Why if a system $AX=Y$ has solutions in a field F then it has solution in a subfield F_1?
System of linear equations having a real solution has also a rational solution.
How are the best director award winners chosen? I do not understand how the best director awards are determined each year. It seems impossible to assess the method of a movie director unless you are part of the crew.  So, how do the nominators/judges assess the directors' performance each year? When a player joins the server, this command block (containing gamemode 2 @a) puts them into gamemode 2/adventure mode. How can I change this so that only non-admin/not "opped" players' gamemode gets changed? I have a Python regular expression that contains a group which can occur zero or many times - but when I retrieve the list of groups afterwards, only the last one is present. Example:  re.search("(\w)*", "abcdefg").groups()  this returns the list ('g',)  I need it to return ('a','b','c','d','e','f','g',)  Is that possible? How can I do it? Can you provide a list of objects that needs to be disposed in SharePoint? Do elevator upgrades increase your passive credit earnings, too? Where is the operator panel in Blender 2.8? How is the nature of logical principles commonly defined in contemporary philosophy? In contemporary philosophy, how exactly is the nature of logical principles defined? For example, the way I've commonly seen logical principles construed are as true propositions which described the most general laws of reality. The previous definition does seem very intuitive to me; if we understand logical principles as being concerned with truth, and if we accept a correspondence theory of truth, then logical principles simply describe fundamental laws regarding various states of affairs occurring in reality. (e.g. If state of affairs S1 obtains and state of affairs S2 obtains, then state of affairs S3 must also obtain given logical law L).  However, I've mostly heard the previous view expressed among laypersons. I'm interested to know how most philosophers would think about the nature of logical principles. How can I get vertex positions from a mesh? I want to read the vertex x and y coordinates with a script or a function.  How can I do that?  The   appears to have no position I could use.  I'd take any alternative approach that could give me x and y coordinates. I was writing a toString() for a class in Java the other day by manually writing out each element of the class to a String and it occurred to me that using reflection it might be possible to create a generic toString() method that could work on ALL classes. I.E. it would figure out the field names and values and send them out to a String.  Getting the field names is fairly simple, here is what a co-worker came up with:  public static List initFieldArray(String className) throws ClassNotFoundException {      Class c = Class.forName(className);     Field field[] = c.getFields();     List&lt;String&gt; classFields = new ArrayList(field.length);      for (int i = 0; i &lt; field.length; i++) {         String cf = field[i].toString();         classFields.add(cf.substring(cf.lastIndexOf(".") + 1));     }      return classFields; }   Using a factory I could reduce the performance overhead by storing the fields once, the first time the toString() is called. However finding the values could be a lot more expensive.  Due to the performance of reflection this may be more hypothetical then practical. But I am interested in the idea of reflection and how I can use it to improve my everyday programming. I just signed up, a day or two ago, and I've received a notification that there's something in my inbox.  Unfortunately, my email client isn't able to show me a link to my Inbox, or the message, because of the format of the email.  I normally use the email address that I gave when I signed up, just with MailWasher, which is intended to safely preview emails, as plain text, to decide whether or not to open them in a normal HTML email client.  I would be nice if that email contained a link to my inbox, which could be shown in MailWasher.  In order to collect that email in a normal email client, I'd also have to collect or delete all the other emails sent to that address.  I'm not entirely clear what's been done in that email to stop MailWasher seeing the link.  I can see a link to leave feedback, and one to unsubscribe, but not one to check my inbox.  Sure, I can live without one, but those emails would be a lot more useful to me if I could just click a link to go to my Inbox. Examples of the Pigeonhole Principle As most of you might know, the Pigeonhole Principle basically  that      If $n$ items are put into $m$ containers, with $n&gt;m$, then at least one container must contain more than one item   It always surprises me how this trivial - and at the same time powerful - idea might be the key in order to solve extremely complicated math olympiad-problems...  Quick and beautiful solutions are characteristic of pigeonhole problems, which are often a three-part process   Recognize that the problem requires the Pigeonhole Principle Figure out what the pigeons and what the pigeonholes might be After applying the pigeonhole principle, there is often more work to be done   I'll illustrate this with an example I've always liked...     (Example-)Problem: Given a $n\times n$ square, prove that if $5$ points are placed randomly inside the square, then two of them are at most $\frac{n}{\sqrt2}$ units apart.   