The question is      Find all polynomials $p(x)$ such that $p(x^2)=[p(x)]^2$.   First of all i saw that  $p(x)=0$ and $p(x)=1$ are two polynomials satisfying the condition. Next I tried putting some values and observed that  Since $p\big((-a)^2\big)=[p(-a)]^2$. Also, $p\big((a)^2\big)=[p(a)]^2$ Hence we can conclude - $p(-x)=p(x)$  Now my doubt is what to do next to find all possible polynomials or whether my approach is right or not .Any help would be appreciated.Thanks in advance.
Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$   Here is what I tried:  First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$.  Let $r$ be a root of $P$ (i.e. $P(r)=0$). It follows that all terms in the infinite sequence $r,r^2,r^4,\dots,r^{2n},\dots$ are roots of $P$. To avoid a polynomial with infinite roots, we have to have the sequence be periodic. So it turns out that either $r=0$, or the roots of $P$ are roots of unity of degree $2n$ for some $n\in\mathbb{N}$.  In the first case, we get $P(x)=xQ(x)$. Plugging in, we get $$x^2Q(x^2)=x^2Q(x)^2\implies Q(x^2)=Q(x)^2 ,$$ i.e. that $Q$ satisfies the same condition as $P$.  In the second case, we have $P(x)=(x^{2n}-1)Q(x)$. Plugging in, we get $(x^{2n}-1)Q(x^2)=(x^{2n}-2x^n+1)Q(x)^2$. I was not sure how to proceed from here.
Which configuration of armor enchantments gives maximum protection? Prior to 1.4.6 it was a well known fact that the protection enchantment on armor was strictly superior to the other enchantments and that simply having a few pieces with protection III/IV would max out this protection, leading to the best possible set of armor.  Now that this has , however, I'm curious to see if anyone has analysed what the optimal armor configurations now are?  Presumably it will be some combination of protection enchantments and other enchantments, going by the description of how enchantments work now on the wiki.  Or, are we forced to choose one or more types of protection to max out?  (This would be the case if there are not enough total enchantments possible to max out all areas).  If this is the case, it would be great to see some analysis of how to optimise that choice as well.  I've looked around but haven't found any good mathcrafting on this question yet.  Furthermore, beyond protection against taking damage, can someone include some explanation of the hidden utility bonuses?  For instance, Blast Protection has been described in wiki sources as reducing the knockback effects from explosions, and Fire Protection is described as reducing burn time.  Do these effects scale and max out at some enchantment level, and does Projectile Protection similarly reduce the knockback effect from projectiles? Is it acceptable to insult god in a user profile? Like always I hang around in Stack Overflow's chatrooms. Today, someone posted a link to his new "reloaded" user profile.  Along the lines I read in Arabic "god is a pig". So I asked him to remove it:     Me: I have a lot of respect towards you because of your knowledge but what you wrote there in Arabic is offensive. Could you remove it please?   OP: if it were in English, would it be equally offensive?   I will write it in Dutch because you ask it so kindly.   Me: yeah, I mean what's your goal? You're just seeding hatred   OP: It is an experiment.   After that, the user changed it to Dutch.  My question is: are such statements acceptable by Stack Exchange's standards? If not, should I flag one of his posts like stated ? If yes, could you state why?  My answers to possible questions:   This happened on SO, why don't you post it on MSO?   This happened on SO but I think it's applicable to all SE sites.  Why didn't you link the user or the chat conversation?   I want the user to remain anonymous unless he or she chooses to reveal him/her self. I will give the user a link to this question  Why do you care?   I find it offensive, not only to my religion but to any religion that believes in god. I posted this question to know what you guys and girls think about this. I live by the rules, so if it's acceptable by SE then so be it and I won't bother you again.  If you don't like it then move on.   I know that this is not a mosque/church/temple and that swearing  to a certain degree. But I don't see the point in insulting god, a religion or a race in a public profile. More over, it's not even built on arguments. What is the most effective way for float and double comparison? What's going on with "compact implies sequentially compact"? I've seen both counterexamples and proofs to &quot;compact implies sequentially compact&quot;, and I'm not sure what's going on. Apparently there are compact spaces which are not sequentially compact; quick googling and wikipedia checks will turn up examples floating around; they tend to be variants of $[0,1]^{[0,1]}$ with the product topology.  Here's a demonstration(?):  $[0,1]^{[0,1]}$ is compact by Tychonoff's theorem.  