I have been trying to prove (the left half of) the following inequality:  $$ \underbrace{\sum_i \sum_j |x_i| \le \sum_i \sum_j |x_i + x_j|}_\textrm{?} \le 2 \sum_i \sum_j |x_i|$$  (All $x_i$s are arbitrary reals and sums are over $1, 2, \dots, n$)  The right half follows by a simple application of the triangle inequality, but the left half isn't as simple.  I tried induction, and I could reduce the above problem to proving the following statement:  $$ \sum_i |x_i + x_{n+1}| + \sum_{i=1}^{n+1} |x_i + x_{n+1}| \ge \sum_{i=1}^{n+1} |x_i| + n|x_{n+1}|$$ or more simply (replacing $x_{n+1}$ by $y$), $$ 2\sum_i |x_i + y| \ge \sum_i |x_i| + (n-1)|y|$$  However, I don't know if the above "reduction" is any simpler than the original problem itself!  Can I continue the above line of thought, or is there an easier way to solve this problem? Any help is appreciated! :)
Let $x_{1},x_{2},\cdots,x_{n}$ be real numbers. Show that     $$\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|.$$   I think this problem may be solved using nice methods, but I can't find them yet; I know this may be of use: $$|a|+|b|\ge |a+b|.$$ But I can't make it work. Thank you everyone.
ArgumentNullException or NullReferenceException from extension method? What would you consider to be the best exception type to throw when an extension method is called on a null instance (where the extension method does not allow it)? Since extension methods are nothing but static methods you could argue that it should be ArgumentNullException, but on the other hand they're used like instance methods so it might be more natural to use the NullReferenceException. Let's take the following example:  public static string ToInvariantString(this IFormattable value, string format) {     return value.ToString(format, CultureInfo.InvariantCulture); }   This way a NullReferenceException will be thrown if the value parameter is null.   The other example would be:  public static string ToInvariantString(this IFormattable value, string format) {     if (value == null) throw new ArgumentNullException("value");     return value.ToString(format, CultureInfo.InvariantCulture); }   EDIT: In some of the answers you have pointed out that an extension methods can be called like a static method and in those cases a null reference exception would be wrong, which is a great point, and actually one of my concerns, not sure why I forgot to mention that in the question in the first place.   Someone also pointed out that it's wrong to throw a NullReferenceException, and yes, it is. That's why I don't throw it, I just let it happen (let the CLR throw it) by not guarding the method.  I think I favor the ArgumentNullException (that's what I've use so far) but I still think there is at least room to argue for an against the NullReferenceException since it seems more natural in most places where the method is going to be used. How can I determine whether a conference is reputable? I recently received an invitation to a conference and I'm trying to determine how reputable it is. Is there a good way to go about this? What is the view on accounts shared by multiple users on Stack Overflow? It has been my understanding that accounts are suppose to be for a specific user. This user, , is a company: . No doubt, this account has provided some quality answers. However, if a specific user cannot be identified, I don't think that the rep gain, rewards, or badges should be given. This should somehow be tagged as a 'Corporate User'. The rep has no real value when you can't be somewhat sure that the same person is giving the answers all the time.  This is where I found an answer by Devart: , peddling a product.    Edit 2 Good points have been brought up and most I can agree with. But we are setting a precedent. Meta Fudgey brought up a good point about the account's ability to use mod tools responsibly and consistently. Maybe this feature is not available to an account that appears to be an umbrella account unless a single user is identified. I think it is important to consider this aspect.    