Example of a group with non equal coset for a fixed element
Conjugate subgroup strictly contained in the initial subgroup?
This morning I upgraded my system to Windows 10 2004, the upgrade went smooth and was through Windows Update. I am using Hyper-V as my virtualization program for virtual machines and to provide extra layer of protection to the system through core isolation. After the upgrade, I noticed that a new Virtual Switch was created with the name &quot;vEthernet (Ethernet)&quot; where &quot;Ethernet&quot; within the parentheses is the name of my primary network adapter. Also noticed this behavior has a specific pattern of creating new vSwitches correspond to the adapters you have, e.g. if you created a new Virtual Switch and restarted the system, a new additional switch will appear with the name &quot;vEthernet (your-new-switch-name)&quot;. Tried to remove Hyper-V and reinstall it but no luck. I am aware of a similar  on SuperUser but has no answers on it. Any clues for the root of the problem? How to reinstall kernels after accidental deletion with a separate boot partition and an encrypted LVM volume? My installation is a Ubuntu 12.04 64 bits (Classic "legacy" MBR - No  EFI/UEFI here) with two partition :   A separate boot partition An encrypted LUKS volume that contains the system Logical Volumes   I seems I accidentaly deleted all the current kernel files with:  sudo apt-get purge 'linux-image-3.8*'    I applied the updates with Ubuntu Software Update, rebooted and then: Ubuntu can't boot.  I tried to boot on a live DVD with a 14.04.1 Ubuntu and can still see my partitions (described later):  /dev/sda1 (boot) /dev/sda2 (Encrypted volume) /dev/sda5 (Logical Volume container)  What are the steps to recover/reinstall the kernel files and dependencies in order to cleanly reboot my installed system?  Thanks in advance I am curious about the proof of these two integrations. It would be great if you could point me in the right direction : $ \int{e^{ax}\cos({bx})dx}= \dfrac{e^{ax}}{a^{2}+b^{2}}[a\cos(bx)+b\sin(bx)] + C $ &amp; $ \int{e^{ax}\sin({bx})dx}= \dfrac{e^{ax}}{a^{2}+b^{2}}[a\sin(bx)-b\cos(bx)] + C $ How to find the kth largest element in an unsorted array of length n in O(n)? I believe there's a way to find the kth largest element in an unsorted array of length n in O(n).  Or perhaps it's "expected" O(n) or something.  How can we do this? Helix in a helix I am trying to work out a "helix in a helix" mathematically.  Intuitively I think of this as a steel cable, which is made up of a number of smaller steel cables all bound together in spiral.  If I wanted to find the length of one of the individual cables, it would be bound in a spiral in the smaller cable and then those cables bound in a larger spiral cable.  I know that if I wanted to do a helix whose ends meet, I would use the parametrization $$((a+b\cos(\omega{t}))\cos{t},(a+b\cos(\omega{t})\sin{t},b\sin(\omega{t})),t=0..2\pi$$ I've been trying to map out in my head how to, instead of curling the helix, making the helix travel in the path of a helix.  I've achieved it partially with $$((a+b\cos(\omega{t}))\cos{t},(a+b\cos(\omega{t})\sin{t},t),t=0..\infty$$ But this doesn't keep the smaller helix in tact, and turns it into a sine wave helix.  I've also tried  $$((a+b\cos(\omega{t}))\cos{t},(a+b\cos(\omega{t})\sin{t},tb\sin(\omega{t})),t=0..2\pi$$ But this gives me sort of a nautilus shape where the helix curls into itself and increases in size and curls around into itself.  What am I missing?  EDIT:  Also, what if we wanted to do this again, like a 'helix in a helix in a ... in a helix'? Is the sum and difference of two irrationals always irrational? If $x$ and $y$ are irrational, is $x + y$ irrational? Is $x - y$ irrational? Enabling GPU rendering for Cycles? The use of weak references is something that I've never seen an implementation of so I'm trying to figure out what the use case for them is and how the implementation would work. When have you needed to use a WeakHashMap or WeakReference and how was it used? I run into this infinite product today. I don't have that much experience evaluating products therefore I don't have any idea of how to tackle this.   Here is the question. Evaluate (if possible) in a closed form the product:  $$\Pi = \prod_{n=1}^{\infty} \frac{1}{1+\pi^{{\large \frac{1}{2^n}}}}$$  The numerical value seems to be $\Pi= 0.534523$ which is very close to $\frac{\pi}{6}$ taking into account that $\frac{\pi}{6} \approx 0.523598$. In the mean time W|A evaluates it to $0$. I'm lost.   Can the community help?  Edit: Based on the answer we have two products. The product I asked tends to zero and the bonus product provided by @you're in my eye (thanks for that) is $\frac{\ln \pi}{\pi-1}$.   Thanks for the quick response. Texture packing algorithm What is a good texture packing algorithm?  Technically,  is , so a heuristic is what I'm really after. Health decreasing for no apparent reason I am playing Dragonborn DLC. I am currently going through one of the black books quests and even though I am not fighting my health bar keeps going down. I can heal back up to full again but it keeps disappearing. If I do not heal I will die in about 30 seconds.  I have no disease or nothing is visible anyway, no clue as to what's wrong. I can leave the black book quest or fast travel to a different location or rest for 24 hours but still the health disappears. I feel like someone made a comment as an answer on a question with a bounty just so they could reap the "auto bounty" since they are the only answer on it.   I've flagged the bad/useless answer (as a non-answer) for moderator review but the bounty will be assigned automatically in less than 24hrs and I feel like this guy/gal is just trying to cheat the system.   All I'm trying to do is ensure the bounty simply expires and the guy/gal that was trying to cheat the system by putting down a useless guess doesn't get it.   I've answered my own question (properly) and marked it as the accepted answer, not sure if that makes a difference though in this person still cheating the system since I'm the OP and created the accepted answer.   Here is the post: Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$ Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$?  That is, if a field is of characteristic 2, then does this field have to be $\{0,1\}$? I use the following command to clear a directory, of files and directories over 30 days old, and move them to an archive directory which I can delete after a few weeks if nobody asks for their files back. The target directory has subdirectories by user name, so will the archive directory.  This is the command I use:  find /path/to/directory/username/ -mtime +30 -exec mv "{}" /path/to/archive/username/ \;   I suggested a modified version of this to answer a question on ask ubuntu, another user edited the code to change the end of line \; for + as it's faster(and more correct?).   However, using + in this way works if the -exec command is ls -lh but not in the actual command that I use. If I try it with + I get an error message:  find: missing argument to '-exec'   I don't understand why it's behaving this way, or what the correct command would be. Please don't just post a command correction, I'd like to understand rather than just follow a suggestion blindly. I need to read a properties files that's buried in my package structure in com.al.common.email.templates.  I've tried everything and I can't figure it out.  In the end, my code will be running in a servlet container, but I don't want to depend on the container for anything. I write JUnit test cases and it needs to work in both. Ubuntu 17.10 GDM Auto-Login not working How to print the full NumPy array, without truncation? Cutting a circular hole in a hollow cylinder without distorting the cylinder Whenever I try to cut out a circular hole in either a cylinder or sphere the polygons around the circular hole always have distorted normals after smoothing. Is there anyway to fix this. I've tried using the remesh modifier but wind up with some pretty ugly results. I am solving the TOEIC workbook, by the way, I faced conflict that doesn't match meaning I knew.  The problem is     The newest branch of the library is located in/on Pine Street.   I think either 'in' or 'on' have the same meaning. But the commentary said that 'be located in' is mainly writing in front of building name and city name.   I don't know what is correct? "All of the ...." or " All the ... "?