Artifacts at particular edges after Subdivision modifier
Inaccurate shading on object
Proving $\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right) \left(1-\frac{x^2}{3^2\pi^2}\right)\cdots$ Proof Verification : Prove -(-a)=a using only ordered field axioms I need to prove for all real numbers $a$, $-(-a) = a$ using only the following axioms:     Thanks to many members of the Mathematics Stackexchange Community, I have the following proof worked out:  Theorem: The Additive Inverse Identity is Unique  $ ( \forall a,b,c \in \mathbb{R} )(a+b=0) \land (a+c=0) \\ ( \forall a,b,c \in \mathbb{R} )(a+b=a+c) \\ ( \forall a,b,c \in \mathbb{R} )(b=c) \\ \text{QED} \\ $  Theorem $\textbf{a} \cdot \textbf{0 = 0}$  $ \begin{align} a \cdot 0 &amp;= a \cdot 0 \\ a \cdot (0 + 0) &amp;= a \cdot 0 \\ a \cdot 0 + a \cdot 0 &amp;= a \cdot 0 \\ a \cdot 0 + a \cdot 0 + (-a) \cdot 0 &amp;= a \cdot 0 + (-a) \cdot 0 \\ a \cdot 0 &amp;= 0 \\ &amp;\text{QED} \\ \end{align} $  Theorem: $-\textbf{1} \cdot \textbf{(a) = (}-\textbf{a)}$  $ \begin{align} a \cdot 0 &amp;= 0 \\ a \cdot \left[ 1 + (-1) \right] &amp;= a + (-a) \\ 1 \cdot a + (-1) \cdot a &amp;= a + (-a) \\ -1 \cdot (a) &amp;= (-a) \\ &amp;\text{QED} \\ \end{align} $  Theorem: $-\textbf{(}-\textbf{a) = a}$  $ \begin{align} 0 &amp;= 0 \\ -a \cdot 0 &amp;= 0 \\ -a \cdot \left[ 1 + (-1) \right] &amp;= 0 \\ -a \cdot 1 + -a \cdot (-1) &amp;= 0 \\ -a + \left[ -(-a) \right] &amp;= 0 \\ -(-a) &amp;= a \\ \text{QED} \\ \end{align} $  Does everything look alright? Have I missed anything?  Also--why is it necessary to show that the additive inverse is unique?  Many thanks in advance. What is the best solution for database connection pooling in python? I have developed some custom DAO-like classes to meet some very specialized requirements for my project that is a server-side process that does not run inside any kind of framework.    The solution works great except that every time a new request is made, I open a new connection via MySQLdb.connect.    What is the best "drop in" solution to switch this over to using connection pooling in python?  I am imagining something like the commons DBCP solution for Java.    The process is long running and has many threads that need to make requests, but not all at the same time... specifically they do quite a lot of work before brief bursts of writing out a chunk of their results.  Edited to add: After some more searching I found  which looks decent, but as I'm relatively new to python I guess I just want to make sure I'm not missing a more obvious/more idiomatic/better solution. Harry is a half-blood and Ginny is a pure-blood, so what does that make his children? I've been told it means they are pure-bloods but I can't be sure. What do you think? I have a residence permit (permesso di soggiorno) for Italy. I am currently in the Netherlands for an exchange. My permit expired on the 30th of November and I applied for a new one but I will get the card in January. I have the documents supporting that I have applied for the renewal of the permit. In the middle I want to travel to Italy for Christmas and come back to the Netherlands.    Would this be a problem? Guarding against cryptanalytic breakthroughs: combining multiple hash functions iMessage doesn't say "delivered" or "not delivered" I sent an iMessage from my iPod touch to my friends iphone on Monday. It didn't say delivered or not delivered. It was just nothing. His phone is on (I know because my other friends are iMessaging him) and my friend said her messages don't say "delivered" at the bottom but they still talk. Mine always say delivered at the bottom. So is this an error, like it didn't go through, or is he just ignoring me? Linked server returned message "Query timeout expired." We have two separate SQL Servers. On one server we have a data warehouse (DWH), on the other we have sales information database.  Now on the DWH server there is an ETL job that collects the information from the sales server. The job runs daily after midnight. The DWH collects the information via linked server from the sales database.  Now, the most time the ETL job runs without any problems. But sometimes it fails because of query timeout. We have found out, that there is a specific pattern: The failure happens every 11th day. So on the 11th day the ETL job fails to collect information.   