Let me re-iterate: the user(s) has provided excellent content and I would want that to continue. However, there is more at stake here than meaningful content and helpful resources.      Edit The view on 'shared' accounts wasn't so bright a year ago. For most of us, we have information in our profiles that identifies us as individuals. No, I don't have absolute knowledge of the activities on this account and how many people might use it. Meh, for all we know it could be a knowledgeable programmer with some good PR sense. On the other hand, it could be a PR person with no programming knowledge getting the answers from an employee. Who knows... Is conservation of momentum and energy valid for non-inertial frames? How to get width of the video and scale another video to that width using ffmpeg? I'm using ffmpeg to stack two videos vertically. They might be different size so I need to scale them to same size. I have found following commands.  First one works for different sizes (but won't scale to proper size):  ffmpeg -i input1 -i input2 -filter_complex '[0:v]pad=iw:ih*2[int];[int][1:v]overlay=0:H/2[vid]' -map [vid] -c:v libx264 -crf 23 -preset veryfast output   Second one works if videos have same width and same pixel format:  ffmpeg -i input1 -i input2 -filter_complex vstack output   How to get width of first input inside complex filter argument so second input could be scaled to same width? What would be the command to do so? I would prefer vstack filter since it should be faster than combination of pad and overlay. This is the terminal message I keep getting when trying to update via: sudo apt-get update  $ sudo apt-get update [sudo] password for petros: Reading package lists... Done E: Could not get lock /var/lib/apt/lists/lock - open (11: Resource temporarily unavailable) E: Unable to lock directory /var/lib/apt/lists/   How can I solve it? Definitionally, we have that  $$ \int_\gamma f\ \overline{dz} = \overline{\int_\gamma \overline{f}\ dz} $$  Now let $\int_\gamma f\ dz = w = x +yi$.  Question 1: Is it not the case that $\int_\gamma \overline{f}\ dz = \bar{w} = x - iy$?  Question 2:  Assuming that $\int_\gamma \overline{f}\ dz = \bar{w} = x - iy$, then is it not the case that  $$ \overline{\int_\gamma \overline{f}\ dz} = \overline{x - iy} = x+iy = \int_\gamma f\ dz? $$  Basically, I'm failing to see how $\overline{\int_\gamma \overline{f}\ dz}$ isn't just $\int_\gamma f\ dz$. I'm from Western Pennsylvania. Until I moved away, I never realized that when I omitted the to be from phrases like needs to be cleaned, my usage was different than what most English speakers are accustomed to. Is it wrong? Calculating the longest distance within polygon in QGIS I want to calculate the "diameter" of polygons in QGIS.  The "diameter" defined as the distance between the two most distant points of the polygon.  I couldn't find a fitting solution in the field calculator. Maybe you have some ideas? Java in Eclipse: Where do I put files on the filesystem that I want to load using getResource? (e.g. images for an ImageIcon) I know the file needs to be where the getClass().getResource(filename) can find it, but I don't know where that is.  I'm interested both in where to put the files on the filesystem itself, and how to go about using Eclipse's functionality to set up the resources. I need to prove that:      if a linear operator $\phi : V \rightarrow V$ on a vector space is nilpotent, then its only eigenvalue is $0$.   I know how to prove that this for a nilpotent matrix, but I'm not sure in the case of an operator. How would I be able to relate $\phi$ to a matrix?  Note: A nilpotent operator $\phi$ has been defined as an operator that satisfies $\phi^{n} = 0$ for some $n \geq 1$. As you know, we can enable the spell checker on single line text boxes in the following way:    We have to enable it every time for checking words.  How can we enable it forever, so it doesn't need to be enabled each time? In chapter 2, the section on bitwise operators (section 2.9), I'm having trouble understanding how one of the sample methods works.  