Cut Point of connected space and it'sseparation
Cut points and connected space
Proving $6^n - 1$ is always divisible by $5$ by induction I'm trying to prove the following, but can't seem to understand it. Can somebody help?     Prove $6^n - 1$ is always divisible by $5$ for $n \geq 1$.   What I've done:  Base Case: $n = 1$: $6^1 - 1 = 5$, which is divisible by $5$ so TRUE.  Assume true for $n = k$, where $k \geq 1$: $6^k - 1 = 5P$.  Should be true for $n = k + 1$  $6^{k + 1} - 1 = 5Q$  $= 6 \cdot 6^k - 1$  However, I am unsure on where to go from here. href to an image label - how to jump to the image instead of the caption below the image? Listings - Only highlight between delimiters I want to implement a function name color highlighting and I use the listings package. My 'regex' for this is def %s( where def (including space) is the first delimiter %s is the function name and ( is the second delimtier. Now my aim is to color only %s in purple. So basically I want to only touch the part between the delimiters and the delimiters should stay untouched.  Here is what I tried so far:  \documentclass{article} \usepackage{xcolor,listings} \usepackage[ngerman]{babel} \usepackage{fontspec} \setmonofont{Courier New}  \begin{document}  \lstset{language=Python,     escapebegin=\color{orange},     keywordstyle=\color{blue}\bfseries,     commentstyle=\color{orange!80!red},     stringstyle=\ttfamily\color{gray},     %moredelim = [is][\color{blue}\bfseries def \mdseries\color{purple}]{def\ }{(},     moredelim = [s][\color{purple}]{def\ }{(},     showstringspaces=false}‎ \lstset{literate=%    *{0}{{{\color{red}0}}}1     {1}{{{\color{red}1}}}1     {2}{{{\color{red}2}}}1     {3}{{{\color{red}3}}}1     {4}{{{\color{red}4}}}1     {5}{{{\color{red}5}}}1     {6}{{{\color{red}6}}}1     {7}{{{\color{red}7}}}1     {8}{{{\color{red}8}}}1     {9}{{{\color{red}9}}}1 }  ‎‎\begin{lstlisting}[basicstyle=\ttfamily,mathescape] def fancy_function_name(page_url = 'tex.stackexchange.com'):     return page_url.split('.')[0] \end{lstlisting}  \end{document}   This code will generate:   You can see that it matches the right place but is too greedy and colors the delimiters too. So I thought about the uncommented line: %moredelim = [is][\color{blue}\bfseries def \mdseries\color{purple}]{def\ }{(}, If I use this one and uncomment the line below I get this result:    It's nearly what I wan't but I cannot add the last, now missing ( in black color.  Does anyone has a solution for this without pretouching the code and add all the @@ symbols? I read that it is a best practice to double quote all of your variable expansions in Bash. I also read that one cannot use a shell glob (wildcard(*)) right after a double quoted variable expansion. These situations conflict when one wants use the value of a regex pattern, expanded out of a variable.   The reason I would sometimes want to combine regex with a wildcard is to keep my regex minimal, neater, in my personal taste.  My particular problem  I downloaded phpmyadmin into my document root directory and unzipped it, but I fail to rename it with mv by a regex pattern I put in a variable, and available when expanding its variable. Here's the exact trace:  userName@compName:/var/www/html# ll total 11336 drwxr-xr-x  3 root root     4096 Feb 14 07:04 ./ drwxr-xr-x  3 root root     4096 Feb 14 06:56 ../ -rw-r--r--  1 root root      612 Feb 14 06:57 index.nginx-debian.html drwxr-xr-x 12 root root     4096 Dec 23 08:50 phpMyAdmin-4.7.7-all-languages/ -rw-r--r--  1 root root 11589684 Dec 23 14:08 phpMyAdmin-latest-all-languages.zip userName@compName:/var/www/html# echo $pma [pP][hH][pP][mM][yY][aA][dD][mM][iI][nN] userName@compName:/var/www/html# mv "$pma"*/ phpmyadmin/ mv: cannot stat '[pP][hH][pP][mM][yY][aA][dD][mM][iI][nN]*/': No such file or directory   If I'll unquote the variable expansion to ${pma} I would indeed be able to combine the variable expansion and the regex, as in ${pma}*, but it is important to me to follow the best practice without exceptions, if I can.  My question  How could I keep the variable expansion double quoted, but still using the extracted value with a wildcard? How to render a background image in 2.8? How to Check If A Plugin Is Enabled Through API? I've encountered different usages of preposition + Home and I want to learn which of them are correct.  