Is $\mathbb{Z}[t,t^{-1}]$ a PID? What about $\mathbb{Z}[\sqrt{2}i]$?  I don't know how to prove that a set IS a PID. I only know how to prove when it is NOT (by proving it is not UFD, for example).  How can I show an ideal can only be generated by a single element? In $\mathbb{Z}$ I understand, there is a minimality argument. But in those sets up there I have no idea how to start.
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?
How do I install pillow in Ubuntu 16.04 LTS? I went to  for installing Pillow since the original PIL is a discontinued project and so now not so good to use. But, in the table support for Ubuntu 16.04 LTS is unmentioned. It appears neither in , or in .     Further, just installing pillow is said to be not enough. There are some dependencies that should be installed first which is also unmentioned for Ubuntu 16.04 LTS. Would the installation steps for Pillow for python 3 given for 14.04 LTS be OK for Ubuntu 16.04 LTS?  If not, How do I install pillow in 16.04 LTS for Python 3.6? (There is support for Python 2.) Transit visa for Indian citizen traveling through Hong Kong Given two coprime integers, find multiples of them that differ by 1 I'll describe the problem with an example.  Find integers $n$ and $m$ such that $n\cdot13 + m\cdot101 = 1$. This is the same as solving the equation $n\cdot13 \equiv 1 \pmod {101}$  I'm revising for a maths exam in a few days and it seems like a lot of the questions and examples rely on an ability to solve this type of problem. I can't seem to find a method in my notes, the solutions just get plucked out of thin air! What is a neat way of doing it on pen and paper with a calculator? $S_3$ permutes the elements of $\mathbb{R}^3$, show that $\sigma (x_1, x_2, x_3) = ( x_{\sigma^{-1}(1)}, x_{\sigma^{-1}(2)}, x_{\sigma^{-1}(3)})$. How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$? Is there any reference where Captain America's shield is made from stolen Wakanda vibranium? I want to disable the internal Microphone on my laptop and the internal Webcam completely that they're no longer selectable, and no longer available?  I will only use my external Webcam, and my external (USB) headset.  The internal audio should still be available for playback. Are the LIGO observations proof of a black hole merger, and what happened to the black holes? Recent reports claim that the gravitational waves detected by LIGO match up with the signal expected from two black holes merging as predicted by general relativity.  Additionally, the masses of both black holes were estimated. Questions:  How certain are we that the detected of gravitational waves are necessarily from two black holes merging?  Since supermassive black holes are typically at the center of a host galaxy, what happened to the galaxies that contained the two merging black holes?  For example, did their host galaxies merge?    Edit After reading the answers, I understand that the signals detected by LIGO are proof of the validity of gravitational wave generational artifacts as predicted by general relativity.  When additionally gravitational waves, caused by the merger of the supermassive black holes (those which ARE indeed located in the centers of the merging galaxies) will be detected, since mergers of galaxies could be observed by the methods independent of gravitational waves such as registering the fact of the quasar appearance (). Additional questions:  Could someone offer me the historical ASTRONOMY precedent when the second degree of inference (from the mathematical model, being associated with the theory of Physics) was accepted as the discovery of the astronomical object?  Could someone give a reference to trustable scientific publication, with the subject of study being discussion of requirements satisfying claim of astronomical discovery of an astronomical object?  Have independent scientific sources (outside of members of the LIGO team) analyzed methodology and results of the LIGO signal detection, and published their conclusions with regards of what was actually discovered? What is this element on the far right of schematic described as "270 OHM"? Is this some kind of inductor? What is its purpose in this device (this is a reference design of dc-dc power supply of sim900 taken from its datasheet) and what would be an exemplary part number? Four tasks in parallel... how do I do that? I have a bunch of PNG images on a directory. I have an application called pngout that I run to compress these images. This application is called by a script I did. The problem is that this script does one at a time, something like this:  FILES=(./*.png) for f in  "${FILES[@]}" do         echo "Processing $f file..."         # take action on each file. $f store current file name         ./pngout -s0 $f R${f/\.\//} done   Processing just one file at a time, takes a lot of time. After running this app, I see that the CPU is just 10%. So I discovered that I can divide these files in 4 batches, put each batch in a directory and fire 4, from four terminal windows, four processes, so I have four instances of my script, at the same time, processing those images and the job takes 1/4 of the time.  The second problem is that I lost time dividing the images and batches and copying the script to four directories, open 4 terminal windows, bla bla...  How do that with one script, without having to divide anything?  I mean two things: first how do I from a bash script, fire a process to the background ? (just add &amp; to the end?) Second: how do I stop sending tasks to the background after sending the fourth tasks and put the script to wait until the tasks end? I mean, just sending a new task to the background as one tasks end, keeping always 4 tasks in parallel? if I do not do that the loop will fire zillions of tasks to the background and the CPU will clog. How does one write on the outside of a circle?  I am looking for sample files/packages which imitate a magazine lay-out. Specifically a page with two columns having a circular picture between the two columns.    Example of the desired layout, created using Word: What are the 'types' of parametric down conversion? Is there a significant difference between "rock" and "stone"? I found this pretty good small research, but I am still concerned about "regular usage" in life, say, when I am talking in the office, especially because of this comment:     To me: Stone is something you can pick up with one hand (for   throwing). A rock takes two hands. I have a number of measurements of the same quantity (in this case, the speed of sound in a material). Each of these measurements has their own uncertainty.  $$ v_{1} \pm \Delta v_{1} $$ $$ v_{2} \pm \Delta v_{2} $$ $$ v_{3} \pm \Delta v_{3} $$ $$ \vdots $$ $$ v_{N} \pm \Delta v_{N} $$  Since they're measurements of the same quantity, all the values of $v$ are roughly equal. I can, of course, calculate the mean:  $$ v = \frac{\sum_{i=1}^N v_{i}}{N}$$  What would the uncertainty in $v$ be? In the limit that all the $\Delta v_i$ are small, then $\Delta v$ should be the standard deviation of the $v_i$. If the $\Delta v_i$ are large, then $\Delta v$ should be something like $\sqrt{\frac{\sum_i \Delta v_i^2}{N}}$, right?  So what is the formula for combining these uncertainties? I don't think it's the one given in  (though I may be wrong) because it doesn't look like it behaves like I'd expect in the above limits (specifically, if the $\Delta v_i$ are zero then that formula gives $\Delta v = 0$, not the standard deviation of the $v_i$). I want to create posters for my poster presentation on a conference. What tools or LaTeX classes are available for preparing posters ? Prove that $\sum_{r=1}^nr(r+1)=\frac{n(n+1)(n+2)}{3}$ using induction $$\sum_{r=1}^nr(r+1)=\frac{n(n+1)(n+2)}{3}$$  could you help me with how exactly I work this out? How to ask great questions As programmers we are daily faced with having to get requirements, test things out, etc.  It's practically half our job to know what questions to ask, and how to ask them to get the most concise, actionable answer possible, especially from reluctant users/managers/customers/etc.  How do you ask questions that get answers? How to quickly and safely defrost chicken? How to show that $\int_0^1 \left(\sqrt[3]{1-x^7} - \sqrt[7]{1-x^3}\right)\;dx = 0$ How to automatically generate N "distinct" colors?