Why does `kill -L` work in `bash` but not in `zsh`
What is the difference between a builtin command and one that is not?
The main effect will be non-significant if the interaction is significant? $\sqrt{1-x^2}$ is not a rational function If $k$ is a field in which $1 + 1 \neq 0$, prove that $\sqrt{1-x^2}$ is not a rational function. Hint. Mimic the classical proof that $\sqrt{2}$ is irrational   It seems rather odd to discuss the square root function in the context of an arbitrary field, but here goes nothing!  Proof:   Suppose that $\sqrt{1-x^2}$ is a rational function in $k(x)$. Then there exist polynomials $p(x)$ and $q(x) \neq 0$ that are relatively prime with $\sqrt{1-x^2} = \frac{p(x)}{q(x)}$; clearly we may take $p(x) \neq 0$. Then $1-x^2 = \frac{p(x)^2}{q(x)^2}$ or $q(x)^2(1-x^2) = p(x)^2$, which says $q^2 | p^2$, and since $q | q^2$, we can infer $q | p^2$. Moreover, since $p|p^2$ and $(p,q)=1$, then $pq|p^2$ or $q|p$, which contradicts the fact that the polynomials are relatively prime. Hence $\sqrt{1-x^2}$ cannot be a rational function.    How does this sound? Aside from the problem of discussing $\sqrt{~~}$ in the context of an arbitrary field, I am worried about not using the fact that $1+1 \neq 0$, at least not explicitly. Where exactly is this assumed used, if at all?  EDIT:  Suppose that $q(x) =1$, and let $f(x) = a_n x^n + ... a_1 x + a_0$. Then $1-x^2 = f(x)^2$. If $x=0$, $1 = a^2_0$ and therefore $a_0$ must be a unit. Letting $x=1$ we get $0 = (a_n + ... + a_1 + a_0)^2$; and letting $x=-1$ we get $0 = (-a_n - ... - a_1  + a_0)^2$. Since we are working in a field, there can be no nonzero nilpotent which means that $a_n + ... + a_1 + a_0$ and $-a_n - ... - a_1  + a_0$ are both zero. Adding the two equations together yields $2a_0 = 0$, and since $1+1 \neq 0$, $a_0 = 0$ which contradicts the fact that $a_0$ is a unit.   How does this sound?  Another attemtpt:  If $f(x)^2 = 1-x^2$, then $2 = \deg(f^2) = 2 \deg (f)$ implies $\deg(f) = 1$. Yet if $x=-1$, we get $f(-1)^2 = 1 - 1 = 0$ and therefore $f(-1)=0$ since there are no nonzero nilpotent elements in a field; similarly, $f(1)=0$. Since $1+1 \neq 0$, $f$ has two distinct roots in $k$ yet is only a $1$-st degree polynomial, a contradiction. Adding/updating field to polygon feature class that lists bordering (neighbor) polygons? I have a dataset of zipcodes.  I want to add a new field "NEIGHBORS" that includes bordering polygons in a comma delimited manner such as attached image: Idiomatic batch processing of text in Emacs? What is the result of $ \lim_{n \to \infty} \frac{ \sum^n_{i=1} i^k}{n^{k+1}},\ k \in \mathbb{R} $ and why? What is the result of the next limit:   $$ \lim_{n \to \infty} \frac{ \sum^n_{i=1} i^k}{n^{k+1}},\ k \in \mathbb{R} $$  Why (theorem)? Rails: What's a good way to validate links (URLs)? Best way to select a subset of items in a long list Eulerian Paths in a Complete Graph $K_5$ As the title suggests, I’m simply looking for a method to compute the number of Eulerian Paths in the complete graph $K_5$, where all paths start from the same node. Since there are 5 nodes, all Eulerian paths must be Eulerian circuits as well.  Thanks. House wiring and grounding. How do you retrofit an old house and ground it? A contractor looked at our house and told me that you can't fix a house (with old wiring) that has some outlets that are not grounded.  He said that you can go on YouTube and search the internet and it will tell you that you can.  He says that the only way to properly fix and ground everything is to do a whole house rewire.  Is he correct?  If, theoretically, you have an old house that has no ground whatsoever, can you run a grounding wire from the panel to all of the outlets to fix this?  updated  We have BX/AC wiring. And also apparently we have "...old-style AC (BX) with cloth insulated wires in a paper overall wrap under the spiral armor (no bonding tape)" Courtesy of @ThreePhaseEel. Does the use of words such as 'fail', 'wrong', 'incorrect' and 'invalid' have a sufficiently negative impact on a users frame of mind to warrant the prohibition of their inclusion in the error messages generated by typical web forms?  i.e. Should we rephrase examples like these to just use positive words?  'date entered incorrectly'  or  'failed to complete registration, please see below for invalid field entries'  (PS, the examples are there to show use of the words in an error message, not to serve as examples of error messages I would use or things that need correcting in answers, I'm not looking for advice on how to write form error messages, but for information about the effect of certain words on a users frame of mind) Start a longtable in Latex at top of next page I want to use the longtable environment for a table. However, I do not know how to place it at the top of the next page, as I can do with the table environment. Do you know how to do it? Loading OpenStreetMap data in QGIS? Graphing y=x^2 and y=x I want to draw the region bounded by y=x^2 and y=x. Any ideas? I was hoping this would be somewhat simple. I've tried used a similar, with \begin{axis}..