The following error occurs:     SQLNCLI11 for linked server “my linked server” returned message “Query timeout expired.”   Note: The job fails usually 10 minutes after start.  We have searched everything and could not found out what this issue causes. Also we know that the amount of data is every time, almost the same. Also there is no any scheduled job that runs all 11 days or something.  The remote query timeout on the linked server is set to 0.  Our next step will be, to turn off the antivirus programm on the sales server, to check if this causes the problem.  Does anyone have any clue or idea, where i can search further to find the problem? My Sophix tablet was working without the SD card, but now the camera will not work. I get the following error message:     Insert a SD card before using. How should one maintain and update a MiKTeX installation? How to make horizontal lists? This might seem like a very simple question, but what is the easiest way to turn a normal vertical list into a horizontal one? How do I resolve samba share to IP address or host name I have a list of hosts in the network providing shares via SAMBA. How can I determine either IP address or the host name of one particular host, e.g. the one with the name “SASAK02”.  The output of smbtree is as follows  WORKGROUP     \\SASAK02                    \\SAURA-PC1                  \\PC-VAN-DAMME How do i make my ubuntu server public? I want to know how to make my ubuntu server ip public. I'm not asking for a domain, if people would be able to access my website by ip, it would be great! (just for now) Does someone have an idea how to do this? Where can I find gedit's backup files and how to open them? So, I just finished off the first part of an important task for school and today when I was going to continue, the document with my notes were gone. The only fragment I've got left of my file is the back up file, which doesn't show in the folder where I saved the document, but I can search for it.  Is there anyway which allows me to open the back up file, save it as a text file and continue my task? I don't really have the time for running restoring tools, which never has helped me in a satisfactory way anyway.  I wrote the document in gedit and my version of Ubuntu is 12.10. Change the passphrase of an ecryptfs directory Why do I need multiple sites for the same property in Google Webmaster Tools? In researching the best practice for adding sites to the Search Console, I've often seen it suggested to add separate properties for different versions of the same website. Even in  it states:     "Similarly, if your site has http:// and https:// versions, you should   add each as a separate site."   I've seen the same suggested for www/non-www. So if a site is accessible at all the variations of http/s and www/non-www we should have four properties.  But what if the site is only accessible at one variant? i.e. https and www are forced via htaccess, so only  can ever be accessed - should we still add all the variants?   In essence, the problem is that I don't know why Google wants us to do this, so I can't work out if it'll be ok just to have one instance. I use ps -ef | grep catalina | grep -v grep to print the tomcat process running on the system:  kshitiz   7099     1  0 May11 ?        00:02:29 /usr/lib/jvm/jdk1.8.0/bin/java -agentlib:jdwp=transport=dt_socket,suspend=y,address=localhost:38156 -Dcatalina.base=/home/kshitiz/Documents/workspaces/ggts/.metadata/.plugins/org.eclipse.wst.server.core/tmp1 -Dcatalina.home=/opt/tomcat-7.0.42 -Dwtp.deploy=/home/kshitiz/Documents/workspaces/ggts/.metadata/.plugins/org.eclipse.wst.server.core/tmp1/wtpwebapps -Djava.endorsed.dirs=/opt/tomcat-7.0.42/endorsed -Dfile.encoding=UTF-8 -classpath /opt/tomcat-7.0.42/bin/bootstrap.jar:/opt/tomcat-7.0.42/bin/tomcat-juli.jar:/usr/lib/jvm/jdk1.8.0/lib/tools.jar org.apache.catalina.startup.Bootstrap start   Then I use ps -ef | grep catalina | grep -v grep | awk -F' ' '{print $2}' to extract the process id:  7099   But when I alias the whole command alias tomcat_id="ps -ef | grep catalina | grep -v grep | awk -F' ' '{print $2}'" and use it via alias, it prints the whole text and awk doesn't seem to work.  type tomcat_id gives:  tomcat_id is aliased to `ps -ef | grep catalina | grep -v grep | awk -F' ' '{print }'' Get Emacs to ignore contents of \Sexpr{} command in Sweave document to prevent incorrect $-based syntax highlighting The following code uses the etoolbox package to convert a list a,b,c into an internal list (called \mylist) and then typesets that with colons: a:b:c:.  