test algorithm to decide if point is inside area here's a geometric problem that I've been unsuccessful to solve:   we have four points A, B, C, D defining an area and two points E, F E is within the boundaries of the polygon ABCD F is outside the borders   We know the (x, y) coordinates of each point.  (see the figure below)    Determine for any point G (x, y) if G is inside or outside ABCD  Any ideas out there?  Thanks Adrien
How can I determine whether a 2D Point is within a Polygon? I'm trying to create a fast 2D point inside polygon algorithm, for use in hit-testing (e.g. Polygon.contains(p:Point)). Suggestions for effective techniques would be appreciated.
Are collective nouns always plural, or are certain ones singular? What is the ideal data type to use when storing latitude / longitude in a MySQL database? Bearing in mind that I'll be performing calculations on lat / long pairs, what datatype is best suited for use with a MySQL database? sliding a tangent line along a curve $f_x$ is Borel measurable and $f^y$ is continuous then $f$ is Borel measurable How do you display the archives of a Custom Post Type by Year &amp; Month? UK Visitor Visa: Required Time or Interval Between Visits? I am a US citizen and have recently visted the UK  with simply a passport which allowed me to vist for 6months I have heard rumors that I am obligated to stay in the US for a Equal amount of time that I have visited the UK before I am able to return, but I am struggling tour no any proof of this obligation .. Am I missing somthing or is it simply not true What is the new turbo boost in the MacBook line? The new MacBook line has a new turbo boost added to the processor:     2.3GHz Quad-core Intel Core i7, Turbo Boost up to 3.3GHz   What exactly does this mean? From what I've heard, the processor can runs above the clocked speed for a short period of time.  How does it work? How do I access a specific element in a vector of vectors clojure Field extensions with(out) a common extension Let $K$ be a field having two field extensions $L\supseteq K$ and $M\supseteq K$. Does there exist a field $N$ along with embeddings $L\to N$ and $M\to N$, such that the diagram $$     \require{AMScd} \begin{CD}     K @&gt;&gt;&gt; L\\     @V  V V @VV  V\\     M @&gt;&gt;&gt; N     \end{CD} $$ is commutative? To put it less formally, do $L$ and $M$ have a common field extension $N$ (with $K$ lying in the intersection)?  If yes, consider this side question (but do leave an answer even if you can only answer the main question!): Does the above property of fields generalise to the following, stronger property? Let $p$ be either $0$ or a prime number. Does there exist a sequence of fields $$ \mathbb F_p = L_0\subseteq L_1\subseteq L_2\subseteq L_3\subseteq L_4\subseteq\cdots$$ (where I use the convention $\mathbb F_0 = \mathbb Q$) such that any field $K$ of characteristic $p$ has an extension field among the $L_\alpha$? This sequence should be understood as enumerated by ordinal numbers. In other words, what I need is a function that assigns to each ordinal number $\alpha$ a field $L_\alpha$ containing all $L_\beta$ with $\beta &lt; \alpha$. (Intuitively, I suspect that this might depend on the Axiom of Choice.)  If the second property holds, it shows that you can essentially only extend a field in one "direction." It also shows that the class (it is obviously not a set) $\mathbb M_p = \bigcup_\alpha L_\alpha$ is a field (class). We can define all the usual field operations (addition, multiplication, division) here since all pairs of elements lie in $L_\alpha$ for a sufficiently large $\alpha$. This "monster field" of characteristic $p$ then contains all other set fields of that characteristic. If $f:U\to V$ is holomorphic and $f'(z)\neq 0$ for all $z\in U$, then$f$ is locally bijective. I am trying to solve the following problem:     Let $f:U\to V$ be a holomorphic function such that $f'(z)\neq 0$ for all $z\in U$. Show that for all $z_0\in U$, there exists a disc $D_\varepsilon(z_0)\subseteq U$ such that $f:D_\varepsilon(z_0)\to f(D_\varepsilon(z_0))$ is bijective.   Atempt:  I am trying to use Rouche's Theorem, but I am stuck on a particular step. Let $z_0\in U$. Since $f'(z_0)\neq 0$, there is a disc $D_r(z_0)\subseteq U$ such that $$f(z)=f(z_0)+(z-z_0)h(z),\quad\forall z\in D_r(z_0)$$ where $h$ is holomorphic on $D_r(z_0)$ and $h(z)\neq 0$ for all $z\in D_r(z_0)$.   Let $0&lt;\varepsilon&lt;r$ to be defined later and fix $w\in D_\varepsilon(z_0)$. To show that $f$ is injective in $D_\varepsilon(z_0)$ is to show that the function $f(z)-f(w)$ has exactly one zero in $D_\varepsilon(z_0)$. But $$F(z):=(z-z_0)h(z)$$ has exactly one zero in $D_\varepsilon(z_0)$, so we might want to apply Rouche's Theorem with $F$ and $$G(z):=f(z)-f(w)-(z-z_0)h(z)=f(z_0)-f(w)=-(w-z_0)h(w)$$ to conclude that $F$ and $F+G$ have the same number of zeros and hence $f$ is injective. But that means that we need to find $0&lt;\varepsilon&lt;r$ such that $|G(z)|&lt;|F(z)|$ on $\partial D_\varepsilon(z_0)$. That is, $$|w-z_0||h(w)|&lt;|z-z_0||h(z)|$$ for all $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$.  Can we find such $\varepsilon&gt;0$?  It seems intuitive since if we expand $h$ in a power series at $z_0$, $h(z)=\sum_{n=0}^\infty a_n(z-z_0)^n$, then $$|w-z_0||h(w)| = |w-z_0|\sum_{n=0}^\infty a_n|w-z_0|^n &lt; |z-z_0|\sum_{n=0}^\infty a_n|z-z_0|^n.$$ for $w\in D_\varepsilon(z_0)$ and $z\in\partial D_\varepsilon(z_0)$. But this is of course not a proof. Can you help me decide, what type of transport can I use to travel from one US town to another (it will be a big journey), without flying?  