It is said that since work is done by the gravitational potential itself , so at a finite distance the gravitational potential energy of the body is negative. Could someone explain why?
Gravitational potential at a point in a gravitational field is defined as work done per unit mass in bringing a small test mass from infinity to the point.  I know that:   By integration, we find that gravitation potential is negative. The maximum value of gravitational potential which is at infinity distance is 0, therefore minimum value should be negative.   But, what I do not understand is force is in the same direction as displacement so why work done is not positive?
I get how a normal fork bomb works, but I don't really understand why the &amp; at the end of the common bash fork bomb is required and why these scripts behave differently:  :(){ (:) | (:) }; :   and   :(){ : | :&amp; }; :   The former causes a cpu usage spike before throwing me back to the login screen. The latter instead just causes my system to freeze up, forcing me to hard reboot. Why is that? Both continually create new processes, so why does the system behave differently?   Both of the scripts also behave differently from  :(){ : | : }; :   which doesn't cause any problems at all, even though I would have expected them to be alike. The bash manual page states that the commands in a pipeline are already executed in a subshell, so I'm led to believe that : | : should already suffice. I belive &amp; should just run the pipeline in a new subshell, but why does that change so much?  Edit: Using htop and limiting the amount of processes, I was able to see that the first variant creates an actual tree of processes, the second variant creates all the processes on the same level and the last variant doesn't seem to create any processes at all. This confuses me even more, but maybe it helps somehow? User and associated Post I have, not one but two microSD cards that my phone (Samsung Galaxy Young, Gingerbread OS) seem to have broken. One is 1GB and the other is 2GB. The 1GB one won't be formatted.  When I put the 1GB one into the computer the computer prompts for a formatting. I don't care for the content so I tried to format it, but to no avail; the format fails and I have no idea what to do to make it work again.  I tried using the SDformatter software, but it can't format the card as it is write protected. I'm googling to solve it but so far no success.  My computer OS is win7 if that's of any relevance. Strange GET requests in logs I'm getting the following GET requests in my apache logs:  109.230.251.14 - - [29/Mar/2011:16:28:18 +0100] "GET http://209.191.92.114/config/pwtoken_get?login=jackmcphee232&amp;src=ygodgw&amp;passwd=e59e2240415e6f1aba3da72b8f189f4e&amp;challenge=9TbU_9yfZhKmzlHtK9X4OkQlesTH&amp;md5=1 HTTP/1.0" 404 1226 "-" "-"   Any idea what it could be and how is that possibly a get request? That IP address seems to point to Yahoo! I'm very confused. Samsung netbook NP300E5A dimming on battery My Samsung netbook NP300E5A screen is continually dimming.  The screen is forty percent dim on battery, but if I plug in the charger, the screen will be normal. I have tried all I know and I could not fix it. What should I do? what am trying to explain  is that the system is dimming like flooricent light  the screen resolution systems are all ok. When I pluging my charge the dimming we stop but when is on battery it look like flashing someone  light when driving I have some troble understanding the Kernel density estimation.  If a consider the next example:  s&lt;-0 z&lt;-density(rnorm(25)) f&lt;-function(i)approx(z$x,z$y,xout=i)$y for(i in seq(min(z$x),max(z$x),length.out = 10000)) s&lt;-s+f(i) s   I expected getting that s is approximately equals to one since I am considering an approximation for the integral of the Kernel density function (that I suppose is also a density). However, I don't get the expected result. Why it doesn's sum one?  The idea behind this is that I wanted to use the Kernel as the density of my data points. I was interpolating to get my explicit function and use with other x values. Or do you know another way to get a density function estimated by fitting? How can I copy from Tmux terminal and paste it other application in my OS? I am using tmux on ubuntu. I read this  for copy and paste inside tmux. But I would like to know how can I copy from TMUX terminal and paste it to other application e.g browser in my OS? Tense change: previous actions on something that's currently true Ethernet: how important is the magnetics topology? I have some Ethernet issues with a set of boards (6/10 are bad). They're possibly just assembly house mistakes... but I'm very concerned about my magnetics jack.  I could have sworn I saw a example schematic with my jack's topology used with my PHY before, but I can't find it. The PHY's datasheet and suggested magnetics jacks are all identical in topology, and very different than what I have.  I'm using a SMSC  PHY, with a Bel  magnetics jack.  The transformer ratios on my jack vs. a suggested jack for this PHY like the  are the same (1:1).  