$g_{1},g_{2} \in G$, then if $g_{1}g_{2}$ has finite order then we have to prove that $g_{2}g_{1}$ also has finite order too.
Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46.
How to get my actual data from ContentNote API in Salesforce using SOQL? The integral $\int_0^8 \sqrt{x^4+4x^2}\,dx$ how to change landscape mode Show that $z^5-1=(z-1)(z^2+2z\cos{\pi \over 5}+1)(z^2-2z\cos{2\pi \over 5}+1)$ Managed metadata columns in document information panel with multiple content types What caused the oathbreakers in the Paths of the Dead to turn into shades? In Return of the King, we are told that the shades who populate the Paths of the Dead are the spirits of oathbreakers who refused to aid Isildur in the first fight against Sauron.  First, we are told of the oath (Book 5, Chapter 2):     But the oath that they broke was to fight against Sauron... For at Erech there stands a stands yet a stone that was brought, it is said, from Númenor by Isildur... and upon it the King of the Mountains swore allegiance to him in the beginning of the realm of Gondor.   Then, when they fail to honor the oath, we are told about Isildur's curse on them:     "...this curse I lay upon thee and thy folk: to rest never until your oath is fulfilled."   And as we see in the following chapters, the Sleepless Dead are indeed subject to this curse exactly as Isildur descibed it. What seems less clear, however, is how Isildur's curse manages to alter the fundamental nature of the oathbreakers' mortality, so that they became undead shades instead of simply dying in disgrace.   Do we have any other examples of Isildur performing similar acts of magic? Based on what we are told in LotR, Isildur appears to have been a great warrior, but not a regular practitioner of magic. Is the stone at Erech somehow magically involved? (If so, do we know anything more about it or how it works? Does its origin in Númenor have any significance?) It is mentioned that the oathbreakers had worshiped Sauron; could this be a clue? My problem is simple. I think that the strange deformations on the top are there because there are so many vertices in one face. How do I fix this while maintaining nice topology?  Although I triangulated the top polygon, the problem still persists. For example, in a novel, if someone is so powerful that no one can stop him, something like this would be said:     He [insert word] in the continent [or some other location].   "Moved unhinderedly" somewhat fits it, but it doesn't really seem like the right term.  The word I'm looking for describes a few things:   Should have an implication that the person himself is powerful That person can go wherever he wishes to, and cannot be stopped because of his strength That person can do whatever he wishes to, and like above, cannot be stopped because of his strength   The other word I have is "swept", i.e.:     He swept through the continent.   I used "swept" because when you sweep the floor, nothing stops you and everything is swept away, similar to the example. However, the dictionary definitions don't match up, so that isn't the word either.  Any suggestions? This is one of the possible ways I come out:  struct RetrieveKey {     template &lt;typename T&gt;     typename T::first_type operator()(T keyValuePair) const     {         return keyValuePair.first;     } };  map&lt;int, int&gt; m; vector&lt;int&gt; keys;  // Retrieve all keys transform(m.begin(), m.end(), back_inserter(keys), RetrieveKey());  // Dump all keys copy(keys.begin(), keys.end(), ostream_iterator&lt;int&gt;(cout, "\n"));   Of course, we can also retrieve all values from the map by defining another functor RetrieveValues.  Is there any other way to achieve this easily? (I'm always wondering why std::map does not include a member function for us to do so.) I read somewhere that companies are not required to pay dividends to shareholders (this is correct, yes?).   If so, then if company A never pays dividends to its shareholders, then what is the point of owning company A's stock? Surely the right to one vote for company A's Board can't be that valuable.   What is it that I'm missing? Why does triply periodic univariate function not exist? A univariate function $f$ is periodic with period $p_1,\ldots,p_k$ if   $$f(z) = f(z + \sum_{i=1}^k n_i \cdot p_i)$$   for all complex $z$ and integers $n_i$. Elliptic function is an example of doubly periodic function.  