Can we paint the plane in two colors such that there is no square of the same color?
Monochromatic squares in a colored plane
I told my wolves to sit down and I accidentally forgot to tell them to stand up and I left them. Now I cannot find them, how do you get back to your wolves if they are lost? Answer draft is lost if you click to sign up I have to resit a calculus exam and for some reason set proofs were never my best friend...  Anyway, on a practice exam I encountered the following proof:  $$A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$$  When I draw a Venn-diagram it seems quite obvious but I couldn't manage to write the proof down properly.  If someone could help me, that'd be great! Render semi-transparent volume with cycles I want to render an image of a fire in cycles, but as a png with a transparent background. Without a transparent background, the output image looks like this:    But as soon as I tick the "Transparent" checkbox under Render->Film, all the semi-transparent parts of the image are not rendered and it looks like this    which is clearly not what I want. I've tried to add a holdout shader or a transparent shader to my smoke domain material nodes, but it didn't help. I've only found answered questions for semi-transparent glass, but none for volumes. My node setup looks like this:    but the transparent shader doesn't do anything.  Has anyone an idea how to fix this?  Thanks  Edit: I've just changed the flow type to "Fire and Smoke" instead of "Fire" and now the semi-transparent parts render too, but I don't want the smoke. Which of the following polynomials are subspaces of $\mathbb{P}_n$ for an appropriate value of n? Definition. A subspace  of a vector space V is a subset H of V that has three properties  (a) The zero vector of V is in H.  (b) $\forall$ u, v $\in H$, we have  u + v $\in H$.   (c) $\forall$ u $\in H$, c scalar, we have c  u $\in H$.  For $n \geq$ 0, denote $\mathbb{P}_n$ to be the set of all polynomials of degree at most $n$.  Which of the following are subspaces of $\mathbb{P}_n$ for an appropriate value of $n$? Support your answer with reason.  (a) All polynomials of the form $p(t) = a + t^2$, where $a \in \mathbb{R}$.  (b) All polynomials of degree at most 3, with integers as coefficients.  Solution:  Both the statements (a) and (b) above are not subspaces of $\mathbb{P}_n$.  Reason:  For part (a), All polynomials of the form $p(t) = a + t^2$, where $a \in \mathbb{R}$, is not a subspace because it failed to meet the standards of the first property stated above, that is, the coefficient of the $t^2$ term is always a 1, therefore the zero vector is not in this subspace of polynomials.    For part (b), All polynomials of degree at most 3, with integers as coefficents is also not a subspace becuase it as well failed to meet the standards of the third property, that is, integer coefficents, when multiplied by a real scalar, will not necessarily be of integer type leading to integer coefficents. Thusly, the resulting polynomial will not be in the subset of polynomials $\mathbb{P}_n$. $\Box$  Can someone please look at this and see if I have the right idea and approach to this question. Thank You "Much of it is" or "Much of it are"? I'm writing an college-entrance essay and for me it's imperative that I understand and implement the correct one of these two expressions:     "Much of it is" or "Much of it are"   If you need context:     I'm having trouble with my writing. The reason I'm having trouble is that much of it is about....   Should there be an "is" or an "are"? Legacy Debian versions and Bash Shellshock We are running Debian Etch, Lenny and Squeeze because upgrades have never been done in this shop; we have over 150 systems running various Debian versions.   In light of the "shell shock" of this week, I assume I need to upgrade bash.  I do not know Debian so I am concerned.   Can I merely execute apt-get install bash on all of my Debian systems and get the correct Bash package while my repository is pointed at a Squeeze entry. If not, what other course of action do I have? I am looking for a word that describes both of these activities, however the word coding does not seem to be suitable.   Likewise with encryption and decryption, cryption does not work well. Clearing NSUserDefaults I'm using +[NSUserDefaults standardUserDefaults] to store application settings. This consists of roughly a dozen string values. Is it possible to delete these values permanently instead of just setting them to a default value? Evolutionary chronology - if any, which species have evolved in the last 100 years I heared somewhere, I cannot remember where, that there has not been an evolution observed in our lifetime. In life science class, right at the start, we was doing the usual MRSGREN (how to tell if something is a living organism), and the tutor said that you can also tell something is a living organism if it evolves. So I asked if there has been an observed evolution in our lifetime. She said that there had been many but could not recite any off the top of her head, so the next week she handed me a piece of paper with 8 or so websites that aparrently corroborated her claim. I have not got the paper anymore but I recall there been on there the peppered moth or something. So I set out to look for a chronological list of evolution but could not find an authoritative one. I could only find something on wikipedia that indicated that species have been dying out for the past thousand years, but nothing saying that any have evolved. How can I deduce/identify an app from a notification icon? How Do I get into the castle walls? bash aliases equivalent for powershell? By default my Windows PowerShell starts in C:\Users\Santosh, my XAMPP installation is in D:\ so the htdocs folder is located at D:\xampp\htdocs. If I have to edit something in htdocs folder then I have to type full cd D:\xampp\htdocs\ (autocompletion is not so kind) then edit that file.  If this PowerShell were a Bash I would do this in .bash_aliases file:  alias htdocs='cd D:\xampp\htdocs'  Is it possible to maintain Bash aliases like file and alias any command in PowerShell? The characteristic and minimal polynomial of a companion matrix of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\mathbb F[x]$-modules.  It is an important fact that the characteristic polynomial and the minimal polynomial of $C(f)$ are both equal to $f$. This can be seen quite easily by induction on the degree of $f$.  Does anyone know a different proof of this fact? I would love to see a graph theoretic proof or a non inductive algebraic proof, but I would be happy with anything that makes it seem like more than a coincidence! Why would using higher ISO and faster shutter speed yield more noise than using lower ISO and slower shutter speed? Simple question about the definition of Brownian motion I have a question concerning the definiton of Brownian motion. Usually (e.g. on Wikipdia) one demands a brownian motion $\lbrace B_t\rbrace_{t\in[0,\infty)}$ to satisfy the following condition:  $\forall 0\leq t_0&lt;t_1&lt;...&lt;t_m: B_{t_1}-B_{t_0},...,B_{t_m}-B_{t_{m-1}}$ are independent random variables.   But today I come across with formally another definition of Brownian motion! Instead of the above condition the book demands the following property:  $\forall t,h\geq 0$, $B_{t+h}-B_t$ is independent of $\lbrace B_u : 0\leq u\leq t \rbrace$.  Now I'm wondering whether both properties are indeed equivalent or not?! Hopefully someone can help me. Highlight tags corresponding to our favorites Each of us have our list of favorite tags.  When a question has tag(s) that match one or more of ours, the background gets highlighted, drawing our attention to the question.  It would also be nice to highlight (perhaps using a bold font) the tags(s) that matched our own, so one could quickly do a further filtering of the question to see if it is of interest at the present time, like the following:    e.g. arduino is in my list of favorites, but the other two in each question are not. Differences between strong and weak in Objective-C Keyboard shortcut to focus on (idle) terminal I would like to know, if there is possible to use keyboard shortcut to focus on first idle (non working) terminal window in unity (13.10)? Something like Ctr+Alt+T, but before opening the new instance of terminal, searching for any idle instance? Can I use my powers for good?