Proving that the intersection of Sylow p-subgroups is normal?
Intersection of all $p$-Sylow subgroups is normal
Can I choose to delay my turn in combat? What wand was used by Mr Graves at the time of Fantastic Beasts? We know that the movie Fantastic Beasts is set in 1926. From the movie, we learn that at this time, Dumbledore is a teacher at Hogwarts and Grindelwald is lose after escaping a previous arrest. In the movie we learn that      Mr Graves is in fact Grindelwald in disguise. Grindelwald is shown to be at least in his forties, maybe fifties so he should already be the owner of the wand since he stole it from Gregorovitch as a teen.   We know that Dumbledore won the wand from Grindelwald after defeating him in their historic duel. That duel is always described as being “the end of the dark wizard Grindelwald” so I’m assuming that it didn’t happen prior to the movie and that he didn’t escape after that duel, please correct me if you have cannon evidence stating the contrary.      In the movie, we see Graves’ wand a few time and it isn’t the (very recognizable) one used in DH as the Elder Wand. This could of course be a prop error but since David Yates was also the director on DH and a lot of the visuals are similar I’m assuming it wasn’t. Graves could of course be using another wand in the fear that someone could recognize it but he would still be the rightful master of the Elder Wand. At the end of the movie, Newt is the one that disarms Graves/Grindelwald.   In all logic, he should have been made the new owner of the wand, whether Grindelwald was actually using it or not. That would have made it impossible for Dumbledore to win it from Grindelwald later.  So my question is: What wand is that character using at that specific time? Is he already the owner of the Elder Wand?      And if so, why wasn’t Newt made the new owner of the wand? Error message in my excel formula Can binary data be ordinal? Binary data is often mentioned as a nominal sub-category, especially in such examples as female/male, smoker/non-smoker, etc. However, binary data with such values as pass/fail, correct/incorrect, absent/present, etc, seems to give some weight to its values. It's not like in the example of the gender, where both values are equal and differ primarily by the nominal and other context-related traits. Instead, this type of binary data clearly indicates that one value means something and the other means nothing.   In case of such distinction, can binary be considered ordinal? If yes, what are statistical tests that are usually used for such data? Also, are there any interesting books or papers on this case? Who built the fortress of Dol Goldur, and when was it built? And was it always a place for evil, or Sauron just occupied it?  Despite the fact that Mordor is much better fortified; artificially by the old fortifications,  and geographically, as it is surrounded by the mountains. Also considering the presence of mount doom that seem to enhance Sauron's power, Sauron chose to start his campaign from Dol Goldur, despite Mordor being ready for quite some time.  Why did he choose this enigmatic fortress, and not go straight for Mordor or even go earlier? Upvoting a downvoted answer will do +2 upvote? I have first downvoted an answer. Then, I decided to leave it the way it was before, meaning to undo my downvote just like it never existed, so I upvote to remove my downvote. Suprise! SO seems to consider it as two upvotes!  Let's say I have an answer with -3, then I decide to downvote it making it -4. After deeper thoughts, I prefer to remove my downvote to let it the way it was before I downvoted, meaning -3, but SO makes it -2 automatically!     Wouldn't it be supposed -3, and not -2? If someone else would have upvoted, it would have made it an upvote +1, not a +2!   Am I missing something here?  In fact, is there a way to tell SO that I only wish to undo my downvote, and not upvote the answer? I may understand why it does behave this way. I only wish to undo my downvote, though. Science fiction short story: beam transporting people into the distant future Can bug reports and feature requests requiring SE Dev attention be reported on per-site metas, or do they need to all go here? PHP function overloading Using Laptop Screen as a monitor How do I recursively grep through compressed archives? I'm trying to find out what modules  in cpan. So I've used  to mirror it. My problem is that I need to iterate through the archives that are downloaded, and grep the files that are in the archives. Can anyone tell me how I might do this? preferably in a way that tells me which file in the archive and what line it's on.  (note: they aren't all tarballs some are zip files) Disable chat migration notification if one of the users has insufficient rep? My PDO Statement doesn't work If we disbelieve $H_0$, why quote a p value calculated assuming $H_0$ was true? How to plot logistic regression decision boundary? I am running logistic regression on a small dataset which looks like this:    After implementing gradient descent and the cost function, I am getting a 100% accuracy in the prediction stage, However I want to be sure that everything is in order so I am trying to plot the decision boundary line which separates the two datasets.  Below I present plots showing the cost function and theta parameters. As can be seen, currently I am printing the decision boundary line incorrectly.    Extracting data  clear all; close all; clc;  alpha = 0.01; num_iters = 1000;  %% Plotting data x1 = linspace(0,3,50); mqtrue = 5; cqtrue = 30; dat1 = mqtrue*x1+5*randn(1,50);  x2 = linspace(7,10,50); dat2 = mqtrue*x2 + (cqtrue + 5*randn(1,50));  x = [x1 x2]'; % X  subplot(2,2,1); dat = [dat1 dat2]'; % Y  scatter(x1, dat1); hold on; scatter(x2, dat2, '*'); hold on; classdata = (dat&gt;40);   Computing Cost, Gradient and plotting  %  Setup the data matrix appropriately, and add ones for the intercept term [m, n] = size(x);  % Add intercept term to x and X_test x = [ones(m, 1) x];  % Initialize fitting parameters theta = zeros(n + 1, 1); %initial_theta = [0.2; 0.2];  J_history = zeros(num_iters, 1);   plot_x = [min(x(:,2))-2,  max(x(:,2))+2]  for iter = 1:num_iters  % Compute and display initial cost and gradient     [cost, grad] = logistic_costFunction(theta, x, classdata);     theta = theta - alpha * grad;     J_history(iter) = cost;      fprintf('Iteration #%d - Cost = %d... \r\n',iter, cost);       subplot(2,2,2);     hold on; grid on;     plot(iter, J_history(iter), '.r');  title(sprintf('Plot of cost against number of iterations. Cost is %g',J_history(iter)));     xlabel('Iterations')     ylabel('MSE')     drawnow      subplot(2,2,3);     grid on;     plot3(theta(1), theta(2), J_history(iter),'o')     title(sprintf('Tita0 = %g, Tita1=%g', theta(1), theta(2)))     xlabel('Tita0')     ylabel('Tita1')     zlabel('Cost')     hold on;     drawnow      subplot(2,2,1);     grid on;         % Calculate the decision boundary line     plot_y = theta(2).*plot_x + theta(1);  % &lt;--- Boundary line      % Plot, and adjust axes for better viewing     plot(plot_x, plot_y)     hold on;     drawnow  end  fprintf('Cost at initial theta (zeros): %f\n', cost); fprintf('Gradient at initial theta (zeros): \n'); fprintf(' %f \n', grad);   The above code is implementing gradient descent correctly (I think) but I am still unable to show the boundary line plot. Any suggestions would be appreciated.    logistic_costFunction.m  function [J, grad] = logistic_costFunction(theta, X, y)      % Initialize some useful values     m = length(y); % number of training examples      grad = zeros(size(theta));      h = sigmoid(X * theta);     J = -(1 / m) * sum( (y .* log(h)) + ((1 - y) .* log(1 - h)) );      for i = 1 : size(theta, 1)         grad(i) = (1 / m) * sum( (h - y) .* X(:, i) );     end  end   EDIT:  As per the below answer by @Esmailian, now I have something like this:  [m, n] = size(x);  x1_class = [ones(m, 1) x1' dat1']; x2_class = [ones(m, 1) x2' dat2'];  x = [x1_class ; x2_class] I have read that positional parameters start at $1 (for example: $1, $2, $3 and so on are positional parameters). But $0 is not a positional parameter.  But why $0 is not a positional parameter?  I think this could be a reason, but not sure:  Positional parameters only take their values when a script is executed. For example: if we do ./myScript Hello, then $1 will have the value Hello. But $0 can take its value on two occasions: when a script is executed (it would have the value of the script name), and when bash itself is executed without a script (it would have the value bash or -bash). Can I re-enter the UK visa free shortly after Tier 2 visa expires? How do I dynamically create the variable name in a PHP loop? I have been wondering what a gold piece is worth, due to the fact there are many things that cost the same, but in reality, are probably worth radically different prices, such as a Goat and a Whip, which both cost 1 gp.  What is a gold piece really supposed to be worth? Like, how would things be priced if they were in familiar modern monetary units instead of "gp"? That would give me something to base my adjustments to abnormal prices on, such as items not listed in the PHB. Chat is down. Where can I find when it'll be back up?