Shouldn't we also see our galaxy clusters in that ring just beyond the shadow of M87?
Using a naked black hole as a mirror?
How do I handle the window close event in Tkinter? How to calculate $\lim_{n \to \infty}\sqrt[n]{\frac{2^n+3^n}{3^n+4^n}}$ I came across this strange limit whilst showing convergence of a series: $$\lim_{n \to \infty}\sqrt[n]{\frac{2^n+3^n}{3^n+4^n}}$$ How can I calculate this limit? How does the Wizard spellbook work? How do I determine what spells are in my spellbook when creating a character? Do I simply have a spellbook with all Pathfinder spells? Or if I'm creating a level 6 wizard, do I start knowing all level 6 spells, and then have to learn the rest as I play the game? Difference between "right of=" and "right=of" in PGF/TikZ In the PGF/TikZ manual, sometimes I see the option right of=somenode instead of right=of somenode. They look very similar, but the effects are different.  The distance between nodes positioned with the latter option is boundary-to-boundary, as stated in the manual. However, with the first option, it seems that the distance is shorter. I couldn't find any explanation of the first option in the manual.  My question is: what is the difference between the two options? Is there any explanation in the manual that I missed? Print numbered paragraph after title with Biblatex My question is related with Biber and Biblatex (latest versions). I need to quote the number of a paragraph of an old Christian text which is published within a book printed nowadays.   More clearly, I'm working with this Biblatex item (it's in Italian):  @INBOOK{Clemente1984Lettera,   author = {Clemente di Roma},   title = {Lettera di Clemente Romano ai Corinti},   year = {1984},   booktitle = {I Padri apostolici},   editor = {Quaquarelli, Antonio},   location = {Roma},   pages = {41-92},   publisher = {Città Nuova} }   I quote it in a common LaTex file with the BibTex key:  \cite[42]{Clemente1985Lettera}   With a humanistic Biblatex style, the final citation is:  Clemente di Roma, Lettera di Clemente Romano ai Corinti, in "I Padri apostolici", edited by A. Quaquarelli, Roma 1985, p. 42.  Anyway, I need to quote specifically the n. 2 of the letter of Clement, so I have to print this number just after the title "Lettera di Clemente Romano ai Corinti". So, I'd like to have this citation:  Clemente di Roma, Lettera di Clemente Romano ai Corinti, n. 2, in "I Padri apostolici", edited by A. Quaquarelli, Roma 1985, p. 42.  Now, I can find no Biblatex field to store the "n. 2" text, and no way to print it after the title. Of course, I could create a specific item in the bibliographic file in which the title field, or another one which is supposed to follow it, contains the "n. 2" indication. Anyway, this solution is very bad to me, because I should fill my archive with items of specific citations, and not of single titles.  What I'm looking for is what I've already implemented in the Bibfilex citations system (), which is moreover far less powerful that Biber: the possibility to add a pipe in the postnote of a BibTex key to separate a possible indication of a number of a paragraph, to be printed just after the title, from the pages of the volume. So a key like that  \cite[2|42]{Clemente1985Lettera}   should print   Clemente di Roma, Lettera di Clemente Romano ai Corinti, n. 2, in "I Padri apostolici", edited by A. Quaquarelli, Roma 1985, p. 42.  Anybody knows how to do that with Biber and Biblatex?  Thanks. I'm having a problem finding out a way to project from top view and the side view so I texture better. It just replaces the current projection with the top view    So is it possible to project from view twice on the same image? How to visualize Reverse Triangle Inequality $||\mathbf{a}| − |\mathbf{b}|| ≤ |\mathbf{a} − \mathbf{b}|$? pp  and ,   illustrates Triangle Inequality. But without referring to Triangle Inequality at all, how to picture  Reverse Triangle Inequality for all $\mathbf{a,b} \in C^n$? I don't want proofs or rigour. User suggested great corrective edit to my answer, but the edit was rejected. What can I do to approve it? Here is my candidate for  the most elementary proof that $\lim_{n \to \infty}(1+1/n)^n $ exists. I would be interested in seeing others.  $***$ Added after some comments: I prove here by very elementary means that the limit exists. Calling the limit "$e$" names it. $***$  It only needs Bernoulli's inequality (BI) in the form $(1+x)^n \ge 1+nx$ for $x &gt; -1$ and $n$ a positive integer, with equality only if $x = 0$ or $n = 1$. This is easily proved by induction: It is true for $n=1$, and $(1+x)^{n+1} = (1+x)(1+x)^n \ge (1+x)(1+nx) = 1+(n+1)x+nx^2 \ge 1+(n+1)x $. (If $-1 &lt; x &lt; 0$, if $1+nx \ge 0$, the above proof goes through, and if $1+nx &lt; 0$, $1+mx &lt; 0$ for all $m \ge n$ so certainly $(1+x)^m &gt; 1+mx$.)  This proof originally appeared in N.S Mendelsohn, An application of a famous inequality, Amer. Math. Monthly 58 (1951), 563 and uses the arithmetic-geometric mean inequality (AGMI) in the form $\big(\sum_{i=1}^n v_i/n\big)^n \ge \prod_{i=1}^n v_i$  (all $v_i$ positive) with equality if and only if all the $v_i$ are equal.  Let $a_n = (1+1/n)^n$ and $b_n = (1+1/n)^{n+1}$.  We will prove that $a_n$ is an increasing sequence  and $b_n$ is an decreasing sequence.  Since $a_n &lt; b_n$, this implies,  for any positive integers $n$ and $m$ with $m &lt; n$  that $a_m &lt; a_n &lt; b_n &lt; b_m$.  For $a_n$, consider n values of $1+1/n$ and $1$ value of $1$.  Since their sum is $n+2$ and their product is $(1+1/n)^n$, by the AGMI,  $((n+2)/(n+1))^{n+1} &gt; (1+1/n)^n$,  or $(1+1/(n+1))^{n+1} &gt; (1+1/n)^n$,  or $a_{n+1} &gt; a_n$.  For $b_n$, consider $n$ values of $1-1/n$ and $1$ value of $1$.  Since their sum is $n$ and their product is $(1-1/n)^n$, by the AGMI,  $(n/(n+1))^{n+1} &gt; (1-1/n)^n$  or $(1+1/n)^{n+1} &lt; (1+1/(n-1))^n$,  or $b_n &gt; b_{n+1}$.  Since $b_n-a_n  = (1+1/n)^{n+1} - (1+1/n)^n = (1+1/n)^n(1/n) =a_n/n $ and every $a_n$ is less than any $b_n$ and $b_3 = (1+1/3)^4 = 256/81 &lt; 4$, $b_n-a_n &lt; 4/n$, so $b_n$ and $a_n$ converge to a common limit.  These proofs do not seem to be really elementary,  since they use the AGMI.  However, they use a special form of the AGMI,  where all but one of the values are the same,  and this will now be shown to be implied by BI,  and thus be truly elementary.   Suppose we have $n-1$ values of $u$ and $1$ value of $v$ with $u$ and $v$ positive.  The AGMI for these values is  $(((n-1)u+v)/n)^n \ge u^{n-1}v$  with equality if and only if $u = v$.  We will now show that this is implied by BI: $(((n-1)u+v)/n)^n \ge u^{n-1}v$  is the same as  $(u+(v-u)/n)^n \ge u^n(v/u)$.  Dividing by $u^n$,  this is equivalent to  $(1+(v/u-1)/n)^n \ge v/u$.  By BI,  since $(v/u-1)/n &gt; -1/n &gt; -1$, $(1+(v/u-1)/n)^n \ge 1+n((v/u-1)/n) = v/u$  with equality only if $n=1$ or $v/u-1 = 0$. Thus BI implies this version of the AGMI. I have a VIP skybag (I believe this does not matter). It has a 3-number TSA lock built into the suitcase. I know the combination to unlock, let's say it's 345. Recently it was locked and while there was a different number on the combination lock the reset/set lock button was pressed.  Since then the reset/set lock button is pressed (stuck) and the lock button also does not unlock. Any pointers on how to unlock it? I am thinking of unscrewing the lock from inside of the suitcase but it seems quite a hassle, as I may have to assemble it back. Let me know if there is any other option to unlock it from the outside. pyvenv vs venv vs python-virtualenv vs virtualenv and python 3 malikarumi@Tetouan2:~$ pip install virtualenv     Collecting virtualenv     Downloading virtualenv-12.0.7-py2.py3-none-any.whl (1.8MB)     100% |################################| 1.8MB 330kB/s   malikarumi@Tetouan2:~$ pip freeze     (a lot of stuff, but not virtualenv)  malikarumi@Tetouan2:~$ virtualenv testvenv1     The program 'virtualenv' is currently not installed. You can install it by typing:     sudo apt-get install python-virtualenv   What is going on here? is python-virtualenv == pyvenv? Isn't that still broken? Does original virtualenv still work with python? if venv (too many name variations!!!) is part of the standard library, , why am I being told to install it?  