If we have two isomorphic groups $A$ and $B$, where $B\subset A$, is it possible that $A/B \not \approx (e)$? Obviously, $A$ and $B$ are infinite.
I need a non trivial  examples of group isomorphism that maps $G\to H$ such that   $G/H \cong G$.
Sorry for asking such a basic question - but is there an actual term for the numbers that appear before and after the decimal point?  Example:  25.18  I know the 1 is in the tenths position, the 8 is in the hundredths position. are there singular terms which apply to all of the numbers that appear both before and after the decimal point?  I'm building a billing system, storing dollars and cents as integers (in separate columns), but for other currencies "dollars" and "cents" are not the correct terms = ).  Thanks! Can anyone explain how we choose one sock from each of finitely many pairs without the axiom of choice?  I mean : To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.  The idea is that the two socks in a pair are identical in appearance, and so we must make an arbitrary choice if we wish to choose one of them. For shoes, we can use an explicit algorithm -- e.g., "always choose the left shoe." Why does Russell's statement mention infinitely many pairs? Well, if we only have finitely many pairs of socks, then AC is not needed -- we can choose one member of each pair using the definition of "nonempty," and we can repeat an operation finitely many times using the rules of formal logic (not discussed here). Correct use of will & would? What would be the correct use of will &amp; would in these sentences?  1) What will happen if I say to my boss that I will not come tomorrow?  2) What will happen if I would say to my boss that I will not come tomorrow?  3) What will happen if I will say to my boss that I will not come tomorrow?  4) What would happen if I would say to my boss that I will not come tomorrow?  There are four sentences which one should I use and in which situation? SQL Server nvarchar(max) vs nvarchar(n) affects performance In Clan Wars, what factors are used to determine the opponent clan? In the clan wars matchmaking, what criteria is used to determine the clan that you are fighting against? Is it based on Clan Total Points (i.e. trophies), or something else?  If you win many wars in a row, does this mean you will get harder opponents? How to properly establish a PostgreSQL connection using QgsCredentials I'm trying to establish a Postgres connection using QgsCredentials as described here:    The only problem is that I cannot establish a working connection unless I "physically" add the layer to the map canvas (which I don't want because I only want a connection to the DB and a reference to the layer which will be edited later on in the code)  When clicking a button the run function reads:  uri = QgsDataSourceURI() connInfo = uri.connectionInfo()  (success, user, passwd) = QgsCredentials.instance().get(connInfo, None, None)  if success:     # Isn't this enough to establish the connection and get a valid reference to the layer?     uri.setConnection("HOST", "PORT", "DB_NAME", user, passwd)     uri.setDataSource("SCHEMA", "TABLE_NAME", "the_geom")     LYR = QgsVectorLayer(uri.uri(), "LYR", "postgres")      # Unless I add the following line, I am not able to use properly the object     # (which asks the user the credentials until he/she inserts the right ones and gets     # a working connection with a valid reference to the layer)     QgsMapLayerRegistry.instance().addMapLayer(LYR)   What am I doing wrong?  Thanks in advance,  m. What are best practices for choosing and pairing fonts? I've always found font selection to be a little bit of a mystery. What factors should be considered when choosing a font? How does one successfully incorporate two fonts into a design?  I realize that some factors will include things like:   Legibility Scalability Appropriateness/Mood/Emotion   However, how does one objectively judge legibility or appropriateness? Are there objective principles that can be applied for font selection? If possible, please list resources or examples in your answers. Studies focused on the effectiveness of different fonts would also be extremely useful. On a series involving harmonic numbers Finding $$\sum^{\infty}_{n=1}\bigg[\bigg(\frac{(-1)^{n+1}}{n+1}\bigg)\bigg(\sum^{n}_{r=1}\frac{1}{r}\bigg)\bigg]$$   Try: Let $\displaystyle H_{n} =\sum^{n}_{r=1}\frac{1}{r},$ then series $\displaystyle \sum^{\infty}_{n=1}\frac{(-1)^{n+1}}{n+1}\cdot H_{n}=\frac{H_{1}}{2}-\frac{H_{2}}{3}+\frac{H_{3}}{4}\cdots\cdots$  could some help me to solve it, thanks Strange behaviour of 'sudo chmod -rwx': Does not remove group write permissions How to calculate such sums of Legendre symbols? Filling in new field in attribute table based on values of another field in ArcGIS Desktop? I have added a new field (Category) in an attribute table of a shapefile and I would like to fill it with numbers, 0 or 1, based on the numbers of another field (Calorific value) of the same attribute table.   That means that if the Calorific value is higher than 15, then I would like to fill in the corresponding cells of the (new) Category field with 1. If the Calorific value is less than 15, then the rest cells of the (new) Category field to be filled in with the number 0.   Is there a way to do this automatically rather than manually? In Avatar: The Last Airbender, Aang's eyes glow when he goes into the Avatar State. This seems to be the case with all Avatars, but Aang and the other airbending Avatars tattoos glow too.      Is there any reason for why airbending Avatars tattoos glow as well as their eyes? Converting mxd to earlier version using ArcPy? You can think this of the following as a 3d QFT where we try to calculate the self-energy of two fields. $I$ is a this external self-energy and let us assume it does not depend on the loop momenta which I denote a $\vec{x}=(x,y)$. Consider the following propagator  $$ P(\vec{r}) = \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} \,\, \gamma_i. $$ Then we want to calculate $$ I(x) = \int \frac{\,dx \,dy\, dz}{(2\pi)^3} \text{Tr} \left( \frac{x-ia}{2i^2[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,}\gamma_x \gamma_x  \right), $$ where Tr$(\gamma_i \gamma_j)=2\delta_{ij}$. Therefore  I am trying to resolve the following integral where $x,y,z \in \mathbb{R}$ and $a,b,c$ are constants (internal self-energies if you wish) in $\mathbb{R}$.  