How to do Water color painting in iOS using CoreGraphics?
Render water-paint in iOS
Let $f(z)$ be entire function. Show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function using Maximum Modulus theorem Let $f(z)$ be entire function. Consider the functions $e^{if(z)}$ and $e^{−if(z)}$ and applying the Maximum Modulus Theorem, show that if $f(z)$ is real when $|z| = 1$, then $f(z)$ must be a constant function.  (We take $f(z)=u(z)+iv(z)$)  I am confused as so far I have $|g(z)|=|e^{if(z)}|=|e^{-v(z)}|$ and then since $f(z)$ is real, $f(z)=u(z)$ and $v(z)=0$ so I assumed it would follow that $|g(z)|=|e^{v(z)}|=1$.  Similarly, $|g(z)|=|e^{-if(z)}|=|e^{v(z)}|=1$.  Using Liouville I assumed one could say that both $g(z)$ and $h(z)$ are bounded entire functions, they are constant and so it follows that $v(z)$ is constant, meaning that both its partial derivatives are equal to 0 and, due to Cauchy Riemann, both of the partial derivatives of $u(z)$ are equal to zero.  It would then follow that $f(z)$ is constant.  I don't know how to go about the question using the Maximum Modulus Theorem, also I feel I am overlooking the importance of $|z|=1$ perhaps?  Any help would be much appreciated!! I want to check if points layer that I have intersect with polygons layer that I have, as Boolean column in the points dataframe. I have two GeoPandas dataframes, the first one is many points and looks like this: &gt;&gt;&gt;ID  geometry 0  12  POINT (5.0279 7.4547) 1  45  POINT (6.6539 12.139) ...  and the second dataframe is one layer of many different polygons  that looks something like this: &gt;&gt;&gt;name     code   geometry 0  Desert   12     POLYGON ((5.52013 13.8902, 5.5265 13.892,...) 1  Water    24     POLYGON ((5.53756 13.88472, 5.5291 13.8791,...) ...  I'm trying to check if there is any intersection between the points layer and the regions layer. For that I have determined the crs and then used intersects : regions=regions.to_crs({'init': 'epsg:4326'}) points=points.set_crs({'init': 'epsg:4326'}) inter=points.geometry.intersects(regions.geometry)  A The script runs with the following warnings:  FutureWarning: '+init=:' syntax is deprecated. ':' is the preferred initialization method. When making the change, be mindful of axis order changes: https://pyproj4.github.io/pyproj/stable/gotchas.html#axis-order-changes-in-proj-6 return _prepare_from_string(&quot; &quot;.join(pjargs)) /opt/conda/lib/python3.8/site-packages/geopandas/base.py:39: UserWarning: The indices of the two GeoSeries are different. warn(&quot;The indices of the two GeoSeries are different.&quot;)  Then when I check the results the only value is False, like all th epoints do not intersect: inter.unique().tolist() &gt;&gt;&gt;[False]  *I saw on QGIS that the there are points that intersect and there are points that do not so there is no way this result is true *I have checked the dtypes - each one of my geodataframes has one column that is geometry and called geometry. My end goal: to add new column in the points geodataframe that will tell if it intersects the regions or not. Trouble downloading packages list due to a "Hash sum mismatch" error When I check for updates, I get a "Failed To Download Repository Information" error.   This is what comes up under details:   W: Failed to fetch gzip:/var/lib/apt/lists/partial/us.archive.ubuntu.com_ubuntu_dists_natty_main_source_Sources  Hash Sum mismatch,  E: Some index files failed to download. They have been ignored, or old ones used instead. Don't you think people need to know what they did wrong exactly so they can fix it and get unbanned from asking questions or posting answers? nodes near coords with logarithmic axis: Wrong values I created a horizontal bar plot to compare benchmark times. Since one of the tested components is really slow, I need a logarithmic y axis for the other values to be distinguishable. Combined with "nodes near coords" this results in the calculated values showing, instead of the original ones. Is there any way to fix this?  