that a triply periodic univariate function cannot exist, but why? Can't we follow the construction of elliptic functions that uses Schwarz-Christoffel formula to map upper-plane onto a triangle, and use the reflection principle to create a lattice which corresponds to three periods?     Which step fails to hold when we mimic the construction of an elliptic function to create a triply period meromorphic univariate function?   Sorry if the question is stupid. I do not have an access to the original proof that a triply periodic function cannot exist, either. So references are welcome! Problem with \mkbibdateapalongextra of biblatex-apa biblatex-apa gives me this:  ! Undefined control sequence. &lt;argument&gt; \mkbibdateapalongextra                                {labelyear}{labelmonth}{labelday}\iffieldu...   I'm using the default (english) language, and have done the:  \DeclareLanguageMapping{norsk}{norsk-apa} \DeclareLanguageMapping{english}{american-apa}   thing.   Here's a short example:  \documentclass[english]{memoir} \usepackage{babel} \usepackage{roffe} \usepackage[backend=biber,date=short,maxcitenames=2,style=apa]{biblatex}  \DeclareLanguageMapping{norsk}{norsk-apa} \DeclareLanguageMapping{english}{american-apa} \addbibresource{Papers2.bib} \addbibresource{R.bib}  \begin{document} \cite{R-base}  \printbibliography \end{document}   This one replicates the problem. Why LastModifiedbyid in Trigger.new gives old value I have noticed the strange behaviour of salesforce due to which even my code broke down in production.  I was trying to stop the user from updating some field..I used before udpate trigger and code like Trigger.new[i].LastModifiedbyId..  Ideally it should give the current user in Trigger.new[i].LastModifiedbyId in  but it gives oldrecordLastModifiedbyId..  Although userinfo.getuserid resolves the problem later.  Any idea about this behaviour of salesforce. I'm going to submit form using cURL, where some of the contents is come from other file, selected using sed  If param1 is line matching pattern from other file using sed, below command will works fine:  curl -d param1="$(sed -n '/matchpattern/p' file.txt)" -d param2=value2 http://example.com/submit   Now, go to the problem. I want show only text between 2 matching pattern excluding the matching pattern itself.  Lets say file.txt contains:  Bla bla bla firstmatch It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. secondmatch The point of using Lorem Ipsum is that it has a more-or-less normal distribution of letters, as opposed to using 'Content here, content here', making it look like readable English.   Currently, lots of "beetween 2 matching pattern" sed command won't remove firstmatch and secondmatch.  I want the result to become:  It is a long established fact that a reader will be distracted by the readable content of a page when looking at its layout. I am trying cmd --> MMC --> Server Manager and then connecting to my Server 2012 no GUI  but i keep getting the following error The WinRM client cannot process the request  I am unable to upload images right now  This is not related to other post "How to Remote Manage Windows Server 2012 in Server Manager?" because title and description of that post is quite different from what i am asking Remove HTML Tags from an NSString on the iPhone There are a couple of different ways to remove HTML tags from an NSString in Cocoa.   is to render the string into an NSAttributedString and then grab the rendered text.   is to use NSXMLDocument's -objectByApplyingXSLTString method to apply an XSLT transform that does it.  Unfortunately, the iPhone doesn't support NSAttributedString or NSXMLDocument. There are too many edge cases and malformed HTML documents for me to feel comfortable using regex or NSScanner. Does anyone have a solution to this?  One suggestion has been to simply look for opening and closing tag characters, this method won't work except for very trivial cases.   For example these cases (from the Perl Cookbook chapter on the same subject) would break this method:  &lt;IMG SRC = "foo.gif" ALT = "A &gt; B"&gt;  &lt;!-- &lt;A comment&gt; --&gt;  &lt;script&gt;if (a&lt;b &amp;&amp; a&gt;c)&lt;/script&gt;  &lt;![INCLUDE CDATA [ &gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt;&gt; ]]&gt; Jeffrey Wooldridge in his Econometric Analysis of Cross Section and Panel Data (page 357) says that the empirical Hessian "is not guaranteed to be positive definite, or even positive semidefinite, for the particular sample we are working with.".  