However, when I load the etextools package (by uncommenting the second line), I get no list output.   \documentclass{minimal} \usepackage{etoolbox} %\usepackage{etextools} \begin{document} A list: \def\mylist{} \forcsvlist{\listadd\mylist}{a,b,c} \def\do#1{#1:} \dolistloop{\mylist} \end{document}   I suspect it is is a bug since etextools claims to depend on etoolbox.  Is this a bug or am I doing something wrong?  In either case, how can I use \dolistloop with  etextools (which I want for goodies like \expandnext)?  Notes   \show\mylist produces the same output a|b|c| in both cases, so the problem is with \dolistloop which expands to \forlistloop\do The \forlistloop command becomes substantially more complicated when etextools is used.  might be relevant (but not obviously). After the performance problems we have run into with  we've decided to make a change, we're moving the network on to .  Here's where to get started:   What works:   All search operators should be in Many changes below from the old search behavior   What's different:   The new search is AND, adding a search term will narrow your results, not expand them. The look of search results is now more unified with the following differences between Q&amp;A:   "Q:" or "A:" prefix Answer counts and tags will be present in questions. The answer count will be highlighted in green if it's accepted. (We may index other things later, that's why the results are more generic/google-y now.)  It's faster, about 5-10x faster for almost all searches on large sites like Stack&nbsp;Overflow Quoted phrases are exact matches except for case-sensitivity, for example,  or . If not in a quoted phrase, words will now be stemmed () Snippets for context will now be more relevant Excludes are now explicitly supported (like -term and -"my phrase") We don't roll up to the question anymore, if you searched for something that matched an answer, we'll show you the answer directly in search results The search per minute limit is effectively gone, it'll block a bot, but no one else should hit it... We'll be monitoring performance though infavorites:mine is back, with infavorites:12345 (any user id) added   What needs work:   What you tell us is broken... Do your worst.   Testing, not yet final:   New range support, here are some examples:   answers:1..1 (exactly one answer) answers:10..20 (10 to 20 answers) created:2008..2009 (created between Jan 1st, 2008 and Dec 31, 2009) created:2010-04..2010-05 (created between April 1st, 2010 and May 31st, 2010) created:2011-01-05..2011-01-06 (to be clear: it goes to end-of-day) This range syntax applies to a new advanced operator: lastactive: and the other numerical ranges, views:, and score:  Additionally, the above syntax is a bit more flexible if you forget it exactly (e.g. score:20-30) in an effort to be intuitive as possible, give it a try - tell us what you think.    I'll try and improve the search as much as possible before it becomes default... At the very least we want it equivalent in relevance and functionality before the switch, but ideally much better. We won't switch until then.  Please give it a try, tell us what you think here, and we'll continually improve it over the holidays as time allows. Bear in mind you may get a search offline page during this test if a change we're making for results necessitates a reindex, and even that should only last about two minutes here on Meta&nbsp;Stack&nbsp;Overflow.  Give it your worst, compare results to the old search, tell us what's better and what's not. We look forward to the feedback.  Update The new search is now the default across the network - we'll be monitoring things over the next few days and would appreciate reports of any crazy behavior you see here.