Prove that if $H$ is a subgroup of group $G$, $H\circ x=H$ if and only if $x\in H$.   Not sure how to start this proof about the right coset.
Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H: hg \not \in H$?  I'm new to abstract algebra and this would appear to be a useful result. However, by definition, a subgroup is nearly closed, and I don't think this directly implies that for any element $g$ from $G$ not in $H$, the composition $gh \not \in H$.  Is this in fact a true statement?
Is there a way to switch the mesh selection mode by a shortcut? Is there a shortcut for switching between these three modes? Sql Server string to date conversion I want to convert a string like this:  '10/15/2008 10:06:32 PM'   into the equivalent DATETIME value in Sql Server.  In Oracle, I would say this:  TO_DATE('10/15/2008 10:06:32 PM','MM/DD/YYYY HH:MI:SS AM')    implies that I must parse the string into one of the , and then convert using one of those codes.  That seems ludicrous for such a mundane operation.  Is there an easier way? Let p be a prime. Consider the equation $\frac1x+\frac1y=\frac1p$. What are the solutions? Write down the set of distinct solutions and prove your list is complete. $x$ and $y$ are positive integers.  I have rewritten it as $\frac{(x+y)}{xy} = \frac1p$, but I don't understand where to go from here? "Open in browser" does nothing when Android app is set as default What is so special about speed of light in vacuum? I will try to be as explanatory as possible with my question. Please also note that I have done my share of googling and I am looking for simple language preferable with some example so that I can get some insight in this subject.  My question is what is so special about $c$? Why only $c$. Its like chicken and egg puzzle for me. Does Einstein reached to $c$ observing light or does he got to light using some number which turned out equal to $c$.   Why is $c$ not relative. If something has zero rest mass like a photon why they only travel at $c$ in vacuum and not with $c+1$ or $c-1$? At what time intervals does ntpd update the time? I'm running the latest ntpd.   When I start ntpd my system time synchronizes with ntp server's ntp. After synchronizing I changed my system time manually using date command date -s '1997-02-22 12:00:00'   My system time got changed as per the date command.   NTPD is still running, I want to know at what time interval my system time will sync with the internet via ntp. Novel about interstellar war: Both sides win, both sides think they lost I recall reading this novel in the &rsquo;90s; my recollection is that the author was better-known for bad fantasy than for SF (but the SF of his that I&rsquo;d read was better than his fantasy).  The story opened in media res just before a battle was joined; there is a dialogue between the captain of one of the ships and one of his crew. We never see the actual battle; we instead see the conquest of Earth by the aliens - and the conquest of the alien homeworld by Earth.  It is slowly revealed that the two sides contrived a way to both avoid the destruction of actual war, and solve their respective worlds&rsquo; respective problems of overpopulation, infrastructure decay, et multae ceterae. We also see that the solution was that the conquering aliens were secretly selecting Earthmen to rule their homeworld, while the humans were selecting aliens to rule Earth.  About halfway through, there is suddenly a third world that must be &lsquo;conquered&rsquo; and brought into the network. The plot gets complicated after that. Applications of ultrafilters I'm looking for some interesting applications of ultrafilters and also everything of interest involving ultrafilters. Do you know some applications or interesting things involving ultrafilters?  I'm at the beginning, so I'd prefer some applications that also a beginner could read. Is there any relation between the chromatic number of a graph $G$ and its complement $G'$ that are always true?  I saw these ones: $\chi(G)\chi(G')\geq n$ and $\chi(G)+\chi(G')\geq 2n$,  but I'm not pretty sure about them. Proving $1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$ This is a follow-up to  question. I came across a tighter potential bound and checked it numerically for $0\le x\le 5$. I think it holds for all positive $x$, can anyone see a proof?  $$1-\exp(-4x^2/ \pi) \ge \text{erf}(x)^2$$  Note: using analysis for previous  you can show that $1-\exp(-k x^2)$ is an upper bound on $\text{erf}(x)^2$ for $k=2$ and a lower bound when $k=1$. The factor $k=4/\pi$ comes out when numerically searching for tightest upper bound. Not only does it seem to give an upper bound, but it also tracks $\text{erf}(x)^2$ very closely.  Dashed graph below is $\text{erf}(x)^2$, red is $1-\exp(-k x^2)$ for $k=4/\pi$, other two graphs are for $k=1$ and $k=2$ No more updates for Google Chrome 32-bit I am using Google Chrome 48.0 32-bit. Today when I launched Google Chrome, I got this message:    This computer will soon stop receiving Google Chrome updates because this Linux system will no longer be supported   I thought Ubuntu 14.04 is supported for five years, what's exactly happening here?  And on 64-bit Ubuntu, I get an error from Update Manager and apt-get update:  Failed to fetch http://dl.google.com/linux/chrome/deb/dists/stable/Release  Unable to find expected entry 'main/binary-i386/Packages' in Release file (Wrong sources.list entry or malformed file)   Some index files failed to download.   