Why is this an incomplete residue system?
Triangular Numbers Modulo $k$ - Hit All Values?
Definition of Statistic I keep seeing conflicting definitions of a statistic. Is a statistic a random variable such that it is a function of the random variables of a random sample? Or is it the value of the function of the random sample, after each random variable of the random sample has taken on a specific value?   $$(1) \ S=f(X_1,X_2...X_n)$$  $$(2) \ s=f(x_1,x_2...x_n)$$  I haven't been able to get any clarification for this and I've seen the term statistic describe both situations How to customise GNOME default background and lock screen image? I am building a Custom ISO of Ubuntu 18.04 LTS using CUBIC and I'd like to change the default background and lock screen image.  I added the backgrounds in /usr/share/backgrounds, changed the background in /usr/share/gnome-background-properties/ubuntu-wallpapers.xml file, but it does not seems to affect new users (for example the OEM user).  Also, I did not find any way to change the lock screen image. How to change desktop background from command line in Unity? I have a script that updates my desktop background every day with the NASA APOD. Since upgrading to 11.10, that script stopped working, presumably because I'm using gconftool-2 to actually update the desktop background. Is there a new way to do this under Unity? Can I have multiple primary keys in a single table? Can I have multiple primary keys in a single table? Do I need an airport transit visa for Dublin, Ireland en route to the UK? Adding coordinate system to layout in QGIS composer? Does the Rule of Two Serve any Actual Purpose? I know Darth Bane came up with the Rule of Two to ensure the survival of the Sith, but, in reality (or in the Star Wars reality), does it (or any Sith rule) serve a true purpose?  And why does each successive Sith bother with it?  Due to the nature of the Sith, Sith Lords tend toward arrogance and self-importance.  Both the Sith master and Sith apprentice seem to regularly break the rule by training others.  Deceit and personal interest over loyalty is a way of life for the Sith.  What is to keep any Sith from following the Rule of Two (or any rule, for that matter) if it is against their own self interest?  If a Sith Master can stay in power longer by training a 2nd apprentice and pitting the two against each other, is he really going to worry about the Rule of Two more than his own plans and survival?  The Rule of Two also assumes that every Sith Master would place the survival of the Sith over his own survival, as opposed to attempting to ensure his own immortality in whatever way he could (like Plagueis).  When you have a group that, by default, is arrogant and self-serving, as well as deceptive, why would they want the Sith to survive themselves?  I can see how the Rule of Two, and any other Sith beliefs would be possible guidelines, but it terms of the nature of ambitious people (and the Sith are ambitious), it's hard to believe that each new Sith Master would follow this rule faithfully.  So does this rule, or any Sith rule, actually work?  Or is it more of a suggestion that is broken when convenient? Database schema documentation for the public data dump and SEDE I tried to find an answer to this a few years ago, but it seemed it wasn't possible then (I suspect it isn't now either! - but what the hell!)   I have a three monitor system, that I mostly use for remote desktop. However occasionally it would be useful to have remote desktop on two monitors and use the other monitor for the local machine.   I know it is possible to have a single monitor full screen, or to stretch an RDP session across monitors, but this loses things like snapping etc.   P.S. this is a duplicate of  thread, but that was a couple of years ago, I'm hoping things have changed. Easy way to switch default sound output device Load CSS/Javascript in frontend conditionally if block is used Assuming I have no infinite scroll or anything else going on in my theme: Is there a way to enqueue my custom Gutenberg block styles and scripts so they are only (lazy) loaded in the front-end when the block is really used? Something like is_active_widget, just for blocks?  If I enqueue them inside a enqueue_block_assets function they're added to any site.  On github, I found this  , but it sounds like it's more on bundling than conditional load, so I still hope they didn't leave that optimization opportunity out - otherwise any other site will soon be junked with 20 additional scripts from numerous Gutenberg block plugins people install and only use once on a page (think of large image gallery block scripts for example).  Thanks &amp; Regards! I know that, whenever I want to see how some control sequence, say \mycontrolsequence, is defined, I can use \show\mycontrolsequence  then compile my code and look up the .log file. However, robust commands seem impervious to \show. For instance, if I compile the following code, \documentclass{article}  \newcommand\mycommand{foo} \show\mycommand  \DeclareRobustCommand{\myrobustcommand}{bar} \show\myrobustcommand  \begin{document} hello \end{document}  my log file contains  \mycommand=\long macro: -&gt;foo. l.4 \show\mycommand  \myrobustcommand=macro: -&gt;\protect \myrobustcommand  . l.7 \show\myrobustcommand   Note that, whereas the definition of \mycommand is exposed (foo), nothing transpires of \myrobustcommand's definition. Of course, in practice, I could always identify in which file a given robust command is defined and look up its definition there. However, finding the file in question may not be obvious. Hence my question: Is there an alternative way of peering into the definition of a robust command? $\|u\|\leq \|u+av\| \Longrightarrow \langle u,v\rangle=0$ Prove that $\langle u,v\rangle=0\Longleftrightarrow \|u\|\leq \|u+av\|$.  