The group $G$ has order 56. Show that it contains normal Sylow p-subgroup. I'm new at this topic, so I'm not sure about whether is my solution acceptable or not. Could you check it?  By Sylow theorem there exist $|P|=7$ and $|Q|=8$. Sylow $p$-subgroup is normal iff it is unique. Denote by $n_p$ amount of different Sylow $p$-subgroups. Suppose $n_i \neq 1$ for $i=2,7$. Since $|G|=n_p \cdot N_G(P)$ and $n_p\equiv 1 \pmod{p}$, we have that $n_7=8$ and $n_2=7$. So there are $8$ conjugates of $P$ which intersect in identity. Thus there are $(7-1)\cdot 8=48$ elements of order $7$ which can't be elements of Sylow 2-subgroups. There left $8$ elements which are obviously not enough to construct $7$ different Sylow 2-subgroups of order $8$. We've got contradiction.
Groups of order $56$ Let $G$ be a group of order $56$. (We do NOT assume the Sylow-$7$ subgroup to be normal.) Then either the Sylow-$2$ subgroup is normal or the Sylow-$7$ subgroup is normal. How to prove?  My idea: consider the case $n_2=7,n_7=8$. Then the number of elements in all Sylow $2$ -subgroups and Sylow $7$-subgroups is at most exactly $(7-1)*8+(2-1)*7+1=56$. This cannot give a contradiction. We need to prove that $G$ is simple. Then $|G||(8!/2)$. To prove $G$ simple, we need to prove there does not exist normal subgroups of order $2$ and order $4$. But I do not know how to prove.
Vertical placing of wide accents in superscript The following code  $\widetilde{aaa} \quad L^{\widetilde{aaa}}$  $\tilde{aaa} \quad L^{\tilde{aaa}}$  $\widehat{aaa} \quad L^{\widehat{aaa}}$  $\hat{aaa} \quad L^{\hat{aaa}}$  $\overline{aaa} \quad L^{\overline{aaa}}$  $\bar{aaa} \quad L^{\bar{aaa}}$   produces the following output:    What is the reason for additional vertical spacing for \widetidle and \widehat in superscript and what can one do about it? Statistics and causal inference? In his 1984 paper , Paul Holland raised one of the most fundamental questions in statistics:     What can a statistical model say about   causation?   This led to his motto:     NO CAUSATION WITHOUT MANIPULATION   which emphasized the importance of restrictions around experiments that consider causation.  Andrew Gelman makes :     "To find out what happens when you change something, it is necessary to change it."...There are things you learn from perturbing a system that you'll never find out from any amount of passive observation.   His ideas are summarized in .  What considerations should be made when making a causal inference from a statistical model? Hibernate computer with a timeout from command line on Windows 7 I am trying to hibernate my computer from the command. I was using shutdown /s /t 20 to shutdown the the computer. I change /s to /h to hibernate and now it just returns the usage text as if it doesn't recognise what I have entered. In this is does say  /h         Hibernate the local computer.   Is there something else that I need to use with /h to get it to hibernate? Windows 10 update broke my MBR / GRUB After I ran Windows 10 update this afternoon my partition table broke... I had dual boot with Windows 10 and Ubuntu 14 with Grub as the boot loader. The error that came up was "Error no such partition, Entering rescue mode... grub rescue>". I googled it,  about the boot-repair ISO, installed it on a USB, ran the repair, got this report, but still no luck:    Now the boot skips Grub entirely and goes to Windows boot manager which fails with "The boot configuration data for your PC is missing or contains errors". So now instead of dual boot I have zero boot. Thanks Microsoft.  Anyway can you guys help me interpret the disk info in the pastebin link? I don't remember my partition configuration but I believe Ubunut had slightlty less disk space than Windows. Questions I have:   Is the Ubuntu partition still there? Could it be the one named "Extended"? (sda4) Where would GRUB be? Gone? Can it be reinstalled? how?   I don't really care about the Windows partition anymore, but it would be nice to rescue Ubuntu. How can I display the current branch and folder path in terminal? What's the meaning "them" here? How does Incense work? In the US, it is common to use the idiomatic expression "knock on wood" to keep "a good thing going."  