I am having trouble with this problem for quite some time. I posted this question before but I still can not figure out this problem.   So far,from the suggestion of user134824, I have tried to define an injective function $f$ from $S_k$ to $\mathbb{N}$ as $f_k({n_1,n_2,...n_k})=p_1^{n_1}p_2^{n_2}...p_k^{n_k}$. I am able to prove that this function is injective. However, after this point, I am stuck. I can either prove that $f$ is bijective by proving it is surjective or I can find an injective function $g:\mathbb{N}\rightarrow S_k$ and use the Schroeder-Bernstein theorem.
(a) $S_k=\{A\subset\mathbb{N}: |A|=k\}$ for $k\in\mathbb{N}$  (b) $S = \bigcup_{k=1}^\infty S_k$   Work: For (a), I am not too sure about what approach I should use. I think finding a bijective function between the $S_k$ and $\mathbb{N}$ is hard to do . I also thought about using the Schroeder-Bernstein theorem and find injective functions $f:\mathbb{N}\rightarrow S_k$and $g:S_k\rightarrow\mathbb{N}$ in order to prove that $S_k$ and $\mathbb{N}$ are numerically equivalent. However, I have a hard time finding and proving these injective functions. I haven't really looked at (b) as I have yet to complete (a).
n-dimensional vector space isomorphic to the n-th product of a field as abelian groups Let $V$ be a finite dimensional vector space (of dimension n) over a field $F$. I need to show that $V$ is isomorphic to $F^n$ as abelian groups. However, I don't really understand what does "isomorphic as abelian groups" mean, my (poor) attempt to solve this was:  Let $f:V\to F^n$, $f((v_1,v_2,\ldots,v_n)):=v_1+\ldots+v_n$, clearly $f$ is a group homomorphism, but it is not biyective since, for example, in the case where $dim(V)=2$ and $F=\mathbb{R}$ we have $f((2,-2))=0=f((3,-3))$, I've also tried it by defining $f$ to be the product of a vector's entries but obviously it didn't work so I'm stuck!   I think this should be an easy problem but since I don't quite understand it, it's giving me problems. Thank you for your help. User group (company) page that shows summary of all user accounts belonging to the same company/user group What is the most straightforward way to pad empty dates in sql results (on either mysql or perl end)? I'm building a quick csv from a mysql table with a query like:  select DATE(date),count(date) from table group by DATE(date) order by date asc;   and just dumping them to a file in perl over a:  while(my($date,$sum) = $sth-&gt;fetchrow) {     print CSV "$date,$sum\n" }   There are date gaps in the data, though:  | 2008-08-05 |           4 |  | 2008-08-07 |          23 |    I would like to pad the data to fill in the missing days with zero-count entries to end up with:  | 2008-08-05 |           4 |  | 2008-08-06 |           0 |  | 2008-08-07 |          23 |    I slapped together a really awkward (and almost certainly buggy) workaround with an array of days-per-month and some math, but there has to be something more straightforward either on the mysql or perl side.   Any genius ideas/slaps in the face for why me am being so dumb?    I ended up going with a stored procedure which generated a temp table for the date range in question for a couple of reasons:   I know the date range I'll be looking for every time The server in question unfortunately was not one that I can install perl modules on atm, and the state of it was decrepit enough that it didn't have anything remotely Date::-y installed   The perl Date/DateTime-iterating answers were also very good, I wish I could select multiple answers! In $\mathbb{R}$ with standard topology the closure of the above set $M$ is $M \cup 0$. I fail showing this by purely topological means. I.e. not using the fact that this holds in the reals as a metric space and deduce it for the induced topology.  I assumed that there exist a closed subset $A$ containing $M$ but not 0. Hence, the complement of $A$ is an open subset containing 0. Does this already yield a contradiction or do I need a different approach?  Could you help me on this? If you open any of localized meta site in browser's private mode, e.g.  you will see untranslated text on the top right corner:        This message is not found in Transifex, hence unable to localize.  Related bug report on ruSO.meta: How can you detect if users have created sock puppet accounts? How many users have created more than one profile on SO to vote themselves on their other original profile? There are some benefits when you have more reputation and sometimes you don't want a single -1 vote on your question or answer. In this case you may have an extra account to convert that -1 vote into 0 vote and  upvote for each of your questions and answers to gain reputation. I think it is not a good thing to do as a good developer.  Is there any way to recognize this situation? I think there should be some stats in each user's profile about how many votes are posted by each user on him/her or something like that.  So how many users have more than one profile? How to handle this situation? I have not created one yet but............ I don't think so :) In mathematics, "A or B" includes "A and B". Does "either" mean "A or B but not (A and B)" or does it include the possibility of "A and B"? The context might be mathematics, formal logic or ordinary language. Style every four posts differently Show that the square of any prime number is the Factof of some integer In the United States, we say that someone lives on a street, whereas I've noticed that British people say in. For instance:     Bubba lives on Washington Street.   Colin lives in Cavendish Avenue.   I believe we both would use at when a number is given. For instance:     Bubba lives at 16 Washington Street.   Colin lives at 7 Cavendish Avenue.   