Probability Convergency / Stochastically bounded
Bounded in Probability and smaller order in probability
Proper use of passive and active voices? Double-clicking a window’s title bar to minimize it in Mavericks After upgrading to Mavericks I’m no longer able to minimize a window to the Dock by double-clicking on its title bar. How do I get this functionality back? Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime. Building a wormhole We regularly get questions about wormholes on this site. See for example  and . Various wormhole solutions are known, of which my favourite is  because it's closest to what every schoolboy (including myself many decades ago) thinks of as the archetypal wormhole.  The trouble is that Visser has pulled the same trick as  of starting with the required (local) geometry and working out what stress-energy tensor is required to create it. So Visser can tell us that if we arrange exotic string along the edges of a cube the spacetime geometry will locally look like a wormhole, but we know nothing about what two regions of spacetime are connected.  My question is this: suppose I construct a Visser wormhole by starting in Minkowksi spacetime with arbitrarily low densities of exotic matter and gradually assembling them into the edges of a cube, how would the spacetime curvature evolve as I did so?  I'm guessing that I would end up with something like Wheeler's bag of gold spacetime. So even though I would locally have something that looked like a wormhole it wouldn't lead anywhere interesting - just to the inside of the bag. I'm also guessing that my question has no answer because it's too hard to do any remotely rigorous calculation. Still, if anyone does know of such calculations or can point me to references I would be most interested. Let S be a finite set.Let F be a surjective function from S to S.  How do I prove that it is injective? Add a message when posting to old questions with more than N answers Can you reject a postdoc offer after the PI has paid a large sum for flights/accommodation for your visit? I am being considered for a very nice industry scientist position but this involves many interviews and assessments and can take some time.  However I have also accepted an invitation to the US (I'm in Australia) to give a presentation in relation to high likelihood of employment as a postdoc in a research institute.  The postdoc supervisor has arranged for sightseeing and lab visits and hotels. I feel bad if somehow I wish to pursue the industry position.  Has anyone had a similar experience and have you turned down something like that before? I have read that positional parameters start at $1 (for example: $1, $2, $3 and so on are positional parameters). But $0 is not a positional parameter.  But why $0 is not a positional parameter?  I think this could be a reason, but not sure:  Positional parameters only take their values when a script is executed. For example: if we do ./myScript Hello, then $1 will have the value Hello. But $0 can take its value on two occasions: when a script is executed (it would have the value of the script name), and when bash itself is executed without a script (it would have the value bash or -bash). I am reading the Dummit and Foote's proof of Cauchy's theorem for abelian groups. On the highlighted line, how do you conclude from $y\notin N$ and $y^p\in N$ that $\langle y^p \rangle \neq \langle y \rangle$ and that $|y^p|&lt;|y|$?     Proposition 21.   If $G$ is a finite abelian group and $p$ is a prime dividing $|G|$, then $G$ contains an element of order $p$.      Proof:   The proof proceeds by induction on $|G|$, namely, we assume the result is valid for every group whose order is strictly smaller than the order of $G$ and then prove the result valid for $G$ (this is sometimes referred to as complete induction).   Since $|G| &gt; 1$, there is an element $x \in G$ with $x \neq 1$.   If $|G| = p$ then $x$ has order $p$ by Lagrange’s Theorem and we are done.   We may therefore assume $|G| &gt; p$.      Suppose $p$ divides $|x|$ and write $|x| = pn$.   By Proposition 2.5(3), $|x^n| = p$, and again we have an element of order $p$.   We may therefore assume $p$ does not divides $|x|$.      Let $N = \langle x \rangle$.   Since $G$ is abelian, $N \trianglelefteq G$.   By Lagrange’s Theorem, $|G/N| = |G|/|N|$ and since $N \neq 1$, $|G/N| &lt; |G|$.   Since $p$ does not divide $|N|$, we must have $p \mid |G/N|$.   We can now apply the induction assumption to the smaller group $G/N$ to conclude it contains an element $\bar{y} = yN$, of order $p$.   Since $y \notin N$ ($\bar{y} \neq \bar{1}$) but $y^p \in N$ ($\bar{y}^p = \bar{1}$) we must have $\langle y^p \rangle \neq \langle y \rangle$, that is, $|y^p| &lt; |y|$.   Proposition 2.5(2) implies $p \mid |y|$.   We are now in the situation described in the preceding paragraph, so that argument again produces an element of order $p$.   The induction is complete.      (Original image .) If I'm playing the Hilltop farm in co-op - what farmer's mining level is used for the hilltop spawns? Getting / setting file owner in C# How to delete ".."