mysql: Add comma each N characters
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Theme Activate Hook Every finite set contains its supremum: proof improvement. Every finite subset of $\mathbb R$ contains its supremum (and its infimum)   Proof Let $A=\{a_1,...,a_n\}$ be a finite subset of $\mathbb{R}$. Since it is non-empty and it is bounded ($\max A$ is an upper bound), it has supremum, that is $\exists \sup A$ and by definition $\forall a \in A \;\, a \leq \sup A$. Let's suppose that $\sup A  \not\in A$ then, since $\max A \in A$ we have that $\max A &lt; \sup A$. But considering that $\mathbb Q$ is dense in $\mathbb R$ we can conclude that $\exists r \in \mathbb Q$ s.t. $\max A &lt; r &lt; \sup A$, but this is absurd since $r$ is an upper bound of $A$ and it is lower than the supremum. Necessarily, $\sup A\in A$.  Is there anything wrong? Is there any way to prove this without using density of $\mathbb Q$ or another property? Thanks in advance. Display solution/file path in the Visual Studio IDE I frequently work with multiple instances of Visual Studio, often working on different branches of the same solution.   used to display the full path of the current source file in its title bar, but Visual&nbsp;Studio 2005 doesn't appear to do this. This makes it slightly more awkward than it should be to work out which branch of the solution I'm currently looking at (the quickest way I know of is to hover over a tab so you get the source file's path as a tooltip).  Is there a way to get the full solution or file path into the title bar, or at least somewhere that's always visible, so I can quickly tell which branch is loaded into each instance? People often use sans, the French word for ‘without’ in English.   What I don’t get, is when should that switch be made. When should I say sans when I really want to say without. Are there any general rules regarding usage, or is it kind of a person-to-person basis/feeling of flowerness. Stack exchange badges It might be useful to encourage existing users of stackoverflow to cross pollinate other stack exchange sites.  While rep-sharing is not useful, as rep is meant to be a measure of expertise in one subject, badges may be one way to encourage participation across many sites.  This was triggered by seeing how few people actually had greater than 1k rep on all three main sites (SO, SF, SU) via Dennis Williamson's  app.  I had tried to do this by hand prior to the API under the assumption that many people would be answering questions on all three sites, but it turns out there are only 33 people, of all the users of stackoverflow, that have a 1k+ reputation on all three sites.  Over time it may be useful to provide badges that appear on all sites one is participating in that are awarded for extraordinary participation in the whole universe of Stack Exchange sites.  For instance, a silver badge might appear on all my profiles if I obtain 1k rep on 5 or more sites, and a gold badge if I have more than 3k rep on 5 or more sites.  I would hesitate extending this much past 5 sites as we don't want to encourage spamming questions and answers on sites where the user has little interest, but across the universe of stackapps I know I'm proficient or expert in at least 10 current sites, and while it's not a big nudge, the idea that I can get extra awards - especially ones that appear on all sites I'm in - would move me to participate a little more on sites that I am registered on, but don't visit regularly.    I expect, eventually as sites are launched, most users of any one site will have some interest or knowledge in at least 5 other sites.  This may provide the extra nudge some sites need to maintain an excellent rating in terms of usage. Is it a good practice to place C++ definitions in header files? My personal style with C++ has always to put class declarations in an include file, and definitions in a .cpp file, very much like stipulated in . Admittedly, part of the reason I like this style probably has to do with all the years I spent coding Modula-2 and Ada, both of which have a similar scheme with specification files and body files. I have a coworker, much more knowledgeable in C++ than I, who is insisting that all C++ declarations should, where possible, include the definitions right there in the header file. He's not saying this is a valid alternate style, or even a slightly better style, but rather this is the new universally-accepted style that everyone is now using for C++. I'm not as limber as I used to be, so I'm not really anxious to scrabble up onto this bandwagon of his until I see a few more people up there with him. So how common is this idiom really? Just to give some structure to the answers: Is it now The Way™, very common, somewhat common, uncommon, or bug-out crazy? How can I embed a View inside a content of a node? How can I embed a view inside a content of a node in Drupal 8?   I found something similar, for example in the question about "" or the  module, but it is all for Drupal 7.  I tried with an Entity reference field pointing to a View but it display only a label and from the Manage Display tab I can choose only from Label and Entity ID. I need something like the "Rendered entity" option that is present on Entity reference fields pointing to other nodes. Does it make sense to Favourite a Question you created? Is this just a redundant act, or are there differences? How long is it safe to drink coffee after brewed? How do I get the distance meter value between two geometries in PostGIS? I have a simple question about calculating distances in PostGIS.  