In his Tractatus logico-philosophicus, Wittgenstein uses an unusual numbering system for paragraphs, here's how he describes it.     The decimal figures as numbers of the separate propositions indicate the logical importance of the propositions, the emphasis laid upon them in my exposition. The propositions n.1, n.2, n.3, etc., are comments on proposition No. n; the propositions n.m1, n.m2, etc., are comments on the proposition No. n.m; and so on.   Example    Ideas  I see two ways of designing such architecture. The first is the one proposed by the author of the plain Tex question that I cite below (that uses up and down). The other is to create a proposition environment.  \begin{proposition}     Proposition 1     \begin{proposition}         Proposition 1.1     \end{proposition}     \begin{proposition}         Proposition 1.2         \begin{proposition}              Proposition 1.21         \end{proposition}     \end{proposition} \end{proposition} \begin{proposition}[0]     Proposition 2     \begin{proposition}         Proposition 2.0         \begin{proposition}              Proposition 2.01         \end{proposition}     \end{proposition} \end{proposition}   Notes   I don't mind about working with commands or environments, I'm just looking for a practical solution. I am aware that  exist but I guess the use of LaTeX can simplify macros.
How can I keep track of a decimal number that indicates the current paragraph's place in a logical hierarchy and print it in the margin of each paragraph?  (To yield a numbering in the style of Tractatus Logico-Philosophicus).  \def\on{} \def\down{} \def\up{}  \on This is the first paragraph, paragraph number 1.  \on This is paragraph number 2.  \down This is paragraph number 2.1  \down 2 This is 2.101  \up This is 2.11  \down 2.111  \up 3 3  \bye   If \down is starred, begin with zero.  If \up is followed by an integer, it should take it as an argument, unless there is a newline between \up and the integer.  An indefinite number of decimals should be possible.  PS.  I apologize for changing the question after (2) answers have been given, but I realize I failed to appreciate a feature of Wittgenstein's numbering: the last decimal is never 0.  Therefore, \down &lt;n&gt; is better than \down*, as now demonstrated above.  I think the 0 is supposed to indicate an additional level of unimportance.
ls: cannot access .gvfs: Permission denied Round function in Excel, worksheet function vs VBA What was the inspiration for the design of R2-D2? Looking at R2-D2, I can't think of any robots I've seen in science fiction that vaguely resemble R2D2's design.  What was the inspiration for the design of R2-D2? Show that $S + T$ and $aT$ are linear Let  $S:  V \to W$ and $T: V \to W$ be linear transformations. Given $a$ in $\mathbb{R}$, define functions $(S+T): V \to W$ and $(aT): V \to W$ by $(S+T)(v)=S(v) + T(v)$ and $(aT)(v)= aT(v)$ for all $v$ in $V$. Show that $S+T$ and $aT$ are linear transformations. How to check if list item already exist when adding it with how to check Title( Country) if exists in the list. if exists it is necessary to do the update if not it is necessary to add it ?  function AddListItem() {      var listTitle = "Test";     //Get the current client context       context = SP.ClientContext.get_current();     var airportList = context.get_web().get_lists().getByTitle(listTitle);          //Create a new record       var listItemCreationInformation = new SP.ListItemCreationInformation();     var listItem = airportList.addItem(listItemCreationInformation);      //Set the values      var country= $("#Country").val();     var client= $("#Client").val();     // var nb1 = $("#nb1").val();      // var nb2 = $("#nb2").val();       // var nb3 = $("#nb3").val();      // var nb4 = $("#nb4").val();       // // var nb5 = $("#nb5").val();        listItem.set_item('Country', Country);     listItem.set_item('Client', Client);     listItem.update();     context.load(listItem);     context.executeQueryAsync(AddListItemSucceeded, AddListItemFailed); } function AddListItemSucceeded() { } function AddListItemFailed(sender, args) {     alert('Request failed. ' + args.get_message() + '\n' + args.get_stackTrace()); } How to enable crash reports/core dumps/stack trace logging globally? Crashing bugs can be the most annoying ones, leading to data loss, downtime and frustrated users. It would be good if applications crashed less.  Because of the complexity of the machine context, crashes often can't be reproduced in reasonable time for an ordinary user. That doesn't mean that the bug is rare - It could simply mean that the thing that triggers it occurs rarely for each user (for example DST changes). Such bugs are unlikely to be fixed unless a lot of users report them. It would be good if more crashes were reported.  To debug crashes, the developers need as much unambiguous context as possible. Generated crash reports are good, because they're usually detailed and accurate. Users can't be expected to zealously observe and report all the context manually, so they often submit sparse and wrong information.  The target audience for a lot of applications is not developers or sysadmins, but rather the general public, at home or at work. Such users can't be expected to know how to collect crash information manually, or to install -dbg packages, but generated reports from such users can still be usable. Some applications have their own crash reporting tools, but in my experience these rarely work, and when they report that they failed to report the error there doesn't seem to be any information on how to do it manually (I've observed this for recent versions of both Firefox and Flash). System-wide generation of crash reports would be good.  