Step 1: This problem can be solved with the Pigeonhole Principle  Step 2: We divide the $n\times n$ square into four $\frac{n}{2}\times\frac{n}{2}$ squares (pigeonholes). Consequently, at least two points (pigeons) are inside the same $\frac{n}{2}\times\frac{n}{2}$ square.  Step 3: The maximal distance between two points in an $\frac{n}{2}\times\frac{n}{2}$ square is the diagonal, which has the length $\frac{n}{\sqrt2}\qquad\square$  Another problem which can be solved with the Pigeonprinciple is the following:     IMO $1972/1$           Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.      At this point, you might have noticed how useful the Pigeonhole Principle can be, if you know how to recognize and use it.     Question: I would like to work on this amazing principle with my students for a week and was, therefore, gathering problems related to the Pigeonhole Principle with beautiful solutions.   Could you suggest some more? Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}  $  Thanks in advance, my professor asked us to this a couple weeks ago, but I was enable to get to the right answer.   Good luck!  Here is what I got up to;  $\frac{(n+1)!}{(n-r)!(r+1)!} = \frac{(n)!}{(r)!(n-r)!} + \frac{(n)!}{(r+1)!(n-r-1)!}    $ Column validation for US phone number format When I use TikZ tree with nodes with rounded corners, the connecting lines do not touch the nodes (near corners) but end where would rectangle end. Is there an easy way around it?  \documentclass{standalone} \usepackage{tikz} \begin{document} \begin{tikzpicture} [sibling distance=100]    \node[rectangle, draw, rounded corners = 3]{ }   child { node { } }   child { node { } }   ; \end{tikzpicture} \end{document} Why does 'g' sometimes read as 'j'? Most words with 'g' read as 'g'   Geek Goose Goods Goal   But sometimes 'g' reads as 'j'   Gymnastics Genesis Genius Genetic   And in the middle of a sentence, compare:   Log Logistics   Or   Frag Fragile check if numbers from command line are powers of 2 Correct way to render crepuscular rays in cycles? While trying to use the new cycles volumetric rending features to create  (with the intention of updating ), I have found this to be a little trickier than I anticipated.    When I create a basic setup with a spot lamp (sun lamps don't seem to work, not sure if that is a result of their one directional nature or what..) and a world material like this:    Which gives me this (500 samples):    The problems with this that make me think that I'm missing something (aside from the rendertime and noise, which I understand to be a part of using bleeding edge new features) are:   The background shader/sky texture isn't visible at all (volume is too dense?) The lamp needed to be set to 50000 strength (this is of course dependent on the distance from the camera, but the kinds of strengths required to get some nice defined rays will blow everything out if volumes are disabled)   I can't seem to get a straight render (no compositing tricks) to look anything remotely like the pretty image in the release notes:        (source: )     Is this just how it works? Or is there some trick I have missed (not counting compositing/renderlayer tricks)? Compare dates in query from IMPORTRANGE and cell for a few days I'm stuck at comparing two dates.  In a big table with dates as columns and products as rows, we want to display the amount of sales per day per product.  What the following query does:   Import our sales via IMPORTRANGE query the specific sales for a given day and product ID return 0 if there is an error (e.g. not found) count the result   This is a working query with the date 2018-03-26 hard coded:  =COUNTA(IFERROR(query(IMPORTRANGE("1OHR0nr_w3oFNNxtMxRwv65xncCBRpK-vV3vvqmHGFPo"; "Sales March!A:K"); "select Col6 where Col6 = '"&amp;$A95&amp;"' and todate(Col11) = date '2018-03-26'"; 0)))  It shows how many sales a specific product has on the specified date.  Now I want the date from a cell in the sheet:  =COUNTA(IFERROR(query(IMPORTRANGE("1OHR0nr_w3oFNNxtMxRwv65xncCBRpK-vV3vvqmHGFPo"; "Sales March!A:K"); "select Col6 where Col6 = '"&amp;$A95&amp;"' and todate(Col11) = date '"&amp;$AR3&amp;"'"; 0)))  But this doesn't work. I tried A LOT of things for the past 3 days but I have no idea if I did that wrong (I'm a beginner) so I'm turning to you. What might be wrong with the date comparison in this section?  "select Col6 where Col6 = '"&amp;$A95&amp;"' and todate(Col11) = date '"&amp;$AR3&amp;"'"  You can see it  in Column AR Row 95 but I can't give edit rights. I'm using the beamer miniframes theme (with subsection=false) and the compress option. Currently the navigation shows the dots for each subsection side-by-side. I would prefer if there would be a horizontal space (e.g. with the width of a dot) between the dots of each subsection. Is this possible? Right now it looks like:  Section 1 oooooo  And I would prefer it to look like:  Section 1 ooo ooo  MWE: \documentclass[compress]{beamer}   \useoutertheme[subsection=false]{miniframes}   \begin{document}   \section{Sec 1}   \subsection{}   \begin{frame}[t]     hi   \end{frame} \subsection{}   \begin{frame}[t]     hi   \end{frame} \end{document}` What is the opposite of "parallel" in architecture? The two towers of the Verrazano Bridge are not parallel: they are slightly [???] to account for the curvature of the earth.    What is that word?