A function $f\colon E\to F$ between finite sets is injective iff it is surjective iff it is bijective
Suppose $X$ is a finite set and $f : X \to X$ is a function. Then $f$ is injective if and only if $f$ is surjective.
First and foremost I am sorry for this; in order to make the questions clear I must first upload the lecture notes from my institution $^\zeta$ for the derivation of Hydrostatic Equilibrium:          $^\zeta$ Lecture notes courtesy of Imperial College London, Astrophysics dept, 2017-2018 edition.    Firstly, in equation $(3.1)$ why is there no negative sign for $F_g$? Last time I checked, Newton's law of gravitation told us that   $$F=-\frac{GMm}{r^2}$$  Nextly, I have a conceptual problem with equation $(3.2)$; when they use the term 'pressure' I am going to assume that they are referring to the radiation pressure that is released due to nuclear fusion reactions taking place in the sun. In that formula it must be the case that $P(r) \gt P(r+\delta r)$ if this is the case then $(3.2)$ should read $$F_p=-\left(\frac{dP}{dr}\right)\delta r\,\delta A $$   Correct me if I'm wrong but Hydrogen nuclei are fused together to generate the more stable Helium and hence lots of radiation pressure acting outwards from the centre of the star. If this is correct then $P(r+\delta r)=0$, since the outside face of that cylindrical volume element in Figure $3.1$ will not 'feel' any of this radiation pressure because this pressure only acts on the inner face. What am I not understanding here?  Moving on to equation $(3.3)$, let's suppose the sign of the gravitational force in $(3.1)$ really is neglected. Does this mean that the negative sign in $(3.3)$ is due to $P(r+\delta r) \lt P(r)$ and hence $$\frac{GM(r)\rho(r)}{r^2}=-\left(\frac{dP}{dr}\right)\,?$$ I know this is the same equation as in $(3.3)$ but I have written it with the negative sign next to the pressure differential as I would like to know where this minus sign originates from. Does anyone know?  My final concern is the sign in $(3.4)$ as last time I checked $$g=-\frac{GM}{r^2}$$ and not $$g=\frac{GM}{r^2}$$  Hence, equation $(3.4)$ should read  $$\left(\frac{dP}{dr}\right)=\rho(r)\,g\,?$$     I forgot to mention that I have done some research and  is the closest derivation I could find, which, needless to say is no less helpful than the notes I have. As a part of self study, I am trying to prove the following statement:  Suppose $X$ and $Y$ are topological spaces and $f: X \rightarrow Y$ is a map. Then $f$ is continuous if and only if $f(\overline{A})\subseteq \overline{f(A)}$, where $\overline{A}$ denotes the closure of an arbitrary set $A$.   Assuming $f$ is continuous, the result is almost immediate. Perhaps I am missing something obvious, but I have not been able to make progress on the other direction. Could anyone give me a hint which might illuminate the problem for me? Can a US citizen leave and enter the US using a foreign passport? I am a dual citizen. I have both a US passport and a non-US passport. I live in the US. My US passport is currently being processed for renewal. I am unsure whether I will receive my passport in time for a trip that I have scheduled. Could I use my non-US passport to leave the US and return? I have never entered the US with my non-US passport. There is no immigration leaving the US but I still need to scan my passport for ticketing. Will the airline refuse me a ticket? I know that in this situation returning with a non-US passport can be problematic as well, but will I be able to leave at all? Note that I am traveling to a country which neither of my passports are from. Why doesn't the last digit method work for divisibility by 4? I wish to explain to a grade 5 kid, the answer to the question - 'When can we know that the observing the last digit of a number will not be sufficient to determine its divisibility by a given N?'  For example - It doesn't work for 3, because we get all sorts of digits, i.e. from 0 to 9, at the units place. The same reason works for 7 and 9.   