icon and subject of lesson
How to look up a symbol or identify a letter from a math alphabet or other character?
I'm trying to prove whether or not the Laurent polynomials $\mathbb{Z}[x, x^{-1}]$ with coefficients in $\mathbb{Z}$ form a principal ideal domain.    I know that $\mathbb{F}[x, x^{-1}]$ is a PID when $\mathbb{F}$ is a field, but clearly $\mathbb{Z}$ is not a field so I cannot appeal to this result. And my intuitions are not serving me very well at the moment. Can anyone provide a hint or direction to take? Can I rent a camper in the US/Canada with a European B license? We are considering a holiday in North America with a camper. I have a European driver's license (B) which allows me to drive personal cars up to 3500 kg, although I can imagine that an average camper easily weighs more.  Does my license qualify me to drive a camper? If not, can I take an American/Canadian driver's test which would allow me to drive a camper? Invalid group name: Group names must begin with a word character I received the following exception when I was using the Regex class with the regular expression: (?'named a'asdf)  System.ArgumentException: parsing \"(?'named a'asdf)\" - Invalid group name: Group names must begin with a word character.   What is the problem with my regular expression? Do these two observations suffice to show that a finite boolean ring must be of the form $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$? Question: If $R$ is a finite boolean ring, then show $R \cong \mathbb Z_2 \times \mathbb Z_2 \times \cdots\times \mathbb Z_2$.   I know that   $\mathrm{char}(R) =2$  $R$ has $2^k$ elements for some $k$  (this is because, if $p\neq 2$ is a prime with $p\mid|R|$, then there exists a non-zero $x \in R$ s.t. $px=0$ and $p $ is odd $\implies x=0$, contradiction).   Does this suffice to answer my question? Can anyone come up with an explicit example of two functions $f$ and $g$ such that: $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective?  I tried the following: $$f:\mathbb{R}\rightarrow \mathbb{R^{+}} $$ $$f(x)=x^{2}$$  and  $$g:\mathbb{R^{+}}\rightarrow \mathbb{R}$$ $$g(x)=\sqrt{x}$$  $f$ is not injective, and $g$ is not surjective, but $f\circ g$ is bijective  Any other examples? I've been trying to play with a rigid body simulations.  I prepared a project of a building made of domino tails. I've placed all objects very tight, but as soon as I start animation it looks like there was some tiny gap between objects, so they all fall down a little bit. For a high building it leads to a construction disaster as the whole building colapses.  Here is an example:  (I'd suggest to use 0.25x speed).  How can I fix it? Inverse images of images, images of inverse images of a set of vectors What can be said about$\ f^{-1}(f(S))$ and$\ S$, where$\ f(S):=\{f(x):x \in S\}$?   I feel as if$\ f^{-1}(f(S))$ will be smaller than or equal in size (number of elements) to$\ S$.  For example, if$\ f(x)=x$, then$\ f(\{-1, 0, 1, 2\})=\{-1, 0, 1, 2\}$and$\ f^{-1}(f\{-1, 0, 1, 2\})=\{-1, 0, 1, 2\}$ (same size).  But if$\ f(x)=x^2$, then$\ f(\{-1, 0, 1, 2\})=\{0, 1, 4\}$ and$\ f^{-1}(f\{0, 1, 4\})=\{0, 1, 2\}$, which is a little bit smaller than$\ S$.   What can be said about$\ f(f^{-1}(U))$ and$\ U$, where$\ f^{-1}(U):=\{x \in X:f(x) \in U\}$?   The same size as each other? Always?? What is the size of a boolean variable in Java? Can any one tell the bit size of boolean in Java? I installed Ubuntu yesterday, so I'm trying to get used to it.  I'm really close to my router, but my wifi signal gets too low. Sometimes I'm in Facebook or Youtube and the signal is high, almost full, but it is usually in one or two bars, and sometimes none.  I can use Internet normally, I can watch videos and stuff, but the problem is when I try to update something or Ubuntu is downloading stuff, the downloads get stuck or get cancelled telling me that I should check my connection.  How can I fix this? I should be getting a really good signal, I'm really close to my router, my phone has full signal, and a notebook with Windows 7 has full signal here too.  Sorry my bad English and I hope you can help me.  $ lsusb Bus 009 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 004 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 008 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 007 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 003 Device 002: ID 0bda:8179 Realtek Semiconductor Corp.  Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 006 Device 002: ID 0458:003a KYE Systems Corp. (Mouse Systems) NetScroll+ Mini Traveler / Genius NetScroll 120 Bus 006 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 005 Device 001: ID 1d6b:0001 Linux Foundation 1.1 root hub Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Query works in SQL Server 2005 but syntax error for SQL Server 2000 $f$ has the form $f(x) =ax^2+bx+c$. Every differentiable function $f:R \rightarrow R$ with the property that $(2h)f′(x) =f(x+h)−f(x−h)$ for all $x \in R$ and all $h$ has the form $f(x) =ax^2+bx+c$.  I would like to get some help on this one. Taking partial derivatives was an idea, but is that legal? Can I get an enters-the-battlefield effect out of a second legendary before I must sacrifice it? If I have a legendary creature out, and I make a copy of it (such as by enchanting it with  or making a ), do I get enters-the-battlefield effects from the creature before I have to sacrifice it to the legend rule? WhatsApp for Ubuntu Phone? Does someone know if there already is a WhatsApp Version for Ubuntu Phone? I decided to install Ubuntu on my Android, but I don't want to use my phone without WhatsApp. I'm going through MSIL and noticing there are a lot of  instructions in the MSIL.  The MSDN article says they take no action and are used to fill space if the opcode is patched. They're used a lot more in debug builds than release builds.  I know that these kinds of statements are used in assembly languages to align later instructions, but why are MSIL nops needed in MSIL?  (Editor's note: the accepted answer is about machine-code NOPs, not MSIL/CIL NOPs which the question originally asked about.) Is every metric space a subspace of some connected metric space? Is every metric space a subspace of some connected metric space? If the space itself is connected then we're done, but if not then I think we can extend our metric space to make it connected.  I'm not sure whether this will work or not, but intuitively I think the answer is yes. Single word describing a person new in something What is a word for someone who is totally new in something or participates in something for the first time? Replacing all values in a table > $T$ OS: UBUNTU 18.04  I had installed Oracle Java 8 earlier. But decided to switch to open JDK 11. it's successfully installed. java -version openjdk version "11.0.2" 2019-01-15 OpenJDK Runtime Environment (build 11.0.2+9-Ubuntu-3ubuntu118.04.3) OpenJDK 64-Bit Server VM (build 11.0.2+9-Ubuntu-3ubuntu118.04.3, mixed mode, sharing)  Then I set the JAVA_HOME sudo nano /etc/environment  File changes: PATH="/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/$ JAVA_HOME="/usr/lib/jvm/java-11-openjdk-amd64/"  Then executed: source /etc/environment  echo $JAVA_HOME /usr/lib/jvm/java-11-openjdk-amd64/  But the moment I exit the terminal and then launch the terminal again: I get this output:  ▶ echo $JAVA_HOME /usr/lib/jvm/java-8-oracle  which you can see is not what is expected. Please help me out Cannot remove block comment in Eclipse after formatting Today I was told a tale by a Unix trainer where the root password got leaked to the students, and one of the fellas removed the execute permission from /usr/bin/chmod itself. How do you recover chmod in this case and make it executable again? Let's say getting it from an external source or recompiling it is not a preferable option, is there some smart hack that can recover this chmod itself?   Note that this happened a long time ago and I'm not looking for a solution for some current problem, just curious about what ways Unix provides us around such an issue.