-folder under Windows? I have a folder on my desktop which is named ".." (which itself contains subfolders with an empty name). The folder can't be deleted the usual ways, even not when using the CLI. Does anyone know how I can remove this folder on Windows Vista 64-bit?  Renaming does not work, either (obviously).  This is what "dir /x" echoes:  09.10.2009  15:04    &lt;DIR&gt;                       . 09.10.2009  15:04    &lt;DIR&gt;                       .. 08.03.2007  11:18    &lt;DIR&gt;                       cgi-bin                0 Datei(en),              0 Bytes                3 Verzeichnis(se), 45.866.037.248 Bytes frei My Granddaughter has lost/forgotten her ID and passwords for her iPad. When the iPad powers on, it goes straight to an iCloud password request and I cannot get past it.  I have installed iTunes on my laptop but it fails to connect to the server (error 310).  I installed iCloud control panel and tried a restore but that failed because of "Find my iPad" was active, which I cannot turn off because of the original password problem!  This is driving me crazy. I just want to reset everything now so that I can hand it back to her. Can anyone help? Should import statements always be at the top of a module? states:  Imports are always put at the top of the file, just after any module comments and docstrings, and before module globals and constants.  However if the class/method/function that I am importing is only used in rare cases, surely it is more efficient to do the import when it is needed? Isn't this: class SomeClass(object):      def not_often_called(self)         from datetime import datetime         self.datetime = datetime.now()  more efficient than this? from datetime import datetime  class SomeClass(object):      def not_often_called(self)         self.datetime = datetime.now() Do you really want to create a new tag? Non-root login on EC2 instance The future of meta.stackoverflow and meta.stackexchange Historically this site, , has been the place to ask questions about not just Stack Overflow, but the functioning of the entire Stack Exchange (2.0) network, including Area 51 and stackexchange.com and careers.  (Yes, there was a , but that dealt with the old legacy Stack Exchange 1.0 sites. It was deprecated a while ago, and currently redirects here.)  Now that we've gotten many, many months under our belt from the SE 1.0 days, I think it's time that we gave Stack Overflow its own, true per-site meta, and moved the higher level network meta discussions to meta.stackexchange.com.  Over time, it's begun to bug me more and more that requests for Stack Overflow, the site, kind of get buried here on meta.stackoverflow under the avalanche of network issues. Retag requests, synonym requests, things truly specific to Stack Overflow, the site, not the network. This is unfair to our vastly largest and flagship Q&amp;A site, and it's unfair to the Stack Overflow users to mix their requests in with network level concerns.  It's also odd that Stack Overflow is the only network site without a proper, dedicated per-site meta, where its meta has its own reputation system not tied to the parent site in any way.  This is mostly a quirk of history more than anything else, and I believe it is now an appropriate time to split into two sites:   , a proper per-site meta for Stack Overflow with integrated rep , a global network meta with its own reputation system   Yes, this will be painful. But I believe it is a good, necessary, and healthy step both for the future of the network and the future of Stack Overflow.  Some things to discuss:   which questions should be migrated from here to meta.so? Obviously site-specific things like the retag-requests, and perhaps questions with the  tag. Anything that is 100% wholly specific to Stack Overflow and not generalizable to the rest of the network should be moved over. This may be a pretty small, narrow list of questions, and that's OK. Should we do some meta spring cleaning in our transition? Which questions / tags should be blown away as no longer relevant, referring to ancient versions of Stack Overflow or issues that have long since ceased to exist and aren't instructive for any future visitors? Your meta reps will generally be unaffected, except insofar as we delete for spring cleaning, or move site-specific things to meta.so. We still plan to have a distinct reputation system for the new meta.se site, just like here.   I do not have a timeline for this change, that is up to Jarrod and David to decide. It may be months away. However, we feel pretty strongly that this is something that we need to do to clean things up and pave the way for the future -- so I expect it will happen in the next few months, giving us lots of time to decide how we want to do it, per the above.  Update  . Do separate autofocus motors exist for entry level Nikon DSLRs? footnotes at the end of the book Is it possible to have all footnotes placed at one specific place in my document instead of bottom of each page? I have multiple footnotes scattered in the article and it would be clearer if all of them are at the very end of the article  Edit (moved from an ``answer'')  is it possible in endnotes to have links in each number where I placed footnote to the actual footnote at the end? What is true for a ring with exactly two right ideals