I was reading about $A_4$ and $S_4$ groups and I got curious about some of the subgroups. And if we have $H:= \{(),(12)(34),(13)(24),(14)(23)\}$, how would I determine and how would the structure look like of the quotient group $A_4/H$ and is $A_4$ a simple group?
Let K={(1),(12)(34),(13)(24),(14)(23)} show that K is a normal subgroup of the alternating group of degree A4 and find all the members of A4/K and write down the group table for A4/K.   i know that K is the Klein four group and i have already proven it is a normal subgroup but i need a start on approaching the A4/K part of this question no direct answers are necessary,just ideas thank you! :)
Cross Reference problem I have noticed after deleting an original page in WordPress with lets say a permalink of your_domain.com/contact/ and then trying to recreate the same page with the same name/slug, my WP is generating a "-2" at the end of the link, like: your_domain.com/contact-2/.  I don't understand why, since the original page was deleted. Why is WordPress treating new pages as if the original still exists? How to I clean WP caches, if such a thing even exists?Thank you ahead of time for clearing this up for me. Selecting all objects that have been specified in Modifiers Could anyone help in making a script for selecting all objects that are specified in the different modifiers that an object has stacked? For example if my object has a Boolean modifier and an Array modifier, which both refer to an empty or another object (in the Array modifier there is for example an Object offset you can specify). Would it be possible to select all those object with one click? Thanks. Resolving ambiguous overload on function pointer and std::function for a lambda using + I am trying to set multiple definition queries for one layer based on one field using python.  ie.  I have a City layer with a field that holds the city name.  I want a copy of the layer in an MXD with the Definition query set for each value in the City field.  So essentially, I have one source layer, with a copy of the layer as many times as there are values in the City field.  The query would look like this: CITY = 'Atlanta' I am considering to use iCloud Photo Library to store a growing number of family photo albums, shared to web or set of AppleIDs.  I dont need to keep original photos on iDevices for 99.9% of photos.  How much space will be used on the photo-originating set of devices (iOS, OS X, Windows)? I.e. what (roughly) is the function from the original size and number of photos, device storage space available, any other variables? I am applying for a UK visa. Can I show 5 months worth of bank statements for the UK visa, or is 6 months a requirement, as its not mentioned on their website?  There are no appointment dates showing in April. Is there any time limit in which we have to book an appointment after paying the fee? Origins of Tension First of all, I have a confusion about the definition and idea of Tension. For example, in my Physics Textbook, the idea of tension is written like this: &quot;Let's say there is a wire with a cross-section area of A and its length is $L_o$. If I hang the wire from a point and hang a weight of W on the other end of the wire, it will generate a repulsive force of T. This T is called the Tension Force.&quot; For me, this only explains to me what force will we call tension force, but it doesn't quite explain to me what is exactly this Tension force or where it originates from. Also, I have been doing pulley problems (Figuring out the acceleration of weights hanging from pulleys)  by first labeling all of the tension forces by $T_1$, $T_2$, and then doing some math with them to get my answer. But It bugs me that I am using these tension force values without even knowing where they came from or what they are. My idea was, maybe Tension force comes from the intermolecular bonds between the molecules of the rope, rubber or wire, etc. But even then, I have some different situational questions. For example:  Why you can't push a rope, copper wire against a wall?  What happens when I pull on two ends of a wire with 2 different forces?  Does the force increase proportionally to the applied force? Why does the wire break after a certain amount of force is exceeded.?  Does the idea of tension apply to everyday objects,(for example, a coffee mug, glass pane, etc). If it does, can I use something like a heat map or 3D equation to describe the tension in every point of the object? [Edit: more concisely, is there a mathematical way to show the various amounts of tension forces on a 2D (like a metal sheet) and 3D objects (like a football ) instead of a 1D object(like a string)].  If I take a metallic object like a paperclip or a thin wire. I can bend It multiple times and it eventually breaks into two pieces. I used to do this to pull of capacitors from old circuit boards. But does it happen with things like ropes or paper? If not, why it doesn't happen? ( Maybe it has something to do with intermolecular forces and the various internal stresses?)  Let's say I have an iron bar and I have thrown it out into space. Does that iron bar have equal stress distribution or still uneven or will the stress on all points become zero? (Is there even such thing as zero stress?) I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ? All  I just install a fresh new Ubuntu 16.04 on my new desktop which has a 8th generation Intel CPU i5 8600K. When OS installed, I found there is only one option in display setting: 1024 x 768 (4:3),  My monitor is a HDMI monitor with 1920x1080 resolution  Do anyone have any idea how to make the resolution setting right? Thanks.  