Here's the method provided:  unsigned int getbits(unsigned int x, int p, int n) {     return (x &gt;&gt; (p + 1 - n)) &amp; ~(~0 &lt;&lt; n); }   The idea is that, for the given number x, it will return the n bits starting at position p, counting from the right (with the farthest right bit being position 0). Given the following main() method:  int main(void) {     int x = 0xF994, p = 4, n = 3;     int z = getbits(x, p, n);     printf("getbits(%u (%x), %d, %d) = %u (%X)\n", x, x, p, n, z, z);      return 0; }   The output is:     getbits(63892 (f994), 4, 3) = 5 (5)   I get portions of this, but am having trouble with the "big picture," mostly because of the bits (no pun intended) that I don't understand.  The part I'm specifically having issues with is the complements piece: ~(~0 &lt;&lt; n). I think I get the first part, dealing with x; it's this part (and then the mask) that I'm struggling with -- and how it all comes together to actually retrieve those bits. (Which I've verified it is doing, both with code and checking my results using calc.exe -- thank God it has a binary view!)  Any help? Modular exponentiation by repeated squaring Consider equations defined in $D_x = (0,\infty) \times (0,\infty)$  1) Find the maximal solution to  $$ \dot y = \frac{3t^2}{5y^4} $$ satisfying the initial value problem $y(1) = 2^{\frac{1}{5}}$  I have found the maximum solution to be $((0,\infty), (t^3+1)^\frac{1}{5})$  2) Find the maximal solution to   $$ \dot y = -\frac{2y}{5t}+\frac{3}{5y^4}$$  satisfying the IVP $y(1) = 2^{\frac{1}{5}}$.  (HINT: Use a transformation of the form $t^{\alpha}y(t)$.   I hope someone can explain how to use the transformation to find the maximal solution. C# Structs: Unassigned local variable? These may be separate questions, but the following MWE:    \usepackage{pdfpages} ...   \addcontentsline{lot}{subsection}{\ref{tab:UserRoles} \textbf{User Roles}}   \includepdf[pages={-}, pagecommand={\label{tab:UserRoles}}, offset=0in -1in, addtolist={9, table, {The User Roles table}, tab:UserRoles}]{UserRolesTable}   ...   has several unexpected behaviors.   I get the following warning:  Package pdfpages Warning: There's something wrong with the entries of `addtolist'. Not all entries were processed. Check `addtolist' [in the MWE]."    Probably due to the \label not being bound to the inserted pdf (in a table environment), my inserted PDF is numbered using the section, and not the expected "Table X" value.   The manually-inserted LoT line (which also uses the section number rather than the incremented table number) is not formatted like the other lines in the LoT - the number lines up, but the table caption itself is not aligned with the others in the list.   The source PDF has no errors and displays correctly in the output PDF.  PDFLaTeX -> BibTex    MWE:  \documentclass[11pt,letterpaper,twoside,final]{article} \usepackage{pdfpages} \begin{document} \listoftables \begin{center} \textbf{Table \ref{tab:UserRoles} - The User Roles Table} \addcontentsline{lot}{subsection}{\ref{tab:UserRoles}  \textbf{User Roles}} % manual entry in ToC  \includepdf[pages={-}, pagecommand={\label{tab:UserRoles}}, offset=0in -1in, addtolist={9, table, {The User Roles table}, tab:UserRoles, lot}]{UserRolesTable} \end{center} \end{document} folder content doesn't get refreshed automatically in ubuntu 14.04 I'm using Ubuntu 14.04 and when I'm doing experiments which creates new files inside a particular folder X. But I can't see these new files inside X without refreshing X.  I have increased the count to 1M in  /proc/sys/fs/inotify/max_user_watches   but still the issue is not fixed.   How to fix it?  EDIT As suggested, output of cat /etc/xdg/autostart/user-dirs-update-gtk.desktop  [Desktop Entry] Encoding=UTF-8 Exec=xdg-user-dirs-gtk-update Name=User folders update Comment=Update common folders names to match current locale Terminal=false OnlyShowIn=GNOME;LXDE;Unity; Type=Application StartupNotify=false X-KDE-autostart-after=panel NoDisplay=true X-Ubuntu-Gettext-Domain=xdg-user-dirs-gtk What does the phrase "begging the question" really mean? And does it even matter if I use it correctly? Almost everyone just uses it as a synonym for "posing the question" these days. Proof that $V = \sum_k U_k$ and $\dim V=\sum_k \dim U_k$ implies $V=\oplus_k U_k$