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Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\in \mathbb{N}^+\setminus\{1\}$   I wonder if it is possible by using only real analysis to demonstrate this "innocent" result?  Edit A more general result showing that  $$\int\limits_{0}^{\infty} \frac{x^{a-1}}{1+x^{b}} \ \text{dx} = \frac{\pi}{b \sin(\pi{a}/b)}, \qquad 0 &lt; a &lt;b$$ can be found in My installation is a Ubuntu 12.04 64 bits (Classic "legacy" MBR - No  EFI/UEFI here) with two partition :   A separate boot partition An encrypted LUKS volume that contains the system Logical Volumes   I seems I accidentaly deleted all the current kernel files with:  sudo apt-get purge 'linux-image-3.8*'    I applied the updates with Ubuntu Software Update, rebooted and then: Ubuntu can't boot.  I tried to boot on a live DVD with a 14.04.1 Ubuntu and can still see my partitions (described later):  /dev/sda1 (boot) /dev/sda2 (Encrypted volume) /dev/sda5 (Logical Volume container)  What are the steps to recover/reinstall the kernel files and dependencies in order to cleanly reboot my installed system?  Thanks in advance Removing OS from Dual Boot computer, and recovering HD space. I have a computer that dual boots Linux Mint, and Ubuntu, I am wondering what the best way would be to remove Linux Mint, then recovering the hard drive space that is resides in so that Ubuntu will be able to access it. Why can’t the Doctor’s companions understand babies and horses? The Doctor claimed to speak both horse and baby. The reason human companions can understand every language is because the TARDIS translates. So why can't anyone else understand babies and horses? Last night I took some night sky shots, and I've noticed that with long exposure times (15s, 30s) the time it took to store the images in the camera memory card (jpeg) took roughly as long?  The actual file size is fairly small at 1 to 3MB per file.  Any idea why this is? Node access based on view mode How can I obtain a Cauchy distribution from two standard normal distributions? I am interested in      Let $X\sim N(0,1), Y \sim N(0,1)$ independently. Show $\frac{X}{X+Y}$   is a Cauchy random variable.   My work:  $f_{X,Y}(x,y)=\frac{1}{2\pi} e^{\frac{-1}{2}(x^2+y^2)}, -\infty&lt;x,y&lt;\infty$ by independence  Let $U=\frac{X}{X+Y},V=X+Y$. (Is there a better $V$ to choose for this bivariate transformation?)  Then, $X=UV, Y=V - UV$. So, $|J|=V$.  $f_{U,V}(u,v)=f_{X,Y}(uv,v-uv)|J|=\frac{v}{2\pi}e^{\frac{-v^2}{2}(2u^2-2u+1)},-\infty&lt;u&lt;\infty$  $f_U(u)=\frac{1}{2\pi}\int_{-\infty}^{\infty}ve^{\frac{-v^2}{2}(2u^2-2u+1)}dv$. Let $y=\frac{v^2}{2}(2u^2-2u+1)$, so $dy=v(1+2u^2-2u)dv$. Then,  $f_U(u)=\frac{1}{2\pi(2u^2-2u+1)}\int_0^{\infty}e^{-y}dy=(\pi[(\frac{u-1/2}{2})^2+1])^{-1}, -\infty&lt;u&lt;\infty$,   which is not exactly a Cauchy distribution. Where did I mess up? More importantly, how would you proceed in solving this problem? In the Wikipedia article on positive definite matrices they claim that if $M$ is positive definite, then the determinant of $M$ is bounded by the product of its diagonal entries. How might we show this?  Ideas: The sum of the diagonal entries of $M$ is the trace, which is also the sum of the eigenvalues. The product of the eigenvalues is the determinant. All the diagonal entries of $A$ must be real and strictly positive. Also we have $|m_{ij}|\leq \sqrt{m_{ii}m_{jj}}\leq \frac{m_{ii}+m_{jj}}{2}$. So far nothing is working...  We might try to write the Gram matrix $m_{ij} = \langle v_i, v_j \rangle$, and then $\prod_i m_{ii} = \prod_i \|v_i\|^2$. Now we'd need to show that the determinant, which is a polynomial in the entries of $v_i$, is less than $\prod_i \|v_i\|^2$ somehow. Add a Texture to an Object using Python and Blender 2.8 i have trouble to get a script working within Blender 2.8. The following function work perfectly for me in Blender 2.79, but not in 2.8.  def CreateMaterialFromImage(matName, imgPath):     '''Creates and returns a material from an Image'''     realpath = os.path.expanduser(imgPath)     try:         img = bpy.data.images.load(realpath)     except:         raise NameError("Cannot load image %s" % realpath)      # Create image texture from image     cTex = bpy.data.textures.new(matName, type='IMAGE')     cTex.image = img      # Create material     mat = bpy.data.materials.new(matName)      # Add texture slot for color texture     mtex = mat.texture_slots.add()     mtex.texture = cTex     mtex.texture_coords = 'UV'     mtex.use_map_color_diffuse = True     mtex.use_map_color_emission = True     mtex.emission_color_factor = 0.5     mtex.use_map_density = True     mtex.mapping = 'FLAT'     return mat   It seems to be that the material.texture_slots were removed.  Under following Link  i found the note:        Texture slots for materials, lamps and world were removed. They remain   for brushes, particles and freestyle linestyles.      So my question is where do i find this property now or how could i change my code to get things done?  Thanks for your help!  EDIT:  This seems to be an interesting link for people run in to issues by updating their scripts to 2.8. Unfortunately there is no solution for my problem listed, yet. I want to change the default location of "Save a screenshot to Pictures" to save it to Desktop.   If that is not possible, is there any other way to do it? I am using Ubuntu GNOME 17.04.  