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Side effects of changing blackboard bold font with \setmathfont
Find all polynomials $P$ such that $P(x^2+1)=P(x)^2+1$ Find all polynomials $P$ such that   $$P(x^2+1)=P(x)^2+1$$ Bundling python library with addon what is the relationship between the dynamical casimir effect and virtual particles? Annoying Viber notification How can I download an older version of Xcode? I am trying to download Xcode from the App Store, but it says that I have to have a newer version of macOS. But I don't have enough space on my computer to upgrade. Its like 8 gigs or something like that and if I only could download Xcode that was compatible with macOS Sierra that would be nice. How to split algorithm2e over two pages let $a,b,c$ be postive real numbers ,and such $$ab+bc+ac=3$$   show that   $$\dfrac{a^2}{a+2b^2}+\dfrac{b^2}{b+2c^2}+\dfrac{c^2}{c+2a^2}\ge 1$$   This problem is from Secrets In Inequalities volume 1  page 30,example 1.24. the comment.the  author this case is a bit more diffcult,But this author can't post solution.Thank you To make it simple, I want to highlight a group of tikz nodes like in the following picture    It was borrowed from the . I would like to know which Tikz commands I can use to do the job in order to automatize this task for several trees I have to make. It is important to keep the blending effect of colors like in node g. Here is the MWE also borrowed from :  \documentclass{article}  \usepackage{tikz} \usetikzlibrary{arrows,backgrounds,calc,trees}  \begin{document} \thispagestyle{empty}  \begin{tikzpicture} \node (f) {f}     child { node (g) {g}       child { node (a) {a}     }       child { node (b) {b}     }   }     child { node (h) {h}       child { node (c) {c}     }   }; \end{tikzpicture}  \end{document} elementary row operation change the eigenvalue of a matrix I know that elementary row operations may change the eigenvalues of a matrix, but if these operations don't change the determinant, why should change the eigenvalues??  Thanks How do we know what type of gauge field to add to a theory? I've been watching  and in one lecture, he discusses a very simple gauge theory.  We have a complex scalar field $\phi(x)$ with Lagrangian  $$\mathscr{L} = \partial_\mu \phi^* \partial^\mu \phi - V(\phi^* \phi)$$  and we want to do the gauge transformation $\phi(x) \to e^{i\theta(x)} \phi(x)$.  But the derivative term in the Lagrangian is not invariant if $\theta$ varies from place to place, so we add a new vector field $A_\mu$, and define it to transform like $$A_\mu \to A_\mu + \partial_\mu \theta$$ under the gauge transformation.  Then we change the Lagrangian to use covariant derivatives $D_\mu = \partial_\mu + iA_\mu$ instead of ordinary ones; now when the gauge transform is done all the derivatives of $\theta$ cancel out, and the result is invariant.  My question is: was there any freedom in choosing to add a vector field, with that specific gauge transformation law?  Are there any other ways we could get gauge invariance here—by using a different transformation law, or with a different type of field (say, a scalar, tensor, or spinor) with some other transformation law?  More generally, how can we tell what type of field is needed and what its gauge transformation law should be, to get some particular Lagrangian to be invariant under some particular gauge transformation? How do I easily switch between windows rather than applications in GNOME Shell? I want to learn more about hopf algebras but I am having trouble finding a down to earth introduction to the subject with lots of motivation and examples. My algebra knowledge ranges from Dummit and Foote, Atiyah-Macdonald commutative algebra, and Hartshorne level algebraic geometry.  What are good places to look at for a hopf algebra beginner? My main motivation is for studying groupoid schemes from an algebraic point of view. Browsing Questions shows the same questions on different pages I have just been looking through the questions to answer those that I can and as I go to the next page (using the Next button at the bottom) I am presented with the same questions.  I imagine this is not a flaw and in fact questions are coming in fast filling up a page pushing my current page to the next one.  An idea to solve would be to set a variable for starting id when I start paging and only clicking an update button will reset the starting id. After my lie down, I posted  on SO. As I suspected would happen, it got moved to programmers. Unfortunately, the kind of people I'm interested in getting answers from (litb,skeet et al - the top SO programmers in their respective disciplines) don't frequent Programmers much. So my question is, where to ask somewhat non-specific questions of high-powered users? Indefinite integral $\int(\sin x)^2/(x\cdot\cos x-\sin x)^2\,dx$ How to find the antiderivative of $\dfrac{(\sin x)^2}{(x\cdot\cos x-\sin x)^2}$? I know that series theory can solve the problem. But I wish to solve the problem by the antiderivative way. The numerator and denominator divide the $x^2$ at the same time, don't they? shortcut key to detach a tab in firefox (move tab to new window) In FF, I can move a tab to a new window by either dragging the tab out of the browser, or right clicking the "move to New window" option from the context menu. is there a shortcut key to move the current tab to a new window without dragging or using the context menu, or a way to set one up?  