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Which is easier, pushing or pulling? It is generally assumed, from a person's perspective, that pushing a cart is more easier than pulling one. But why? Is there any difference in terms of force required to achieve the same amount of displacement?  Or is it merely human perception?  Why is it that almost all automobiles transfer torque to the back axle. But then, why do trains have engines in the front? I am watching Susskinds's lecture , in which he introduced square angular momentum operator $L^2$, which can be written as $$ L^2=L_z^2+L_-L_++L_z $$ where $$ L_\pm=L_x\pm iL_y $$  Besides, he argued that the spectrum of $L$ can only be $m,m-1,\cdots,-m+1,-m$ in which $m$ is an integer or half-integer. If we denote $|m\rangle$ the eigenvector corresponds to eigenvalue $m$, we have $$ L_\pm|l\rangle=|l\pm 1\rangle\;\text{or}\;0 $$  In the lecture he showed that $L^2|m\rangle=m(m+1)|m\rangle$ for $m$ the highest eigenvalue, by $$ \begin{align*} L^2|m\rangle&amp;=L_z^2|m\rangle+L_-L_+|m\rangle+L_z|m\rangle\\ &amp;=m^2|m\rangle+0+m|m\rangle\\ &amp;=m(m+1)|m\rangle \end{align*} $$  Then he left an exercise to show that for any $-m\leqslant l&lt; m$, we also have $|l\rangle$ is an eigenvector of $L^2$ with eigenvalue $m(m+1)$.  In the lecture, he hinted by showing that it suffices to show that $L^2$ commutes with $L_i$ for $i=x,y,z$, which gives us $$ L^2|m-1\rangle=L^2L_-|m\rangle=L_-L^2|m\rangle=L_-m(m+1)|m\rangle=m(m+1)L_-|m\rangle=m(m-1)|m-1\rangle $$  I believe it is true since I find the   online.   However, my question is that, why the following proof is wrong.  I tried to mimic the proof for $|m\rangle$ to have that  $$ \begin{align*} L^2|k\rangle&amp;=L_z^2|k\rangle+L_-L_+|k\rangle+L_z|k\rangle\\ &amp;=k^2|k\rangle+L_-L_+|k+1\rangle+k|k\rangle\\ &amp;=k^2|k\rangle+|k\rangle+k|k\rangle\\ &amp;=(k^2+k+1)|k\rangle \end{align*} $$  which shows that $|k\rangle$ does not have eigenvalue $m(m+1)$ but $k^2+k+1$ for $k&lt;m$. I'm a bit overwhelmed by the available options in regards to the three major photo editing apps from Adobe.  Does anyone know of an article or site that lists the features of Lightroom, PS Elements, and PS, side by side?  I know that Lightroom revolves around edits and workflow, and that PS is the big kahuna of image editing.  Where does Elements fit in to this understanding? What if I run a virus/trojan Windows EXE on Ubuntu with Wine? I'm not quite sure how this happened, but somehow a completely empty hierarchy of directories has ended up in my repository:  com/ com/companyname/ com/companyname/blah/ com/sun/ com/sun/java/ com/sun/java/jax_rpc_ri/   I think what happened was that these directories did have files in them, but then a developer realized he/she shouldn't have checked them in in the first place since these are by-products of the build process, so he/she removed the files but somehow the empty directories are left in the repository as ancient relics.  How can I remove this from CVS? The only results I seem to be able to find on google say that there shouldn't be a need to remove empty directories as CVS won't keep them around in the first place, and that the -P (prune) options to cvs update should remove them from the working directory  - which is zero help if you actually have empty directories in your repository.  A cvs remove and cvs commit doesn't seem to take care of this situation:  $ cvs remove -Rf com cvs remove: Removing com cvs remove: Removing com/companyname cvs remove: Removing com/companyname/blah cvs remove: Removing com/sun cvs remove: Removing com/sun/java cvs remove: Removing com/sun/java/jax_rpc_ri $ cvs commit com cvs commit: Examining com cvs commit: Examining com/companyname cvs commit: Examining com/companyname/blah cvs commit: Examining com/sun cvs commit: Examining com/sun/java cvs commit: Examining com/sun/java/jax_rpc_ri $ ls -l com total 24 drwxrwxr-x  2 matt matt 4096 Oct 15 14:38 CVS drwxrwxr-x  9 matt matt 4096 Oct 15 14:38 companyname drwxrwxr-x  4 matt matt 4096 Oct 15 14:38 sun   It's still there!  