Are there uncountably many injective functions from $\mathbb N$ to $\mathbb N$?
Explicit bijection between $\Bbb R$ and permutations of $\Bbb N$
Finding $\lim_{n\to\infty} a_n^n$ where $a_n=\frac{(n!)^2}{(n+k)!(n-k)!}$ In the Avengers movie, why is Loki sweating and weak when he first arrives on Earth, in the SHIELD base? Group of order $8p$ is solvable, for any prime $p$ Consider the following question:     Show that a group $G$ of order $8p$ is solvable, for any prime $p$.   I am kind of stuck, but here are my first attempts:  I chose the series of subroups $G&gt;H_8&gt;H_4&gt;H_2&gt;H_1=\{e\}$ where $H_k$ has order $k$. All of these subgroups exist due to Sylow's Theorems. We have the quotients $H_8/H_4\cong H_4/H_2 \cong H_2/H_1 \cong \mathbb Z/2\mathbb Z$, so $G&gt;H_8\rhd H_4\rhd H_2\rhd H_1=\{e\}$. Only the factor $G/H_8 \cong \mathbb Z/p\mathbb Z$ is causing me a headache, because although its of prime order (and hence abelian) I don't know whether $H_8$ is normal in $G$ (unless $p=2$).  If $p\ne 2$, then the number $k$ of Sylow $2$-subgroups is $1$ mod $2$. Since $k$ divides $|G|$ we might have $k=1$ or $k=p$. If we had $k=1$ then $H_8$ would be the only Sylow $2$-subgroup (and hence normal in $G$). But how could we show this?  Or is there even an easier way to approach this problem?  $\ $  edit: I am going to try a different approach: The case $p=2$ is clear. When $p=3$ or $p=7$ the group has order 24 or 56 and since the smallest simple non-abelian group has order 60, the group must also be solvable in these exceptional cases (as Mariano pointed out).  In any other case we get a Sylow $p$-subgroup $H_p$ of order $p$, which is is normal in $G$. The quotient $G/H_p$ has order $8=2^3$; and prime power order implies solvable. So $G/H_p$ is solvable and $H_p$ is also solvable. And since the composition factors of $G$ are those of $H_p$ together with those of $G/H_p$, we conclude that $G$ is solvable iff $H_p$ and $G/H_p$ are solvable, which is the case. Hence $G$ is solvable if $|G|=8p$. C++ map access discards qualifiers (const) The following code says that passing the map as const into the operator[] method discards qualifiers: #include &lt;iostream&gt; #include &lt;map&gt; #include &lt;string&gt;  using namespace std;  class MapWrapper { public:     const int &amp;get_value(const int &amp;key) const {         return _map[key];     }  private:     map&lt;int, int&gt; _map; };  int main() {     MapWrapper mw;     cout &lt;&lt; mw.get_value(42) &lt;&lt; endl;     return 0; }  Is this because of the possible allocation that occurs on the map access? Can no functions with map accesses be declared const? MapWrapper.cpp:10: error: passing const std::map&lt;int, int, std::less&lt;int&gt;, std::allocator&lt;std::pair&lt;const int, int&gt; &gt; &gt; as this argument of  _Tp&amp; std::map&lt;_Key, _Tp, _Compare, _Alloc&gt;::operator[](const _Key&amp;)  [with _Key = int, _Tp = int, _Compare = std::less&lt;int&gt;,  _Alloc = std::allocator&lt;std::pair&lt;const int, int&gt; &gt;] discards qualifiers A relative managed to change her password, yet can not input the correct password.  She does not want to reset the phone to avoid loss of data (pictures, contacts, messages, etc.).  Will the phone eventually stop accepting logon attempts? recover data after using dd command Is it possible to activate an iPhone 4 without a SIM? Which predictions of The Last Jedi: Darths & Droids Bingo were correct? Parametric vs non-parametric machine learning methods I looked-up many references and websites and researched on how to determine if a method is between parametric or non-parametric. I came up with below definitions,     A parametric algorithm has a fixed number of parameters. In contrast, a non-parametric algorithm uses a flexible number of parameters, and the number of parameters often grows as it learns from more data.   From .  Moreover, I found,     A parametric model, we have a finite number of parameters, and in nonparametric models, the number of parameters is (potentially) infinite.    From .  And many other methods of determination, though the problem is none of them can helo ones to determine if a certain hypothetical method is parametric or not. (For instance, why k-means' number of parameters are constant but KNN is variable or basically what do we call a parameter and what we do not?) How do you obtain the Poké Flute to wake up Snorlax in FireRed? I'm trying to get past a Snorlax and I need a Poké Flute to wake him up—how do I get one? Dual boot UEFI Windows 7 and Ubuntu 12.04 (both 64 bits). W7 entry doesn't appear in GRUB I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical.  In ZFC, we can prove that the (second-order) Peano axioms have a model, call it $\mathbb{N} = (N,0,S)$. Furthermore, $\mathbb{N}$ is unique up to isomorphism. Thus, it would seem that we've pinned down $\mathbb{N}$.  However, if ZFC is consistent, then it has some very peculiar models. In particular, it has a model $(V_0,\in_0)$ whose "native" $\mathbb{N}$, let call it $\mathbb{N}_0$, is a model for the sentence "ZFC is consistent," and another model $(V_1,\in_1)$ whose "native" $\mathbb{N}$, lets call it $\mathbb{N}_1$, is a model for the sentence "ZFC is inconsistent."  But in reality, in other words, for the "real" $\mathbb{N}$, only one of these can be sentences can be true. Furthermore, the objects $\mathbb{N}_0$ and $\mathbb{N}_1$ aren't isomorphic, since the set of sentences they satisfy are different.  