Does 'incomplete' mean the same thing in formal arithmetic compared to Godel's work?
Completeness and Incompleteness
Duplicate explanation should encourage improvement of original The new closed-as-duplicate banner says:     This question has been asked before and already has an answer. If those answers do not fully address your question, please .   I think it should instead say     This question has been asked before and already has an answer. If answers there do not fully address your question, please  or  to the pre-existing one.   &mdash; to discourage re-re-asking the same question. (I also changed "those answers", which has no plural , to "answers there", which doesn't need one.)    As of September 19, 2013, the original text quoted above has returned here. If I call a command using  in Ruby, how do I get its output?  system("ls") If $p$ and $q$ are distinct primes , prove that $$p^{q-1}+q^{p-1} \equiv 1 \pmod {pq}$$     Using Fermat's Theorem we can get $$p^{q-1} \equiv 1 \pmod q, \qquad q^{p-1} \equiv 1 \pmod p.$$  After that I have no idea at all. Can anyone help me please I installed vsFTPd for running an FTP server on  (Wheezy). I checked the vsFTPd version was 2.3.5, and I configured it like so:  listen=YES local_enable=YES write_enable=YES chroot_local_user=YES pasv_min_port=15000 pasv_max_port=15200 allow_writeable_chroot=YES   I followed these articles for solving this problem:       And many others on Google and forums, but my problem was not solved.  NOTE: I have solved this problem on  (Precise Pangolin), but that solution does not work on Debian 7.3.  I really mixed up on it?! UTF-8 all the way through I'm setting up a new server and want to support UTF-8 fully in my web application. I have tried this in the past on existing servers and always seem to end up having to fall back to ISO-8859-1.  Where exactly do I need to set the encoding/charsets? I'm aware that I need to configure Apache, MySQL, and PHP to do this — is there some standard checklist I can follow, or perhaps troubleshoot where the mismatches occur?  This is for a new Linux server, running MySQL 5, PHP, 5 and Apache 2. The weighted sum of two independent Poisson random variables Using wikipedia I found a way to calculate the probability mass function resulting from the sum of two Poisson random variables. However, I think that the approach I have is wrong.  Let $X_1, X_2$ be two independent Poisson random variables with mean $\lambda_1, \lambda_2$, and $S_2 =  a_1 X_1+a_2 X_2$, where the $a_1$ and $a_2$ are constants, then the probability-generating function of $S_2$ is given by $$ G_{S_2}(z) = \operatorname{E}(z^{S_2})= \operatorname{E}(z^{a_1 X_1+a_2 X_2}) G_{X_1}(z^{a_1})G_{X_2}(z^{a_2}). $$ Now, using the fact that the probability-generating function for a Poisson random variable is $G_{X_i}(z) = \textrm{e}^{\lambda_i(z - 1)}$, we can write the probability-generating function of the sum of the two independent Poisson random variables as $$ \begin{aligned} G_{S_2}(z) &amp;= \textrm{e}^{\lambda_1(z^{a_1} - 1)}\textrm{e}^{\lambda_2(z^{a_2} - 1)}  \\ &amp;= \textrm{e}^{\lambda_1(z^{a_1} - 1)+\lambda_2(z^{a_2} - 1)}. \end{aligned} $$ It seems that the probability mass function of $S_2$ is recovered by taking derivatives of $G_{S_2}(z)$ $\operatorname{Pr}(S_2 = k) = \frac{G_{S_2}^{(k)}(0)}{k!}$, where $G_{S_2}^{(k)} = \frac{d^k G_{S_2}(z)}{ d z^k}$.  Is this is correct? I have the feeling I cannot just take the derivative to obtain the probability mass function, because of the constants $a_1$ and $a_2$. Is this right? Is there an alternative approach?  If this is correct can I now obtain an approximation of the cumulative distribution by truncating the infinite sum over all k? Is there a software utility to adjust screen gamma/brightness/contrast? What spell targets count for the Graviturgy Magic wizard's Gravity Well feature? The  feature from the  (EGtW, p. 185) triggers &quot;whenever you cast a spell on a creature&quot;, and has an effect &quot;if the target is willing to move, the spell hits it with an attack, or it fails a saving throw against the spell.