$x_n=\sin(x_{n-1}),x_{0}\in (0,\pi)$, find limit $x_{n}\sqrt{n}$
Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)$
A coin is tossed three times. The probability of zero heads is 1/8 and the probability of zero tails is 1/8. What is the probability that there is at least one head and at least one tail?  So, if P(zero heads)= 1/8 , then that should be the same of p(all tails)?  We would use the complement rule and Multiplication Rule?   P(at least one head) = 1 - P(no heads) = 1 - 1/8= 7/8 P(at least one tail) = 1 - P(no tails) = 1 - 1/8= 7/8  would I multiply the two values? Can the weak force create a bound state? Custom cross-reference formats (e.g. "Equation 1" instead of just "1") How long does the new dwellers queue last? When clicking the radio room icon that attracts a new dweller to your vault, how long will they stay queued out in front of the vault for before giving up?  Or will they?  I'm at max population and would rather not kill existing dwellers to accommodate them in my vault. How to handle calendar TimeZones using Java? I have a Timestamp value that comes from my application. The user can be in any given local TimeZone.  Since this date is used for a WebService that assumes the time given is always in GMT, I have a need to convert the user's parameter from say (EST) to (GMT). Here's the kicker: The user is oblivious to his TZ. He enters the creation date that he wants to send to the WS, so what I need is:  User enters: 5/1/2008 6:12 PM (EST)  The parameter to the WS needs to be: 5/1/2008 6:12 PM (GMT)  I know TimeStamps are always supposed to be in GMT by default, but when sending the parameter, even though I created my Calendar from the TS (which is supposed to be in GMT), the hours are always off unless the user is in GMT. What am I missing?  Timestamp issuedDate = (Timestamp) getACPValue(inputs_, "issuedDate"); Calendar issueDate = convertTimestampToJavaCalendar(issuedDate); ... private static java.util.Calendar convertTimestampToJavaCalendar(Timestamp ts_) {   java.util.Calendar cal = java.util.Calendar.getInstance(       GMT_TIMEZONE, EN_US_LOCALE);   cal.setTimeInMillis(ts_.getTime());   return cal; }   With the previous Code, this is what I get as a result (Short Format for easy reading):  [May 1, 2008 11:12 PM] USB stick mysteriously become write protected So, somehow over the weekend, my 1GB USB flash drive has managed to become write-protected.  There is no switch on the stick, so I'm concluding that something has corrupted at some point.  Obviously I can't format it or remove.  The output from dmesg | tail is:  Buffer I/O error on device sdb1, logical block 328168 lost page write due to I/O error on sdb1 sd 73:0:0:0: [sdb] Unhandled sense code sd 73:0:0:0: [sdb] Result: hostbyte=DID_OK driverbyte=DRIVER_SENSE sd 73:0:0:0: [sdb] Sense Key : Data Protect [current] Info fld=0x0 sd 73:0:0:0: [sdb] Add. Sense: Write protected sd 73:0:0:0: [sdb] CDB: Write(10): 2a 00 00 05 08 00 00 00 01 00 Buffer I/O error on device sdb1, logical block 328200 lost page write due to I/O error on sdb1   I think the second and final lines are a giveaway to corruption, but I don't know how to interpret this output.  Could someone help to see if this output gives any indication?  I usually eject the drive before removing it, but I may have been lax once or twice.  The data isn't critical, it's just bugging me.  Any suggestions on how to fix it (if the drive even can be) would be greatly appreciated. WIth Vimeo Plus account, how to hide/remove the "play" button in the middle of an embedded video? What is a StackOverflowError, what causes it, and how should I deal with them? Is it possible to cross the Wagan border by car? I was planning a trip by car when I found something a bit weird in :if you try to cross from Lahore to India, it tells that you have to 6457 km detour.  Is it possible to cross this border driving? If not, is there another way to cross without having to make more than 6000 km? Getting image dimensions without reading the entire file How to fix "W: Duplicate sources.list entry"? I keep getting this warning whenever I try to run sudo apt-get update.  