $\binom{n}{i} \rightarrow$ the number of subsets with $m$ members from set $n$  In case $ n \leq m \implies \binom{n}{i}=2^n$  So this is what I came up with I still can't see why those expression are equal.  Any ideas? Thanks. Helpless after exhausting daily limit for reviewing Suggested Edits Playing a remote movie on the remote computer Factor 2 in Heisenberg Uncertainty Principle: Which formula is correct? Some websites and textbooks refer to $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ as the correct formula for the uncertainty principle whereas other sources use the formula $$\Delta x \Delta p \geq \hbar.$$ Question: Which one is correct and why?  The latter is used in the textbook "Physics II for Dummies" (German edition) for several examples and the author also derives that formula so I assume that this is not a typing error.  This is the mentioned derivation:     $\sin \theta = \frac{\lambda}{\Delta y}$      assuming $\theta$ is small:      $\tan \theta = \frac{\lambda}{\Delta y}$      de Broglie equation:      $\lambda = \frac{h}{p_x}$      $\Rightarrow \tan \theta \approx \frac{h}{p_x \cdot \Delta y}$      but also:      $\tan \theta = \frac{\Delta p_y}{p_x}$      equalize $\tan \theta$:      $\frac{h}{p_x \cdot \Delta y} \approx \frac{\Delta p_y}{p_x}$      $\Rightarrow \frac{h}{\Delta y} \approx \Delta p_y \Rightarrow \Delta p_y \Delta y \approx h$      $\Rightarrow \Delta p_y \Delta y \geq \frac{h}{2 \pi}$      $\Rightarrow \Delta p \Delta x \geq \frac{h}{2 \pi}$   So: Which one is correct and why? Can "it" be used with plural subject? Several years ago I heard of something called dummy subjects in high school. It was then stated that, for example, it is a dummy subject when it starts many instances of sentences, e.g.        It is rainy today.   It is hard to imagine how hard is to live alone.   It is not my university degree that matters here.     My question is if we can use it with plural subjects? For example in       If some think that in Mahmoud Dowlatabadi’s works it is only social themes that matter and the writer is ignoring fictional arrays and artistic techniques, at least, endings of his stories show wrongness of this idea.     My question deals with the italicized part. If this usage is OK, can we also write they are social themes that matter? I mean can they also play the role of a dummy subject? How to find and compare flights carried out by a specific type of plane? Modular arithmetic: division, fractions, solving linear congruences Why are PATH variables different when running via sudo and su? On my fedora VM, when running with my user account I have /usr/local/bin in my path:  [justin@justin-fedora12 ~]$ env | grep PATH  PATH=/usr/kerberos/sbin:/usr/kerberos/bin:/usr/local/bin:/usr/bin:/bin:/usr/local/sbin:/usr/sbin:/sbin:/home/justin/bin   And likewise when running su:  [justin@justin-fedora12 ~]$ su - Password:  [root@justin-fedora12 justin]# env | grep PATH PATH=/usr/kerberos/sbin:/usr/kerberos/bin:/usr/local/bin:/usr/bin:/bin:/usr/local/sbin:/usr/sbin:/sbin:/home/justin/bin   However, when running via sudo, this directory is not in the path:  [root@justin-fedora12 justin]# exit [justin@justin-fedora12 ~]$ sudo bash [root@justin-fedora12 ~]# env | grep PATH PATH=/usr/kerberos/sbin:/usr/kerberos/bin:/sbin:/bin:/usr/sbin:/usr/bin   Why would the path be different when running via sudo? WPF TextBox and Scroll behavior I have a problem. I need to host grid with controls in ScrollViewer to prevent textbox from being either truncated or collapsed to zero-with at the UI. Also I want the with of textbox to be expanded when user change windows width. I'm setting Window's content to following code  &lt;DockPanel&gt;     &lt;TreeView DockPanel.Dock="Left" Width="150"/&gt;     &lt;ScrollViewer VerticalScrollBarVisibility="Auto" HorizontalScrollBarVisibility="Auto"&gt;         &lt;Grid&gt;             &lt;Grid.ColumnDefinitions&gt;                 &lt;ColumnDefinition Width="Auto"/&gt;                 &lt;ColumnDefinition /&gt;             &lt;/Grid.ColumnDefinitions&gt;             &lt;Grid.RowDefinitions&gt;                 &lt;RowDefinition Height="Auto"/&gt;             &lt;/Grid.RowDefinitions&gt;             &lt;TextBlock Text="Name"                         Margin="5"                         VerticalAlignment="Center"/&gt;             &lt;TextBox Grid.Column="1"                      Text="Some Name"                      Margin="5"                      VerticalAlignment="Center"                      MinWidth="200"/&gt;         &lt;/Grid&gt;     &lt;/ScrollViewer&gt; &lt;/DockPanel&gt;   All work fine, but when user types very long text in TextBox it is being expanded and horizontal scroll appears. Is there any easy way to limit TextBox maximum width and allow it to be expanded only when user changes window size. Who decides which half of the universe should be destroyed? Although my question is based on the comic mini-series Infinity Gauntlet issues #1 through 6, I'm open to any and all answers (i.e Movies, other comic books etc. etc.) across the Marvel multi-verse.   It was elegantly presented in the Infinity Gauntlet series how Thanos, after collecting all the soul gems snaps his fingers and destroys half the universe.  