I need to sketch the image under the stereographical projection of sphere where the spherical cap $A &lt; X &lt; 1,$ with center lying on the equator, for fixed $ A. $ Separating the cases according to various ranges of $A.$  Here $X^2 + Y^2 + Z^2 = 1$ and $ (X,Y,Z) $ are points the unit sphere.
Let $\mathbb{S} = \{ (X,Y,Z) \in \mathbb{R}^3 :X^2 + Y^2+Z^2=1\}$ denote the Riemann sphere. The following problem is taken from Gamelin "Complex Analysis" chapter one:     What is the image of the spherical cap $ C := \{(X,Y,Z) \in \mathbb{S} : A \leq X \leq 1\}$, where $A$ is some fixed real number between $-1$ and $1$, under the stereographic projection $ \pi(X,Y,Z) = \frac{X+iY}{1-Z}$?   Here is my work so far:  ( The case $A=1$ is trivial, so assume $A &lt;1$)  We want to find $\pi(C) = \{z \in \mathbb{C} \mid \exists(X,Y,Z) \in C: \pi(X,Y,Z) = z \}= \{z \in \mathbb{C} \mid \pi^{-1}(z) \in C\}$, where the last equality holds by $\pi$ being a bijection. Write $z=x+iy$.  Here, $ \pi^{-1}(z) \in C\iff A\leq \frac{2x}{1+x^2+y^2} \leq 1 \iff (\star) :A \leq \frac{2x}{1+x^2+y^2}$, since the inequality $2x \leq 1+x^2+y^2$ holds for all $x,y \in \mathbb{R}$.  $A=0$: $( \star)$ reduces to $ 0 \leq x$, so in this case the image is $\{z \in \mathbb{C} \mid \Re(z) \geq 0 \}$.   $A &gt;0$: Since $A \neq 0$ in $(\star)$ we have,  $ \frac{2x}{1+x^2+y^2} \geq A \iff Ax^2-2x+Ay^2+A \leq 0 \iff (x-\frac{1}{A})^2 +y^2 \leq \frac{1}{A^2}-1$. Thus the image is the interior and boundary of the circle centered at $\frac{1}{A}$ with radius $\sqrt{\frac{1}{A^2}-1}$. Note $\frac{1}{A^2}-1 \geq 0$ as $A \in (0,1)$.  $A &lt;0$ : By symmetry and the above, here the image is $ \mathbb{C} / \{ $ interior and boundary of circle centered at  $  \frac{1}{A}$ with radius $\sqrt {\frac{1}{A^2}-1} \}$.  Is my working correct? Many Thanks!
I want to add a feature class within a Geodatabase to a map in ArcGIS Pro (1.4.1) using arcpy. I am converting some python code from ArcGIS Desktop to ArcGIS Pro.   This issue is very similar to:    and      However, instead of adding a feature class (or layer) to a map using the Python Window I am doing this from a  Script within a Toolbox. I have included a code snippet below.  lyrTest = r"C:\data\test.gdb\Layer1"  aprx = arcpy.mp.ArcGISProject("CURRENT") aprxMap = aprx.listMaps("MainMap")[0]  lyrFile = arcpy.mp.LayerFile(lyrTest) aprxMap.addLayer(lyrFile)   I get the following error message:  Traceback (most recent call last): File "C:\data\AddLayers.py", line 5, in &lt;module&gt;     aprxMap.addLayer(lyrTest) File "c:\users\user\appdata\local\programs\arcgis\pro\Resources\arcpy\arcpy\utils.py", line 191, in fn_ return fn(*args, **kw) File "c:\users\user\appdata\local\programs\arcgis\pro\Resources\arcpy\arcpy\_mp.py", line 1048, in addLayer return convertArcObjectToPythonObject(self._arc_object.addLayer(*gp_fixargs((add_layer_or_layerfile, add_position), True))) ValueError: C:\data\test.gdb\Layer1 Failed to execute (AddLayers). $\mathbb Z_{p^n}$ is not the direct product of any family of its proper subgroups I'm trying to solve this question of Hungerford's Algebra book:     $S_3$ is not the direct product of any family of its proper subgroups.   The same is true of $\mathbb Z_{p^n}$.   The first claim is easy: we note that every subgroup of $S_3$ is of order $2$ or $3$ by Lagrange, since its subgroups are of prime order, they are cyclic, then abelian, but $S_3$ is not abelian, contradiction because $S_3$ is not abelian while the direct product of abelian groups has to be abelian.  My problem is with the second claim, I need help. Correcting for multiple kruskal wallis tests counting number of specific string in each row I have a SNP table looks like this  A N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N C C N N N N N N N N N N C C N N N C C N N N N N N N N N N N N N N N N N   I want to count total number of N in each row? any idea how to do that Does a non-trivial solution exist for $f'(x)=f(f(x))$? Shapefile layer no longer editable (toggle edit option greyed out) in QGIS I'm using Ubuntu 12.04.3 with NVidia drivers (319) for my Quadro K2100M graphic card, and when I switch my main 3200x1800 laptop screen to mode 1920x1080 (using nvidia-settings), the display goes blurry...  I am using two screens:    My main screen is a LCD with max resolution 1920x1200. My laptop screen (original resolution 3200x1800) is sitting to the right of that.   The main problem is that, on Ubuntu, the font is far too small on my laptop screen. Because of that, I wanted to change the resolution of my laptop to 1920x1080.  Reading , I tried to  and here is the DPI configuration which I should have:   94x94 for my main LCD screen 1920x1200 235x236 for my 3200x1800 laptop screen   Is there a way to have dual DPI resolutions with an extended desktop? SElinux: allow httpd to connect to a specific port My system is running CentOS 6.4 with apache2.2.15. SElinux is enforcing and I'm trying to connect to a local instance of redis through my python/wsgi app. I get Error 13, Permission denied. I could fix this via the command:  setsebool -P httpd_can_network_connect  However, I don't exactly want httpd to be able to connect to all tcp ports. How can I specify which ports/networks httpd is allowed to connect to? If I could make a module to allow httpd to connect to port 6379 ( redis ) or any tcp on 127.0.0.1, that would be preferable. Not sure why my paranoia is so strong on this, but hey...  