The inductance is the same, 350uH. However, the choke on mine is on the circuit side, not network side. Also, while the TLA has four 75 ohm resistors from taps tied to a single 1nF/2kV cap, mine has an extra set of 1nF caps between the taps and the 75 ohm resistors.  The main difference is that my jack is a POE jack, and I thought I found an example that used one exactly like it. So, the topology is obviously different just because of that feature.  At the moment, I have 4 good boards, 1 'sometimes' board, and 5 boards that just blink a LED at 11.68Hz representing some error that no datasheet explains.  How important is this? Of course there's no other jack with that footprint.  EDIT  I added some detail on the magnetics jacks above. I was just recommended the SI-60152-F, so I'll compare to that one. Of course like all the other recommendations... it also has no stock at anyone on FindChips.  LATE EDIT  It turns out my problem was the crystal layout that fed the Ethernet PLL.  On the respin, I went with a more standard non-POE jack. List of lists changes reflected across sublists unexpectedly I needed to create a list of lists in Python, so I typed the following: my_list = [[1] * 4] * 3  The list looked like this: [[1, 1, 1, 1], [1, 1, 1, 1], [1, 1, 1, 1]]    Then I changed one of the innermost values: my_list[0][0] = 5  Now my list looks like this: [[5, 1, 1, 1], [5, 1, 1, 1], [5, 1, 1, 1]]    which is not what I wanted or expected. Can someone please explain what's going on, and how to get around it? Let $D \subset \mathbb{C}$ be an open, connected set and let $\{ u_n \}$ be a sequence of harmonic functions with $u_n: D \longrightarrow (0, \infty)$. Show that if $u_n(z_0) \rightarrow 0$ for some $z_0 \in D$, then $u_n \rightarrow 0$ uniformly on compact subsets of $D$.  If you could offer a hint or a helpful question to get me started I'd appreciate it.   Progress: I now see why showing it for the unit disk suffices (any compact subset in $D$ can be covered by finitely many disks each which is contained in $D$). It seems that through various related questions here, there is consensus that the "95%" part of what we call a "95% confidence interval" refers to the fact that if we were to exactly replicate our sampling and CI-computation procedures  many times, 95% of thusly computed CIs would contain the population mean. It also seems to be the consensus that this definition does not permit one to conclude from a single 95%CI that there is a 95% chance that the mean falls somewhere within the CI. However, I don't understand how the former doesn't imply the latter insofar as, having imagined many CIs 95% of which contain the population mean, shouldn't our uncertainty (with regards to whether our actually-computed CI contains the population mean or not) force us to use the base-rate of the imagined cases (95%) as our estimate of the probability that our actual case contains the CI?   I've seen posts argue along the lines of "the actually-computed CI either contains the population mean or it doesn't, so its probability is either 1 or 0", but this seems to imply a strange definition of probability that is dependent on unknown states (i.e. a friend flips fair coin, hides the result, and I am disallowed from saying there is a 50% chance that it's heads).  Surely I'm wrong, but I don't see where my logic has gone awry... Proving an identity relating to the complex modulus: $z\bar{a}+\bar{z}a \leq 2|a||z|$ Show that $z\bar{a}+\bar{z}a \leq 2|a||z|$  I was able to check this using some examples which is not ideal for a mathematician way of proving a problem/case/theorem. I couldn't generalize it (prove it). I will be glad if I can be given the hint/proof for this problem. Thank you. Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq n} \frac{1}{k}$, and failed. But should we necessarily fail?      More precisely, is it known that $H_n$ cannot be written in terms of the elementary functions, say, the rational functions, $\exp(x)$ and $\ln x$? If so, how is such a theorem proved?    Note. When I started writing the question, I was going to ask if it is known that the harmonic function cannot be represented simply as a rational function? But this is easy to see, since $H_n$ grows like $\ln n+O(1)$, whereas no rational function grows logarithmically.  Added note: This  asks a similar question for “elementary integration”. I guess I am asking if there is an analogous theory of “elementary summation”. If $x\sqrt{1+y}+y\sqrt{1+x}=0$ find $y'$ Find $\frac{dy}{dx}$ if $x\sqrt{1+y}+y\sqrt{1+x}=0$ for $-1\leq x\leq 1$   My Attempt $$ x\sqrt{1+y}=-y\sqrt{1+x}\implies x^2(1+y)=y^2(1+x)\implies x^2+x^2y=y^2+xy^2\\ 2x+2xy+x^2\frac{dy}{dx}=2y\frac{dy}{dx}+y^2+2xy\frac{dy}{dx}\\ \frac{dy}{dx}\Big[ x^2-2y-2xy \Big]=y^2-2x-2xy\\ \frac{dy}{dx}=\frac{y^2-2x-2xy}{x^2-2y-2xy} $$ How do I proceed further and find the derivative ? How to give different permissions to multiple groups of users? I installed Ubuntu 12.04 server and want to give full folder access say example: "Folder01" to group "Group01' and Limit "Group02" for read access only. How I can do this. How do I compare strings in Java? How does delete[] know it's an array? Simple examples of uncorrelated but not independent $X$ and $Y$