Limit of product with unbounded sequence $\lim_{n\to\infty} \sqrt{n}(\sqrt[n]{n}-1)=0$
How to prove $\lim_{n \to \infty} \sqrt{n}(\sqrt[n]{n} - 1) = 0$?
Asynchronous shell exec in PHP I've got a PHP script that needs to invoke a shell script but doesn't care at all about the output. The shell script makes a number of SOAP calls and is slow to complete, so I don't want to slow down the PHP request while it waits for a reply. In fact, the PHP request should be able to exit without terminating the shell process.  I've looked into the various exec(), shell_exec(), pcntl_fork(), etc. functions, but none of them seem to offer exactly what I want. (Or, if they do, it's not clear to me how.) Any suggestions? Difference between "I'm home" vs "I'm at home" What's the difference between these two sentences? And which one's grammatically correct? I'm stuck with the proof of the following:  Suppose that $(s_n)$ converges to $s$, $(t_n)$ converges to $t$, and $s_n \leq t_n \: \forall \: n$. Prove that $s \leq t$.   I've tried starting with $s_n \leq t_n \: \forall : n$ and the definitions of each limit (i.e. $|s_n - s| \leq \epsilon \: \forall \: n &gt; N_1$), but I'm not really getting very far. Any help is appreciated! I wonder whether it is possible to draw the three following quadric surfaces (cone, hyperboloids)  x^2+y^2-z^2=0  x^2+y^2-z^2=1  x^2+y^2-z^2=-1  (and the axis)   using pgfplots 3d or TikZ as it is done in the following image What are the uses of "using" in C#? How to install applications to a separate hdd? I was so good with windows, now I’m feeling like a complete idiot.  I have an SSD for ubuntu and want to install the majority of my programs to a separate HDD but I have no clue how. All searches I have tried only turn up with dual-boot installations.   (RESOLVED -> ) I am also having trouble locating "/etc/". Specifically, to modify Conky. Searching for this term only turns up instances where people are telling you to access this file, not how to get to it. How do I give players already enchanted Tools/Armour/Weapons Why square the difference instead of taking the absolute value in standard deviation? In the definition of standard deviation, why do we have to square the difference from the mean to get the mean (E) and take the square root back at the end? Can't we just simply take the absolute value of the difference instead and get the expected value (mean) of those, and wouldn't that also show the variation of the data? The number is going to be different from square method (the absolute-value method will be smaller), but it should still show the spread of data. Anybody know why we take this square approach as a standard?  The definition of standard deviation:  $\sigma = \sqrt{E\left[\left(X - \mu\right)^2\right]}.$   Can't we just take the absolute value instead and still be a good measurement?  $\sigma = E\left[|X - \mu|\right]$ How to batch import Wavefront OBJ files? Is it possible to batch import Wavefront .obj to blender? I'm working on 2.69. If I select multiple .obj files, only one gets imported. Also if I hit A on file selection window to select all files (.obj and .mtl) I get an error, nothing gets imported. I have to import several hundreds of .obj into the blender file, is this possible? I have a cron job that runs every minute on an Ubuntu 12.04 box  $sudo crontab -e * * * * * mylogin /pathto/file.sh &gt; /log/filelog   If I run file.sh, the bash script does some stuff and echos out runs:  $ ./file.sh  runs   If I check the log of cron tab it shows that the job is running:  Jul 10 12:41:01 localhost CRON[1811]: (root) CMD (mylogin /pathto/file.sh &gt; /log/filelog) Jul 10 12:41:01 localhost CRON[1810]: (CRON) info (No MTA installed, discarding output) Jul 10 12:42:01 localhost CRON[1813]: (root) CMD (mylogin /pathto/file.sh &gt; /log/filelog)   However, the script is not running. It is not doing it's work and not echoing runs to /log/filelog.  $cat /log/filelog  #shows nothing   What other steps can I take to debug this problem? It seems like to deleted too much files by rm -r and now I have the error of "the system is running a low-graphic mode" in Ubuntu and I have to work in Windows because of that.  However I can load it in recovery mode. I need to copy the files from Downloads folder to a flash drive. Downloads folder has them as I can see by ls command.   