Can a group be non-empty by definition of 'group'?
Is the empty set a group?
What prevents photons from getting mass from higher order Feynman diagrams Categories other than "off-topic" should allow custom close reasons Custom close reasons are a great boon to Stack Exchange sites, allowing the community to provide more tailored advice to site-specific situations. Their big problem (aside from being limited to three by default) is that they are all lumped under "off-topic". This is often confusing.  I suggest that all close categories but "exact duplicate" should support customized close reasons &mdash; not just the "off-topic" category.  For example, the recommendation close reason     Questions asking us to recommend or find a tool, library or favorite off-site resource are off-topic for Stack Overflow as they tend to attract opinionated answers and spam.   This would make more sense as a child of primarily opinion-based.    Similarly, the "why doesn't this work" close reason     Questions seeking debugging help ("why isn't this code working?") must include the desired behavior, a specific problem or error and the shortest code necessary to reproduce it in the question itself. Questions without a clear problem statement are not useful to other readers. See: .   This would make more sense as a child of unclear what you're asking.  Similar examples can be given for pretty much any of the Stack Exchange sites. When is the Killing form null? Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ otherwise. Additional horizontal space when the last line of an environment is full I have created an environment env which just adds a \medskip and some text (say "Blablabla") after the text inside \begin{env} ... \end{env}. The problem is that if the last line is full, then I get an additional horizontal space between the text and "Blablabla".  Here is an MWE:  \documentclass{article} \newenvironment{env}{ ENVIRONMENT:\par }{ \medskip \par\noindent Blablabla }  \begin{document}  \section*{First case: the last line is not full}  \begin{env} Lorem ipsum dolor sit amet, consectetur adipiscing elit. Nulla malesuada porta tortor non vestibulum. Vivamus vel tincidunt purus. Curabitur quam arcu, suscipit sed diam sed, ornare efficitur felis. Duis at arcu iaculis, viverra ligula sit amet, hendrerit eros. Proin finibus venenatis sollicitudin. Nunc ullamcorper sem ut lacinia posuere. Integer pulvinar nulla lorem, ut pulvinar ipsum egestas at. Duis quam ipsum, posuere ac pulvinar vitae, auctor non leo. \end{env}  \section*{Seconde case: the last line is full}  \begin{env} Lorem ipsum dolor sit amet, consectetur adipiscing elit. Quisque suscipit vitae neque eu gravida. Quisque et enim quam. Integer elementum diam a ex laoreet, ac egestas lacus volutpat. Nunc ullamcorper non urna tincidunt ultrices. Etiam eu ultricies diam. Proin mollis, nunc sit amet volutpat varius, ipsum tellus bibendum nulla, in eleifend lacus felis pellentesque nisl. Phasellus laoreet, libero quis auctor congue, nulla nulla aliquam metus, in hendrerit arcu justo quis tortor. Nunc nec nisi euismod, consectetur tellus nec, suscipit libero.  \end{env}  \end{document} One-byte-off pointer still valid in C? A certain element which makes functionals positive Suppose $X$ is a possibly non-separable Banach space and let $X^*$ be its dual. Also, let $(f_n)_{n=1}^\infty$ be a sequence of [EDIT: linearly independent] norm-one functionals in $X^*$. Does there exist an element $x\in X$ such that $f_n(x) \in (0,\infty)$ for all $n$? Express solutions of equation $ \tan x= x $ in closed form I know that the equation $$ \tan(x) = x $$ can be solved using numerical methods, but I’m looking for a closed form of the solutions. In my opinion, having only numerical solutions means that we don’t know the problem, and sooner or later, we’ll be able to find a closed-form solution or at least a power-series solution. I’m looking for an explicit form of a sequence $ (x_{n})_{n \in \mathbb{Z}} $ in $ \mathbb{R} $ such that  $ 0 &lt; x_{0} &lt; \dfrac{\pi}{2} $, $ \tan(x_{n}) = x_{n} $ for all $ n \in \mathbb{Z} $, and $ (2 n - 1) \dfrac{\pi}{2} &lt; x_{n} &lt; (2 n + 1) \dfrac{\pi}{2} $ for all $ n \in \mathbb{Z} \setminus \{ 0 \} $.  The existence of such a sequence follows from the continuity of $ x \mapsto \tan(x) - x $ over its domain. Its uniqueness follows from the monotonicity of $ \tan(x) - x $ over the intervals $$ \left( (2 n - 1) \frac{\pi}{2},(2 n + 1) \frac{\pi}{2} \right), \quad n \in \mathbb{Z}. $$ I don't understand the proof to the a/m claim. How we know that $\eta &lt; \kappa$ and $(\alpha,\beta)&lt;_{cw} (0, \eta) $ and hence $h: \mu \to \eta \times \eta$ is injective? Appreciate if anyone could advise.  