Does SVN have this weird behavior too? Network button missing gone from Nautilus? I'm using Ubuntu 16.04.3 LTS.  I remember having a link in Nautilus that I used to browse network shares with in the past. I don't remember exactly but I think it was "Windows network" or "Browse network".  Now I can't find this on my current installation, there's only "Connect to server", but this is not what I want. Is there a way to put that link there? Solving "mv: Argument list too long"? I have a folder with more than a million files that needs sorting, but I can't really do anything because mv outputs this message all the time  -bash: /bin/mv: Argument list too long   I'm using this command to move extension-less files:  mv -- !(*.jpg|*.png|*.bmp) targetdir/ input() error - NameError: name '...' is not defined For a simple circuit with a battery supplying a voltage $V$ to a capacitor, let us assume that the charge on the capacitor is $Q$. Now, the work done by the battery or the energy supplied is given by the relation:  $$W=QV$$  But the energy stored in the capacitor is given by:  $$U = \tfrac12 QV$$  The value of $Q$ as well as that of $V$ should be the same in both the equations.  Now my question is, where is the other half of the energy that the battery supplied? \foreach not behaving in axis environment Am hesitant to ask yet one more embarrassing question, but I can't seem to see the problem with the \foreach in the MWE. It should produce several horizontal lines but only produces the one with the yValue as specified in the \newcommand{\yValue}{0.3}:  Code: \documentclass{article} \usepackage{pgfplots}  \newcommand{\yValue}{0.3}  \begin{document} \begin{tikzpicture} \begin{axis} [  ymin=0,  ymax=1,  xmin=0,  xmax=1, ]  \foreach \yValue in {0.00,0.1,...,1.00} {     \draw [red] (axis cs:0,\yValue) -- (axis cs:1,\yValue); }  \end{axis} \end{tikzpicture} \end{document} Quotient of a local regular ring How can I prove this:     Let $A$ be a local regular ring with maximal ideal $\mathfrak m$ and $x \in \mathfrak m-\mathfrak m^2$. Then $A/(x)$ is a regular ring. Prove also that if $x\in\mathfrak m^2$, $x\ne 0$, the result does not hold anymore.   I don't know how to begin! Thanks to everyone who helps!  But if $d=\infty$? Have a web-part display only sub-folders? I am new to SharePoint 2013. I was curious how to put a WebPart for a Document Library on a page with SharePoint 2013, not 2010.   This WebPart would then show the content of a specific library folder and not start at the root of the folder. So say I have a library with multiple folders like "Marketing", "Finance" etc,   I want to be able to create a web part that would only show the contents within the “Marketing subfolder and all of it contents. Is this possible? What do \makeatletter and \makeatother do? Many LaTeX “hacks” begin with \makeatletter and end with \makeatother. What do these commands do? Output of getent group Virtualbox: OS cannot use the new extended disk space (.vdi) Is there Ubuntu's analogue of snapshot.debian.org? Why am I getting a NoClassDefFoundError in Java? What is a deterministic method for evaluating a sensible buffer pool size? I'm trying to come up with a sane way to understand if the max server memory (mb) setting is appropriate (either should be lower, or higher, or stay the way it is).  I am aware that max server memory (mb) should always be low enough to leave room for the operating system itself, etc.    The environment I'm looking at has several hundred servers; I need a reliable formula I can use to determine if the present size of the buffer pool is appropriate since RAM is costed per GB allocated to each server.  The entire environment is virtualized, and "physical" RAM allocated to a VM can easily be changed up or down.  I have a particular SQL Server Instance I'm looking at now with a PLE of 1,100,052 seconds, which equates to 12.7 days (the amount of time the server has been up).  