Hello Academia Community,  I hold a PhD and would like to pursue a research project independent of an university (outside of academia on my own or in collaboration with other researchers). First of all, is that even possible? Second, what governing body do I submit my proposal to for review? Third, could you please outline the process?....I wonder whether if it is as simple as  conducting the research and submitting to a journal?  I appreciate your reply,  M
Assume you have already completed your undergrad and have been working for a number of years. Does one need to be part of a university or a learning institution to publish papers?
Show that $f_n(z)=z^n$ converges locally uniformly on $|z|&lt;1$ but not uniformly.   I am really confused. I saw a seemingly reliable reference stating that $z^n$ does not converge uniformly to $0$ on $\{|z|&lt;1\}$ but it does converge uniformly on $\{|z|\le r,r&lt;1\}$. What is the nuance between the two former sets? With $r_n&lt;1,r_n\to 1$, shouldn't one conclude that $f_n(z)$ uniformly converges on $\{|z|&lt;1\}$? And besides, this source tells me what I have to prove for Homework is disproved, which really makes it more frustrating.  I would appreciate any help. Is it ever revealed what Crookshanks is? When Hermione first got her pet thing Crookshanks, there was a mention that it looked kinda like a cat, but wasn't.  Was it ever revealed what Crookshanks really was? Is there a JavaScript equivalent of Java's class.getName()? Clicking on a notification on Android app opens an empty feed When I receive a message on an SE post, I get a notification like this:    If I double tap this notification, the SE application opens, but my feed is empty approximately half the time. It looks like this:    If I refresh the feed (slide from the top), the feed does refresh and shows the new notification.   The notification, above, leads to  comment.  I haven't figured out the pattern yet, but this occurs roughly half the time I open a notification. It's occurred for chat messages, comments and answers - but not consistently. I have received notifications for the same post (for example, the two comments after the one linked above) that do not show an empty feed. But, a chat message barely 10 minutes later will show the empty feed again.  I am utilizing the beta app - version 1.0.72 on Android 5.0.1. The process substitution &lt;(cat) doesn't work. Why?  head &lt;(cat)  cat: -: Input/output error   I can use an extra file descriptor, but I don't understand why the above doesn't work.  # Using file descriptor 3 to reroute the input (head &lt;(cat &lt;&amp;3)) 3&lt;&amp;0 Segmentation fault error occurs for a simple function I have coded Combining values in one column (or part) when values in another column (or part) match I just edited an answer on Unix&amp;Linux and when scrolling down, the semi-transparent right sidebar boxes "How to edit" and "How to format" kept their fixed positioning and overlapped with the page footer.     While posting this, I have seen several (not) duplicate questions about this issue, and although from my personal taste, I would prefer the boxes to stop scrolling before the footer is reached, I can accept the development teams decision to let it overlap.   However, on all those other posts, the boxes had an opaque background, so that the footer was hidden and the box content was readable. Here the boxes are transparent though, so the texts get mixed and are no longer well readable. Use early 2009 20" iMac as external display for 2015 rMBP Moving SQL Server to different domain Computing the Lie bracket on the Lie group $GL(n, \mathbb{R})$ Consider the Lie group $GL(n, \mathbb{R})$. Since $GL(n, \mathbb{R})$ is an open subset of the space $M_{n,n}(\mathbb{R})$ of $n \times n$ matrices, we can identify the tangent space (Lie algebra) $T_{e}GL(n, \mathbb{R})$ with $M_{n,n}(\mathbb{R})$, where $e$ is the identity matrix. I'm trying to understand the following:  For all $x, y \in M_{n,n}(\mathbb{R}) = T_{Id}GL(n, \mathbb{R})$ we have $[x, y] = xy -yx$ where $[\cdot, \cdot]$ is the Lie algebra bracket.  According to the question , this seems to be the case for matrix Lie groups in general.  So far, I've worked out the following: Let $x \in M_{n,n}(\mathbb{R}) = T_{Id}GL(n, \mathbb{R})$ and let $X \in \Gamma^{inv}(TG)$ be the left invariant vector field corresponding to $x$, i.e. with $X(e) = x$. By openness, we can choose $\epsilon &gt; 0$ such that the curve $(-\epsilon, \epsilon) \ni t \mapsto e + tx$ through the identity is contained in $GL(n, \mathbb{R})$. For $g \in GL(n, \mathbb{R})$ let $L_g$ be the left multiplication by $g$ in $GL(n, \mathbb{R})$. Using left invariance of $X$ calculate: $$X(g) = ((L_g)_{*}X)(g) = D_eL_g(X(e)) = D_eL_g(x) = \frac{d}{dt}|_{t=0}g(e+tx) = gx$$ Now, the above implies for $f \in C^{\infty}(GL(n, \mathbb{R}))$ and $g \in GL(n, \mathbb{R})$: $$X(g)(f) = (D_gf)(gx)$$ where we identify the tangent vector $X(g)$ with the corresponding derivation at $g$.  Now how do I calculate $[x,y] \stackrel{def.}{=} [X, Y](e) \stackrel{\text{WHY?}}{=}xy-yx$ from this? Last two digits of $2^{1000}$ via Chinese Remainder Theorem? I bumped across the aforementioned question in my notes while studying today and I have completely forgot how to do this. I remember using CRT to solve a problem like this on one of my tests, too bad they didn't give back my solutions :(.  Since $\gcd(100,2) = 2$, we can't use the usual Euler's theorem to solve via $\pmod {100}$. So $100 = 5^2 2^2$, and applying Euler's on the 25 gives me $2^{20} \equiv 1 \pmod {25}$. However since $\gcd(2,4) = 2$, we can't do the same for $\bmod 4$. Accordingly how do I set up the other modulo congruence so I can apply CRT? When you tap a row in a UITableView, the row is highlighted and selected. Is it possible to disable this so tapping a row does nothing? amsmath align environment: row spacing How can I stretch the spacing between consecutive lines in an align* environment? I have a few lines with \fracs in which both the numerator and the denominator consist of textual phrases, not math symbols, and the result looks too cluttered.  \begin{align*} X &amp;= \frac{\textit{Some text here}}{\textit{And some text here}}\\ Y &amp;= \frac{\textit{Will make everything}}{\textit{Look too close together}} \end{align*} How to find out which (not installed) Debian package a file belongs to? On Debian systems (and derivatives):  $ dpkg --search /bin/ls coreutils: /bin/ls   That is, the file /bin/ls belongs to the Debian package named coreutils.  But this only works if the package is installed. What if it's not? How to perform hard / cold reboot of iPhone or iPad? Should I sign, watermark or otherwise label a framed print? And if so how? How can I enable Ctrl+Alt+Backspace? Ctrl+Alt+← can usually be set to restart X, however I can't find the option to allow this (it is disabled by default).  I have looked in System-&gt;Preferences-&gt;Keyboard and System-&gt;Preferences-&gt;Keyboard Shortcuts but I can't find an option to enable it.  Where can I change this setting? Am I Using JDBC Connection Pooling? How to prove that the only subgroup of the symmetric group $S_n$ of order $n!/2$ is $A_n$?  Why isn't there other possibility?  Thanks :)