Proving $r^{k}$ is the only nonidentity element of $D_{2n}$ that commutes with all elements for even $n$ I am working on proving that $z = r^{k}$ is the only nonidentity element of $D_{2n}$ ($n = 2k$) which commutes with all elements of $D_{2n}$. I have been able to show that any other power of $r$ cannot commute with $s$. To do this I showed that $$r^j s = sr^{n-j}$$ I understand that $r^j \neq r^{n-j}$, but how can it be assured that $r^j s \neq sr^{n-j}$? I have been reading a solution of this problem that claims from here it suffices to show that $sr^j$ does not commute with $r$ for any integer $j$. Why is this sufficient? How did they choose $sr^j$?
The center of the dihedral group How to prove that the center of the dihedral group $D_{2n}$ is $\{1,r^{n}\}$ and the center of $D_{2n-1}$ is $\{1\}$?  I don't know how to prove it in this general case.
Draw power from a portable mobile charger to power circuits I usually power my beginner prototype circuits using mb102 power supply board, but it needs an external elimeter to draw power and then regulate it.  I was looking for a portable solution for my 5 volt and 3.3 volt circuit needs and came up with using a portable battery charger for mobile phones.  I tried powering up a simple transistor and LED circuit with it but it wouldn't power-up.   I feel that there is some load sensing circuit in the portable charger which doesn't detect these low power circuits.  I also tried connecting different resistors to the charger, the charger outputs 5 Volts for a few seconds and then again shuts down and doesn't stay ON continuously.  It would be really comfortable to use a portable charger since it could be charged using a phone charger and would be very handy.  What can I do to make the charger output power on low loads? Limit point compactness implies compactness. I have a question about the following counterexample, why  $U_n = \{n\} \times Y$ has not finite collection covering $X$?   Is it because if you take a finite set $S\subset U_n$ then it's impossible to have $X\subset \bigcup S$ since X is infinite?  Counterexample:  Let $Y$ consist of two points; give $Y$ the topology consisting of $Y$ and the empty set. Then the space $X = \mathbb{Z}_{+} \times Y$ is limit point compact, for every nonempty subset of $X$ has a limit point. It is not compact, for the covering of $X$ by the open set $U_n = \{n\} \times Y$ has not finite collection covering $X.$ What would be your preferred way to concatenate strings from a sequence such that between every two consecutive pairs a comma is added. That is, how do you map, for instance, ['a', 'b', 'c'] to 'a,b,c'? (The cases ['s'] and [] should be mapped to 's' and '', respectively.)  I usually end up using something like ''.join(map(lambda x: x+',',l))[:-1], but also feeling somewhat unsatisfied. For example:  $ sudo apt-get install curl Reading package lists... Done Building dependency tree Reading state information... Done You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: bsh : Depends: libjline-java but it is not going to be installed groovy : Depends: libjline-java but it is not going to be installed rhino : Depends: libjline-java but it is not going to be installed E: Unmet dependencies.   Try 'apt-get -f install' with no packages (or specify a solution).   I get the same or similar errors when I attempt to install clojure1.3, leiningen, and several other packages.   When I try the suggestion made in the error message, this is what happens:  $ sudo apt-get -f install  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following packages were automatically installed and are no longer required:   diffstat linux-headers-3.2.0-26-generic linux-headers-3.2.0-26 dh-apparmor dkms html2text libmail-sendmail-perl libsys-hostname-long-perl Use 'apt-get autoremove' to remove them. The following extra packages will be installed:   libjline-java Suggested packages:   libjline-java-doc The following NEW packages will be installed:   libjline-java 0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded. 23 not fully installed or removed. Need to get 0 B/72.0 kB of archives. After this operation, 129 kB of additional disk space will be used. Do you want to continue [Y/n]? Y (Reading database ... 226243 files and directories currently installed.) Unpacking libjline-java (from .../libjline-java_1.0-1_all.deb) ... dpkg: error processing /var/cache/apt/archives/libjline-java_1.0-1_all.deb (--unpack):  trying to overwrite '/usr/share/java/jline.jar', which is also in package scala 2.9.2-400 Errors were encountered while processing:  /var/cache/apt/archives/libjline-java_1.0-1_all.deb E: Sub-process /usr/bin/dpkg returned an error code (1)  $ sudo apt-get upgrade Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run 'apt-get -f install' to correct these. The following packages have unmet dependencies:  bsh : Depends: libjline-java but it is not installed  groovy : Depends: libjline-java but it is not installed  rhino : Depends: libjline-java but it is not installed E: Unmet dependencies. Try using -f. Was Luke aware the Emperor was a Sith Lord? Similar to , I'm asking the more specific question, was Luke aware that the Emperor was a Sith Lord? I don't recall it being mentioned in any of the films.  