Here is an excerpt of the proof:  For an infinite ordinal $\kappa,$ the canonical well-ordering of $\kappa \times \kappa,$ denoted by $&lt;_{cw}$ is defined as follows: $(\alpha_1, \beta_1)&lt;_{cw} &lt; (\alpha_2, \beta_2)$ iff either one of the following holds.  $(1): \max\{\alpha_1,\beta_1\} &lt; \max\{\alpha_2,\beta_2\}$  $(2): \max\{\alpha_1,\beta_1\} =\max\{\alpha_2,\beta_2\}$ and $\alpha_1&lt;\alpha_2$   $(3): \max\{\alpha_1,\beta_1\} =\max\{\alpha_2,\beta_2\}$ and $\alpha_1 = \alpha_2$ and $\beta_1 &lt; \beta_2.$  Claim: For all infinite cardinals $\kappa, (\kappa \times \kappa, &lt;_{cw}) \cong (\kappa, \in).$  Proof: (It has been show that it is true for $\kappa = \omega$) Assume there is a counterexample  $\kappa.$ Let $A= \{\lambda \in \kappa + 1| \lambda\geq \omega \ is \ a \ cardinal \ and \ (\lambda \times \lambda, &lt;_{cw}) \not\cong (\lambda, \in)\}.$ Then, $\kappa \in A.$ Let $\mu$ be least element of $A.$ Thus, $\mu &gt; \omega$ and for all $\lambda,$ if $\omega \leq \lambda &lt; \mu$ and $\lambda$ is a cardinal, then $(\lambda \times \lambda, &lt;_{cw}) \cong (\lambda, \in).$  By Comparability theorem, $(\mu, \in)$ is isomorphic to initial segment of $(\mu \times \mu, &lt;_{cw})$ since $|\mu \times \mu| \geq \mu$ and $\mu$ is cardinal. Let $(\alpha, \beta) \in \mu \times \mu$ such that $(\mu, \in)$ is isomorphic to initial segment of $(\mu \times \mu, &lt;_{cw})$ given by $(\alpha, \beta).$ Let $h$ be the isomorphism. Let $\eta= \max\{\alpha, \beta\}+ \omega.$ Then $\eta &lt; \kappa$ and $(\alpha,\beta)&lt;_{cw} (0, \eta).$ HENCE, $h: \mu \to \eta \times \eta$ is injective. How can I unlock my phone? I locked it using Android Device Manager, but it claims I entered a wrong password I have an Samsung Galaxy S3 Neo.  Bus 002 Device 042: ID 04e8:6860 Samsung Electronics Co., Ltd GT-I9100 Phone [Galaxy S II], GT-I9300 Phone [Galaxy S III], GT-P7500 [Galaxy Tab 10.1]   The keyboard &amp; language setting is português.  I today installed Android Device Manager and to test it's functionality I locked the phone with the password: "abcd" (4 letters were required).  After entering the password and confirming with the "OK" button of my keyboard, the phone showed me that the password is wrong. "Senha incorreta".  I reset the password after some tries to "123456" and the phone reloaded the lock screen. Again the password showed up to be wrong. I also tried the alphabetic password: "abcdefg" - same error.  I changed the language of my Google account from German to Portuguese and reset the password to "123456" again and I tried to really reboot my phone and unlock the SIM card again. But anyways I can not get through my lock of the Android Device Manager.  I tried to enter the password 35 times wrong with hope to get a chance to authenticate with my Google Account login.  I also tried to connect the phone with adb but it does not show up in this state with:  adb devices   So what can I do to get access to my phone again? What is the word for act of monotonously demanding for something (many be a toy) repeatedly? Since it seems to me, that a lot of people still need sidebar workarounds, I would be interested to learn about the use-cases, experienced challenges and (most important) possible strategies to avoid them by using different approaches. This is related to the methods discussed here:   In my experience there are a few extremely important and constantly repeating patterns, which to my knowledge are still very hard to be solved using more official techniques. I will also contribute my own top-N use-cases, which are mostly related to UX and UI concerns. You are very welcome to add your own scenarios as answers and add comments on how you would avoid a workaround. My dream would be to team-elaborate strong and repeatable solution-patterns for the most common use-cases.  Finally I would be interested in the experiences of all of you who participate in the Aura pilot, which hopefully could make this kind of workarounds obsolete. Why are floating point numbers inaccurate? Where to put the periods when using a parenthetical sentence? I have two complete sentences; the second is in parentheses, essentially a comment on the first.  Where do I place the periods?  I would use this:     They played opera. (I covered my ears.)   