What it looks like:   What the values(!) should look like:   Clearly, I'm overlooking something terribly obvious. I'm just a beginner at PGFPlots, so please don't kill me. :)  MWE:  \documentclass[fontsize=12pt]{scrartcl}  \usepackage{pgfplots} \pgfplotsset{compat=1.8}  \begin{document}   \begin{tikzpicture}   \begin{axis}[      width=12.5cm,     xbar=0.1cm,     xmin=0,     y=0.6cm,     bar width=0.4cm,     xlabel={Time (ms)},     xmode=log,     major x tick style = transparent,     major y tick style = transparent,     symbolic y coords={       {SpreadsheetGear},       {Aspose.Cells},       {NPOI},       {C1.Excel},       {Spire.XLS},       {GemBox.Spreadsheet},       {SmartXLS},     },     ytick=data,     y dir = reverse,      nodes near coords,     nodes near coords align={horizontal},   ]    \addplot coordinates {       (3078.6,{Aspose.Cells})       (10059.2,{C1.Excel})       (28794.8,{GemBox.Spreadsheet})       (7437079.8,{SmartXLS})       (19975.2,{Spire.XLS})       (573.2,{SpreadsheetGear})       (10025.2,{NPOI})};   \end{axis}   \end{tikzpicture} \end{document} Novel about a dragon lord who has to wear gloves at all times How to disable the grouping of windows on the launcher? For example, I ran Firefox and opened . Then opened some link in a new window. As a result, I have two windows of Firefox and one icon of Firefox on the launcher. I want to have an icon for each opened window.  How do I do? Markov's paper on insolubility of the homeorphy problem I know this may be a tough or confusing question, but I have numerous existing materials I created and reuse for a project, which were all prepared for Blender Internal.  Now, I want to start the transition to Cycles, but I've hit a roadblock. How can I transition my materials without completely starting over? Or should I just forget it and start with fresh materials? Compiling with g++ using multiple cores UK Visa Refusal: Provenance of funds/parking I tried to prove this question but without a success: Let $K_1$ and $K_2$ be definable sets, prove that $K_1\cup K_2$ is definable.  What I tried to do is to assume: $K_1=Ass(X)=\{ v|v \vDash X \}$ and $K_2=Ass(Y)=\{ v|v \vDash Y \}$ and to say that $K_1\cup K_2=Ass(X \cap Y)$. but I couldn't prove that: $K_1\cup K_2 \supseteq Ass(X \cap Y)$ .   I would like to get help with this proof Everybody has to obey their own parents I am new here and I don't really know how to post, so please forgive my mistakes. My question is this: Is a sentence like "everybody has to obey their own parents" nowadays completely acceptable under a grammatical point of view or it is still preferable, in written form, to use "everybody has to obey his/her own parents"? Thank you very much and sorry if this has been asked before. How to make a figure span on two columns in a scientific paper? How to use Font as a variable? Why can Naruto still use Rasenshuriken? When Naruto first used Rasenshuriken against Kakuzu the effect was massive. He destroyed   two of Kakuzu's hearts, but there were a huge after effect to him. If I'm not mistaken, Tsunade advised Kakashi to ask Naruto never to use the technique again due to the huge risk. So why is Naruto still able to use the technique and have no side effects after that battle? I may have missed out on something. I'm at roughly 67% completion of the Alola Dex, with ~200 caught, ~290 seen. After scanning QR codes, I've caught a Swinub and Cyndaquil. However, neither were registered in the Pokédex, since neither are Pokémon in the Alola Dex. This brought up the following question:  When will I unlock the National Dex?   Do I need to have seen all the Pokémon in the Alola Dex? Do I need to complete some specific quest, like catching/defeating the Ultra Beasts? What Does it Really Mean to Have Different Kinds of Infinities? As a result, I am able to be understanding and accommodating to the people that I work with, including clients, which I believe is an important asset for a professional to have.   I wanted to make sure it is absolutely known that I mean to acknowledge clients as a class of people I can work with. Is this the right way to include it with commas? How do I add myself into the sudoers group? Possible Duplicate:          I've added myself back into the admin's group I think, by following   But I still can't do commands like:  sudo aptitude update   I get the message  oshirowanen is not in the sudoers file.  This incident will be reported.   How do I add myself into the sudoers group now that I am in the admin group?