This particular statement was one of the options in a question which was to be chosen if Right and I was unable to think how to reason it . So  I am asking here.  If G is cyclic then Aut(G) is cyclic or not?  CAn you please tell how to prove it or Give some counteexample if it can't be proved. Thanks!!
A question I'm struggling with is to prove or disprove that the automorphism group of a finite cyclic group must be cyclic. I think the statement is false but have not been able to come up with a counterexample.  Thank you for any assistance you can provide.
Is it acceptable to nest parentheses? Is it acceptable to nest parentheses (for example, if I (meaning myself) write like this)? Does slowing light violate relativity? What does it mean that GUIkit is being deprecated? How to find chromatic number of the $n$-dimensional hypercube $Q_n$? I haven't used regular expressions at all, so I'm having difficulty troubleshooting. I want the regex to match only when the contained string is all numbers; but with the two examples below it is matching a string that contains all numbers plus an equals sign like "1234=4321". I'm sure there's a way to change this behavior, but as I said, I've never really done much with regular expressions.  string compare = "1234=4321"; Regex regex = new Regex(@"[\d]");  if (regex.IsMatch(compare)) {      //true }  regex = new Regex("[0-9]");  if (regex.IsMatch(compare)) {      //true }   In case it matters, I'm using C# and .NET2.0. How do I make environment variable changes stick in Python? From what I've read, any changes to the environment variables in a Python instance are only available within that instance, and disappear once the instance is closed.  Is there any way to make them stick by committing them to the system?  The reason I need to do this is because at the studio where I work, tools like Maya rely heavily on environment variables to configure paths across multiple platforms.  My test code is  import os os.environ['FAKE'] = 'C:\\'   Opening another instance of Python and requesting os.environ['FAKE'] yields a KeyError.  NOTE: Portability will be an issue, but the small API I'm writing will be able to check OS version and trigger different commands if necessary.  That said, I've gone the route of using the Windows registry technique and will simply write alternative methods that will call shell scripts on other platforms as they become requirements. Use Listing in newenvironment I am trying to do a new environment or command (i tried both) where i can put in the name, description and code of a Method in c++ and it gives me a new section with the MethodName, the code listed and the description below. But i am not able to use the listings environment in a new command or new environment.. Consider my MWE   \documentclass[a4paper,11pt]{article} \usepackage{listings}  \newcommand{\method}[3] { \section{#1}\label{#1} \begin{lstlisting} #2 \end{lstlisting} #3 }   \begin{document}  \method{MethodName}{Method Void MethodName()}{Description of the Method blablabla}  \end{document}   How can I fix this?   I would prefer to do it using a macro since I don't need an ending. Let random vector $x = (x_1,...,x_n)$ follow multivariate normal distribution with mean $m$ and covariance matrix $S$. If $S$ is symmetric and positive definite (which is the usual case) then one can generate random samples from $x$ by first sampling indepently $r_1,...,r_n$ from standard normal and then using formula $m + Lr$, where $L$ is the Cholesky lower factor so that $S=LL^T$ and $r = (r_1,...,r_n)^T$.  What about if one wants samples from singular Gaussian i.e. $S$ is still symmetric but not more positive definite (only positive semi-definite). We can assume also that the variances (diagonal elements of $S$) are strictly positive. Then some elements of $x$ must have linear relationship and the distribution actually lies in lower dimensional space with dimension $&lt;n$, right?  It is obvious that if e.g. $n=2, m = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, S = \begin{bmatrix} 1 &amp; 1 \\ 1 &amp; 1\end{bmatrix}$ then one can generate $x_1 \sim N(0,1)$ and set $x_2=x_1$ since they are fully correlated. However, is there any good methods for generating samples for general case $n&gt;2$? I guess one needs first to be able identify the lower dimensional subspace, then move to that space where one will have valid covariance matrix, then sample from it and finally deduce the values for the linearly dependent variables from this lower-dimensional sample. But what is the best way to that in practice? Can someone point me to books or articles that deal with the topic; I could not find one. If perimeter of a triangle is $2d$, what is the length of sides so the triangle has maximal area?    I found some solution using circle and angles, but I think I have to use derivatives.    I need help. Frequency of the sound when blowing in a bottle I'm sure you have tried sometime to make a sound by blowing in an empty bottle. Of course, the tone/frequency of the sound modifies if the bottle changes its shape, volume, etc.   I am interested in understanding how the tone/frequency of this sound can be (approximately) calculated from the shape of the bottle. Is there some formula?  If you have any references about this please post them in your answers.  Being a mathematician I keep hearing that the principal tone is given by the first eigenvalue of the Laplace operator, and I'd like to understand the physical explanation. Making Ctrl+C copy text in gnome-terminal? Is it possible for me to make Ctrl+C perform a copy command if there is highlighted text in the terminal? Otherwise, it should retain its normal behavior.  (If there is a terminal that can do this other than gnome-terminal, that's probably fine too.) Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers. Given an example of a non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i, \forall a, b \in G$ for two consecutive integers.   This is question 5 from Herstein Page 35. I have proved that conclusion holds for three positive integers.Thanks. Method defined as testMethod do not support web service callouts, test skipped I have been struggling with this error for a few days now. I get this error every time I try to run a test for a method that has an httpCallout. I followed the architecture from  , but to no avail. I thought it could be something I was messing up in my code, so I copied all three of the classes from the documentation, and I am still getting the error when I try to run the test. All of the classes are on Salesforce API 29. Has anyone run into this before? A standard proof for press needs crop marks and color bars as below    With the package crop, the crop marks can be easily added.  But how about the color bar ?  Is there a way or a package to make it ?  An example is   \documentclass[b5paper,10pt]{article} \usepackage[utf8]{inputenc} \usepackage[cross,a4,center,info,axes]{crop}  \title{Test} \author{User} \begin{document} \maketitle \section{Fox} The quick brown fox jumps over a lazy dog. \end{document} Is it true that for algebraic sets $V,W$ we have $I(V \times W ) =I(V) + I(W)$? This is a follow up question to my previous question  Let $k$ be a field and $V \subseteq \Bbb{A}^n$ and $W \subseteq \Bbb{A}^m$ be algebraic sets. Then it should be true that $I(V \times W ) = I(V) + I(W)$ where by $I(V)$ here we mean the extension of $I(V)k[x_1,\ldots,x_{m+n}]$. Now I believe I have proven this (see the proof at the bottom of my question) but when I look at Martin's answer , it is instead claimed that we  actually have  $$I(V \times W )= \sqrt{I(V) + I(W)},$$  and for $I(V) + I(W)$ to be a radical ideal we need $k$ to be algebraically closed.           My question is: What's going on here? I believe my claim is true even without the assumption that $k$ is algebraically closed.           Here's a proof user Sanchez told me of, which I have simplified:       First it is clear that we always have $I(V) + I(W) \subseteq I(V \times W)$. For the reverse inclusion consider a polynomial $f \in I(V \times W)$. Then we can always write $$f = \sum_{i=1}^n f_ig_i$$ where $f_i \in k[x_1,\ldots,x_n]$ and $g_i \in k[x_{n+1},\ldots,x_{m+n}]$. Now take any $b' \in W$. If for all $i$ we have $g_i(b') = 0$ then since $b'$ is arbitrary, $g_i \in I(W)$ for all $i$. Then $f \in I(W) \subseteq I(V ) + I(w)$ and we are done. Otherwise suppose there exists $b \in W$  and $i$ such that $g_i(b) \neq 0$. Then wlog we may suppose that  $g_1(b) \neq 0$.   Next, $\sum f_ig_i(b) = 0 $ on all of $V$ by assumption of $f \in I(V \times W)$. So $\sum f_i g_i(b) = p$ for some $p \in I(V)$. Now write $$f_1 = \frac{ p - g_2(b) f_2 + \ldots g_n(b)f_n}{g_1(b)}.$$  Substituting this for $f_1$ in $\sum f_ig_i$, followed by taking things mod $I(V)$ we get an  expression with only $n-1$ terms $\mod{I(V)}$. Continuing this process we will finally get an expression with $0$ terms $\mod{I(V)}$ so that $f \in I(V)$. This shows $I(V \times W) \subseteq I(V) + I(W)$ which completes the proof. I have noticed that the confidence interval for predicted values in an linear regression tends to be narrow around the mean of the predictor and fat around the minimum and maximum values of the predictor. This can be seen in plots of these 4 linear regressions:    I initially thought this was because most values of the predictors were concentrated around the mean of the predictor. However, I then noticed that the narrow middle of the confidence interval would occur even if many values of were concentrated around the extremes of the predictor, as in the bottom left linear regression, which lots of values of the predictor are concentrated around the minimum of the predictor.  is anyone able to explain why confidence intervals for the predicted values in an linear regression tend to be narrow in the middle and fat at the extremes? How to calculate $\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$? Find the expected value of $\frac{1}{X+1}$ where $X$ is binomial The problem:  X is a binomial random variable, find $E[\frac{1}{X+1}]$ n and p are not given  PDF for a binomial distribution is $\binom{n}{k}p^k(1-p)^{n-k}$  Expected value is $\sum{x_ip(x_i)}$  But this is where I get stuck, I'm really rusty on my statistics and I'm not sure exactly how to structure it in the next step? I think I want to get the form of the following out of the summation  $\sum _{k=0}^{n} \binom{n}{k}p^k(1-p)^{n-k} = (p + 1 - p)^n = 1$  But I'm not sure if it should look like  $\sum \frac{1}{xp(x)+1} $  and if it should where to go from here? Why is it bad to log in as root? I've often come across posts on forums or other websites where you see people joking in such a manner about running/logging in as root as if it's something awful and everyone ought to know about it. However, there isn't much that a search reveals on the matter.   It may be widely known to Linux experts, but I really don't know why. I remember always running as root when I first tried Linux years ago (Redhat and Mandrake) and don't remember running into any problems because of that.   There are actually some distros that have a bright red background with alert signs all over it as wallpaper for the root user (SuSe?). I still use the "Administrator" account for regular use on my Windows installation and haven't ever run into any problems there either. Standard error of regression coefficient without raw data