Find all polynomials $p(x)\in\mathbb{C}[x]$ satisfying $p(x)p(-x)=p(x^2)$.  We can see that if $x_0$ is a root of $p$, then so is ${x_0}^2$. If $0&lt;|x_0|&lt;1$ (or $|x_0|&gt;1$), then we have $|x_0|^2&lt;|x_0|$ (or $|x_0|^2&gt;|x_0|$). So repeating this process will give an infinite number of distinct roots, a contradiction. Hence any root $x_0$ of $p$ must have $|x_0|=0$ or $|x_0|=1$. Experimenting with lower degree polynomials, we find solutions: $p(x)=1, 0$ $p(x)=-x, 1-x$ $p(x)=x^2, -x(1-x), (1-x)^2, x^2+x+1$. Further, we can verify that the general form $f(x)=(-x)^p(1-x)^q(x^2+x+1)^r$ will work. I am unsure if these capture all possible solutions, and if so how do we show it? I've been able to show that any root $x_0=e^{i\theta}\neq 1$ must satisfy $\theta=\frac{2^n p}{q}\pi$, where $m\geq 1$ and $p,q$ are coprime integers.
Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$ ?  I have noticed that if $a_n$ is the leading coefficient of $p(x)$ then $a_n=(-1)^n $, so the leading coefficient cannot be imaginary; also $p(0)^2=p(0)$ so the constant term also is real. Please help. Thanks in advance.
Person, Persons, People, Peoples Calculate $E(X^{2n})$ where $X$ is normal (0,1) I need help proving the following:  Let $X$ be normally distributed with parameters $\sigma=0$ and $\mu=1$. Let $n$ be a positive integer. Show that:  $$E(X^{2n})=\frac{(2n)!}{2^nn!}=:(2n-1)!!$$  I've tried the change of variables $Y=X^2$ and then calculating the integral, but got nowhere. Any hints?  Thanks in advance! I want to make a long shadow just like in this image.  I can create this shadow with shapes and by deleting the unwanted shadows after rasterizing the layer. But it's not very reusable. I can't resize the layer after rasterizing it.  Is there any easy way to make a long shadow like in the image below? Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$. Add a dependency in Maven Best way to remove an event handler in jQuery? I have an input type="image". This acts like the cell notes in Microsoft Excel. If someone enters a number into the text box that this input-image is paired with, I setup an event handler for the input-image. Then when the user clicks the image, they get a little popup to add some notes to the data.  My problem is that when a user enters a zero into the text box, I need to disable the input-image's event handler. I have tried the following, but to no avail.  $('#myimage').click(function { return false; }); What are the summary rules for when to use image stabilization and when not to? When I got my DSLR, I took a lot of pics and all of them came excellent, even though I had VR at the "Off" position.   Since those came out so well, I wonder when should I turn VR on, and when I should just leave it off.   I found  which answer this question in detail, but that's too much for me to understand easily. Can someone explain the strict rules in short?  Should I use VR with tripod? When should I use it when I'm not using a tripod, and when should I leave it off? Assume that $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$ and $f$ is positive on the interval $(0,1)$ and $0$ at the endpoints.    I want to prove that $$\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx &gt;4. $$  I could not make any progress. Since the unit interval is compact, we know that the continuity of $f, f'$ and $f''$ are uniform and they attain their maximum and minimum values. Moreover, I tried to apply the Mean Value Theorem but I failed. Height and minimal number of generators of an ideal 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 find the most common name in passwd file font set problems in beamer i want to set the main font in my beamer as "Consolas" i do like this :  \documentclass{beamer} \usepackage{fontspec} \usetheme{Warsaw} \begin{document} \setmainfont{Consolas} \begin{frame} we discuss the ambiguities of existing security definitions and then provide a general framework for defining AKE security when involve strong adversary capabilities \end{frame} \end{document}   but after compile it , the font doesn't change, it is the same with that before the main font was set. what's the reason? Is there a difference between single and double quotes in Java? Should I load an entire html page with AJAX? Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d  $ roots.  Thoughts so far:  I feel like I might be missing something here. If $ R $ is an integral domain, then so is $ R[X] $. The factor theorem gives a correspondence between roots and factors. Clearly (is this clear?) a polynomial of degree $ d $ can't have more than $ d $ irreducible factors. /etc/shadow date of last password change—UTC or local time? man 5 shadow gives the following description of the third field in each line:  The date of the last password change, expressed as the number of days since Jan 1, 1970. The value 0 has a special meaning, which is that the user should change her pasword the next time she will log in the system. An empty field means that password aging features are disabled.  Does this refer to times in UTC or the local timezone? In particular, if I want to calculate comparable numbers using something like $(( $(date +%s) / 86400 )), do I need to pass date the -u option? Integral of $\int \frac{dx}{\sqrt{x^2 -9}}$ What is an intuitive explanation of the Expectation Maximization technique? Avoiding errors due to unexpanded asterisk In bash I often use for-loops such as the following  for file in *.type; do    sommecommand "$file";  done;   to perform an operation for all files matching *.type. If no file with this ending is found in the working directories the asterisk is not expanded and usually I will get an error message saying that somecommand didn't find the file. I can immediately think of several ways to avoid this error. But adding a conditional does not seem to be very elegant. Is there a short and clean way to achieve this? Evaluate definite interval of normal distribution I know that an easy to handle formula for the CDF of a normal distribution is somewhat missing, due to the complicated error function in it.  However, I wonder if there is a a nice formula for $N(c_{-} \leq x &lt; c_{+}| \mu, \sigma^2)$. Or what the "state of the art" approximation for this problem might be. How many passes do you need to wipe/shred your files to make them not undeletable?