I'm trying to find a simple and easy to understand example of a free module that has bases of differing sizes. We were told about the existence of them in class but no example was given.
Does somebody has an example where the left $R$-modules $R^m$ and $R^n$ are isomorphic for all positive integers $m$, and $n$?
I've already defeated the Big Bug.   I see the hint that the small bugs disguised as cursors are vulnerable to the three words "Charity", "Uganda", and "Bram".  I think I have all keys available, I've tried undoing with the bugs on the three words, I've tried motions which include the three words in their range, I have each of the three words stored in a named register, but I haven't figured out how to defeat the three small bug minions underground in Level 14 of Vim Adventures.  I suspect I should have defeated them before defeating Big Bug, maybe even before finding 'undo'. Is there instead something I should do in the 'lorem' buffer where I found the three cursors (Blue, White, and Pink) at the beginning of the level?  How does one defeat the three small bug minions in Level 14? Having different horizontally  distributed equations inside an align environment, only lines are numbered, but not individual equations.    How can the left equations be numbered using subequations? See also the question   \documentclass{article} \usepackage{amsmath} \begin{document}   \begin{subequations}     \begin{align}        %\label{a}         a &amp;= a ,&amp;       \label{b}         b &amp;= b ,\\       %\label{c}         c &amp;= c ,&amp;       \label{d}       d &amp;= d .     \end{align}   \end{subequations} \end{document}     The solution should be able to either:    attach a label to the left equations, or modify the labels on the right to read (1a,b) and (1c,d)    I couldn't find anything in the documentation of amsmath or other packages. How can I fix apt error "W: Target Packages ... is configured multiple times"? Can compact sets completey determine a topology? Suppose that $\tau_1$ and $\tau_2$ are two topologies on a set $X$ with the property that $K\subset X$ is compact with respect to $\tau_1$ if and only if $K$ is compact with respect to $\tau_2$. Then is this enough information to determine whether $\tau_1 = \tau_2$?  If not (which is suspect to be the case) is there a nice counterexample, and what is the minimum amount of extra information required for $\tau_1 = \tau_2$?  I feel like there would be an issue regarding 'points at infinity.' Give an example of a continuous function $f : [0, ∞) \mapsto [0, ∞)$ such that $\int_{0}^{\infty}f(x)dx$ exists but $f$ is unbounded. Give an example of a continuous function $f : [0,  ∞) \mapsto [0, ∞)$ such that  $\int_{0}^{\infty}f(x)dx$ exists but  $f$ is unbounded.   I have been thinking about this. And I have come to the conclusion that I will need to construct a function, $f$, such that $f$ is a sequence of triangles of increasing height, but decreasing base. I obviously need $f$ such that both the height of the triangles, and sum of the bases tends to infinity. But I also need that the $\sum (\text{height} \times \text{base}) \leq \infty $ This is probably easier to show than tell...   On the recent question , I see the following:    As you can see, the first five links render properly, but after that, they stop converting.  Unfortunately, this isn't obvious to the person writing the post, which I discovered when I went to edit the post to fix the "broken" links. In the edit window, it renders all of the links just fine:    I understand that this behavior is expected based on balpha's answer :        Links to questions on the same site are unlimited.   Links to other sites are limited to             a) 5 linked-to sites and     b) N questions per target site, where N is 200 on meta sites and 10 on non-metas (this is about the site that the post containing the link is posted on, not the site that is being linked to)         So if this rule is still going to stay in effect, please fix the edit view to also not automatically convert more than five linked-to sites to help people writing posts know that they need to manually title the links to the other questions.  That said, I'd prefer that (at least on MSE) this limitation be removed or raised. Even in the answer, balpha says:     The restrictions are currently as follows, but note that they do look a bit contrived, resulting from the fact that they have grown over time (from single site to cross-site, from "there are no subsites" to "facebook.SO", etc.), so it's not unlikely that we'll revisit them.   So, if it's not hurting performance, can we please increase the limit or remove it? NVMe PCIe x4 SSD on M.2 PCIe x2 slot? i'm about to get a new SSD for my PC, and before spending the money i have some doubts about my decision i would like to clarify First, my mobo is a B450 Aorus Pro Wifi, it has 2 M.2 slots, one of them full length (22110) compatible with PCIe 3.0 x4 and the other one standard length (2280) and support only for PCIe 3.0 x2 About the drive, i'm aiming for a 1tb XPG SX8200 Pro , which is a PCIe 3.0x4 SSD My first M.2 slot is already taken by the main SSD, which i use for the OS and other stuff, i can't really use it. Will a PCIe x4 SSD work on a PCIe x2 slot? How much performance will be lost if it does? Will it be worth it? Constructing $g$ holomorphic differentials on $\omega^3=(z-1) \cdots (z-8)$ Let us have a surface $X$ which satisfy the equation $\omega^3 = (z-1)(z-2)\cdots (z-8)$ It's stated in the task, that there are exactly $g$ holomorphic differentials on the curve. How can I find $g$ and construct them?  