How can I open the Google Chrome Console from JavaScript?
Can I programmatically open the devtools from a Google Chrome extension?
This is a follow up question to my previous question  Let $k$ be a field and $V \subseteq \Bbb{A}^n$ and $W \subseteq \Bbb{A}^m$ be algebraic sets. Then it should be true that $I(V \times W ) = I(V) + I(W)$ where by $I(V)$ here we mean the extension of $I(V)k[x_1,\ldots,x_{m+n}]$. Now I believe I have proven this (see the proof at the bottom of my question) but when I look at Martin's answer , it is instead claimed that we  actually have  $$I(V \times W )= \sqrt{I(V) + I(W)},$$  and for $I(V) + I(W)$ to be a radical ideal we need $k$ to be algebraically closed.           My question is: What's going on here? I believe my claim is true even without the assumption that $k$ is algebraically closed.           Here's a proof user Sanchez told me of, which I have simplified:       First it is clear that we always have $I(V) + I(W) \subseteq I(V \times W)$. For the reverse inclusion consider a polynomial $f \in I(V \times W)$. Then we can always write $$f = \sum_{i=1}^n f_ig_i$$ where $f_i \in k[x_1,\ldots,x_n]$ and $g_i \in k[x_{n+1},\ldots,x_{m+n}]$. Now take any $b' \in W$. If for all $i$ we have $g_i(b') = 0$ then since $b'$ is arbitrary, $g_i \in I(W)$ for all $i$. Then $f \in I(W) \subseteq I(V ) + I(w)$ and we are done. Otherwise suppose there exists $b \in W$  and $i$ such that $g_i(b) \neq 0$. Then wlog we may suppose that  $g_1(b) \neq 0$.   Next, $\sum f_ig_i(b) = 0 $ on all of $V$ by assumption of $f \in I(V \times W)$. So $\sum f_i g_i(b) = p$ for some $p \in I(V)$. Now write $$f_1 = \frac{ p - g_2(b) f_2 + \ldots g_n(b)f_n}{g_1(b)}.$$  Substituting this for $f_1$ in $\sum f_ig_i$, followed by taking things mod $I(V)$ we get an  expression with only $n-1$ terms $\mod{I(V)}$. Continuing this process we will finally get an expression with $0$ terms $\mod{I(V)}$ so that $f \in I(V)$. This shows $I(V \times W) \subseteq I(V) + I(W)$ which completes the proof. Recover data from SD card I have a 2&nbsp;GB Kingston  card which is about three years old. I put it in a reader today in my Windows Vista computer, wrote a 32&nbsp;MB file onto it, safely removed it, and then tried to read it elsewhere. Nothing. Putting it back in the Windows Vista computer it now says:     You need to format the disk in drive F: before you can use it.   What should I do? I have access to many computers and OSes if your recommendations need that. I would be very sad if I lost all the contents of the card. Most of the data is backed up, but there are a few things that aren't. :(  Doing a   # dd if=/dev/sdg of=~/tmp/sd.bin   gives me a 2&nbsp;GB file, and  the file it seems like lots of my data is still there. How can I put it back together? It appears that in PHP objects are passed by reference. Even assignment operators do not appear to be creating a copy of the Object.  Here's a simple, contrived proof:  &lt;?php  class A {     public $b; }   function set_b($obj) { $obj-&gt;b = "after"; }  $a = new A(); $a-&gt;b = "before"; $c = $a; //i would especially expect this to create a copy.  set_b($a);  print $a-&gt;b; //i would expect this to show 'before' print $c-&gt;b; //i would ESPECIALLY expect this to show 'before'  ?&gt;   In both print cases I am getting 'after'  So, how do I pass $a to set_b() by value, not by reference? When applying to a PhD program in the US, how does the admissions process work? If an applicant is weak in a particular area, is it possible to offset that by being strong in a different area?  Note that this question originated from this . Please feel free to edit the question to improve it. Correct usage of "see" vs. "watch" $f$ bounded but $f'$ isn't I know about beamer's  feature \setbeamercovered{invisible} and \setbeamercovered{transparent} to hide or grey out upcoming item's . But is there some way to have past items greyed (transparent) and future items hidden (invisible), so that only the current item is black/highlighted? And the best would be if on or 1 slide after the last all the item's could be visible highlighted. Something like this:   [grey] past 1 [grey] past 2 current [hidden] future 1 [hidden] future 2 [hidden] future 3   Is there any easy way of doing this? How to manage a redirect request after a jQuery Ajax call I'm using $.post() to call a servlet using Ajax and then using the resulting HTML fragment to replace a div element in the user's current page. However, if the session times out, the server sends a redirect directive to send the user to the login page. In this case, jQuery is replacing the div element with the contents of the login page, forcing the user's eyes to witness a rare scene indeed.   