Are buses available, and is there any more suitable method of traveling? I stumbled upon an example, which needed to be compiled with the additional option --shell-escape. Using this option it worked just fine, but I'm curious:     Why or when is this option needed and what does it actually do? I installed the latest pgAdmin 4 (v2.1) but I found out that (), for the moment, the only way to establish a connection through an SSH tunneling is using another outside process, an  says:  I setup a tunnel as follows: ssh -f dpage(at)hostname(dot)enterprisedb(dot)com -L 5999: hostname.enterprisedb.com:5432 -N  (...) I was running on Mac, but the same procedure should work on Linux. On Windows, there's a tunnelling app in Putty iirc, but I don't know how to use it offhand.  I have found a simple . The connection is established but it seems, when I try to connect through pgAdmin it fails or is refused (using 127.0.0.1:3306 or 127.0.0.1:5432 as the tunnel's destination). Why? reversing plot axis for Plot, LogPlot, LogLogPlot Adding CSS Reference on Site Page So I have created a Site Page that has the "New Experience" look and I would like to change its style a little. Nothing major, just the height, width, color, etc. However, I can not seem to figure it out.   For example, I tried to change the height of the .pageHeader to a smaller value.   .pageHeader{ height: 150px !important } //for example of CSS in file   Here is a reference image of the page i'm trying to change    I tried adding   &lt;link href="../SiteAssets/file.css" rel="stylesheet" type="text/css"&gt;   to the .aspx page using SharePoint Designer and it didn't work.    I also tried adding   &lt;SharePoint:CssRegistration ID="CssRegistration" Name="SiteAssets/file.css" runat="server" /&gt;   to the .aspx page and it didn't work    I tried to use the "Embed" feature to add the CSS; however, it complains and says,     This embed code doesn't seem to work. We only support iframe based embeds.   Question:   How can I add custom CSS to this type page?  I can't seem to figure it out and I'm not sure exactly why the page will not load the file.  Is this even possible? I have a simple question but I am not sure on how to proceed. I would like to put 10 figures on the same page: a grid of 5 rows and 2 columns.  What should I type in order to have the figure inserted like this and with equal alignement/size ? Eventually with custom captions... ;) Seeking copy of 19th century South African birth certificate? I'm trying to find the birth certificate of my wife's great grandmother who was probably born in Durban, South Africa in 1884. I'm currently unable to find any online resources for this.  If such a thing does not exist online, is there any other method of retrieving a copy or at least just the details? $D_8$ as a derived subgroup Every undergraduate student knows that there are (exactly) two non-abelian groups of order 8: Dihedral ($D_8$) and Quaternion ($Q_8$). The group $Q_8$ has many interesting properties; simple of them are: it has unique element of order $2$, all subgroups are normal, etc. A property, which is strange and not obvious, the group $Q_8$ has, is that there is no group $G$ such that $G/Z(G)$ is isomorphic to $Q_8$. On the other hand, $D_8$ has no such property; there are (infinitely many) groups $G$ such that $G/Z(G)$ is isomorphic to $D_8$; example - $D_{16}$.   The question, I would like to ask, is related to a property of $D_8$, which is not shared by $Q_8$. There are (infinitely many) groups whose derived subgroup is $Q_8$; example- $SL_2(3)$. For groups $G$ of order &lt;100, one can check on GAP that $D_8$ is not a derived subgroup of any group. The natural question is then:  Question 1: Does there exist a finite/infinite group whose derived subgroup is equal to $D_8$?  Question 2: Does there exist a finite/infinite group whose derived subgroup is equal to (general) dihedral group $D_{2n}$ of order $2n$ ($n&gt;2$)? Getting secondary citations right I have some questions regarding secondary citations (in an computer-science research paper that should be published in a journal). I know that they should be avoided when possible, but it is not always that easy. More specifically, my situation is this:  Article A says:     The average value of ... in ... is ... [B]. [C]’s study says that in ... the value is ...   This statement is exactly what i need for my paper. If I read [B] and [C], I see that [A] cited them correctly and if I would have found these two other articles myself, I would have written the same.  Now how to cite this in a correct way?      [B] and [C] say (cited in [A]) ...   and do a “bad” secondary quote;    [B] and [C] say ...   and neglect the investigation done by [A] and peform citation plagiarism;    [A] says ...   and neglect that the data was the achievement of [B] and [C].    [A] says based on the findings of [B] and [C] ...   and do a secondary quote again. Install on usb external hard drive I am new to ubuntu and i would like some help. I have a laptop with windows 7 installed but i would love to have ubuntu too because it's a great OS. So i would like to install ubuntu on an external usb hard drive (320 - 500 GB) and i have some questions about that :   Do i need a usb external hard drive which will be compatible with ubuntu - linux ? or i can do it with any hard drive  If i make a partition on it for ubuntu staff will i be able to use it and save other staff in the usb hard drive (movies,songs,documents etc) ? If i have ubuntu installed on the usb hard drive will i be able to boot ubuntu from any pc ? Are my changes going to be saved normally or my settings will be lost after shut down ?   Waiting for your answers, if something is not explain well tell me to try explain it better . Thank you!