How to insert the "X is a subset of Y, but is not equal to Y" symbol?
How to look up a symbol or identify a letter from a math alphabet or other character?
What is an quick way to create a new blank file in Lion? In Windows, one can create a new file using Right Click -> Add New -> Type of File. Is there an analogous action in Lion?  I just want to create a new blank file on my Desktop. Serializing anonymous delegates in C# Evaluating a sum involving binomial coefficient in denominator I came across the following sum: $$\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}\frac{4^k}{{2k \choose k}}$$    I thought that this can be evaluated using the expansion of $\dfrac{\sin^{-1}x}{\sqrt{1-x^2}}$ but I couldn't make any use of it. Then I tried to use the following: $$\frac{1}{(2k+1)}\frac{1}{{2k \choose k}}=\frac{\Gamma(k+1)\Gamma(k+1)}{\Gamma(2k+2)}=\int_0^1 x^k(1-x)^k\,dx$$ but this didn't help either and now I am stuck.   Any help is greatly appreciated. Thanks!    EDIT:  The sum originated from the following definite integral:  $$\int_0^{\pi/2}\tan^{-1}(\sin x)\,dx$$ I know that I can use  to make cross-references and hyperlinks clickable.  That makes the clickable areas outlined in fluorescent green, however.  How can I make the green boxes go away? I had a Minecraft world on Minefold.  I archived my Minecraft world just before Minefold closed about a year ago.  I'm trying now to open the world in my local Minecraft installation on OS X, but it doesn't show up in the worlds list.  To import it, I simply unzipped the archive from Minefold into my ~/Library/Application Support/minecraft/saves folder.  I'm running Minecraft 1.8, and I don't have any mods installed.  There's no error; the world just doesn't show up in my list of worlds after I click [Singleplayer] (or [Multiplayer]).  Now, the big clue is that the directory structures are different between my locally-created worlds and the world created on Minefold, but I can't figure out how to adapt the files to make them match:  This is a screenshot of my saves folder. I've circled in blue "local" and "minefold", which contain data files for the local and Minefold worlds, respectively.    I bracketed similar filenames in both worlds, but I'm flummoxed in that the Minefold version has three (3) sets of data files, whereas the local world has only one (1) set of files (even though I have visited Nether in the local world).  How do I load my old Minefold world locally? I already have the "Edit Questions And Answers" privilege. Yet, if I approve someone else's edit, I'm told that more votes will be required before the edit is accepted.  This seems a bit peculiar to me, since I could just copy the edited version and paste it myself. Proving for all integer $n \ge 2$, $\sqrt n < \frac{1}{\sqrt 1} + \frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots+\frac{1}{\sqrt n}$ So first let me state my homework problem:  Let $X$ be a set, let $\{A_k\}$ be a sequence of subsets of $X$, let $B = \bigcup_{n=1}^{+\infty} \bigcap_{k=n}^{+\infty} A_k$, and let $C = \bigcap_{n=1}^{+\infty} \bigcup_{k=n}^{+\infty} A_k$. Show that (a) $\liminf_k\; {\xi_A}_{_k} = \xi_B$, and $(b)$ $\limsup_k \;{\chi_A}_{_k} = \chi_C.$  I know that, in the context I am familiar with, that $$\liminf_{k\rightarrow +\infty}\; X_k = \bigcup_{k=1}^{+\infty} \bigcap_{n=k}^{+\infty} X_n$$ and $$\limsup_{k\rightarrow +\infty}\;X_k = \bigcap_{k=1}^{+\infty} \bigcup_{n=k}^{+\infty} X_n.$$   I also know that the characteristics (indicator) function is defined as $\chi_A(x) = \begin{cases} 1, &amp; x \in A \\ 0, &amp; x \notin A .\end{cases} $  So I wrote out $B$ in some of its `glory': $B= (A_1 \cap A_2 \cap A_3 \cap \cdots) \cup (A_2 \cap A_3 \cap \cdots) \cup (A_3 \cap A_4 \cap \cdots) \cup \cdots$, and as the first argument is the smallest, with increasing size to the right, the last term in the expression for $B$ would be $B$, which would be the largest.  So if I replace the $X_k$'s above with $\chi$'s I still don't see how I can get the correct answer - though it looks pretty clear from the definition of $B$ and that of $\liminf$ being basically the same, except in this case for the $\chi$.  Any direction would be greatly appreciated. By the way, I have checked out  but I was somewhat confused by the topology, the meets/joins, etc.  Thanks much, Nate Area51 proposals are being advertised now all over the network? Connecting GSM SIM900A module with Ardunio I have bought an SIM900A Mini V4 GSM module. Here is the module,   I want to configure with arduino to send and receive SMS.   Do I have to connect TXD of this module--Tx of Arduino &amp; RXD of this module--Rx of Arduino or other way around?  There is no datasheet or documents on this new module.  When I search about RXD &amp; TXD, I found this in      TxD means "Data to be transmitted over the comms link" - which will clearly be an input to the modem;    Similarly, RxD means "Data that has been received from the comms link" - which will clearly be an output from the modem.    