Let $G$ be an infinite group and $H\le G;\;K\le H$. Show that $K\le G$ and $[G:K]=[G:H][H:K]$.
Understanding the difference between two theorems related to $|G\,:\,K|=|H\,:\,K|\cdot|G\,:\,H|$.
I've been reading a Wikipedia article that describes the symptoms of rabies in various species, and I want to add a clarification note about which species (or species) a specific paragraph refers to. How can I make it clear that I'm referring either to the singular or plural of the word "species" when the singular of this word is the same as its plural? "Species or species" is the only grammatically correct way that I can think of, but it seems redundant and confusing. Is there any better way to contrast the singular of "species" from the plural of "species"? Algorithms for computing multiplicative order modulo $n$ If I take the $k \in \mathbb{N}$ power of 10 and mod it by a large prime, I notice that the remainders repeat at some point.  For instance $10^k mod~7$ seems to repeat every $8$th value of $k$.  Given $k$ (the power of ten), and a large prime number $p$, how can I compute the size of the cycle?  I tried writing a python script that computes the first instance of a repeat. However, I don't know if this works since a single instance of a repeat may not necessarily mean a full cycle. Where is "the" used? Is there a rule? Why no definite article in "Shops are open late in summer"? Notice these sentences:   Shops are open late in summer.   Summer is a traditionally viewed as a slow season for games.   At the summer solstice, the days are longest and the nights are shortest.   it's time to start thinking about what 2016's Song of the Summer will be.    In English class, they told us that "the" is used for the things that are known.  One example for this was a phrase like "The man who killed his wife".  So when we are speaking of that man, we know what he did. Somehow we know something more than nothing about him. He's not just every man in the world.     So Now I'm confused about why "the" is not used in the first and second sentences. Can anyone please explain it for me? Why does QgsSpatialIndex.nearestNeighbor(point, 1) sometimes return 2 lines? I'm trying to get the closest line to a point using PyQGIS in a plugin. The active code file is this:   When I request the closest line (only 1)  nearestIds = self.spIndex.nearestNeighbor(self.feat2.geometry().centroid().asPoint(),1)   I sometimes get a list with two lines and the first one is not the closest but as far as I can see the second one is. Why is this?   Does QgsSpatialIndex use only boundingboxes for lines? How do I get the real closest line always, not only most of the time? Contribute to bounty? How should tiny $p$-values be reported? (and why does R put a minimum on 2.22e-16?) For some tests in R, there is a lower limit on the  calculations of $2.22 \cdot 10^{-16}$. I'm not sure why it's this number, if there is a good reason for it or if it's just arbitrary. A lot of other stats packages just go to 0.0001, so this is a much higher level of precision. But I haven't seen too many papers reporting $p &lt; 2.22\cdot 10^{-16}$ or $p = 2.22\cdot 10^{-16}$.   Is it a common/best practice to report this computed value or is it more typical to report something else (like p &lt; 0.000000000000001)? I quite often add a question as a favourite when there is something of interest to me in it or its answers. Looking back through my favourites, I cannot remember why I added some of them as a favourite.  What I'd like to be able to do is to add a note/comment when I add a favourite as a reminder as to why I've added it. This will show on my favourites page, but nowhere else - and possibly to no one else.  In addition, it would be good to be able to favourite an answer to a question, thereby adding a favourite to both the question and a particular answer with something of note in it. This would only show as a single favourite, but when clicking its link would go to the favourited answer.  Possible? Or is there another way of doing this? I've already got too many favourites to remember why I added a lot of them. AlwaysOn AG: Current Host Server - Owner Node - Primary Instance I'm building up a SQL Server AlwaysOn AG and I'm on the step "failover", manual and automatic:  I noticed that in the Windows Server Failover Cluster a "Current Host Server" exists. My knowledge says that this is the "master" of the cluster that the cluster hosts. But now I noticed that the "Owner Node", which hosts the Availability Group(-role) a other node could be. (The owner node is also the primary instance of the AG, right?)  Now for me there exist two questions for the normal operating of an AlwaysOn AG:  Q1: Should the "Current Host Server" the same as the Owner Node of the AlwaysOn AG?  Q2: Imagine if I want to maintain a host. Shall I first failover the AG (via SSMS or with ALTER AVAILABILITY GROUP...) to the second node/another node and at second a failover of the WSFC-Current Host?  Maybe someone could help me with this questions and give me some advice.  Thanks Why is there a temperature gradient on Earth? I was thinking about why the poles are colder and I came out with three possible explanations.  1. The atmosphere blocks light beams coming to poles    The Sun is seen only slightly over the horizon at the poles, so the light beams have longer way to the poles. If only this case were true, then the total difference between the poles and the equator would be the same as the energy that was absorbed by the atmosphere.  2. Sun is only slightly over the horizon at the poles.    Poles are often in shadow of the parts of the Earth which are closer to the equator.  3. Poles are farther from the Sun    The longer is the way to Sun the lower is the chance to catch a light from the Sun.  Please, correct me wherever I'm wrong. I'd like to know   Which of these three is most correct?  How significant are the described effects of the other explanations? Inequality from Chapter 5 of the book *How to Think Like a Mathematician* This is from the book How to think like a Mathematician,  How can I prove the inequality $$\sqrt[\large 7]{7!} &lt; \sqrt[\large 8]{8!}$$  without complicated calculus? I tried and finally obtained just $$\frac 17 \cdot \ln(7!) &lt; \frac 18 \cdot \ln(8!)$$ What is the difference between a framework and a library?   I always thought of a library as a set of objects and functions that focuses on solving a particular problem or a specific area of application development (i.e. database access); and a framework on the other hand as a collection of libraries centered on a particular methodology (i.e. MVC) and which covers all areas of application development. Is it completely against the Java way to create struct like objects?  