And when I did try to install it, I got:  malikarumi@Tetouan2:~$ sudo apt-get install python-virtualenv     Reading package lists... Done     Building dependency tree            Reading state information... Done     The following extra packages will be installed:     python-colorama python-distlib python-html5lib python-pip python-setuptools   At which point I aborted, because installing python3-pip just before that had given me   Setting up python3-setuptools (3.3-1ubuntu1) ... Setting up python3-pip (1.5.4-1) ...   And I wasn't sure if the extra packages would overwrite them or go onto python 2.7.  I read that python 3 is supposed to be the default going forward. Since both 2.7 and 3.4 are there, and have separate commands, how do I not only make sure Python 3 is my default, but that anything I install goes there and is used by it instead of 2.7?   BTW, I tried sudo apt-get python3-virtualenv and got: E: Unable to locate package python3-virtualenv I want to convert a std::string to lowercase. I am aware of the function tolower(). However, in the past I have had issues with this function and it is hardly ideal anyway as using it with a std::string would require iterating over each character. Is there an alternative which works 100% of the time? Drupal unexpected "Fatal error: Class name must be a valid object or a string [...] common.inc" on every page "Fatal error: Class name must be a valid object or a string in C:\xampp\htdocs\uc\includes\common.inc on line 7506"  The error above showed up unexpectedly. I was adding a "Long text" field to a content-type, but when I expected to be brought to a saved field configuration page, I instead got a Fatal error on every page.  I had a look at includes\common.inc:  /**  * Get the entity controller class for an entity type.  */ function entity_get_controller($entity_type) {   $controllers = &amp;drupal_static(__FUNCTION__, array());   if (!isset($controllers[$entity_type])) {     $type_info = entity_get_info($entity_type);     $class = $type_info['controller class'];     $controllers[$entity_type] = new $class($entity_type); // Line 7506   }   return $controllers[$entity_type]; }   However, I just don't understand Drupal at a low enough level to understand the relationships between these variables and functions.  I also searched Drupal for people with a similar problem:  (1)  who claims that I have to install the submodule "Relation Endpoints Field", but searching Drupal, .  (2) :  DELETE FROM cache; DELETE FROM cache_filter; DELETE FROM cache_menu; DELETE FROM cache_page; DELETE FROM watchdog;   (via , misleading name, I know.)  Clearing those tables from the cache did not help.  (3) And another user . Because I can't find a module specifically pertaining to old search facets, I went into MySQL system table, and marked the Search module's line with a 0 status, indicating to stop loading the module altogether.  On that same link as (3), one user suggested enabling the , so I dropped all the tables, imported the latest stable .sql, and installed the module, but after recreating the problem (trying to add a field to a content-type), that as well concluded fruitless.  Still, the problem remains  The Fatal error shows up on every page, making the site completely non-functional. Any help or direction would be immensely appreciated. Thank you for looking. BAV99 forward voltage I am reading the  From the table on ELECTRICAL CHARACTERISTICS (page 2) (TA = 25°C unless otherwise noted) (Each Diode)     Forward Voltage mVdc     IF = 1 mAdc 715mV   IF = 10 mAdc 855mV   IF = 50 mAdc 1.0V   IF = 150 mAdc 1.25V     On page 3 there is the following VI curve graph:  Clearly at 25°C - 100mA is Vf = 0.9V. There is huge difference between the table and the IV curve.  