Anyone know? Comparisons: "so [adjective/adverb] as" or "as [adjective/adverb] as" Assume   $$\lim_{n\to \infty}{x_ny_n}=0$$  then  $$\lim_{n\to \infty}{x_n}=0$$ or $$\lim_{n\to \infty}{y_n}=0$$  I couldnt find two series who disprove that.. just hint please thanks Find $n$ such that $n/2$ is a square, $n/3$ is a cube, and $n/5$ a fifth power Awesome WM menu missing Firefox After sudo apt-get install awesome awesome-extra and logging into it (with no previous configuration that I know of), there's no menu entry for Firefox. All my other stuff is there - Mutt, Meld, jEdit, BOINC, etc., only Firefox is missing. I have no ~/.config/awesome/ directory, so there should be no override in place. Cannot add custom launcher to Ubuntu Dock ('Add to Favorites' option does not show up) I have a custom .desktop file: [Desktop Entry] Type=Application Terminal=false Icon[en_US]=/home/alex/bin/idea-IC-173.4127.27/bin/idea.png Name[en_US]=IntelliJ Exec=/home/alex/bin/idea-IC-173.4127.27/bin/idea.sh Name=IntelliJ Icon=/home/alex/bin/idea-IC-173.4127.27/bin/idea.png  Output of ls -lha .local/share/applications/intellij.desktop: -rwx--x--x 1 alex alex 248 Dez 30 10:54 .local/share/applications/intellij.desktop  Answers posted to a previous question () don't work for me. I can find this launcher in Activities and I can double click on that file and IntelliJ will start. Nevertheless, I cannot add this launcher as a favorite to the dock. The option does not show up. Script for display wrapper class Account list of records? Hi i am trying to write script for display list of records from wrapper class.   @remoteaction     public static wrappercls getaccounts(){         string account = 'select id,name,accountnumber from account';         listacc = database.query(account);         count = listacc.size();         system.debug('#####'+count);         wrappercls wra = new wrappercls(count,listacc);         return wra;     }      public class wrappercls{         public integer count11;         public list&lt;account&gt; account;          public wrappercls(integer count11,list&lt;account&gt; account){             this.count11 = count11;             this.account = account;         }     }    This class returns list of account records and record count parameter.Need to display these records in page.Is it possible if possible Please help me on this..  Script  &lt;script&gt;   function getallaccounts(){     exremote.getaccounts(         function(result,event){             if(event.status){              }             else {              }        },{escape:true}     ); } &lt;/script&gt;   Thanks in Advance... Looking for a Pre-1975, young adult, sci-fi, colored circle dimension crossing Looking for the title and or author of a long ago favorite; I have searched the web for it on and off for years.  I read a paperback novel about 40 years ago; it was probably a used book at that time. Probably pre-1975 publication.  In my memory it was written in the style of Andre Norton I have looked through her , but I am not finding it. It was a young adult sci-fi novel.  Plot; a young woman (gender was not strong, but I remember the main character being female) can see different colored circles floating around. When she looks at them just right, she can jump through to another universe.  In the other universe, things change.  She might become a lion, or maybe she is carrying a seed that becomes a net in the other universe.   Two bad guys are chasing her around the different dimensions, I recall one scene where they are all sliding down an icy slope and she jumps through a portal, to escape the bad guys.   There were a lot of complexities about, dimensions, what became what, different colors of portals and such. How do I extract text that lies between parentheses (round brackets)? At what point is a creature's toughness calculated? If I have  where I can sacrifice a creature and put X 1/1 token creatures onto the battlefield, where X is the sacrificed creature's toughness.  If I have a creature that is 4/4 and cast  to give it +3/+3 then sacrifice it? Do I get 4 tokens or 7?  Also what if my 4/4 creature was dealt 2 combat damage before I sacrificed it? Do I get 2 or 4 creatures?  Finally, what about counters? If my 4/4 had a +1/+1 do I get 4 or 5 tokens? Hot chicken soup takes hours to cool I made a big pot of a very hot chicken soup. I put it in my small extra fridge at 11 PM, and it was still warm 9 hours later, at 8 AM.  Do I need to throw it away? If $[G' : G'']\leq p^2$, then $G'$ is abelian. Problem : Let $G$ be a p-group and $G'$ denote the commutator subgroup of $G$. If $[G' : G'']\leq p^2$, then $G'$ is abelian.  It is easy to prove it for the case of $[G' : G'']=1$ since G is solvable. But it is too hard to prove the other cases. Please give me any advice or solutions. Thanks in advance. Unit ball in $C[0,1]$ not sequentially compact This question is taken from Saxe K -Beginning Functional Analysis.   Show that the closed unit ball in $C[0,1]$ is not compact by proving that it is not sequentially compact.   (It's assumed that we are using the uniform norm).  I've been working on this for ages but I could not come up with any sequence $\{f_n\}$ in the unit ball such that there exists $N\in \mathbb{N}$ such that for all $m,n\geq N$ we have that $d(f_n,f_m)&gt;c$. Should be a nice example of this, please help me!