In my set theory book (A book of set theory, Charles,C,Pinter) domain and range of graph $G$ is defined by  $dom G=\{x:\exists y \ni (x,y)\in G\}$ $ran G=\{y:\exists x \ni (x,y)\in G\}$  But I don't know what $\exists x \ni (x,y)$ exactly means. Ordered pair is defined with $(x,y)=\{\{x\},\{x,y\}\}$ so $x\in \{x\}\in \{\{x\},\{x,y\}\}=(x,y)$. I think we should write $x \in (x,y)$ instead of $x \ni (x,y)$ What makes that definition possible?
What does the $\ni$ (backwards element of) symbol mean?  It doesn't appear in the , and a Google search for "backwards element of" or "backwards epsilon" turns up contradictory (or unreliable) information.  It seems it can mean both "such that", or "contains as an element".  Is this correct, and if so, which is the more common usage?
I can read the MySQL documentation and it's pretty clear. But, how does one decide which character set to use? On what data does collation have an effect?  I'm asking for an explanation of the two and how to choose them. What is the difference between an Intel i7 and a Xeon quad core processor? I know the  processors have been around for a long time and are mostly used in servers, but I am curious, why do people not use the Xeons in a high performance desktop?    As far as I know about the best desktop processor out there now is the  line. The i7's and Xeons are both quad-core processors, what is the main difference in these? I just saw that the MacBook Pro's are using the quad core Xeons instead of the i7's. Just recently, the right-click context menus in Windows 7 have been acting in a kind of left-handed way. That is, when possible, the initial menu appears to the left of the mouse pointer. When I hover over a submenu option, the next level of context menu appears to the left of the first and so-on.  Basically, what normally happens when you're right at the right-hand edge of the screen is happening all the time - except when I'm at the left-hand edge. Perhaps it's some kind of left-handed users option, but that doesn't seem likely - more like a right-to-left language option, except that I've never even tried to use a right-to-left language.  I haven't checked every app. for the issue - probably only Windows Explorer, definitely not in Firefox, for the moment that's as far as I've checked.  Scratch that - this affects menu bars as well as context menus. The menus dangle down to the left of the menu-bar item instead of the right, and submenus open to the left too. This seems to apply to most applications, but not too Firefox or Thunderbird.  I find this extremely annoying.  I have no idea how this occurred, though it isn't the only weird glitch I've had in Windows 7 in the last few days - see   I assume there's a setting somewhere that has got changed, but I can't find anything. There's also a chance that I've suffered some kind of registry corruption, so I might need to apply a fix in the registry - not something that scares me, but I need to know what to fix and how.  So - does anyone know what setting has been changed, and how I can fix it?  Also, beyond CCleaner (which I already have), are there any tips on making sure that the registry stays clean and sane?  Finally, both Microsoft Security Essentials and Malwarebytes Antimalware seem happy that I don't have a malware problem. Should my recent glitches make me doubt that? Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$ Can an iOS user play with pc users? I have a Mac only, so I am not able to get Terraria through Steam. I was wondering if I can play Terraria on my iPad Mini with a friend who has it on his PC. I was wondering this, because I saw that Deepworld on iOS is compatible with the Mac version. My question is not about (pseudo) philosophical debate; it concerns mathematical operations and experimental facts.  What is an observer? What are the conditions required to be qualified of observer, both mathematically and experimentally? C non-blocking keyboard input I'm trying to write a program in C (on Linux) that loops until the user presses a key, but shouldn't require a keypress to continue each loop.  Is there a simple way to do this? I figure I could possibly do it with select() but that seems like a lot of work.  Alternatively, is there a way to catch a ctrl-c keypress to do cleanup before the program closes instead of non-blocking io? Are there some good overviews of basic facts about images and inverse images of sets under functions? I'm trying to install the Jupiter power settings app but, I read it is no longer compatible with the new kernel. Is there a similar program or a way to manage the power settings on 13.04? Would the Animate Objects spell end an ongoing spell affecting the targeted object(s)? Grammar of "supposed to": part of speech / word class, active or passive? If I say     I'm supposed to go.   Is supposed to here an adjective in a passive construction? Proving $ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction Entering and leaving the UK on the day of visa expiry Prove $\lim_{n\to\infty} \frac{a_n}{n}$ exists for positive sequence where $a_{n+m} \leq a_n + a_m$ Let $\{a_n\}_{n=1}^\infty$ be a positive sequence of real numbers such that $a_{n+m} \leq a_n + a_m.$  Prove that $\lim_{n\to\infty} \frac{a_n}{n}$ exists by showing that $\lim_{n\to\infty} \frac{a_n}{n} = \inf_{n \geq 1} \frac{a_n}{n}.$   A similar question has been posted previously, see  , and I have found that this is the general result of "Fekete's Subadditive Lemma" for sequences, but I am unhappy with the few proofs I've seen.  Perhaps the clearest one I have found is  , but this seems a bit tricky for what is expected on this qualifier question.  We are given a hint "treat $\liminf$ and $\limsup$ separately".  Does anyone know of a more direct way to show that this limit exists? Why can't this be done with Master Theorem Apparently recurrences like this cannot be solved with the Master Theorem:  $T(n) = 2T\left(\frac{n}{2}\right) + \frac{n}{\log(n)}$  Because $n^{\log_b(a)} = n^1$ is not a polynomial multiple of $f(n) = \frac{n}{\log(n)}$ since $\frac{n}{\frac{n}{\log(n)}} = \log(n)$, which is less than $n^\epsilon$ for any positive $\epsilon$.  