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What do the different colors mean in ls? Finite subgroups of the multiplicative group of a field are cyclic In Grove's book Algebra, Proposition 3.7 at page 94 is the following     If $G$ is a finite subgroup of the multiplicative group $F^*$ of a field $F$,   then $G$ is cyclic.   He starts the proof by saying "Since $G$ is the direct product of its Sylow subgroups ...". But this is only true if the Sylow subgroups of $G$ are all normal. How do we know this? Should operator<< be implemented as a friend or as a member function? That's basically the question, is there a "right" way to implement operator&lt;&lt; ? Reading  I can see that something like:  friend bool operator&lt;&lt;(obj const&amp; lhs, obj const&amp; rhs);   is preferred to something like   ostream&amp; operator&lt;&lt;(obj const&amp; rhs);   But I can't quite see why should I use one or the other.  My personal case is:  friend ostream &amp; operator&lt;&lt;(ostream &amp;os, const Paragraph&amp; p) {     return os &lt;&lt; p.to_str(); }   But I could probably do:  ostream &amp; operator&lt;&lt;(ostream &amp;os) {     return os &lt;&lt; paragraph; }   What rationale should I base this decision on?  Note:   Paragraph::to_str = (return paragraph)    where paragraph's a string. Can I use PCA (or should I use regression) for testing the effect of multiple variables on one dependent variable? I have 2000 soil property measures and 14 different variables like rainfall, temperature, slope, etc. I want to check the effect of those 14 variables on soil property measures, including which variable affects my soil property the most. Is PCA a suitable solution to this? Or should I try multiple regression instead? How can I run blender from command line or a python script without opening a GUI? I'm interested in using Blender for producing images, 3d files, or calculating geometry on a server.  Can I make blender run a python script without opening a GUI?  If not, can I incorporate Blender's python API into my own python script without running a GUI? Merge flash site into one executable Why do extract recipes call for a late addition of the bulk of the extract? My recipe calls for a pound of DME at the beginning of the boil and 6 pounds at 15 minutes to go.  What's the point of that? Electrolytic Capacitor Voltage Rating vs capacitance I have always believed that it was ok to replace a capacitor with a slightly higher voltage rated one. eg 16v with 25v.Going higher than this would risk the capacitance value being out at the working voltage. However I just read the following quote from this       Electrolytic capacitors are non-ferroelectric with a very low dielectric constant. Their capacitance is derived from a very high surface area and nanometer thick dielectric layers. Their capacitance is not a function of applied voltage.‐Design Solutions For DC Bias In Multilayer Ceramic Capacitors (August 2010 Electronic Engineering Times)   So does this rule not apply to aluminum electrolytics? Eg will I have no issues replacing an electrolytic 25v 100uf with a 50v 100uf assuming both are low esr, similar footprint and good quailty? Is it possible/sensible to add a suspension fork to an old mountain bike? I have a ~1993 Giant mountain bike.  It's not great, but I am a very new and casual rider.    The bike has a solid fork and it makes for the predictable rough trail ride.  The trails are generally fairly modest (I live in North Texas, a 100' change in altitude makes our ears pop here) and the surface is generally dirt (very few rocky areas).  So, I am thinking that it would be very nice to have a suspension bike.  And seeing that a modest fork can be purchased for a modest price, I am wondering if it makes sense to add a fork to this bike.   asked about going the other direction, and there were some cautionary words about changing steering geometry.  So, on balance, is this something that I would be best off to avoid?  I suspect that if I have to replace a bunch of steering parts, the cost will quickly approach that of a new low end bike. wp_customize_image_control default value I am trying to create a custom theme using Wordpress 3.4's customization options. I would like to create an option to change the theme's logo, but I would like to also display a default logo image.   I am using the following code in my function.php page:  $defaultbranding = "get_bloginfo('template_directory') . '/images/logo.png";      $wp_customize-&gt;add_setting( 'change_branding', array(         'default' =&gt; get_bloginfo('template_directory') . '/images/logo.png',     ) );      $wp_customize-&gt;add_control( new WP_Customize_Image_Control( $wp_customize, 'change_branding', array(         'label'   =&gt; 'Image Control',         'section' =&gt; 'theme_settings',         'settings'   =&gt; 'change_branding',     ) ) );   And in the theme's header.php page:  src='&lt;?php echo get_theme_mod( 'change_branding' , 'default' ) ?&gt;'   I would like the 'default' value on the front end to be:  get_bloginfo('template_directory') . '/images/logo.png'   Is there some way I can achieve this?  Thanks in advance for any help. How can I convert the binary string $x_bin="0001001100101" to its numeric value $x_num=613 in Perl? Synonym for "Fellow Sufferers"? I have to do proof of $(a+b)^n$ and $(a+b)^{n+1}$ with mathematical induction.  I finished the first one $(a+b)^n = a^n + na^{n-1}b+\dots+b^n$.  