How can I prove that the following series does not converge uniformly.
$\sum_{n=1}^\infty\frac{z^n}{n}$ does not converge uniformly on $\mathbb{D}$.
Maximum Likelihood Estimator for $\theta$ when $X_1,\dots, X_n \sim U(-\theta,\theta)$ Exercise :     Calculate a Maximum Likelihood Estimator for the model $X_1,\dots, X_n \; \sim U(-\theta,\theta)$.   Solution :  The distribution function $f(x)$ for the given Uniform model is :  $$f(x) = \begin{cases} 1/2\theta, \; \; -\theta \leq x \leq \theta \\ 0 \quad \; \; , \quad\text{elsewhere} \end{cases}$$  Thus, we can calculate the likelihood function as :  $$L(\theta)=\bigg(\frac{1}{2\theta}\bigg)^n\prod_{i=1}^n\mathbb I_{[-\theta,\theta]}(x_i)= \bigg(\frac{1}{2\theta}\bigg)^n\prod_{i=1}^n \mathbb I_{[0,\theta]}(|x_i|) $$  $$=$$  $$\bigg(\frac{1}{2\theta}\bigg)^n\prod_{i=1}^n \mathbb I_{[-\infty,\theta]}(|x_i|)\prod_{i=1}^n \mathbb I_{[0, +\infty]}(|x_i|)$$  $$=$$  $$\boxed{\bigg(\frac{1}{2\theta}\bigg)^n\prod_{i=1}^n \mathbb I_{[-\infty,\theta]}(\max|x_i|)}$$  Question : How does one derive the final expression in the box from the previous one ? I can't seem to comprehend how this is equal to the step before.  Other than that, to find the maximum likelihood estimator you need a $\theta$ sufficiently small but also $\max |x_i| \leq \theta$ which means that the MLE is : $\hat{\theta} = \max |x_i|$. Why does (0 < 5 < 3) return true? Where does gravitational waves' energy go? Following the measurement of gravitational waves, many sources described them and explained they carry energy away. What I don't get is how this energy will get transfered back to anything else.  If the fabric of space-time itself is vibrating, it would seem to be in impossible for any physical object to gain this energy.    What am I missing?  How would one hypothetically get energy out of gravitational waves?  If impossible, does the universe end up with nothing but GW? Can I look at the upgrade log after a distribution upgrade? I've upgraded to Ubuntu 12.04 and did it . I didn't look at the "packages to remove" section before clicking next - I've done Ubuntu upgrades many times and assumed it was just the normal set of obsolete library packages etc. But I do tend to watch the terminal text go by and see what it's doing, and I saw it removing a number of packages I wanted. I guess these are packages that aren't on the CD that can't co-exist with the new Ubuntu.  Anyway, I want to find which packages have been removed. So does the update log get saved somewhere?  Then I can start doing some grepping on it and then reinstall the packages I did want. Rewriting a custom-post-type permalink with taxonomy term? I know how to CALCULATE dot product, but still don't understand MEANING of it.  I mean, If I have to explain vector addition I would say that "Sum of two vectors give you the third vector that has the same effect as other two combined"  But I don't know how to explain dot product. For example we have two vectors: v=(3,1) and w=(2,4). Dot product of these two is 3*2 + 1*4 = 10. What does this result (10) tell us? I have an object at a fixed position in 3D space. I would like the camera to circle around it while facing it. I want to path of the camera to be circular in 2D, meaning that it does not move up or down.  I need this to be pretty accurate. I don't want to manually set a keyframe every frame for position and rotation. I would like to do this maybe with an expression, not with manual keyframing. I am not sure how to go about this but I think it is going to involve Python as well as some trigonometry. Identifier for win64 configuration in Qmake Is there a "win64" identifier in Qmake project files?  documentation does not mention other than unix / macx / win32.  So far I've tried using:  win32:message("using win32") win64:message("using win64") amd64:message("using amd64")   The result is always "using win32".  