How can I manage a redirect directive from an Ajax call with jQuery 1.2.6? How to make a file (e.g. a .sh script) executable, so it can be run from a terminal I have a script.sh file and type of this file is shellscript file. I want to make this file as application/x-executable file. How can I make it? Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A I ran across this problem in my Analysis class and can't seem to come up with a good solution. Here's the question: Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals $A$. $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$.  I tried applying the Mean Value theorem like this: There exists a $c \in (a, x)$ st $f(x)-f(a)=f'(c)(x-a) \implies \frac{f(x)-f(a)}{x-a}=f'(c) \implies f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{c\to a}f'(c)=A$  However, I think this is wrong. I am pretty sure the Mean Value theorem is needed, but my application of it seems to be incorrect. Would somebody be able to point me in the right direction? How to redirect users to custom lostpassword page? I need to keep a session alive for 30 minutes and then destroy it. I am designing a poster, which has some geometrical shapes (TikZ paths) surrounded by some text. I would like the text to flow neatly around these shapes. I bet there's some clever TikZ magic that could do this.  EDIT: This is the sort of effect I want, sorry to not be clear:    \documentclass[a4paper]{article} \usepackage{tikz} \begin{document} \begin{tikzpicture} \draw (0,0) to (2,-2) to (2,-4) to (4,-4) to [out=up, in=down] (6.5,0); \node [text width=6cm, anchor=north west] at (0,0) { \hspace{5pt}  This is the sort of effect I want to \\ \hspace{23pt} achieve, with text automatically \\ \hspace{36pt} constrained by some \\ \hspace{46pt} geometrical form. Here \\ \hspace{55pt} it's just a curve, \\ \hspace{57pt} but this could \\ \hspace{57pt} also be the \\ \hspace{57pt} text label \\ \hspace{57pt} of a node.}; \end{tikzpicture} \end{document} DC Motor as a Generator Voltage and Current Plotting complex numbers as an Argand Diagram I apologize for the subjective "best" keyword.  My friend and I have started creation of a 2D adventure game. It will be top-down in the style of pokemon or zelda (just the perspective). We have been discussing methods of creating a large world-map that the player can traverse without straining the memory capabilities of our machine.  Our first impulse was to create a large map and a circle around the player in which content will be loaded. We figured this won't hold for long and decided to partition the map into sections. First we had four large sections, but realized we could simply break it down into many tiny sections.  I played some Zelda from the SNES and saw that, during the shift of a map, the content could be loaded just then. What I mean is, instead of just checking a rectangular area for data to load, we simply section the map into many tiny chunks that load and de-load data as we move from map-portion to map-portion.  Today, he told me he wanted to create a simply 2D array map[WIDTH][HEIGHT] that contains data about every grid in the game and is a constant save-to-disk operation for data that we don't need.  I'm not sure about these ideas and thought that I might as it here. Any links, resources or tutorials on the subject would be greatly appreciated as well as direct answers to our question on how to do it efficiently. Detecting Mouse clicks in windows using python Understand the meaning of `$_` Could you explain the following sentences from Bash manual about $_, especially the parts in bold,  maybe with some examples?      At shell startup, set to the absolute pathname used to   invoke the shell or shell script being executed as passed in the environment   or argument list.      Subsequently, expands to the last argument to the previous   command, after expansion.      Also set to the full pathname used to invoke each   command executed and placed in the environment exported to that command.     When checking mail, this parameter holds the name of the mail file. Why do my LED lights stay on, even when the switch is off? Keyboard shortcut to switch between applications in Mission Control Does Apple have keyboard shortcuts to switch between applications in Mission Control? (Instead of hovering over with the cursor)?  e.g., which key combination cal I press to get me from "preview" to "app store?"