Print a file, skipping the first X lines, in Bash
How can I remove the first line of a text file using bash/sed script?
Polynomial ring $R[X] $ has zero Jacobson radical, for $R$ a domain How can I calculate how long it will take to transfer a file over LAN? I want to know about the method with which we can calculate data transfer speed.   If I am sending a 1 GB file through a 1 Gb/s LAN connection, from one computer to another, how much time will it take to transfer that file? I have an Ubuntu 14.04 server that has no external monitor connected.  I use NoMachine to remote control the machine.  When I do so, the Unity/Gnome interface doesn't see any monitors connected, so I can only use 800x600 when connecting with NoMachine.  If I plug in a monitor, I can set the resolution to whatever size the NoMachine window is on the remote computer.  Is there a way to create a "fake" monitor device on Ubuntu so I can set a desktop resolution in Unity/Gnome? How to prove $\lim\limits_{n \to \infty} (1+\frac1n)^n = e$? How closely can we estimate $\sum_{i=0}^n \sqrt{i}$ By looking at an integral and bounding the error? Every ordered field has a subfield isomorphic to $\mathbb Q$? I'm going through the first chapter in a text on real analysis, which contains preliminaries on ordered fields, the real numbers, etc. Supposedly I had learned about such things already, in calculus, but I thought it wouldn't hurt to go over it again.  Up until the paragraph which is the subject of my question, everything is thoroughly proven and examples are provided. However the following excerpt just goes through facts, whose proof I cannot conceive:      Although $\mathbb Q$ is an archimedean field, these properties cannot be used to define $\mathbb Q$ since $\\$ there are many Archimedean ordered fields. What distinguishes $\mathbb Q$ from the other $\\$Archimedean ordered fields is that $(\mathbb Q,&lt;)$ is the $smallest$ ordered field in the following sense: $\\$ if $(X,\prec)$ is an ordered field, then $X$ contains a sub-field which is (field) isomorphic to $\mathbb Q$.$\\$ Furthermore such an isomorphism preserves the order relation.   I've concatenated the paragraph a bit, so the text isn't idem from the book (if anyone wishes to know it's Phillips, An Introduction to Analysis and Integration Theory, Dover Publications). Also $\mathbb Q$ contains no proper subfields, and this can be verified by the apparently easy-to-prove fact that, if a field has characteristic zero, then it contains a subfield isomorphic to $\mathbb Q$. Then it remains to prove the preservation of order.  I'm guessing that some concepts from abstract algebra or field theory would easily suffice, but at the moment such topics are a bit over my head, so all I can think of doing is actually constructing the isomorphism $\phi:\mathbb Q\to\hat{\mathbb Q}$, where $\hat{\mathbb Q}$ is a certain subfield of  $X$ in such a way that field operations are preserved, but I really can't come up with anything. So the question is: how do I construct this isomorphism, or, if there's a better way of proving this, how is it done? Thanks for any help.  Edit: Assume the existence of $(\mathbb Q,&lt;)$ as an archimedean ordered field. "the better" vs "the best" of two players How far into space does one have to travel to see the entire sphere of earth? Virgin Galactic will take passengers aboard SpaceShipTwo  above the surface of the earth. But from this altitude, passengers will only be able to see a certain segment of the curvature of the earth through .  How much further into space would SpaceShipTwo have to travel to give passengers a view of the entire sphere of earth through one of these windows? $n$ choose $k$ where $n$ is less than $k$ CUDA Driver API vs. CUDA runtime When writing CUDA applications, you can either work at the driver level or at the runtime level as illustrated on this image (The libraries are CUFFT and CUBLAS for advanced math):   (source: )    I assume the tradeoff between the two are increased performance for the low-evel API but at the cost of increased complexity of code. What are the concrete differences and are there any significant things which you cannot do with the high-level API?  I am using CUDA.net for interop with C# and it is built as a copy of the driver API. This encourages writing a lot of rather complex code in C# while the C++ equivalent would be more simple using the runtime API. Is there anything to win by doing it this way? The one benefit I can see is that it is easier to integrate intelligent error handling with the rest of the C# code. PHP Error - Uploading a file Add different materials to different parts of a mesh? How do I get started creating an Ubuntu Desktop App? I would like to find all ellipses which contain 2 given points and has one focus at origin (zero). All in 2D plane.   