The interior of interior of a set is its interior
Don't understand proof that interior of a set is open
I'm very new to latex, and I've noticed that using a book class causes the text on one page to be on the right and then left on the next and so on. I'd like for it to all to be on the left or centered. How can I change this? How is the width of a slit related to the intensity of light passing through it? Here's a question I got in my final exam this morning. &quot;If in Young's double-slit experiment setup, the ratio of the intensity of the bright spot to the dark spot is 25:9, what is the ratio of the width of the slits?&quot; Here's what I did. Since the ratio of intensity at the bright and dark spots is 25:9, the ratio of amplitudes there must 5:3. Which means the amplitude of one wave is 4 times the other. Now, knowing that the amplitude of light through the wider slit is 4 times the amplitude of light through the narrower slit, how can I determine the ratio of the slits' width? PlotLegends won't generate automatically more than 15 labels in v10 Is there any way to include an animated GIF directly in either PDFLaTeX or XeLaTeX? I realize the animate package can include animations in a PDF, but it doesn't support animated GIFs and you have to split them up manually into EPS or PNG files as far as I can tell. How can I create light trails without a blurry subject? I would like to achieve this effect using my Nikon P510:      Are there any suggestions or tips that would help me do that? Is there an easier or more 'correct' way to open gedit as root? I have ubuntu 12.04 running in a virtual machine on virtual box.  Its existence is mainly as a 'toy' to play about with Ubuntu, so for this reason I wanted the GUI (is that what 'Dash' is in ubuntu 12.04?  Anyway...)  One thing I am currently using this machine for is a LAMP server.     I use 'gedit' to edit my files, but because I haven't correctly set permissions yet I just open it with sudo in terminal.  Just now, to try to simplify this process I created a shell script on the desktop with a very simple job - sudo gedit  I can't help thinking there must be an even more obvious way to open gedit as root from the GUI than this.  So is there?  Edit:  I just 'improved' my script by replacing sudo with gksudo.  Now I don't have to have a terminal window open while gedit is open.   So I guess this is quite convenient for now :)   It would be good, for the sake of knowledge, to know if this is anything close to how the more seasoned ubuntu users would do this type of thing. How to split a large file into two parts, at a pattern?  Given an example file.txt:  ABC EFG XYZ HIJ KNL   I want to split this file at XYZ such that file1 contains lines up-to XYZ and rest of the lines in file2. I am using Sql Server 2008 on Small Business Server 2003; client is using WinXP.   I added a user to my Active Directory Security Group; Why can't this user immediately access the database? It seems there is a delay before the user is recognized in Sql Server.   I am using AD Security Groups for permissions expressly so that I don't need to add individual users in Sql Server. So effectively I don't need to do anything but add the user to the AD Security Group in order to grant access.  But for some reason Sql Server doesn't immediately recognize the addition. I've seen this a number of times. I add the user to the group but that user can't access data until the next day. It seems that it doesn't query the Active Directory in real time. Can you confirm that is so?  What do I need to do so that Sql Server "refreshes" the list of users from Active Directory? I'm having some problems with listings and UTF-8 in my document. Maybe someone can help me? Some characters work, like é and ó, but á and others appear at the beginning of words...  \documentclass[12pt,a4paper]{scrbook} \KOMAoptions{twoside=false,open=any,chapterprefix=on,parskip=full,fontsize=14pt}  \usepackage[portuguese]{babel} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{listingsutf8}  \usepackage{inconsolata} \lstset{     language=bash, %% Troque para PHP, C, Java, etc... bash é o padrão     basicstyle=\ttfamily\small,     numberstyle=\footnotesize,     numbers=left,     backgroundcolor=\color{gray!10},     frame=single,     tabsize=2,     rulecolor=\color{black!30},     title=\lstname,     escapeinside={\%*}{*)},     breaklines=true,     breakatwhitespace=true,     framextopmargin=2pt,     framexbottommargin=2pt,     extendedchars=false,     inputencoding=utf8 }  \begin{document} \begin{lstlisting} &lt;?php  echo 'Olá mundo!'; print 'Olá mundo!'; \end{lstlisting}  \end{document} \end{lstlisting} I have a shop created with drupal commerece modules,I enabled 2 currency in my shop , but only show one of them, I want to show price in 2 currency       something like this price :  10$  / 20 IRR I use multi currency module to achieve this job, but  I my problem is in payment  phase, I use multi currency module to achieve this job,in multicurrency  module's setting I don't want  use  synchronize method, if you want have two field in my product , one form $ and another for IRR, I did this job by this method     first enable IRR currency in  admin/commerce/config/currency  then in  admin/commerce/config/currency/handling under Enable dedicated price fields for Product check Iranian Rial  by doing this  another field added you my product ( ** Price for IRR** ), this is good , after this I enable  *currency selector * block ,in checking out process by switching the currency I don't see any change in current currency of billing and cart. :( why changing currency not changing cart currency? The relationship of ${\frak m+m=m}$ to AC Uniform continuity of $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0.