How is $\mathrm dU=C_V\,\mathrm dT$ always true?
$\Delta U$, $C_p$, $C_v$ for an ideal gas process
I'm planning to visit a summer school in the Schengen zone. What kind of visa should I apply for? I will be attending a summer school which will last 10 days.  I want to know: for applying for a Schengen visa, is the purpose of my trip considered as touristic or study? I'm putting some type on a path, a circle; but when I do it, the type is upside down. How can I flip it so it reads the right way up? See image below: My usb mouse sleeps after 5 seconds of inactivity when running on battery power. It wakes uo again when i press any button on the mouse. This is really annoying.  I have made lsusb, and I noticed my mouse is   Bus 001 Device 012: ID 04d9:2083 Holtek Semiconductor, Inc.  I have checked my /sys/bus/usb/devices/usb1/power/autosuspend but it is already set to 0, and the directory usb1 has many subdirectories which also have autosuspend files, and i dont know how to edit them.  Also, I would like a permanent solution, which would work for any mouse, on any of my 3 usb ports. Any ideas?  Using: Ubuntu 12.04 64bit Do I need to have antivirus software installed? I thought that there was no need to scan for viruses either in Ubuntu or any Linux distros until I found virus scanner packages clamtk and klamav in Ubuntu software center yesterday.  This leads to the following questions:   How do viruses differ between Linux and Windows? How do the strategies for protection differ between Linux and Windows? Should a virus scanner package be installed on my system? If so, which would be a better option? Proving that a list of perfect square numbers is complete Well, I have a number  $n$ that is given by:  $$n=1+12x^2\left(1+x\right)\tag1$$  I want to find $x\in\mathbb{Z}$ such that $n$ is a perfect square.  I found the following solutions:  $$\left(x,n\right)=\left\{\left(-1,1^2\right),\left(0,1^2\right),\left(1,5^2\right),\left(4,31^2\right),\left(6,55^2\right)\right\}\tag2$$     Is there a way to prove that this a complete set of solutions? So I mean that the solutions given in formula $(2)$ are the only ones?     My work:   We know that: $$ 1 + 12x^2 \left(1+x \right) \ge 0   \space \Longleftrightarrow \space   x \ge -\frac{1+2^{-2/3}+2^{2/3}}{3}   \approx -1.07245 \tag3 $$ So we know that for $x&lt;-1$ there are definitely no solutions. I typically have two needs:  Scenario A. Change a single PDF page.  In this case I have a PDF but not the original source file used to create the PDF. I don't want to try to recreate the document from scratch. I'd like to open the PDF and change a few things. A good example of this scenario: I was responsible for planning a big event at a campground site, I had a PDF of the site. I wanted to start with that document, highlight some parts, add some labels, remove some parts that weren't relevant.   or   Scenario B. Combine PDFs or extract page(s) from a PDF  This scenario usually arises because I want a single PDF deliverable that is made up of parts that are best created in different programs. In this case I have the source files for all the documents but they don't play well enough together to easily create a single PDF deliverable. For part of it, I may want to use Libre Office Writer. For another page I may want to use Gimp. Still another page I may use Libre Office Calc. I could use Writer as the master document and embed images or the Calc object into that, but for ultimate control, you can't beat separate PDF documents that are then combined.  What are the best tools / processes for editing PDFs in Ubuntu? I would like to port Ubuntu Touch to a device that is currently not supported. What are the requirements for doing this and where do I begin? Creating mask file from shapefile using ArcGIS for Desktop? I have a shapefile of Lakes over my study area. Each polygon has attribute information of each lake, name, shape etc. Actually I want to eliminate all lakes from analysis. In other word, I want lakes to have 0 value and the area that out of boundary of lake to have their orginal pixel values. when I do subset using this shapefile, the output is not I want because my shapefile deoesn't have any value outer boundary of lake.   How can I build a proper mask file (inner part of lake has 0 value, outer is 1) using this shapefile in ArcGIS Desktop? How can I get textures on edge of walls like in Super Metroid and Aquaria? Games like Super Metroid and Aquaria present the terrain with the other facing parts having rocks and stuff while deeper behind them (i.e. underground) there's different detail or just black.      