Argument for the Smallest Division of Time? Okay so a couple of months ago I was watch Vi Hart's video on how .999999... is equal to 1.  Some really interesting arguments that I had never heard before.  Yesterday, I came across an article on Planck time, and it got me thinking about where those two ideas might work together.   So ignoring that we already know about Planck time (and the other units), could we come up with an argument/proof showing that there must be a smallest, discrete unit of time using the same methods that we can use to prove that .99999... is equal to 1, only in reverse?  Yes, I've already googled this, but I could only fight people talking about the idea, not actually doing it.
Does the Planck scale imply that spacetime is discrete? On a quantum scale the smallest unit is the , which is a discrete measure.  There several question that come to mind:   Does that mean that particles can only live in a discrete grid-like structure,  i.e. have to "magically" jump from one pocket to the next? But where are they in between? Does that even give rise to the old paradox that movement as such is impossible (e.g. )? Does the same hold true  for time (i.e. that it is discrete) - with all the ensuing paradoxes? Mathematically does it mean that you have to use difference equations instead of differential equations? (And sums instead of integrals?) From the point of view of the space metric do you have to use a discrete metric (e.g. the ) instead of good old Pythagoras?   Thank you for giving me some answers and/or references where I can turn to.  Update: I just saw this call for papers - it seems to be quite a topic after all: Is Reality Digital or Analog? FQXi Essay Contest, 2011. , , . One can find some pretty amazing papers over there.
Find $\lim \limits_{n \to \infty}{1*4*7*\dots(3n+1) \over 2*5*8* \dots (3n+2)}$ I am trying to find  $$\lim \limits_{n \to \infty}{1*4*7*\dots(3n+1) \over 2*5*8* \dots (3n+2)}$$  My first guess is to look at the reciprocal and isolate factors:  $${2 \over 1}{5 \over 4}{8 \over 7} \dots {3n+2 \over 3n+1}= {\left(1+1\right)}\left(1+{1 \over 4}\right)\left(1+{1 \over 7}\right) \dots \left(1+{1 \over 3n+1}\right)$$ Now we take the natural log: $$\ln{ + \ln\left(1+1\right)} + \ln\left(1+{1 \over 4}\right) +\ln\left(1+{1 \over 7}\right) \dots +\ln\left(1+{1 \over 3n+1}\right)$$ Using $\ln(1+x) \le x$, we get: $$\ln{ + \ln\left(1+1\right)} + \ln\left(1+{1 \over 4}\right) +\ln\left(1+{1 \over 7}\right) \dots +\ln\left(1+{1 \over 3n+1}\right) \le 1 + {1 \over 4} + {1 \over 7} + \dots + {1 \over 3n+1}$$ Now, I'm stuck. I suppose I might use the fact that the RHS is similar to the harmonic series and show that it converges to some log, but I'm not sure how to do that. Do you have any clues? I know it has been discussed that it is not possible to start an application from an usb drive upon connection due to limitations of autorun (or autoplay??) in Win 7. But it is possible to create a scheduled task having an event type trigger. Surely there must be an event occurring when the drive -- or any USB device, for that matter -- is connected.    Does anybody have the slightest idea which Event ID should I use? Or at least what type of event? Where can I find the event in event viewer? Make display of reputation in revisions list consistent with the user card display The user card in the revision list, for example , shows the exact amount of reputation:  This is not consistent with how the flair works on (all?) other places in the site, e.g. under a post. There, the amount is displayed in units of thousands (k):  (or, in , millions (m).) It used to be consistent, see e.g. : What can I do if I forgot my Windows password? if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian Prove that if $R$ is a commutative ring in which all the prime ideals are finitely generated, then $R$ is Noetherian.   Here is what I been told to do: Suppose that $R$ is not Noetherian, and use Zorn’s lemma to obtain a maximal element $I$ in the collection of all ideals of $R$ that are not finitely generated.  Then use the following proposition: Let $I$ be an ideal of a commutative ring $R$, and let $r ∈ R$. If the ideals $I +rR$ and {$s ∈ R : sr ∈ I$} are finitely generated, then $I$ is a finitely generated ideal.  Can anyone help please. thanks a lot. Two kinds of Duplicate closing are too confusing - suggestion for a possible fix In Gmail, what is the safest way to download, save & print all emails with a given label? Attempting to securely download all the emails with a given label in Gmail, save them to a single file, and print that file. Is there any way to hide taskbar labels and not combine taskbar buttons? There are three options for customizing taskbar buttons in Windows 7:   Always combine, hide labels Combine when taskbar is full Never combine   Is there any way to hide labels and not combine taskbar buttons? Or even better, to exclude only certain applications from combining?  My main problem is that I switch a lot between two instances of a single application, and having to hover over combined icon and then choosing the correct one. Alt-Tabbing is not a good alternative when I'm switching between more than three applications; clicking on taskbar only once is what I'm looking for. Limits, find a limit that exist in absolute value but not outside the absolute value. Let I be an open interval that contains the point c and let f be a function that is defined on I except possibly at the point c. Suppose that lim |f(x)| as x->c exists. Give an example to show that lim f(x) as x->c may not exist.   Not really sure of an example for this? Term for group of guys attempting to show who is alpha? Similar to pissing contest - but not in an aggressive manner. Just each guy attempting to out talk one another to show they are the leader / in charge.   