Must I use a separate project-file for x32 and x64 projects, so they would compile against correct libraries? Is there any other way to identify between 32-bit and 64-bit environments? Making a Comment on a page without being on that page? I have a built a Wordpress site that enables comments on pages and I am using them as reviews, what I would like to do is for someone to be able to make a comment on a page without being on that page.    On the link above, the top right, there is a link that says have your say about a university, it would be good to have a HTML form and then a dropdown box of all the child pages of university-guide; which is where all the universities are stored, and then I can put my comments.php file into this.  So my HTML form appears and when a page is picked from a dropdown box, the comment will then be assigned to that page.  Any help or direction?    New Question for Toscho:  Below is your tweaked plugin to fit into my website, as you can see I have added the select option yearSelect and need this to appear when the comments appear. How would I go about doing this? - Could you provide ay help please?      &lt;?php /* Plugin Name: haveyoursay */  /**  * Call the form in your template with:  * do_action( 'wpse_67527_comment_form' );  */ add_action( 'wpse_67527_comment_form', 'wpse_67527_comment_form' );  /**  * Create a review form.  *  * @return void  */ function wpse_67527_comment_form() {     $select_id = 'page_select';     $url = 'http' . ( is_ssl() ? 's' : '' ) . '://' . $_SERVER['HTTP_HOST'] . $_SERVER['REQUEST_URI'];     $url = esc_url( $url );     ?&gt; &lt;form action="&lt;?php echo site_url( '/wp-comments-post.php' ); ?&gt;" method="post"&gt; &lt;h3&gt;Leave us your student perspective &lt;img src="/wp-content/themes/blue-and-grey/images/hmepage-tri.png"&gt;&lt;/h3&gt;     &lt;p&gt;         &lt;label for="author"&gt;&lt;?php _e( 'Name' ); ?&gt;&lt;/label&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;         &lt;input id="author" name="author" type="text" size="40" /&gt;     &lt;/p&gt;     &lt;p&gt;         &lt;label for="email"&gt;&lt;?php _e( 'Email' ); ?&gt;&lt;/label&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;         &lt;input id="email" name="email" type="text" size="40" /&gt;     &lt;/p&gt;     &lt;p&gt;         &lt;label for="&lt;?php echo $select_id; ?&gt;"&gt;             &lt;div class="clear"&gt;&lt;/div&gt;             Pick a university from the list: &amp;nbsp;&amp;nbsp;         &lt;/label&gt;         &lt;?php         // Ordered by page name automatically         wp_dropdown_pages(             array (                 'name' =&gt; 'comment_post_ID',                 'id'   =&gt; $select_id,                 'child_of' =&gt; 2,             )         );         ?&gt;       &lt;/p&gt;     &lt;div class="clear"&gt;&lt;/div&gt;     &lt;p style="margin:10px 0px 0px 0px;"&gt;Please select your year of study: &amp;nbsp;&amp;nbsp;     &lt;select name="yearSelect"&gt;     &lt;option value="1st"&gt;First Year&lt;/option&gt;     &lt;option value="2nd"&gt;Second Year&lt;/option&gt;     &lt;option value="2nd"&gt;Third Year&lt;/option&gt;     &lt;/select&gt;     &lt;/p&gt;     &lt;br /&gt;     &lt;p&gt;         &lt;textarea name="comment" cols="45" rows="8"&gt;&lt;/textarea&gt;     &lt;/p&gt;     &lt;p&gt;         &lt;?php         wp_nonce_field( 'unfiltered-html-comment_' . get_the_ID(), '_wp_unfiltered_html_comment_disabled', FALSE );         ?&gt;         &lt;input type="hidden" name="redirect_to" value="&lt;?php echo $url; ?&gt;" /&gt;         &lt;input type="submit" style="margin:15px 0px 0px 0px; padding:4px 10px;" /&gt;     &lt;/p&gt; &lt;/form&gt;     &lt;?php }  add_filter( 'comment_post_redirect', 'wpse_67527_redirect' );  /**  * Strip '#comment-number' from redirect url.  *  * @param  string $url  * @return string  */ function wpse_67527_redirect( $url ) {     $parts = explode( '#', $url );     return $parts[0]; } ?