Proof by induction for recursive sequence $a_n =a_{\lfloor n/2 \rfloor} + a_{\lceil n/2 \rceil} +3n +1$
Induction with floor, ceiling $n\le2^k\implies a_n\le3\cdot k2^k+4\cdot2^k-1$ for $a_n=a_{\lfloor\frac{n}2\rfloor}+a_{\lceil\frac{n}2\rceil}+3n+1$
Are IP addresses with and without leading zeroes the same? What does "apostrophe s" stand for in "What's it all mean"? I heard someone on a TV show say:     What's it all mean?   As far as I know, 's could be the short form of has and is. But in this case, it seems more like to be the short form of does. Am I correct? btrfs with multi devices fails automount in fstab I'm trying to automount btrfs  on 12.04-desktop-amd64. During bootsplash, fstab failed to mount btrfs and asked skip or manually prepare.  On the rootshell, btrfs device scan and mount -a could automount properly. It seems "btrfs device scan" was missing from bootstrap.  Where should I add the line to prepare bootstrap? Why did I just get the Announcer Badge and the Booster badge? Today I got both the Announcer badge and the Booster badge for  that I asked last year.   I don't remember ever sharing a link to it, and it seems odd to me that I would get both badges in one day, implying that there were at least 276 people who viewed it today from a link I don't remember ever sharing.  What's going on? Was Luke Skywalker the leader of the Rebel forces on Hoth? Rewatching Star Wars: The Empire Strikes Back, I noticed the opening crawl declares Luke Skywalker as the leader of the Rebel forces on Hoth:     Evading the dreaded Imperial Starfleet, a group of freedom fighters   led by Luke Skywalker has established a new secret base on the remote   ice world of Hoth.   Is that correct? I though Princess Leia or General Rieekan commanded the Rebel base on Hoth.  I can't find any references to this discrepancy on the Internet. Maybe in an earlier version of the script, Luke Skywalker was the commander and the crawl wasn't updated. Or perhaps "led" is meant in the sense Luke was a moral leader of the Rebel fighters? TZInfo::DataSourceNotFound error starting Rails v4.1.0 server on Windows What is a good package for displaying algorithms? I found this page with some examples:   I'm wondering which one to use. Any suggestions? I'm using other packages too so whichever is less likely to interfere with them is better for me. Photons travel at the fastest speed in our universe, the speed of light.  Do photons have acceleration? But when I try to run it using  tar zxvf jre-8u151-linux-i586   I get tar (child): jre-8u151-linux-i586: Cannot open: No such file or directory tar (child): Error is not recoverable: exiting now tar: Child returned status 2  What is wrong? The name and place matches prefectly. Non-ASCII characters in external links I have designed a circuit where I want to give a pulse of very short duration using a switch. How can I implement it in spice. How to know if Google Sheets IMPORTDATA, IMPORTFEED, IMPORTHTML or IMPORTXML functions are able to get data from a resource hosted on a website? Google Sheets has several import functions   IMPORTDATA IMPORTFEED IMPORTHTML IMPORTXML   Sometimes the above functions returns errors like #N/A Imported content is empty and would like to be sure that there isn't a problem with the resource content to be imported.  How do I know if these functions are able to get the content that I looking to import?  I know that there also exist IMPORTRANGE but that function only is able to import content from Google Sheets spreadsheets Can you release your zanpakuto if you are mute? So, this sounds like a stupid question I know but in the bleachverse 'names' are very important and you can only unleash the power of your zanpakuto if you can call out its name. Therefore, if a shinigami is mute, will they ever be able to achieve shikai/bankai? If so then how? Only one case that I know of is for shinigamis who have already achieved bankai to be able to release their shikais without a command.  Answers based on LN are welcome, but if you are citing from a dubious source or a source whose canon status can be question, please specifically mention it. I am mostly interested in an answer based on a  canon source. Mathematical operators, such as function names, should be set in roman type, not italics. Latex already has commands for some operators, including \max, \min, and \log.  How can I define additional such commands? How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.  This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated. What happens if no one survives the Hunger Games? It is possible that the remaining last two tributes might get killed at the same time, for example, by a land mine explosion, and in that case the games will end with no victor standing.  Is there a reference in the original books as to what happens if nobody emerges victor in the Hunger Games? How can I make Ubuntu 18.04 / 18.10 desktop use Unity (be like Ubuntu 14.04)? I recently installed Ubuntu 18.04 and I'm not keen on the desktop environment. I want the desktop env. I used to have in Ubuntu 14.04. How can I get it?  I've attempted to use , but when I try to install the Unity's 7.2.5 version using the command:  apt install unity=7.2.5   it gives me   E: Version '7.2.5' for 'unity' was not found   For sure I need to add a PPA for this, but which one!? Dimensioning of a technical drawing in TikZ There is a nice property of normal spaces, namely, closed disjoint subsets can be separated by continuous functions into $\mathbb R$.   Then you ask yourself, what about Hausdorff spaces?, are all Hausdorff spaces completely Hausdorff?   Well, if we choose our separating space $X$, that is, the space such that for any Hausdorff $Y$ and any distinct $a,b\in Y$ there is some continuous $f:Y\rightarrow X$ with $f(a)\neq f(b)$, to be completely regular and Hausdorff, this cannot happen by an example of E. Hewitt, Hewitt’s Condensed Corkscrew, call it $\mathbb H$; as any continuous function from $\mathbb H$ into $\mathbb R$ is constant, and any Hausdorff completely regular space $X$ can be embedded into a product of closed intervals $\prod_{i\in I}[0,1]$, and thus any continuous $f:\mathbb H\rightarrow X\subseteq \prod_{i\in I}[0,1]$ is constant, as each $\pi_i\circ f$ is constant, so our candidate to be the separating space $X$ cannot even be $[0,1]$.  My question is, does there exist a Hausdorff space $X$ such that for all Hausdorff spaces $Y$, and any distinct $a,b\in Y$ there exists a continuous $f:Y\to X$ such that $f(a)\neq f(b)$?  Thanks. Where in the DNS process are NS records used? I register a new domain, mynewdomain.com, at a hosting company.   In the advanced DNS settings there is an A record which maps mynewdomain.com to a ip address.  In my webhost control panel, it shows me the nameservers for the domain: ns1.mywebhost.net &amp; ns2.mywebhost.net  When I type in mynewdomain.com into a browser and hit enter I believe the following happens.   ISP resolver searches "." nameserver ISP resolver then told to search ".com" nameserver instead. ISP resolver finds mynewdomain.com, and searches the zone file to get the A record.     So my question is, where does my webhost's nameservers come into play? I believe the flaw in my logic comes from the fact that my zone file (in which the A record is kept) is stored on ns1.mywebhost.net). But if that is the case, how does my ISP resolver know to look here? (because I didn't type this into my address bar)