Is there any sort of crash report generation* which can be enabled globally** without installing a ton of -dbg packages, reading the documentation of each and every application, or slowing a normal machine down to a crawl?  * Logs, core dumps, stack traces, whatever  ** Not necessarily for init, but at least for a significant subset of the applications running on a typical desktop Linux installation. In my experience GUI applications crash 100+ times more frequently than shell applications, so GUI applications would naturally be the focus. Prove that a limit doesn't exist In my assignment I have to prove that the following function doesn't have a limit:  $$f(x)=\frac{x}{x-\lfloor{\sin x}\rfloor}$$ when $${x \to 0}$$  In other words, I have to prove that for every $\delta$ such that $0&lt;|x-{x_0}|&lt;\delta,$ $|f(x)-L|&gt;\epsilon$.  In fact, I don't even know how to start. Do I choose my $\delta$? I know that the sin function has certain values that's supposed to help me. How do I use it?  I have to prove this by definition, so I can't use limit arithmetic for example  Thanks. Maximum Champion Growth size I was reading about Lulu (new champion) and saw this:     Wild Growth (Ultimate) - Lulu enlarges an ally, knocking enemies away   from them and granting them a large amount of bonus health. For the   next few seconds, that ally gains an aura that slows nearby enemies.   Then I start wondering: what is the max size that a champion reach ? Is there a Maximum size ?  Example: Cho'Gath's Feast +  +  + Wild Growth + Something that make champions grow in size. Cho'Gath will grow every time ? or it will stop at some point ? Is there any way to erase personal information from my phone that has already been stolen? My phone has been stolen. I do not want the thief to have access to any of my personal information such as calls, texts, and information saved in the phones contents. Is there any possible way I can get this information erased from my phone? How to copy only file attributes (metadata) without actual content of the file? How do you get a submit button in the Windows Phone keyboard? Between vs. Among - "Difference [between | among] one thing and other things." Consider the following two versions of a question:1   Version 1: What is the difference in meaning between "[to be] not invited" and similar negation forms?  Version 2: What is the difference in meaning among "[to be] not invited" and similar negated forms?    Personally, I think Version 1 might subtly suggest a Star Topology (see below) with the target sentence "[to be] not invited"at the center.  And Version 2 might subtly suggest a more free-form comparison like the Mesh or Fully Connected, with the target sentence simply "one among many":   Between approx equal "Star" and among approx equal "Fully Connected"?  Don't get me wrong: I do think this is quite subtle, not always applicable, and I don't think anyone would intuit such a "network topology" meaning without further explanation.   So the main question is, are both grammatically correct?  If yes, is there a semantic difference to choose one over another, perhaps as I suggested?  Or does the plurality aspect of between and among trump other semantic and stylistic issues?    1. In my question , another user provided a very helpful edit. However, the change of the word between to among prompted this question. Is there a convergent, alternating series that fails the AST? The alternating series test (AST) says, briefly, that if   $a_k&gt;0$ $a_k \geq a_{k+1}$, and $a_k \to 0$ as $k \to \infty$   then $\sum_k (-1)^k a_k$ converges.  This seems to be a one-way test (that is, if an alternating series fails the test, we don't know that it diverges).   document says so explicitly. It then gives an example of an alternating series which fails the AST. That doesn't prove the series is divergent (but it turns out to be divergent, anyway, by $n$th term test).  Is there a convergent, alternating series that fails the AST?  Of course, if the series fails condition 3, then it also fails the $n$th term test, and must diverge. And if it fails the first condition, then it's not strictly alternating, anyway.  So it must be a series that is not getting smaller (condition 2), but still converges. Can I switch the alt and ctrl keys on my keyboard? Change default saving folder Implementing INotifyPropertyChanged - does a better way exist? I would like to prove inductively that $${2n\choose n}=\sum_{i=0}^n{n\choose i}^2.$$  I know a couple of non-inductive proofs, but I can't do it this way. The inductive step eludes me. I tried naively things like $${2n+2\choose n+1}={2n+2\over n+1}{2n+1\choose n}=2\cdot {2n+1\over n+1}{2n\choose n},$$  But I don't think it can lead me anywhere. I would like the proof to be as simple as possible. Problems when adding a fillable PDF form if the page is rotated To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$ My question is, how would I make the visible HDRI/Enviroment Map be image 1 and then have reflections as Image 2? If someone can post a node setup, or a .py file that would be great &lt;3 I was thinking maybe using the Is Camera Ray node but I'm not confident it would work, and have no idea where to start.