Sample sentence:     Well, that meeting was more of a X than being productive with Greg trying to show he was in control. Is there a conventional wisdom to what gear a bike should be left in, when used infrequently? Was Tracy Mills a part of John Doe's plan from the beginning? In , John Doe is shown as being extremely meticulous and patient while carrying out his plan of murdering and/or torturing people in accordance with the seven deadly sins.  At the end of the film, it is revealed that John Doe had killed and decapitated Tracy Mills, the wife of Detective David Mills.  Was Tracy Mills his intended target from the beginning?  Did John Doe decide to kill Tracy only after Detective Mills and Somerset discovered his residence? Link to the recommended job is broken in the mobile apps How to handle a deliberate duplicate question I got  question in review. First sentence is:     I know there is a  question, but no answers.   And indeed the two questions are virtually identical. I started to flag as duplicate - but the original question had no answer at the time, so it wouldn't be correct and apparently . (I didn't go so far as to try to submit the flag).  What is the proper action here? How do we want such a question to be handled? Should both (or more) questions stay open indefinitely until either is answered and somebody gets the good idea to do the duplicate linking?   As the new question is essentially a bump to the front page (perhaps with re-tagging), especially in the case where the OP knows that a previous question exist, perhaps something should be done to enable bumping re-emphasizing without writing the new question.  As I wrote this,  actually got an answer. So the old one should now be closed as a duplicate of the new - but my question remains for the case where all known/found duplicate questions have no answer?  Ironically, just before asking this, the "Questions that may already have your answer" gave me ... (Although the suggestion has no answer). So it has been asked before - the comment there even suggest the ability to re-emphasize. LINQ's Distinct() on a particular property I am playing with LINQ to learn about it, but I can't figure out how to use Distinct when I do not have a simple list (a simple list of integers is pretty easy to do, this is not the question). What I if want to use  on a list of an Object on one or more properties of the object?  Example: If an object is Person, with Property Id. How can I get all Person and use Distinct on them with the property Id of the object?  Person1: Id=1, Name="Test1" Person2: Id=1, Name="Test1" Person3: Id=2, Name="Test2"   How can I get just Person1 and Person3? Is that possible?  If it's not possible with LINQ, what would be the best way to have a list of Person depending on some of its properties in .NET 3.5? Is there a way to indicate to  how/where to find (paths to)  tools such as python, nmap, php and so on?  I ask this because when I schedule a job using crontab -e, I always need to specify the full path of python, nmap and whatever other tool I want to use. \end{abstract} conflicts with arabtext package I am using elsarticle  Everything was find before I used \usepackage{arabtext}.   When I compiled to PDF, I got the following error message.    And if I type "s", everything became fine again but I don't want to type an "s" every time I compile PDF preview.   Please anyone advise.  Thank you very much in advance for your help.  Here is the code  \documentclass[12pt, authoryear]{elsarticle}    % use "amsart" instead of "article" for AMSLaTeX format \usepackage[colorlinks=true,linkcolor=black, citecolor=blue, urlcolor=blue]{hyperref} \usepackage{geometry}                       % See geometry.pdf to learn the layout options. There are lots. \geometry{a4paper}                          % ... or a4paper or a5paper or ...  %\geometry{landscape}                       % Activate for rotated page geometry %\usepackage[parfill]{parskip}          % Activate to begin paragraphs with an empty line rather than an indent \usepackage{graphicx}               % Use pdf, png, jpg, or epsÂ§ with pdflatex; use eps in DVI mode                             % TeX will automatically convert eps --&gt; pdf in pdflatex         \usepackage{amssymb}  \usepackage{amsmath} \usepackage{tipa} \usepackage{arabtex} %SetFonts  %SetFonts   \title{Translating from English to Bahasa Indonesia: Which procedures should be taught to EFL students preparing to be a translator?} \author[mkl]{Mohammad Kholid}  \author[fai]{Faisal Mustafa\corref{cor1}} \ead{faisal.mustafa@unsyiah.ac.id}  \cortext[cor1]{Corresponding author}  \address[mkl]{Syiah Kuala University, Banda Aceh, Indonesia} \address[fai]{Syiah Kuala University, Banda Aceh, Indonesia} \date{}                         % Activate to display a given date or no date  \begin{document} %\section{} %\subsection{}  \begin{frontmatter}  \begin{abstract}   Abstract content is here.  \end{abstract}  \begin{keyword}  translation procedures \sep English to Bahasa Indonesia \sep teaching translation \end{keyword} \end{frontmatter} How do you set the expiry date or a maximum age in the HTTP headers for static resources in IIS What is the plural form of "Software"? Displaying attachments on visual force page I need to display attachments on a visual force page for the Case object.  This is the code I am using. I had it working for awhile, but it somehow broke down. It gets all the attachments, but it will not show the images.   &lt;apex:repeat value="{!Case.attachments}" var="att"&gt;             &lt;apex:panelgrid columns="1"&gt;                 &lt;h1&gt;Image/h1&gt;                 &lt;apex:image url="/servlet/servlet.FileDownload?file={!att.Id}"/&gt;             &lt;/apex:panelgrid&gt; &lt;/apex:repeat&gt;