Is it the same in the UK, or do they use a different idiomatic expression? Why crontab scripts are not working? Evaluating $\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx$ using complex analysis I'm taking a complex analysis course and doing some practice computing residues &amp; evaluating integrals. I pulled out an old book called     On page 196-197, there are some interesting integrals to evaluate. I'm at 5.4.3.10.: I was able to do question 1, but was stumped at how to even begin with question 2:     Evaluate the integral   $$\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx ,$$   where $n$ is an natural number.   The answer is simply $\frac{2 \pi}{n}$. Any hints? Terminal not displaying colors for directory listings I don't know why my bash session doesn't show colors. I have tried Terminal as well as iTerm.  I have tried:   ls -G Setting export CLICOLOR=1 and export LSCOLORS=GxFxCxDxBxegedabagaced Using different themes such as solarized and something else   The versions of my software are:   bash: GNU bash, version 3.2.51(1)-release (x86_64-apple-darwin13) osx: 10.9.2   I have Homebrew installed  I don't know what other information might be helpful. I spend a lot of time on the shell, this problem is becoming very annoying :)  edit: in terminal preferences, I have 'Declare terminal as: xterm-256color' and 'Display ANSI colors' along with 'Use bright colors for bold text' Is "do something by date X" inclusive? Proving the cross product matrix tranformation identity with an alternative solution I'm solving a problem from the book, Mathematics for 3D Game Programming and Computer Graphics, Third Edition, by Eric Lengley. The problem goes:  Let $N$ be the normal vector to a surface at a point $P$, and let $S$ and $T$ be tangent vectors at the point $P$ such that $S \times T = N$. Given an invertible 3 $\times$ 3 matrix $M$, show that $(MS) \times (MT) = (\text{det}M(M^{-1})^{T}(S \times T)$, supporting the fact that normals are correctly transformed by the inverse transpose of the matrix $M$.  The author provided a hint stating we can represent $(MS) \times (MT)$ as $$ (MS) \times (MT) = \begin{bmatrix} 0 &amp; -(MS)_{z} &amp; (MS)_{y} \\  (MS)_{z} &amp; 0 &amp; -(MS)_{x} \\  -(MS)_{y} &amp; (MS)_{x} &amp; 0  \end{bmatrix} MT $$ We then find a matrix $G$ such that $$ GU = \begin{bmatrix} 0 &amp; -(MS)_{z} &amp; (MS)_{y} \\  (MS)_{z} &amp; 0 &amp; -(MS)_{x} \\  -(MS)_{y} &amp; (MS)_{x} &amp; 0  \end{bmatrix} M $$ where $$ U = \begin{bmatrix} 0 &amp; -S_{z} &amp; S_{y} \\  S_{z} &amp; 0 &amp; -S_{x} \\  -S_{y} &amp; S_{x} &amp; 0  \end{bmatrix} $$ and show that $G = (\text{det}M)(M^{-1})^{T}$ to solve the problem. I am aware that there is an  to this problem, but I would like to solve it through the hints provided. Unfortunately, I am only able to go as far doing: $$ G = \begin{bmatrix} 0 &amp; -(MS)_{z} &amp; (MS)_{y} \\  (MS)_{z} &amp; 0 &amp; -(MS)_{x} \\  -(MS)_{y} &amp; (MS)_{x} &amp; 0  \end{bmatrix} M U^{-1} $$ At this point, I do not know how to proceed with showing that $G = (\text{det}M)(M^{-1})^{T}$. How would you proceed? I'd like to ask for hints on solving the problem. Ball motion with air resistance coupled differential equation for fourth-order Runge-Kutta I've created a MATLAB function for solving coupled differential equation with the fourth-order Runge-Kutta method based on what is provided  (Simultaneous Equations of First Order). Here the function:  function [y] = rk4_c(f, g, h, x, y, z, n) % Runge-Kutta % Implementation of the fourth-order method for coupled equations % h = dt % x is the time here  for ii=1:(n-1)     k1 = h * f(x(ii), y(ii), z(ii));     l1 = h * g(x(ii), y(ii), z(ii));     k2 = h * f(x(ii) + 0.5*h, y(ii) + 0.5*k1, z(ii) + 0.5*l1);     l2 = h * g(x(ii) + 0.5*h, y(ii) + 0.5*k1, z(ii) + 0.5*l1);     k3 = h * f(x(ii) + 0.5*h, y(ii) + 0.5*k2, z(ii) + 0.5*l2);     l3 = h * g(x(ii) + 0.5*h, y(ii) + 0.5*k2, z(ii) + 0.5*l2);     k4 = h * f(x(ii) + h, y(ii) + k3, z(ii) + l3);     l4 = h * g(x(ii) + h, y(ii) + k3, z(ii) + l3);     y(ii+1) = y(ii) + (1/6)*(k1 + 2*k2 + 2*k3 + k4);     z(ii+1) = z(ii) + (1/6)*(l1 + 2*l2 + 2*l3 + l4); end   I would like to use this function for solving motion equations for a ball with air resistance. So starting with the basic equations (I hope they are correct): $$md^2x/dt^2 = F_{D_x}$$ and $$md^2y/dt^2 = F_{D_y}-mg$$ with $$F_{D_x} = -Dvvcos(\theta)$$ and $$F_{D_y} = -Dvvsin(\theta)$$ where $D$ is a drag coefficient. From my understanding, and from what I read, both second-order equation can be converted to two coupled equations that could be used with my coupled Runge-Kutta method function. Can someone help me understand how to convert my equation to a set of coupled equations that could be used with my MATLAB function?  