Find all functions $f:\mathbb N_0\to \mathbb N_0$ such that $f(a^2+b^2)=f(a)^2+f(b)^2$
If $f(1)=1$, then is it true that $f(n)=n$ for all $n \in \mathbb{N}\cup\{0\}$.
How can I boot with an older kernel version? The latest kernel is causing problems with my sound, which worked fine with an older version. As I have only Ubuntu installed, Grub is not getting displayed while booting. How can I manually choose my kernel version while booting? Attaching event handlers in Visual Basic .NET Inaccurate shading on object Why can a 1 element set be a member of another set but not a subset of it? I have came across this in a textbook:     $\{2\}\nsubseteq\{\{2\}\}$ but $\{2\}\in\{\{2\}\}$   I understand that $\{2\}$ is an element (member) of the other set but considering $\{2\}$ is a set itself, why is it not a subset? I am programming a tile based game and I have some basic tiles (grass, dirt, etc..), but I can't figure out how to make good random map generation, because when I do some really random selection, if the tile should be grass/dirt, I get this:    I understand why this is happening, but what I want is to create some random continuous areas of grass or dirt. Something that would make more sense, like this: How to set the default value of an exposed filter rather than -Any-? I have a view with exposed filters in Drupal 7 and I need to set the default value of a filter to current year. I also need to unset/remove the - Any - option from the select list. Cube of an integer $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=k$ and $x, y, z, k$ are integers.  Prove that $xyz$ is cube of some integer number.    I was wondering about giving a parametrization for the rational points on the elliptic curve in $\mathbb{P}^2(\mathbb{R})$ $$ x^2 y + y^2 z + z^2 x = k xyz$$ but I am not too confident in the topic, so I am asking you to shed some light.  Thank you for your help. Inline suggested edit reviews are truncated in medium-width windows When reviewing a suggested edit inline (not from the queue) on a site with the new “responsive” design, at a window width where normal content does fit, the diff window is truncated on both sides, as is the edit window after cliking “Improve Edit” or “Reject and Edit”.    This is from  which happened to have a pending suggested edit at the time, after clicking the edit(1) button. Note how the scroll bar is all the way to the left, but there's content that vanishes to the left of the window. That's in Chrome 67.0.3396.87 on Linux at zoom level 100%, with a window width of 800.  Note that I'm not complaining that the user interface isn't responsive yet (although now that this has been deployed to SO I'd expect it to be), but that the interface is completely unusable even with scrolling. In sites with the old UI, you can see all the content even if it requires scrolling.  Reviews done from the suggested edit review queue are ok (not responsive, but you can see the content by scrolling). Inline edits are ok. The problem is specific to inline suggested edit reviews. Can this identity be derived from the binomial theorem?  $$n(n+1)2^{n-2} = \sum_{i=1}^{n}  i^2 \binom{n}{i}$$    I tried starting from $2^n = \displaystyle\sum_{i=0}^{n} \binom{n}{i}$ and dividing it by $4$ in order to get $2^{n-2}$. What is a session 0? What is a "Session 0"?   When is it done? What does it contain? Is it recommended, if not mandatory? What is each person's role (DM/GM, players, etc.)? What topics are best discussed there?   Usually with my groups, we go directly to the meat of the game right after the character creation, but I see a lot of answers that basically say "you should have a session 0", but it's never explained what it is made of, so it's kind of an enigma for me. I have this alias set in my system /etc/bashrc file:  alias root="sudo !!"   The intention of this being to run the last used command using sudo of course. When used, it of course appears to substitute the last command in history to the bashrc file upon shell initialization, and not the actual command that you would get if you were to run sudo !! in an interactive shell. I've also tried alias root="sudo fc -s" to no avail.  I realize this is probably something to do with how the BASH does command substitutions, but can someone explain why this is, and provide a usable substitute?  I'm running BASH version 3.2.51(1)-release (x86_64-apple-darwin13). Let $ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}$ . I would like to know whether the given sequence converges.   I see  that,  $ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}= \sum_{k=1}^{n} \frac{1}{n+\frac{k}{n}}.$ When $n$ gets sufficiently large the contribution by the $ \frac{k}{n} $ term is diminishing and $ a_{n} &lt; \sum_{k=1}^{n} \frac{1}{n} = 1 $.   Thank you. Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$ JavaScript OR (||) variable assignment explanation What is the concept of erasure in generics in Java? What is the concept of erasure in generics in Java? How many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$? I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$.     I am interested in counting how many such $p(x)$ there exist (that is, given $n\in\mathbb{N}$, $n\ge 1$, how many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$).   I don't have a counting strategy and I don't expect a closed formula, but maybe we can find something like "there exist $X$ irreducible polynomials of degree $n$ where $X$ is the number of...".  What are your thoughts ? Functions which are Continuous, but not Bicontinuous What are some examples of functions which are continuous, but whose inverse is not continuous?  nb: I changed the question after a few comments, so some of the below no longer make sense. Sorry. How to prove the set $S$ is closed under the binary operation $a*b=a+b+kab$? Let $k$ be any nonzero real number and let $S$ be all real numbers except ${-1\over k}$. Let $a,b \in S$. Define * on S by $a*b=a+b+kab$. Prove that $S$ is closed under *.  All I can think to do is set $a*b$ equal to ${-1\over k}$, but I don't know where to go from there. Is that the extent of the proof? adjective describing middle distance (neither near nor far) How to import a .max model into Blender? I've tried to import but I don't see the (.max) file as an option when importing. Is there another way to import the 3DMax model into Blender for rigging and animation?