What is the longest discrepancy between death and publication?
One of my advisors suddenly passed away while I was in graduate school.  We had some discussions and ideas about future publications, but he passed away before any of the work was completed.  When the work was finally completed and published, I and my co-authors were therefore presented with an ethical dilemma about how best to acknowledge his contributions to the ideas behind the paper.  Should we list him as a co-author?  Put him in the acknowledgements?  Listing him as an author would give credit for the original idea, however, we would have no way of knowing if he actually approved of—and would want his name attached to—our methods and writing.  In the end my co-authors and I decided to list him as a co-author with a footnote stating that he passed away before publication.  I’m interested to hear from others who have been in similar situations and/or suggestions on what constitutes “co-authorship” when one of one’s collaborators passes away before the publication or work is complete.
How to evaluate environment variables into a string in Python? I have a string representing a path.  Because this application is used on Windows, OSX and Linux, we've defined environment variables to properly map volumes from the different file systems.  The result is:  "$C/test/testing"   What I want to do is evaluate the environment variables in the string so that they're replaced by their respective volume names.  Is there a specific command I'm missing, or do I have to take os.environ.keys() and manually replace the strings? On my headless NAS I have sdf1 (a flash-card) mounted as / while /home is mounted from lv00 (an LVM volume backed by a software RAID). To be able to access the machine when the RAID fails, I have a copy of my ssh public key, etc. in /home/foo/.ssh on the file-system from sdf1. To update the files that are hidden by the mounted /home I normally remount lv00 in /mnt/home, do what I have to do, and then move lv00 back in place. Is there a way to achieve this without unmounting /home? Is the composition of two injective functions injective? Why is this sentence correct? “She suggested that he go to the cinema.” $$\lim_{n\to\infty} (1+\frac{2}{n})^{n^2} e^{-2n} = ? $$         $e$   $e^2$   $e^{-1}$   $e^{-2}$      My answer is 1. Since $$\lim_{n \to \infty} \left(1+\frac{2}{n}\right)^{n} =e^{+2}.$$  Therefore $$\lim_{n\to\infty} \left(1+\frac{2}{n}\right)^{n^2} =e^{2n}.$$  Where am I wrong? Consider a "wavefunction" $\psi(x)$, which has a Fourier transform $\tilde \psi(p)$  Suppose that we know, for each $x$, $|\psi(x)|^2$, and that we know, for each $p$, $|\tilde \psi(p)|^2$.  Have we enough information to reconstruct the "wavefunction" $\psi(x)$, that is, obtain the phase of the "wavefunction" for all $x$ (up to a global phase, we are only interesting to the relative phases between the $x$)? How can one prove that the series $\displaystyle \sum\limits_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number? There's no need to use Taylor expansion, integrals or any advanced/professional techniques. It can be proved using only basic techniques. My first attempt was to assume that the series does converge to a rational number $ \frac{p}{q}$ and then it follows by definition that for every positive epsilon there exists an integer k such that $ {\sum_{n=1}^{k} \frac{1}{(n!)^2} -\frac{p}{q}}&lt;=\epsilon$ I tried to get a contradiction but I failed. I understand there is a method for solving simultaneous modular equations. For example; $$x = 2 \mod{3}$$ $$x = 3 \mod{5}$$ $$x = 2 \mod{7}$$ We find numbers equal to the product of every given modulo except one of them - giving $5 \cdot 7$, $3 \cdot 7$ and $3 \cdot 5$. We then find the multiplicative inverses of these numbers with modulo equal to the number missing from the product. The numbers found are then 2, 1 and 1 in this case. The value of x is then given by: $$x = 2 \cdot (5 \cdot 7) \cdot 2 + 3 \cdot (3 \cdot 7) \cdot 1 + 2 \cdot (3 \cdot 5) \cdot 1 = 233 = 23 \mod{3\cdot5\cdot7}$$  But I do not understand how this method correctly gives the value of $x$. I understand that the Chinese remainder theorem proves that there is a unique value of $0\le x \lt 3\cdot5\cdot7 \mod{3\cdot5\cdot7}$ but can someone please explain why this method finds this value of x? Disable cache for some images Why must $a$ and $b$ both be coprime when proving that the square root of two is irrational? Suppose we wish to prove that the square root of two is irrational. We begin by assuming that it is rational.  Namely, where both $a$ and $b$ are integers $$\frac{a}{b} = \sqrt 2  % MathType!MTEF!2!1!+- % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0x % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa % aiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamOyaaaacq % GH9aqpdaGcaaqaaiaaikdaaSqabaaaaa!31D3! $$  Why is it so important that both $a$ and $b$ be coprime and the fraction be irreducible?  That is, if $a$ and $b$ are not coprime and the fraction is indeed reducible, then$$\frac{a}{b} \ne \sqrt 2  % MathType!MTEF!2!1!+- % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0x % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa % aiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamOyaaaacq % GHGjsUdaGcaaqaaiaaikdaaSqabaaaaa!3294! $$ How to make read-only file system writable? At some point, the filesystem on my digital audio player has become read-only. I cannot copy files into it or remove files on it.  Are there some possible reasons for the player's file system to change permissions in this way?  I tried using chmod:  $ sudo chmod a+rw SGTL\ MSCN/  chmod: changing permissions of `SGTL MSCN/': Read-only file system   SGTL MSCN is the mount point of the digital audio player.  What else can I try to make it writeable? What is the importance of eigenvalues/eigenvectors? What is the importance of eigenvalues/eigenvectors? Creating an arbitrary state of the quantum simple harmonic oscillator Why does $1+2+3+\cdots = -\frac{1}{12}$? What IDE to use for Python? Convergence of Ratio Test implies Convergence of the Root Test In Elias Stein and Rami Shakarchi's Complex Analysis textbook, we have the following exercise:     Show that if $\{a_n\}_{n=0}^\infty$ is a sequence of complex numbers such that $$\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}=L,$$ then $$\lim_{n\to\infty}|a_n|^{1/n}=L.$$    I've been trying to prove this with no luck. The only thing I've thought of doing is $$\lim_{n\to\infty}\left(\frac{|a_{n+1}|^n}{|a_n|^n}\right)^{1/n},$$but this hasn't lead me anywhere except dead ends. Will someone provide a hint for me about how to proceed? Thanks!  Minor update: I don't know if it's helpful yet, but I know we can write the limit as $$\lim_{n\to\infty}\left(\frac{|a_{n+1}a_n\cdots a_0|}{|a_n\cdots a_0|}\cdot\frac{1}{|a_n|}\right).$$This reminds me a lot of the geometric mean, which even has the exponents I'm trying to get... Order of redirections I don't quite understand how the computer reads this command.  cat file1 file2 1&gt; file.txt 2&gt;&amp;1  If I understand, 2&gt;&amp;1 simply redirect Standard Error to Standard Output.  By that logic, the command reads to me as follows:    concatenate files file1 and file2.  send stdout from this operation to file.txt. send stderr to stdout. end?   I'm not sure what the computer's doing. By my logic, the command should be   cat file1 file2 2&gt;&amp;1 &gt; file.txt   but this is not correct. Static nested class in Java, why? Consider the Fibonacci numbers $F(0) = 0; F(1)=1; F(n) = F(n-1) + F(n-2)$.  Prove by induction that for all $n&gt;0$,  $$F(n-1)\cdot F(n+1)- F(n)^2 = (-1)^n$$  I assume $P(n)$ is true and try to show $P(n+1)$ is true, but I got stuck with the algebra. How do we reach $P(n+1)$ from $P(n)$? Also, strong induction may be used instead. Can a git submodule be created retroactively?