EDIT  I was able to use my rk4_c function, however it only works for solving the velocities. I'm now trying to see how I could solve for the positions x and y. I'm kind of learning about Runge-Kutta methods at the same time, so sorry for the questions that may sound trivials.  % Runge Kutta code to solve projectile motion with quadratic drag % dVx/dt = -(D/m)*vx*sqrt(vx^2+vy^2) % dVy/dt = -(D/m)*vy*sqrt(vx^2+vy^2) - g  clc clear all  % Constant D = 0.24; % m = 2; % kg g = 9.80665; % m/s^2  % Define function handles fVx = @(t,vx,vy) -(D/m)*vx*sqrt(vx^2+vy^2); fVy = @(t,vx,vy) -(D/m)*vy*sqrt(vx^2+vy^2) - g;  % Initial conditions v0 = 200; % m/s theta = 30*pi/180; % rad t(1) = 0; vx(1) = v0*cos(theta); vy(1) = v0*sin(theta);  % Step size h = 0.01; % s tFinal = 2; N = ceil(tFinal/h);  % RK4 simultaneous coupled loop for ii = 1:N     % Update time     t(ii+1) = t(ii) + h;      % Update vx and vy     k1 = h * fVx(t(ii), vx(ii), vy(ii));     l1 = h * fVy(t(ii), vx(ii), vy(ii));     k2 = h * fVx(t(ii) + 0.5*h, vx(ii) + 0.5*k1, vy(ii) + 0.5*l1);     l2 = h * fVy(t(ii) + 0.5*h, vx(ii) + 0.5*k1, vy(ii) + 0.5*l1);     k3 = h * fVx(t(ii) + 0.5*h, vx(ii) + 0.5*k2, vy(ii) + 0.5*l2);     l3 = h * fVy(t(ii) + 0.5*h, vx(ii) + 0.5*k2, vy(ii) + 0.5*l2);     k4 = h * fVx(t(ii) + h, vx(ii) + k3, vy(ii) + l3);     l4 = h * fVy(t(ii) + h, vx(ii) + k3, vy(ii) + l3);     vx(ii+1) = vx(ii) + (1/6)*(k1 + 2*k2 + 2*k3 + k4);     vy(ii+1) = vy(ii) + (1/6)*(l1 + 2*l2 + 2*l3 + l4); end  % Plot the solution figure(1) plot(t, vx, t,vy) xlabel('Time (s)') ylabel('Velocities (m/s)') legend('Vx', 'Vy') How do I flatten this vector logo leaving only the visible black paths? My textbook explains the proof, which I don't understand:  "Consider two bases $v_1...v_p$ and $w_1...w_q$ of V. Since the vectors $v$ are linearly independent and the vectors $w$ span V..."  How exactly does $w$ span V?   The book then says the same for vectors $v$, that $v$ spans V and hence $p=q$, but I don't really understand how you can assume that given two sets of vectors that are basis, one set must span V. Prove that for any $x_0 \in X$ and any $r&gt;0$, the open ball $B_r(x_o)$ is open.   My attempt: Let $y\in B_r(x_0)$. By definition, $d(y,x_0)&lt;r$. I want to show there exists an $r_1\in\mathbb{R^+}$ s.t. $B_{r_{1}}(y)\subseteq B_r(x_0)$. Let $a\in B_{r_{1}}(y)$. Then, $d(a,y)&lt;r_1$. For $a\in B_{r}(x_0)$, $d(a,x_0)&lt;r$. I want to show $d(a,y)&lt;r_1$ implies $d(a,x_0)&lt;r$. By triangle inequality, $d(a,y)\leq d(a,x_0) + d(y,x_0) \rightarrow$  $d(a,y)&lt;r_1\leq d(a,x_0)+d(y,x_0)&lt;2r...$  I'm a little stuck after this point. Capillary action and conservation of energy When I dip a paper towel in a cup of water the water gets drawn up due to . How is this reconciled with conservation of energy, as it would seem on the surface that the potential energy of the system has gone up with no work being done? Only squares instead of letters and numbers are displayed in my calculator and other apps in Ubuntu 18.04 Unmet dependencies unable to fix I'm running ubuntu 14.04. I cannot use software center to install anything, even a local deb package so I have to rely on Terminal.  Recently I was trying to install mysql-server which generated error which I could pin-point was the required version of libstdc++6 . My system had 4.8.x while mysql 5.6 required >=4.9. Trying the usual upgrading process and unsuccessful, I consulted my sys admin, who tried installing the 4.9 package manually(using a local .deb package), which failed as well citing unmet dependencies.   At present, what I have is a system with error saying no new software can be installed.  The sudo apt-get install -f gives following output:  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... failed. The following packages have unmet dependencies:  libstdc++6 : Depends: gcc-4.9-base (= 4.9.2-10) but 4.9.1-0ubuntu1 is installed  libstdc++6:i386 : Depends: gcc-4.9-base:i386 (= 4.9.2-10) but 4.9.1-0ubuntu1 is installed E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.   I checked for held packages using  dpkg -l | grep ^h dpkg -l | grep hold   Result: nothing.  Also tried to find the broken package, if any but the file  /var/log/dist-upgrade/apt.log doesn't exist.  So, what should/can I do now.