Can someone please explain (with an example or two) what is meant by “a polynomial $p(x_1, ..., x_n)$”? Stillwell starts referring to them without defining them and they are beyond my elementary mathematics education. He usually adds “with integer coefficients.”
Let's consider a polynomial ring of single variable. One can define them informally by saying $P(X)=\sum_{i=1}^n a_n X^n$ while $X$ is an indeterminate variable. However, since mathematics is based on first-order logic, one can only talk about something that actually exists (as a set). So as its name says, indeterminate variable is not a sentence in $ZFC$. Nevertheless, we can formally define a polynomial ring $R[X]$ as a subset of $R^\omega$ whose support is finite. ($R$ is a commutative ring with unity) Just like a single variable, I want to know what would be the formal definition of a polynomial ring of several variables. What would be a formal definition? === I have figured out a candidate. That is,  Candidate for a &quot;Definition of a polynomial ring $R[X_1,...,X_n]$&quot;. ($R$ is a commutative ring with unity) Let $R[X_1,...,X_n]$ be the set of all functions $f:\omega^n \rightarrow R$ whose support is finite. Define $(f+g)(a,b,c)=f(a,b,c)+g(a,b,c)$ and $(f•g)(a'',b'',c'')=\sum_{a+a'=a''} f(a,b,c)•g(a',b',c')$ (Note that in this way, $+,•$ are well defined.) Then, it can be checked that $(R[X_1,...,X_n],+,•)$ is a ring.  Is this a formal definition of a polynomial ring?
Scala 2.10 + Json serialization and deserialization Meaning of "stackexchange" What does stackexchange mean? I have checked out this  to read about the Stack Exchange Network. Can I use a one time pad key twice with random plaintext? I understand the basics of OTP: $|\text{key space}| = |\text{plaintext space}|$ implies perfect security, key reuse destroys this.  Cryptanalysis on the $N$-Time Pad for $N &gt; 1$ involves finding patterns in the ciphertext; this, however, all seems based on the premise that patterns exist in the plaintext (e.g. English language).  My question is: if $m_1$ and $m_2$ are two truly random strings, and they are both encrypted with the same OTP key to yield $c_1$ and $c_2$, is it possible to recover $m_1$ and $m_2$ from $c_1$ and $c_2$?  In other words: while "perfect security" (aka perfect indistinguishability) is obviously lost (constructing a distinguisher for two messages is trivial), is the plaintext still safe? I know, it's been asked before but I couldn't really get a satisfying answer for some details/backgrounds and I guess it's an important question for many.    Some application seem only to be running with Adobe Flash 15 (or at least I know it does, because I have that version on my Windows Partition).  The official Adobe website says:     NOTE: Adobe Flash Player 11.2 will be the last version to target Linux as a supported platform. Adobe will continue to provide security backports to Flash Player 11.2 for Linux.   Some search on the web like "Ubuntu Adobe Flash 15" or similar does not return any good websites.  I found something that says   (using a "pepper-based version"), and  .   Is it impossible to use this for Firefox too? Would it be a privacy risk?  Are there no alternative projects (I know it's closed source but there are so many so very bright people out there using something similar to wine but just for the plugin....)?  How about the Mac-Version? Shouldn't someone be able to port that one to Linux? Distance between footnote and the line above it How do I get a consistent byte representation of strings in C# without manually specifying an encoding? How do I convert a string to a byte[] in .NET (C#) without manually specifying a specific encoding?  I'm going to encrypt the string. I can encrypt it without converting, but I'd still like to know why encoding comes to play here.  Also, why should encoding even be taken into consideration? Can't I simply get what bytes the string has been stored in? Why is there a dependency on character encodings? When I've worked a bit with my source code, I did my usual thing commit and then I pushed to a remote repository. But then I noticed I forgot to organize my imports in the source code. So I do the amend command to replace the previous commit:  &gt; git commit --amend   Unfortunately the commit can't be pushed back to the repository. It is rejected like this:  &gt; git push origin To //my.remote.repo.com/stuff.git/  ! [rejected]        master -&gt; master (non-fast forward) error: failed to push some refs to '//my.remote.repo.com/stuff.git/'   What should I do? (I can access the remote repository.) Prove that $\lim_{x \to +\infty} \frac{f(x)}{x} = L$ if $\lim_{x \to +\infty} [f(x+1) - f(x)] = L \space$ Let $f:[0, +\infty) \rightarrow \mathbb{R} $ be a bounded function in each bounded interval. If $$\lim_{x \to +\infty} [f(x+1) - f(x)] = L$$ then $$\lim_{x \to +\infty} \frac{f(x)}{x} = L$$   I tried using the definition on the first limit; then, I attempted to use that inequality to apply the triangular inequality to arrive at the definition that gives me the second limit. It is supposed to be simple, but I'm not feeling safe as in how I should write it. Can I use SPI for asynchronous serial output? I need a second serial async output channel on the MSP430G2553 and would like to use the SPI USCI to do this.  Is the USCI SPI baud rate stable enough to do this at 300 baud, 2400 baud or 9600 baud? I am currently using the 8 MHz clock.  Is there any other option that can save me from bit-banging?  