Let $P$ be a $n$-degree real polynomial having $n$ simple roots $x_1, x_2, \cdots,x_n$.  The problem asks to prove that   $$ \displaystyle\sum_{k=1}^n \frac{P''(x_k)}{P'(x_k)} =0$$  I tried writing $P$ as the product of $x−x_i$ and differentiating it twice but the calculations are too hard to carry out. Thanks in advance.
Let $P(x)$ be a degree $n$ polynomial with distinct roots $r_1, r_2, \cdots r_n$. Prove that   $$\displaystyle\sum_{i=1}^{n}\frac{P''(r_i)}{P'(r_i)}=0$$  My proof: We can equivalently rewrite the polynomial as $(x-r_1)(x-r_2)\cdots(x-r_n)$ and if we plug in the roots in the polynomial, we get that all of them are equal to $0$ so their first derivatives are equal to $1$ and then their second derivatives are equal to $0$ which means that all quotients are equal to $0$ and there for the sum is as well, $Q.E.D$  Don't show the solution please, but if you have a hint, feel free. Thank you.
Where are all islands and discoverable locations? After taking off in Simpleplanes and flying around over the ocean for a while, I've found a lots of tiny islands all over the place, and a few really big ones that unlock new spawn locations once found.  Large islands appear to also have many discoverable 'locations' around them as well, like airports, carriers, or landmarks.  These locations automatically become available as start points once discovered.  Where can I find all of these major islands, and once there, where should I look to unlock all of the locations? Working with a huge file. Illustrator is super slow. What can I do to ease the load? The size of the file is 168mb and it's nearly impossible to work on the file because every action takes like five minutes or so to complete. The artwork is of a scene filled with characters and I so far outlined the entire thing with a brush and afterwards expanded all the strokes thinking it would help ease the load. However, the file is still way too big. Is there something I can do to make it easier to work on the file?  Edit: I flattened transparency and got it reduced to 33mb. Huge difference, but would like it even smaller if possible. I've got an MP4 video file and I want to add another AAC audio track to it. I've tried YAMB and MeGUI (frontends for MP4Box) and it plays correctly in Zoom Player, but it picks the wrong track in WMP and plays both at once in Quicktime. I think this might have to do with designating the default audio track somehow.  Does anyone know how to specify the default audio track with YAMB/MeGUI or know of another way of adding a track to an MP4 file? Where M is a matrix calculate a formula for M^n Let $$M = \begin{bmatrix} -7 &amp; 8 \\ -8 &amp; -7 \end{bmatrix}.$$ Find formulas for the entries of $M^n$ where $n$ is a positive integer.  (Your formulas should not contain complex numbers.)  Your answer should be in the form of a matrix.  I diagonalized to the form $M = P D P^{-1}$  and $M^n = P D^n P^{-1}$ where $P$ is my matrix of eigenvectors and $D$ is my matrix of eigenvalues.  My final answer after diagonalization was $$M^n = \begin{bmatrix} .5((-7+8i)^n+(-7-8i)^n) &amp; (i/2)(-(-7+8i)^n+(-7-8i)^n) \\ (.5/i)(-(-7+8i)^n+(-7-8i)^n) &amp; .5((-7+8i)^n+(-7-8i)^n) \end{bmatrix}$$  I can't seem to find an answer not in terms of complex numbers.  Can someone show me what I'm missing? Chat is an offshoot of the main site, and using it is  during their time here.  Chat is our place for:        Real-time collaboration   Meeting fellow members of the community   Less structured, casual (but still roughly on-topic) conversation      Our chat system exists to serve the main site; users of each site can gather and commune with each other.  When in chat, we expect all users to abide by our  policy, which means that we expect community members to treat each other with respect.    While it is a place where the community can go to interact on a more casual level, chat still needs to be moderated by the community and there are tools in place to help with the moderation. This raises a few questions, that are addressed here: Disclaimer: I'm pretty novice at sysadmin stuff.  I'm trying to set up port forwarding in an AWS EC2 instance, this has to be done in the command-line because I don't want to go in and edit anything, it has to be automatic (it's part of a build process).   sudo echo "net.ipv4.ip_forward = 1" &gt;&gt; /etc/sysctl.conf      Permission denied   The weird thing is I've been (successfully) using sudo for pretty much every command that required su privileges. If I do sudo su before the command (trying it out by hand in an ssh session), then it works.  Reasons behind this? Possible solutions that don't involve sudo su or manual edits? Strange math notation, vertical bar with parentheses. So I was reading "Resistance Distance" (D. J. Klein, M. Randić) (Journal of Mathematical Chemistry 12(1993)81-95) when I came up with strange notation.  From the paper:  The graph adjacency matrix is defined as: $$A_{xy} = (x|A|y) =  \begin{cases} 1/r_{xy} &amp; x \sim y \\ 0 &amp; \text{otherwise}  \end{cases}$$  The graph degree matrix of a graph is defined as:  $$\Delta_{xy} = (x|\Delta|y) = \delta(x,y) \sum_{z}^{\sim x} 1/r_{xy}$$  $$ |\phi) \equiv \sum_x |x)  $$  My main question is what does the combination of the vertical bar parentheses mean. Thanks in advance. Some class features, monster traits, and feats allow you to take a −5 to your attack roll in order to gain a +10 to your damage roll. At first glance this looks really good, but when will it deal more damage than not using it on average?  For instance if you have a total attack bonus of +8 and you deal 1d12 damage. You have to roll a 7 to hit an AC of 15. If you take the −5 that makes it where you have to roll a 12. That's going to reduce your average damage by a lot. Hyphen or no hyphen when modifying an adjective with a quantity? I have a sentence which has an object that is described with an adjective:     We need to inform our interested patrons of this change.   