Results that seem trivial, but are not
"It looks straightforward, but actually it isn't"
How can I chain implicits in Scala? durbinWatsonTest() in R for non time-series data I am trying to understand whether the Durbin-Watson test is meaningful at all when applied to regression data that have no temporal order (eg. blood pressure ~ bodyMassIndex + exercise).   I would say no, as the autocorrelation should obviously vary with the order of the subjects (which is random), but I wonder whether the durbinWatsonTest in R involves a bootstrapping of the data, where order of the bootstrapping data is shuffled each time, and then perhaps an average autocorrelation is computed over the bootstrapping samples (but then again, shouldn't this average autocorrelation be zero for randomly sampled data?).  On the other hand, the value of the statistics seems to be quite stable over multiple runs of the function, so I am even more confused...  I came across it in Andy Field's book "Discovering Statistics with R", where it is used is an example with non-time series data, although in another, more recent book of the same author ("Adventures in Statistics") it does say that the test is applicable only to time series data. How do we know the universe is expanding, and not that its contents are shrinking? Is there a physical reason why not to think that instead of space expanding, all physical constants and parameters are shrinking (including of course the instruments we use to measure the constants) and space is static, or is it a case of Occam's razor? Let $n&gt;0$ and assume that $\chi^{(n)} \in \mathrm{Irr}(G)$ for every $\chi \in \mathrm{Irr}(G)$. Show that $G = H \times A$, where $A$ is abelian and $(|H|, n) = 1$.  $\chi^{(n)}$ is defined by $\chi^{(n)}(g) = \chi(g^n)$. $\mathrm{Irr}(G)$ is the set of all irreducible characters of $G$.  Here are the hints in the book:   Let $d = (|G|,n)$. Show that it is no loss to assume that $(|G|/d,n) = 1$. Let $A = \bigcap_{\chi \in \mathrm{Irr}(G)} \mathrm{ker} \chi^{(n)}$. Show that $A = \{g \in G | g^n=1\}$ and $|A| = d$. Let $H = \bigcap \{\mathrm{ker} \chi | \chi \in \mathrm{Irr}(G), \chi^{(n)} = 1_G\}$. Show $|G : H| = d$.   Hint 1 and 2 are easy, and it is easy to deduce the result from the hints. As for hint 3, I have proven that $\{ g \in G | (o(g), n) = 1 \} \subseteq H$, and that $|G : H| \mid d$. And I am stuck here.  Can anyone help me? Does the 14 day self-isolation period in the UK end if a travel-corridor starts during the 14-day self-isolation period? What do you recommend as an optimal point and orientation for mounting a rear-view mirror? I'm open to handlebar-, glasses-, or helmet-mounted.  Some of the considerations, as I understand it, are:   Ease of mounting. View provided. Vibration. Likelihood of breaking. Robustness for repeated off and on (for glasses-mounted).   I've got a hybrid bike with flat handlebars.  I'd also welcome specific product recommendations and recommendations about other aspects to look for in a mirror. How can I reliably and accurately identify the passive voice in writing or speech? How to use \graphicspath? I am using MikTex 2.9 and TeXstudio for my TeX preparation. I would like to set the path to my figures folder, which contains all the figures, plots etc. required for my work.   I used \graphicspath to set the directory, but it gives me an error message:  \documentclass[12pt]{article}  \usepackage{graphicx}  \graphicspath{D:/LATEX/Reports@IIT/figures}  \begin{document} \includegraphics[width=4.5cm,height=4cm,angle=0]{logo.jpg}  \end{document}      Error: Package pdftex.def Error: File `logo.jpg' not found   However, when I prepend the filenames of the images with the same path in the \includegraphics command, it works just fine:  \documentclass[12pt]{article}  \usepackage{graphicx}  \begin{document}  \includegraphics[width=4.5cm,height=4cm,angle=0]{D:/LATEX/Reports@IIT/figures/logo.jpg}  \end{document}   Why is \graphicspath not setting the path? $f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $ Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f(0) = 0$ and $f'(x) = [f(x)]^{2}$, $\forall x \in \mathbb{R}$. Show that $f(x) = 0$,  $\forall x \in \mathbb{R} $.  I first (unsuccessfully) tried using the Mean Value Theorem, but this is in the Integrals chapter so the solution probably involves them. Can't really see where integrals come in here though.  