(Q,+) and (C*,+) has no finite index subgroup
Subgroup of $\mathbb{Q}$ with finite index
Why are American-style options worth more than European-style options? Why are American-style options worth more than European-style options?   I realize I can exercise American-style options anytime before the  expire, but I can only exercise European-style options during their  "exercise period" (usually right when they expire, but no earlier).   So it makes sense that an American option is worth at least as much as  a European option.   But why is it worth more? If I exercise my American option early, I  may make more money than waiting until expiration, but I also may make  less. Mathematically, there's no advantage, since I'm equally likely  to make as much money by waiting for expiration.   Mathematically speaking, is there ever a good reason to exercise an  American option early?   I realize people don't always behave logically, but even the formulas  that valuate options show that American options are worth more.   EDIT (answering comments):    Many of the comments people made would apply to lookback options  (), but not to American  options.  American options can do everything European options can and more. I  understand that, and that means American options can't be worth any  LESS than European options, but I'm still not clear on how you would  mathematically calculate this extra value.  @Aaronaught: "The difference between an American and European  option is the difference between getting N chances to get it right (N  being the number of days 'til expiration) and getting just one  chance. It should be easy to see why you're more likely to profit  with the former, even if you can't accurately predict price  movement."    REPLY: I don't think you really get N chances. Once you exercise the  option, that's it, no more chances. And, if you decide not to exercise  the option today, and the underlying's price falls, you can't go back  in time and exercise it yesterday.    @jdsweet: Think of it this way, if you traveled back through time  one month - with perfect knowledge of AAPL's stock price over that  period - which happens to peak viciously then return to its old price  at the end of the period - wouldn't you pay more for an American  option?    REPLY: Well, no. I'd simply buy a cheaper European option that expires  right at the spike. Or, I'd buy a European option that expires later,  and sell it when the spike occurs. American options don't give you  precognition, so you're still guessing as to when to sell/exercise.    @jdsweet: "one of the more common reasons people exercise (as  opposed to sell) an American option before expiration is if an  unexpected dividend (larger than remaining time value of the option)  was just announced that's going to be paid before the option contract  expires"    REPLY: This makes sense. I'm actually trading FOREX options, so there  is no dividend. In that scenario, are European and American options  worth the same? How do you get a submit button in the Windows Phone keyboard? Encopresis: how can we encourage our child to clean himself or ask for help immediately instead of ignoring the dirty underwear? I'm running badblocks to check for bad segments on an external drive, and it's been about an hour and it has not yet finished. Now, I need to go and consider cancelling. Is this somehow risky? Should I avoid this?   Clearly, I will need to start again from scratch; I just want to know if this is somehow risky to abort midway. Prove $\sum_{d \leq x} \mu(d)\left\lfloor \frac xd \right\rfloor = 1 $ Complete references in between the document When we do   \bibliographystyle{style} \bibliography{References.bib}   We get all the references at the end of the document, with some kind of reference/id in the document.   What I need is the complete reference in the document itself (not at the end) and then to be able to write what I want below this reference. For example;  Chaminade, T., Hodgins, J., &amp; Kawato, M. (2007). "Anthropomorphism influences perception of computer-animated characters' actions." Social Cognitive and Affective Neuroscience 2(3): 206-216      - Some text..........      - Some more text.......  Cowley, S. J., Kanda, T. (2005). "Friendly machines: interaction-oriented robots today and tomorrow." Alternation 12(1a): 79-106.      - Some text.....      - Some more text....   How can I do it? Thanks. I'd like to do the equivalent of the following in LINQ, but I can't figure out how:  IEnumerable&lt;Item&gt; items = GetItems(); items.ForEach(i =&gt; i.DoStuff());   What is the real syntax? Is it possible to create an instance of a generic type in Java? I'm thinking based on what I've seen that the answer is no (due to type erasure), but I'd be interested if anyone can see something I'm missing:  class SomeContainer&lt;E&gt; {     E createContents()     {         return what???     } }   EDIT: It turns out that  could be used to resolve my issue, but it requires a lot of reflection-based code, as some of the answers below have indicated.  I'll leave this open for a little while to see if anyone comes up with anything dramatically different than Ian Robertson's . To me, using Present Perfect form means the event can occur again. So, saying      someone has died   may not be grammatically correct.  