I don't think it matters if it's a road, avenue, street, circle, or lane — as far as I know, in America, we always live on a street. Does it vary in British English, or is it always in? If it varies, what are the rules? Error message "sudo: unable to resolve host (none)" Write a formula in an Excel Cell using VBA I'm trying to use VBA to write a formula into a cell in Excel. My problem is that when I use a semicolon (;) in my formula, I get an error:     Run-time error 1004   My macro is the following :   Sub Jours_ouvres()     Dim Feuille_Document As String     Feuille_Document = "DOCUMENT"             Application.Worksheets(Feuille_Document).Range("F2").Formula = "=SUM(D2;E2)"     End Sub example.com is hosted on IP address c.c.c.c at host1, no access any windows server admin at all blog.example.com is  on IP a.a.a.a at host2, full access to Linux server admin, using WordPress   So basically, I can fiddle with subdomain all I want.  I can't 301 redirect because the content on host 1 is an eCommerce site that I have no server control over. Can't put WordPress on it, can't run scripts, can't do anything on the server period.  Is there a method to "mask" all the URLs on root blog.example.com so they appear as example.com/blog/ folder?  They can't actually be sent to example.com/blog/ as that doesn't exist. It just needs to look to the surfer as if they are there.  Can this be done with some kind of masking or something?  Will this be better for SEO as hopefully search engines will see the blog in a directory on the main domain? Easiest way to import access db (.accdb) into SQL Server 2016? I have a .accfb file from an access 2016 database and am trying to import this database into a SQL Server 2016 developer edition instance.   While I found posts online discussing using import/export data tools in SQL Server and also migration assistant for access, neither have worked out for me.   Does anyone know how to easily move a database from access 2016 to sql server 2016? More specifically an "accdb" file type?  All suggestions are greatly appreciated! When IE10 is set as my default browser, double-clicking an internet shortcut (*.url file) opens it in a new tab of the existing IE window. However, when it is not the default, every way of opening a *.url that I've tried brings up a new IE window per file. Is it possible to keep opening files in new tabs of the same IE window when it's not set as the default browser, and how?  What I've tried and it didn't work:   "C:\Program Files (x86)\Internet Explorer\iexplore.exe" "C:\Dir1\file.url" (opens a new window every time) C:\Windows\System32\rundll32.exe "C:\Windows\System32\ieframe.dll",OpenURL C:\Dir1\file.url (opens the default browser) classexec "C:\Dir1\file.url" --class htmlFile (using  utility, opens a new window every time)   I use Windows 7 x64. In the book , at page 27 we find the exercise asserting that       Exercise I.XXXVI.   The underlying space of a zero-dimensional scheme is discrete; if the scheme is Noetherian, it is finite.     Thoughts I thought that a zero-dimensional scheme is one such that, in every local ring $\mathscr O_{X,p}$, the only prime ideal is its unique maximal ideal $\mathfrak m_{X,p}.$ Hence, supposing that $X=\operatorname{Spec} R$ is affine, we deduce that every prime ideal is maximal, so each ponit in $X$ is closed. But how does this imply the discreteness? It only implies the Hausdorff-ness of $X$, right? Furthermore, I cannot perceive what the "Noetherianity" has to do with the finiteness claimed here. In the affine case, it seems to be claiming that there are only a finite number of prime ideals in a Noetherian ring, if its scheme is zero-dimensional? For example, $\mathbb Z$ is a Dedekind domain, hence its scheme is zero-dimensional, but it has infinitely many primes: does this contradict the statement?    Thanks in advance for any reference or hint, and point out any inappropriate point if any is presented. Edit I see why my example cannot work now: it is one-dimensinoal: don't forget the pirme $(0)$. But I am still wondering why the statement is true... My issue is that I'm trying to write variable names in italic, but as some of them have underscores in their names, LaTeX assumes that I am writing an equation and it messes up the rest of my document after that point. Here is a sample of my code: As variable \textit{sensor.linear_acceleration} is a matrix  The underscore in linear_acceleration is what makes my whole document math modded, and is a matrix renders as if it was an equation. How can I display that underscore in italic without LaTeX making everything an equation? A commercial airport being any airport that has regular scheduled passenger flights.  The mathematics of  got me thinking: is the network of scheduled flights a connected graph?  That is, is there a pair (or triplet or even larger set) of airports such that they only have scheduled flights with each other?  Or can you get from any airport to any other?  Ignore vagaries like cargo and general aviation. I'm looking for two airports that I can shuttle between, but would have to drive (or sail) to get to one or the other of them. Is there a max array length limit in C++? Is there a max length for an array in C++?  Is it a C++ limit or does it depend on my machine? Is it tweakable? Does it depend on the type the array is made of?  Can I break that limit somehow or do I have to search for a better way of storing information? And what should be the simplest way?  What I have to do is storing long long int on an array, I'm working in a Linux environment. My question is: what do I have to do if I need to store an array of N long long integers with N > 10 digits?  I need this because I'm writing some cryptographic algorithm (as for example the p-Pollard) for school, and hit this wall of integers and length of arrays representation. Can I export Shotwell pictures database saving all the tags? I've imported lots of pictures into Shotwell and spent some time setting tags. Are those tags locked-in in Shotwell or I can export them (and import again or use with other software)?