Have there been cases of errors in math papers, that were undetected for so long, that they caused subsequent errors in research, citing those papers. ie: errors getting propagated along. My impression is that this type of thing is extremely rare.  What was the worst case of such a scenario? Thanks.
I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of time before someone found a hole in the argument. Does this happen anymore now that we have computers? I imagine not. But it seems totally possible that this could have happened back in the Enlightenment.   Feel free to interpret this how you wish!
Permanent PATH variable How will I make this /media/De Soft/mongodb/bin PATH variable permanent?  Everyone is saying "export PATH=$PATH:media/De\ Soft/mongodb/bin to your ~/.profile, or .bashrc, or .zshenv depending on your shell".   I don't know what is ~/.profile, or .bashrc, or .zshenv. What do they do actually?  How will I add export PATH=$PATH:my/path to my .profile/.bashrc/.zshenv?   I'm using 64 bit Ubuntu 14.04 LTS with default terminal. How to format a USB flash drive? In Windows there is a straightforward "Format" option in a flash drive's right-click pop-up menu. Where's that in Ubuntu? Two particles with initial positions and velocities $r_1,v_1$ and $r_2,v_2$ are interacting by the inverse square law (with G=1), so that  $$ {d^2r_1\over dt^2} = - { m_2(r_1-r_2)\over |r_1-r_2|^3} $$ $$ {d^2r_2\over dt^2} = - { m_1(r_2-r_1)\over |r_1-r_2|^3} $$  (the inverse square law along the line of separation). What is the complete solution of these differential equations? What is the position of the two objects as a function of time?  After reading a lot on Wikipedia, I've come to the definition of center of mass and relative coordinates:   $$R(t) ~=~ \frac{m_1 r_1 + m_2 r_2}{m_1+m_2}$$  $$\ddot{r}(t) r(t)^2 ~=~ (m_1+m_2)G$$  Where $R$ is the center of mass, and $r$ is the displacement between the particles... Is this correct? How do I proceed to solve the differential equation? Graph theory: Tree holds two sub-tree with an amount of vertices Difference between a Structure and a Union Is there any good example to give the difference between a struct and a union? Basically I know that struct uses all the memory of its member and union uses the largest members memory space. Is there any other OS level difference? Using notify-send with cron Then what is the probability that out of $n$ men $A_1,A_2,....,A_n$ each aged $x$ years,$A_1$ will die and will be the first to die Let $p$ be the probability that a man aged $x$ years will die in a year time.Then what is the probability that out of $n$ men $A_1,A_2,....,A_n$ each aged $x$ years,$A_1$ will die and will be the first to die?  The probability that a man aged $x$ years will die in a year time$=p$  The probability that a man aged $x$ years will not die in a year time$=1-p$  But from here on,i got stuck.Some hints and suggestions are welcome. Objective-C Static Class Level variables Converting Degree Minutes to decimal degrees using Excel formula? I have location points collected from a garmin device stored in an excel sheet in Degree Minutes format --- W00208.172,N1046.977.  I am looking for an Excel formula to convert either to Decimal Degrees or Degrees Minutes seconds Format ? Why didn't Palpatine publicly change his name once he became Emperor? Anakin Skywalker was known as Darth Vader (this was already his Sith name) after the creation of the Galactic Empire.  So why did Palpatine remain "Emperor Palpatine"? Why didn't he embrace his persona as Darth Sidious?  I started wondering after reading one of the answers to Connection pooling in PHP Is it possible to cache database connections when using PHP like you would in a J2EE container?  If so, how? How to use hook_node_presave to change field's value Basically, I want to put first name &amp; surname together into $node->field_par_contact_name  But it gives me warning. (See below)   function partnership_node_presave($node) {   if($node->type == 'partnership') {     $first_name = $node->field_par_contact_first_name['und'][0]['name'];     $surname = $node->field_par_contact_surname['und'][0]['name'];      // https://drupal.org/project/field_hidden     // Hidden variable     $node->field_par_contact_name['und'][0]['value'] = $first_name. ' '. $surname;      // Warning after save: array_flip(): Can only flip STRING and INTEGER values! in DrupalDefaultEntityController->load() (line 173 of .../drupal-7.23/includes/entity.inc   } }   Updated, please see image: css background color with floating elements What is the word for act of monotonously demanding for something (many be a toy) repeatedly? Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. Strange floating-point behaviour in a Java program In my program I have one array with 25 double values 0.04 When I try to sum these values in a loop I get following results:  0.0 + 0.04 = 0.04 0.04 + 0.04 = 0.08 0.08 + 0.04 = 0.12 0.12 + 0.04 = 0.16 0.16 + 0.04 = 0.2 0.2 + 0.04 = 0.24000000000000002 0.24000000000000002 + 0.04 = 0.28 0.28 + 0.04 = 0.32 0.32 + 0.04 = 0.36 0.36 + 0.04 = 0.39999999999999997 0.39999999999999997 + 0.04 = 0.43999999999999995 0.43999999999999995 + 0.04 = 0.4799999999999999 0.4799999999999999 + 0.04 = 0.5199999999999999 0.5199999999999999 + 0.04 = 0.5599999999999999 0.5599999999999999 + 0.04 = 0.6 0.6 + 0.04 = 0.64 0.64 + 0.04 = 0.68 0.68 + 0.04 = 0.7200000000000001 0.7200000000000001 + 0.04 = 0.7600000000000001 0.7600000000000001 + 0.04 = 0.8000000000000002 0.8000000000000002 + 0.04 = 0.8400000000000002 0.8400000000000002 + 0.04 = 0.8800000000000002 0.8800000000000002 + 0.04 = 0.9200000000000003 0.9200000000000003 + 0.04 = 0.9600000000000003   Why on earth could that happen?! How do I return the response from an asynchronous call? A function takes every function value twice - proof it is not continuous I want to prove the following nice statement I've found:  A function $f: [0,1] \rightarrow \mathbb{R}$ takes every function value twice - proof it is not continuous  I've already found an answer to my question but one thing is still not clear to my mind. The answer I found was :   Whole post:  "Assume that you have a function $f$ that does take every real value exactly twice. Let $f(a) = f(b) = 0$ with $a &lt; b$. Let $c \in (a,b)$ be given, and assume without loss of generality $f(c) &gt; 0$. Then for every $0 &lt; \epsilon &lt; f(c)$ there exist $d \in (a,c)$ and $e \in (c,b)$ such that $f(d) = f(e) = \epsilon$. So on the intervals $[0,a)$ and $(b,1]$ our function cannot go any higher than zero. That means that $f$ should take every positive real value on the interval $[a,b]$. However the restriction of $f$ to $[a,b]$ is continuous from a closed interval to $\mathbb{R}$ and thus bounded. Contradiction."  Now everything is clear here except for:  "However the restriction of $f$ to $[a,b]$ is continuous from a closed interval to $\mathbb{R}$ and thus bounded. Contradiction."  Well I think know what Arthur is telling there: Because of the boundary there is always at least one function value that can't be taken twice by the function, e.g. just like the peak of $-x^2$.  But that is exactly where I was stuck before I started researching that issue. Is it really possible to just say it the way Arthur did? I thought that I'd have to further somehow prove the statement that boundary leads to that contradiction.    I hope that it is clear what I was trying to express here...  Anyways as always - thank you for your help!  FunkyPeanut What is the difference between .pem, .csr, .key and .crt and other such file extensions? I'm new to SSL / TLS and I want to work with the OpenSSL toolkit.  I don't know what .pem and .csr stands for?  I do know that .key is the private key and .crt is the public key. I haven't seen this behavior out in the wild much, but it happened again recently and I was wondering if a flag could be automatically raised when this happens:   User posts answer Answer is down voted User deletes answer User posts exact same text as a new answer User effectively ditched down votes   Got down voted again? Wash, rinse and repeat. Profit (perhaps after modifying the post, which is what you should have done to begin with)    I feel that this should raise a duplicate content flag, with slightly stronger wording. Would this be possible to implement? Or, perhaps, don't let users post new answers if they have a deleted answer to a question. Rather, force them to edit and restore the deleted answer instead, unless locked by a moderator.  I do have a link, but I don't want a lot of people to descend on it and create an even bigger disruption. It happened exactly as I described.