Evaluation of $$\int^{1}_{0}\bigg(\frac{1}{1-x}+\frac{1}{\ln x}\bigg)dx$$   Let $$I = \int^{1}_{0}\bigg(\frac{1}{1-x}+\frac{1}{\ln x}\bigg)dx = \int^{1}_{0}\frac{(1-x)+\ln x}{(1-x)\ln x}dx$$  Now How can i solve after that , Help required, Thanks
What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &amp;=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &amp;=\int_0^\infty \left(\frac{1}{xe^x} - \frac{1}{e^x-1} \right)\,dx\tag2\\ &amp;=-\int_0^1 \left(\frac{1}{1-x} + \frac{1}{\ln x} \right)\,dx\tag3\\ &amp;=\int_0^\infty \left( e^{-x} - \frac{1}{1+x^k} \right)\,\frac{dx}{x},\qquad k&gt;0\tag4\\ \end{align}$$  This is not homework problems and I know that the above integrals equal to negative of the Euler–Mascheroni constant. I got these integrals while reading this Wikipedia page: . According to Wikipedia, the Euler–Mascheroni constant is defined as the limiting difference between the harmonic series and the natural logarithm: $$\gamma=\lim_{N\to\infty} \left(\sum_{k=1}^N \frac{1}{k} - \ln N\right)$$ but I don't know why can this definition be associated to the above integrals?  I can obtain the equation $(1)$ using substitution $t=\ln \left(\frac{1}{x}\right)\rightarrow x=e^{-t} \rightarrow dx=-e^{-t}\,dt$ and I know that $$\int_0^\infty e^{-x} \ln x\,dx=\Gamma'(1)=\Gamma(1)\psi(1)=-\gamma$$ but I can't obtain the rest. Any idea? Any help would be appreciated. Thanks in advance.
I want to use some constants, height and width, when creating an ellipse but it won't work with \newcommand. Here is a minimal example:  \documentclass[12pt, a4paper]{article}  \usepackage{amsthm, tikz} \usetikzlibrary{arrows, shapes, trees, positioning} \begin{document}  \begin{figure} \centering \begin{tikzpicture}       \newcommand*{\width}{2}%     \newcommand*{\height}{0.5*\width}%      \draw[fill=gray!40] (0, 0) ellipse (\width and \height); \node (Y) at (0,0) {\(Y\)};      \end{tikzpicture} \end{figure}  \end{document}    I get the following error:  error: line 14:  Package PGF Math Error: Unknown operator `a' or `an' (in '2and 0.5*2'). ...ay!40] (0,0) ellipse (\width and \height) How to show this equality of $\rho(X,Y)$ Let I have $(S,\Sigma,\mu)$ be a probability space, then $X,Y \in \Sigma$. Define $\rho (X,Y)$ by $\rho (X,Y)$ = correlation between random variable $I_X$ and $I_Y$, where $I_X$ and $I_Y$ are the indicator function of $X$ and $Y$. Express $\rho (X,Y)$ in term of $\mu (X)$, $\mu (Y)$, $\mu(XY)$. Conclude that $\rho(X,Y)=0$, if and only if $X$ and $Y$ are independent.      How to show that: $$\rho(X,Y)= \frac{\mu(XY)\,\mu(X^cY^c)-\mu(XY^c)\,\mu(X^cY)}{(\mu(X) \,\mu(X^c)-\mu(Y) \,\mu(Y^c))^{1/2}};0&lt;\mu(X)&lt;1;\,0&lt;\mu(Y)&lt;1$$ How to read high voltages on microcontroller? How to create my own list of things What ultimates can be damage boosted by Mercy? What ultimates can be damage boosted by Mercy?  We already know that , but are there other ultimates that can be damage boosted by Mercy? Can all ultimates be damage boosted by Mercy? How do Android tablets use magnet to auto-lock the device? How to combine two QUERY results into one column I have the following QUERY  =query(XX!A2:A;" select A where A&gt;'50' ")   and  =query(YY!A2:A;" select A where A&gt;'50' ")   how do I combine these two data sources so I don't have to have two separate columns they process data to?   best would be if the 1st row from XX is followed by the 1st row of YY, 2nd row followed by 2nd row... and so on If the above option is not easily possible, it is OK for one query to be followed by the second one (if I expect correctly that the start of the second will be moved with newly added data into the first one) How do I run a command? I am a new user, who has just been told to run / execute a command. The answer has some text with a grey background like this text here or  sometimes this text here.   However, I don't know what "run" or "command" means in relation to Ubuntu.  How do I do this?    This is intended for use as a Q&amp;A you can link to when you tell someone to run a command. How to stop creating .DS_Store on Mac? strcmp on a line read with fgets How to make overlay still work inside lstlisting environment? Problem  I want to explain C# programming language step by step using overlay in beamer.cls. But it does NOT works as shown in the following figure.  How to solve this problem?    