W: Duplicate sources.list entry http://archive.ubuntu.com/ubuntu/ precise-updates/main i386 Packages (/var/lib/apt/lists/archive.ubuntu.com_ubuntu_dists_precise-updates_main_binary-i386_Packages) W: You may want to run apt-get update to correct these problems   Below is the output from /etc/apt/sources.list file:  deb http://archive.ubuntu.com/ubuntu precise main restricted deb-src http://archive.ubuntu.com/ubuntu precise main restricted  deb http://archive.ubuntu.com/ubuntu precise-updates main restricted deb-src http://archive.ubuntu.com/ubuntu precise-updates main restricted  deb http://archive.ubuntu.com/ubuntu precise universe deb-src http://archive.ubuntu.com/ubuntu precise universe deb http://archive.ubuntu.com/ubuntu precise-updates universe deb-src http://archive.ubuntu.com/ubuntu precise-updates universe  deb http://archive.ubuntu.com/ubuntu precise multiverse deb-src http://archive.ubuntu.com/ubuntu precise multiverse deb http://archive.ubuntu.com/ubuntu precise-updates multiverse deb-src http://archive.ubuntu.com/ubuntu precise-updates multiverse  deb http://archive.ubuntu.com/ubuntu precise-security main restricted deb-src http://archive.ubuntu.com/ubuntu precise-security main restricted deb http://archive.ubuntu.com/ubuntu precise-security universe deb-src http://archive.ubuntu.com/ubuntu precise-security universe deb http://archive.ubuntu.com/ubuntu precise-security multiverse deb-src http://archive.ubuntu.com/ubuntu precise-security multiverse   How do I fix it? How can I copy a custom Dictionary on Word 2000? NVMe PCIe x4 SSD on M.2 PCIe x2 slot? i'm about to get a new SSD for my PC, and before spending the money i have some doubts about my decision i would like to clarify First, my mobo is a B450 Aorus Pro Wifi, it has 2 M.2 slots, one of them full length (22110) compatible with PCIe 3.0 x4 and the other one standard length (2280) and support only for PCIe 3.0 x2 About the drive, i'm aiming for a 1tb XPG SX8200 Pro , which is a PCIe 3.0x4 SSD My first M.2 slot is already taken by the main SSD, which i use for the OS and other stuff, i can't really use it. Will a PCIe x4 SSD work on a PCIe x2 slot? How much performance will be lost if it does? Will it be worth it? Maximum power transfer proof I have the following homework problem.  Consider a power supply with fixed emf $ε$ and internal resistance $r$ causing current in a load resistance $R$. In this problem, $R$ is fixed and $r$ is a variable. The efficiency is defined as the energy delivered to the load divided by the energy delivered by the emf.  When the internal resistance is adjusted for maximum power transfer, what is the efficiency?  The answer is 50% but I'm confused how this is calculated. Here is my thought process so far.  The power dissipated in the load is $I^2R$ and $\displaystyle I = \frac{ε}{r+R}$ so $\displaystyle P = \frac{ε^2R}{(r+R)^2} = \frac{ε^2}{\frac{r^2}{R} + 2r + R}$ and so power transferred to the load will be maximum when $\displaystyle \frac{r^2}{R} + 2r + R$ is a minimum. Taking the first derivative $\displaystyle \frac {d}{dr}(\frac {r^2}{R} + 2r + R) = (\frac{2r}{R} + 2)$ and this equals 0 when r = -R. The second derivative is $\displaystyle \frac{2}{R} &gt; 0$ so this point would be a minimum and therefore max power transferred when r = -R.  But having $r = -R$ doesn't make sense to me. From what I read, for maximum power transfer $r$ should equal $+R$. So I've probably done something wrong in my calculations or assumptions in the previous paragraph. Also wouldn't max power be transferred when the internal resistance is as close to zero as possible?  Can someone please show me a proof that shows why $r = R$ for max power transfer and then how to calculate the efficiency. Removing nodes from an XmlDocument The following code should find the appropriate project tag and remove it from the XmlDocument, however when I test it, it says:  The node to be removed is not a child of this node.  