So, we demonstrate failure of sequential compactness. Choose a unique binary representation for each $x\in [0,1]$. For each $n\in\mathbb{N}$, let $f_n : [0,1]\to[0,1]$ (an element of $[0,1]^{[0,1]}$) be the function which maps each $x$ to its $n$-th place digit in its binary expansion. Let $f_{n_k}$ be a subsequence of this sequence. Let $x'\in [0,1]$ be such that the $n_{2m}$-th digit is $0$ and the $n_{2m+1}$-th digit is $1$, for all $m\in\mathbb{N}$. Then $f_{n_k} (x')$ does not converge (it alternates between $0$ and $1$), and hence $f_{n_k}$ cannot converge. Thus we have found a sequence in $[0,1]^{[0,1]}$ without any convergent subsequence and so it is not sequentially compact.  (Aside: this is apparently based off of an example in Steen's Counterexamples in Topology, according to  ) Nevertheless, there's also some proofs(?) floating around that compactness of a space implies sequential compactness, along these lines (this proof(?) is of the contrapositive):  Suppose $X$ is not sequentially compact. By definition, this means there is some sequence $(x_n)$ over $X$ with no convergent subsequence. If any $x\in X$ had for each of its neighborhood $U$ infinitely many $n$ for which $x_n \in U$, then we could define a convergent subsequence of $(x_n)$, contradicting our assumption.  (Presumably this is done by choosing for each neighborhood a sufficiently-large-indexed term in that neighborhood.) Thus, for each $x \in X$ we can select an open set $U_x$ such that $x\in U_x$ but with $x_n \in U_x$ for only finitely many $n$. The collection $\mathcal{U}=\{U_x : x\in X\}$ is clearly an open cover of $X$. If $\mathcal{U}$ had a finite subcover $\{U_1, \dots, U_k\}$ then the union $U_1 \cup \cdots \cup U_k$ would contain all of $X$ but only contain $x_n$ for finitely many $n$, which is impossible. Thus $X$ is not compact, since we have found an open cover without a finite subcover.  (Aside: this proof is essentially the same as the proof appearing in Rudin's Principles of Mathematical Analysis for Thm 2.37 that infinite subsets of compact spaces have limit points.) So, what's going on here?  It can't be that what both the counterexample and the proof are telling us is correct.  Is there some subtle (or more embarassingly for me -- glaring) flaw in the proof? Heisenberg famously derived his uncertainty principle by considering the disturbance that a measurement would have on a small enough system.  Of course in the mathematical formalism of Quantum Mechanics the relationship is derived from more basic principles.  How does String Theory account for it? Heuristically, and mathematically? Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups How does the simulation hypothesis relate to the ethics of simulated violence? Say, for the sake of this question,  is real. Are we then hurting "persons"/conscious beings in "our simulations" - video games - when we fight them? Why (not)? If so, is this unethical? Think for example about shooting an alien in the game No Man's Sky.  Edit/answer on the comment of Joseph Weissman: My reasoning is like this:   It is unethical to harm a person (with a person being everything that has ). By the simulation hypothesis (assuming it's true), everything around us (everything that we know of), is simulated. By 1 and 2, all persons we know of are simulated. So it's unethical to harm the simulated persons around us. A game is a simulation just like we are in a simulation, so the game has simulated persons. Therefore -  - it's unethical to harm the persons in a video game.   I know that the premises - besides the simulation hypothesis itself - are debatable, but that's the reason I asked the question in the first place.  I was also thinking that the characters in the video game differ because they have far less to no intelligence, that much of it is scripted. But I neither have a proof that we ourselves aren't scripted, with free will being an illusion. Also, how intelligent does a character in a simulation/video game has to be to have personhood then? I think this is a case of the .  Just to be sure, please note that this question is not about whether the simulation hypothesis is true or how likely it is to be true. It is about (one of the) consequences if the simulation hypothesis is true. It's also not a question about violence in games in general. To me, using Present Perfect form means the event can occur again. So, saying      someone has died   may not be grammatically correct.  Also, I noticed (it might be just coincidence):     passed away   is used more often than     has passed away   Is using Present Perfect correct here? Question Close Updates This week we rolled out a series of changes to  of the question close workflow.  You can read more about the details of these changes .    