Let $G$ be a non-abelian group and $H$,$K$ be two abelian subgroups of $G$.   Then will $$HK=\{hk:h\in H, k\in K\}$$ be a subgroup of $G$ ?
Suppose $G$ is a non-abelian group and $H,K$ are two abelian subgroups of $G$. Then must $HK$ be an abelian subgroup of $G$?  I know an example, but I am confused. Thus I just want to check that.
When to pull loose teeth My 5 year old is starting to have some teeth become loose. Specifically her lower two front teeth. I just noticed today that one of them is really loose. However, I don't know how loose it should be before pulling it out.  I'm also curious about how to pull them (or should I just let them fall out, which seems bad). My father pulled mine with pliers but I think my wife would freak out if I tried to do that.  So, how loose is lose enough and how to pull them out (technique, tools, etc.). Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$  Apart from induction, I tried with  to check the validity, but I can't think of an easy (manual) alternative.  Please suggest an intuitive/easy method. Short story about survivors on Earth after the atmosphere has frozen This is a short story I read maybe in the '80s.  Earth has been thrown away from the Sun, probably due to a very large object passing through the Solar System, and the atmosphere has frozen in layers. There is a small family - father, mother, two children, who live in a room they have made airtight by hanging many layers of material to slow the loss of air. They keep warm with a 'camp-fire'. The father goes out every day in a suit of some sort to find food and mine frozen oxygen. It ends when they are found by another, larger group of survivors.  Does anybody know the title and author? Man undergoes radical surgery to research aliens What is the best way to find out a researcher's current affiliation? Let's say I know a name and a field, but not much more.   A google search seems like the most obvious approach, but unfortunately university personnel or research group pages often seem to be quite poorly maintained, and it is not uncommon that someone is listed even if they left more than a year ago.  Early-career researchers are particularly difficult to track down since they often have less of a "presence" on institution webpages and are likely to be moving relatively frequently. Linked server returned message "Query timeout expired." We have two separate SQL Servers. On one server we have a data warehouse (DWH), on the other we have sales information database.  Now on the DWH server there is an ETL job that collects the information from the sales server. The job runs daily after midnight. The DWH collects the information via linked server from the sales database.  Now, the most time the ETL job runs without any problems. But sometimes it fails because of query timeout. We have found out, that there is a specific pattern: The failure happens every 11th day. So on the 11th day the ETL job fails to collect information.   The following error occurs:     SQLNCLI11 for linked server “my linked server” returned message “Query timeout expired.”   Note: The job fails usually 10 minutes after start.  We have searched everything and could not found out what this issue causes. Also we know that the amount of data is every time, almost the same. Also there is no any scheduled job that runs all 11 days or something.  The remote query timeout on the linked server is set to 0.  Our next step will be, to turn off the antivirus programm on the sales server, to check if this causes the problem.  Does anyone have any clue or idea, where i can search further to find the problem? How do I definitively show that there are only two propagating degrees of freedom in the Lorenz Gauge $\partial_\mu A^\mu=0$ in classical electrodynamics.  I need an clear argument that   involves the equations of motion for just the potentials $A^0$ and $\mathbf{A}$, and not the electric and magnetic fields.   includes sources $\rho$ and $\mathbf{J}$ in the equations of motion.  This is to justify the assertion that there are degrees of freedom that decouple from the rest of system. does not critically rely on quantum field theoretic arguments (although any supplementary remarks are welcome).     To illustrate the level of clarity I expect, I provide an argument in the Coulomb gauge $\nabla\cdot \mathbf{A} = 0$:  Of the four field degrees of freedom, the gauge condition $\nabla\cdot \mathbf{A} = 0$ removes one degree of freedom (the longitudinally polarized EM waves).  To show that among the remaining three degrees of freedom only two are propagating, consider the field equations of motion in the Coulomb gauge:  \begin{align} \nabla^2 A^0 &amp;= -\rho/\epsilon_0,\\ \big[\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2\big]\mathbf{A}&amp;= \mu_0 \mathbf{J}-\nabla \frac{1}{c^2}\frac{\partial}{\partial t}A^0. \end{align}  The first equation is NOT a wave equation for $A^0$, and thus does not propagate.  The final equation IS a wave equation, and describes the propagation of two degrees of freedom (Gauss' law in first eq. can be solved, and then inserted into the second equation to show that $\mathbf{A}$ only couples to the Solenoidal part of the current). Who are the diamond moderators, and what is their role? Some people have a diamond after their username (e.g. )    What special privileges do diamond moderators have? How can I become a diamond moderator? Who are the diamond moderators?  How many are there?      For more information, see  in the .     Note: not to confuse with certain Stack Exchange employees, for example  members or "community managers", who often have diamonds as well, but are not ordinary moderators. Could there be an eighth Horcrux? Why is CBS.log file size 20 GB I've built a current sense differential amplifier that measures the voltage across a 1 Ohm resistor, to give me a reading for the current through it.  