What's the difference between `=` and `<-` in R?
What are the differences between "=" and "<-" assignment operators in R?
I just visited Ask Ubuntu (askubuntu.com) while logged into Stack Exchange, and noticed that "join this community" doesn't appear:    It is similarly broken on Math Overflow (mathoverflow.net). It works, however, on all sites that are subdomains of stackexchange.com, like Mathematics (math.stackexchange.com) and Academia.  I'm using Google Chrome version 46.0.2490.86 on Windows 8.1 Update 1.  (I don't know if this can actually be fixed, given how cookie security works, but I figured Stack Exchange should be aware of the issue.) How can I shorten my command line (bash) prompt? Currently it is:   michael@Castle2012-Ubuntu-laptop01:~/Dropnot/webs/rails_v3/linker/spec/controllers$   Outside of renaming my machine and directory structure...  How could I make it be something more like:  michael:controllers$ Which OpenType Math fonts are available? What OpenType math fonts are there available for usage in TeX/LaTeX, e.g. with the unicode-math package?  Here is a MWE which you can use to show off the font.  \documentclass{article} \pagestyle{empty}  \usepackage{amsmath} \DeclareMathOperator\Res{Res} \newcommand*\diff{\mathop{}\!\mathup{d}}  \usepackage{amsthm} \newtheorem{theorem}{Theorem}  \usepackage{unicode-math}  %%% % Set up you text and math fonts %%%  \begin{document}  \begin{theorem}[Residue theorem]   Let $f$ be analytic in the region $G$ except for the isolated   singularities $a_1,a_2,\dots,a_m$. If $\gamma$ is a closed   rectifiable curve in $G$ which does not pass through any of the   points $a_k$ and if $\gamma\approx 0$ in $G$, then   \[     \frac{1}{2\pi i} \int\limits_\gamma f\Bigl(x^{\mathbf{N}\in\mathbb{C}^{N\times 10}}\Bigr)     = \sum_{k=1}^m n(\gamma;a_k)\Res(f;a_k)\,.   \] \end{theorem}  \begin{theorem}[Maximum modulus]   Let $G$ be a bounded open set in $\BbbC$ and suppose that $f$ is a   continuous function on $G^-$ which is analytic in $G$. Then   \[     \max\{\, |f(z)|:z\in G^- \,\} = \max\{\, |f(z)|:z\in \partial G \,\}\,.   \] \end{theorem}  First some large operators both in text: $\iiint\limits_{Q}f(x,y,z) \diff x \diff y \diff z$ and $\prod_{\gamma\in\Gamma_{\bar{C}}}\partial(\tilde{X}_\gamma)$;\ and also on display \[   \iiiint\limits_{Q}f(w,x,y,z) \diff w \diff x \diff y \diff z   \leq   \oint_{\partial Q} f'\Biggl(\max\Biggl\{   \frac{\Vert w\Vert}{\vert w^2+x^2\vert};   \frac{\Vert z\Vert}{\vert y^2+z^2\vert};   \frac{\Vert w\oplus z\Vert}{\vert x\oplus y\vert}   \Biggr\}\Biggr)\,. \]  \end{document}   Here is a similar MWE for ConTeXt.  The images in the answer will only be shown for LaTeX output but the ConTeXt output looks very similar.  %%% % Set up you text and math fonts %%%  \setupmathlabeltext[Res=Res] \definemathcommand[Res][nolop]{\mfunctionlabeltext{Res}} \definemathcommand[diff]{\mathop{}\!\mfunctionlabeltext{diff}}  \defineenumeration   [theorem]   [text={Theorem},    style=italic,    title=yes,    titlestyle=normal,    distance=0pt,    titleleft={(},    titleright={).~},    alternative=serried,    width=fit]  \starttext    \starttheorem[title={Residue theorem}]     Let $f$ be analytic in the region $G$ except for the isolated     singularities $a_1,a_2,\dots,a_m$. If $\gamma$ is a closed     rectifiable curve in $G$ which does not pass through any of the     points $a_k$ and if $\gamma\approx 0$ in $G$, then     \startformula       \frac{1}{2\pi i} \int\limits_\gamma f\Bigl(x^{{\mathbf N}\in\mathbb{C}^{N\times 10}}\Bigr)       = \sum_{k=1}^m n(\gamma;a_k)\Res(f;a_k)\,.     \stopformula   \stoptheorem    \starttheorem[title={Maximum modulus}]     Let $G$ be a bounded open set in $\mathbb{C}$ and suppose that $f$ is a     continuous function on $G^-$ which is analytic in $G$. Then     \startformula       \max\{\, |f(z)|:z\in G^- \,\} = \max\{\, |f(z)|:z\in \partial G \,\}\,.     \stopformula   \stoptheorem    First some large operators both in text:   $\iiint\limits_{Q}f(x,y,z) \diff x \diff y \diff z$   and   $\prod_{\gamma\in\Gamma_{\bar{C}}}\partial(\tilde{X}_\gamma)$;\   and also on display   \startformula     \iiiint\limits_{Q}f(w,x,y,z) \diff w \diff x \diff y \diff z     \leq     \oint_{\partial Q} f'\Biggl(\max\Biggl\{     \frac{\Vert w\Vert}{\vert w^2+x^2\vert};     \frac{\Vert z\Vert}{\vert y^2+z^2\vert};     \frac{\Vert w\oplus z\Vert}{\vert x\oplus y\vert}     \Biggr\}\Biggr)\,.   \stopformula  \stoptext Cardinal of quotient rings of gaussian integers. It is known that $\mathbb{Z}[i]$ is a PID and that $\mathbb{Z}[i]/(a+bi)\mathbb{Z}[i]$ is finite for all $(a,b) \in \mathbb{Z}^2\backslash \{(0,0)\}$.  