Additionally, in the recent movie Avengers: Infinity War,       Thanos does it again! He snaps his fingers and destroys half the universe.   However, one thing is unclear to me. In both cases,      Half the universe is destroyed (or obliterated, or erased from existence).   But who decides which half of the universe should not exist anymore? Does the wielder of the Gauntlet choose, or is the Gauntlet (along with the stones) sentient enough to make the decision? "The product key entered does not match any of the Windows image" I have a HP with Windows 7 installed. I want to do a clean installation of Windows 8 from a bootable DVD.   I purchased my Windows 8 32-bit core from the Microsoft store and I received my product key. I then burned the .ISO file to a blank DVD using imgburn. When I restart and change my BIOS to boot from internal CD/DVD and then boot from my internal DVD, I start the installation process.  After I pass the screen where I select the language and location, and click Install it stays there for a minute and then it says:     The product key entered does not match any of the Windows image   I did my research and learned that I need to add a PID.txt file to my source file containing the Windows 8 .ISO but the thing is: I do not know how to add files to the .ISO. The guide I researched is for a bootable USB and I am using a DVD. Recently I have installed an older version of Ubuntu on my old machine. Whenever I try to install any software, I get an error saying it couldn't be found:  $ sudo apt-get install vlc Reading package lists... Done                Building dependency tree        Reading state information... Done     E: Couldn't find package vlc How do I mimic Pathauto transliterated output for an alias in my custom PHP code? When writing my custom function I need to manually append node name to a taxonomy term link. To get the taxonomy term part done I use:    $link_uri = taxonomy_term_uri($term);   $path_alias = drupal_get_path_alias($link_uri['path']);   This works ok, now - I want to append node name ("/" . $node_name), but instead of raw string, I want the Pathauto transliterated output, as set in the URL alias configuration, for that node context.  What function should I use to mimic that output? I looked at hook_pathauto_alias_alter(&amp;$alias, array $context) but I'm not sure that's what I need. I have just discovered that my local bank, Nordea, has decided to remove the only good thing about their entire operation: the so-called "simplified login" feature on their website, which allows (soon allowed) the customer to log in to a kind of very limited "read-only" mode which basically just allowed you to see your current balance, the last few transactions (limited descriptions) and see whether or not there are any new "customer support messages", but not read those messages, not make any kind of payments, edit accounts, etc.  Now, they will only allow "full" logins, which 100% prevents any kind of automation and requires you to sit with a hardware device and enter numbers into it and then into the webpage before it allows you to see as much as your account balance. Other than being very obnoxious for every customer, it also blocks me and other "advanced" users out from our custom-programmed bots which have always regularly logged in with that "simplified" login method just to regexp out the current balance/new messages and then notify us via whatever mechanism we use privately.  I have previously asked them if they have any kind of JSON/RSS API of any kind, but no. They don't even have as much as e-mail notifications! Nothing except their spyware "smartphone app" which I would never install, nor would I ever allow such a surveillance device to track me 24/7 in the first place. Only by being a registered corporation with special, way more costly account types do you get any kind of "API" access from them, even to do this most basic imaginable task.  Before you tell me to switch banks, the others are at least just as bad (even worse from what I can tell, and for a longer time), and the act of switching banks is practically impossible for me personally.  What I'm asking is if maybe somebody knows of some sort of "hidden standard" or something which all banks in the world support, but don't have documented, for people like me to automate basic things such as knowing my account balance. Needless to say, I have looked for this countless times on their website, asked them about it, and asked other people about it. Nobody seems to know anything, or, if they do, won't tell me.  I know that there are a bunch of "Nordea-supporting" PHP libraries and whatnot on Github, but they are all ancient and even if they were up to date, they were most likely using the same method as I have been using: automating the login process through their "simplified login" mechanism, which is "soon" to be gone. (Probably at new year; they didn't even mention any dates or time frames.)  