Suppose $Y$ is an affine variety in $\mathbb{A}^n$ of $\dim r$. Then height of the prime ideal $\mathcal{I}(Y)$ is $n-r$. We have a result:     "Let $A$ be a Noetherian ring, $x_1,\dots ,x_r\in A$. Then every minimal prime ideal $\mathfrak{p}$ belonging to $(x_1,\dots ,x_r)$ has height $\leq r$." (Atiyah-Macdonald, Corollary $11.16$, page $121$).   Now suppose $\mathcal{I}(Y)$ is generated by $m$ elements. Then, using the result, $n-r\leq m$. So $\mathcal{I}(Y)$ is minimally generated by $n-r$ elements.  $\underline{\text{My question}}$: Is it always possible to find $n-r$ elements which will generate $\mathcal{I}(Y)$ or there are some examples for which it is not possible to find that minimal number of elements ? Also, in particular, if $Y$ is a linear variety then as @Georges pointed out we can always find $n-r$ elements which will generate the ideal $\mathcal{I}(Y)$. I need an explanation for this last part.   Thank you.
How can I determine the height and the least number of generators of the ideal $ I=(xz-y^2,x^3-yz,z^2-x^2y) \subset K[x,y,z] $?      I tried to calculate the dimension of the vector space $I/I\mathfrak m$ with $\mathfrak m=(x,y,z)$, but I'm not able to find it.  I think the height is $2$, because the affine variety should be a curve, and also I know that the ideal is prime, so I need a chain $0\subset \mathfrak p \subset I$. Maybe $(xz-y^2)?$ Is it prime?
"How to [...]?" and "Where to [...]?" Questions that are not questions. Is this defensible? Adding a question mark to the end of a "How to" or "Where to" sentence appears to be quite common. Here are two examples from this very site:  How to punctuate a list of questions? ()  Where to put the periods when using a parenthetical sentence? ()  In my mind, the question mark doesn't belong at all, and it bothers me.  Am I being unreasonable? Is this practice defensible? Inequality ${n \choose k} \leq \left(\frac{en}{ k}\right)^k$ Why is the Java plugin (JRE) disabled in Chrome? Why is the Java plugin (JRE) is disabled in Chrome? It is some security concern?  From official Java website:     Chrome no longer supports NPAPI (technology required for Java applets)   The Java plug-in for web browsers relies on the cross platform plugin architecture NPAPI,   which has been supported by all major web browsers for over a decade.   Google's Chrome version 45 (scheduled for release in September 2015) drops support   for NPAPI,   impacting plugins for Silverlight, Java, Facebook Video   and other similar NPAPI based plugins.   But anyone knows why? How it could be dangerous for Chrome user with latest version of Java JRE installed? I have written a small script to count number of occurrences of logs in a particular file. The following is my code:   #!/bin/sh count=1 while read line do     found=$( grep M $line )     OCCURENCE=$( grep M /log.txt)     if [ "$OCCURENCE" ]; then         echo $count")" $line &gt;&gt; filesave.txt     fi     count=$((count+1)) done &lt; log1.txt   but I am getting this error:  ./count.sh ./count.sh: syntax error at line 10: `found=$' unexpected Curved space-time VS change of coordinates in Minkowski space Understanding the definition of the order of an entire function in Ahlfors's Complex Analysis Let $f: \mathbb C \to \mathbb C$ be an entire function. The order of $f$ is defined by $$\lambda=\limsup_{r \to \infty} \frac{\log \log M(r)}{\log r}, $$ where $$M(r)=\max_{|z|=r} |f(z)| .$$  Ahlfors in his Complex Analysis claims that     "According to this definition $\lambda$ is the smallest number such that       $$M(r)\leq e^{r^{\lambda+\varepsilon}} $$ for any given $\varepsilon &gt; 0$ as soon as $r$ is sufficiently large."   Why is this true?  My attempt:  We know that $$\lambda=\lim_{\rho \to \infty} \sup_{r \geq \rho} \frac{\log \log M(r)}{\log r}. $$  From the definition of the limit we have that for any $\varepsilon&gt;0$, there exists some $\rho_0&gt;0$, such that $$\left\lvert \sup_{r \geq \rho} \frac{\log \log M(r)}{\log r}-\lambda \right\rvert \leq \varepsilon ,$$ for every $\rho \geq \rho_0$. In other words $$\frac{\log \log M(r)}{\log r} \leq \lambda+\varepsilon $$ for every $r \geq \rho_0$. From here it is easy to see that $$M(r)\leq e^{r^{\lambda+\varepsilon}}, $$ for all $r \geq \rho_0$. I cannot see why $\lambda$ is the smallest number with this property.  