My question :      Is there any result on the cardinal of  $\mathbb{Z}[i]/(a+bi)\mathbb{Z}[i]$ ? I need a non trivial  examples of group isomorphism that maps $G\to H$ such that   $G/H \cong G$. Show that $B$ is diagonalizable if If $AB=BA$ and $A$ has distinct real eigenvalues We were asked to prove the following:  Let $ A $ be an $n \times n$ matrix with $n$ distinct real eigenvalues. If $AB=BA$, show that $B$ is diagonalizable.   It was suggested I show that an eigenvector of $A$ is also an eigenvector of $B$. I am both having trouble doing this and failing to see how I would complete the proof after. Any help would be appreciated.  Thanks I have a column in an Excel spreadsheet with several hundred entries, and some of these entries are duplicated.  I would like to remove all duplicate entries from the column so that each entry remaining is unique.   Is there a way to accomplish this automatically? What is the motivation for including the compactness and semi-simplicity assumptions on the groups that one gauges to obtain Yang-Mills theories?  I'd think that these hypotheses lead to physically "nice" theories in some way, but I've never, even from a computational perspective. really given these assumptions much thought. How to get the file name from a full path using JavaScript? It is possible to press ⎇ Backspace on macOS in order to delete the previous word (left of the cursor), and ⌘ Backspace to delete the previous line (left of the cursor). If you have an external keyboard with a "delete" key (or the "forward delete key"), or are able to emulate the delete key via software, it is possible to press ⎇ Delete to delete the word to the right of the cursor. However, perhaps surprisingly, it is not possible to press ⌘ Delete in order to delete the line to the right of the cursor.  Now, I have only been able to make this possible by emulating a ⌘⇧→ and then a Backspace with BetterTouchTool. However, I want to avoid doing something like this because BTT does not work at a "low enough level" for me – that is to say, the action is not entirely reliable.  Is there any official macOS way to delete the line to the right of the cursor? Prove that if W is a subspace of a finite dimensional vector space V, then dim(W) ≤ dim(V). My attempt at a solution:  Proof  Given a vector space $V$, let $S={v_1,...,v_n}$ be a basis for $V$. Consequently $\dim(V)=n$. Now given $W$, a subspace of $V$, it will be shown that $\dim(W)\le \dim V$   First, consider the case where $W=${0}. Then $dim(W)=0 \le\ n$,   Subsequently, consider the case for $W=V$. Then it follows that $dim(W)=dim(V)=n$.  Finally, consider the case where $W \subset V$ and without loss of generality assume $v_n \notin W$. Let $T={v_1,...,v_k}$ where $k\lt n$ be a basis for the subspace W. Now, since S is a basis for V, for every $v_i \in S$,  $c_1v_1+...+c_nv_n=0$ has only the trivial solution, thus none of the vectors in S can be expressed as linear combinations of the other vectors in S. It follows that the set T is not a basis for V as $v_n$ cannot be generated from T. The $dim(W)$ is given by the number of vectors in the basis. Therefore, $dim(W)=k \lt n$  Therefore, it has been shown that if $W \subseteq V$ then $dim(W) \le dim(V)$ What evidence is there supporting an alternative parentage for Jon Snow? I've heard speculation that Jon Snow's actual parents may be Rhaegar and Lyanna Stark, but in reading the books I'm not sure what evidence could be used to support this conclusion.  I thought there might have been something in the conversation Davos had before being smuggled into White Harbor, but I couldn't stretch anything that far.  What evidence do proponents of this theory cite?  Does any of it involve stuff in the texts or just sweeping generalizations of the characters' personalities (e.g. "Ned Stark is too honorable to father a bastard"). Do the following negations mean the same thing? I don't think you understood me. / I think you misunderstood me.    Do these senteces mean the same thing? If not, what's the difference?  Edit: I just realized that I asked something different from what I originally wanted. So I am also interested in the following comparison:     I don't think you understand me. / I think you don't understand me. How do you run as a different user from the Start Menu in Windows 10? Running an application as a different user (e.g. domain admin account) from the start menu (by shift + right-clicking the application) used to be an option in Windows 7 &amp; XP.  However, I can't find that option in Windows 10. The workaround seems to be either 1) to find the application in Windows Explorer (shift + right-click) or 2) use runas.exe from the command line.  However, in order to use those workarounds, I have look up the executable name first. It's a bit difficult because I don't have the name of every RSAT tool or executable name memorized.  (e.g. "Active Directory Users and Computers" is dsa.msc, "Routing and Remote Acces" is rrasmgmt.msc)  Is there a simpler way to do this? Does a size large or larger creature killed by Negative Energy Flood rise as a medium sized zombie? How can I suspend/hibernate from command line? How can I suspend or hibernate my laptop using command line, without installing additional software? Chat messages should use consistent icon in iOS app It'd be nice if the chat notifications in the feed used the default Stack Exchange icon like the ones in the inbox do*:     Since items in the feed are grouped so that multiple items from the same site , the site icon helps delimit grouped items a bit less subtlely than the grey divider lines, and chat replies' lack of one causes them to sort of blend in to the previous group.  *Ideally it'd use the icon of the site the room was associated with, but I don't think that information is readily available In Star Trek TOS, we see the Defiant disappear to unknown parts  ('The Tholian Web'), and Star Trek: Enterprise revealed that the Defiant went back in time to the mirror universe ('In a Mirror Darkly', parts 1 &amp; 2).   Because of this, one might wonder why (in-universe) the mirror universe didn't obtain (and maintain) technological superiority over the main timeline. (Granted, the out of universe answer is that TOS depicted the two universes as having essentially the same level of tech)  So does any work (whether STO, or a pocket books work) address why the mirror universe didn't simply leap frog the main universe's technology?   (Or to put it another way, why did the mirror universe's technology appear to stagnate for a century after obtaining access to the Defiant's advanced technology?) Comparing mixed-effects and fixed-effects models (testing significance of random effects) Given three variables, y and x, which are positive continuous, and z, which is categorical, I have two candidate models given by:  fit.me &lt;- lmer( y ~ 1 + x + ( 1 + x | factor(z) ) )   and  fit.fe &lt;- lm( y ~ 1 + x )   I hope to compare these models to determine which model is more appropriate. It seems to me that in some sense fit.fe is nested within fit.me. Typically, when this general scenario holds, a chi-squared test can be performed. In R, we can perform this test with the following command,  anova(fit.fe,fit.me)   When both models contain random-effects (generated by lmer from the lme4 package), the anova() command works fine. Owing to boundary parameters, it is normally advisable to test the resulting Chi-Square statistic via simulation, nonetheless, we can still use the statistic in the simulation procedure.  When both models contain only fixed-effects, this approach---and, the associated anova() command---work fine.  However, when one model contains random effects and the reduced model contains only fixed-effects, as in the above scenario, the anova() command doesn't work.  More specifically, I get the following error:   &gt; anova(fit.fe, fit.me)  Error: $ operator not defined for this S4 class   Is there anything wrong with using the Chi-Square approach from above (with simulation)? Or is this simply a problem of anova() not knowing how to deal with linear models generated by different functions?  In other words, would it be appropriate to manually generate the Chi-Square statistic derived from the models? If so, what are the appropriate degrees of freedom for comparing these models? By my reckoning:  $$ F = \frac{\left((SSE_{reduced}-SSE_{full})/(p-k)\right)}{\left((SSE_{full})/(n-p-1)\right)} \sim F_{p-k,n-p-1} $$  We are estimating two parameters in the fixed effects model (slope and intercept) and two more parameters (variance parameters for the random slope and random intercept) in the mixed-effects model. Typically, the intercept parameter isn't counted in the degrees of freedom computation, so that implies that $k=1$ and $p=k+2=3$; having said that I'm not sure if the variance parameters for the random-effects parameters should be included in the degrees of freedom computation; the variance estimates for fixed-effect parameters are not considered, but I believe that to be because the parameter estimates for fixed effects are assumed to be unknown constants whilst they are considered to be unknowable random variables for mixed effects. I would appreciate some assistance on this issue.  Finally, does anybody have a more appropriate (R-based) solution to comparing these models? What is a complete list of the usual assumptions for linear regression? What are the usual assumptions for linear regression?   Do they include:   a linear relationship between the independent and dependent variable independent errors normal distribution of errors homoscedasticity   Are there any others?