Consider a system having spherical symmetry filled with gas. If it expands or compress, there is some work done by system or surroundings, respectively, according to the equation $W=nRT\log(v_2/v_1)$. But the coordinates of centre of mass of system is same at the centre of sphere, so according to original definition of work ($W$ equal to force times displacement) it should be zero. I can't understand where is the mistake.
Take a sponge ball and compress it. The net force acting on the body is zero and the body isn't displaced.  So can we conclude that there is no work done on the ball?
Inconsistent accessibility error with the following c# code. Why? Why is this CW post saying 36% unknown? Looking at         it says "36% unknown". No user by the name of 'unknown' appears in the  ()  What madness is this? Who is this 'unknown' you speak of? Dropping one of the columns when using one-hot encoding $ \int_{0}^{ \infty} \int_{0}^{ \infty} \frac { e^{-(x+y)}}{x+y} dx dy $ Is the unit sphere in $\Bbb R^4$ path connected? Can sweetened condensed milk be substituted for whole milk and sugar? I am making a fudge recipe that calls for 1cup whole milk and 5 cups sugar. It also uses 2 sticks of butter and 25 marshmallows. I'm wondering if I can omit the milk and cut back on the sugar by using sweetened condensed milk. What is the difference between UNION and UNION ALL? What is the difference between UNION and UNION ALL? Radius of Convergence of a given Power series Mount cifs Network Drive: write permissions and chown I have access to a cifs network drive. When I mount it under my OSX machine, I can read and write from and to it.   When I mount the drive in ubuntu, using:  sudo mount -t cifs -o username=${USER},password=${PASSWORD} //server-address/folder /mount/path/on/ubuntu   I am not able to write to the network drive, but I can read from it. I have checked the permissions and owner of the mount folder, they look like:  4.0K drwxr-xr-x  4 root root    0 Nov 12  2010 Mounted_folder   I cannot change the owner, because I get the error:  chown: changing ownership of `/Volumes/Mounted_folder': Not a directory   When I descend deeper into the network drive, and change the ownership there, I get the error that I have no permission to change the folder´s owner.  What should I do to activate my write permission? Is there something like python's interactive REPL mode, but for Java? Is the density operator a mathematical convenience or a 'fundamental' aspect of quantum mechanics? In quantum mechanics, one makes the distinction between mixed states and pure states.  A classic example of a mixed state is a beam of photons in which 50% have spin in the positive $z$-direction and 50% have spin in the positive $x$-direction.  Note that this is not the same as a beam of photons, 100% of which are in the state $$ \frac{1}{\sqrt{2}}[\lvert z,+\rangle + \lvert x,+\rangle]. $$ It seems, however, that at least in principle, we could describe this beam of particles as a single pure state in a 'very large' Hilbert space, namely the Hilbert space that is the tensor product of all the $\sim 10^{23}$ particles (I think that's the proper order of magnitude at least).  So then, is the density operator a mathematical convenience, or are there other aspects of quantum mechanics that truly require the density operator to be a 'fundamental' object of the theory?  (If what I mean by this is at all unclear, please let me know in the comments, and I will do my best to clarify.) How can slightly blurred photos be improved in post processing? How can I dual boot Windows and Ubuntu? I currently have Windows 7 on my computer and I want to install Ubuntu alongside Windows safely.  Could someone please provide clear instructions? Why are Only Real Things Measurable? Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with operators in quantum mechanics, why must measurable quantities be Hermitian, and consequently real? Your content must have a ListView whose id attribute is 'android.R.id.list' Connection between rank and matrix product I have a problem understanding the following:     Let $A$ be an $m \times n$ matrix and let t $\in \mathbb{N}$. Prove that   $\operatorname{rank}(A)\leq t$ if and only if there exists an $m \times t$ matrix $B$ and a $t \times n$ matrix $C$ so that $A = BC$.   I know what a rank is but I can't make a connection between the rank and the existence of two matrices such that $A = BC$. Let $(X,d)$ be a metric space and let $S$ be the set of Cauchy sequences in $S$. Define "~" in $S$ by "$\{s_k\}$ ~ $\{t_k\}$" to mean that $d(s_k,t_k) \to 0$ as $k \to \infty$. Let $X'$ denote the set of equivalence classes of $S$ and let $s'$ denote the equivalence class of $s = \{s_k\}$.   I know $\phi(s',t') = \lim\limits_{k \to \infty} d(s_k,t_k)$ is a metric on $X'$.   Show $(X',\phi)$ is complete.  For $x \in X$, define $x'$ to be the equivalence class of the constant sequence ${x,x,...}$. Show that the function $x \to x'$ is an isometry of $X$ onto a dense subset of $X$.   If anyone can share a proof, or steer me in the right direction, maybe tell me what it is that I need to prove specifically, that would be great. How do I get a block with data to place myself? Since the Earth has a magnetic field, can it, in theory, be run through a conductive metal coil to create electricity? boolean variable values in PHP to javascript implementation