Is this punctuation correct? My dconf/gsettings installation is broken. How can I fix it without Ubuntu reinstall? I run Ubuntu 12.04.1 LTS x64 in VirtualBox. After one very unfortunate misclick (reset saved state instead of load saved state) I got a very annoying problem.  Almost all applications (unity, synaptic, gedit, etc.) print on start:  Using the 'memory' GSettings backend.  Your settings will not be saved or shared with other applications.   And all GUI settings reset after reboot.   Another symptom:  $ GSETTINGS_BACKEND=dconf dconf-editor (dconf-editor:2353): GLib-GIO-WARNING **: Can't find module 'dconf' specified in GSETTINGS_BACKEND GLib-GIO-Message: Using the 'memory' GSettings backend.  Your settings will not be saved or shared with other applications   But /usr/lib/x86_64-linux-gnu/gio/modules/libdconfsettings.so is present.     What I tried (and it didn't help):   sudo apt-get install -f --reinstall  dconf-tools libdconf0 libdconf-dbus-1-0 dconf-service Build dconf-0.5 from sources and make install it Create empty user profile and start programs there   I have to keep current Ubuntu installation so a complete reinstall is not an option for me.   How can I fix it? Python HTTPS requests (urllib2) to some sites fail on Ubuntu 12.04 without proxy I have an little app I wrote in Python and it used to work... until yesterday, when it suddenly started giving me an error in a HTTPS connection. I don't remember if there was an update, but both Python 2.7.3rc2 and Python 3.2 are failing just the same.  I googled it and found out that this happens when people are behind a proxy, but I'm not (and nothing have changed in my network since the last time it worked). My syster's computer running windows and Python 2.7.2 has no problems (in the same network).  &gt;&gt;&gt; url = 'https://www.mediafire.com/api/user/get_session_token.php' &gt;&gt;&gt; response = urllib2.urlopen(url).read()   File "/usr/lib/python2.7/urllib2.py", line 126, in urlopen     return _opener.open(url, data, timeout)   File "/usr/lib/python2.7/urllib2.py", line 400, in open     response = self._open(req, data)   File "/usr/lib/python2.7/urllib2.py", line 418, in _open     '_open', req)   File "/usr/lib/python2.7/urllib2.py", line 378, in _call_chain     result = func(*args)   File "/usr/lib/python2.7/urllib2.py", line 1215, in https_open     return self.do_open(httplib.HTTPSConnection, req)   File "/usr/lib/python2.7/urllib2.py", line 1177, in do_open     raise URLError(err) urllib2.URLError: &lt;urlopen error [Errno 8] _ssl.c:504: EOF occurred in violation of protocol&gt;   What's wrong? Any help is appreciated.  PS.: Older python versions don't work either, not in my system and not in a live session from USB, but DO work in a Ubuntu 11.10 live session. Commutator of a group I must be missing something with regards to setting up PostgreSQL. What I'd like to do is create multiple databases and users that are isolated from each other so that a specific user only has access to the databases I specify. However, from what I can determine, any created user has access to all databases without any specific grants being given.  Here is what I do on an Ubuntu Server 12.04:   apt-get install postgresql sudo -u postgres createuser -DRSP mike1 (Specifying the password for the new user) sudo -u postgres createdb data1 psql -h localhost -U mike1 data1 (Specifying the password for the user mike1 to login)   It seems that new user "mike1" has no problem connecting to database "data1" and creating tables etc. And this without running any GRANT command at all (and the owner of "data1" is "postgres" since I didn't specify an owner in step 3). Is this really how it is supposed to work?  What I'd like to do is grant mike1 full access to data1 and then repeat this for more users and databases, making sure that the users only have access to one (or possibly several) databases of my choice. Can anyone enter as a student at the Citadel ? Or must one be recommended by Lords or a powerful institution?   Aemon Targaryen was a son of a King, so the Citadel would let him access without questioning. Samwell Tarly was sent by Lord Commander Jon Snow, from the Night's Watch. , a novice of the Citadel. A common born but it is not known how he entered the Citadel. We can notice that he is not a good student, but has not been fired.    I don't remember how the other Maesters were recruited, if they are always relatives of Lords, Kings... Can any lowborn commoner just "knock the door", and enter the Citadel? Distribution of orbital velocities in a disk galaxy for N-body simulation?