If $U$ is uniformly distributed on $S^{2}$, then its first component is uniformly distributed on $(-1,1)$.
Random point uniform on a sphere
How to enable top bar background opacity? There is a way to change top bar background opacity in ubuntu 20.04 lts.  I installed the hide top bar extension but is not a option to change it. Which is correct: "what if there was" or "what if there were"? Possible Duplicate:          Is this correct grammar?     What if there was a Stack Overflow on…    Shouldn't it be "what if there were a Stack Overflow on…"? I read in several places, including Wikipedia, that a tessellation of the plane by a single, convex, $n-$sided polygon is not possible for $n\geq7$. I was not able to locate a proof, or a paper that discusses this. Maybe it is too trivial, but I am not able to figure it out myself.   So I would like to ask you for help. Where can I find a proof for this?  Also, I am interested in the mathematical background behind plane tiling (especially periodic), If you can suggest papers on the topic it would really help. Assigning values to divergent series I have been looking at divergent series on wikipedia and other sources and it seems people give finite "values" to specific ones. I understand that these values sometimes reflect the algebraic properties of the series in question, but do not actually represent what the series converges to, which is infinity. Why is it usefull to assign values to divergent series?  The only theory I could come up with, is this:  Say you have 2 divergent series, series' A and B, and you assign each a value,   Series ($A= \sum_{n=0}^\infty a_n$), which I assigned the value Q  and series ($B= \sum_{n=0}^\infty b_n$ ), which I assigned the value P  But it just so happens that series $C=A-B= \sum_{n=0}^\infty (a_n-b_n)$ converges. Could that imply that the actual value of series $C$ is the difference of the two assigned values to $A$ and $B$, that is $\sum_{n=0}^\infty (a_n-b_n)=Q-P$ ?  If so, then that would make some sense to me, as to why people sometimes assign values to divergent series. Why not use "which"? What to use then? Are actions and bonus actions interchangeable? Is there an analog to the Runge-Lenz vector for a 3D spherically symmetric harmonic potential? Do small title edits constitute trivial edits? Derivatives of $ \frac{1}{r} $ and Dirac delta function I am trying to understand the formula \begin{equation} \nabla^2\left(\frac{1}{|{\bf r}-{\bf r}'|}\right) = - 4 \pi \delta(\bf{r}-\bf{r}'), \qquad\qquad {\rm (I)} \end{equation} where ${\bf r}=(x,y,z)$. This is something heavily used in electrostatics and the steps to 'show' this is often the following: The first derivative reads \begin{equation} \nabla \frac{1}{| {\bf r} - {\bf r}' |} = - \frac{ {\bf r} - {\bf r}'}{| {\bf r} - {\bf r}'|^3} \end{equation} And taking the second derivative gives zero, except for the singularity at ${\bf r} = {\bf r'}$. Then from the divergence theorem we have \begin{equation} \int dV \,  \nabla^2 \frac{1}{| {\bf r} - {\bf r'}|} = \int dS  \,\,{\bf n} \cdot ( \nabla\frac{1}{|{\bf r} - {\bf r}'|}) = -4 \pi \end{equation} where the integration is performed over a sphere centered at ${\bf r}'$. Q1: Is there a more direct proof for equation (I)? Then my main question is about the separate second-order differentials. For instance, we can obtain, by direct computation \begin{equation} \partial_x^2 \, \frac{1}{| {\bf r} - {\bf r}'|} = \frac{ 3 (x-x')^2 }{| {\bf r} - {\bf r'} |^5} - \frac{1}{| {\bf r} - {\bf r}'|^3} \end{equation} Q2: Should there be a $\delta$ function on the r.h.s of this equation? How can I ensure my trip to the UK will not have to be cancelled because of Brexit? I've got a trip planned to the UK in June. Due to brexit, I'm worried what changes may be introduced before my arrival.  I'm traveling on a EU / Polish passport. I'm going for a couple weeks of holiday.  Is there anything I can do right now to make it less likely any changes won't stop me at the immigration? For example, applying for some tourist visa which does not rely on the EU status? Why split the  tag when writing it with document.write()? Easy curves in TikZ I find myself needing to draw lots of elegantly curved paths in TikZ. Ideally, I'd just specify a series of points, and TikZ would calculate the extra data itself to draw a nice series of curves passing smoothly through these points, perhaps with an optional "looseness" parameter that I could specify. But the only way I can find to draw nice curves is by explicitly giving control points, or by manually specifying in and out angles.  I can think up a simple algorithm to do this, which would certainly be within TikZ's power to perform: just choose the in and out angles in a simple fashion based on the relative angles between each adjacent pair of line segments.  Is something like this already built-in? Or can someone cook something up that does the job?  Edit: Jake has provided an answer using the plot [smooth] functionality. This is almost perfect! But it can't do what I need, because it doesn't let me specify tangent angles manually where needed, which is especially important at the beginning and end of the curve. I would have thought this would be a natural and straightforward addition to the existing plot [smooth] algorithm: for every coordinate, an optional angle should be able to be specified as an argument, which if supplied is treated as the tangent angle for the curve at that point. And while we're at it, it wouldn't hurt to also allow the tension to be modified along the path.  