How do I prove that there is no isomorphism between $\Bbb Z$ under addition and $\Bbb Q$ under addition?   They both are infinite order. I thought they might be isomorphic.  Help would be appreciated!  Thanks in advance!
Prove that the additive groups of $\Bbb Z$ and $\Bbb Q$ are not isomorphic.    It is hard to create a map that will show homomorphism.
C++ handling very large integers I am using the RSA Algorithm for encryption/decryption, and in order to decrypt the files you have to deal with some pretty big values. More specifically, things like   P = C^d % n   = 62^65 % 133   Now that is really the only calculations that ill be doing. I have tried using Matt McCutchen's BigInteger Library, but I am getting a lot of compiler errors during linking, such as:  encryption.o(.text+0x187):encryption.cpp: undefined reference to `BigInteger::BigInteger(int)'  encryption.o(.text+0x302):encryption.cpp: undefined reference to `operator&lt;&lt;(std::ostream&amp;, BigInteger const&amp;)'  encryption.o(.text$_ZNK10BigIntegermlERKS_[BigInteger::operator*(BigInteger const&amp;) const]+0x63):encryption.cpp: undefined reference to `BigInteger::multiply(BigInteger const&amp;, BigInteger const&amp;)'   So I was wondering what would be the best way to go about handling the really big integers that come out of the RSA Algorithm.  I heard that a possibility would be to declare your variables as a double long, so...  long long decryptedCharacter;   but I'm not sure exactly how big of an integer that can store.    Well for example, I try to compile and run the following program using dev C++:  #include iostream  #include "bigint\BigIntegerLibrary.hh"  using namespace std;  int main() {     BigInteger a = 65536;     cout &lt;&lt; (a * a * a * a * a * a * a * a);     return 0; }   then I get those errors.  Derek, I thought that by including the BigIntegerLibrary.hh file, that the compiler would go through and compile all the necessary files that it will use.  How should I try and compile the program above in order to resolve the linking errors? Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$? How do I set up a "secure" open resolver? This is a  about securing public DNS resolvers   Open DNS servers seem pretty neat and convenient, as they provide IP addresses that we can use consistently across our company regardless of where they are located. Google and OpenDNS provide this functionality, but I'm not sure that I want these companies to have access to our DNS queries.  I want to set up something like this for use by our company, but I hear a lot about this being dangerous practice (particularly in regards to ) and I want to make sure that we do this right. What things do I need to keep in mind when building this type of environment? During Winter Bash, users can earn hats. But sometimes it seems that a hat was awarded even though its requirements weren't (yet) met, or vice versa.  What could be the reason for this? Ideals in ring of continuous functions $\mathcal{C}[0,1]$ ... NBHM- Algebra I would like to compile all questions I have encountered with Ideals in the ring $\mathcal{C}[0,1]$ of all continuous real valued functions and ask if there are any gaps.  Question is to see if :     the ideal $\mathcal{I}=\{f\in \mathcal{C}[0,1] : f(0)=0\}$ is maximal ideal in the ring $\mathcal{C}[0,1]$ of all continuous real valued functions.   What I have done so far is :  consider $\eta :\mathcal{C}[0,1]\rightarrow \mathbb{R}$ where $f\rightarrow f(0)$   $Ker (\eta) = \{ f\in \mathcal{C}[0,1] : f(0)=0\}$  I could see that for each $r\in \mathbb{R}$, I would set $f$ such that $f(x)=x+r$  So, $\eta$ is surjective.  So, I would get $\mathcal{C}[0,1]/\mathcal{I}\cong \mathbb{R}$  As $\mathbb{R}$ is a field so ia the quotient $\mathcal{C}[0,1]/\mathcal{I}$ which means that $\mathcal{I}$ is an ideal in $\mathcal{C}[0,1]$.    Now Another Question on same idea :  Question is to see if :     the ideal $\mathcal{I}=\{f\in \mathcal{C}[0,1] : f(0)=f(1)=0\}$ is maximal ideal in the ring $\mathcal{C}[0,1]$ of all continuous real valued functions.   I thought of using same ideas as above but i am unable to choose perfect $\eta$  I believe that this is not even a prime ideal..  I set  $f(x)=x$ and $g(x)=x-1$ and consider $f(x)g(x)=(x)(x-1)$ then,   $(fg)(0)=f(0)g(0)=0.(-1)=0$   $(fg)(1)=f(1)g(1)=1.0=0$  So, $fg\in \mathcal{I}=\{f\in \mathcal{C}[0,1] : f(0)=f(1)=0\}$ but the neither $f$ nor $g$ is in $\mathcal{I}$.    I have two questions:   I see that $0$ does not play any role in the question $\mathcal{I}=\{f\in \mathcal{C}[0,1] : f(0)=0\}$ I mean I would have same case if i replace $0$ by any real number.  So, I would like to say that $$\mathcal{I}=\{f\in \mathcal{C}[0,1] : f(r)=0\}$$  is a maximal ideal in the ring $\mathcal{C}[0,1]$ of all continuous real valued functions.  And one more thing I would claim is that      If the collection $\mathcal{I}$ is such that all elements of $\mathcal{I}$ has more than $1$ common zero then $\mathcal{I}$ is not a maximal ideal and not even a prime ideal..   