For context, the overall effort to improve the question close experience will be optimized for Stack Overflow. Given Stack Overflow’s scale, many of the pain points we hear about with question closing and reopening are unique to Stack Overflow and won’t make sense for other network sites. However some  changes are general improvements that should improve usability and guidance overall. Wherever possible, we plan to roll these changes out network wide as well. Any time we roll out a change network wide, we’ll make sure to cross-post here. This first phase is being rolled out to the network and is why I’m posting today.  Let us know if you run into any issues or have questions regarding the network here. Is there a way to offset an edge? (extrude edge loop perpendicular to each edge) I recently figured out how to do it with faces, but with edges, I can't seem to get it to work. I have a complex flat plane and i need to extend an edge loop. 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My styleSheet is referred in VF page as below  apex:stylesheet value="{!URLFOR($Resource.SLDS221,      'assets/styles/salesforce-lightning-design-system-vf.min.css')}"   Page works absolutely fine if it is not rendered as PDF.  Any suggestions? I have a question about making an ending symbol.  It is a LaTex-default that every proof environment has an ending symbol. I want to make every theorem-alike environment such as the example environment have also an ending symbol. But how to manually set this up? Specifically, I want to make all these environments ended by a square symbol.  \newtheorem{thm}{Theorem} \newtheorem{lem}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \newtheorem{pro}[thm]{Proposition} \newtheorem{assump}{Assumption} \newtheorem{axiom}{Axiom} \newtheorem*{conj}{Conjecture} \theoremstyle{definition} \newtheorem{defi}[thm]{Definition} \newtheorem{exam}{Example} \newtheorem{exer}{Exercise} \newtheorem*{rem}{Remark} \newtheorem*{notation}{Notation} \newtheorem*{note}{Note}   I am looking for, with gratitude, a comprehensive setup. How do you photograph rain? What are recommended settings for my Nikon Coolpix P1 to best portray rain?  Here's what I got. No retouching was applied.   The picture I took when the rain was heavy with the automatic settings:  The picture I took with the "Sports" program; by then, however, the rain was not as hard. Ugh, ugly gray strip at the bottom, that was the flat surface I put my camera on to get a stable picture.   My main problem is that the actual rain and hailstorm is very hard to see in the picture. The overall effect is, instead, of fog.    Follow-up from the thread, courtesy of more rain, now with less hailstorm and more thunder.  The feedback amounted to flash and faster shutter. I took a few photos and I'm having some trouble to reconstruct what I did to get each result. Two representative results:   Fast shutter, with some guest starring:I'm talking about that horrible building obviously. Faster shutter and flash:Kinda difficult to say those pictures were taken sixty seconds of each other. :)   I'm not sure, however, that either of this is really an improvement of the original results. Left pedal coming loose On my bike, which I just built out of spare parts, the left pedal seems to constantly be coming loose. It seems to come out about 5m every time I ride ~10km.  It's an old cotter pin crank, and the pedal is a left pedal (there's an L stamped into the end of the screw that goes into the crank). The pedal has a crack in one side of the axle compartment, but I don't think that would affect it.  I thought that these pedals had left-hand threads so that the action of pedalling would work to tighten the screw? If so, why would this be happening, and is there anything I can do about it short of replacing the crank and pedal? Auto-fit TextView for Android How map multiple inputText to an array property? Why does this identity hold for Fejér Kernels? What is the Physical Meaning of Commutation of Two Operators? I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable (self-adjoint operator) in quantum mechanics?   E.g. an operator $A$ with the Hamiltonian $H$? I have a script that does a number of different things, most of which do not require any special privileges. However, one specific section, which I have contained within a function, needs root privileges.  I don't wish to require the entire script to run as root, and I want to be able to call this function, with root privileges, from within the script. Prompting for a password if necessary isn't an issue since it is mostly interactive anyway. However, when I try to use sudo functionx, I get:  sudo: functionx: command not found   As I expected, export didn't make a difference. I'd like to be able to execute the function directly in the script rather than breaking it out and executing it as a separate script for a number of reasons.  Is there some way I can make my function "visible" to sudo without extracting it, finding the appropriate directory, and then executing it as a stand-alone script?  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