From Page 177 of differential equations demystified (2005):  We must evaluate the integral I: (1)  $$ I = \int_{\infty}^{0} e^{- s^{2}} $$    observe that: (2)  $$I \cdot I = \int_{0}^{\infty} e^{- s^{2}}\, ds \cdot \int_{0}^{\infty} e^{- u^{2}}\, du = \int_{0}^{\infty} \int_{0}^{\pi / 2} e^{- r^{2}} r d\theta dr $$   Could somebody explain these steps in better detail.  I'm not really sure how they get from (1) to the double integral in polar coordinates in (2)
How to prove  $$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
How do I see a C/C++ source file after preprocessing in Visual Studio? Convergence of $\sqrt{n}x_{n}$ where $x_{n+1} = \sin(x_{n})$ Why my DAC works without GND and VCC? This is the most interesting thing I have seen! I have built a  that sends data (D0-D7) to a DAC (ADV2175). The design worked fine and I just have a low frequency alias noise (). This noise was present on all my ground planes (This is in prototyping stage and that's not unexpected from the test PCB).  I started tracing the noise path by disconnecting wires to see when the noise is gone. Surprisingly even after disconnecting all VCC and GND connections (+ all VREF and logics gone to 0 or 1) the noise was present yet. The more unexpected behavior is when the DAC is just connected to DATA port+ CLK pin (all coming from the source FPGA, NO VCC or GND), it continues working and a sine wave is produced (but with the same noise)!!!  First I thought it may receive its VCC/GND form the logic 1/0 fed into it but if it was correct, it should not have produce 0x00 and 0xFF levels, but it makes those levels correctly (I changed 0xFF data stage to another level and a notch appeared on top of the waves. this shows it can build oxFF correctly).    This is a big challenge for my PCB design, as if this device really takes its GND from the data port, instead of the dedicated pins, I will have a hard time for designing a reliable ground plane for it (the return voltage should go back to a logic noisy environment and analog-digital ground plane isolation may be impossible).  Can anyone explain what is happening there? How to install and configure Wine? I am looking for a step by step instruction to guide me in the ways of installing, configuring and debugging many of the issues new users have when dealing with Wine in Ubuntu.  What is the recommended way of installing Wine, some of the key issues that it has and how to solve them?  If I have a specific problem with an App in Wine, how do I go about solving it? What are the recommended steps to ensure that I can fix a problem related to an app I wish to use? What is it called when a non-verb is used as a verb? This is in very common use on the internet. I just read this:     "[The] hotel employee walks in, I say "uh, puppy" and she just NOPEd the [heck] out of the room."   This is often done by adding 'd to the word. Isometries of the sphere $\mathbb{S}^{n}$ Got this as homework and I don't know how to tackle this. Help please!  Prove that the isometries of $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$, with the induced metric, are restrictions to $\mathbb{S}^{n}$ of the linear orthogonal transformations. jQuery ui tabs Use of “f ” instead of “s” in historic, printed English documents I was at a museum in London yesterday, and one of the items on exhibit is a document from the eighteenth century. It uses the letter f a lot where s should be used—for example, in Majefty.     Did the letter f once have the same sound as s today, or were some of those words actually spoken with an f ? What countries allow me to pass their borders visa-free?  Am I allowed to visit and work in EU and Schengen countries? Why aren't all the badges shown on the track menu? I like using the 'Track Badges' menu on my profile a lot (I know what you're going to say, I just have a knack for these things), but I keep noticing something a bit unusual.  Many of the Badges featured on the actual list do not get featured on this menu. For example, I've been hoping to be able to track progress for the Pundit badge (ten comments with score of five or more), but it doesn't show up in the menu.  Why not? Distance between footnote and the line above it How can I use Gmail (in Chrome) as the default email handler in Windows 8?  I used to use GmailNotifier in Windows 7, but that doesn't seem to work anymore. Clicking on a mailto: link simply opens Chrome with a blank page.  I am not interested in using the Mail app built into Windows 8. Python: List vs Dict for look up table I have about 10million values that I need to put in some type of look up table, so I was wondering which would be more efficient a list or dict?  I know you can do something like this for both:  if something in dict_of_stuff:     pass   and  if something in list_of_stuff:     pass   My thought is the dict will be faster and more efficient.  