If yes, at what point does he learn this? If not why did Yoda or Obi-Wan not warn him?  Perhaps he could have avoided being defenseless and struck by Force Lightning if he'd been made aware? Why is it that when you add normally distributed random variables the variance gets larger but in the Central Limit Theorem it gets smaller? Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $ In comics does Batman have a rule against killing? Inspired by  question, I have to admit I don't know a lot about the Comic Book Batman, but I know a fair amount about his other incarnations.   I know for example, that the 1989 movie had him kill the Joker. He vehemently wanted the Joker dead and had no compunctions about taking a life. I also know that he had a rule against killing in The Animated Series that he never broke, and that he very nearly broke it in Batman Beyond.  But as far as the comic books go, when did Batman develop a rule against killing his foes? Does he ever kill them rather than deliver them to the proper authorities? Best way to strip punctuation from a string Solving "MySQL server has gone away", mysql error 2006 I have a site at hostgator, using their "Business account". It's cheap, and comes with a private SSL cert. HOWEVER, their MySQL settings are such that any db connection goes stale fairly quickly, and then the hostgator staff try to convince you to use a VPS or dedicated server at multiple times the expense.  I'm creating and deploying WebAPIs, which can have a 'conversation' which exceeds 60 seconds. When this occurs, the db connection handle is stale, causing any db operations after my timeout to fail.  I'm in the process of implementing the following solution, but am curious if anyone else has solved this issue (where they can not modify their my.ini or my.cnf to give MySQL more resources) in a similar or completely different manner.  last part of a hook_update:  $ret = db_query( "UPDATE {my_table} set field1 = '%s', field2 = %d ... WHERE vid - %d", $node-&gt;field1, $node-&gt;field2, ... $node-&gt;vid ); if (!$ret) {        // get private storage directory for this node:        $cexStorage    = _cex_front_getCexStore();        $nodeFilesPath = DRUPALROOT.'/'.$cexStorage.'/'.$node-&gt;storeDir;        $fname         = $nodeFilesPath.'/recover.node';         $readyToWrite  = serialize( $node );        file_put_contents( $fname, $readyToWrite );        if (file_exists( $fname )) {          _dbgReport( 'seems to have worked!' );        }     }   Then this is called in my hook_form, right after I determine that node->nid is valid:  function _cex_front_recover_node( &amp;$node ) {   if (!isset($node-&gt;storeDir)) {      _dbgReport( '**************************** _cex_front_recover_node: no storeDir!!!' );      return;   }    // get private storage directory for this node:   $cexStorage    = _cex_front_getCexStore();   $nodeFilesPath = DRUPALROOT.'/'.$cexStorage.'/'.$node-&gt;storeDir;   $fname         = $nodeFilesPath.'/recover.node';   if (file_exists( $fname )) {     $recovered = file_get_contents( $fname );     if ($recovered) {       // no longer needed:       unlink( $fname );       $recoveredNode = unserialize( $recovered );       // restore the key values I need to proceed with this node:       $node-&gt;stage = $recoveredNode-&gt;stage;       $node-&gt;rnid  = $recoveredNode-&gt;rnid;       $node-&gt;iris  = $recoveredNode-&gt;iris;       cex_front_update( $node );     }     else {       _dbgReport( '********* failed to read "'.$fname.'"' );     }   } }   I'll be pairing the logic down to only store the key fields I need, but this works! Sure beats upgrading my server at 3x the expense. Anyone else doing something like this? Chrome phishing warning on main I just got a Chrome phishing warning on main for   Is it just me, Chrome screwing things up or do you people have work to do?  The message in English:     Known phishing site detected.   Google Chrome has denied the access to stackoverflow.com. This site is reported as phishing site.      