The genus $g$ of this curve (I tried to use Riemann-Hurwitz theorem)  The map $X(z,\omega) \to CP^1(\omega)$ has degree 3. and we have 8 branching points with 3 leaves at each. $\chi(Domain) = DegreeOfMap * \chi(Target) - \sum_{points}(BranchingLeaves - 1) $ $2-2g = 3 *(2) - 8*2 $ hence $ 12 = 2g$ and  $g = 6$ I know that in case of hyperelliptic curve there are $g-2$ (If i'm not mistaken) holomorphic differetials and they are of sort $\frac{z^k dz}{w}$. But the form of the curve equation was heavily used. UPD: there is a branhing point at $\infty$, because if we go around all points $ 1 \cdots 8$ at once, we get that cubic root of the right side is multiplied by $\exp{4 \pi /3}$ $2-2g = 3 *(2) - 9*2 $ hence $ 12 = 2g$ and  $g = 7$ (I hope that this is true) Why is a finite integral domain always field? This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse:   let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$,  then as $R$ is closed under multiplication $\prod_{n=1}^k\ x_i=x_j$,  therefore by canceling $x_j$ we get $x_1x_2\cdots x_{j-1}x_{j+1}\cdots x_k=1 $,  by commuting any of these elements to the front we find an inverse for first term, e.g. for $x_m$ we have $x_m(x_1\cdots x_{m-1}x_{m+1}\cdots x_{j-1}x_{j+1}\cdots x_k)=1$, where $(x_m)^{-1}=x_1\cdots x_{m-1}x_{m+1}\cdots x_{j-1}x_{j+1}\cdots x_k$.   As far as I can see this is correct, so we have found inverses for all $x_i\in R$ apart from $x_j$ if I am right so far. How would we find $(x_{j})^{-1}$? "PuTTY key format too new" when using ppk file for PuTTY SSH key authentication I wanted to connect to my Unix server using SSH keys. For that I had converted my id_rsa file from ./ssh directory to a ppk file using PuTTYgen, but after loading my .ppk key I get this warning:  PuTTY key format too new   If I use my default id_rsa file, then I get error:  OpenSSH SSH-2 private key (old PEM format)  Either way I am getting stuck. Can anyone please help me? When I tried to install Matlab using the Software Center, it prompted me for the Location of MATLAB installation(s). Since I am trying to install it, I don't know what to do. Should I assign dwellers to the storage room? I've played Fallout Shelter for a while now and normally use dwellers with appropriate SPECIAL stats for each room (Strength in the reactor, Perception in the water supply, etc).  However, I have a question regarding Endurance. The only room which uses this is the storage room. It doesn't produce anything; it has no cooldown; it just does nothing...  Even if I just build and upgrade the warehouse, my stock capacity is increased. Is there any good reason for me to waste my dwellers by assigning them to storage rooms? If I call finalize() on an object from my program code, will the JVM still run the method again when the garbage collector processes this object?  This would be an approximate example:  MyObject m = new MyObject();  m.finalize();  m = null;  System.gc()   Would the explicit call to finalize() make the JVM's garbage collector not to run the finalize() method on object m? How to get ASCII value of string in C# I want to get the ASCII value of characters in a string in C#.  If my string has the value "9quali52ty3", I want an array with the ASCII values of each of the 11 characters.  How can I get ASCII values in C#? Let $\mathbf{FinVec}$ denote the category of finite-dimensional real vector spaces and linear maps, let $F\colon\mathbf{FinVec}\rightarrow\mathbf{FinVec}$ be a functor. For each pair of objects $V,W$, the Hom-set $\operatorname{Hom}(V,W)$ is itself a finite-dimensional vector space and we equip it with the unique structure of a topological space/of a smooth manifold that is compatible with its vector space structure. The composition maps are then continuous/smooth. In this way, we view $\mathbf{FinVec}$ as enriched over the category $\mathbf{Top}$/$\mathbf{Diff}$ respectively. It then makes sense to ask whether the functor $F$ is enriched over $\mathbf{Top}$/$\mathbf{Diff}$. That is just to ask whether the induced map $F\colon\operatorname{Hom}(V,W)\rightarrow\operatorname{Hom}(FV,FW)$ is continuous/smooth for all $V,W$. (Irrelevant to the rest of the question, but the reason this is a useful notion is that it is the natural hypothesis assuring that the functor induces another functor on the category of topological/smooth vector bundles, applying the original functor fiberwise.) In the case $V=W$, by functoriality, we can restrict $F$ to a group homomorphism $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$. If the functor is continuous/smooth, so is this map. Note that $\operatorname{GL}(V)$ is a Lie group. It is a general result that a measurable group homomorphism between Lie groups is automatically smooth. This begs the following questions:  What's an argument for the existence of functors that aren't continuous? Can we construct one? Given the above, it either happens that the restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$ are continuous and continuity fails elsewhere, or the induced maps will be non-measurable, so that the construction will necessarily have to involve choice to some extent. Are there continuous functors that aren't smooth? Given the above, a continuous functor will have smooth restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$, but smoothness may fail elsewhere. @username auto-completion for comments was , and is already active on MSO.  