So it seems to me that $\mathbb{N}$ cannot truly be pinned down. Furthermore, its categoricity is, in some sense, illusory. Is this correct, or am I missing something? Basis of cotangent space Can't install MySQL on Mountain Lion ++ is dangerous for C programmers C# 4.0 optional out/ref arguments Sending a message to nil in Objective-C As a Java developer who is reading Apple's Objective-C 2.0 documentation: I wonder what "sending a message to nil" means - let alone how it is actually useful.  Taking an excerpt from the documentation:     There are several patterns in Cocoa   that take advantage of this fact. The   value returned from a message to nil   may also be valid:         If the method returns an object, any pointer type, any integer scalar   of size less than or equal to   sizeof(void*), a float, a double, a   long double, or a long long, then a   message sent to nil returns 0.   If the method returns a struct, as defined by the Mac OS X ABI Function   Call Guide to be returned in   registers, then a message sent to nil   returns 0.0 for every field in the   data structure. Other struct data   types will not be filled with zeros.   If the method returns anything other than the aforementioned value   types the return value of a message   sent to nil is undefined.      Has Java rendered my brain incapable of grokking the explanation above?  Or is there something that I am missing that would make this as clear as glass?  I do get the idea of messages/receivers in Objective-C, I am simply confused about a receiver that happens to be nil. Currently my spouse and I rent a small apartment.  Eventually we'd like to expand the family, and a one bedroom rental won't cut it forever - even if we continue renting we'll be expanding to a detached single family home eventually.  In our area the rent ratios work out to about 16, which is under the cut for buying being a better financial decision over the long haul - i. e. I don't think it'd be economically an awful decision to buy (NYT calculator shows 10 years at the current prices and rents for buying to make sense).  I've also read enough of the rental propagandists to know that home ownership isn't the best way to grow my money.  My reasons for wanting to own are the intangibles: knowing we aren't going to have to move due to a landlord, being able to modify the place to our own liking, and that intangible sense of "this is mine" (even when it's the bank's for a while).  The things we do have:   Around (at or over) a 30% down payment depending on purchase price, and we won't be under 30% Closing costs separate from that fund Around 1% of the home value in reserve for emergency repairs separate from those funds A six month emergency fund separate A fund to replace an ailing car that will probably die before we can save up after this expense   At some point in the year or so after the purchase we plan to have enough saved up to pay down the principal and change our monthly payment somewhat, but not significantly.  So it seems like we've got the financial ducks in a row.  We can continue to fully fund our retirement accounts and make the monthly payments (PITI).  The only hesitation we have is that we can't make our payments comfortably on one income alone - it'd be over 30% of one income to do so, by just a bit.  So neither of us can completely stop working, although the second income does not need to be large.  Are we missing any important financial data about this step?  Should we be concerned about job loss (even though no signs of it are imminent for either of us)? What follows if we fail to reject the null hypothesis? What conclusions can we draw if $p&gt;\alpha$? Does not rejecting the $H_0$ mean anything? Recover Data from Card that cannnot be read I am trying to help a friend who is very concerned that she has just lost all of the data on her 32GB SanDisk microSD card.  I told her I would post this and see if anyone could offer help.  Here are the details:   The SanDisk microSD card is less than two years old.  Probably only about a year old.  The store will take it back, but that doesn't really help.  She wants her data. She used the card in her Samsung Galaxy S4 smartphone for over a month.  The phone was never exposed to water or temperature extremes. Today, her phone suddenly said something like "Your SD card is blank or an unsupported format". Her phone was then unable to read the SanDisk microSD card at all.   Here's what she has tried:   Rebooting her phone.  Had no effect. Taking out the SanDisk microSD card and putting it inside a full-size SD card adapter, and inserting it into a Windows 7 SP1 computer.  The computer did not recognize its presence at all. Putting the SanDisk microSD card into a USB 2.0 external card reader and inserting the card reader into a USB port on a Windows 7 SP1 computer.  The computer recognized the card reader and installed drivers for it.  The card reader shows up as 'Removable Disk' in Windows Explorer.  Clicking on 'Removable Disk' in Windows Explorer results in the error "Insert disk: Please insert a disk into the Removable Disk". After trying #3, tried opening the 'Removable Disk' in FreeCommander.  Doing so results in the error "The device is not ready".   She does not have access to a linux box, but does have access to her Android phone.  What can she do to recover the data on her 32GB SanDisk microSD card?