Code Snippet  \documentclass[dvipsnames,cmyk]{beamer}  \usepackage{listings}  \begin{document}  \defverbatim[colored]\Lst{% \begin{lstlisting}[language={[Sharp]C}] using System; public delegate void Foo(object o); \uncover&lt;1&gt;{public class Foo} \uncover&lt;2&gt;{\{} \uncover&lt;3&gt;{public static void Main()} \end{lstlisting}}  \begin{frame}{MyListing} \Lst \end{frame}  \end{document} Looking around for rim brake pads, I have noticed a wide variety:   low end-ish: cheap, rubber pad with metal stem, unknown brand high end-ish: expensive, rubber pad with metal stem, known brand cool end-ish: dedicate pad sections (1 for dry, 1 for mud, 1 for rain)   Is there any good reason to go for an high end-ish or for the cool end-ish, or is it just plain old marketing tricks? How to secure my laptop so that hacking by physical access is not possible? I messed up my system earlier, I was greeted with a black screen, when booting in to Ubuntu. When I started up my laptop, I selected the recovery option from the grub menu, and chose fallback at root terminal. I saw that I was able to use the add user command, with it, I probably could use to create a privileged user on my machine.   Isn't that a security issue?  One could have stolen my laptop and at startup chose recovery and add another user, I'm fudged then. Including my data.  Come to think of it, even if you somehow remove that entry, one could boot from a live-CD, get a chroot up and running and then add another user, with the right privileges that allows it to see all my data.   If I set the BIOS to boot at my HD only, no USB, CD/DVD, Network startup, and set a BIOS password, it still wouldn't matter, because you'd still have that grub recovery startup entry.  I am fairly certain that someone from China, Russia can't hack my Ubuntu Trusty Tahr, from the network, because it's secure like that. But, if one has physical access to my - your - machine, then, well, that's why I'm asking this question. How can I secure my machine so that hacking through physical access is not possible?    Bug Report: Removing control chars (including console codes / colours) from script output I can use the "script" command to record an interactive session at the command line.  However, this includes all control characters and colour codes.  I can remove control characters (like backspace) with "col -b", but I can't find a simple way to remove the colour codes.  Note that I want to use the command line in the normal way, so don't want to disable colours there - I just want to remove them from the script output.  Also, I know can play around and try find a regexp to fix things up, but I am hoping there is a simpler (and more reliable - what if there's a code I don't know about when I develop the regexp?) solution.  To show the problem:   spl62 tmp: script Script started, file is typescript spl62 lepl: ls add-licence.sed  build-example.sh  commit-test         push-docs.sh add-licence.sh   build.sh          delete-licence.sed  setup.py asn              build-test.sh     delete-licence.sh   src build-doc.sh     clean             doc-src             test.ini spl62 lepl: exit Script done, file is typescript spl62 tmp: cat -v typescript Script started on Thu 09 Jun 2011 09:47:27 AM CLT spl62 lepl: ls^M ^[[0m^[[00madd-licence.sed^[[0m  ^[[00;32mbuild-example.sh^[[0m  ^[[00mcommit-test^[[0m         ^[[00;32mpush-docs.sh^[[0m^M ^[[00;32madd-licence.sh^[[0m   ^[[00;32mbuild.sh^[[0m          ^[[00mdelete-licence.sed^[[0m  ^[[00msetup.py^[[0m^M ^[[01;34masn^[[0m              ^[[00;32mbuild-test.sh^[[0m     ^[[00;32mdelete-licence.sh^[[0m   ^[[01;34msrc^[[0m^M ^[[00;32mbuild-doc.sh^[[0m     ^[[00;32mclean^[[0m             ^[[01;34mdoc-src^[[0m             ^[[00mtest.ini^[[0m^M spl62 lepl: exit^M  Script done on Thu 09 Jun 2011 09:47:29 AM CLT spl62 tmp: col -b &lt; typescript  Script started on Thu 09 Jun 2011 09:47:27 AM CLT spl62 lepl: ls 0m00madd-licence.sed0m  00;32mbuild-example.sh0m  00mcommit-test0m         00;32mpush-docs.sh0m 00;32madd-licence.sh0m   00;32mbuild.sh0m          00mdelete-licence.sed0m  00msetup.py0m 01;34masn0m              00;32mbuild-test.sh0m     00;32mdelete-licence.sh0m   01;34msrc0m 00;32mbuild-doc.sh0m     00;32mclean0m             01;34mdoc-src0m             00mtest.ini0m spl62 lepl: exit  Script done on Thu 09 Jun 2011 09:47:29 AM CLT Normality test with large data set I think I am dealing with this issue   I have large data set(10k data points) that slightly diverges from normal, and I get p-value of 0. I am interested in having perhaps a more crude test that tells me if data is extremely divergent from normal, versus looking somewhat normally distributed. I am currently trying Kolmogorov-Smirnoff, and in both cases I just get p-value of 0. Any alternatives?  