&quot; It's not clear to me what kinds of spells can actually trigger this feature. Based on the singular phrasing of &quot;on a creature&quot; and the later use of the word &quot;target&quot;, I would normally read this as only applying to spells that explicitly target creatures (and perhaps only those with a single target, in contrast to the phrasing of the  feature of the  wizard, for example). However, spells that target a location can use the word target in a different sense (see ). What's actually going on here? Does this feature only work on targeted spells? Or does it function for area of effect spells as well? Or is this just an ambiguous phrasing that will rely on the DM's ruling? How to disable the touchpad? I want my touchpad to be disabled when I use my mouse. How can I do that? touchpad-indicator has stopped working on 11.10. It used to work on 11.04. Gnome3 is not a solution as I don't like it and find it buggy. Solving Hardy Weinberg problems I really fail to understand Hardy-Weinberg equilibrium and can't find an easy enough source of information.  Can you help me to understand Hardy-Weinberg equilibrium?  My goal is to be able to solve the following kind of problem     In a population with two alleles for a certain locus, B and b, the allele frequency of B is 0.7. What is the frequency of heterozygotes if the population is in Hardy-Weinberg equilibrium? According to the Wikipedia page for the , "Use of the comma is consistent with conventional practice" and "Use of the comma is inconsistent with conventional practice." Did the Oxford Comma come before its omission, or was the Oxford Comma traditionally omitted?  It makes logical sense that every item in a list would be separated in the same way: by a comma. If the Oxford Comma is conventionally correct, when and why did people begin to omit it? Laptop battery charge management I work on a HP laptop plugged in most of the time, and after a few months the battery went off-duty.  Of course, I could remove the battery each time the laptop is plugged and replace it each time I must move to where no sockets are around. But I think that:   It would be really BORING! The computer would not avail itself of the useful backup power supply a battery can provide in case of black out.   It should simply switch off the battery charging when a safe power level is reached (i.e. 95%) and start recharging it, not every five minutes, but only under a certain safe threshold (i.e. 40% - 50%).  In its energy-saving profiles, Windows 7 only allows for setting up screen brightness, screen fadeout, screen deactivation and the suspension, not managing any charging issue.  Is there a way to accomplish this? An irreducible polynomial in a subfield of $\mathbb{C}$ has no multiple roots in $\mathbb{C}$ How can I extract a page range / a part of a PDF? Do you have any idea how to extract a part of a PDF document and save it as PDF? On OS X it is absolutely trivial by using Preview. I tried PDF editor and other programs but to no avail. I would like a program where I select the part that I want and then save it as a PDF file with a simple command like CMD+N on OS X. I want the extracted part to be saved as PDF and not as JPEG, etc. How can I find the version of Ubuntu that is installed? Proper submodules of Noetherian modules are intersections of indecomposable modules If $M$ is Noetherian, any submodule $N \neq M$ can be written as a finite intersection of intersection-indecomposable ones.   I just don't know how to begin with the proof of this proposition.   Very kind of you for reading or answering. Is it possible to view Google Chrome bookmarks and history from the terminal Is it possible to view Google Chrome bookmarks and history by using command line? Evaluate  $$\int_0^\infty \frac{\log x \; dx}{x^{2} + 2x + 2}$$  by integrating a branch of $(\log z)^{2}/(z^{2} + 2z +2)$ along a keyhole contour.  The thing I have trouble with is why I should be examining the square of the log - I guess it has something to do with ln x in fact NOT being the real part of log z, since log z is (or can be) defined on the entire negative real axis and ln x can't. But looking at the square would definitely not have been my first plan of attack :( The app forgets that I voted when I scroll the question How do I boot into a root shell? When booting to the graphical desktop fails we sometimes are recommended to boot into a "root shell" or to boot in "recovery mode".  How do I do that?