\end{axis}, but it doesn't like that. Did I not declare a package or something?   Also, if I wanted to show the solid this region creates by rotating it about the x-axis, is there a way to did this in latex? (If you had cal 2, think volume of a solid rotated about ...). Is starting a group with 5 newbies (DM included) possible? Some friends and I want to start playing D&amp;D (5e), but none of us, DM included (not me) have played before to any great extent. The DM and I have some cursory experience with 4e, but have never been able to play to any great extent. Is this advisable, or should we play with existing groups first? We live in a relatively remote area, making it hard to find experienced players. What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity? I tried and got this   $$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$ $$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$ where $m$ is an integer. $$\lim_{n\to\infty}n\sin(2\pi en!)=\lim_{n\to\infty}n\sin\left(2\pi n!\sum_{k=0}^n\frac{1}{k!}\right)=\lim_{n\to\infty}n\sin(2\pi m)=\lim_{n\to\infty}n\cdot0=0$$  Is it correct? Prove: If a sequence converges, then every subsequence converges to the same limit. I need some help understanding this proof:  Prove: If a sequence converges, then every subsequence converges to the same limit.  Proof:  Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k \geq k$ for all $k$. This easy to prove by induction: in fact, $n_1 \geq 1$ and $n_k \geq k$ implies $n_{k+1} &gt; n_k \geq k$ and hence $n_{k+1} \geq k+1$.  Let $\lim s_n = s$ and let $\epsilon &gt; 0$. There exists $N$ so that $n&gt;N$ implies $|s_n - s| &lt; \epsilon$. Now $k &gt; N \implies n_k &gt; N \implies |s_{n_k} - s| &lt; \epsilon$.   Therefore: $\lim_{k \to \infty} s_{n_k} = s$.   What is the intuition that each subsequence will converge to the same limit I do not understand the induction that claims $n_k \geq k$ Adding an error on a dynamic referenced field Is there a way to add an error on a dynamically referenced field? Let's take this simple example:  opp.get('Name').addError('Invalid name');   This is not working, I'm looking for a similar minded way.  Thanks A topological space $X$ is simply connected if it is pathwise connected and each closed path $u : I = [0,1] \to X$ is path homotopic to the constant path at $x_0 = u(0) = u(1)$. Recall that  A closed path $u$ is one with $u(0) = u(1)$. A closed path will also be called a loop and we write $\mathcal L'(X)$ for the set of all loops in $X$. If we want to be more precise, we say $u$ is a loop based at $x_0 \in X$ provided $x_0 = u(0) = u(1)$. By  $\mathcal L'(X,x_0)$ we denote the set of all loops in $X$ which are based at $x_0$. Also note that each constant path is a loop.  Two paths  $u, v : I \to X$ with the same initial point $x_0 = u(0) = v(0)$ and the same terminal point $x_1 = u(1) = v(1)$ are path homotopic, $u \simeq_p v$, if there exists a  homotopy $H : I \times I \to X$ such that $H(t,0) = u(t)$ and $H(t,1) = v(t)$ for all $t$ and $H(0,s) = x_0, H(1,s) = x_1$ for all $s$ (such a homotopy $H$ is stationary on the boundary $\partial I = \{0,1\}$; it is called a path homotopy). It is well-known that $\simeq_p$  is an equivalence relation on the set $\mathcal P(X)$ of all paths in $X$. Clearly $\simeq_p$ restricts to an equivalence relation on $\mathcal L'(X)$ and to an equivalence relation on each $\mathcal L'(X,x_0)$.  The set of equivalence classes $\mathcal L'(X,x_0)/\simeq_p$ is nothing else than the fundamental group $\pi_1(X,x_0)$ with base point $x_0$.   Question: Find a list, as complete as possible, of properties characterizing topological spaces which are simply connected. Each of these properties can then be taken as an alternative definition of simply connected. Note that many questions in this forum deal with aspects of simple connectedness. The answer to the present question will hopefully cover most of them. If a drive doesn't know about partitions and files systems why moving a file from one partition to another takes relatively long time, unlike the instantaneous move on the same partition? Is that file system inefficiency? In this case, NTFS. It seems like the file is being copied and then original is erased, instead of removing the entry about that file in one partition and making it in another? How to get the path of a running JAR file? My code runs inside a JAR file, say foo.jar, and I need to know, in the code, in which folder the running foo.jar is.  So, if foo.jar is in C:\FOO\, I want to get that path no matter what my current working directory is.