A minimal extension to the algorithm would just accept two optional parameters, for the curve tangent at the beginning and end. We have two sequences , $(a_i)_{i=1}^{2n}$ and $(b_i)_{i=1}^{2n}$ such that  $1\leq a_i, b_i\leq n$ for every $i$.  Show that there are two sets of indexes $I, J \subseteq \left \{ 1,2, ... 2n \right \}$ such what $\sum_{i\in I}a_i=\sum_{j\in J}b_j$.  Well, the question didn't say anything about those sets being empty but I believe that's not what they meant. I don't know that to do with questions like these. There are obviously much more subsets that possible sums ($2^{2n}-1$ compared to $2n^2$) but it doesn't really help.  I'd be glad to hear ideas, hints or solutions. What is likely to happen when you plug two ends of a network cable to a single switch/router? Will this create problems on the network, or just be ignored? 'Country have' vs 'country has' 1) Australia has won the World Cup Cricket.      2) Australia have won the World Cup Cricket.   Which sentence is correct? Also, I would like to know more contexts where 'country has' and 'country have' are used. came in Low Quality Posts review which was  as "This is commentary on another post, not an answer" by *. But the comment was added 5 times automatically on behalf of that user.        First four comments' time is 2013-07-15 10:23:18Z while the last one's time is 2013-07-15 10:23:19Z. Those comments are still exist. 10K+ users can see that.  Is this a bug or something else?  * Because the comment is added on behalf of that user. In Drupal 8 you can load an entity with:  $node = \Drupal::entityManager()-&gt;getStorage('node')-&gt;load(123);   This looks for the entity definitions and finds that node is defined by Drupal\node\Entity\Node - so (I guess) Drupal\node\NodeStorage will instantiate a new Drupal\node\Entity\Node instance.  What I'd like to achieve is subclassing Drupal\node\Entity\Node and being able to instantiate this subclass when it's appropriate. For example if I have a node bundle article, there would be a class:  namespace Drupal\my_module\Entity\Article; class Article extends Drupal\node\Entity\Node { }   And I'd call:  $node = \Drupal::entityManager()-&gt;getStorage('node_article')-&gt;load(123);   And the return would be my Article subclass.  I can achieve this by creating a new entity type and wiring it back to another existing entity definitions, for example the node-article example would be this class:  namespace Drupal\my_module\Entity; use Drupal\node\Entity\Node; /**  * @ContentEntityType(  *   id = "node_article",  *   label = @Translation("Content"),  *   bundle_label = @Translation("Content type"),  *   handlers = {  *     "storage" = "Drupal\node\NodeStorage",  *     "storage_schema" = "Drupal\node\NodeStorageSchema",  *     "view_builder" = "Drupal\node\NodeViewBuilder",  *     "access" = "Drupal\node\NodeAccessControlHandler",  *     "views_data" = "Drupal\node\NodeViewsData",  *     "form" = {  *       "default" = "Drupal\node\NodeForm",  *       "delete" = "Drupal\node\Form\NodeDeleteForm",  *       "edit" = "Drupal\node\NodeForm"  *     },  *     "route_provider" = {  *       "html" = "Drupal\node\Entity\NodeRouteProvider",  *     },  *     "list_builder" = "Drupal\node\NodeListBuilder",  *     "translation" = "Drupal\node\NodeTranslationHandler"  *   },  *   base_table = "node",  *   data_table = "node_field_data",  *   revision_table = "node_revision",  *   revision_data_table = "node_field_revision",  *   translatable = TRUE,  *   list_cache_contexts = { "user.node_grants:view" },  *   entity_keys = {  *     "id" = "nid",  *     "revision" = "vid",  *     "bundle" = "type",  *     "label" = "title",  *     "langcode" = "langcode",  *     "uuid" = "uuid",  *     "status" = "status",  *     "uid" = "uid",  *   },  *   bundle_entity_type = "node_type",  *   field_ui_base_route = "entity.node_type.edit_form",  *   common_reference_target = TRUE,  *   permission_granularity = "bundle",  *   links = {  *     "canonical" = "/node/{node}",  *     "delete-form" = "/node/{node}/delete",  *     "edit-form" = "/node/{node}/edit",  *     "version-history" = "/node/{node}/revisions",  *     "revision" = "/node/{node}/revisions/{node_revision}/view",  *   }  * )  */ class Article extends Node { }  // Results my Article sub type. $node = \Drupal::entityManager()-&gt;getStorage('node_article')-&gt;load(123);   This works fine (as much as I can see); however, it smells. It adds a new entity type, which is not true, and might cause other troubles in the future.  How do I define a subclass for an entity bundle so that loading the entity would return an object of that class? Best books for mathematical background? What are the best textbooks to read for the mathematical background you need for modern physics, such as, string theory?  Some subjects off the top of my head that probably need covering:   Differential geometry, Manifolds, etc. Lie groups, Lie algebras and their representation theory. Algebraic topology. Careers tag search should only search tags From what I understand, midi-chlorian count is a strong indicator of ability in the Force (though there are other factors as well). So it seems like when Anakin loses his arm or Luke loses his hand, their midi-chlorian counts (and thus their Force ability) should slightly diminish.  Is this the case?