The localisation of the ring $\mathbb{Z}$ at the prime ideal $(p)$ is PID
Localization of a UFD that is a PID
Why does the road look like it's wet on hot days? Often, I'll be driving down the road on a summer day, and as I look ahead toward the horizon, I notice that the road looks like there's a puddle of water on it, or that it was somehow wet. Of course, as I get closer, the effect disappears.  I know that it is some kind of atmospheric effect. What is it called, and how does it work? I am writing a small KOMA-Script scrartcl document with multiple sections. In the table of contents, LaTeX fills the current line with dots between the end of the section name and the page number. Is there any way to customize this behavior? I would like to reduce the space between the single dots (i.e. use more dots in the same space). How can we prove that $D(f)\backsimeq\operatorname{Spec} A_f$? Let $$D(f)=\operatorname{Spec}A - V(f)=\{p\in \operatorname{Spec} A:f\notin p\}$$ be the principal open subsets of $\operatorname{Spec}A$ with the Zariski topology. I'm trying to prove that $D(f)\backsimeq \operatorname{Spec} A_f$, where $A_f$ is the localization of $A$ at $f$.  If we consider the canonical homomorphism $\varphi:A\to A_f$ given by $\varphi (a) = \frac {a}{1}$, we can define $\varphi':\operatorname{Spec} A_f\to \operatorname{Spec} A$, given by $\varphi'(p)=\varphi^{-1}(p),\forall p\in \operatorname{Spec}A_f$.  I'm having trouble proving that this map is surjective.  Thanks. In Star Trek II: The Wrath of Khan (the one from 1982, not the recent remake), Khan yells "This is Ceti Alpha V!" and they're all like this is a big surprise, we didn't know.  If they had previously charted the planet, wouldn't the computer have told them the planets weren't quite where they should be based on earlier visits? I mean, we can predict the next hundred transits of Venus, shouldn't 23rd century explorers be able to predict where the Ceti planets are less than twenty years after their first visit? When cutting/copying some cells to paste them into another sheet/page, sometimes Excel freezes/locks up and fades out. At the top toolbar it says in brackets "not responding".  Eventually, I must click 'X' to close the program. It offers to wait for the program to respond, but never does – it just does nothing until I finally close it, where it offers to recover files etc.   Is there an issue with memory here? What can I do to stop it locking up? I had a problem with the laptop lid so I am using it with an external monitor. I was wondering if there is any way to power on the laptop without opening the lid, that is through ports.  Laptop: Dell Inspiron 1525 How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? How can I determine whether a conference is reputable? I recently received an invitation to a conference and I'm trying to determine how reputable it is. Is there a good way to go about this? There's a problem statement:     For each $m \in \mathbb{N}$, we construct a sequence $m_0$, $m_1$, $m_2,\dots$ denoted $S_m$, recursively via $m_0=m$ and      $$m_{i+1} = m_i + \left\lfloor \sqrt{m_i} \right\rfloor$$      for all $i \ge 0$.   Here, $\lfloor x \rfloor$ is the floor of $x$, the greatest integer less than or equal to $x$.   Hence, we have $\left\lfloor \sqrt{10} \right\rfloor=3$ and $\left\lfloor \sqrt{29} \right\rfloor=5$.      Show that for each positive integer $m$, the sequence $S_m$ contains the square of some integer.   I'm pretty certain that this can be proved with induction. I am just not quite sure what to induct on. Examining examples shows that $S_m$ always results in a perfect square eventually, though I'm not sure how to prove it. Let $A$ be a $m \times n$ matrix with entries from some field $F$. Define the determinant rank of $A$ to be the largest possible size of a nonzero minor, i.e. the size of the largest invertible square submatrix of $A$. It is true that the determinant rank is equal to the rank of a matrix, which we define to be the dimension of the row/column space.  It's not difficult to see that $\text{rank} \geq \text{determinant rank}$. If some submatrix of $A$ is invertible, then its columns/rows are linearly indepedent, which implies that the corresponding rows/columns of $A$ are also linearly indepedent.  Is there a nice proof for the converse? Do tag edits count for Archaeologist? How does a magnet work? Given, $\{x_n\}$ and $\{y_n\}$ are two real valued sequences such that $\lim_{n\to\infty} (x_{n}^{3}-y_{n}^{3})=0 $ Then is it always true that $\lim_{n\to\infty} (x_n-y_n) =0 $ ? I tried by splitting $ (x_{n}^{3}-y_{n}^{3}) $, but that didn't help as $\lim_{n\to \infty} (x_{n}^{2}+x_{n}y_{n}+y_{n}^{2}) $ can be $0$ or non zero or doesn't exist. How do I check if a file exists in Java? This is what happens when I launch the terminal  There was an error creating a child process  So I have been using the Ubuntu terminal on my Raspberry Pi 4 model B 8 GB with Ubuntu for a couple moths when Ubuntu 20.10 came out for the Pi. But now I have a problem where I launch the terminal and it gives that message. I have gone for alternatives and tried to fix it but all the solutions were outdated or they were on a different distro. Also the Ubuntu commands did not fix the issue because it only triggers upon a certain command. Windows 7 low and critical battery levels cannot be set lower than 10% reason for exec in wrapper scripts Free GIS workshops, tutorials, and applied learning material This Q&amp;A lists free as in $0 workshops in GIS and related fields that have their material available to view or download online. The list is not limited to FOSS GIS, as GISers usually use a combination of open and closed source tools to accomplish their tasks. Some of these tools are easier than others and some are better documented. This list will be a great benefit to the community especially for new comers to the field. General GIS   from   from   from   from   Open-source GIS   from   from   from   from   from   ESRI Products   from   from   Transportation These links include either theoretical or applied transportation knowledge in transportation planning          Disaster Management    Databases   from   Geo Statistics    from I have a long line of code that I want to break up among multiple lines. What do I use and what is the syntax?  For example, adding a bunch of strings,  e = 'a' + 'b' + 'c' + 'd'   and have it in two lines like this:  e = 'a' + 'b' +     'c' + 'd' existence of complex structure in dimension 2 Does an orientable closed 2 manifold always admit a complex structure?﻿  I know that an almost complex doesn' t necessarily give rise to a complex structure(the almost complex structure should be integrable) I am studying 2 manifolds over R, and I wonder if an almost complex structure always exist? This almost structure may be chosen integrable?  I ask because there are certain known results on the different possible geometries in dimension 2.I don' t know exactly how these manifolds (closed and orientable) are classifyied, but I know there are three cases. May be I am confused, and the complex structure is irrelevant to this classification. Some help please!