I would like to learn more about this kind of ideals in $\mathcal{C}[0,1]$.  I would be thankful if some one can assure that what  have done is sufficient to conlcude what i have concluded and I would be thankful if some one can suggest some material to read regarding this.  Thank you full-screen browser window on load document How to say that you have plans to go to the movies tonight (but you are not sure yet, maybe you will have a pizza)? I mean there are options:     I am going to the cinema tonight.   Way too much certainty in this. For me it means that you have already bought tickets, the least.     I am going to go to the cinema tonight.   Sounds a bit awkward.     I will go to the cinema tonight.   Sounds too formal to me. What is the difference between image size and image quality? I have a Casio EX-S12. It has a setting for image size, but also a separate setting for what it calls "image quality". There are three image quality values: "Fine", "Normal", and "Economy".  The Casio manual is not clear on what image "quality" actually means -- specifically how it's different from image size. All they say is that the "Fine" setting "helps to bring out details when shooting a finely detailed image of nature that includes dense tree branches or leaves, or an image of a complex pattern." I'm confused because isn't this what image size is all about (more pixels = more detail)?  In terms of memory usage, the manual also says that for a 5 megapixel picture, a "Fine" image takes up 2.99 MB, a "Normal" image takes up 1.62 MB, and an "Economy" image takes up 1.12 MB, so this "image quality" setting of theirs is certainly having a significant impact on memory usage.  My question is, what exactly is "image quality", if it's not image size? What is the "thing" that is taking up additional memory?  Thank you. How to install CUDA 8.0 on Ubuntu 16.04 with Nvidia GeForce GTX 1080 I am attempting to install CUDA 8.0 within Ubuntu 16.04 for the Nvidia GeForce GTX 1080. I had CUDA 7.5 installed, but it did not appear to be compatible with the GTX 1080. I downloaded the .run installer from Nvidia's developer page, but the installation fails and outputs the following to a nvidia-installer.log     An NVIDIA kernel module 'nvidia-drm' appears to already be loaded in     your kernel.  This may be because it is in use (for example, by an X     server, a CUDA program, or the NVIDIA Persistence Daemon), but this     may also happen if your kernel was configured without support for     module unloading.  Please be sure to exit any programs that may be     using the GPU(s) before attempting to upgrade your driver.  If no     GPU-based programs are running, you know that your kernel supports     module unloading, and you still receive this message, then an error     may have occured that has corrupted an NVIDIA kernel module's usage     count, for which the simplest remedy is to reboot your computer.   I have rebooted the computer, but that did not resolve the problem. How do I unload the 'nvidia-drm' module so that I can install the 8.0 CUDA? I'm using Linux Mint. My login shell (cat /etc/passwd | grep myUserName) is bash.  After I start my graphical desktop environment and run a terminal emulator from it, I can see that .bash_profile is not sourced (environment vars that are exported in it are unset). But if I log in from a text console (ctrl+alt+F1) or manually run bash -l from the terminal emulator, .bash_profile works fine.  Am I wrong when I think that .bash_profile should be sourced when X starts and all export'ed vars should be available in the terminal, running from X?  P.S. Placing everything in .bashrc and sourcing it from .bash_profile is not good idea (): environment stuff should be sourced only once. $\lim_{h\to 0}\int \frac{f(x+h)-f(x)}{h}\,\mathrm{d}x = f(b)-f(a)$ Let $f:[a,b]\to  R$ be integrable and continuous on $(a,b)$. Prove that: $$\lim_{h\to 0} \int_a^b\frac{f(x+h)-f(x)}{h} \,\mathrm{d}x = f(b)-f(a).$$  What I've tried so far: make a change of variables, which didn't help. I also thought of using the intermediate value theorem inside the integral, but the function being differentiable is not specified. Any help, tips? Thanks in advance! Google Chrome Address Bar Font has gone too large, how can we reduce it Running Unity on my desktop (2*20" LCD displays), I initiated a standard daily update on my Ubuntu 14.04 LTS today and Google Chrome updated itself to Version 35.0.1916.114.  The URL address bar input now shows IMO a too larger font size which then proceeds to auto show in green. I prefer the smaller font size and have no need for 'gooey slime green' auto complete font colours. Any advice on how to revert back to smaller, generic Address font styling.  I use both FF and Chrome for numerous tasks, Chrome being my primary browser and I've checked my system Unity 'default' theme (Ambiance! LOL.. yes I'm tired of that also).  