Convergent sequences
Inequality between $\ell^p$-norms
How do I solve this limit: $\lim _{x \to 0} \left(\frac{ \sin x}{x}\right)^{1/x}$? Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times? If not, how could I prove that such a function does not exist? Evans PDE Problem 5.15: Poincaré inequality for functions with large zero set I have trouble proving the following problem (Evans PDE textbook 5.10. #15). Could anyone kindly help me solving the problem? I know that I should somehow use Poincaré inequality but I still cannot solve it.   Fix $\alpha&gt;0$ and let $U=B^0(0,1)\subset \mathbb{R}^n$. Show there exists a constant $C$ depending only on $n$ and $\alpha$ such that $$ \int_U u^2 dx \le C\int_U|Du|^2 dx,  $$ provided that $u\in W^{1,2}(U)$ satisfies $|\{x\in U\ |\ u(x)=0\}|&gt;\alpha$. efficient and accurate approximation of error function I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z  e^{-t^2} \,\mathrm dt$$ Conditional formatting of an entire row based on multiple cells I'm trying to make a conditional formatting rule in Google Spreadsheet that would highlight a row green based on text in multiple cells.   If cells L2, M2, and N2 = APPROVED; and cell O2 = X; then row 2 = highlighted green. I have interesting bug that I have experienced before but has gone away on its own. This one is not going away though.  I have a giant stripe or chunks missing from my desert biome, I would say probably 7 chunks long.  I'm guessing the chunks are corrupted or something, what was hoping to figure is whether it can repaired or reverted.  Here is a picture of the problem. After update, get "error: file not found." followed by "grub rescue> _" How would gravity change on a planet rotating around itself very fast? Let's take a planet identical to Earth, but with rotation speed multiplied by ten thousand. What would happen with the gravity if it was spinning madly around itself? Would the centrifugal force make objects seem lighter than on normal Earth? Would the people on that planet be able to jump higher? Why do investors pay trillions for minority stakes in companies, when their only potential payback is modest uncertain dividends? I'm struggling to understand the core reason to invest in stocks as a minority shareholder. Majority shareholding makes more sense as you effectively control the company, so you can pay yourself dividends or even sell the entire company at will. You can guarantee your own payback.  However, as a minority shareholder, you don't get to make any of these decisions. You're basically just hoping to eventually, some day get paid, if the majority holders decide so.  Why do so many investors agree to pay now huge sums of money (trillions in total) for something that may eventually pay dividends, and even in the best case, these dividends are only a fraction of the price you paid to acquire these stocks?  The average annual dividend rate for the US stock market is 2%. That means if you bought 100 million worth of shares, you'd have to wait 50 years to recover your investment.  Of course, the most probable way you'd eventually get liquidity for these shares is by selling them to some other starry-eyed investor - which makes the whole thing seem like a giant Ponzi scheme.  Given all that, it's unclear to me why the public is willing to spend so much money to acquire minority shares in companies. The Megahertz Myth became a promotional tactic due to differences between the PC's INTEL 8086 processor and Apple's Rockwell 6502 processor.  The 8086 ran at 4.77MHz while the 6502 ran at 1MHz.  However, instructions on the 6502 needed fewer cycles; so many fewer, in fact, that it ran faster than the 8086.  Why do some instructions need fewer cycles?  And why can't the instructions of the 6502, needing fewer cycles, be combined with a fast cycling processor of the 8086?  Wikipedia's article for instructions per cycle (IPC) says     Factors governing IPC   A given level of instructions per second can be achieved with a high IPC and a low clock speed...or from a low IPC and high clock speed.   Why can't you have both high instructions per cycle and high clock speed?  Maybe this has to do with what a clock cycle is?  Wikipedia mentions synchronization of circuits?  Not sure what that means.  Or maybe this has to do with how a pipeline works?  I'm not sure why instructions in a short pipeline are different from instructions in a long pipeline.  Any insight would be great!  Just trying to understand the architecture behind the myth.  Thanks!  References: Accessing private member variables from prototype-defined functions How to split bibliography into "works cited" and "works not cited"? I need two bib-like sections: a "References" giving works explicitly cited in my text, and a "Bibliography" (or "Further reading" if you prefer) giving everything from my bibfile that didn't already appear in "References". The first is of course completely standard, but the second... I'm stumped.  