There are several possible approaches but I'm not sure which is the best - both looks quite difficult to solve algebraically.   using polar equation relative to focus with $(R_1,\phi_1),(R_2,\phi_2)$, being coordinates of points $$ R_1 = \frac{a(1-e^2)}{1-ecos(\phi_1-\theta)} $$ $$ R_2 = \frac{a(1-e^2)}{1-ecos(\phi_2-\theta)} $$ then for given $\theta$ solve for semimajor axis $a$ and eccentricity $e$ using deffinition of elipse as a set of points of the same distance from both foci. Given 2 points of cartesian coordinates $(x_1,y_1),(x_2,y_2)$ and one focus in origin $(0,0)$. For each given distance parameter $L$ solve for coordinates of second focus $(x_f,y_f)$, $$ L = \sqrt{x_1^2 + y_2^2} + \sqrt{(x_1 - x_f)^2 + (y_1 - y_f)^2} $$ $$ L = \sqrt{x_2^2 + y_2^2} + \sqrt{(x_2 - x_f)^2 + (y_2 - y_f)^2} $$ I can also first rotate the coordinate system (or my input points) by given angle (which is my arbitrary parameter) and then use some simplified equation of ellipse which has major axis parallel to x-axis which has just 2 degrees of freedom. But even after this rotation I don't see much simplification of algebraic solution.    Or is there any better way?    The resulting equations are quite difficult to solve. I wonder if there is some trick to simplify it and make it more elegant.  Also, because I would later to implement it into computational code where it would be computed manytimes per second, I would prefer solution inf form which is fast to evaluate numerically ( prefferably without goniometric or any transcedental functions )    Why I'm interested ? - I need it for some optimalization of orbital transfers for spaceship orbiting the Sun. I asked also in physics.stackexchange, and people recomand me that I should rather go to math section. ffmpeg is not even cutting this and other mp4s to the nearest second I have a file  Other - Analogue Clock-JEJqy1Wlovw.mp4  which is just a download from     It's 2 minutes 13 seconds long.  I had some issues cutting it to the second.  I was able to use this command   ffmpeg -ss 0 -i infile.mp4 -c copy -t 60 output.mp4  to make a file that was 60 seconds, so no problem so far.  That's the file exactly 1 minute, no issue there so far.  C:\vids\a&gt;dir blahhclo.mp4   9,590,540 bytes  C:\vids\a&gt;   C:\vids\a&gt;mediainfo blahhclo.mp4 General Complete name                            : blahhclo.mp4 Format                                   : MPEG-4 Format profile                           : Base Media Codec ID                                 : isom File size                                : 9.15 MiB Duration                                 : 1mn 0s Overall bit rate                         : 1 278 Kbps Writing application                      : Lavf58.3.100  Video ID                                       : 1 Format                                   : AVC Format/Info                              : Advanced Video Codec Format profile                           : High@L4.0 Format settings, CABAC                   : Yes Format settings, ReFrames                : 3 frames Codec ID                                 : avc1 Codec ID/Info                            : Advanced Video Coding Duration                                 : 1mn 0s Bit rate                                 : 1 146 Kbps Width                                    : 1 920 pixels Height                                   : 1 080 pixels Display aspect ratio                     : 16:9 Frame rate mode                          : Constant Frame rate                               : 25.000 fps Color space                              : YUV Chroma subsampling                       : 4:2:0 Bit depth                                : 8 bits Scan type                                : Progressive Bits/(Pixel*Frame)                       : 0.022 Stream size                              : 8.21 MiB (90%)  Audio ID                                       : 2 Format                                   : AAC Format/Info                              : Advanced Audio Codec Format profile                           : LC Codec ID                                 : 40 Duration                                 : 1mn 0s Bit rate mode                            : Constant Bit rate                                 : 126 Kbps Channel count                            : 2 channels Channel positions                        : Front: L R Sampling rate                            : 44.1 KHz Compression mode                         : Lossy Delay relative to video                  : 40ms Stream size                              : 920 KiB (10%)    C:\vids\a&gt;   But I do have an issue when I try to cut it at the following times, and it's whether I try to cut the 1 minute one or the whole 2 min 13sec one.  