And yes, I have a Bitcoin wallet and have that fully automated. The problem is that virtually nobody else in the world cares about my Bitcoin, so "switching to Bitcoin" remains a pipe dream which I would be ready to do, but the world clearly isn't. So many interesting questions get asked on Stack Overflow and then immediately closed (or worse yet, closed and re-opened for days on end) because they're subjective, argumentative, and fundamentally unanswerable due to our lack of omniscience: without the ability to see into the future and/or know the secret motivations of companies and their developers, we are left to make wildly varying (un)informed guesses. The obvious solution to this is to add a  feature to the Ask a Question page. Similar to the current &quot;Related Questions&quot; feature, this could be used to answer otherwise-unanswerable questions before they're asked, thus saving everyone time and frustration. I propose a simple system whereby a web cam is aimed at an ordinary  device, long recognized as one of the most accessible and accurate fortune-telling mechanisms available to modern society. A simple  could be placed under this rig and triggered in response to the user's typing. Once focus leaves the Title input field, a short delay will be triggered, followed by a bit of logic to capture the 8-ball's result and feed it through a simple OCR routine to produce the result.  So far as I'm aware, SO would be the first major Q&amp;A site to implement such a feature...  Examples:            ...another UV resurrection... I'm developing a Custom Post Type plugin, which works fine in dev. Now that I've moved it to production, I'm getting a 404 when I try to display the template for the CPT.   So, two questions:  1- Is there some mechanism to determine the path WP took to get through the hierarchy?  2- Why wouldn't it just give me the single or index page?  I've dumped $wp_query and a stacktrace at the end of the 404. Here's part of what I found:  [query] =&gt; Array     (         [page] =&gt;          [pagename] =&gt; refletters/proximity     )  [request] =&gt; SELECT   wp_posts.* FROM wp_posts  WHERE 1=1  AND (wp_posts.ID = '0') AND wp_posts.post_type = 'page'  ORDER BY wp_posts.post_date DESC  [posts] =&gt; Array     (     )  [queried_object] =&gt;  [queried_object_id] =&gt; 0   And the stack trace:  2013-04-27 22:16:41 - 24.52.197.40 - TRACE - index.php - 404.php[39] -      wordpress/wp-content/themes/notoriousconsultant/404.php[39] - aaLog-&gt;logtrace     wordpress/wp-includes/template-loader.php[50] - include     wordpress/wp-blog-header.php[16] - require_once     wordpress/index.php[17] - require   Thanks for your help. Why is find not accepting '-exec cp {} dir +'? I have a directory, dir1 which contains many files whose names end in either .jpg or .png. I want to copy all the .png files to dir2 which is empty.  This command works:  find dir1 -name '*.png' -exec cp {} dir2 \;   but this command doesn't:  find dir1 -name '*.png' -exec cp {} dir2 + find: missing argument to `-exec'   I also tried:  find dir1 -name '*.png' -exec cp {} -t dir2 + find: missing argument to `-exec'   and:  find dir1 -name '*.png' -exec cp {} dir2 \+ find: missing argument to `-exec'   After looking at , I even tried:  find dir1 -name '*.png' -exec cp {} dir2 {} + find: Only one instance of {} is supported with -exec ... +    says that:     -exec {} + was added in [version] 4.2.12 in 2005   My version of find is 4.4.2.  What am I doing wrong? Can more than one antibody bind the same antigen I'd like to clear something up about antibodies that I'm not sure I've understood in the articles I've read. Looking at concepts such as "affinity maturation", "monovalent antigens" and "polyvalent antigens" it seems as if there are multiple antibodies which can bind to the same antigen. (Although I think I'm right in saying that individual antibodies can only bind to one antigen?) However, when I read sentences like "It has been estimated that humans generate about 10 billion different antibodies, each capable of binding a distinct epitope of an antigen" on the Wikipedia page (for antibody) it does make it sound like it's one antibody per antigen? Is this the case? Or is it one antibody per epitope. Either way, to put this question in a nutshell:  Is it true to say that the same substance could bind with (and, therefore, be recognised by) more than one antibody, or not?  Also, if the answer is no, does that mean that antibodies with different binding affinities are classed as the same antibody or not? Problems with wifi range on laptop. Wifi card: realtek rtl8723be So I just started using linux/ubuntu today and I've been getting problems with my wifi, ive tried the solutions on this page:  When im really close to my router, its fine, i can see wifi signals, but when i go just a few steps back it disappears right away. Im using the latest version of ubuntu 17.10  Any suggestions would be nice! :) If banks offer a fixed rate lower than the variable rate, is that an indication interest rates may head down? This is one of the major banks in Australia:   At present time, the variable rate is 7.80% but the fixed rate is 6.99% for one year.  And it seems most banks in Australia, if you fix the rate (at the moment), you will get a lower rate.  Is this an indication that in the eyes of the banks, that they believe the interest rates are going to down? I just ask because it seems most people in the media think the rates are going up and they've risen about 6 times in a row now.  Could it be that the banks are secretly preparing for rates to drop?