Some examples:     Back home   Back at home   Back to home Is there a way to run Python on Android? How do I get HTML5 h.264 Video working on Firefox? According the the , h.264 video should now be working in Linux.  It does also say the appropriate gstreamer plug-ins need to be installed.  Does anyone know what these are?    I used this  and the  to verify that h.264 support works in Google Chrome on my Ubuntu 12.04 64-bit, but does not work on Firefox 26. Using OpenSUSE leap 15.1 10.2.29-lp151.2.9.1-x86_64 from vendor OpenSUSE (installed)  CREATE TABLE testtab2(     my_id2 INT AUTO_INCREMENT,     name2 VARCHAR(10),     comment2 VARCHAR(20),     PRIMARY KEY (my_id2) );   testtab.txt:  |Roger |hurray   | |George|today    | |Paul  |yesterday| |John  |last year| |Ringo |tomorrow |   LOAD DATA LOCAL INFILE 'testtab.txt' INTO TABLE testtab FIELDS TERMINATED BY '|';   works fine with linenumbers (my_id) 1-5  INSERT INTO testtab (name,comment) VALUES ('Beatles','band');   this gets linenumber (my_id) 8 - should be 6.  If testtab.txt have 8 records and using INSERT INTO 1 record this gets linenr (my_id) 16. This happens in other tables too - one moved from 199 to 256.  Anyone who can solve the problem? Beamer: how to justify the whole text? Proving $2$ generates $(\mathbb{Z}/19\mathbb{Z})^*$ by only looking at a few powers How to calculate square root or cube root? I was reading Richard Feynman biography when I read that one time he was able to calculate the  cube root of large number in his brain by just using simple facts of everyday life. So my question is there a way to calculate square root or cube root by using just a pen a paper, I mean a systematic way not by trial and error. Why is this sentence correct? “She suggested that he go to the cinema.” I have read enough threads on QQplots here to understand that a QQplot can be more informative than other normality tests. However, I am inexperienced with interpreting QQplots. I googled a lot; I found a lot of graphs of non-normal QQplots, but no clear rules on how to interpret them, other than what it seems to be comparison with know distributions plus "gut feeling".   I would like to know if you have (or you know of) any rule of thumb to help you decide for non-normality.  This question came up when I saw these two graphs:    I understand that the decision of non-normality depends on the data and what I want to do with them; however, my question is: generally, when do the observed departures from the straight line constitute enough evidence to make unreasonable the approximation of normality?  For what it's worth, the Shapiro-Wilk test failed to reject the hypothesis of non-normality in both cases. Assessing approximate distribution of data based on a histogram Suppose I want to see whether my data is exponential based on a histogram (i.e. skewed to the right).   Depending on how I group or bin the data, I can get wildly different histograms.   One set of histograms will make is seem that the data is exponential. Another set will make it seem that data are not exponential. How do I make determining distributions from histograms well defined? How can I enable AHCI on a SATA drive after installing Windows? Can I enable AHCI after installing Windows 7 in a SATA drive that was on IDE mode? Favorite Tag Buttons partly sticking off page I appear to have found a bug, when adding Favorite tags, I find that the buttons are half off the page (this is of the  site):    Although on other sites it looks more like this:    I am running Ubuntu GNOME 15.04 with Firefox 41.0.1+build2-0ubuntu0.15.04.2. Is there some trick to burning a cd with the CD/DVD creator?    I'm installing k3b because that's a known entity.  What is this, and where is the "burn" button?  Help just tells me this is Nautilus 3.4.1, but I don't see a .  There's no option to burn in the Files menu, nor does clicking on the whitespace indicate an option to burn anything. What is $\lim_{n\to \infty}\sum_{k=1}^n \left(\frac{k}{n}\right)^n$? I'm asked to find $\displaystyle\lim_{n\to \infty}\sum_{k=1}^n \left(\frac{k}{n}\right)^n$.  It seems that the sum converges.  I managed to prove that $\forall n,1&lt;\sum_{k=1}^n \left(\frac{k}{n}\right)^n&lt;e$  But I don't know how to prove convergence, nor find the value of the limit.