But what would be a convincing argument for 4 and 8? Even though the last digits form a pattern - 0, 2, 4, 6 and 8 - but we're still not sure about its divisibility. many Servers of SQL server 2012 needs mirroring Difference between >>> and >> I'm trying to make airsoft chronograph/chronometer (a device that measures airsoft bullet speed). The idea is to measure time in which bullet is traveling from point A to point B, and then I can calculate bullet speed. Bullet is shot into plastic pipe with diameter of 32mm. I've tried to put on one side IR LED and on the other side phototransistor (BPW 40), both at the start and end of the tube. Something like this:  Blue circles are phototransistors, red circles are IR LEDs and white circles are airsoft bullets. When bullet passes through area where bullet 2  is drawn (just in between IR LED and phototransistor), everyting works perfectly. But when the bullet passes where bullets 1 and 3 are drawn, it isn't detected. That is expected, but unwanted behavior. So my question is: how can I detect bullets regardlessly of it's position ? I've thought about putting IR LEDs and phototransistors around the pipe (not only in one place), like this:  but this solution isn't cheapest: 5 IR LEDs + 5 BPW 40 = cca $12 times 2 (because I need it on both sides of the pipe) = $24. Is there any cheaper solution ? Is BPW 40 good choice for phototransistor ? I don't know which IR LEDs I'm using (and salesman in the store also doesn't know - he told me these are generic ones for remote controllers, like TV or DVD controller). Diameter of the pipe is 32mm and it will be long about 14cm (sensors will be 10cm apart). Diameter of airsoft bullet is 6mm. EDIT: I'll go for my second idea. I just have one more question: is it better to arrange IR emitters and photo transistors this way:  or this way:  Thanks !! Using round function in QGIS expression? Using Google Music Manager in Linux from the command line My goal is to use Google Music Manager on a Linux server.  I'm wondering if the music manager can operate from the command line and with no GUI.  If I could set the software up to watch a particular directory on the server and upload from there, that would be the only necessary functionality.  Thank you. Why do corticosteroids harm COVID-19 patients? states that steroids could do more harm than good in treating coronavirus (COVID-19), referring to .      Understanding the evidence for harm or benefit from corticosteroids in   2019-nCoV is of immediate clinical importance.   It's unclear to me how exactly the use of corticosteroids (which is a common treatments for Asthma patients) can do harm other than the vague explanation of "steroids also impair the immune system's ability to fight viruses".   How exactly does the use of steroids harm the body in case of fighting a virus such as SARS-CoV-2 (2019-nCoV) that causes COVID-19? Why is median age a better statistic than mean age? If you look at       Or this Wikipedia page      Clearly  seems to be the statistic of choice when it comes to ages.  I am not able to explain to myself why  would be a worse statistic. Why is it so?  Originally posted  because I did not know this site existed. Show only hidden files (dot files) in ls alias What is the differance between Linked and Appended data from external .blend files? Why do light disappears the moment we switch off the source (inside the wooden box)? Why does my Mac OS X 10.6 kernel run in 32-bit mode? I have a MacBook Pro (5,1) with 4GB of memory running 10A432 - but it's running the 32-bit kernel.  Any ideas why it wouldn't be running in 64-bit mode, and how I can get it to do so? Changing Posts Per Page and offset with pre_get_posts I have a filter using pre_get_posts to change the number of posts per page from 4 on the news landing page to 8 on all other pages (page 2, 3, etc). However I don't seem to have got the offset correct, or something, because my get_next_posts_page_link doesn't show a page 3, even though it does exist if I go to the url manually. The current offset is something I somehow managed to conjure up accidentally, I thought it worked as page 2 was correct but as I added more test posts, I realised it didn't work.  function page_2_posts_per_page_filter( $query ) {      $paged = (get_query_var('paged')) ? get_query_var('paged') : 1;      if ( !is_admin() &amp;&amp; $query-&gt;is_paged &amp;&amp; $query-&gt;is_main_query() ) {          $offset = 4 + (($paged - 2) * 8);         $query-&gt;set( 'offset', $offset );         $query-&gt;set( 'posts_per_page', 8 );      } else if (!is_admin() &amp;&amp; $query-&gt;is_main_query()) {          $query-&gt;set( 'posts_per_page', 4 );      }     return; } add_action( 'pre_get_posts', 'page_2_posts_per_page_filter', 1 );   EDIT: SOLUTION  function page_2_posts_per_page_filter( $query ) {      $paged = (get_query_var('paged')) ? get_query_var('paged') : 1;     $offset = 4;      if ( !is_admin() &amp;&amp; $query-&gt;is_paged &amp;&amp; $query-&gt;is_main_query() ) {          $offset = (($paged - 1) * 8) - $offset;         $query-&gt;set( 