$ I am just curious that the following format looks good to you or not? The preamble I used is just   \documentclass[12pt,a4paper]{article}  \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb}   I don't feel it's so good because the first paragraph has no indent but the second does. If the first paragraph is much longer, than it's fine. But if not, it looks strange. Should I leave it, or you have suggestions for this? Footnote to a footnote How can I make a footnote to a footnote? E.g.,  Playfair's axiom\footnote{so called in honor of mathematician John Playfair, etc, etc., but was actually discovered by Proclus\footnote{Greek mathematician, etc, etc.}} Why are angles dimensionless and quantities such as length not? So my friend asked me why angles are dimensionless, to which I replied that it's because they can be expressed as the ratio of two quantities -- lengths.   Ok so far, so good.  Then came the question: "In that sense even length is a ratio. Of length of given thing by length of 1 metre. So are lengths dimensionless?".   This confused me a bit, I didn't really have a good answer to give to that. His argument certainly seems to be valid, although I'm pretty sure I'm missing something crucial here. How do I get my players to form a PC party without just forcing them to? I believe that it is generally a good thing that all the PCs travel together, as a "party". This way the GM has to manage one environment – not, say, five. The problem is how the GM can ensure such a relationship between the characters.  One simple and really great solution is to require the players to design their characters so that they know each other. This has the added advantage of cross-backstories – as the players try to adjust their stories to fit together, they get and create hooks for more narrative!  However, I am now experimenting with giving the players complete freedom at character creation and later meeting in-game. No luck so far.  It is not easy, as trusting someone with your life (what happens during combat) is not a trivial thing. Furthermore, people generally like others with the same behaviour/interests/skin colour. Fantasy is all about diversity – you do not character-generate a strong man with a sword, no, you generate a half-ogre with an axe never before seen!  How can the GM get players, with diverse characters who are strangers to each other, decide to have their PCs act as a coherent group. I absolutely want to avoid asking my players to just solve this through metagaming ("Hey Joe, think of a way to get together with those, I can't story-tell two separate groups at the same time!").  I'm not having the stereotypical problem where a bunch of PCs meet for the first time and are automatically so paranoid of strangers that they start trying to kill each other. My players have just created a bunch of characters who have their own legitimate interests, and are faithfully following those in completely different directions that don't result in a classic "party". I put the PCs in great peril together and everyone escaped by their own devices, but now I have to figure out get the guy who ran for the hills back together with the couple that entered the city. And, I want to do it by resorting only to reasonable in-game events, not out-of-game metagame suggestions. How do I join my partitions? Matrix representation of linear transformation Preview: PDF Highlighting & Search Stop Working until Restart A problem has recently become much more prevalent for me when using Preview to search PDFs for text (using Command + F or the search box in the upper right).  At first, everything generally seems to go fine:    (as it should be) a list of thumbnails appears in the sidebar, indicating all of the pages that contain the search term. Generally, though not always, the term I search for will be highlighted on the first page that it occurs on.   But, after this, things quickly start malfunctioning:   Generally speaking, if I click on thumbnails lower on the list, Preview will take me to that page in the PDF (like it used to).  But, it will no longer highlight the searched for terms. Instead, Preview will make a low-pitched beeping noise of the sort that occurs when one tries to perform some un-permitted action in a variety of mac situations. Furthermore, once this occurs, I cease being able to make any highlights at all in the document I am viewing with Preview.  In fact, I can't even select any text any more.   Below is a list of unsuccessful efforts to resolve this:   Closing the side bar does not work to resolve this. x-ing out the search box in the upper right does not resolve this. Selecting the "Tools" menu and then "Text Selection" does not work either, nor does clicking on the text selection icon in the edit panel. If I quit preview and then re-open the PDF, I am able to highlight again.  But, the problem will re-surface as soon as I try performing another search. Working with the same files in other PDF programs (e.g. Skim, Acrobat, etc.) poses no problems.  I'm quite certain it isn't a problem with the underlying PDFs. None of the responses to  resolve this.  (indeed, I supplied one such response myself a while ago, but that trick and the others no longer work).   I've used Preview for many years without problem.  This issue started cropping up relatively recently for me, I believe within the past 3-4 months.  I had hoped that upgrading to Catalina might resolve it, but it did not.    I'd like to think I am fairly sophisticated with computers (I have a reputation of close to 8,000 on the main stackoverflow page), but I am baffled by this.  I also don't have a good handle on how I can get diagnostic information or how I can create a reproducible example.    Any leads on how to tackle this would be greatly appreciated. Can toddlers be taught to safely navigate stairs?