Let $\bar m$ be the isometry sending a point $P$ to its opposite point $-P$, show that $\bar m$ commutes with any isometry of $S^2$  I know I can use the fact that isometries preserve distances, but do I need to individually apply this to each type of isometry?
"The antipodal map of $S^2$ is the isometry $\bar m$ sending each point $P$ to its opposite point $-P$" (from textbook)  Show that antipodal points remain antipodal under any isometry of $S^2$. In other words, for each isometry f of $S^2$, if $P$ and $P'$ are antipodal, then so are $f(P)$ and $f(P')$  I am unsure how I can define the $P$ and $P'$ so that they are opposites of each other in the antipodal map, and am I supposed to apply the traditional isometries (translation, reflection and rotation) to the points? I don't understand this problem.
Why doesn't \makeatletter work inside \newcommand? After hours of banging my brain against my skull, I narrowed down an issue I'm having to the fact that \makeatletter doesn't work inside \newcommand. Commands that I call after \makeatletter are registered as undefined, as if \makeatletter didn't do anything.  Why is this the case?  In the example below, I define two commands. Calling one of them from the document body causes the undefined error, while the other doesn't.  \documentclass[11pt]{book} \usepackage{kvsetkeys}  \newcommand{\commandThatFails}{     \makeatletter \comma@parse{}{} \makeatother }  \makeatletter     \newcommand{\commandThatWorks}{         \comma@parse{}{}     } \makeatother  \begin{document}  Hi there.  \commandThatFails \commandThatWorks  \end{document} Joining Excel table to shapefile in ArcMap? I have a shapefile of lower layer super output areas (LSOA) for England and Wales and I have a spreadsheet of the indices of deprivation for each of the LSOA's in England. I want to join the deprivation data to the shapefile so it can be mapped in ArcMap. I tried joining the table through the Join and Relates > Join option and put in the fields for the LSOA data for both shapefile and spreadsheet (the data that matches). the attribute table added the datafields to the attribute table from the excel spreadsheet but all of the data is blank!!   I have tried looking at the field "type" as ive read that it has been an issue before but it seems to be OK.  Is there anything anyone can suggest? Invert colors of an image in CSS or JavaScript More than once I've accidentally run a number of commands and polluted my bash history. How do I close my terminal without saving my bash history? I'm using Fedora. Normality of dependent variable = normality of residuals? Why does a star beyond a certain mass limit (Chandrasekhar limit) only become a black hole? Why does a star beyond a certain mass limit () only become a black hole?   A star is first made of hydrogen, it undergoes nuclear fussion reaction combining into helium and releasing a large amount of energy. This process continues till star is made up of iron core as iron has largest value of binding energy per nucleon, after this if mass of the star is above the value of Chandrasekhar limit it becomes a black star, what is the reason for this and why is certain mass limit required? What would happen if the earth would stop spinning? How much heavier would we be? I mean absolutely stop spinning. How much does the centrifugal force affect us?  If you give technical answers (please do), please explain them in laymen's terms, thanks.  Edit: I am asking what would be different if the earth were not spinning. Nevermind the part about it stopping. Enable USB debugging (under settings/applications/development) programmatically from within an app How do I remove all specific characters at the end of a string in PHP? Can a script modify an environment variable of its calling shell? I am setting my JAVA_HOME in my ~/.bashrc file.  Now, I need to create a script that will change the JAVA_HOME just for the current terminal, so that I can start an app that needs java 5.  I have created this script to do this task, but after finishing, I can see that JAVA_HOME is not updated  export JAVA_HOME=/usr/lib/jvm/java-5-oracle/ export PATH=$PATH:$JAVA_HOME       Here is the result  $ ./javaHome5.sh $ echo $JAVA_HOME /usr/lib/jvm/java-6-oracle/   I think that the reason this is not being applied, is that a script is executing in it's own terminal, so when the script ends, the current terminal will not be affected.  Currently, the only way I've found around this is:   Edit my ~/.bashrc and change the JAVA_HOME var Run source ~/.bashrc to apply the changes in current terminal. Which again cannot be applied in a script, as the source command needs to be run in the current window.   Needless to say, this change applies to all new terminal windows, so I practically need to do this twice: One before starting my app, and one more time right after this, just to restore the environment vars to their default. That's not really convenient.  Do you have any ideas how can I change this var using a script? Converting to upper and lower case in Java I was just looking at the statistics for migrated questions in the  mod tools, and noticed this oddity:    The actual statistics, as visible to any 10k+ SFF user , are: 6 migrated to Worldbuilding, 2 to Literature, 2 to SFF meta, 1 to Movies &amp; TV, 1 to Writers, and 1 to Aviation.  Why is one of the meta migrations showing up with a blank icon rather than with the other?  Both migrated questions were closed as duplicates on meta. Both were migrated by 5 community members without mod involvement. One of the stubs on main has since been deleted, but so have the stubs for some of the other migrated questions. I can't think of anything to explain this discrepancy. Is it possible to allow usage for an app or program for a specific times on a Mac? I was trying to achieve this using a second account (which is the admin account) and Parental Controls, but it seemed this isn't possible to do so.  I know there are some programs which you can download that help you with this, (Cold Turkey, for example). But, it would be much easier if I could do this using only tools already available on macOS. Print total free memory from `free -m` free -m currently puts out something like this.     I would like to know using grep/awk how we can get the total free amount. i.e. 9083 Supersymmetry in Quantum Field Theory Let $a,b,c,x,y,z&gt;0$ and $x+ y+ z= 1$. Show that $$ax+ by+ cz\geq  a^{x}.b^{y}.c^{z}$$ I have found the generalization of this inequality: For 2 sets of numbers $\left \{ a_{i} \right \}, \left \{ b_{i} \right \}&gt; 0$ and $\sum_{1}^{n}b_{i}= 1$, we have: $$\sum_{1}^{n}\left ( a_{i}b_{i} \right )\geq  \prod_{1}^{n}\left ( a_{i}^{b_{i}} \right )$$ First solution Simply compare the logs: as the natural logarithm is a concave function, one has $$\ln\bigl(a^x b^y c^z)=x\ln a+y\ln b+z\ln c\le \ln(ax+by+cz)$$ for all non-negative $x,y,z$ such that $x+y+z=1$. How about another solution? I really hope to see that. Thanks! When scripting in bash or any other shell in *NIX, while running a command that will take more than a few seconds, a progress bar is needed.  For example, copying a big file, opening a big tar file.  What ways do you recommend to add progress bars to shell scripts? What determines speciation at a molecular level?  At what point does a scientist determine two lineages are different enough to be considered separate species?  Does it have a margin of error? Transitive subgroup of symmetric group $S_n$ containing an $(n-1)$-cycle and a transposition clerics have the ability  (PHB, p. 62):     When a creature within 5 feet of you that you can see hits you with an attack, you can use your reaction to cause the creature to make a Dexterity saving throw. The creature takes 2d8 lightning or thunder damage (your choice) on a failed saving throw, and half as much damage on a successful one.   It doesn't specify what the DC of the saving throw is.   I assume it uses the spell save DC, even though it technically isn't a spell?     Spell save DC = 8 + your proficiency bonus + your Wisdom modifier