How does one pile two lines under a union or intersection symbol in display mode? For example $\bigcup_{a,b \in \mathbb{R}; a+b=x} ... $ displays everything in one line, as it should, but I would like to put those two statements one above the other. I'm sure it's been asked before and I apologise, but I just don't know what to look for.
I have the following equation, where there is a lot of stuff under the sum symbol:  \begin{equation} d(\vec{x},\vec{y})=   \sum_{Z_{xy}\in\vec{Z}_{xy},\forall x\in\vec{x},\forall y\in\vec{y}}      f(Z_{xy}) \end{equation}   In the resulting document, I find it kind of hard to read. Is there a way to write the equation to make the result more readable, e.g. putting the stuff under the sum symbol on different lines?
is_active_sidebar() Always Returns False What's the arc length of an implicit function? While an explicit function $y(x)$'s arc length $s$ is easily obtained as  $$s = \int \sqrt{1+|y'(x)|^2}\,dx,$$   is there any formula for implicit functions given by $f(x,y) = 0$? One can use the implicit differentiation $y'(x) = -\frac{\partial_y f}{\partial_x f}$ to obtain  $$s = \int\sqrt{1 + |\partial_y f / \partial_x f|^2}\,dx,$$  but that still requires (locally) solving for $y(x)$. Is there any formulation that does not require this, e.g. another implicit equation involving $s$?    Thoughts so far:  One could rewrite $s$ as  $$s = \int |\nabla f|\, |\partial_x f|dx,$$  or symmetrize to  $$s = \int |\nabla f|\, \underbrace{(|\partial_x f|dx + |\partial_y f|dy)}_{(*)}/2$$  where $(*)$ might be strongly related to $|df|$ I guess (though it's not identical due to the $|\cdot|$), but then? Why is my text pixelated? Within my game I have some cubes with text which render fine. In addition to the cubes I want to display some static text, e.g. the current score. The score text has the camera as parent and a copy rotation constraint on the camera. While this concept works fine in principle, the score text is displayed very pixelated. I already increased the font size to 50 and made the text smaller by scaling it down, but it still doesn't show nicely. The score text is about 11 Blender units away from the camera. If I move the score text further away, it displays better, but then the text interferes (collides) with the cubes. What could I do to improve the quality of this text? Download . Open the SSCCE in Blender 2.70. Hover your mouse over the 3D viewport. Press P to start game engine. MySQL Hashing Function Implementation I know that php has md5(), sha1(), and the hash() functions, but I want to create a hash using the MySQL PASSWORD() function.  So far, the only way I can think of is to just query the server, but I want a function (preferably in php or Perl) that will do the same thing without querying MySQL at all.  For example:  MySQL hash -> 464bb2cb3cf18b66  MySQL5 hash -> *01D01F5CA7CA8BA771E03F4AC55EC73C11EFA229  Thanks! Is there a way to set an "Adventure Mode Boundary" in Minecraft? Is there a way to set an "Adventure Mode Boundary" in Minecraft?  I am trying to create this village where players can spawn in, and then set out to explore the world and build and do normal minecraft stuff. I don't want them to destroy this village, so I want them to be in adventure mode when they are in the village and then switch them to survival when they leave, and vice-versa. When they return to the village, they get switched back to adventure mode.  In other words, I want to set an "Adventure Mode Boundary" to protect buildings. I am working without mods. I am hoping there is an simple way using command blocks. How to negate a specific symbol? I want to negate the symbol whose code is \models (this would be specific to Formal Logic). How can I do that? I tried a number of combination, but didn't find how this could be done. I tried \neqmodels, \nemodels, \notmodels, \nmodels  but all of them are wrong.  