class SomeData1 {     public int x;     public int y; }   I can see a class with accessors and mutators being more Java like.  class SomeData2 {     int getX();     void setX(int x);      int getY();     void setY(int y);      private int x;     private int y; }   The class from the first example is notationally convenient.  // a function in a class public int f(SomeData1 d) {     return (3 * d.x) / d.y; }   This is not as convenient.  // a function in a class public int f(SomeData2 d) {     return (3 * d.getX()) / d.getY(); } Why does the experience-to-next-level not change between some levels? At every level from 1st to 11th, the experience needed to reach the next level (from the current level) increases every level. However, this pattern gets broken for most of the later levels:   Going from 10th to 11th takes 21,000xp (= 85k - 64k), but 11th to 12th takes only 15,000xp! (= 100k - 85k). Both 13th and 14th level only take extra 20k experience each, i.e. the amount needed doesn't increase between those two levels. Levels 16 and 17 similarly only need 30k experience each, and 18 and 19 only need 40k experience each.   Why does the last half of the experience chart have some of these levels maintain the same experience costs, and even briefly decrease the experience cost for 12th level? Help with packages in java - import does not work When trying to install CrossOver deb I get "Error: Dependency is not satisfiable libpng12-0" I have tried to install libpng12-0 but it still doesn't work.  Here:    How do I fix this? Help finding this set Lets define the following:  Let A be a set. A is innumerable if and only if there exists a bijective function from A to $\mathbb{N}$  Proof that there exists an innumerable set $B \subseteq \mathcal P \left({\mathbb{N}}\right)$ such that it has the following properties:  1) $X\cap Y =\emptyset$ for all $X,Y \in B$ such that $X \neq Y$  2) $\mathbb{N}= \bigcup B$  3) All elements of B are innumerable.  Any help is appreciated. I don´t know how to find this set ... Are there any banned subjects on Area 51? I know of the following rules for creating a proposal on Area 51:  The scope must be reasonably well-defined, not too unclear (e.g. &quot;cool stuff&quot;), too opinion-based (e.g. &quot;car reviews&quot;), too broad (e.g. &quot;life on Earth&quot;), or too narrow (e.g. &quot;The 1995 academic misconduct scandal at East Oak Mews High School&quot;). One  come to Area 51 with an already existing community rather than hope to build one on Area 51. Proposals that are subsets of existing sites are not acceptable (e.g.  — deleted). These get closed as &quot;This subject is already well-covered by a live Stack Exchange site. We generally do not split off subjects simply to give them their own space.&quot; Proposals for non-English sites .  Other than the criteria above, are there any per se restrictions on site content for an Area 51 proposal? If so, what topics are currently banned? I would expect a proposal for a &quot;Let's organize a lawsuit against Stack Exchange&quot; or &quot;[Specific SE staff member] sucks, amirite?&quot; would get closed for obvious reasons, but I'm interested if there are any non-obvious topics that are not currently allowed in proposals regardless of how well-defined, well-scoped, original, and well-supported by a community they are. Having a list of banned topics would be helpful to potential proposers as well as the communities that back them. For example, I wouldn't want to go to all the effort of organizing a community, bringing them to Area 51, etc., only to have my proposal closed with a curt &quot;Sorry, due to the political climate, we are not currently accepting proposals for sites about real-life weapons.&quot; custom closure. To be clear, I am asking solely about topic restrictions (e.g. &quot;no sites about fishing, hunting, or weapons&quot;). I'm not asking about clarity requirements, the requirement to have a pre-existing community, not duplicating an existing site, or the requirement to not use sock puppets to artificially inflate proposal stats. Those requirements are already well-defined elsewhere or else are common sense. Also to be clear, I don't have a specific proposal idea in mind right now or the community to make it happen. I'm asking in general. The question could be rephrased as &quot;What are those topics where I shouldn't even bother to try to define, gather a community, etc. because any attempt to propose a site will be summarily closed due to covering a banned topic?&quot; This is not a duplicate of  because that question is about the appropriateness of expanding a proposal to cover things that are acceptable proposal topics but arguably don't make sense in combination with the original proposal scope. My question is about topics that are completely banned from proposals regardless of whether they are included in the original proposal description or tacked on by the community later. I have several input and option elements on my page, each (well almost) have an event attached to update some text on the page once they change. I use jQuery which is really really cool :)   I also use Microsofts  framework, utilizing the UpdatePanel. The reason why I do that is that certain elements are created on the page based on some server-side logic. I don't really want to explain why I use the UpdatePanel - even if it could (it can with quite some effort) be rewritten to use only  I still want that UpdatePanel.  You probably guessed it - once I have a postback on the UpdatePanel the jQuery events stops working. I actually was expecting this, since the "postback" is not really a new postback so my code in document.ready that binds the events won't be fired again. I also confirmed my suspicion by reading up on it in the jQuery help libraries.  Anyway I'm left with the problem of rebinding my controls after the UpdatePanel is done updating the . I preferably need a solution that does not require adding more .js files (jQuery plug-ins) to the page but something as simple as being able to catch the UpdatePanel's 'afterupdating' where I can just call my method to rebind all the form elements. Closed form for $\sum_{k=0}^{m} {\binom {m}{k}} a^{k} (b+ck)^N$ Is there a closed form for the following?   $$\sum_{k=0}^{m}  {\binom {m}{k}} a^{k} (b+ck)^N$$      how about a pretty limit for large $b$.   I have tried using the binomial expansion for the $(b+ck)^N$ to convert the above expression to double sum and then tried to simplify the double sum which was not successful. Any idea if I have find a closed form for it? How can I reconnect to a ssh session after a broken pipe?