&gt; Why particle hole symmetry and chiral symmetry are called symmetries? $PHP^{-1}=-H$ (particle-hole symmetry) and   $\Gamma H \Gamma^{-1}=-H$ (chiral symmetry)   I understand why we get the negative signs but im just a bit confused as to why such equalities mean $H$ is particle hole and chiral symmetric.   When we say $H$ is symmetric under some operation, like time reversal,  don't we usually mean that $H$ is invariant under the said transformation?  $THT^{-1}=H$ if $H$  is time reversal symmetric.  On the contrary, we say $H$ is symmetric under particle hole and chiral operations when $H$ picks up a negative sign. This seems to contradict our usual convention of what we mean by $H$ being symmetric(invariant). I've heard that exposing database IDs (in URLs, for example) is a security risk, but I'm having trouble understanding why.  Any opinions or links on why it's a risk, or why it isn't?  EDIT: of course the access is scoped, e.g. if you can't see resource foo?id=123 you'll get an error page. Otherwise the URL itself should be secret.  EDIT: if the URL is secret, it will probably contain a generated token that has a limited lifetime, e.g. valid for 1 hour and can only be used once.  EDIT (months later): my current preferred practice for this is to use UUIDS for IDs and expose them. If I'm using sequential numbers (usually for performance on some DBs) as IDs I like generating a UUID token for each entry as an alternate key, and expose that. The below statement is a true/false exercise.  Statement: For all square matrices A and B of the same size, it is true that $(A + B)2 = A^2 + 2AB + B^2$.  My thought process: Since it is not a proof, I figure I can show by example and come to a valid conclusion based on such example.  My work:  Come up with a square matrix A and B let both be a 2 by 2 matrix(rows and cols must be same).  Matrix $A$:  $A = \begin{array}{ccc} 3 &amp; 5 \\ 4 &amp; 6 \\ \end{array} $  Matrix $B$:  $B =  \begin{array}{ccc} 5 &amp; 8 \\ 9 &amp; 4 \\ \end{array} $  $A + B =  \begin{array}{ccc} 8 &amp; 13 \\ 13 &amp; 10 \\ \end{array}$  $(A + B)^2 =  \begin{array}{ccc} 233 &amp; 234 \\ 234 &amp; 264 \\ \end{array}$  $A^2 =  \begin{array}{ccc} 29 &amp; 45 \\ 36 &amp; 56 \\ \end{array}$  $(AB) =  \begin{array}{ccc} 60 &amp; 44 \\ 74 &amp; 56 \\ \end{array}$  $2(AB) =  \begin{array}{ccc} 120 &amp; 88 \\ 234 &amp; 112 \\ \end{array}$  $B^2 =  \begin{array}{ccc} 97 &amp; 72 \\ 81 &amp; 88 \\ \end{array}$  $A^2 + 2AB + B^2 =  \begin{array}{ccc} 246 &amp; 205 \\ 265 &amp; 256 \\ \end{array}$  Based my above work, the answer is false.   Is there another way to approach the problem? It seems like a lot of work needed to be done for a true/false question which raised my suspicion about whether there is a better way to look at the problem. How does "curved space" explain gravitational attraction? They say that gravity is technically not a real force and that it's caused by objects traveling a straight path through curved space, and that space becomes curved by mass, giving the illusion of a force of gravity.  That makes perfect sense for planetary orbits, but a lot less sense for the expression of gravity that we are the most familiar with in our day-to-day lives: "what goes up must come down."  Imagine that I hold a ball in my hand, several feet off the ground, with my fingers curled around it, and my hand is above the ball.  Then I open my fingers, releasing my grip on it, being very careful to not impart any momentum to the ball from my hand as I do so.  An object at rest remains at rest unless acted upon by an outside force.  If the ball is not moving (relative to my inertial reference frame), it has no path to travel that's any different from the Earth's path through space.  It should remain at rest, hanging there in the air.  And yet it falls, demonstrating that an outside force (gravity) did indeed act upon it.  