It is well known that $\pi_2(G)$ is trivial for any Lie-group $G$. Is there an elementary proof of this, say, that can be understood with minimal homotopy theory?  Also, who gave the first proof of this theorem?
I try to understand why for a connected Lie group $G$ the first fundamental group $\pi_1(G)$ is abelian, and mainly why the second fundamental group is trivial $\pi_2(G)=0$?  Thanks for anyone who give me references for a 'simple proof' of these results
To be honest I find it hard to keep up with all the changes, and very frequently I have to go look at the privileges list to see whether a user (that I'm looking at, so I have his reputation value) has a certain privilege or not.  However, I still don't know what that user will actually see or how the site actually behaves for that user.  There's quite a number of posts here on meta about "where did this go" and "why can't I" and I can't really tell either way how a site behaves for a user with a reputation level of 2500.  So, I propose the following functionality:   For all users, be able to view the site as though you had less reputation than you actually have Bonus-feature: For all users, be able to use the site at that reputation level, unless some features of low-level users conflict with your actual functionality For mods, be able to view the site as a specific users (be able to see all he sees, exactly the way he sees it), but of course not pose as that user (ie. things I do would still be me)   The UI could be merged into the mod menu and the privileges menu:    (Note, I chose a user randomly, Olivier Pons is in no way related to this question or flags, or whatever that I know of.) I am writing a cover letter and CV using moderncv. Everything works fine so far, but the text in the cover letter is not justified. I think it is good that it is not hyphenated, but it would look much nicer if the text was justified.  Here is a minimal example:  \documentclass{moderncv} \moderncvstyle{classic}  \usepackage{lipsum} % just for dummy text  \firstname{John} \familyname{Doe}  \begin{document} \recipient{Prof.\ Dr.\ Foo Bar}{} \opening{Dear Professor Dr.\ Bar,} \closing{Yours sincerely,} \enclosure[Attached]{curriculum vit\ae}  \makelettertitle \lipsum[1] \makeletterclosing  \clearpage  \makecvtitle  \section{Education} \cventry{2009--present}{PhD Student}{here}{and}{there}{} \end{document} Abnormal behavior with iwlwifi driver: the device is shown as not working Recently I've installed Kubuntu in a newly bought laptop with an Intel WiFi 6 AX200 network card, I can connect to the wifi without apparent problems but when I open the driver manager in the settings I face this weird situation:  Intel Corporation: Wi-Fi 6 AX200 This device is not working.  [ ] Using iwlwifi driver backport in DKMS format from backport-iwlwifi-dkms (open source)  [ ] Continue using a manually installed driver  [x] Do not use the device   Note that "Continue using a manually installed driver" is the only voice selectable in the list.  Here is some additional information:  $ lspci -nnk | grep 0280 -A3 02:00.0 Network controller [0280]: Intel Corporation Wi-Fi 6 AX200 [8086:2723] (rev 1a)         Subsystem: Intel Corporation Wi-Fi 6 AX200 [8086:0084]         Kernel driver in use: iwlwifi         Kernel modules: iwlwifi  $ rfkill list all 0: hci0: Bluetooth         Soft blocked: yes         Hard blocked: no 1: phy0: Wireless LAN         Soft blocked: no         Hard blocked: no  $ dmesg | grep iwl [    2.888227] iwlwifi 0000:02:00.0: enabling device (0000 -&gt; 0002) [    2.945179] iwlwifi 0000:02:00.0: Direct firmware load for iwlwifi-cc-a0-50.ucode failed with error -2 [    2.945217] iwlwifi 0000:02:00.0: Direct firmware load for iwlwifi-cc-a0-49.ucode failed with error -2 [    2.947244] iwlwifi 0000:02:00.0: TLV_FW_FSEQ_VERSION: FSEQ Version: 43.2.23.17 [    