Phishing sites are designed to mislead you, so that you disclose your login credentials, password or other personal data. These sites occur as sites which you probably trust. Geometric meaning of the determinant of a matrix Masking NetCDF time series data from shapefile using Python I have a 3-D time-series precipitation data (187 x 1800 x 3600), stored in a NetCDF file. I need to obtain the precipitation data for a shapefile. import matplotlib.pyplot as plt from netCDF4 import Dataset, num2date,  from matplotlib.pyplot import figure from datetime import datetime, date, timedelta import numpy as np import xarray as xr import pandas as pd import geopandas as gpd       MSWEP_monthly = 'D:\G3P\DATA\Models\MSWEP\MSWEP_monthly.nc4'  MSWEP_monthly = Dataset(MSWEP_monthly, 'r') Pre_MSWEP = MSWEP_monthly.variables['precipitation'][:]   MSWEP_monthly2 = xr.open_dataarray('D:\G3P\DATA\Models\MSWEP\MSWEP_monthly.nc4')  Lon_MSWEP = MSWEP_monthly2.lon Lat_MSWEP = MSWEP_monthly2.lat  Africa_Shape = gpd.read_file('D:\G3P\DATA\Shapefile\Africa_SHP\Africa.shp')    from osgeo import gdal,osr,ogr  def makeMask(lon,lat,res):     source_ds = ogr.Open(shapefile)     source_layer = source_ds.GetLayer()       # Create high res raster in memory     mem_ds = gdal.GetDriverByName('MEM').Create('', lon.size, lat.size, gdal.GDT_Byte)     mem_ds.SetGeoTransform((lon.min(), res, 0, lat.max(), 0, -res))     band = mem_ds.GetRasterBand(1)       # Rasterize shapefile to grid     gdal.RasterizeLayer(mem_ds, [1], source_layer, burn_values=[1])       # Get rasterized shapefile as numpy array     array = band.ReadAsArray()        mem_ds = None     band = None     return array  shapefile = 'D:\G3P\DATA\Shapefile\Africa_SHP\Africa.shp' source_ds = ogr.Open(shapefile)  # calculate the cellsize cellsize = Lon_MSWEP[:][1] - Lon_MSWEP[:][0]   # create the mask mask = makeMask(Lon_MSWEP,Lat_MSWEP,cellsize)  Now if I implement the following code, the precipitation data for the first day (from the time-series) can be obtained with shape of 1800x3600: precip = np.ma.masked_where(mask==0,Pre_MSWEP[0,:,:])  I tried to use a for loop to mask the precipitation data for the entire time series (time, lon, lat) over the area of interest. however, the below code gives me a 2-D data, probably for the last day. Why is that? for i in range(len(Pre_MSWEP)):     precip = np.ma.masked_where(mask==0,Pre_MSWEP[i,:,:]) (Reference : Fraleigh, A first course in abstract algebra)     Prove that a nonempty set $G$, together with an associative binary operation * on $G$ such that $a*x=b$ and $y*a=b$ have solutions in G, $\forall a,b\in G$, is a group.   Because * is an associatie operation, we have to prove that: there exists $e\in G$ such that $e*a=a=a*e$ for any $a\in G$, and that for any $a\in G$ there exists $\alpha \in G$ such that $a*\alpha=e=\alpha *a$.  This is very confusing, first I tried to prove the identity exists. If we consider a=b, then $a*x=a=y*a$, and because of the hypothesis $x,y\in G$ always exists, now we have to show that $x=y$, but I just can't do this with out inverses, so I tried to prove that the inverses exists, however I have no idea how the identity looks like, I'm stuck.  And I tried to get unstuck, so I thought particular cases, first if $G$ had only one element, then what would happend if it had just two elements... but I remembered that we don't know if G is finite or not, and I think this makes things a little bit more complicated.  What can I do? Can one construct a non-measurable set without Axiom of choice? Is axiom of choice required to show the existence of non-measurable sets? Is there a Lebesgue non-measurable set that can be constructed without axiom of choice?  Related question on MO says it is consistent: \begin{eqnarray*} M&amp;=&amp; \max{\left \{a,b,\frac{a+b}{2},\right}\\ &amp;&amp; a+b,a+b/2\Bigg\} \end{eqnarray*} I'm trying to condense the space around examples in gb4e, with the goal of removing unnecessary whitespace around examples in an abstract. Here's what gb4e gives me now:    What I'd like is for the examples to essentially look like part of the running text, as follows:    In sum, I'd like to remove the left indent before the top level item numbers, remove the vertical space between examples, and remove the vertical space above and below the exe environment.  I think \vspace commands might work here, but it would get very tedious to do this every time, and I'd also like to do it the 'right' way.  (I also think linguex might be able to do this without too much fuss, but I'm used to using gb4e so I'd prefer to stick with it if there's a way to do this.)  MWE:  \documentclass{article}  \usepackage{gb4e} \usepackage{lipsum}  \begin{document}     \lipsum[1]     \begin{exe}         \ex Blah \begin{xlist}             \ex Blah             \ex Blah         \end{xlist}         \ex Blah         \ex Blah     \end{exe}     \lipsum[1] \end{document} What is the meaning of the third derivative of a function at a point (Originally  by AJAY.)  What is the geometric, physical, or other meaning of the third derivative of a function at a point?  If you have interesting things to say about the meaning of the first and second derivatives, please do so. Difference between staticmethod and classmethod What is the difference between a function decorated with  and one decorated with ? Add transparency to an existing PNG I'd like to use this image on a web site.    What I need to do is change the background from black to transparent, so I can use the arrow on my site maintaining the page background.  I used Gimp to convert the image to .png, but how do I proceed now?