I would like to do something similar using polygons. Terrain is created in my current level as a set of overlapping square boxes. I'm not sure if this rendering method will work such a system for creating terrain but if anyone has ideas I'd love to hear them. Otherwise I'd like to know how I should re-write the terrain rendering system so it actually works to draw terrain in this manner... A generalized version of the second Borel-Cantelli lemma says     Theorem 5.3.2. Second Borel-Cantelli lemma, II. Let $\mathcal F_n, n \ge 0$ be a filtration   with $F_0 = \{\emptyset, \Omega\}$ and $A_n , n \ge 1$ a sequence of events with $A_n ∈ \mathcal F_n$ . Then   $$ \{A_n \,i.o.\} = \left\{\sum_{n \ge 1} P (A_n |\mathcal F_{n−1}) =\infty \right\}. $$   An exercise of this lemma is     Exercise 5.3.6. Show    $\sum_{n \ge 2} \mathbb P (A_n | \cap_{m=1}^{n−1} A_m^c ) = \infty$ implies $P (∩_{m \ge 1} A_m^c ) = 0$.   I think we can actually prove that $\cap_{m \ge 1} A_m^c = \emptyset$. My proof is like this: Let $B_n = \cup_{m = 1}^n A_n$. Let $\mathcal F_n = \sigma(A_1,\dots,A_n)$, which forms a filtration. It is easy to verify that $$ \mathbb P (A_n | B_{n-1}^c ) = \mathbb P (A_n | \mathcal F_{n-1})(\omega) $$ when $\omega \in B_{n-1}^c$.  Assume there exist $\omega \in \cap_{m \ge 1} A_m^c$. It follows from the previous lemma that $$ \omega \in \left\{\sum_{n \ge 1} P (A_n |\mathcal F_{n−1}) =\infty \right\} = \{A_n \,i.o.\}. $$ In other words, $\omega \in \cap_{m \ge 1} \cup_{n \ge m} A_n$, which contradicts $\omega \in \cap_{m \ge 1} A_m^c$.  Is this the right proof? Is it possible to "upgrade" from a 32bit to a 64bit installation? As per title, I wish to know if I can, after a successful 32bit setup and consequent software installations, update to a 64bit version.  I know how to partition (actually one of the solutions is to set /, /etc, /home, /var/www, and /opt as separate partitions) and I know that a clean install is way better than a dirty one, yet I would like to know if/how it's possible to do that. Doubly stochastic matrix proof A transition matrix $P$ is said to be doubly stochastic if the sum   over each column equals one, that is $\sum_i P_{ij}=1\space\forall i$.   If such a chain is irreducible and aperiodic and consists of $M+1$   states $0,1,\dots,M$ show that the limiting probabilities are given by   $$\pi_j=\frac{1}{1+M},j=0,1,\dots,M$$   I have no idea how to prove it but   If a chain is irreducible then all states communicate i.e $$P_{ij}&gt;0\space \text{and}\space P_{ji}&gt;0\space\forall i,j$$ If $d$ denotes the period of any state, if a chain is irreducible aperiodic, then $d(i)=1\forall i$   If $P_{(M+1)\times (M+1)}$ matrix and $\pi$ is the stationary distribution  $$\pi_j=\sum_iP_{ij}\pi_i\space j=0,1,\dots,M+1$$  but how I can get this expression? \documentclass{minimal} \usepackage{tikz} \begin{document} \pgfmathparse{random(-5,5)} \pgfmathresult \end{document}   Works, but I have a .0.0 that not supposed to be here. I wish to display a map in QGIS (world country shapefiles) showing all countries but centered on the Pacific area.   I am not familiar with Proj4, so is there any way this can be done in QGIS? How can I have  for Ubuntu? Java: Difference between the setPreferredSize() and setSize() methods in components I'm trying to show the following:  Let $f:X\to Y$ be a proper holomorphic map between connected, non-empty Riemann Surfaces. Show that a map $g:Y\to\mathbb{C}$ is  holomorphic if and only if  its composition with f is holomorphic.  So far I know that $f$ is surjective, since it's proper and holomorphic. My next step was to look at the charts, but it's there, I get stuck. Access nested dictionary items via a list of keys? How to create a file system as a non root user? I'd like to create a filesystem image as a non-root user, the problem I'm facing is that the way I'm currently doing it, it needs a mount and as a non-root can't do a mount.  This is what I'm now doing:  #!/bin/sh  IMG="app.ext4" LBL="PIHU_APP" MNT="/mnt/PIHU_APP" APP="/home/buildroot/board/arctura/PIHU_APP/application"  #64MB: dd if=/dev/zero of=$IMG bs=4096 count=16384  mkfs.ext4 $IMG e2label $IMG $LBL  mkdir -p $MNT mount -o loop $IMG $MNT cp -r $APP $MNT sync umount $MNT   I have full root access, so I can setup/prepare anything, but the script will be executed from a non-root account.  However, I have a feeling there may be a better way that might not even need mounting, but I'm not sure.. Find $|(\mathbb{Z} \oplus \mathbb{Z}) / \langle(3,6)\rangle|$ and determine if its cyclic