I would like to get the distance between two geometries. I am using this sid : 4269 in meters what i am doing now is this : ST_Distance((a.geom,b.geom)) FROM ... but I am getting result in degrees. I think that I should work with geography but how can I cast a geom to a geography? I tried with (a.geom::geography) but i am getting an error.  What can I do to get my result in meters?   Thank You Here's my attempt at it:  $query = $database-&gt;prepare('SELECT * FROM table WHERE column LIKE "?%"');  $query-&gt;execute(array('value'));  while ($results = $query-&gt;fetch())  {     echo $results['column']; } In ZFC, a good way of ordering sets by cardinality is by leveraging the notion of an injection. We define: $$X \lesssim Y \leftrightarrow \mbox{ there exists an injection } X \rightarrow Y.$$  Alternatively, we can leverage the notion of a partial surjection, by which I mean a partial function that hits every point of its codomain at least once.  $$Y \succsim X \leftrightarrow \mbox{ there exists a partial surjection } Y \rightarrow X.$$  Now in ZFC, these approaches are equivalent, in the sense that $$X \lesssim Y \iff Y \succsim X.$$  But if we just consider ZF (that is, we drop choice) then the equivalence breaks down. However, we've still got the forward implication, because the converse of an injection is always a partial surjection.  A specific question. Define that two sets are weakly equipotent iff $X \succsim Y$ and $Y \succsim X$. Question: does weak equipotence imply genuine equipotence? I'm guessing "no."  A general question: In broad terms, how would the structure of the cardinal numbers in ZF be altered by adopting $\succsim$ rather than $\lesssim$ as the fundamental order relation, and taking equality of cardinal numbers to correspond to weak equipotence of sets? I'm guessing this would hide a lot of the (usually very complicated) structure of the cardinal numbers in ZF.  Remark. The order symbols can be written \lesssim and \succsim. Game resource timers - general implementation I'm interested in understanding how resource generation timers work on many of the mmorts type games out there. I can't wrap my head exactly around how the timers are synched with servers, especially with multiple users. If I need to provide more details, please let me know.  Thanks! Windows 7 won't boot after installing Ubuntu 11 meaning of the word "in" in context It is from . It is at around 8 minute and 39 second. Here it goes:     Culinary cultural fusion is all the rage; more novels translated from languages then ever before, although few are read, and in the surest sign of cultural globalization, football, the world's game, finally has reached America.   I guess the sentence would make sense if the preposition in were dropped out. Could you please tell me what did the host use it for? Reprojecting MODIS Swath data to WGS84 using GDAL I can write  VAR=$VAR1 VAR=${VAR1} VAR="$VAR1" VAR="${VAR1}"   the end result to me all seems about the same. Why should I write one or the other? are any of these not portable/POSIX? Why isn't apt full-upgrade getting an updated software key signature for skype After a recent apt update &amp;&amp; apt full-upgrade I started getting the following error when running apt full-upgrade subsequently: Get:4 https://repo.skype.com/deb stable InRelease [4,502 B]                                         Err:4 https://repo.skype.com/deb stable InRelease   The following signatures were invalid: EXPKEYSIG 1F3045A5DF7587C3 Skype Linux Client Repository &lt;se-um@microsoft.com&gt; W: An error occurred during the signature verification. The repository is not updated and the previous index files will be used. GPG error: https://repo.skype.com/deb stable InRelease: The following signatures were invalid: EXPKEYSIG 1F3045A5DF7587C3 Skype Linux Client Repository &lt;se-um@microsoft.com&gt; W: Failed to fetch https://repo.skype.com/deb/dists/stable/InRelease  The following signatures were invalid: EXPKEYSIG 1F3045A5DF7587C3 Skype Linux Client Repository &lt;se-um@microsoft.com&gt; W: Some index files failed to download. They have been ignored, or old ones used instead.  Is there a way to correct this, or is it down to Microsoft/Canonical to produce/update the necessary key signatures? Why does adding a new value to list overwrite previous values in the list CSS Display an Image Resized and Cropped I want to show an image from an URL with a certain width and height even if it has a different size ratio.  So I want to resize (maintaining the ratio) and then cut the image to the size I want.  I can resize with html img property and I can cut with background-image.  How can I do both?  Example:  This image:     Has the size 800x600 pixels and I want to show like an image of 200x100 pixels   With img I can resize the image 200x150px:  &lt;img      style="width: 200px; height: 150px;"      src="http://i.stack.imgur.com/wPh0S.jpg"&gt;    That gives me this:    &lt;img style="width: 200px; height: 150px;" src="https://i.stack.imgur.com/wPh0S.jpg"&gt;       And with background-image I can cut the image 200x100 pixels.  &lt;div      style="background-image:            url('https://i.stack.imgur.com/wPh0S.jpg');      width:200px;      height:100px;      background-position:center;"&gt;&amp;nbsp;&lt;/div&gt;    Gives me:        &lt;div style="background-image:url('https://i.stack.imgur.com/wPh0S.jpg'); width:200px; height:100px; background-position:center;"&gt;&amp;nbsp;&lt;/div&gt;      How can I do both?  Resize the image and then cut it the size I want?