Thanks for sharing. Linux Equivalent of Windows Ctrl to show Mouse Pointer In Windows, it is possible to set pressing Ctrl to display the current location of the mouse pointer, as per .  Is there a way of setting a key or key combination to perform the same action in Ubuntu? Note, this is the opposite of turning this function off, as detailed at . queries with db_select don't respect integers When I check "SHOW PROCESSLIST" in phpMyAdmin I can see lots of queries where argument-integers are being inserted as strings (with single quotation marks).  I'm wondering if this is due to the way phpmyadmin reports the queries (if it puts every argument between quotation marks) of if it is db_select that puts every argument between quotation marks.  An example of my code where integers are being used as arguments:  $select-&gt;join('taxonomy_index', 't_'.$key, 't_'.$key.'.nid = n.nid AND t_'.$key.'.tid = :tid_' . $key, array(':tid_'.$key =&gt; (int)$tid)); $select-&gt;condition('t_'.$key.'.tid', (int)$tid);   As you can see I explicitly write (int)$tid but in my query log I can see the following:  ... WHERE (t_0.tid = '10') ...   The big question is: How can I be sure that integers are being treated as integers in my queries? I don't want MySQL having to transform strings to integers. Where [if anywhere] should the apostrophe go in "three days work"? Which is correct?     1 Three day's work      2 Three days' work      3 Three days work   I would probably guess (2) is right, since the work belongs to the three days ("three days of work"). But I'm not sure. Would it be different if it was only one day? We have a SBS 2011 server with 16GB memory that runs Exchange, WSUS and File Sharing for approx 20 users. The server also gives 2GB memory to a virtual machine running in Hyper-V so 14GB left for the rest.  When the server first boots Store.exe consumes about 500MB memory, WSUS and IIS consume some as well, but we generally are only at 60% capacity. During the course of the working day Store.exe consumes 8GB memory and the server sits at approx 97% capacity.  This makes using the server very slow, logging on via RDP is slow, general server admin is slowed down.  Is this normal? I have read that Exchange will consume lots of memory that it does not neccessarily need and then releases it back to other applications that need it. I have also checked that all mailboxes are opened in cached mode. We do have some external uses who connect via Outlook Anywhere or OWA.  If it's deemed to be "normal" is there some way I can restrict it using ALL available memory and leave a little bit for SBS to use for server admin? I have seen some articles regarding setting a high and low memory flag but rumours that this does not work with update rollup 5 which we have installed.  Am I expecting a single server to do to much with only 14GB memory?  Thanks Prove $\alpha \in R[[x]]$ is a unit iff $a_0 \in R$ is a unit Prove $\alpha \in R[[x]]$ is a unit iff $a_0 \in R$ is a unit.   $"\Rightarrow"$ suppose $\alpha \in R[[x]]$ is a unit, where $\alpha = \sum_{n=0}^{\infty} a_nx^n$  then there exists an inverse $\beta = \sum_{n=0}^{\infty} b_nx^n$   stuck at this spot,  $(\sum_{n=0}^{\infty} a_nx^n)(\sum_{n=0}^{\infty} b_nx^n) = \sum_{n=0}^{\infty}(\sum_{k=0}^{n}(a_kb_{n-k}))x^n$  A little mixed up on how this becomes $1 = \sum_{n=0}^{\infty}e_nx^n$ where $e_0 =1, e_{n&gt;0} = 0$ and why that would make $a_0b_0 = 1$ which would make $a_0 \in R$ a unit. I haven't tried the other direction yet.  EDIT  Maybe I get it, because we assumed this sum is a unit then its inverse exists and so the $x^0=1$ and all $a_ib_i=1$ thus $a_0b_0=1$ which gives $a_0$ is a unit in $R$ Discriminant analysis vs logistic regression I found some pros of discriminant analysis and I've got questions about them. So:     When the classes are well-separated, the parameter estimates for   logistic regression are surprisingly unstable. Coefficients may go to   infinity. LDA doesn't suffer from this problem.      If number of features is small and the distribution of the predictors   $X$ is approximately normal in each of the classes, the linear   discriminant model is again more stable than the logistic regression   model.    What is stability and why is it important? (If logistic regression provides a good fit that does its job, then why should I care about stability?)      LDA is popular when we have more than two response classes, because it   also provides low-dimensional views of the data.    I just don't understand that. How does LDA provide low-dimensional views? If you can name more pros or cons, that would be nice. MySQL Date Between What is an idiom/slang for "someone who pretends to be good when they're not"? This is not from real life, but from a movie on my local TV. A character in the movie is really bad, but when she talks with others, she pretends to be an innocent/ good woman. I want to know how to describe this kind of person.