When I plug in a flash drive, there will be some messages saying that everything is ok (as I understood) and it's plugged. But when I go to /media/my_name and do ls then I don't see anything. I don't see anything in /run/media/my_name either(I'm not sure if should though).  I can't connect to the Internet for some reason. Ubuntu was installed by wubi.exe file.  How do I save the content of my Downloads folder to a flash drive? Java Command line arguments My model has a clear plastic (or glass) dome with objects inside.  The objects appear dark.  I've tried the glass shader with color 100% RGB, with very small IOR, and using just a refraction shader.    I know Cycles has problems with caustics, but I just want broad illumination to fall clean through the glass.  The light-to-column path can ignore refraction, but is blocked or tremendously attenuated. The camera-column path needs refraction, and seems to work fine anyway.  This demo image offers all optical paths of interest:   light upon object, seen directly light upon object, seen through glass (looks okay) light through glass onto object, seen directly. This is the shadow on the lit side of the column just below the glass tubing.  I expect full illumination, but there is shadow.  There is some light, but it's diffuse light from the floor. light through glass onto object, seen through glass. This     I did at one point try non-progressive rendering, and find extremely coarse light where there shouldn't be shadow, but this is not a practical way to make nice images. I've even tried cranking up certain sampling settings to many thousands, but still get bad noise though less, at the cost of long render times. I  would like to finish my creations sometime before "Star Trek, the Generation After the Next" becomes real life, if you know what I mean.  The next image is an example. I think Transmission in Sampling was 5000, but not sure.    I'm using Cycles in Blender 2.67 on a quad core Linux machine, 16GB RAM. Machine limits have never been a problem for 3D rendering.  Possibly, I screwed up some setting when I made my startup file, and just need to fix it. OTOH maybe this is just a limitation of path tracing, or of the way Cycles does it. If that's the case, what to real-world Blender artists do when they need to produce images on a budget and schedule?  What is the practical real-world fix? Though tangentially related to another post on MathOverflow (), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...    Suppose $\Sigma$ is a sigma-algebra over a set, $X$. For any given topology, $\tau$, on $X$ denote by $\mathfrak{B}_X(\tau)$ the Borel algebra over $X$ generated by $\tau$.     Question 1. Does there exist a topology, $\tau$, on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$?   If the answer to the previous question is affirmative, it makes sense to ask for the following too:     Question 2. Denote by ${\frak{T}}_X(\Sigma)$ the family of all topologies $\tau$ on $X$ such that $\Sigma = \mathfrak{B}_X(\tau)$ and let $\tau_X(\Sigma) := \bigcap_{\tau \in {\frak{T}}_X(\Sigma)} \tau$. Is $\Sigma = \mathfrak{B}_X({\frak{T}}_X(\Sigma))$?   Updates. Q2 was answered in the negative by Mike (). Subsetting a 2D numpy array What book features people living in a tree who move up it in a bubble? Markov/Chebychev inequality If Z is a square integrable random variable such that $E[Z]&gt;0$, then prove that  $$P(Z\geq k) \geq \frac{(E(Z)-k)^2}{E(Z^2)} \forall k \in [0,E(Z)]. $$  This is a homework question and I don't even understand why the inequality is reversed comparing to Markov/Chebychev inequality. Should Cauchy-Schwartz inequality be used? If yes, then how to begin? Thanks a ton! Salesforce REST API authentication? I feel like this should be really easy to do, but I can't find anything about it on the web. Is there a way, through either javascript or a java web servlet, to skip the subsequent authentication?  We are trying to use the REST Api in production, but the problem is, that each time you refresh the page in javascript, or delete the cookies in java, you have to login again to salesforce.  Is there some way to do an ajax call or the login in the background or whatever, so that you can have your application connecting to salesforce right away? Is there a continuous function such that $\int_0^{\infty} f(x)dx$ converges, yet $\lim_{x\rightarrow \infty}f(x) \ne 0$? Interpreting the Kretschmann scalar