Then, after simplifying the propagator $P(x,y)$ we end up with the following expression for the $x$ component:  $$ I = \frac{1}{2i^2} \int \frac{dxdy}{(2\pi)^2} (x-ia) \int \frac{dz}{2\pi} \frac{1}{[\,(x-ia)^2+(y-ib)^2+(z-ic)^2]\,} $$ but it can be shown that  $$ \int \frac{dq}{(q+iv)^2+w^2} = \frac{\pi}{|w|}\theta(|w|-|v|).  $$ (e.g. see here ) Thus we get that for $|(y-ib)^2+(x-ia)^2|&gt;|a|$, $\theta=1$ (for this is what we are interested about) and then we are left with $$ I = \frac{1}{4i^2} \int \frac{dxdy}{(2\pi)^2} \frac{(x-ia)}{|(x-ia)^2+(y-ib)^2|}.   $$ The question is wether I can make the transformation $$ (x-ia) = q \cos \theta  $$ $$ (y-ib) = q \sin \theta $$ in polar coordinates. Then $dx dy = qdqd\theta$ and the previous integral becomes, assuming $|q|&gt;0$, $$ I = \int_0^{\infty} dq \int_0^{2\pi} d\theta \frac{q \cos \theta}{|q|} =0. $$ Is this correct? I am confused since one can find complex poles in the $I$. If not, what would the correct way to resolve this be? I have a suspicion that this integral should indeed give zero but how can we find this? continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$ Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$.   $g(0)=1$. If $a=g(1)$,then $a&gt;0$ and $g(r)&gt;a^r$ for all $r$ in $\mathbb Q$.   Show that the function is strictly increasing if $g(1)$ is greater than $1$, constant if $g(1)$ is equal to $1$ or strictly decreasing if $g(1)$ is between zero and one, when $g$ is continuous. How do I install Ubuntu Server (step-by-step)? I'd like to install Ubuntu server on my machine, is there a step-by-step instructions on how to accomplish that or a guide that will just go through the basic steps of installing Ubuntu Server from beginning to end.  Instructions originally for 13.04. This is a  about DNS/Hostnames resolution to IPs/Ports   Example 1  I'm running a web server on port 80 and another on port 87. I would like to use DNS so that www.example.com goes to port 87. How can I accomplish this using DNS only?  Example 2  I'm running a service on my server on a non-standard port. How can I get clients to connect to this non-standard port automatically? Can I use DNS? Is there some application specific support where DNS could indicate the IP and Port?   Example 3  Do some application protocols specifically support hostname awareness, and allow special actions to be taken based on this information? Are there other questions on Server Fault that cover some of these?  Commandeering: This question was originally asking about running IIS and Apache on the same server, but the same concepts can be applied to any server software receiving connections from clients. The Answers below describe the technical problems and solutions of using DNS and application protocol support to assign a port number for a client to connect. I have just upgraded my system but it boots to Low Graphics mode by default. I have seen questions on how to solve this problem and they all suggest to run a couple of commands from a command-line interface.  But there isn't any command-line console by default in Low Graphics mode and hence, I cannot run those commands. And, the answers I saw to solve the problem doesn't explain this situation either.  How to switch to a command-line console in order to run the commands in Low Graphics mode? I am currently using a macbook pro but the operating system is debian 8.2 not os x  I am using a macbook pro but there's this one device that's just draining a lot of power. This USB device: Card Reader (Apple)   32.5 W    100.0%                      Device         USB device: Card Reader (Apple)   2.60 W    100.0%                      Device         Radio device: btusb   209 mW     19.7 pkts/s                Device         Network interface: wlan0 (wl)   This device really drains my battery, I noticed that my laptop wouldn't even get 2 hours when not plugged in, I installed powertop. Made all the recommended changes but this device, I can't find out how to disable it. I don't even use that reader and forgot that it was there.  I configured all my devices to use limited power in my rc.local  echo '0' &gt; '/proc/sys/kernel/nmi_watchdog' echo 'min_power' &gt; '/sys/class/scsi_host/host0/link_power_management_policy' echo '1' &gt; '/sys/module/snd_hda_intel/parameters/power_save' echo '1500' &gt; '/proc/sys/vm/dirty_writeback_centisecs' echo 'auto' &gt; '/sys/bus/usb/devices/2-4/power/control' echo 'auto' &gt; '/sys/bus/usb/devices/1-12/power/control' echo 'auto' &gt; '/sys/bus/usb/devices/1-8.3/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1f.3/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:01:00.1/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.3/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.4/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.2/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1b.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:16.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:14.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:01.1/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:01.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:01:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:08:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:06.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:05.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:04.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:03.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:06:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:05:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:04:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:03:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1f.0/power/control'   How can I get this device to act sanely or disable it completely because I  haven't used it. Please have a look at this sentence:     We haven't gotten a lot of Finnish groups that we liked here at Bananas but I'm pretty happy with this four tracker from Helsinki, apparently recorded in the same place where the Ramones played in the 70's.     The first verb is in present perfect (up to now) and the second is in simple past tense. I don't understand the simple past for the second verb.  Why "like" is not in its present perfect form as the first verb is?