I however have trouble with the second one, I don't know what to start exactly. Any help/hints? Making sense of principal component analysis, eigenvectors & eigenvalues In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues.   I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like a machine. But I didn't understand it. I didn't get the purpose of it. I didn't get the feel of it.        I strongly believe in the following quote:     You do not really understand something unless you can explain it to your grandmother. -- Albert Einstein   Well, I can't explain these concepts to a layman or grandma.   Why PCA, eigenvectors &amp; eigenvalues? What was the need for these concepts? How would you explain these to a layman? How can I compile different .tex file on click on typeset in TeXShop? (MacTeX 2011) My problem is that I want to have different .tex files for the preamble, the presentation aroundings (my target is a beamer PDF file) and the content of the presentation.  Mostly I'll be working on the content file. I even think about partitioning this content .tex file into several chapter files for a better overview. My problem with this solution is that I have to open the presentation.tex file to typeset the whole beamer PDF for looking at the results of my previous changes on the content files. (My presentation.tex file includes the preamble and the content.)  Is there any way to get TeXShop to always typeset one particular file (my presentation.tex) although I'm currently working on a different one (e.g. the content.tex)?  A shortcut? A command in the .tex file? An option in TeXShop? Anything? If $$Y=\sum_{i=1}^{N}X_i^2$$ where $X_i \sim \mathcal{N}(0,\sigma^2)$, i.e. all $X_i$ are i.i.d. normal random variables of zero mean with same variances, then $$Y \sim \Gamma\left(\frac{N}{2},2\sigma^2\right).$$  I know the chi-squared distribution is a special case of the gamma distribution, but could not derive the chi-squared distribution for random variable $Y$. Any help, please? I have a custom search page for use in Customer Portal.  VF Page:    &lt;form name="frmSearch"&gt;      &lt;input type="text" name="keyword" id="keyword" value="{!portalSearchModel.searchTerm}" onkeypress="return noenter(event);" style="width:25%; height:25px;margin:0; padding: 0px 6px 0px;" placeholder="What are you looking for?" /&gt;      &lt;input type="button" id="btnSearch" name="btnSearch" value="Search" onclick="searchTerm()" class="go-we3" /&gt;   &lt;/form&gt;   &lt;script type='text/javascript'&gt;       function searchTerm(){         var searchTerm = document.getElementById("keyword").value;          var url="/apex/PortalSearch?s="+searchTerm;          window.location = url;          return false;     }          &lt;/script&gt;   Is there a way to call this apex method in Controller on button click along with the JavaScript?      public void logSearchTerm(){         portalSearchModel.searches();         Search_Log__c sl = new Search_Log__c();          sl.search_term__c = portalSearchModel.searchTerm;         sl.user__c = UserInfo.getUserId();              insert sl;  } How to interpret the margin of error in a poll? Recently the media reported on a political poll that stated that "46% of Republican voters in Mississippi think that interracial marriage should be illegal".  One example story (of many around the 'net) is this one from the NY Daily News: .  From that article:     A survey conducted last month shows 46% of GOP voters in the state   believe interracial marriage should be illegal - a plurality of the   people questioned   Given that I know many people in interracial marriages (including in my own family) I was motivated to track down the full story rather than fall to the obvious hysteria.  I located the press release on the poll from Public Policy Polling  (PDF)  which has:     46% of these hardcore Republican voters believe interracial marriage   should be illegal, while 40% think it should be legal. With Barbour   included, Huckabee gets more support (22%) from the former than the   latter (15%), as does Palin (13-6). The support for Bachmann (10-2),   Gingrich (13-8), and Pawlenty (4-1) works the opposite way.   And at the end of the first page they state:     PPP surveyed 400 usual Mississippi Republican primary voters from   March 24th to 27th. The survey’s margin of error is +/-4.9%.   Now what I don't understand is how to interpret the results especially the margin of error value.    From the raw numbers, 46% (~50%) of 400 voters is 200 voters who responded who think that interracial marriages should be illegal.  But from stats of Mississippi in the 2008 Federal election, McCain polled about 725,000 Republican votes (to Obamas 556,000). My gut feeling is that you can't extrapolate that 46% number to the other 724,600 Republican voters and still retain a 5% margin of error.  So is the media misrepresenting the numbers (really? the media would do that?!?!?!) Or are there statistics at work that I don't understand?  Thanks for your help! How can I have 2 different mod folders with the new Minecraft launcher? What are the pros and cons of flip lock vs. twist lock on tripods? I am in search of a tripod. While doing a bit of research myself, I came to know that there are two broad variants of leg locks, viz. flip and twist lock. In many forums, I find people expressing conflicting views preferring each one over the other. This seems to confuse me even more while not being aware of what each type of lock operates. What are the prime considerations in choosing a particular lock system?