Further loose constraints: I'm somewhat committed to biblatex and the biblatex-apa style. But I'll happily accept an answer not related to them if there's any chance I can adapt it. This is a  about understanding and remediating the Heartbleed security issue.   What exactly is CVE-2014-0160 AKA "Heartbleed"? What is the cause, what OSs and versions of OpenSSL are vulnerable, what are the symptoms, are there any methods to detect a successful exploit?   How can I check to see if my system is affected? How can this vulnerability be mitigated? Should I be concerned that my keys or other private data have been compromised? What other side effects should I be concerned about? I have to design a circuit in which I have to transfer data through lasers (serial transmission of data). Now what I was thinking was to check if the laser light is on then '1' is passed and when the laser is off '0' would be passed. Now I need to check if a 1 or a 0 was passed.   How would I check for this and another problem I'm facing is if it's on it may keep on passing 1's. How would I check for this? I though of using LDR's (Light Dependent Resistors)? Would this be okay or it there a better method? Functional Analysis. Continuous extensions of linear functionals. I would like to show that a continuous functional on a subspace $W$ of a normed space $(V,\|\cdot \|)$ has a unique continuous extension to $V$ iff $W$ is dense in $V$.  I have proved $(\Leftarrow$). But the converse is currently eluding me! Help would be much appreciated! How did nobody at Hogwarts (especially Dumbledore) figure out what killed Myrtle? The information is all clearly there, since Harry and Ron, two 12-year-olds, were able to figure it out (with a bit of help from a petrified Hermione), thought at the time Aragog was much younger and smaller and wouldn't have been as easy to find, so maybe the "spiders flee before it" clue would have been lost, although to be honest it's not exactly as vital a piece of the puzzle as the others.  Dumbledore himself is obviously a very capable and intelligent Wizard, and after Myrtle died she came back as a ghost and was able to provide (admittedly limited) details of her own death.  I don't know about you, but I'd have thought at least someone would have tried to piece it together at the time, and it wouldn't have taken them long to figure out "Oh, she saw a pair of big yellow eyes after hearing a boy speak a funny language in the bathroom and dropped dead? Sounds like she might have heard somebody order a Basilisk to look her in the eyes in parseltongue", especially if the investigator in question was someone like Dumbledore.  Do we have any info on why nobody seemed to have found out anything in 50 years prior to Harry Potter (and, probably more importantly, Hermione Granger) turning up at Hogwarts? Clothing store access in Pokémon X & Y Once the player obtains access to Lumiose City for the first time, they'll be able to visit Boutique Couture from the city's Vernal Avenue (South fork, connecting from South Boulevard). Upon entering the store without performing any other actions, the player will be turned away and informed they "aren't stylish enough" and that they should visit more places in Lumiose City before visiting again.  This description is dubious, and doesn't inform the player of what they need to accomplish to gain access. Specifically:  Even beating the game (completing the Pokémon League) fails to allow access, as does modifying the player's look, visiting most shops in all areas of Lumiose City, and performing all hair stylist, PR studio, and associated actions. Completing a set of triple battles in Restaurant Le Wow did set the event flag that allowed access in my copy of Y, but it is not clear if this is the only requirement for access.  So other players may benefit, and so I know how to repeat this process if I obtain a copy of X or the future release of Z, what exact requirements yield access to this store for the first time? Straight lines - product of slope of perpendicular lines. Let $l$ be a line with slope $\frac{1}{0}$, and $m$ a line with slope $0$.  Now, line $l$ is perpendicular to line $l$, therefore the product of the slope of line $l$ and the slope of line $m$ equals $-1$. This implies that $\frac{1}{0} \times 0 = -1$! Counting homomorphisms between $\Bbb Z$ and $\Bbb Z_n$ and also between $\Bbb Z_n$ and $\Bbb Q$ How do I disable DVD auto eject in Windows 7? I just discovered an incredibly annoying "feature" of Windows 7 tonight.  If you click on your optical drive in Explorer, and there is no disc in the drive, Windows 7 is kind enough to open the tray for you.  I'm finding it hard to contain my rage over this idiotic design decision.  WTF?  According to , this behavior was apparently introduced in Vista.  The author posted a workaround, but I don't want a workaround.  I want it to stop doing this.  Like he, I have a case with a door that covers the drive bays.  If it's closed, and I accidentally click on that optical drive, it knocks on that door 10 or 12 times before giving up.  Can this behavior be turned off in Windows 7?