2.947250] iwlwifi 0000:02:00.0: Found debug destination: EXTERNAL_DRAM [    2.947252] iwlwifi 0000:02:00.0: Found debug configuration: 0 [    2.947697] iwlwifi 0000:02:00.0: loaded firmware version 48.4fa0041f.0 op_mode iwlmvm [    3.145585] iwlwifi 0000:02:00.0: Detected Intel(R) Wi-Fi 6 AX200 160MHz, REV=0x340 [    3.159226] iwlwifi 0000:02:00.0: Applying debug destination EXTERNAL_DRAM [    3.160060] iwlwifi 0000:02:00.0: Allocated 0x00400000 bytes for firmware monitor. [    3.336456] iwlwifi 0000:02:00.0: base HW address: 50:eb:71:d7:0b:40 [    3.353553] iwlwifi 0000:02:00.0 wlp2s0: renamed from wlan0 [    4.457920] iwlwifi 0000:02:00.0: Applying debug destination EXTERNAL_DRAM [    4.621843] iwlwifi 0000:02:00.0: FW already configured (0) - re-configuring [    4.629999] iwlwifi 0000:02:00.0: BIOS contains WGDS but no WRDS   Someone have idea why this happened? Run TTY script every time I open a GUI terminal When I log in to a TTY I get the following nice text:  Ubuntu 16.04.03 LTS Log-in: username Password: Last log-in date/time Welcome to Ubuntu 16.04.03 LTS   * Documentation:  * Management:  * Support:  9 packages to update 4 updates are security updates   I would like the last 2 lines repeated every time I open a GUI terminal for all users in the adm group (but will settle for having the entire thing displayed each and every time I open a GUI Terminal for all users)  I tried the obvious:  fab-root@fab-ux-predator:~ $ cd /etc/update-motd.d/ fab-root@fab-ux-predator:/etc/update-motd.d $ ./90-updates-available  fab-root@fab-ux-predator:/etc/update-motd.d $ cat /var/lib/update-notifier/updates-available cat: /var/lib/update-notifier/updates-available: Permission denied fab-root@fab-ux-predator:/etc/update-motd.d $   What am I missing?  P.S. Obviously I can chmod o+r the file, but how future-proof would that be? Floating point arithmetic not producing exact results Trouble with Vakil's FOAG exercise 11.3.C I'm having trouble with the exercise in the title, even with part (a), which asks to prove that if $X$ is a closed subset of $\mathbb{P}^n_k$ of dimension at least 1 and $H$ is a non-empty hypersurface, then $X$ and $H$ meet.   Following the hint, I've done the following: By assumption, $X$ is given by $V(I)$ for some homogenous prime ideal $I\subset k[x_0,\dots x_n]=:A$ and $H$ by $V(f)$ for some homogenous polynomial $f\in A$. We wish to prove that there is a homogenous prime ideal $\mathfrak{p}$ (that's not the irrelevant one) containing both $I$ and $f$.  Consider the affine cone $\mathrm{Spec}\ A/I$ of $X$ in $\mathrm{Spec}\ A$. Being homogenous, both $I$ and $f$ are contained in the ideal $(x_0,\dots x_n)$. By Krull's principal ideal theorem, every minimal prime ideal containing $f\mod I$ has codimension one. $X$ having dimension at least one implies that there are homogenous prime prime ideals $\mathfrak{q}\supsetneq \mathfrak{r}\supset I$. This implies that $(x_o,\dots x_n)$ has codimension at least two. Thus there are prime ideals containing $f\mod I$ that are contained in $(x_o,\dots x_n)$. However, I'd need to show that there are homogenous such prime ideals, right? I'd rather get hints than full solutions. SQL - an error occurred during the pre-login handshake Which is faster/best? SELECT * or SELECT column1, colum2, column3, etc On  (but I remember seeing it in other cases) the diff shows the same tag being both added and removed. Is there a possibility to change the font-style in the table of contents? Everything I wrote down in my document was written with Times. However, the ToC does not have the same font family. Is there a way to change the font of the ToC to Times New Roman? Thank you : )    \documentclass[a4paper, fontsize=11pt]{scrbook} \usepackage{geometry} \geometry{a4paper, top=25mm, left=25mm, right=25mm, bottom=20mm, headsep=10mm, footskip=12mm}  \usepackage[ngerman]{babel}  \usepackage[T1]{fontenc}        \usepackage[latin1]{inputenc} \usepackage{acronym}      \usepackage{anyfontsize}  \begin{document}  \tableofcontents  \chapter{Test} Is there a possibility to change the font-style in the table of contents? Everthing I wrote down in my document was written with Times. However, the ToC does not have the same font family. Is there a way to change the font of the ToC to Times New Roman?  \end{document} I start with knowing that two numbers are coprime if:  $n*k + m*j = 1$  So, setting $k = a$ and j = $a+1$ I can solve as follows:  $n*a + m*a + a$  Then,  $a(n+m) + m = 1$  Where can I go from here? Is there a word that contains all the vowels? "Jealousy" contains most of the vowels but lacks 'i'. Is there a word in the English language that contains every vowel (and preferably 'y')? Get display resolution from the command line for Linux Desktop find command: -or option doesn't work I want to execute a command for an each file that matches a pattern. But the command that seems to be right, doesn't work for me, I have no idea why.  $ find . -type f -name '*.c' -or -name '*.h' -or -name '*.cpp' -exec ls {} \; $ ls script.sh  test.c  test.h Does the use of words such as 'fail', 'wrong', 'incorrect' and 'invalid' have a sufficiently negative impact on a users frame of mind to warrant the prohibition of their inclusion in the error messages generated by typical web forms?  i.e. Should we rephrase examples like these to just use positive words?  'date entered incorrectly'  or  'failed to complete registration, please see below for invalid field entries'  (PS, the examples are there to show use of the words in an error message, not to serve as examples of error messages I would use or things that need correcting in answers, I'm not looking for advice on how to write form error messages, but for information about the effect of certain words on a users frame of mind) I am familiar with the set-theoretical definition of functions.   A function $f : A \rightarrow B$ from a set $A$ to a set $B$ is a subset $f \subseteq A \times B$ of the Cartesian product $A \times B$ such that the following conditions hold:   $\forall a \in A : \exists b \in B : (a,b) \in f$ $\forall a \in A : \forall b, b' \in B : (a,b), (a,b') \in f \rightarrow b=b'$   Is there a purely set-theoretical definition of the function evaluation $f(a)$ for every $a \in A$? I get as far as defining  $f(a) = \{ b \in B : (a,b) \in f \}$ but that defines rather the set of values of a multi-valued function. Block IPs if they access a resource show all my unanswered questions from SE Any way how to get    my all un-answered questions  from across all SE sites? I have shown that a smooth solution of the problem $u_t+uu_x=0$ with $u(x,0)=\cos{(\pi x)}$ must satisfy the equation $u=\cos{[\pi (x-ut)]}$. Now I want to show that $u$ ceases to exist (as a single-valued continuous function) when $t=\frac{1}{\pi}$.  When $t=\frac{1}{\pi}$, then we get that $u=\cos{(\pi x-u)}$.  With single-valued function is it meant that the function is 1-1 ?  If so, then we have that $\cos{(2 \pi-u)}=\cos{(4 \pi -u)}$, i.e. for two different values of $x$, we get the same $u$, and so for $t=\frac{1}{\pi}$, $u$ is not 1-1.  But if this is meant, how are we sure that for $t \neq \frac{1}{\pi}$ the function is single-valued? Determine the size of a pipe without calling read() I need a function called SizeOfPipe() which should return the size of a pipe - I only want to know how much data is in the pipe and not actually read data off the pipe itself.   I thought the following code would work:  fseek (pPipe, 0 , SEEK_END); *pBytes = ftell (pPipe); rewind (pPipe);   but fseek() doesn't work on file descriptors. Another option would be to read the pipe then write the data back but would like to avoid this if possible. Any suggestions?