But I couldn't send or receive SMS using this configurations.  As I am new to this, It would be very helpful if anyone help on this.  Thank you in advance. Splitting lines with points while maintaining other dissolved lines? I am on ArcGIS Desktop 10.2. I have a line feature class of a stream network. I also have a point feature class along the main stream (line) where a catchment enters that stream. I have dissolved my whole streams layer into one line feature and want to split this feature at the points.    However, when I use the  tool, the lines that don't have a point splitting them are still being split at the intersection of two streams.    Why did this line not stay as one feature with the other lines on that side of the splitting node and how can I prevent this from happening? Prove that for all integers $n$, $n \geq 1$,   $$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$   How would I go about proving this? std::string formatting like sprintf Identifying a resistor with a weird band combination Acne underneath skin Introduction  Okay, I've almost won my war. The battle, its almost won... However, I must still claim victory in my final battle. The battle against the little enemies, the ones that have fought with my blade of benzoyl peroxide every single day.  Problem  Through spot treatments I have been able to fell every single one of these spots, but only the ones on my face. I've been using a spot treatment with benzoyl peroxide in, which has worked really well for me. However, my issue comes here:     There are spots underneath my skin, I think. The areas where they come through are a bit red.   So, because there is still slightly red skin on my face, and I think this is causing my acne to constantly return. I need to stop this, and end my war once and for all.  What I've tried   Washing my face with a rag and sponge every day, three times a day Changing my pillow case every day Using a benzoyl peroxide spot treatment Minimizing how much I touch my face Not interfering with the spots with my fingers, not picking at them, not playing with them. Just letting spot creams eradicate them.   Conclusion  Seriously, this driving me up the wall. I spot cream in the evening, by the time tomorrow has ended and its time for a shower, they've all come back. Its like they're coming in waves, damn waves! I must put an end to this.  So, how can I prevent acne from coming back within the space of a single school day? How can I get rid of these once and for all? What types of measures could I try? I have no reason to visit my doctor, its only acne, but I have no idea what I can try now.  Is there even a point in me trying? I acknowledge that I am a teenager, but no others that I know seem to have acne/spots like I do. I realize this is probably going to come off as a bit silly, but in Iron Man II how could James Rhodes easily don the Mark II armor that was designed to fit Tony Stark?  As you can see in this image (which isn't an "in universe" shot) Tony and Rhodey are roughly the same height, but it appears that Rhodey has a smaller frame than Tony, which would seem to make for an ill fit of the armor, especially for flight.  This issue could of course have been addressed by the Air Force during the retrofit with Justin Hammer. From time to time I lose all edits because of undo in Object Mode. If I enter immediately after Undo in Edit Mode I can't redo, and switching again in Object results in losing all edit history.  How can I resolve this? Separating undo history by mode or linear/unified will both be acceptable.  Edit:I think the most preferable solution will be to disable/skip edit undos when in object mode through an addon. Superman has been exploited by Poison Ivy in Hush, and he's definitely mad in Injustice. What about Batman? Has Batman ever had to be brought to justice/stopped/killed by his fellow superheroes? (That one time when Superman dreams of Batman snapping Joker's neck doesn't count)  Edit: The Dark Knight Returns doesn't count: Batman wasn't really mad, and he wasn't wrong either. I have fought the pumpkin moon and died like 3 times and made it to wave 7 or 6 whenever the pumpking first appears. I have full turtle armor and my primary melee weapon is a demon sickle. I also have the possessed hatchet, magnet sphere, and sniper rifle. I also have other gear like the Uzi. What should I replace, what should I take out, and what should I add? Thanks! In the current era of cricket, we see batsmen using switch hits to play a ball. That is, while playing a ball they change their batting side (left to right handed and vice versa) just before the ball comes to them.  My question is, can a bowler bowl a switch ball? Meaning, can a left hand bowler bowl right handed to a batsman? If it is not permitted, then is it not unfair to the bowlers that batsmen can change their batting style but not the bowlers? Or are there are any rules against a switch hit?