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What are the left and right ideals of matrix ring? How about the two sided ideals? Closing Streams in Java Desktop folders won't open with clean install of 18.04 How do you find out your very first chat in a particular chatroom? I want to know my very earliest/first chat post in a chatroom. Is there perhaps a way to search/view all my posts and sort the results? Tools for automating document compilation A lot of people write makefiles that say something like  paper.pdf: paper.tex         pdflatex paper         bibtex paper         pdflatex paper         pdflatex paper   To handle re-running TeX to get new/changed references and so forth. Is there a better way to do this? If $ a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}<1$ If $\displaystyle a,b,c\in \left(0,\frac{\pi}{2}\right)\;,$ Then prove that $\displaystyle \frac{\sin (a+b+c)}{\sin a+\sin b+\sin c}&lt;1$   $\bf{My\; Try::}$ Using $$\sin(a+\underbrace{b+c}) = \sin a\cdot \cos (b+c)+\cos a\cdot \sin (b+c)$$  $$ = \sin a\cdot (\cos b\cos c-\sin b\sin c)+\cos a(\sin b\cos c+\cos b\sin c)$$  $$ = \sin a\cos b\cos c-\sin a\sin b\sin c+\cos a \sin b\cos c+\cos a\cos b\sin c$$  Now how can i solve it after  that , Help required, Thanks I'm currently unable to access anything via the Stack Exchange app under iOS; is this a known issue? C# and USB HID Devices Defining coordinate reference system with rotation in GeoServer? I am using GeoServer and have a layer in EPSG:900913 ("Google Mercator").  I need to "rotate" the map around certain point (say, 1500000, 7000000) by certain degree (say, 30 degrees clockwise). How could I define such a coordinate system based on EPSG:900913?  GeoServer's  does not work for my purposes as I need to tile the map later on.  As far as I understand this, my only option is to define an own coordinate system. For GeoServer I'd need to . The configuration seems to be straightforward, but I have a difficulty defining my rotated CRS in .  I am wondering how to apply a rotation around certain point onto a CRS like Google Mercator:  PROJCS["WGS84 / Google Mercator",    GEOGCS["WGS 84",      DATUM["World Geodetic System 1984",        SPHEROID["WGS 84", 6378137.0, 298.257223563, AUTHORITY["EPSG","7030"]],        AUTHORITY["EPSG","6326"]],      PRIMEM["Greenwich", 0.0, AUTHORITY["EPSG","8901"]],      UNIT["degree", 0.017453292519943295],      AXIS["Longitude", EAST],      AXIS["Latitude", NORTH],      AUTHORITY["EPSG","4326"]],    PROJECTION["Mercator_1SP"],    PARAMETER["semi_minor", 6378137.0],    PARAMETER["latitude_of_origin", 0.0],    PARAMETER["central_meridian", 0.0],    PARAMETER["scale_factor", 1.0],    PARAMETER["false_easting", 0.0],    PARAMETER["false_northing", 0.0],    UNIT["m", 1.0],    AXIS["x", EAST],    AXIS["y", NORTH],    AUTHORITY["EPSG","900913"]]   My questions, specifically:   How to write a WKT which transform an existing CRS? My guess would be that I need a new PROJCS wrapping an existing one and adding a PROJECTION clause. How would I found out the projection id (like Mercator_1SP above) and the required parameters (the PARAMETER clauses)? Can I "reference" EPSG:900913 in CRS WKT instead of copy-pasting the whole PROJCS clause? Parallel evaluation produces wrong result Is two-thirds plural? Is 2/3 always, sometimes or never plural?  E.g.     1a) 2/3 of the pizza were eaten.     1b) 2/3 of the pizza was eaten.      2a) 2/3 of the visitors were men.     2b) 2/3 of the visitors was men.   I feel that example 1 could go either way but example 2b sounds very wrong.  Is there a rule for this? Say if I have a dropdown in a form and I have another nested class inside of this class . Now what's the best way to access this dropdown from the nested class? How does Python evaluate the expression 1+++2?  How many ever + I put in between, it is printing 3 as the answer. Please can anyone explain this behavior  And for 1--2 it is printing 3 and for 1---2 it is printing -1 Very basic question: how do I write a short literal in C++?  I know the following:   2 is an int 2U is an unsigned int 2L is a long 2LL is a long long 2.0f is a float 2.0 is a double '\2' is a char.   But how would I write a short literal?  I tried 2S but that gives a compiler warning. Prove that   $\dfrac {\left( 5^{125}-1\right)}{\left( 5^{25}-1\right)}$ is composite number using number theory. Do not use calculator or Wolfram alpha or anything like that. Proving $\sum_{k=0}^n\binom{2n}{2k} = 2^{2n-1}$ I'm undergraduate student of mathematics.  I need to prove: $$\sum_{k=0}^{n} \binom{2n}{2k}= 2^{2n-1}$$ Can you please help me Why can we flag/vote a question on Meta as duplicate of another question which has no accepted/upvoted answers? When we flag/vote-to-close a question as a duplicate on Stack Overflow, and that potential duplicate doesn't have any upvoted or accepted answers, we are alerted that we can't use that question as a duplicate and can't continue. This same check does however not seem to apply to Meta questions. Why not? I am trying to do a new environment or command (i tried both) where i can put in the name, description and code of a Method in c++ and it gives me a new section with the MethodName, the code listed and the description below. But i am not able to use the listings environment in a new command or new environment.. Consider my MWE   \documentclass[a4paper,11pt]{article} \usepackage{listings}  \newcommand{\method}[3] { \section{#1}\label{#1} \begin{lstlisting} #2 \end{lstlisting} #3 }   \begin{document}  \method{MethodName}{Method Void MethodName()}{Description of the Method blablabla}  \end{document}   How can I fix this?   I would prefer to do it using a macro since I don't need an ending. Function in Table How to solve "sudo: /etc/sudoers.d is world writable"