Is there a general way to negate symbols? Are there any open source alternative to Landscape? As a disclaimer, I must admit this may benefit me (I haven't checked), but probably not as significantly as most others. This issue is also something that has prevented me from answering questions by users with 1 rep in the past, especially if they are poorly fashioned. The Problem New users with a programming question find SO from various methods, such as google. They ask a question with haste, hoping for a quick answer -- which they usually receive. Many of these users have no interest in Stack Overflow itself, but are simply interested in getting the solution they require. As a result of this, they'll check the site regularly for the first day or two until a sufficient answer has been posted. Once they get what they need, they bail; without the required rep or understanding that they should up-vote and select a response as the answer. The Result A great many questions on SO are abandoned by users with no intention of ever coming back. The answer they require has been provided, but for one or more reasons they have neglected their duty of contributing back by way of awarding reputation. The Reach (Since Jeff loves Datum) Some queries on the data dump examine the reach of this 'problem'. The query I used is based on extensions of the following base query; SELECT p1.Id, p1.Score, p1.OwnerUserId, p2.OwnerUserId, p2.Score FROM posts p1  INNER JOIN posts p2 ON (p1.Id=p2.ParentId) LEFT JOIN users ON (p1.OwnerUserId=users.Id) WHERE p2.Score &gt;= 2  AND p2.Score = ( SELECT MAX(p3.Score) FROM posts p3 WHERE p3.ParentId=p1.Id ) AND users.Reputation &lt; 15 AND p1.PostTypeId = 1 AND p1.AcceptedAnswerId = 0  Result: 3,288 Questions by users with under 15 rep which have answers with 2+ up votes that have no accepted answer. Adding: AND (users.LastAccessDate-p1.LastActivityDate) &lt; 604800 // 1 week AND p1.CreationDate &lt; 1249084800 // 1249084800 is 1 week before latest dump data  to the where clause further reveals that 877 questions that have answers with 2+ up votes have no accepted answer, and the owner of the question has not returned since the week the question was asked. Modifying there where clause, and removing the left join: AND p1.OwnerUserId = 0    Reveals a staggering 3,675 questions with answers of 2+ votes by deleted users.  These are questions that were probably answered but the users who did the work can never be recognized. Building off the previous and using an even more restrictive WHERE clause WHERE p2.Score &gt;=5  Still 1,178 posts that can never have an accepted answer. How many posts are there with over 5 up votes that don't provide the correct answer? Modifying the where clause for Reputation to AND users.Reputation = 1  Still reveals 388 questions by users with 1 rep have been abandoned. Solution Similar to how bounty questions work, if a user has abandons the site (not active since x time after the question was asked) with a question that has answers with 2+ up votes, force the answer with the most up votes to be the selected answer. This will reduce the amount of questions that are unanswered, and also give users more of an incentive to answer questions by new users without extra hesitation.  Update Added more statistics for deleted users.  Please feel free to add/remove rules to make this a truly practical feature. When clicking between "Week", "Month", "Quarter", "Year" and "All" on the users page, the name I have typed in to filter on disappears. I feel like it should stay there, so that I can type in my name, and see how I've done in a week/month/etc. Let $U\subset \mathbb{C}$ be a nonempty connected open set such that for every $z\in U$, $\overline z\in U$.  Let $f$ be analytic on $U$. Suppose $f(x)\in\mathbb R$ for every $x\in U\cap\mathbb R$. Prove that $f(\overline{z})=\overline{f(z)}$ for any $z \in U$.    By definition, I know that $f$ analytic on $U$ means that for every $z_0 \in U$, there exists $r&gt;0$ and a sequence of complex numbers $\left(a_n\right)_{n=0}^\infty$ such that $f(z)=\sum_{n=0}^\infty a_n\left(z-z_0\right)^n$ on the disc $D(z_0,r)\subseteq U$. I see that $f$ takes points without imaginary components to other points without imaginary components. But I don't see how this implies a symmetry that $f(\overline{z})=\overline{f(z)}$ for $z \in \mathbb C \setminus \mathbb R$. It seems like a very strong conclusion and I'm not sure how I would prove it.  I have also shown in the preceding question that $U\cap\mathbb R$ contains an open interval, however I am not sure if that detail is meant to be helpful to this question. Getting "System program problem detected" pops up regularly after upgrade Which version of NFS is my NFS server using? I'm mounting a NFS filesystem on my machine.  How do I figure out what version of the NFS protocol the server uses?  I don't have access to the NFS server machine, but I do have root on my client machine.  