So far I can get the $\Longrightarrow$ very easily, but I need some help with the $\Longleftarrow$ implication, any hints would be greatly appreciated.  Please no answers, I just want a small nudge to get me in the right direction, I've been stumped on this for a little bit.  Here's what I have so far:  ($\Longleftarrow$)  Assume $\|u\| \leq \|u+av\|$  $\|u+av\|^2=\langle u+av,u+av\rangle=\langle u,u\rangle+ \langle u,av\rangle+\langle av,u\rangle+\langle v,v\rangle=\|u\|^2+\bar{a} \langle u,v\rangle+a\langle v,u\rangle+\|v\|^2=\|u\|^2+ \bar{a}\bar{\langle v,u\rangle}+a\langle v,u\rangle+\|v\|^2$  Thoughts/ideas? Reference book on measure theory I post this question with some personal specifications. I hope it does not overlap with old posted questions.  Recently I strongly feel that I have to review the knowledge of measure theory for the sake of starting my thesis.  I am not totally new with measure theory, since I have taken and past one course at the graduate level. Unfortunately, because the lecturer was not so good at teaching, I followed the course by self-study. Now I feel that all the knowledge has gone after the exam and still don’t have a clear overview on the structure of measure theory. And here come my specified requirements for a reference book.   I wish the book elaborates the proofs, since I will read it on my own again, sadly. And this is the most important criterion for the book. I wish the book covers most of the topics in measure theory. Although the topic of my thesis is on stochastic integration, I do want to review measure theory at a more general level, which means it could emphasize on both aspects of analysis and probability. If such a condition cannot be achieved, I'd like to more focus on probability. I wish the book could deal with convergences and uniform integrability carefully, as Chung’s probability book.    My expectation is after thorough reading, I could have strong background to start a thesis on stochastic integration at an analytic level.  Sorry for such a tedious question.  P.S: the textbook I used is Schilling’s book: measures, integrals and martingales. It is a pretty good textbook, but misprints really ruin the fun of reading. How to start Skype 4 with Ubuntu 12.10? I've installed Skype and it's running all OK but I want it to start automatically when I log with my user.  How can I do that? What is recordSetVar and where is it used? I was looking at some code and in the 'apex:page' tag, an attribute called recordSetVar was mentioned. I'm new to VisualForce so could someone please tell me when and why is it used? I wanna transfer some files from this PC with Ubuntu to a PC with Windows 7 I don't care how to do it (ftp, samba or whatever) I wanna know the steps, I searched a lot but I can't find anything that works. Thank you. KeyListener not working Natural (human alpha-numeric) sort in Microsoft SQL 2005 We have a large database on which we have DB side pagination. This is quick, returning a page of 50 rows from millions of records in a small fraction of a second.  Users can define their own sort, basically choosing what column to sort by.  Columns are dynamic - some have numeric values, some dates and some text.  While most sort as expected text sorts in a dumb way. Well, I say dumb, it makes sense to computers, but frustrates users.  For instance, sorting by a string record id gives something like:  rec1 rec10 rec14 rec2 rec20 rec3 rec4   ...and so on.  I want this to take account of the number, so:  rec1 rec2 rec3 rec4 rec10 rec14 rec20   I can't control the input (otherwise I'd just format in leading 000s) and I can't rely on a single format - some are things like "{alpha code}-{dept code}-{rec id}".  I know a few ways to do this in C#, but can't pull down all the records to sort them, as that would be to slow.  Does anyone know a way to quickly apply a natural sort in Sql server?    We're using:  ROW_NUMBER() over (order by {field name} asc)   And then we're paging by that.  We can add triggers, although we wouldn't. All their input is parametrised and the like, but I can't change the format - if they put in "rec2" and "rec10" they expect them to be returned just like that, and in natural order.    We have valid user input that follows different formats for different clients.  One might go rec1, rec2, rec3, ... rec100, rec101  While another might go: grp1rec1, grp1rec2, ... grp20rec300, grp20rec301  When I say we can't control the input I mean that we can't force users to change these standards - they have a value like grp1rec1 and I can't reformat it as grp01rec001, as that would be changing something used for lookups and linking to external systems.  These formats vary a lot, but are often mixtures of letters and numbers.  Sorting these in C# is easy - just break it up into { "grp", 20, "rec", 301 } and then compare sequence values in turn.  However there may be millions of records and the data is paged, I need the sort to be done on the SQL server.  SQL server sorts by value, not comparison - in C# I can split the values out to compare, but in SQL I need some logic that (very quickly) gets a single value that consistently sorts.  @moebius - your answer might work, but it does feel like an ugly compromise to add a sort-key for all these text values. what are all the open subgroups of $(\mathbb{R},+)$