Also, I noticed (it might be just coincidence):     passed away   is used more often than     has passed away   Is using Present Perfect correct here? I know this sounds silly but I would like some objects to follow physics while others not to collide each other.   I tried to achieve them by setting their position exclusively. But then it will ignore all physics.   Is what I am trying to do even possible? How to add a description to the user name input field in the registration form? I want a description under the user name input field in the Wordpress registration form like in the next example. How can I do this?  //create an extra field in login page for invitation code add_action( 'register_form', 'add_invitation_code_field' );  function add_invitation_code_field() {    ?&gt;       &lt;p&gt;          &lt;label for="invitation_code"&gt;             Invitation code             &lt;input id="invitation_code" class="input" type="text" size="20" value="" name="invitation_code"/&gt;             &lt;span class="description"&gt;&lt;?php _e("Description text"); ?&gt;&lt;/span&gt;          &lt;/label&gt;       &lt;/p&gt;    &lt;?php } I am not sure what caused this error, but here is what the whole error says, and also this is sitting as a notification, and preventing me from updating any software using Update Manager - Please provide some assistance or tell me how to figure out what to do to fix it.  Could not calculate the upgrade  An unresolvable problem occurred while calculating the upgrade.  Please report this bug against the 'update-manager' package and include the following error message: 'E:Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.' If $ \operatorname{Tr}(M^k) = \operatorname{Tr}(N^k)\;\forall 1\leq k \leq n$, then how do we show the $M$ and $N$ have the same eigenvalues? Forgotten the pattern and phone not showing the "forgot pattern" button. I upgraded my Moto G2 to Android Lollipop using over-the-air upgrade. It was awesome but I set a pattern lock which I forgot. Usually after 5 wrong attempts it shows the "forgot password" button but it is not showing that button even after 100 wrong attempts. How to unlock the phone? How do I recover my accidentally lost Windows partitions after installing Ubuntu? I have a Toshiba satellite A-200 laptop with a Vista OS on it with 4 NTFS partitions  (C:) Vista  (D:) Entertainment  (E:) Work  (F:) Sources   and I wanted to start using Ubuntu instead. So I tried it first from the live CD and everything was OK and all the partitions were shown and working and so I decided to install Ubuntu to replace Vista on the (C:) drive. After I did that I can no longer find my folders and files on the (D:), (E:), (F:) partitions and the only file system that is shown is one 198 GB although my HDD is 320 GB.  I can't access the lost data on the remaining 120 GB which I hope is still there and not totally lost  I am now working from the live CD but I am unable to install testdisk. Can I recover the Vista partitions by the product recovery CD to get my laptop back to the factory settings? Can I recover the NTFS partitions using a recovery program for Windows or will that make the problem worse?   I need these data badly as I don't have a backup for them. How to put a '+' sign inside a square like xor What space does the spell Cloud of Daggers fill? I am trying to workout what template  (PHB, p. 222) would be on a grid. The spell states:     You fill the air with spinning daggers in a cube 5 feet on each side, centered on a point you choose within range.   So does that mean it would look like this:     Or like this:    From what I've read online, there seems to be some suggestion that it is a single square (tile) though this does stack with the spell description of "a cube 5 feet on each side".  Clearing this up for my group would be great, and I am sure our DM will be pleased as his monster were nicely diced by this spell. Thanks. How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$ I need help to prove this in real analysis. I think it uses IMVT, but not sure how to do it.     Let $f(x)$ be a real valued continuous function on $[0,1]$. Show that   $$ \lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1) $$ Does "non-" prefixed to a two word phrase permit another hyphen before the second word?  If I want to refer to an entity which is defined as the negation of another entity by attaching "non-" it seems strange to attach the "non-" only to the first word when the second one is really the word naming the entity. For example,     non-control freak   looks like a person obsessed with not being in control rather than one who is simply not obsessed with control.     non-control-freak   actually looks better because the "freak" is attached to the "non-" as much as it is to "control-", without the space implying the presence of a phrase break.    and its answers make it sound like totally detaching the "non" could be an option, but if I were following the convention of hyphenating "non-" could I add a hyphen after the next word to resolve this problem? Is there a way to add some custom font on a website without using images,  or some other graphics?  For example, I was working on a wedding website, and I found a lot of nice fonts for that subject. But I can't find the right way to add that font on the server. And how do I include that font with CSS into the HTML? Is this possible to do without graphics?