But when is this really even an issue?   For example why wouldn't the time complexity be $n \log n$ using case 2 of the Master Theorem? Unrolling the recursion:  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{\log_2(2^k)}$$  $$T(n) = n\sum_{k=1}^{\log_2(n)} \frac{1}{k}$$  Doesn't this suggest $\Theta(n \log n)$ even though it supposedly cannot be done with Master Theorem? Is abstract algebra (mostly?) restricted to $2$-ary operators? I have 41 GB of space on my E:/ drive. Is it sufficient to install Ubuntu? My Windows XP is installed in C:/ drive. Is it OK for dual boot? Should I install Ubuntu (I am installing using Live CD). I don't want to lose my Windows option. Environment variables in Mac OS X Update: The link below does not have a complete answer. Having to set the path or variable in two places (one for GUI and one for shell) is lame.  Not Duplicate of:     Coming from a Windows background where it's very easy to set and modify environment variables (just go to System Properties > Advanced > Environment Variables), it does not seem to be that straight forward on Mac OS 10.5. Most references say I should update /etc/profile or ~/.profile. Are those the equivalent of System Variables and User Variables? For example, where should I set my JAVA_HOME variable?    EDIT:  I want to be able to access the variable from the terminal as well as an app like Eclipse. Also, I hope I don't have to restart/logout to make this take effect. STM32 send from UART3 and receive on UART2 I thought to create a simple test to send and receive a buffer via UART. I succeeded in sending bytes from my PC to the nucleo bord and receiving them in blocking and non-blocking mode. But for some reason I can't succeed to send data from UART3 and receive the bytes on UART2.  I have a nucleo-f103rb and have the following pins setup:   UART2: tx=PA2;rx=PA3 UART3: tx=PB10;rx=PB11   And then the following code:    while (1)   {     /* USER CODE END WHILE */     /* USER CODE BEGIN 3 */     if (HAL_UART_Transmit(&amp;huart3, send, 2, 1000) == HAL_OK)     {         if (HAL_UART_Receive(&amp;huart2, receive, 2, 1000) == HAL_OK)         {             HAL_GPIO_TogglePin(GPIOA, LD2_Pin);         }     }   }   All the boiler plate is generated using cube mx. So the UART 2 and 3 are all default, same for the button pins.    I connected the transmit pin of uart3 to the receiving pin of uart2 I connected the receive pin of uart3 to the transmitting pin of uart2 (while not used in this trivial example though)   What am I missing here?  Update  As comments and answers suggested, I shouldn't be using the blocking mode approach. So I rewrote it slightly to rely more on interrupts.  I attached a callback to the nucleo blue button  void HAL_GPIO_EXTI_Callback(uint16_t GPIO_Pin) {     if (GPIO_Pin == B1_Pin)     {         HAL_UART_Transmit_IT(&amp;huart3, (uint8_t *)send, 2);     } }   And I added a callback for receiving stuff over uart  void HAL_UART_RxCpltCallback(UART_HandleTypeDef *huart) {   /* Prevent unused argument(s) compilation warning */   UNUSED(huart);    /* NOTE : This function should not be modified, when the callback is needed,             the HAL_UART_RxCpltCallback can be implemented in the user file    */       // Toggle the pin to see some proof that I received something      HAL_GPIO_TogglePin(LD2_GPIO_Port, LD2_Pin);       // Resubscribe uart2      HAL_UART_Receive_IT(&amp;huart2, (uint8_t *)receive, 2); }   In the main function I call the interrupt so that the callback is active.  while (HAL_UART_Receive_IT(&amp;huart2, receive, 2) != HAL_OK);  __NOP(); while (1) { }   Again, when debugging through I do end up in the button released callback, but the callback on the uart receive isn't triggered. When I send stuff from the PC, it does. So it seems that sending stuff via uart3 isn't working. So I guess that must be the wiring.     But I really can't see what I am missing here, the pins are clearly defined in the pinout here: .  I hope the photo is clear enough, but this is how I connected them. This post is an expansion of  For many graduated sites (eg Physics), moderators do not want to apply the unilateral close vote in all cases--it is better to let some community votes pile on first.  We only use the unilateral in obvious cases then. This renders the CV queue useless to us. If we &quot;skip&quot; something, we can't get back to it. If we say &quot;do not close&quot;, it's removed from the queue and our community can't see it. If we use &quot;close&quot;, we are using a unilateral mod vote--we don't want to do that. What I propose is a set of &quot;next/previous&quot; buttons, that let us navigate the queues, looking for closeable questions, and coming back to the others later. Thoughts? Update and bounty reason Due to the recent Great Flag Overhaul, mods no longer see close flags in the flag queue. Which means that the only way to get to them is via the review queue 1 Shog's solution is ideal in this case; let mods view the close queue as a list. (In fact, a list similar to the old flag queue -- which shows a list of mods who have seen the post and what mod actions have already been taken -- would be great) On Physics, the majority of the flags that we get are VTC flags. And the community isn't always able to close things on their own, so we have to step in. However, powering through the close queue isn't an option; because &quot;skip&quot;ing a post2 leads to it being dumped into the void (I can't get back to it unless I have a link). Extremely inconvenient. Could the workflow here be improved? 1. The /tools listing for posts with outstanding close votes doesn't work because it doesn't list posts which have been flagged but not VTCd 2. which I may want to take a closer look at later; gotta be careful with the binding votes.