The server has a max server memory setting of 2560MB (2.5GB), of which only 1380MB (1.3GB) is actually committed.    I've read several items including one by Jonathan Keheyias () and another by Paul Randal (), and several others.  Jonathan advocates monitoring for a PLE below 300 per 4GB of buffer pool as being too low.  For the SQL Server Instance above, 300 * (2.5 / 4) = 187 results in a really really low target PLE below 300.  This instance has 290GB of SQL Server data (not including log files), and is only used for integration testing.  Assuming the last 12 days is representative of typical usage for this server, I'd say the max server memory (mb) setting could be lowered.  At the other end of the scale, I have another integration-test server with a PLE of 294, that has a max server memory (mb) setting of only 1GB.  This server has only 224MB of SQL Server data not including logs, and is running some BizFlow databases.  This server might benefit from a higher max server memory (mb) setting.  I'm thinking a good starting place for targets that might be assigned too much memory could include looking at:  SELECT      RamMB = physical_memory_in_bytes / 1048576     , BufferPoolCommittedMB = bpool_committed * 8192E0 / 1048576     , BufferPoolCommitTargetMB = bpool_commit_target * 8192E0 / 1048576     , PercentOfDesiredSizeMB = CONVERT(INT,(CONVERT(DECIMAL(18,2),bpool_committed)                              / bpool_commit_target) * 100) FROM sys.dm_os_sys_info;   If BufferPoolCommitTargetMB / BufferPoolCommittedMB is greater than 1, the server is not using the entire buffer pool.  If the machine in question also has a PLE of greater than "x" then it might be a good candidate for a decrease in max server memory (mb).  Since the Buffer Manager:Lazy writes/sec performance counter tracks the number of times SQLOS has written pages out to disk between checkpoints due to memory pressure, this might be another good thing to look at.  DECLARE @WaitTime DATETIME; SET @WaitTime = '00:00:15'; DECLARE @NumSeconds INT; SET @NumSeconds = DATEDIFF(SECOND, 0, @WaitTime); DECLARE @LazyWrites1 BIGINT; DECLARE @LazyWrites2 BIGINT;  SELECT @LazyWrites1 = cntr_value  FROM sys.dm_os_performance_counters dopc WHERE (         dopc.counter_name LIKE 'Lazy writes/sec%' COLLATE SQL_Latin1_General_CP1_CI_AS     )     AND dopc.object_name = 'MSSQL$' + CONVERT(VARCHAR(255),                SERVERPROPERTY('InstanceName')) + ':Buffer Manager';  WAITFOR DELAY @WaitTime;  SELECT @LazyWrites2 = cntr_value  FROM sys.dm_os_performance_counters dopc WHERE (         dopc.counter_name LIKE 'Lazy writes/sec%' COLLATE SQL_Latin1_General_CP1_CI_AS     )     AND dopc.object_name = 'MSSQL$' + CONVERT(VARCHAR(255),                SERVERPROPERTY('InstanceName')) + ':Buffer Manager';  SELECT LazyWritesPerSecond = (@LazyWrites2 - @LazyWrites1) / @NumSeconds;   The above code assumes the server is under load during the 15 seconds it takes to run, otherwise it will report 0; which might be a misleading false-negative.  Should I also look at PAGELATCHIO_* wait stats or some other wait type as an indicator of memory pressure, or lack thereof?   My question is, how can I reliably determine a "good" target value for PLE and max server memory (mb)? Render just part of a final image from Cycles? How is an empty set truly "empty"? In a  question, an answerer says:  an empty bag is a bag with nothing inside it.  Makes sense, but I'm reading a textbook right now that says:  The empty set has only one subset (namely, itself).  Which seems odd to me. If a set is empty, why would it contain a subset? Regardless of whether or not the subset is also an empty set, the &quot;empty set&quot; still contains a subset, so how is it really &quot;empty&quot;? To go back to the empty bag analogy, it just sounds like my text book is saying:  an empty bag is a bag with nothing but an empty bag inside it.  Which doesn't make sense because then the bag isn't empty. I can't wrap my head around this concept.