Suppose elements $a$ and $b$ in a domain satisfy $a \mid b$ and $b \mid a$. How do I see that $a = bu$ for some unit $u$?
Let $x$ and $y$ be nonzero elements of an integral domain $D$. Then $x$ and $y$ are associates if and only if $x = yd$ for some unit $d \in D$.    I am done proving the $\Leftarrow$ part. For $\Rightarrow$ what I did was: If $x$ and $y$ are associates then $x \mid y$ and $y \mid x$, so $x \mid y$ implies that $xs = y$ for some $s \in D$ and $y \mid x$ implies that $yt = x$ for some $t \in D$. That is, $yts=y$ which implies $ts=1$. Therefore $s$ is a unit and $t$ is a unit.    Am I on the right track? Can I say that the proof is complete?
How to concatenate characters in java? How do you concatenate characters in java? Concatenating strings would only require a + between the strings, but concatenating chars using + will change the value of the char into ascii and hence giving a numerical output. I want to do System.out.println(char1+char2+char3... and create a String word like this.  I could do   System.out.print(char1); System.out.print(char2); System.out.print(char3);   But, this will only get me the characters in 1 line. I need it as a string. Any help would be appreciated.  Thanks Is it possible to make LaTeX mark overfull boxes in the output? Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? I'm preparing for my exam, but I have difficulties in perceiving why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? What is object serialization? So I was writing up a simple binary tree in Python and came across [...]  I don't believe this to be related to the Ellipsis object, more it seems to have something to do with an infinity loop (due to Python's shallow copy?). The source of this infinity loop and why it doesn't get expanded while expanding when accessed is something I'm completely lost to, however   &gt;&gt;&gt; a [[[[[], [], 8, 3], [[], [], 3, 2], 6, 3], [], 1, 4], [[], [], -4, 2], 0, 0] &gt;&gt;&gt; Keys(a)#With a+b [0, 1, 6, 8, 3, -4] &gt;&gt;&gt; Keys(a)#With [a,b] [8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...]] &gt;&gt;&gt; Keys(a)[1]#?? [8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...], 8, [...], [...], 3, [...], [...], 6, [...], [...], 1, [...], [...], -4, [...], [...], 0, [...], [...]]     Version using a+b  def Keys(x,y=[]):     if len(x):y+=[x[2]]+Keys(x[0],y)+Keys(x[1],y)#Though it seems I was using y=y[:]+, this actually outputs an ugly mess     return y   version using [a,b]  def Keys(x,y=[]):     if len(x):y+=[x[2],Keys(x[0],y),Keys(x[1],y)]     return y   So what exactly is [...]? Would Gollum qualify, as a Ringbearer, to go to the Undying Lands? I received an undesirable status for overstaying a SA visa while hospitalized for injuries sustained in Lesotho. I had to cross into SA from Lesotho. Now I am banned entry into SA until Feb 2016 and though I have submitted a waiver request to remove the undesirable status, I hear from some the waiver request may take 4-5 months or even take longer than the time of banishment.  I have a pre-paid safari planned in October in which I need to fly through Johannesburg to get to Botswana. The Jburg border control requires tourists to pass through SA immigration even when simply changing planes.  1) Will SA grant me "transit" access if I don't leave the airport?  2) Will SA allow me to spend a night in the airport as the layover is 12 hours>  Thanks for passing on any info. SQL Azure and Full-Text Catalogs, Indexes and Searching Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial The problem:  X is a binomial random variable, find $E[\frac{1}{X+1}]$ n and p are not given  PDF for a binomial distribution is $\binom{n}{k}p^k(1-p)^{n-k}$  Expected value is $\sum{x_ip(x_i)}$  But this is where I get stuck, I'm really rusty on my statistics and I'm not sure exactly how to structure it in the next step? I think I want to get the form of the following out of the summation  $\sum _{k=0}^{n} \binom{n}{k}p^k(1-p)^{n-k} = (p + 1 - p)^n = 1$  But I'm not sure if it should look like  $\sum \frac{1}{xp(x)+1} $  and if it should where to go from here? tl;dr  Due to the special nature of Meta and the high rate of "lost souls", should we try out the Ask a Question Wizard here on Meta Stack Exchange too?  General idea  In case anyone doesn't know yet, . It guides new users towards writing better questions.  I think that such a question wizard would be useful on Meta Stack Exchange too, for three main reasons:   . There are four types of questions on Meta, each represented by the different required tags: , ,  and . Especially bug reports can often be split into a certain structure (what's wrong, how to reproduce, why I think it's wrong, examples). The same holds true for feature requests. Such a structure is ideal for a question wizard. Quite a few users  and . While the Meta community is well prepared to deal with these posts quickly, they lead to an less-than-ideal first impression by the new users and they distract the community from more important issues. Site-specific questions are often off-topic. This  since nowhere is it hinted at in the help pages (and there are a lot of old, not-yet-closed SO-specific questions from a prior era of this site) and can lead to unnecessary drama. If we add an honeypot into the wizard to catch this case (and ideally redirect them to the proper meta site), we could prevent this.   