I am not sure about my answer. I think that the cup of tea cools down by the following modes:   Radiation between the surface of the tea and the air molecules. Conduction between the tea and the cup itself and between the cup and the table or whatever the surface is.   Do you think it is correct?
For a while now I've noticed that if I take a cup of hot tea and pour it into two cups and leave it then both cups will cool faster than the single cup.  How is it that nature cools two cups of tea in parallel more quickly? It's still the same amount of tea but it gets cooled quicker if I have two cups.
Prove $\sqrt{2}$ is between $\dfrac{a}{b}$ and $\dfrac{a+2b}{a+b}$ Let $a$ and $b$ be positive integers. Show that $\sqrt{2}$ always lies between $\dfrac{a}{b}$ and $\dfrac{a+2b}{a+b}$.  Please give the easy solution as possible. Is there any way to get a hat wider than widehat?  Why doesn't  \widehat{abcdefghijklmnopqrstuvwxyz}   really go over all of it? If a factor variable (e.g. gender with levels M and F) is used in the glm formula, dummy variable(s) are created, and can be found in the glm model summary along with their associated coefficients (e.g. genderM)  If, instead of relying on R to split up the factor in this way, the factor is encoded in a series of numeric 0/1 variables (e.g. genderM (1 for M, 0 for F), genderF (1 for F, 0 for M) and these variables are then used as numeric variables in the glm formula, would the coefficient result be any different?  Basically the question is: does R use a different coefficient calculation when working with factor variables versus numeric variables?  Follow-up question (possibly answered by the above): besides just the efficiency of letting R create dummy variables, is there any problem with re-coding factors as a series of numeric 0,1 variables and using those in the model instead? What happens in electron-electron collisions? Why have a letter in a word when it’s silent in pronunciation, like the b in debt?  Can anyone please clarify my uncertainty here? Does anybody know which files I need to save in order to have a simple backup?   Reason for my question is, that my Galaxy Android smartphone lost all calendar data without any interaction. I found tons of apps, tons of sync stuff, but NO simple list of files to save. I do NOT want to save my data remotely on Google.  There must be at least one local file (database?) on my phone containing the synced calendar file (on my server, using ICSSync) why $1+\frac{1}{1+\frac{1}{\frac{1}{1+\dots}}}$ cannot be equal to $\frac{1-\sqrt{5}}{2}$  I know that some of you will answer because it is positive.I know it but I don't know how to prove it.Maybe the second part is positive how do you know that?Thanks in advance. Prove that, if A and B is similar, then $A^T$ is similar to $B^T$.  Attempt: I try to use the defn of similarity, and then the rule to transpose in different order, but get nowhere. This should be simple...  (Note to self: page 92.) Cause of black artifacts in render? 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 Is is possible to restore the MFT of a NTFS partition after a format? Automatically put PGFPlots xtick scale label in x axis label I'm trying to put xtick scale label (defined by pgfkey /pgfplots/xtick scale label code/.code) in x axis label, between variable name and unit.  I've tried executing using the option xlabel = {$ x $ (\pgfkeysvalueof{/pgfplots/xtick scale label code/.code} m)} to put in but this doesn't work.  I've also tried the following xtick scale label code/.code={\pgfkeyssetvalue{/pgfplots/xlabel}{#1}} to force PGFPlots to change my xlabel but it doesn't work as well.  I know I can do this by hand, but I'm interested in automatic approach. Is there a workaround or a solution to this?  My MWE is:  \documentclass{article} \usepackage{pgfplots} \pgfplotsset{compat=1.14}  \begin{document}      \begin{tikzpicture}     \begin{axis}[         scaled x ticks=true,         scaled y ticks = false,         y tick label style={/pgf/number format/fixed},         %xtick scale label code/.code={\pgfkeyssetvalue{/pgfplots/xlabel}{#1}},         xlabel = {$ x $ (\pgfkeysvalueof{/pgfplots/xtick scale label code/.code} m)},         ylabel = {$ y $},     ]      \addplot[line width=3pt] coordinates {(0.0,0.0)(100,1)(400,2)(10000,0)};      \end{axis}     \end{tikzpicture}  \end{document}   Output from this code is: Will stationary charge radiate E.M waves if observer is accelerating? How to Select Hyperlink Text in Google Chrome? I am running Google Chrome 5. How can I select hyper link text using the mouse with, for example, CTRL or ALT keys? What is the gradient of $f(x, y, z) = \sqrt{x^2+y^2+z^2}$?  I know that $\nabla f = \left\langle\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right\rangle$ or equivalently $\nabla f = \frac{\partial f}{\partial x}\mathbf{i} + \frac{\partial f}{\partial y}\mathbf{j} + \frac{\partial f}{\partial z}\mathbf{k}$.  Eventually, I am going to evaluate this gradient at a point and then dot it with some given unit vector to find a directional derivative. I think maybe what is troubling me is the radical in $f$. Will I have to use the chain rule in order to find the three partial derivatives?  I tried this:  $$\frac{\partial f}{\partial x} = \frac{1}{2}(x^2 + y^2 + z^2)^{-\frac{1}{2}}2x$$ $$= \frac{x}{\sqrt{x^2 + y^2 + z^2}}$$  That result looks ugly to me. However, if I did that partial derivative correctly, then I think I know what I am doing. I updated MacOS Sierra's Apache and PHP, and now whenever I $ sudo apachectl restart I get this:  AH00557: httpd: apr_sockaddr_info_get() failed for My-MacBook-Pro.local AH00558: httpd: Could not reliably determine the server's fully qualified domain name, using 127.0.0.1. Set the 'ServerName' directive globally to suppress this message   Things I've tried so far that didn't work:   Setting ServerName localhost in /etc/httpd.conf Setting ServerName localhost:80 in /etc/httpd.conf Setting ServerName 127.0.0.1 in /etc/httpd.conf Setting ServerName 127.0.0.1:80 in /etc/httpd.conf Ensuring 127.0.0.1       localhost is set in /etc/hosts Repeatedly restarting Apache Nearby objects won't move independently? Bash terminal not wrapping text correctly I am using the gnome-terminal with bash, and when I have a command that should go over the line, it just stays on the same line. So instead of:  echo "abcdefghijklmnopqrstuvwxyz"   I would get  opqrstuvwxyz""abcdefghijklmn   Or something similar to that. However, if I press enter the command will still work. I believe it has something to do with my PS1 variable, but I don't know what is wrong with it. Here it is:  export PS1='\e[1;35m[\u:\w]\$\e[0;35m ' How to properly adjust monitor brightness and contrast? How do I evaluate these fork() calls with the && and || operators in C?