Does anyone know the proper way to do this?  public void DeleteProject (string projectName) {     string ccConfigPath = ConfigurationManager.AppSettings["ConfigPath"];      XmlDocument configDoc = new XmlDocument();      configDoc.Load(ccConfigPath);      XmlNodeList projectNodes = configDoc.GetElementsByTagName("project");      for (int i = 0; i &lt; projectNodes.Count; i++)     {         if (projectNodes[i].Attributes["name"] != null)         {             if (projectName == projectNodes[i].Attributes["name"].InnerText)             {                                                                 configDoc.RemoveChild(projectNodes[i]);                 configDoc.Save(ccConfigPath);             }         }     } }   UPDATE  Fixed. I did two things:  XmlNode project = configDoc.SelectSingleNode("//project[@name='" + projectName + "']");   Replaced the For loop with an XPath query, which wasn't for fixing it, just because it was a better approach.  The actual fix was:  project.ParentNode.RemoveChild(project);   Thanks Pat and Chuck for this suggestion. Is it true that $aH = bH$ iff $ab^{-1} \in H$ Let $H$ be a subgroup of a group $G$. The I know that for $a,b\in G$ we have $aH = bH$ if and only if $a^{-1}b \in H$.  My question is if it is also true that $aH = bH$ if and only if $ab^{-1} \in H$? Does it matter where the $-1$ goes?  I am guessing that this is not true since we keep putting the $-1$ on the left element, but I am also thinking that it might be true because they look like the same.  If $G$ is abelian, then it doesn't matter, so a possible counter example would have to involve a non-abelian group. If $H$ is a normal subgroup, then it should also be true because then $a^{-1}bH = Ha^{-1}b$. Proving the limit of a function of a sequence is equal to the function of the limit of that sequence Suppose $f$ is a continuous function at $x = c$ in $[a,b]$.  Prove that for any sequence ${x_n}$ in $[a,b]$ converting to $c$, the sequence $\{f(x_n)\}$ converges to $f(c)$.  That is, $$ \lim_{n\to\infty}f(x_n)= f\left(\lim_{n\to\infty}x_n\right)$$    This proof seems simple but there are a few things that I need to know first. If $\{x_n\}$ converges to $c$, is it sufficient to substitute $c$ in for $\lim_{n\to\infty}x_n$?  Also needing some guidance on the structure of this proof.  Thanks! Prove: If a sequence converges, then every subsequence converges to the same limit. I need some help understanding this proof:  Prove: If a sequence converges, then every subsequence converges to the same limit.  Proof:  Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k \geq k$ for all $k$. This easy to prove by induction: in fact, $n_1 \geq 1$ and $n_k \geq k$ implies $n_{k+1} &gt; n_k \geq k$ and hence $n_{k+1} \geq k+1$.  Let $\lim s_n = s$ and let $\epsilon &gt; 0$. There exists $N$ so that $n&gt;N$ implies $|s_n - s| &lt; \epsilon$. Now $k &gt; N \implies n_k &gt; N \implies |s_{n_k} - s| &lt; \epsilon$.   Therefore: $\lim_{k \to \infty} s_{n_k} = s$.   What is the intuition that each subsequence will converge to the same limit I do not understand the induction that claims $n_k \geq k$ Intuition behind the definition of a measurable set Is the EmDrive, or "Relativity Drive" possible? In 2006, New Scientist magazine published an article titled 1 [] about the  [] which stirred up a fair degree of controversy and  that New Scientist was engaging in pseudo-science.   Since the original article the inventor claims that a "Technology Transfer contract with a major US aerospace company was successfully completed", and that papers have been published by Professor Yang Juan of The North Western Polytechnical University, Xi'an, China. 2   Furthermore, it was  that the Chinese were going to attempt to build the device.  Assuming that the inventor is operating in good faith and that the device actually works, is there another explanation of the claimed resulting propulsion?  Notes:  1. Direct links to the article may not work as it seems to have been archived. 2. The abstracts provided on the  website claim that they are Chinese language journals which makes them very difficult to chase down and verify.