'posts_per_page', 8 );         $query-&gt;set( 'offset', $offset );      } else if (!is_admin() &amp;&amp; $query-&gt;is_main_query()) {          $query-&gt;set( 'posts_per_page', 4 );      }     return; } add_action( 'pre_get_posts', 'page_2_posts_per_page_filter');  function found_offset( $found_posts, $query ) {      $offset = 4;      if( !is_admin() &amp;&amp; $query-&gt;is_main_query() ) {         $found_posts = $found_posts + $offset;     }     return $found_posts; } add_action( 'found_posts', 'found_offset', 10, 2 ); How to enable numlock at boot time for login screen? I am trying to force numlock to be on upon initial boot at the login screen on Ubuntu 12.04. The only solutions I have found so far switch numlock on only after initial login.  I'm looking to force numlock to be on when the login screen is displayed, and before the user has logged in. Can anyone assist? What is the maximum possible length of a .NET string? What is the longest string that can be created in .NET? The docs for the String class are silent on this question as far as I can see, so an authoritative answer might require some knowledge of internals. Would the maximum change on a 64-bit system?  [This is asked more for curiosity than for practical use - I don't intend to create any code that uses gigantic strings!] Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer? If $n&gt;1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer.  If you know , then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from $\frac 1p$, every other term $\frac 1k$ has $k$ divisible only by primes smaller than $p$. We can combine all those terms to get $\sum_{k=1}^n\frac 1k = \frac 1p + \frac ab$, where $b$ is not divisible by $p$. If this were an integer, then (multiplying by $b$) $\frac bp +a$ would also be an integer, which it isn't since $b$ isn't divisible by $p$.  Does anybody know an elementary proof of this which doesn't rely on Bertrand's Postulate? For a while, I was convinced I'd seen one, but now I'm starting to suspect whatever argument I saw was wrong. Lighting in a Minecraftian World Minecraft is a game that is largely based on a heightmap and uses that heigtmap information to flood the world with light.  From my understanding the highest point in the heightmap is the end of the sunlight influenced area.  Everything above that is lit by sunlight, everything below that just is influenced by light nearby in a radius of 8 blocks.  Thus if you have a floating island on the top of your world everything below that will be seen essentially as a cave.  When two lights influence the same point the brighter light wins (unsure about that).  Either way there are a couple of problems with minecrafts lighting model: first of all, if your world does not have a heightmap it becomes trickier to figure out what exactly is supposed to emit sunlight and what not.  A simple way would be to assume that the world is (in my case) a floating rock and then traverse each axis from both directions and figure out where the rock starts and ends.  But this does not fully eliminate the problem as dents in the rock are not supposed to be in darkness.  Minecraft itself will cache the light information in its chunks together with the information about the material of a block.  Thus only if the world is modified the lighting has to update.  Unfortunately that process is still pretty slow on updates and on quick light changes one can see the lighting lag behind.  That's especially true if a lot of blocks change (TNT, sunset etc.) and you're not running the fastest computer (Or Java on Mac).  From my still limited understanding of 3D graphics lighting a world like minecraft shouldn't be the biggest issue.  How would you tackle the problem?  I think the basic requirements for lighting in a voxel world would be   update fast enough that it could happen in a single frame.  One might be able to do the lighting in the graphics device and download the changed light information to the main RAM. light information must be quickly available for the main game logic so not entirely based on the graphics device: reasoning: light affects the growth of grass, spawning of monsters etc. light updates would have to be local to a chunk or have some other limit so that one does not have to relight the whole world which might be very large in size.   The main idea would be to make the light updates fast, not necessarily more beautiful. For general light rendering performance improvements one could easily add SSAO on top of that which should result in much nicer worlds.