My current thinking for 300 bd Tx is as follows:   set prescaler to 26666 LSB first load byte into Txbuf load 0x7F into Txbuf to generate stop and start bits repeat 2-3 send final stop bits 0xFF   It might also be possible to send ten bits of data on other members of the MSP430 family that contains a SPI.  Thanks for your responses. Why is my membership length only "Six months"? I was viewing the  and I noticed that I've apparently only had my account for six months, which is news to me... I opened my first site account here in February of 2014.  My account is much older than this, what's causing this bug?  To add some detail... On the per-site list, it does line up the &quot;member for X&quot; text with my actual membership time on that site. For example, when I look at the Meta Stack Exchange leaderboard, it shows me as a member for 2 years, 10 months. This is expected.  Update. With some work, today, I've managed to exceed my Interpersonal Skills hat count of with my Meta Stack Exchange hat count. This has changed my account age to match my MSE account age...  Still not reflective of my actual age... but better. Ugly "Next Privilege" text on user activity page Automatic updates are not auto-downloading I have my Xbox One set up to automatically download and install updates, the rest of my settings are shown below:    Is there a reason my games will show an update symbol on their tile, but they still don't download the update until I start the game manually? Accidentally ejected External drive while downloading file, now having problems reading I'm hoping someone can help! I recently downloaded a torrent directly onto an external hard drive, (using a 2009 MacBook Pro and 1TB Seagate Expansion portable Drive) somehow during the download the drive was accidentally ejected, since this incident when I plug the drive back in it’s having problems reading.  The drive works fine for about 10 minutes, I can read/copy/paste and access files but then starts to wreak havoc; apps hangs, it hangs the Finder, etc.  I’ve tried repairing it in Disk Utility, which reads and repairs this disk but problems inevitably continue as described above.  I’ve deleted the Torrent file also. All my files on this external drive so re-formatting is not an option until I can retrieve them. Automate fixes for issues found by Sonar Calculate Limit 0f nested square roots Make flash drive bootable with MULTIPLE Windows Installers I have seen many ways to make a USB an installer for about any Windows OS. But how can I (using grub or something like that) make it bootable to all editions on Windows XP and 7?   I have tried Googling it and researching, I've even tried to do it, But I dont understand much about GRUB or linux at all.  Please keep in mind that I am not very good with linux, so please use as many details as possible. Area of an ellipse. (Calculus) Here is the problem: From any IP address not belonging to your mail server:  telnet me.myemailserver.com 25    helo me.someserver.com mail from: &lt;yourusername@mydomain.com&gt; rcpt to: &lt;yourusername@mydomain.com&gt; data This is spam.  Buy my stuff. .   I'm using Postfix.  I'm having a problem finding a solution to requiring SMTP-AUTH for email claiming to be from mydomain.com.  Googling around, this guy has identified the same problem (where I cut-n-paste with some modifications) the above example from:   This link   gets close to a solution but it has a side effect of requiring SMTP-AUTH for mail not from mydomain.com.  For mail not claiming to be from mydomain.com, I would do the usual RBL and Spam filtering.  In short, I want to reject mail to local domains (mydomain.com) from outside/unauthenticated clients claiming to be from local domains (mydomain.com).  This is what I tried: I've tried both permit and reject as the default.  Here is exact excerpt from my main.cf:  smtpd_recipient_restrictions = reject_unauth_pipelining,                            permit_sasl_authenticated,                            check_recipient_access pgsql:/etc/postfix/pgsql-recipient.cf,                            reject_unauthenticated_sender_login_mismatch,                            reject_unauth_destination,                            reject_unlisted_recipient,                            check_sender_access pgsql:/etc/postfix/pgsql-sender.cf,                            reject_unlisted_sender,                            reject_invalid_hostname,                            reject_non_fqdn_hostname,                            reject_non_fqdn_sender,                            reject_non_fqdn_recipient,                            reject_unknown_sender_domain,                            reject_unknown_recipient_domain,                            reject_rbl_client cbl.abuseat.org,                            reject_rbl_client sbl.spamhaus.org,                            reject_rbl_client sbl-xbl.spamhaus.org,                            reject_rbl_client bl.spamcop.net,                            reject_rbl_client dnsbl.njabl.org,                            reject_rbl_client blackholes.wirehub.net,                            reject_rbl_client relays.mail-abuse.org,                            reject_rbl_client dialups.mail-abuse.org,                            reject_rbl_client blackholes.mail-abuse.org,                            reject_rhsbl_sender dsn.rfc-ignorant.org,                            (reject and permit both tried here) Is it worth encrypting email addresses in the database? How to accumulate values of each two rows across a line in linux?