Is there anything I can run on my client machine to identify what version of the NFS protocol is being used by the server, or what versions it supports?  I wasn't able to find any useful information in /var/log/messages or kernel debugging output (dmesg).  I have tried running nfsstat, but I'm not sure if it is giving me any useful information.  However, when I run nfsstat -s to request information about the server, I don't see anything useful:  # nfsstat -s Server rpc stats: calls      badcalls   badfmt     badauth    badclnt 0          0          0          0          0          When I run nfsstat -c to request information about the client, I do see some information about Client nfs v3, but I'm not sure how to interpret this.  Does this tell me anything about the protocol being used between my client machine and the NFS server?  Does it mean I am currently using v3 of the NFS protocol?  Does it tell me anything about what versions of the NFS protocol the server supports, e.g., NFS v4? A friend wants to use my account for a wire transfer. Is this a scam or is it legitimate? I have a friend since July online and Exxon owe him for a balance on a 2014 contract off the shores of Liverpool. He is presently in the Gulf of Mexico completing a contract that last for one year. During that year he cannot get off the boat there fore he cannot receive the money.  Exxon has notified him that his check has been deposited in Samba Bank in London. If he doesn’t receive his money within two weeks it goes back to Exxon and he said that there I not telling nohow long before he will receive the money again. He said that with him being off shore that he needs me to be his beneficiary to receive the money for him. He wants me to keep the money in my account and when he gets off the boat in 6 weeks he is relocating to where I live.  I am to buy a house before he gets here and then he wants us to start a business together. He talked with me for quite a while before he told me how much he makes. He knows I do not have any money to mention. I have stock in a formal shop but not any money.  He said if this money had not come in when it did he would have had to postpone his plan until he gets paid what Shell Oil owes him for this contract that he will finish October 15th. He said he had given enough of his life to the gas and oil industry and is not going to sign another contract.  What do you think about the safety of this? Add table row in jQuery What is the best method in jQuery to add an additional row to a table as the last row?  Is this acceptable?  $('#myTable').append('&lt;tr&gt;&lt;td&gt;my data&lt;/td&gt;&lt;td&gt;more data&lt;/td&gt;&lt;/tr&gt;');   Are there limitations to what you can add to a table like this (such as inputs, selects, number of rows)? Can I read the hash portion of the URL on my server-side application (PHP, Ruby, Python, etc.)? An arithmetic function is defined as follows:$f(1)=1$,  $f(2k)=k$ and $f(2k+1)=f(k)+f(k+1)$. When (for which $k$) is $f(k)$ even?  While it is obvious that $f(4n-1)=f(4n)=2n$, therefore $f(k)$ is even for $k=4n$ and $k=4n-1$, which is possible to prove using induction (isn't it?), I don't know how to prove the converse.  I tried to solve the recurrence or make use of constant difference between certain elements of the sequence $\ldots$ What am I missing?  Thank you. In eevee, shadows are clipped by camera distance Gaussian integral evaluation Asked a question to evaluate the Gaussian Integral,  $$\dfrac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty x^2 \exp(-x^2/2) dx  $$  using the the following approximation,  $J=\Bbb E[X^2] \sim J_N = 1/N \sum_1^NX_n^2 $  where $x_n \sim N(0, 1)$  Examin $J_N$ for different values of $N$ and plot an error graph (difference between exact and numerical values)  I can simulate and generate the approximations but I am stuck at getting the exact value to compare to. As far as I am aware there are no analytical solutions for gaussian integrals. The other way I can generate comparable numbers is to go use numerical integration techniques but those are approximate values too.  Any help will be greatly appreciated! Why "a user" instead of "an user"? I am getting confused while using an article an. Because an is used while referring to vowels (a,e,i,o,u).  But today I read an article where for user they used a user.  My question is, why was an not used?  Could someone explain this with an example? Strange artifacts along window borders after waking computer from sleep mode