BTW. I do not install any graphic card, I use the internal GPU of Intel Why is a ConcurrentModificationException thrown and how to debug it I am using a Collection (a HashMap used indirectly by the JPA, it so happens), but apparently randomly the code throws a ConcurrentModificationException. What is causing it and how do I fix this problem? By using some synchronization, perhaps?  Here is the full stack-trace:  Exception in thread "pool-1-thread-1" java.util.ConcurrentModificationException         at java.util.HashMap$HashIterator.nextEntry(Unknown Source)         at java.util.HashMap$ValueIterator.next(Unknown Source)         at org.hibernate.collection.AbstractPersistentCollection$IteratorProxy.next(AbstractPersistentCollection.java:555)         at org.hibernate.engine.Cascade.cascadeCollectionElements(Cascade.java:296)         at org.hibernate.engine.Cascade.cascadeCollection(Cascade.java:242)         at org.hibernate.engine.Cascade.cascadeAssociation(Cascade.java:219)         at org.hibernate.engine.Cascade.cascadeProperty(Cascade.java:169)         at org.hibernate.engine.Cascade.cascade(Cascade.java:130) There aren't any new or popular questions. There are tons of questions in both of those tabs but the question count is 0 for both of those when I try to view them in the expanded view. I am doing , and am having a few problems with function notation. I understand the form f(x).  What does the form f: x ↦ y mean? Could you also give one or two examples?  And, if possible, state your source. Thank you. Show that a continuous function has a fixed point Question: Let $a, b \in \mathbb{R}$ with $a &lt; b$ and let $f: [a,b] \rightarrow [a,b]$ continuous. Show: $f$ has a fixed point, that is, there is an $x \in [a,b]$ with $f(x)=x$.  I suppose this has to do with the basic definition of continuity. The definition I am using is that $f$ is continuous at $a$ if $\displaystyle \lim_{x \to a} f(x)$ exists and if $\displaystyle \lim_{x \to a} f(x) = f(a)$. I must not be understanding it, since I am not sure how to begin showing this... Should I be trying to show that $x$ is both greater than or equal to and less than or equal to $\displaystyle \lim_{x \to a} f(x)$ ? Does light pollution affect sleep quality while already asleep? Many articles on the internet write about melatonin and its effects, making you tired. A lot of articles also point out how the use of devices before going to bed in modern days likely decrease melatonin production. Therefore, making it harder to fall asleep. But this information is only about the process of getting to sleep, before we are actually sleeping.  A recent Harvard Medical article 1 states:     High exposure to bright, artificial outdoor lights during the night may result in sleepless nights for older adults.   Referring to a publication on Nov. 15, 2018, in the Journal of Clinical Sleep Medicine.  Considering that:    many people nowadays, use their mobile devices in bed prior to going to sleep; modern cities or houses have (street) lights that are always enabled at night; ignoring the phase of falling sleep; focusing only on the time we are asleep already, so during the actual sleep cycles.   Does a darker room with less or no (artificial) light pollution improve sleep quality, with all associated health benefits as a result? In other words: Does light pollution, while asleep decrease the sleep quality or does light pollution only negatively affect the phase of “falling asleep”?  I wasn't able to find any studies or information regarding sleep quality in relation to light pollution during sleep. Were such studies performed? Are there specific moments when light pollution affects sleep quality the most, for example during REM-sleep or in a specific sleeping phase?    1 Can you make an opportunity attack against a moving object? Group cohomology is one thing I don't seem to get my head around.  I understand $H^0$ as being the "fixed points" but when it comes to anything higher I have no idea what the notions are meant to be capturing...if anything.  For example I can see that 1-cocycles are homomorphisms with a specific property but that property doesn't seem to mean anything to me...and I think it should!  Is there an intuition or example that explains the meaning or is it just something that "stuck" and seems to work in lots of places? \includegraphics: Dots in filename \includegraphics cannot handle filenames that contain more than the one dot, separating the filename from the extension. Apparently it uses everything after the first dot as extension and then, of course, complains about an unknown graphics extension.  This is annoying as I very often have filenames that contain parameter values, e.g. plot_a0.4_b0.6.pdf.  Is there a way to teach LaTeX to interpret only the string behind the last dot as extension? Given $S \hookrightarrow T$ construct $U ≈ T$ disjoint from $S$ in Z set theory? Does any one know the exact expression of the integral,  $$E_n(s)=\int_0^\infty \frac{r^{n-1}}{(1+r^2)^{\frac{s}{2}}}\,dr~~~~s&gt;n, n\in \mathbb{N}$$ or more generally, $$E_a(s)=\int_0^\infty \frac{r^{a-1}}{(1+r^2)^{\frac{s}{2}}}\,dr~~~~s&gt;a, a\in \mathbb{R}$$ For the special case $s=n, n+2$ I find out by induction that $$ E_{n-1}(n)=\frac{\omega_{n-1}}{2\omega_{n-2}}~~\text{and}~~E_{n-1}(n+2)=\frac{\omega_{n-1}}{2n\omega_{n-2}}. $$ where $\omega_{n-1} = \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}$ is the  surface measure of the n-dimensional sphere of $\mathbb{R}^n$. Further result is welcome