I will be honest about it: the following integral is involved in my current investigations about the interplay between Fourier-Legendre series expansions, hypergeometric functions of the $\phantom{}_3 F_2$ and $\phantom{}_4 F_3$ kind and polylogarithms. Significative contributions will be rewarded through a proper mention of the author.  The value of $\phantom{}_3 F_2\left(\tfrac{1}{2},\tfrac{3}{4},1;\tfrac{5}{4},\tfrac{3}{2};-1\right)$ is related to the integral  $$ \mathcal{J}_1 = \int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx, $$ which on its turn (by Feynman's trick) is related to  $$ \mathcal{J}_2 = \int_{0}^{1}\frac{1-\phantom{}_2 F_1\left(-\frac{1}{4},1;\frac{1}{4};-x^2\right)}{x^2}\,dx.$$ I would be glad to know how to convert $\mathcal{J}_1$ (or $\mathcal{J}_2$) into a combination of standard mathematical constants and polylogarithms. It most certainly can be done since $\arctan(x)=\text{Im}\log(1+ix)$, but my version of Mathematica is not collaborating with me as I wished.  I will be away from keyboard for a few days (time to work for the Italian most celebrated mathematical contest), I hope to find some nice insight after my return.
When playing with the complete elliptic integral of the first kind and its Fourier-Legendre expansion, I discovered that a consequence of $\sum_{n\geq 0}\binom{2n}{n}^2\frac{1}{16^n(4n+1)}=\frac{1}{16\pi^2}\,\Gamma\left(\frac{1}{4}\right)^4 $ is:     $$\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \tfrac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2\tag{A}$$   which might be regarded as a sort of Ahmed's integral under steroids.  I already have a proof of this statement (through Fourier-Legendre expansions), but I would be happy to see a more direct and elementary proof of it, also because it might have some consequences about the moments of $K(x)$ of the form $\int_{0}^{1}K(x)\,x^{m\pm 1/4}\,dx$, which are associated with peculiar hypergeometric functions.
I'm entering the world of Linux and at work I'm using grep more and more.  By doing that I'm figuring out that sometimes it's not adequate for what I want.  I was struggling with grep a few days ago and a colleague of mine who is a senior Linux admin, told me to use awk. I was stunned by how fast I got a result.  So my question is when do you choose to use one over the other? What questions can I ask myself before going to work with grep and spending a lot of time, when I could have done it with awk and saved time? Escape velocity from Earth After converting to nodes in order to render using Cycles I've no idea where to enable the z-transparency. I guess the image textures alpha channel needs to be pluged somewhere but I've no idea which node would be suitable.   The question is how could I correctly setup the nodes to have the same effect as when using z-transparency in BI?      The materials original settings for BI are:    Which renders like:    I had no luck with DarenWs suggestion: Why do people call their characters "toons"? I've seen that in games like MMORPGs and the Diablo series, where the player has an "arsenal" of several characters, people refer to their characters as "toons".  What is the origin of this term? Reflection of a line in a plane The line $l_1$ has the equation $r=(6i+2-2k)+\lambda(4i+5j-k)$ and the plane $\pi_1$: $2x-y+4z=4$, the line $l_2$ is the reflection of $l_1$ in the plane $\pi_1$. Find the exact vector equation of line $l_2$   So the line intersects the plane when $\lambda=-2$, giving the point $(-2,-8,0)$ which will be common on $l_1$ and $l_2$. But I am unsure on how to find the direction vector for $l_2$. Any help would be appreciated. Independence of Gaussian variables What is the best possible way to check if a string can be represented as a number in Python?   The function I currently have right now is:  def is_number(s):     try:         float(s)         return True     except ValueError:         return False   Which, not only is ugly and slow, seems clunky. However I haven't found a better method because calling float in the main function is even worse. Let $f:[0,+\infty)\to\Bbb R$ be a function bounded on each finite interval. I want to show that if $\lim\limits_{x\to+\infty}[f(x+1)-f(x)]= L,$ then also $\lim\limits_{x\to+\infty}\dfrac{f(x)}x = L$ Command Options on Command Line - Details and Functions? Where are the evolutionary "inbetweeners"? The problem I've always had with evolution is the actual lack of variation between animals. More specifically, the lack of observable gradual change between species.  Take for example the hammerhead shark. It obviously has a strong connection with a normal shark, but where's the "half-hammerhead" shark? Where's the species that only has a SLIGHTLY wider head?  I really doubt that a single mutation on a normal shark resulted in a head shaped like that, it had to happen gradually. Right? As far as I understand, evolution works by random mutation in a child being somehow slightly beneficial to that animal so it get's to live longer and reproduce more, eventually resulting in it's genetic material in dominating the inferior ones before it. But if it was for whatever reason beneficial to have a slightly wider head, why don't all the sharks have slightly wider heads now? On the other hand, if it wasn't beneficial, how come the hammerhead still exists?  