Often, I like to look at 2 tabs side by side, so I drag one out, then hit WINDOWS KEY+→ to put the detached tab on the right, then ALT+TAB to tab back to the main window, then WINDOWS KEY+← to put the main window on the left. It would be great if I could do the whole process on the keyboard.  I checked  documentation and did not find any shortcut key for detaching a tab: Show all inbox items (notifications) in app The SE app currently only shows some inbox items:   Answers Comments Chats   When I see those I assume I'm all caught up, but it turns out there are more on the web which aren't displayed in the app:   Bounty expiration Elections Edits on your own posts Maybe others?   I was about to tag this  but realized it's probably intentional.  I imagine the reason for not showing them is because they're not something you can act on from within the app. Bounties can't be offered or awarded1 and elections can't be voted on.  But then again chats can't be replied to either. Badge descriptions can't be viewed (on the Achievements side). Selecting those launches the browser. Why then shouldn't bounties or elections launch the browser too? Hiding them just means I'll miss it if I haven't visited in browser recently.  1 Though accepting answers does award the bounty Last night, I was using Chrome just nicely, but this morning, I can't even start Google Chrome as it immediately crashes. I have already tried a clean uninstall and reinstall of Google Chrome but still the same error pops up (it does not happen while I'm opening Google Chrome. I woke up and only then realized I can't open my Chrome and it keeps popping up saying that Google Chrome has stopped working.)  OS : Win 7 64-bit  Here is the detail of the crashes  Problem signature:  Problem Event Name:    APPCRASH   Application Name:    chrome.exe    Application Version:    35.0.1916.114 Application Timestamp:    53726019    Fault Module Name:    chrome.dll   Fault Module Version:    35.0.1916.114   Fault Module Timestamp:    53725d18    Exception Code:    c0000005    Exception Offset:    00728bc8      OS Version:    6.1.7600.2.0.0.768.3    Locale ID:    17417    Additional Information 1:    0a9e    Additional Information 2:    0a9e372d3b4ad19135b953a78882e789    Additional Information 3:    0a9e    Additional Information 4:    0a9e372d3b4ad19135b953a78882e789     Any help that will solve this problems is greatly appreciated. Subgroups of finite abelian groups. For every subgroup $H$ of a finite abelian group $G,$ there exists a subgroup $N$ of $G$ such that $G/N \cong H.$ I need to prove this or give a counter example.  I am aware of isomorphism theorems and classification of abelian groups, direct products etc. Sanity check question about correlation between log-normal random variables Recently I came across the following result in a paper, which I can't seem to find right.  Anyway the authors had cited  by Johnson &amp; Kotz, page 20 as the source. Since I do not have access to this book I am unable to check on the derivation of this result. I was hoping someone could shed some light on how to prove this.  Let $X$ and $Y$ denote a pair of bivariate log normal random variables with their means and variances given by the ordered pairs $(\zeta_{X}, \eta_{X})$ and $(\zeta_{Y}, \eta_{Y})$ and with correlation coefficient $\rho_{(X, Y)}$, derived from the bivariate normal with marginal distributions having parameters $(\xi_{X}, \nu_{X}^2)$ and $(\xi_{Y}, \nu_{Y}^2)$ and correlation coefficient $\rho_{N}$.  The expression for $\rho_{(X, Y)}$ is given as \begin{equation} \rho_{(X, Y)} = \dfrac{\mathrm e^{(\rho_{N} \nu_{X} \nu_{Y})} - 1}{\sqrt {(\mathrm e^{\nu_{X}^2} - 1)(\mathrm e^{\nu_{Y}^2} - 1)}} \end{equation}  The below is a manipulation I came up with, any error in that would be completely mine.  Upon rearranging we get \begin{equation}\begin{aligned} \rho_N &amp;= \dfrac{\ln (1 + \rho_{(X, Y)} \sqrt {(\mathrm e^{\nu_{X}^2} - 1)(\mathrm e^{\nu_{Y}^2} - 1)})}{\nu_{X}\nu_{Y}}\\ &amp;= \dfrac{\ln (1 + \rho_{(X, Y)} \frac{\eta_{X}\eta_{Y}}{\zeta{X}\zeta{Y}})}{\sqrt {\ln ((\frac{\eta_{X}}{\zeta_{X}})^2 + 1)\ln ((\frac{\eta_{Y}}{\zeta_{Y}})^2 + 1)}}. \end{aligned} \end{equation}  I am very interested in learning if this result would hold true between $X$ and $Y$'s correlations if we had a quad-variate situation as in there were 2 more random variables which are also correlated with $X$ and $Y$ would this result still be true ? I suspect it should be, but I am newbie and due to lack of a credible source I was wondering if some one would be kind to confirm it.