This seems wrong to me as (numerical problems apart) the Hessian must be positive semidefinite as a result of the definition of the M-estimator as the value of the parameter which minimizes the objective function for the given sample and the well-known fact that at a (local) minimum the Hessian is positive semidefinite.  Is my argument right?  [EDIT: The statement has been removed in the 2nd ed. of the book. See comment.]  BACKGROUND Suppose that $\widehat \theta_N$ is an estimator obtained by minimizing $${1 \over N}\sum_{i=1}^N q(w_i,\theta),$$ where $w_i$ denotes the $i$-th observation.  Let's denote the Hessian of $q$ by $H$, $$H(q,\theta)_{ij}=\frac{\partial^2 q}{\partial \theta_i \partial \theta_j}$$  The asymptotic covariance of $\widehat \theta_n$ involves $E[H(q,\theta_0)]$ where $\theta_0$ is the true parameter value. One way to estimate it is to use the empirical Hesssian  $$\widehat H=\frac{1}{N}\sum_{i=1}^N H(w_i,\widehat \theta_n)$$  It is the definiteness of $\widehat H$ which is in question. scope of clause within complex sentence If I said, &quot;I will run, not if it is snowing, if I my shoes are comfortable.&quot; Does the &quot;if my shoes are comfortable&quot; modify the &quot;I will run&quot; clause? I agree the sentence would be clearer with the addition of &quot;and&quot; in place of the comma. However, without the &quot;and&quot; is it still clear that the &quot;if my shoes are comfortable&quot; clause modifies the &quot;I will run&quot; clause? Transfer subtype Descriptions form long integer value into a string at new diffrent field WP List Table custom quick edit box - post meta data missing and columns change on submit I have extended class WP_List_Table to create a custom sortable table of posts along with a quick edit box with custom meta boxes.  The meta boxes populate and save but when I click submit the field values disappear and the defaults ( cb, title, tags, cats, etc) which are not even included in my class get added back to the column saved. This throws off the other columns on the page.  The saved custom field values return when the page is refreshed.  It seams like on submit the $post_object is missing.  I'm using a hacked get_inline_data function that includes the meta_values in my quick edit form instead of calling get_inline_data in my class.  My save function for the meta boxes:  add_action( 'save_post', 'save_inline_edit_meta' ); function save_inline_edit_meta( $post_id  ) {     global $pagenow;      if ( $pagenow == 'admin.php' ) {      if ( isset( $_POST[ 'is_quickedit' ] )  &amp;&amp; isset( $_POST[ 'standinghead' ] ) )         update_post_meta( $post_id, '_wnd_standing_head', $_POST[ 'standinghead' ] );      if ( isset( $_POST[ 'is_quickedit' ] )  &amp;&amp; isset( $_POST[ 'headline' ] ) )         update_post_meta( $post_id, '_wnd_alt_title', $_POST[ 'headline' ] );      if ( isset( $_POST[ 'is_quickedit' ] )  &amp;&amp; isset( $_POST[ 'deck' ] ) )         update_post_meta( $post_id, '_wnd_deck', $_POST[ 'deck' ] );     }   return $post_id;  }   My Javascript save function:  save : function(id) {         var params, fields, page = $('.post_status_page').val() || '';          if ( typeof(id) == 'object' )             id = this.getId(id);         $('table.widefat .inline-edit-save .waiting').show();         params = {             action: 'inline-save',             post_ID: id,             post_type: 'post',             edit_date: 'true',             post_status: page         };         fields = $('#edit-'+id+' :input').serialize();         params = fields + '&amp;' + $.param(params);          $.post('admin-ajax.php', params,             function(r) {                 $('table.widefat .inline-edit-save .waiting').hide();                  if (r) {                     if ( -1 != r.indexOf('&lt;tr') ) {                         $(inlineEditPost.what+id).remove();                         $('#edit-'+id).before(r).remove();                         $(inlineEditPost.what+id).hide().fadeIn();                     } else {                         r = r.replace( /&lt;.[^&lt;&gt;]*?&gt;/g, '' );                         $('#edit-'+id+' .inline-edit-save .error').html(r).show();                     }                 } else {                     $('#edit-'+id+' .inline-edit-save .error').html(inlineEditL10n.error).show();                 }             }         , 'html');         return false;     },