Furthermore, I feel like some characteristics of a certain animal can't really have gradual benefits. There's a lizard that shoots blood out of it's eyes. The original ancestor certainly didn't, and I'm sure that if it's child only bled slightly from it's eyes it wouldn't really help if much in life. So for it to actually become beneficial, it seems to me that there had to be a ton of lizards that had a disadvantage of bleeding slightly. In fact, there had to be so many that they evolved even further in the same direction and kept doing that until they bled so much it's an actual benefit (sorry for this turning so gross). So why did the inbetweeners survive long enough to evolve even more, and why did they THEN die out? Postfix Connection Timed out Eigenspace of the companion matrix of a monic polynomial How do I prove that the eigenspace of an $n\times n$ companion matrix  $$ C_p=\begin{bmatrix} 0 &amp; 1 &amp; 0 &amp;\cdots &amp; 0\\ 0 &amp; 0 &amp; 1 &amp;\cdots &amp; 0 \\ \vdots&amp;\vdots &amp;\vdots&amp;\ddots&amp;\vdots\\ 0 &amp; 0 &amp; 0 &amp;\cdots &amp;1 \\ -\alpha_0 &amp;-\alpha_1 &amp;-\alpha_2 &amp;\cdots&amp;-\alpha_{n-1} \end{bmatrix} $$ equals $\operatorname{Span}\{v_{\lambda} \} $ where $v_{\lambda}$ is an eigenvector of the companion matrix w.r.t. the eigenvalue $\lambda$: $$ v_{\lambda} =  \begin{bmatrix} 1 \\ \lambda\\ \lambda^{2} \\ \vdots\\ \lambda^{n-1} \end{bmatrix}. $$ What would be a good way to go about attacking a Quagmire 3 cipher?  I understand that it is polyalphabetic with a key and an indicator. I have started by taking the cipher text, taking every n-th letter and putting that in a string, computing the index of coincidence for the n strings I get and then averaging that. I would assume that whatever n value corresponds with the closest average index of coincident to English (0.066) would be the number of alphabets, i.e. the length of the indicator.  Where do I go from there? Or is my approach incorrect? Put a where clause around a stored procedure I cannot edit Finite Cartesian Product of Countable sets is countable? Is the finite Cartesian Product of Countable sets countable?  If possible, could you give a bijection that would work for any example?  Any help would be appreciated. Placement of images in row using subfigure I want to put four figures in a single row. The code I am trying to use is taken from .  This puts three figures in a row. When I add a fourth one, I get three images in row and the last one appears in the second row. But I want all of them in one row. Here is my code.  \documentclass[12pt,twoside]{report}% Use this line for the print version of the thesis   \usepackage{graphicx} \usepackage{caption} \usepackage{subcaption} \begin{document}  \begin{figure}         \centering         \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{gull}                 \caption{A gull}                 \label{fig:gull}         \end{subfigure}%           \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{gull}                 \caption{A gull2}                 \label{fig:gull2}         \end{subfigure}%          ~ %add desired spacing between images, e. g. ~, \quad, \qquad etc.           %(or a blank line to force the subfigure onto a new line)         \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{tiger}                 \caption{A tiger}                 \label{fig:tiger}         \end{subfigure}         ~ %add desired spacing between images, e. g. ~, \quad, \qquad etc.           %(or a blank line to force the subfigure onto a new line)         \begin{subfigure}[b]{0.3\textwidth}                 \centering                 \includegraphics[width=\textwidth]{mouse}                 \caption{A mouse}                 \label{fig:mouse}         \end{subfigure}         \caption{Pictures of animals}\label{fig:animals} \end{figure} \end{document}   As I have mentioned above, it works for three figures.  But I can not use it for four figures. What should I change?   What does  width=\textwidth mean?   I changed    \begin{subfigure}[b]{0.3\textwidth}   to  \begin{subfigure}[b]{0.2\textwidth}   but it didn't help.  Then, I changed     \includegraphics[width=\textwidth]{mouse}   to  \includegraphics[width=0.5\textwidth]{mouse}   Still no solution. Where did the "wheel" group get its name? The wheel group on *nix computers typically refers to the group with some sort of root-like access. I've heard that on some *nixes it's the group of users with the right to run su, but on Linux that seems to be anyone (although you need the root password, naturally). On Linux distributions I've used it seems to be the group that by default has the right to use sudo; there's an entry in sudoers for them:  %wheel ALL=(ALL) ALL     But that's all tangential; my actual question is: Why is this group called wheel? I've heard miscellaneous explanations for it before, but don't know if any of them are correct. Does anyone know the actual history of the term? So I'm trying to install Visual Studio 2015 Community on my Windows Server 2012 R2 box which will end up being a Jenkins build server. Attempting to do so gives an error saying that I need to install KB 2919355.    So I attempt to download the right KB from the . The two files that I would assume to work are either Windows8.1-KB2919355-x64.msu and Windows8.1-KB2919355-x86.msu (just to try) and neither update work, saying that the update is not applicable.    I've not been able to get this working and I haven't been able to find a proper way to fix this. Can anyone offer any suggestions or solutions? Thanks! Can I disable auto logon on windows 10 home I recently setup a new computer for my parents with windows 10 home. I made two local accounts for them - no microsoft accounts. I also would like to keep them without a password. Problem is now that windows automatically signs in with the last used account. No lock/login screen.  I tried a number of answers from similar questions here but none worked. Since it's a home version there are no group policies and the method using task scheduler does not work because accounts with no password are not allowed to create such tasks.  Are there other options to force windows to show a user list on startup without setting passwords for the accounts? Does SOQL join not respect OWD /sharing settings?