Which source of information I should trust? Is this (typical) and (maximum) where the table guarantees the diode won't exceed Vf 1.0v at 50mA while the graph is just typical values? I have my CIFS setup in fstab and they are working as they are supposed to on boot. They mount as they should and work for a while. Out of nowhere it seems (could be after unlocking machine etc) I get "Host is down" error trying to access it. I have multiple and they are all down. They are also shared from the same server. At this time i check on a windows computer and an outdated 14.04 machine and they are up and functioning as they are supposed to. After clicking around on the shares in nautilus and getting repeat errors they will just start working again. To access a share that is "down" takes about 2-3min of randomly clicking different mounts and going back to the first one when automagically it shows the data in the mount point.  I do not have this problem on 14.04 machines that have not been updated in a while. All of those machines are fully functional and the CIFS never go "down". On 16.04 they were not a problem until more recently.  I have made sure to update every other day and have cleaned old linux headers (in hind sight i probably should have reverted). I do this because im begging for a fix to just appear but its been weeks of battling CIFS mounts without any solution. Continuous weighted random distribution, biased toward one end I am currently contributing to a particle system for our game and developing some emitter shapes.  My uniform random distribution along a line or along a rectangular area works fine - no problem.  But now I would like to have something like a 1 dimensional gradient in this distribution. This would mean for example lower values are more common than higher values.  I don't know what would be appropriate mathematical terms for this problem, so my search skills are rather useless with this one. I need something that is computationally simple, as the particle system needs to be efficient. What does closed form solution usually mean? This is motivated by  and the fact that I have no access to Timothy Chow's paper  indicated there by Qiaochu Yuan.  If an equation $f(x)=0$ has no closed form solution, what does it normally mean? Added: $f$ may depend (and normally does) on parameters.   To me this is equivalent to say that one cannot solve it for $x$ in the sense that there is no elementary expression $g(c_{1},c_{2},\ldots ,c_{p})$ consisting only of a finite number of polynomials, rational functions, roots, exponentials, logarithmic and trigonometric functions, absolute values, integer and fractional parts, such that   $f(g(c_{1},c_{2},\ldots ,c_{p}))=0$. \foreach not behaving in axis environment Am hesitant to ask yet one more embarrassing question, but I can't seem to see the problem with the \foreach in the MWE. It should produce several horizontal lines but only produces the one with the yValue as specified in the \newcommand{\yValue}{0.3}:  Code: \documentclass{article} \usepackage{pgfplots}  \newcommand{\yValue}{0.3}  \begin{document} \begin{tikzpicture} \begin{axis} [  ymin=0,  ymax=1,  xmin=0,  xmax=1, ]  \foreach \yValue in {0.00,0.1,...,1.00} {     \draw [red] (axis cs:0,\yValue) -- (axis cs:1,\yValue); }  \end{axis} \end{tikzpicture} \end{document} How do I run a sudo command needing password input in the background? I have recently disabled  so that it now prompts me for a password every time.  And though this is good for security, it has caused one slight problem which I haven't been able to figure out a solution for, I am unable to run commands which go along the lines of:   sudo &lt;command&gt; &amp;   In the past I would have run a sudo command before that, it would have cached my password and allowed me to run sudo commands without prompt for the next few minutes, and thus it would allow me to run the command.  But when I run it now as there is no caching before hand and as it starts a new thread immediately and sudo doesn't even prompt me for a password, I am unable to run it in this way.  So unless I run sudo -i before it I am unable to run a command in this format which is becoming rather annoying. So I was wondering if there is any way to get around this and still run programs and commands in this way?  I am running Ubuntu GNOME 15.10 with GNOME 3.18, and specifically the program I want to run in this fashion is etherape if that makes any difference, but I would really like the solution to work for all programs and commands. how to get the base url from jsp request object?