What I've got so far:   (i) Since $f$ is differentiable, thus it is continuous and, hence,  integrable. Therefore $f^2$ is also integrable and as $f'=f^2$, $f'$ is too.   (ii) $f' \geq 0$, $\forall x \in \mathbb{R}$ In a Poisson model, what is the difference between using time as a covariate or an offset? I recently discovered how to model exposures over time using the log of (e.g.) time as an offset in a Poisson regression.  I understood that the offset corresponds to having time as covariate with coefficient 1.  I'd like to better understand the difference between using time as an offset or as a normal covariate (therefore estimating the coefficient). In which situation should I want to use one method or the other?  UPGRADE: I don't know if it is interesting, but I ran a validation of the two methods using randomly split data repeated 500 times and I noticed that using the offset method leads to a larger test error. Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = \gcd(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$. Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = hcf(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.   We know that if $a|c$ and $b|c$ then $a\cdot b\cdot s=c$ (for some positive integer $s$). $(ab|c)$  Then doesn't $ab|dc$ since $ab|c$?  I feel like I'm misunderstanding my givens.    Can we say $\operatorname{lcm}(a,b)=c$,  $\operatorname{hcf}(a,b)=d$, and $\operatorname{lcd}(a,b) \operatorname{hcf}(a*b)=a*b$? Thus, $ab|dc$  as $dc = ab$. Is it possible to load some package from a .sty file only if some option is selected?  (Naive approach \DeclareOption{something}{\RequirePackage{something}} doesn't work.) How do I backup data (SMS/contacts) from a device with a broken screen? I just broke the screen of my (unrooted) Galaxy S.  How do I get back my SMS and contacts?   PS: I tried Kies but it won't detect my phone for some reason. (Kies works fine with another Galaxy I possess but not this one) "To mentor someone during a project" vs. "to mentor someone on a project" ..., whom I mentored during his final semester's project.   ..., whom I mentored on his final semester's project.      Which of these two is grammatically correct? Since I am not talking about the final semester but the final semester's project, I am not very sure about during. Can someone help me with this? Why is there an "electric" potential drop across the resistor? An electric potential difference is created when two charges are separated. In a capacitor, there is a clear accumulation of opposite charges on the two separated plates, therefore work has to be done in moving an unit charge from one plate to another, against the electric field.   But, , a potential difference is created in a resistor due to the heat produced by the collision and increased lattice vibrations. So there is an energy loss across the resistor, yes, but how does this energy loss due to heat create an electric potential difference. In other words, do electrons accumulate on one or both sides of the resistor? If not, why should we do electric work in moving from one end of the resistor to another. If the difference in energy is not due to electric work, then that's not electric potential, is it? Is there an Extended Backus–Naur form (EBNF) that covers all of English? ColA                  Col B              ColC Project A            A.pdf               A.pdf Project B                                No Link   I have a list of project records. One column (e.g., B) may have a reference to a pdf file. If I create a formula   =if(isblank(B2),"No Link",hyperlink(folderURL + B2),B2)   What I want is to place a hyperlink on only those rows where column B contains a pdf reference, and insert text "No Link". I get the proper text values, but in all rows have the hyperlnk default appearance and behavior -- Excel seems to be creating "phantom links" out of the "No Link" cells.     Clicking on the "No Link" value in Col C activates an attempt to open the file listed in Col B, which returns a No file found error. Show that $\left(1+\dfrac{1}{n}\right)^n$ is monotonically increasing No human players, just AI and have the ability to scroll around and see what the AIs are doing with the turn management being also automated, so the AIs just end turn when they are done. Oh, and all of this hopefully without any sort of fog of war.  This way, I can see how the AI manages resources and units and maybe learn a thing or two as a civ newb. Recovering broken or deleted NTFS partitions