Thanks in advance. What is $\mathbb{E}[W(s)\mathrm{e}^{W(S)}]$ where W(S) is a standard Brownian Motion? I know that since it is a Brownian motion that $W(s)\sim{N(0,s)}$. One approach I tried, was to see that $W(s)-W(0)\sim N(0,1)$ and to let $X = W(s)$ and then solve $\mathbb{E}[X\mathrm{e}^X]$ but I end up with some awful integral. I was wondering whether there was an easier way? Or perhaps a simple way to solve this integral: $\int_{-\infty}^{\infty} \frac{1}{\sqrt{2{\pi}s}}x\mathbb{e}^{x}\mathbb{e}^{-\frac{x^2}{2s}}$ This is a  about IPv6 Subnetting.      Related:               I know a lot about , and as I prepare to (deploy|work on) an IPv6 network I need to know how much of this knowledge is transferable and what I still need to learn. IPv6 seems at first glance to be much more complex than IPv4. So I would like to know:   IPv6 is 128 bits, so why is /64 the smallest recommended subnet for hosts? Related to this:   Why is it recommended to use /127 for point to point links between routers, and why was it recommended against in the past? Should I change existing router links to use /127? Why would virtual machines be provisioned with less than a /64 worth of addresses? Are there other situations in which I would use a subnet smaller than /64?  Can I map directly from IPv4 subnets to IPv6 subnets? For instance, does an IPv4 /24 correspond directly to an IPv6 /56 or /120? My interfaces have several IPv6 addresses. Must the subnet be the same for all of them? Why do I sometimes see a % rather than a / in an IPv6 address and what does it mean? Am I wasting too many subnets? Aren't we just going to run out again? In what other major ways is IPv6 subnetting different from IPv4 subnetting? Update: (Originally this was a comment, but I thought it was worth sharing here at the top.)   In the original question, I asked if there was a polite, socially-acceptable way to ask an Indian co-worker not to use the phrase "do the needful", as I didn't care for it.   In the years since I asked this question I've asked many people about the phrase. To the Indians I've asked in-person, it's not rude in any way. To Americans I've asked it varies anywhere from "I don't like it, but I don't mind it" to "It's very rude and makes me angry when I see it".   Through a long series of edits the question morphed and I wasn’t able to delete it because it had upvoted answers. The question, as it stands now, has no other answer.  I guess, though, if you're non-Indian and you find it rude, or if you're Indian and you've never realized someone might find it rude, this post may still have merit. Hopefully it does. It certainly has gotten a lot of views. Best wishes, folks.  End Update. See below for what's left of the actual question.    "Do the needful."   It's a phrase that I've only seen used in email, and I find it . . . presumptuous (maybe even rude). Regardless of prefacing with "please", one is commanding rather than asking for assistance.  I've only seen it used by those of Indian origin, so I've simply not mentioned it at all rather than worry about any cross-cultural offense that may come of bringing it up.  But still, I don't understand why it's used. Why not request rather than make two statements, one factual, one imperative?  For example, why would one use:        I'm told you have Jane's email address. Please do the needful.      instead of,        Would you send me Jane's email address? Division algorithm for polynomials in R[x], where R is a commutative ring with unity. Is there a division algorithm for polynomials in R[x], where R is a commutative ring with unity?    All the algebra books I read mention division algorithm for polynomials in F[x], where F is a field. Since the leading coefficient of a non zero polynomial in R[x] is not necessarily invertible, I find it difficult to proceed the same way we do for polynomials in F[x].  Can I get some help? I tried to write out the aura if statement but I am getting stuck if someone can explain it to me in detail. thank you  &lt;aura:if isTrue=&quot;{!lessthan(v.Tl.time, v.Tl.End_Date_Time1__c)}&quot;&gt;                            &lt;/aura:if&gt; If I've got a file (or files) selected in Windows Explorer, the only way I've found to "unselect" them is click with the mouse on some empty space.  I thought Esc would do this, but it doesn't. Also changing focus with Tab or Shift + Tab doesn't work; the file(s) are still selected even if focus goes to the address bar for instance.  