NOTE: I don't want to change the location with dconf-editor as it changes the location of screenshot applet and not of the shortcut I require. how to use common modulus attack? I am struck with the following problem:  Let Alice, Bob, Chris and Eve communicate over a public network. They encrypt all messages they send using RSA system. Bob and Chris have the RSA modulus $n_B$ and $n_C$ respectively with $n_B$ = $n_C$ But different public encryption exponents: $e_B \neq e_C$. Suppose $\gcd(e_B, e_C)=1$, and that Alice sends the same secret message to Bob and Chris. Show how Eve can decipher the message. In Bash, when to alias, when to script, and when to write a function? It's taken me almost 10 years of Linux usage to ask this question. It was all trial and error and random late-night internet surfing.  But people shouldn't need 10 years for this. If I were just starting out with Linux, I'd want to know: When to alias, when to script, and when to write a function?  Where aliases are concerned, I use aliases for very simple operations that don't take arguments.   alias houston='cd /home/username/.scripts/'   That seems obvious. But some people do this:  alias command="bash bashscriptname"   (and add it to the .bashrc file)   Is there a good reason to do that? I'm trying really hard, but I genuinely can't think of any circumstances in which I'd want to do that. So, if there is an edge case where that would make a difference, please answer below.  Because that's where I would just put something in my PATH and chmod +x it, which is another thing that came after years of Linux trial-and-error.  Which brings me to the next topic. For instance, I added a hidden folder (.scripts/) in the home directory to my PATH by just adding a line to my .bashrc (PATH=$PATH:/home/username/.scripts/), so anything executable in there automagically autocompletes.  If I needed to.  I don't really need that, though, do I? I would only use that for languages that are not the shell, like Python.  If it's the shell, I can just write a function inside the very same .bashrc:  funcname () {   somecommand -someARGS "$@" }   As I stated, I found a lot of this out through trial and error. And I only truly saw the beauty of functions when my computer died and I was forced to use the computers of the people around me when they weren't using them.   Instead of moving a whole directory of scripts from computer to computer, I ended up just replacing everyone else's .bashrc with my own, since they had never even made a single modification.  But did I miss anything?   So, what would you tell a beginning Linux user about when to alias, when to script, and when to write a function?   If it's not obvious, I'm assuming the people who answer this will make use of all three options. If you only use aliases, or only use scripts, or only use functions—or if you only use aliases and scripts or aliases and functions or scripts and functions—this question isn't really aimed at you. Can I get a \widebar without using the mathabx package? One can use \bar to put a bar over a letter in math mode, but often the bar isn't wide enough. An alternative is \overline, but this isn't an accent, so it doesn't take into account the skew of the letter. In the following example, the bar extends too far to the left:    Now the mathabx package offers a \widebar accent, but this has several drawbacks:   mathabx changes a lot of other things, The \widebar sits slightly higher than \bar and \overline, In previewers (checked with kpdf and acroread), the \widebar appears fuzzy, which \bar and \overline don't.   So my question is: How can I put a wide bar over a letter in math mode without the above drawbacks? Exercise $1.8$ of chapter one in Hartshorne. What is the fastest car in Saints Row 4, and where can I find it? I found an Attrazione in Salander, which has traditionally been fastest in previous games, but I don't like sinking upgrades money into a car unless it's for keeps. What car is fastest in Saints Row IV (before and after upgrades), and how can I get it? Is there any way to get stats on the car in-game? Converting Linux into a Router- Traffic from eth0:0 to eth0 So I have been trying to convert my linux box which is running Mint into an router. eth0 has a public address that has 71.xx.xx.xx for an IP addr. I created an eth0:0 and set it an IP address , 10.0.0.100. I started a DHCP server that is on eth0 ( because it wont use eth0:0) and that's giving out addresses between 10.0.0.101 and 10.0.0.200. I can ping between eth0 and eth0:0 on the same machine. I have a another laptop connected to the linux via a switch and its able to receive an ip addr via the DHCP server. Problem is how do I route traffic from and to eth0:0 to eth0 and vice versa so that computers on the private network that is 10.0.0.x can access internet via the static IP on eth0. Linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T.$ Let $T$ be a linear transformation $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $T^{2}=\lambda T$ for some $\lambda\in\mathbb{R}.$ Then which of the following is/are true?  $1.\|T(x)\|=|\lambda|\|x\|$ for all $x\in\mathbb{R}^n.$  $2.$ If $\|T(x)\|=\|x\|$ for some non-zero $x\in\mathbb{R}^{n}$, then $\lambda=\pm 1.$  $3.T=\lambda I$ where $I$ is the identity transformation on $\mathbb{R}^{n}.$  $4.$ If $\|T(x)\|&gt;\|x\|$ for a nonzero vector $x\in\mathbb{R}^{n},$ then $T$ is singular.  According to me since $T^{2}=\lambda T$ we have $T$ satisfies the polynomial $x^{2}-\lambda x.$ So choices for its minimal polynomial are $x,x-\lambda,x(x-\lambda).$ So $3$rd option is clearly incorrect and $4$th option is not satisfied if $T=I$ and hence correct one. I am confused in first two options . Please help me. Thanks in advance. What kind of components are black blobs on a PCB? Cross-platform space remaining on volume using python I need a way to determine the space remaining on a disk volume using python on linux, Windows and OS X. I'm currently parsing the output of the various system calls (df, dir) to accomplish this - is there a better way? Ubuntu 20.04 still supporting Python2