How does curved space explain this? sum of polynoms of given property Let $\{X_\alpha\}_{\alpha\in J}$ be a family of topological spaces and let $X=\prod_{\alpha\in J}X_\alpha$ be endowed with the product topology. Show that $X$ is path connected if and only if each $X_\alpha$ is path connected.  I know this is true if path coennected is changed by connection but I am very confused with this problem, I have tried to do the following:  $\Rightarrow $ let $x,y \in X_\alpha$ be then $(x_\alpha), (y_\alpha)\in \prod X_\alpha$ where $x_\alpha=x, y_\alpha=y$ and $x_\beta=0, y_\beta=0$ for any $\beta\neq\alpha$, and since P is path connected then there is continuous $f:[0,1]\rightarrow \prod X_\alpha$ such that $f(0)=(x_\alpha)$ and $f(1)=(y_\alpha)$, then if $\pi_\alpha:\prod X_\alpha \rightarrow X_\alpha$ is the projection, which is continuous, we have that $\pi_\alpha f:[0,1]\rightarrow X_\alpha$ is continuous and $\pi_\alpha (f(0))=x$ and $\pi_\alpha (f(1))=y$.  $\Leftarrow$ I do not know how to do this address, could someone help me please? Thank you very much. LWC URL reading parameters I am trying to authorize another sandbox/production to create connection. When I first time authorize the connected app, it is returning the Auhorization Code in the URL, however when it gets redirected to LWC app, code is disappeared because the parameter is 'code' not like 'c__code'. So, pagereference state is unable to read the param value. It was working till the previous release, with Spring release it started behaving differently.  Original URL: not working    I tested with adding 'c__' to the param manually and it worked fine.  Modified URL: Working    is there a way we can rename the redirected URL params dynamically?  'code' to 'c__code' How to get coloured terminal over ssh? When I connect over ssh to remote hosts everything is just a single font/color. I would like to have colours like I do locally e.g. green for executable and blue for symlinks etc. And such that when I run $ git diff on the ssh host it shows me diff with colours =) A characterisation of quadratic extensions contained in cyclic extensions of degree 4 Let $D$ be a squarefree integer. I am trying to prove that $\mathbb Q[\sqrt D]$ is contained in a Galois extension of $\mathbb Q$ with Galois group $\mathbb Z/4$ if and only if $D$ is the sum of two squares, $D = a^2 + b^2$ with $a,b\in \mathbb Q$.   The hint in the exercise in Dummit and Foote suggests considering the extension $\mathbb Q[\sqrt{s + s\sqrt D}]$ for the forward direction. However, I am unsure how to show that this extension is Galois, much less that its Galois group is cyclic of order 4.   For the reverse direction, I expressed the given Galois extension as $\mathbb Q[\sqrt{a+b\sqrt D}]$ for some $a,b\in \mathbb Q$. I showed that this extension has $\mathbb Q[\sqrt{c^2 - d^2 D}]$ as a subfield for some $c,d \in \mathbb Q$. From this, I deduced that the $\sqrt D$ and $\sqrt{c^2 - d^2 D}$ generate the same extension, so $\sqrt D = x \sqrt{c^2 - d^2 D}$ for some $x$, and hence that $D = \frac{x^2 c^2}{1 + x^2 d^2} $. It seems that there should be a way to decompose this as a sum of two squares, but I did not see how.   I did not tag this as homework as I am studying it independently. Show that the function $g(x) = x^2 \sin(\frac{1}{x}) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$ Show that the function $g(x) = x^2 \sin\left(\frac{1}{x}\right) ,(g(0) = 0)$ is everywhere differentiable and that $g′(0) = 0$. Will Windows 10 Technical Preview be upgradable to final release? Is there any information about if Windows 10 Technical Preview will then be upgradable to the final release ?  I'm asking that because I would like to use it on my primary PC (I'm aware of the risks but I accept them), in place of Windows 8.1 But I'd like to avoid to reinstall all after the release of the final version.