I've connected it as follows, all resistors are matched 10k to give a gain of 1:    (Image source: )  I am powering it from a single rail 12 volt source. The problem is even though this is a rail to rail opamp (Output Low swing from 5-20mV) I'm getting a constant error of 0.62V and it seems the opamp is not able to go below this value.   I have tried to resolve this by connected Vout to ground using a 10k resistor, but that makes no difference whatsoever.  I tried it with dual supply configuration (+8V, -8V) and that seems to resolve the issue. However I want to run this off a single supply rail, so this is not a viable option.  How can I resolve this? I'm trying to measure the voltage drop across a 1 ohm resistor to give me a reading for current through the circuit (0-1V represents 0-1 A) so I can set up a current limit. A better way (or practice) to doing this is also welcomed. The initial (but not ideal) fix I can think of, is to offset the readings by a few volts?  Edit:  Changed R1-R4 from 10k to 100k and I'm down to about 30-50mV! What is "it" in the following sentence: It is clear that Bob likes doughnuts Where to locate "sockets" I am new to this, so sorry if its obvious. I am running a Debian server and installing the likes of UWSGI, NGinx etc on there. The configurations keep talking about pointing to "sockets".  In the build options I seem to be able to specify where the sockets for each program go. By default it looks like most of them go in /tmp (not all of them).  Is this a good place for them to go? I'm trying to keep things as organized as possible but just bunging them in /tmp doesn't seem like the best option. Sharing a Bluetooth Keyboard and Mouse sequentially between two Macs Can the effects of a person's mass upon the local gravitational field be detected and measured remotely? Why doesn't Neo teleport at random? Why do birds sitting on electric wires not get shocked? I dont have a clue about the proof in the reverse implication on the iff statement.  I will show what I have by now:  1) A cofinite topology implies that any infinite subset in $X$ is dense. Let $U\in\tau$ and is cofinite topology so $U=X-\bigcup_{k=1}^{n}\{x_k\}$.  Let $V\in X$ an infinite subset then $$V\cap U=V\cap(X-\bigcup_{k=1}^{n}\{x_k\})=(V\cap X)-(V\cap \bigcup_{k=1}^{n}\{x_k\})=V-(V\cap \bigcup_{k=1}^{n}\{x_k\})$$  and cause $|V|\ge\aleph_0$ and $|\bigcup_{k=1}^{n}\{x_k\}|=n\in\Bbb N\,$ we have that $|V\cap \bigcup_{k=1}^{n}\{x_k\}|\le n$. And finally is obvious that $V-(V\cap \bigcup_{k=1}^{n}\{x_k\})\ne\emptyset$ so any infinite subset is dense on $X$.  2) A cofinite topology implies a $T_1$-space. Let $x, y\in X$ and $U\in\tau: U=X-\{y\}$ so $x\in U\land y\notin U$. We can define too $V\in\tau:V=X-\{x\}$ so $x\notin V\land y\in V$ then a cofinite topology is a $T_1$-space.  But my problem is that I dont have a clue to prove the reverse implication i.e. if we have a $T_1$-space where every infinite subset is dense then it is a cofinite topology. Some hint? Some advice or correction in the proof of above? Thank you in advance.    I will complete the proof with the advice of @Cameron Buie.  1) A $T_1$-space implies that any finite subset is closed. Let the singleton $\{x\}\subset X$. Then $\overline{\{x\}}=\{x\}\cup\{x\}'$ where $\{x\}'=\{y:(\forall U\in\tau : y\in U)\land x\in U\}$, but because we are in a $T_1$-space exist some $U$ for any points $x,y\in X: x\in U\land y\notin U$ so $\{x\}'=\emptyset$ then $\overline{\{x\}}=\{x\}$ and is closed.  This means that for every $T_1$-space it cofinite sets are open.  2) If any infinite subset is dense on $X$ it means that the closure of any infinite subset is $X$ so the only closed sets are finite (and $X$ itself), so the topology is cofinite. Customizing citations and the list of publications depending on the author Say I use a bibliography file in my document, relying on Biblatex and Biber for citation management.  Given a citation command such as \cite{A, B, C} (or any other standard Biblatex citation command), I'd like the resulting citations to be typeset differently depending on whether a given field (e.g. the author field) in the corresponding bibliography entry contains a given string (e.g. someone's lastname).   Besides, I'd like to generate two different bibliography sections in the resulting document, depending on the same condition as above.  Typically this would be used to differentiate, both at the citation level in the main text and in the appendix, the publications that I co-authored from others.  EDIT: adding a MWE:  \documentclass{article} \usepackage{filecontents} \usepackage[backend=biber,style = alphabetic]{biblatex} \usepackage[colorlinks=true, breaklinks, allcolors = blue]{hyperref} \DeclareSourcemap{%   \maps[datatype=bibtex]{%     \map{%       \step[fieldsource=author, match=Doe, final]%       \step[fieldset=keywords, fieldvalue=doe]%     }%   }% } \begin{filecontents}{\jobname.bib} @Book{p1,   author =       "A. U. Thor and John Doe",   title =        "A Scientific Article 1",   year =         "1975"   } @Book{p2,   author =       "John Doe",   title =        "A Scientific Article 2",   year =         "1985"   } @Book{p3,   author =       "Joe Bar",   title =        "A Scientific Article 3",   year =         "1995"   } @Book{p4,   author =       "Jane Baz",   title =        "A Scientific Article 4",   year =         "2005"   } \end{filecontents} \addbibresource{\jobname.bib} \begin{document} % Requirement 1: should make citations of Doe bold In \parencite{p4, p1, p2, p3}, ``TD75'' and ``Doe85'' should appear in bold (the semicolon should remain plain).  % Requirement 2: separating bibliographies. IT WORKS. \section{Co-authored Publications} \nocite{*} \printbibliography[keyword=doe,heading=none]  \section{Other Publications} \nocite{*} \printbibliography[notkeyword=doe,heading=none]  \end{document} Getting filesystem path of class being executed