Does anyone know of a keyboard shortcut to do this? I want to synchronise two folders in real time under Windows 7.  Basically, I want to monitor a folder and synchronise each change (new files, changed files, deleted files) to another drive. It has to be in real time, so it gets synchronised instantly when a change happens. A one-direction synchronisation is enough.  I tried Microsoft's , but it does only syncing by hand or scheduled (thus not satisfying the "real-time" requirement).  Can this be achieved with Windows 7 itself? If not, are there other methods of doing this? Novel about a teleportation device and a con artist alien I read two (maybe three) books about a protagonist who communicates with an alien who provides the instructions on how to create a device to teleport. The device is flawed in that it purposefully uses more energy (much more) by taking it from the rotation of the Earth.  The main character and two others continue to fight, commit intrigue, etc. The alien then uses them to recruit other races to do the same thing. How to install Microsoft fonts on Ubuntu 16.10? Would this look natural:     StackExchange is a helluva site! I found out about this site at Facebook.    Or should we use the preposition on:     StackExchange is a helluva site! I found out about this site on Facebook.    Or are both okay?   (Inspired by a ) Stopping a Thread in Java? I'm in the process of writing a piece of code that connects to a server spawns a bunch of threads using that connection and does a bunch of "stuff".  There are certain instances where the connection fails and I need to stop everything and start from scratch with a new object.  I wanted to clean up after the object but calling thread.stop on the threads, but this method is seemingly deprecated.  What is the recommended alternative to doing this? Should I write my own cleanup and exit method for each of the threads? Set the thread to null? or something else? Global variables in AngularJS Can linear momentum be conserved before and after collision in the presence of an external force? I'm a physics tutor for 11th and 12th grade students. I taught my students that momentum is conserved only when the net external force on the system is zero. But in the books we are following, momentum was conserved in the following cases in spite of presence of an external force:   A body (mass m1) is dropped from a tower (height h) and another body (mass m2) is projected vertically upward with initial velocity u. Find the velocities just after collision. (Gravitational force is the external force) Two bodies projected towards each other on a rough table surface, with all required parameters given asked to find the velocities after collision. (frictional force is the external force) Body explodes in mid air. Momentum is conserved before and after explosion. (Again gravitational force is the external force)   I explained to my students in all these cases though an external force is present it has negligible effect. But I'm not satisfied with my answer.  EDIT: Thank you for the comments and answer. It is clear that contribution of external force to change in momentum is negligible. Can someone do the calculation of taking change in momentum into consideration and showing that it is negligible? A theorem about Cesàro mean, related to Stolz-Cesàro theorem Original Title: Tauberian theorems and Cesàro sum  Theorem (Landau-Hardy, From Rudin's Principle of Mathematical Analysis Exercise 3.14)  $\newcommand\abs[1]{\left\lvert#1\right\rvert}$ If $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\frac{s_0+s_1+\dotsb+s_n}{n+1}\qquad(n=0,1,2,\dotsc)$$ Put $a_n=s_n-s_{n-1}$ for $n\ge1$. Assume $M&lt;+\infty$ and $\abs{na_n}\le M$ for all $n$, and $\lim_{n\to\infty}\sigma_n=\sigma$, then $\lim_{n\to\infty}s_n=\sigma$.  The outline of the proof  If $m&lt;n$, then $$s_n-\sigma_n=\frac{m+1}{n-m}(\sigma_n-\sigma_m)+\frac1{n-m}\sum_{k=m+1}^n(s_n-s_k)\tag{*}$$ Notice that $\abs{s_n-s_k}\le(n-m-1)M\,/\,(m+2)$, fix $\epsilon&gt;0$ and associate with each $n$ the integer $m$ that satisfies $$m\le\frac{n-\epsilon}{1+\epsilon}&lt;m+1$$ Then $(m+1)\,/\,(n-m)\le1/\epsilon$ and $\abs{s_n-s_k}&lt;M\epsilon$. Hence $$\limsup_{n\to\infty}\,\abs{s_n-\sigma}\le M\epsilon$$  Questions and thoughts  It seems that the equation (*) comes out strangely. I wonder how to discover such kind of strange identities. So is there any observation, even deeper, to look through that equation?  Thanks!