They have been ignored, or old ones used instead. I'm looking for the name of a sci-fi series, intended for children. Robots have control over the humans. There is a squad of humans who attack them. Every member in this squad has a badge on their suit, and when they touch this badge the suit becomes armed. This series was not cartoon series, it was performed by real people. The main robot character is very similar in style to Darth Vader in Star Wars. The series aired in the 1990s. I am sorry but I can't recall many details. Using LINQ to concatenate strings What is the most efficient way to write the old-school:  StringBuilder sb = new StringBuilder(); if (strings.Count &gt; 0) {     foreach (string s in strings)     {         sb.Append(s + ", ");     }     sb.Remove(sb.Length - 2, 2); } return sb.ToString();   ...in LINQ? What happens to the temperature when an ideal gas is compressed? I am rather confused about this. I know from Charles' law that under constant pressure, the volume of a gas is directly proportional to its absolute temperature i.e.  $$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$  Thus accordingly, during compression, temperature of the gas would decrease. But in Lectures of physics ,vol 1 by Feynman, it is written:     Suppose that the piston moves inward, so that the atoms are slowly compressed into a smaller space. What happens when an atom hits the moving piston? Evidently it picks up speed from the collision. [...] So the atoms are "hotter" when they come away from the piston than they were before they struck it. Therefore all the atoms which are in the vessel will have picked up speed. This means that when we compress a gas slowly, the temperature of the gas increases.   (Constant pressure?) So, this is contrary to Charles' law. Why does this happen? Who is right? Or are they both correct? I am confused. Help. What is the difference between the dashed and solid attack bars? I was flipping through my Pokemon list to see which kind of attacks they had, and I notice there was a small different between the attacks. Some attacks contained a dashed blue bar where as others contained a solid blue bar.   What is the difference and significance between the two? Heisenberg famously derived his uncertainty principle by considering the disturbance that a measurement would have on a small enough system.  Of course in the mathematical formalism of Quantum Mechanics the relationship is derived from more basic principles.  How does String Theory account for it? Heuristically, and mathematically? Turning off Twitter Bootstrap Navbar Transition animation Word for someone that always has to be right Is there a word for someone that always has to be right? The person gets angry if they are not. Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$?  I remember seeing such a formula, but can not remember where. Any details, (and possibly references) for such a result would be greatly appreciated. Very large log files, what should I do? ( deals with a similar issue, but it talks about a rotated log file.)  Today I got a system message regarding very low /var space.   As usual I executed the commands in the line of sudo apt-get clean which improved the scenario only slightly. Then I deleted the rotated log files which again provided very little improvement.  Upon examination I find that some log files in the /var/log has grown up to be very huge ones. To be specific, ls -lSh /var/log gives,   total 28G -rw-r----- 1 syslog            adm      14G Aug 23 21:56 kern.log -rw-r----- 1 syslog            adm      14G Aug 23 21:56 syslog -rw-rw-r-- 1 root              utmp    390K Aug 23 21:47 wtmp -rw-r--r-- 1 root              root    287K Aug 23 21:42 dpkg.log -rw-rw-r-- 1 root              utmp    287K Aug 23 20:43 lastlog    As we can see, the first two are the offending ones. I am mildly surprised why such large files have not been rotated.  So, what should I do? Simply delete these files and then reboot? Or go for some more prudent steps?  I am using Ubuntu 14.04.  UPDATE 1  To begin with, the system is only several months old. I had to install the system from scratch couple of months back after a hard disk crash.  Now, as advised in ,  I first checked the offending log files using tail, no surprise there. Then, for deeper inspection, I executed this script from the .  for log in /var/log/{syslog,kern.log}; do    echo "${log} :"   sed -e 's/\[[^]]\+\]//' -e 's/.*[0-9]\{2\}:[0-9]\{2\}:[0-9]\{2\}//' ${log} \   | sort | uniq -c | sort -hr | head -10 done   The process took several hours.  The output was in the line of,   /var/log/syslog : 71209229  Rafid-Hamiz-Dell kernel:  sda3: rw=1, want=7638104968240336200, limit=1681522688 53929977  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device 17280298  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device    1639  Rafid-Hamiz-Dell kernel:  EXT4-fs warning (device sda3): ext4_end_bio:317: I/O error -5 writing to inode 6819258 (offset 0 size 4096 starting block 54763121030042024)        &lt;snipped&gt;  /var/log/kern.log.1 : 71210257  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device 71209212  Rafid-Hamiz-Dell kernel:  sda3: rw=1, want=7638104968240336200, limit=1681522688    1639  Rafid-Hamiz-Dell kernel:  EXT4-fs warning (device sda3): ext4_end_bio:317: I/O error -5 writing to inode 6819258 (offset 0 size 4096 starting block 954763121030042024)    (/dev/sda3 is  my home directory. As we can find,   lsblk /dev/sda NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT sda      8:0    0 931.5G  0 disk  ├─sda1   8:1    0 122.1G  0 part / ├─sda2   8:2    0   7.6G  0 part [SWAP] └─sda3   8:3    0 801.8G  0 part /home    Why a process will want to write beyond the limit is actually outside the scope of my comprehension. Perhaps I will want to ask a different question in this forum if this continues even after a system update.)  Then, from  (you may want to check  for a deeper understanding), I executed,   sudo su - &gt; kern.log &gt; syslog   Now, these files have zero sizes. The system is running fine before and after a reboot.   I will watch these files (along with others) in the next few days and report back should they behave out-of-line.  As a final note, both  the offending files (kern.log and syslog), are set to be rotated, as inspection of the files (grep helped) inside  /etc/logrotate.d/ shows.  UPDATE 2  The log files are actually rotated. Looks like the large sizes were attained on a single day.