I also looked at this:   So, taking all this into consideration, is there any kind of test I can perform that distinguishes between data that's roughly normal, and not normally distributed at all?  I am using scipy. Add review queues to Android app Onomatopoeia for disgust Show $100+t$ is reducible in $\mathbb{Z}[[t]]$ $\mathbb{Z}[[t]]$ is the ring of formal power series with integer coefficients, and the problem asks to show that $100+t$ is reducible. This means I need to find two power series, $\sum_{i=0}^\infty a_i t^i$ and $\sum_{j=0}^\infty b_j t^j$ with $a_i,b_j\in\mathbb{Z}$, that when multiplied give $100+t$. Since the integers are an integral domain, I know that neither of these power series may be finite, but I'm having a really hard time figuring out how to come up with two series that will telescope properly. Any suggestions? Can I use a higher rank tensor field as the metric field? What a metric should necessarily do is it should give me a way to associate a frame invariant number with a given pair of spacetime events. Now, if I use a higher rank tensor field (say, for example, a tensor field of rank 3: $g_{\mu \nu \rho}$) then also I can certainly produce a frame invariant scalar out of a given displacement $\vec{A}$ in this trivial way: $ I := g_{\mu \nu \rho} A^{\mu}A^{\nu}A^{\rho}$. Since the components of the higher rank tensor and that of the displacement vector are going to transform in a covariant and a contravariant manner respectively, the cooked up quantity is certainly a scalar.   Another crucial property, which I think has more direct physical content than the previous one, is that between two inertial frames, at least one such transformation should exist that leaves at least one metric invariant. i.e., There should exist at least one combination of transformation matrix $\displaystyle\frac{\partial{x^{\alpha}}}{\partial{x^{\mu '}}}$ and metric $g_{\alpha \beta \gamma}$ that satisfies the following equation:      $\displaystyle\frac{\partial{x^{\alpha}}}{\partial{x^{\mu '}}}\displaystyle\frac{\partial{x^{\beta}}}{\partial{x^{\nu '}}}\displaystyle\frac{\partial{x^{\gamma}}}{\partial{x^{\rho '}}} g_{\alpha \beta \gamma} - \delta_{\mu '}^{\alpha}\delta_{\nu '}^{\beta}\delta_{\rho '}^{\gamma} g_{\alpha \beta \gamma} =0$    The last thing I can think of that can put a restriction on the choice of a tensor as a metric is the existence of a possibility of finding a metric compatible symmetric connection field. Following the usual procedure of finding the expression for a metric compatible symmetric connection field, I reached following condition (unlike the case of the usual two rank metric where we get a full-fledged expression) for the connection in the terms of the metric:      $g_{\nu \rho k_1} \Gamma^{k_1}_{\mu \lambda} - g_{\lambda \mu k_2} \Gamma^{k_2}_{\rho \nu} = \displaystyle\frac{1}{2} (\partial_{\nu}g_{\rho \lambda \mu} + \partial_{\rho}g_{\lambda \mu \nu}  - \partial_{\lambda}g_{\mu \nu \rho}  - \partial_{\mu}g_{\nu \rho \lambda}) $   My question is that if it is possible to satisfy the two highlighted conditions then can we use such a 3 rank (or even higher rank tensors with similarly produced conditions) tensor fields as metric fields?   PS: This is NOT a proposal for a new home-production theory of gravity (or that of anything for that matter) but rather it is just that I am trying to understand why a two rank tensor is used in General Relativity as the metric. Thank you. Finding the maximum likelihood estimators for this shifted exponential PDF? Consider a random sample $X_1, X_2, \dots, X_n$ from the shifted exponential PDF $$f(x; \lambda, \theta) = \begin{cases}\lambda e^{-\lambda(x-\theta)} ;&amp; x \geq \theta\\                     \theta ; &amp;\text{Otherwise}\end{cases}$$ Taking $\theta = 0$ gives the pdf of the exponential distribution considered previously (with positive density to the right of zero). a. Obtain the maximum likelihood estimators of $\theta$ and $\lambda$. I followed the basic rules for the MLE and came up with: $$\lambda = \frac{n}{\sum_{i=1}^n(x_i - \theta)}$$ Should I take $\theta$ out and write it as $-n\theta$ and find $\theta$ in terms of $\lambda$?