If I cut with -ss 0 -t 1,   it produces a file that is 1s 22ms.  So that's not too bad that's fairly close to one second. 1.022 seconds is quite good for an attempt to cut one second.  If I cut with -ss 0 -t 2, it produces a file that is 3s 19ms   That is terrible! it's meant to be one second!  If I cut with -ss 0 -t 30 then it is fine it does 30s 1ms.  Or this video for example, also an mp4   C:\blah&gt;ffmpeg -ss 189 -i vid.mp4 -c copy -t 6 someoutput.mp4  C:\blah&gt;mediainfo someoutput.mp4   produces a duration of 8s 383ms  That is so wrong though 'cos it should be 6 seconds! So that is two seconds too long!  C:\blah&gt;ffmpeg -ss 191 -i vid.mp4 -c copy -t 6 someoutput2.mp4  C:\blah&gt;mediainfo output2.mp4   shows that ffmpeg produced a duration of 10s 403ms  which is crazily wrong 'cos the ffmpeg command specified 6 seconds!  So that's not just one second out, that is 4 seconds out!  That's almost 50% longer than it should be! Is there a list of Ubuntu versions with default corresponding Linux kernel versions somewhere?  I would specifically like to know the most recent version of Ubuntu that still used Linux Kernel 2.x. Finding an angle within an 80-80-20 isosceles triangle The following is a geometry puzzle from a math school book. Even though it has been a long time since I finished school, I remember this puzzle quite well, and I don't have a nice solution to it.  So here is the puzzle:    The triangle $ABC$ is known to be isosceles, that is, $AC=BC$. The labelled angles are known to be $\alpha=\gamma=20°$, $\beta=30°$. The task is to find the angle labelled "?".  The only solution that I know of is to use the sine formula and cosine formula several times. From this one can obtain a numerical solution. Moreover this number can be algebraically shown to be correct (all sines and cosines are contained in the real subfield of the 36th cyclotomic field). So in this sense I solved the problem, but the solution is kind of a brute force attack (for example, some of the polynomials that show up in the computation have coefficients > 1000000). Since the puzzle originates from a book that deals only with elemetary geometry (and not even trigonometry if I remember correctly) there has to be a more elegant solution. Hiding downvoted questions silences important discussions on Meta I just noticed a particular MSE post today on a topic I care about that was posted almost a week ago, which I entirely missed before. I'm a pretty avid MSE user, and this happens quite rarely to me.     was created by an SE employee, and is essentially the final answer to , which is the 10th most upvoted MSE post with a bit more than 900 upvotes right now. So it's kind of an important post, and on a topic I care about, but I still missed it entirely until it was linked in a comment somewhere else.  The reason I missed it is because it's heavily downvoted. This means that as it has below -8 score, it's hidden from the home page entirely, drastically reducing the visibility of the post. In particular, the negative score removes all visibility a question usually receives from being bumped due to edits, new answers and bounties.   Many of the recent heavily-downvoted Meta posts by SE have been featured manually, which grants them a lot of visibility beyond the particular meta site. This masked the problem pretty efficiently, but it is more of a workaround, and one that isn't applicable to all situations.  Voting means something different on meta sites, and not every downvoted question should be hidden from view. Closed downvoted questions can certainly still be affected by the usual rules on the home page, and that should cover the vast majority of questions that actually have a purpose being on the home page. But open questions should be displayed on the home page on meta sites even if they're heavily downvoted. In this sentence:  "We would also accept an international transfer, but in that case we would ask you to pay your bank's charges at the time of the instruction to ensure that we received the full amount"  is it correct to use "received" or should it be in present tense? I'm using Ubuntu 16.04.3 LTS, on a 64-bit system.   There are two available updates in "Updates" section of the Ubuntu Software, named OS Updates and Software respectively:      I usually update through Software Updater utility, (and occasionally used sudo apt update and sudo apt upgrade as recommended by some software installation tutorials)- anything that I think is required. So why is Ubuntu Software showing these updates? Should I install them?  As far as I know, these types of updates, if you look at the list, aren't ever available in the Software Updater.