Thanks for your help.  EDIT 1  Little more info on what I'm trying to do.  .  I'm making a look up table to see if a value calculated has all ready been calculated.   EDIT 2  Efficiency for look up.  EDIT 3  There are no values assosiated with the value...so would a set be better? What kind of behaviour causes an interrupted exception? I'm relatively new to Threading in Java and I've noticed that everytime I use Thread.sleep() I have to catch InterrupetdException.  What kind of behaviour causes this, and in simple applications where I have a monitor thread can I just Ignore the exception? Can we get a flag history for chat's moderator attention flags? I checked my  to see whether I can find something similar to the main-site flag-history for chat-flags, but didn't find anything. I recently had the (dis)pleasure of regularly having to deal with a user, that has been deleted multiple times. Whenever they show up (with the same nick-name) again I have to flag them again. I'd like to get an overview of how often I had to do that, and whether my flags were helpful for the moderators. As such I propose the implementation of a flag-history as it is already in place for the main-site for chat-flags, that request moderator intervention. To be completely clear I am not talking about the 10k user flags, which are acted on in a matter of sometimes seconds with instantly visible results, but about moderator attention flags from this menu: I find today that I can no longer mount my Netgear ReadyNas drive because apparently it is using SMB1 and Windows 10 have disabled since release 1709. This is very problemaic because I need to access these files and because I have lost the admin password for the ReadyNas itself so I cannot login in order to update/reconfigure it to use SMB2 instead.  Microsoft refers me  to  which in turn refers me to  if I really want re-enable SMB1  And I successfully did re-enable SMB1 by running  Enable-WindowsOptionalFeature -Online -FeatureName smb1protocol   from Powershell as Adminstrator.  However even after reboot I still cannot re-establish connection to my Netgear from Windows Explorer, it gives the same error  What else do I need to do ?  PS C:\WINDOWS\system32&gt; Get-WindowsOptionalFeature –Online –FeatureName SMB1Protocol   FeatureName      : SMB1Protocol DisplayName      : SMB 1.0/CIFS File Sharing Support Description      : Support for the SMB 1.0/CIFS file sharing protocol and the Computer Browser protocol. RestartRequired  : Possible State            : Enabled CustomProperties :                    ServerComponent\Description : Support for the SMB 1.0/CIFS file sharing protocol and the Computer                    Browser protocol.                    ServerComponent\DisplayName : SMB 1.0/CIFS File Sharing Support                    ServerComponent\Id : 487                    ServerComponent\Type : Feature                    ServerComponent\UniqueName : FS-SMB1                    ServerComponent\Deploys\Update\Name : SMB1Protocol   Solution  Thanks Bob, SMB 1.0/CIFS File Sharing Support was enabled but within it SMB 1.0 Client was not enabled, checking that box and rebooting fixed the issue. Problem numbering environment Prove that if $0\le p_n&lt; 1$ and $S=\sum p_n&lt; \infty$ then  $\prod (1-p_n)&gt;0$. Hint given: First show that if $S&lt;1$, then $\prod (1-p_n)\ge 1-S$.   Attempt: I was able to show the hint by using recursion setting $A_n=\prod_{i=1}^{n}(1-p_i)$ re expression the $\prod (1-p_n)$ as $1-S+\sum_{n_1, n_2=1,n_1&lt;n_2 }^{\infty} p_{n_1}p_{n_2}-..... $ and observed that every subsequent term is less than the previous term hence  $\prod (1-p_n)\ge 1-S$ which is $&gt;0$ if $S&lt;1$ but am unsure about how to extend it to $S\ge1$. A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$.  Is there a characterization of the groups in which this property holds?  If this question is too broad, I might ask if such a characterization exists for $p$-groups.    History: I originally posed the opposite question, regarding groups for which $\exists N\unlhd G\,:\, \not\exists H \unlhd G\, \text{  s.t. } H \cong G/N$, and crossposted this to .  I received an answer there to the (now omitted) peripheral question about probability, which shows that most finite groups probably have this property.  After this, I changed the question to its current state, as this smaller collection of groups is more likely to be characterizable. Is it normal for DC wall adapter power supplies to provide voltages above the nominal ones? When I measure a 9V DC wall adapter power supply with a multimeter it shows 11.8V. When connected to my board this voltage lowers to 10V. Is this normal? Should I be afraid of toasting my  supplied board when using this adapter?  Edit: This regulator accepts up to 12V inputs. So that will be no problem. A second "9V" adapter yields 15V to the multimeter. Will this one burn my circuit?