However, it only seems to auto-complete the usernames of people who have commented on the post. According to , the list of users who can be explicitly notified also includes:   Anyone who edited the post Anyone who started a bounty on the question The &diams; moderator or  who cast a binding close vote on the question   Why not auto-complete those, as well? I'm using X11 forwarding over ssh to run Linux apps on my Windows box, and when the network drops it loses everything that was running. Is there anything similar to screen for X11? $\require{enclose}\enclose{horizontalstrike}{\rm Greatest}\!$ Least prime factor of $n$ is less than square root of $n$, proof I remember reading this somewhere but I cannot locate the proof. Equation System with 3 variables "Find all the possible solutions $(x,y,z) \in$  $\Bbb R$ that satisfies the following equation system".  My teacher gave me this problem and i got some clues in how to solve it. I evaluated $1$ and $-1$ and i got these solutions $(1,-1,1)$ and $(-1,-1,-1)$. I think these are the only ones but i can't prove it.  Am i wrong?, are there more solutions?.   Thanks  $$ \left\{  \begin{array}{c} x+y-z=-1 \\  x^2-y^2+z^2=1\\  -x^3+y^3+z^3=-1 \end{array} \right.  $$ How to get rid of update notifications for a given app in Google Play Store? General question I have an app X installed on my device in version Y. I don't want to update it to any newer version (Y+z) for some reason. Still, the Google Play Store app always lists updates for X, which keeps me from a simple &quot;update all&quot;. My question, put to a single sentence:  How to hide updates for X permanently from the &quot;available updates&quot; list?   Specific examples First, I know there are alternatives as e.g. the , which can handle this quite well (skip this update, skip all future updates). I prefer a &quot;generic solution&quot;, which does not depend on a 3rd party's market app. Second, I know that  includes a &quot;Market doctor&quot;, which can be used to disconnect an app from Google Play.1 However, in my specific case that only worked for a few hours: I disconnected GTalk, as I don't want it replaced by Hangouts. The update notification disappeared for a few hours, then it was back. No idea what restored the &quot;link&quot;. Third, just to make sure: I'm not talking about the &quot;auto update&quot; feature as discussed e.g. in . I was rather hoping for something close to , but without the need to &quot;freeze&quot; the app in question (which I still want to use, but in the version installed, not in a &quot;future version&quot; available). Fourth: No, uninstalling the app in question is no option here, though it would perfectly hide any future updates from being displayed :) Nor is &quot;freezing&quot;, as the app is used regularly. Fifth: There is a related question for my GTalk example:  However, I want the opposite: No update.  It might very well be that &quot;Google Apps&quot; (and other pre-installed apps) are handled in some specific way, and restore their &quot;market links&quot; themselves somehow. For this case, my question includes: How to prevent that from happening?  Update There's a  discussing the very same issue. Until recently, a work-around for &quot;user installed apps&quot; was to uninstall them and then sideload the .apk – but even that stopped working. Well, re-signing the .apk with a different key might work for user-apps. But both would fail for pre-installed apps (can't sideload them to override; different signature would refuse that install completely). Moreover, there's an  on this by some &quot;joel.bou..&quot;. Not hard to guess, though blanked out: that would be Joël Bourquard. If it's really him, then that's really an issue (for those who don't know: Joël is the dev behind Titanium Backup). Let me quote from the issue:  For the record: in the Android Market days it was possible to &quot;Attach&quot; or &quot;Detach&quot; apps (on rooted devices) with a system tool such as Titanium Backup. But nowadays (with the Google Play Store app) this has become nearly impossible to do, because the complete app list seems to sync from Google's servers and the user has no control over that.  Which explains why detaching with apps like Titanium Backup is no longer permanent. And I'm afraid I've asked the impossible. But I thought the same in the past with other questions, and surprisingly got solutions here. So I do not yet give up hope on this one!  1: what that does is explained by eldarerathis  (thanks to  for digging that up!)