If I modify "interested" with "more" or "less", do I connect the words with a hyphen or not?  Example sentence:     We need to inform our less-interested patrons of this change.     Also, please let me know if I have not used the correct terms for sentence parts in this question. An incorrect answer for an integral As the authors pointed out in  (p. 2), the following evaluation which was in Gradshteyn and Ryzhik (sixth edition) is incorrect (and has been removed). $$ ''\int_{0}^{\infty}\frac{1}{\left(1+x^{2}\right)^{3/2}} \frac{1}{\sqrt{1+  \frac{4 x^{2}}{3\left(1+x^{2}\right)^{2}}+\sqrt{1+  \frac{4 x^{2}}{3\left(1+x^{2}\right)^{2}}}}} \mathrm{d}x = \frac{\pi}{2\sqrt{6}}'' \qquad (*) $$ A numerical evaluation gives $0.6663771 \cdots$ on the left hand side and $0.64127491 \cdots $ on the right hand side.  I have not succeeded to correct $(*)$.  Do you have any idea on how to evaluate the above integral? How to perform hard / cold reboot of iPhone or iPad? I understand that when I press and release the power button on the top of my iPhone or iPad, it stays on but turns off the screen.  Whereas if I press and hold the power button, a "slide to turn off" control appears on the screen.  Sliding this, as I was told, does turn the unit off, but it does so by suspending the RAM to disk (flash memory, actually.) So, when turning on the iPhone or iPad, it isn't so much rebooting the OS as it is resuming it from flash storage.  How to force the OS to completely reboot on these devices? "invalid EFI file path" while booting Windows Suppose $X_1,\ldots,X_n$ are a random sample of $U(-\theta,\theta)$ with $\theta&gt;0$. How can find $$ E\left(|X_1|\ \big|\max |X_i| \right) $$ that $1\leq i\leq n$? So I wish to create and object and set it's shading to 'smooth' using python. It seems simple enough but after trying every method I could find on the internet none of them worked for me. Here's a short example script where the object needs to have smooth shading:  import bpy from mathutils import Vector  verts = [(0,0,0),(0,5,0),(0.25,5,0),(0.25,0,0),(0,0,5),(0,5,5),(0.25,5,5),(0.25,0,5)] faces = [(0,1,2,3), (4,5,6,7), (0,4,5,1), (1,5,6,2), (2,6,7,3), (3,7,4,0)]  grat1mesh = bpy.data.meshes.new("grat1") grat1obj = bpy.data.objects.new("grat1", grat1mesh) grat1obj.location = (5, -5, 0) bpy.context.scene.objects.link(grat1obj) grat1mesh.from_pydata(verts,[],faces) grat1mesh.update(calc_edges=True)  grat2obj = grat1obj.copy() grat2obj.data = grat1mesh.copy() bpy.context.scene.objects.link(grat2obj) grat2obj.location += Vector((-0.125, 1.25, 1.25)) grat2obj.scale = (2, 0.5, 0.5)  bm = grat1obj.modifiers.new("Chop", 'BOOLEAN') bm.object = grat2obj bm.operation = 'DIFFERENCE' bpy.context.scene.objects.active = grat1obj bpy.ops.object.modifier_apply(modifier=bm.name) bpy.context.scene.objects.unlink(grat2obj)   I found this online and tried to place it at the end of the script:  smoothings = [] smoothings.append(face.use_smooth) me_ob.faces.foreach_set("use_smooth", smoothings)   However this just returned an error on the second line, perhaps as it was from an older website. Obviously using a simple bpy.ops.object.shade_smooth() didn't work and I'm kind of stumped as of what to do here! Permanently removing apache2 My understanding of the Stern–Gerlach experiment is that neutral (0 total charge) particles are sent through a non-homogeneous magnetic field, with the expectation that the field will push that particle's path higher or lower on a detector because of the collective spin of that particle. While the detector can detect particles in a two-dimensional surface, the results of the experiment are that particles appear in only two localized areas directly above the path of the particle stream, or directly below the path of the particle stream - spin up or spin down. The conclusions from these measurements are that the particle, when measured, will always have about the same magnitude.  Why does this not follow from classical mechanical theory related to magnetism? If you shot a magnet through a similar apparatus, I would expect the magnet to be rotated to align with the magnetic field in some way which, at high enough field strengths relative to the mass of the magnet, would cause us to measure basically the same magnitude as if the magnet entered the apparatus pre-aligned with the field.  How is my explanation incorrect? Is there a simple way to add exFAT support? When I insert a USB stick or camera flash card I get the following error:  unknown filesystem type 'exfat'   Is there a simple way to add exFAT support to Ubuntu without kernel patching? How can I use Web.debug.config in the built-in visual studio debugger server? See the revision history on , where the  tag was edited in (and subsequently edited out after I started this discussion).  I think, by adding this tag, you're making assumptions about the OP which they could find a little bit offensive.  Particularly with this question, there's no mention of the level of PHP knowledge — it's more a math question than it is a programming question. Someone could be excellent at a particular language whilst not being particularly great at math, so isn't it a bit insulting to make the assumption they're a beginner? It's also interesting to note  tag, which currently stands at 7,784 on SO.  Compare this to , which only has 18 questions, most of them being &quot;discussiony.&quot;  Why do users retag questions as &quot;beginner&quot; but not &quot;expert&quot;? UPDATE, , .  Related   (thanks @Ralph) Solve $$\int \limits_{0}^{\infty} \frac{\cos(x)}{\cosh(x)} dx$$ without complex integration.  This integral can be very easily solved with contour integration, but how can you solve it without taking a tour in the complex plane?