As Sonic said, there's an  new users need to read and confirm. However this doesn't really prove effective right now and we might need to try stronger tools (e.g. question wizard).  Therefore I think that we should test the Ask Question Wizard here on Meta too.  Implementation details (help welcome)  I think the wizard should have six main entry points:   I want to discuss a specific post or action from any Stack Exchange site. (-> honeypot, redirects to the appropriate per-site meta) I have a question regarding the use of Stack Exchange sites (-> support mode) I think I found a bug while using the Stack Exchange sites  (-> bug mode) I have an idea for improving the Stack Exchange sites  (-> feature request mode) I have a discussion I want to start about Stack Exchange communities or their rules  (-> discussion mode) I have some other (e.g. programming) question  (-> honeypot, redirects to )   Coming from these choices, the user would either be redirected to a landing page explaining, why this is not the right community and offering some alternative choices or they would follow the guided assistant.  For 1. the user might be asked on which site the post/action is. Maybe the system could show the three sites where the user was recently active/has the highest reputation on.  support mode would ask the user for these details:   what do you want to achieve? what have you tried? what is your current problem?   bug mode would ask the user for these details:   bug description how to reproduce the problem why do you think that this is wrong? (possibly) examples   feature request mode would ask the user for these details:   what is your goal? how do you think could this be achieved? why will the community benefit from this change? (possibly) details   discussion mode would ask the user for these details:   which rule do you want to propose/clarify (optionally) examples/why is this rule required?   Note: the above implementation details probably require some revision. Feel free to edit/suggest improvements for both wording and details. I've been working a bash script to automate the process of opening and closing an LVM over LUKS external hard drive. However I continue to receive errors on line 28:   $(vgchange -a n ; cryptsetup luksClose vol)   The error I receive:  $ sudo ./encryption.sh -l Locking System.............. ./encryption.sh: line 28: 0: command not found System Locked!   Here is the rest of the code for reference:  if [ $1 = -u ] ; then     $(cryptsetup luksOpen /dev/sdb2 vol)     echo "Unlocking.................." | pv -qL 10     $(mount /dev/mapper/vol-lvroot /mnt ; mount /dev/mapper/vol-lvvar /mnt/var ; mount /dev/mapper/vol-lvhome /mnt/home ; mount /dev/sdb1 /mnt/boot)     echo "System Mounted!" | pv -qL 10 #   exit 0 fi  if [ $1 = -l ] ; then     $(umount /mnt/var ; umount /mnt/home ; umount /mnt/boot ; umount /mnt/)     echo "Locking System.............." | pv -qL 10     $(vgchange -a n ; cryptsetup luksClose vol)     echo "System Locked!" | pv -qL 10     exit 0 fi   Other information:   I receive no apparent error when running the command in question by itself interactively. I'm new to writing bash, so any recommendations and constructive criticism is appreciated. The process appears to complete successfully, but I don't understand the error or how to make them go away. Upon further research when running the troubled code alone in interactive mode it does provide output after being executed.   This is given after posting the troubled code interactively.   0 logical volume(s) in volume group "vol" now active "DataBinding: 'index+NewsItem' does not contain a property with the name 'Link'", but property exists (Not a Typo) How do you disable anonymous login? Why don't we include $\rho g h$ in a hydraulic jack below? Can I ask questions about software application recommendations anywhere in the stackexchange system?  For example, I have a requirement, which will involve software and hardware.  I want to type this requirement up somewhere where possible readers will be able to recommend hardware and software solutions to my requirements? Python CSV parsing not returning proper rows Let $A$ and $B$ be $n\times n$ matrices such that $ABAB=0_{n\times n}$.  Can we conclude that  $BABA$ must be  $0_{n\times n}$?   I fail to give a counterexample when $n=2$， so I guess the answer is "yes" Jamaica Inn episode 5 from BBC learning English drama section:  The dialogue says:     Mary: My aunt is in bed.   She doesn't say -      My aunt is in the bed.   That section is for teaching English.  Are they using incorrect English? Is there a way to mix LIKE operator with IN ? Something Like  SELECT Id FROM Account WHERE Name LIKE IN:nameList   Where nameList is list of String? Is this is possible ? (I can make a dynamic query by doing OR with all the elements, but was looking for a more elegant way to do this) I am in a situation where when I get an HTTP 400 code from the server, it is a completely legal way of the server telling me what was wrong with my request (using a message in the HTTP response content)  However, the .NET HttpWebRequest raises an exception when the status code is 400.   How do I handle this? For me a 400 is completely legal, and rather helpful. The HTTP content has some important information but the exception throws me off my path.