Can we paint $\mathbb{R}^2$ with two colors (say, black and white) so that any horizontal line pass through countable black points and any vertical line pass through countable white points?    Update. I was amazed that this is a classical problem! Actually what I wanted to do is using set theory to prove that there exist incomparable Turing degrees.
The problem is to prove the following equivalence:  $$2^{\aleph_0}=\aleph_1 \iff \exists\ A,B\in\mathbb{R}^2:\\ \textrm{a)}A\cup B=\mathbb{R}^2 \\\textrm{b)}\ \forall\ \textrm{lines } l\ \textrm{parallel to x-axis, } |A\cap l|\leq\aleph_0 \\\textrm{c)}\ \forall\ \textrm{lines } l\ \textrm{parallel to y-axis, } |B\cap l|\leq\aleph_0$$  I think I know how to prove "$\impliedby$":  Let's assume the RHS and that $2^{\aleph_0}\geq\aleph_2$. Now let's take a sequence $\langle x_{\alpha}:\ \alpha&lt;\omega_1\rangle$. Since $\bigcup_\limits{\alpha&lt;\omega_1}|B\cap x_{\alpha}|\leq \aleph_1\cdot\aleph_0=\aleph_1$ there is such a $y$-coordinate that neither of $(x_{\alpha},y)$ is in $B$. But hence we obtain $\aleph_1$ point which all lie on the same line parallel to $x$-axis. Since $A\cup B=\mathbb{R}^2$ they all must be contained in $A$ $-$ a contradiction as $A$ has at most countably many of those points.  But what about the other implication? Do I contruct these sets by induction? Over all lines or something else? I haven't come up with anything yet, perhaps you could help me.
What is the dimension of $R[[X]]$ where $R$ is a Noetherian ring? If $R$ is a Noetherian ring then we know that the Krull dimension of the polynomial ring $R[X]$ is $\rm dim(R)+1.$ Is there any formula for the Krull dimension of the power series ring $R[[X]] $ When $R$ is a Noetherian ring ? Can one construct a non-measurable set without Axiom of choice? Is axiom of choice required to show the existence of non-measurable sets? Is there a Lebesgue non-measurable set that can be constructed without axiom of choice?  Related question on MO says it is consistent: Where does a layman go to get a basic understanding of AKS primality testing. I am not talking about the optimal choice of $r$ (which I am told is the hardcore number-theoretic part of the algorithm). I mean basic things, like what does $\bmod(n, x^r-1)$ mean (how can something be mod two things?) And what does binomial expansion have to do with anything?  I can't find anything in Wikipedia that is appropriate for my level. Mathematica not plotting whole logarithmic function Forgotten PostgreSQL Windows password How do I get the UTC time of "midnight" for a given timezone? The best I can come up with for now is this monstrosity:  &gt;&gt;&gt; datetime.utcnow() \ ...   .replace(tzinfo=pytz.UTC) \ ...   .astimezone(pytz.timezone("Australia/Melbourne")) \ ...   .replace(hour=0,minute=0,second=0,microsecond=0) \ ...   .astimezone(pytz.UTC) \ ...   .replace(tzinfo=None) datetime.datetime(2008, 12, 16, 13, 0)   I.e., in English, get the current time (in UTC), convert it to some other timezone, set the time to midnight, then convert back to UTC.  I'm not just using now() or localtime() as that would use the server's timezone, not the user's timezone.  I can't help feeling I'm missing something, any ideas? Math mode macro in TeXstudio I want to set up a macro in TeXstudio for math mode which displays $*$ and places the cursor at position *. So far I have tried using $%|$ but upon triggering, this macro does not allow text insertion at the placeholder.  EDIT: The method above works fine if the macro shortcut is set to the default Shift+F1. However, changing the shortcut to another combination, as I would like to do, seems to be the cause of the problem. I started thinking about the number of associative (binary) operations on a set with $n$ elements today.  Looking online I found  which indicates only $113$ of the possible $19,683$ operations on a three-element set satisfy association.  So, $50\%$ of binary operations on a two-element set satisfy association, and less than $0.6\%$ of all binary operations on a three element set satisfy association.  For an $n$-element set, where $n$ denotes a natural number, is the ratio $R_{n}=A_{n}/B_{n}$, of the number of associative binary operations $A_{n}$ to the number of all binary operations $B_{n}$, in general small?  I don't mean the following questions as equivalent, but since they seem more concretely answerable in principle, does   $$ \lim_{n \to +\infty} \frac{A_{n}}{B_{n}} = 0 ? $$       Also, is $F : n \to R_{n}$ a monotonically decreasing function?  Some Background of the Question: In his Linear Algebra Problem Book 1995 on p. 6 Paul Halmos writes "The commonly accepted attitudes toward the commutative law and the associative law are different.  Many real life operations fail to commute; the mathematical community has learned to live with that fact and even to enjoy it.  Violations of the associative law, on the other hand, are usually considered by specialists only."  For all I know, Halmos might only have written that to motivate the study of Linear Algebra and doesn't quite literally mean what he appears to say.  But, if he means what he appears to say, and if $F : n \to R_{n}$ is monotonically decreasing, or $R_{n}$ is generally small, I think there's something amiss with what Halmos says, since non-associative operations seem so common that one may as well enjoy them. C structure initialization? London Heathrow airport - do I need a transit visa? I am traveling from the US to India via London Heathrow on November 6. I will be at LHR on 7th at 6:20 am and departing at 9:25 pm. I'm an Indian national and have a valid US visa.  Do I need a transit visa to stay airside?  I have been scared that British Airways have stopped a few people from boarding flights (no details just rumors).  Do I need a transit visa to stay landside? I want to do some sightseeing. What are the chances of getting past officials using transit without visa concession?  Real stories will help a lot to calm the butterflies in my stomach. How to determine the genre of a song? I listen to many songs and I always wonder how the genre of a song is determined. Is it by listening to the song, or by the song's info like the artist, year released and so on? Alternate Row Coloring in Django Template with More Than One Set of Rows Django templates offer the builtin tag cycle for alternating between several values at different points in a template (or for loop in a template) but this tag does not reset when it is accessed in a scope outside of the cycles definition.  I.e., if you have two or more lists in your template, the rows of all of which you'd like to use some css definitions odd and even, the first row of a list will pick up where the last left off, not with a fresh iteration from the choices (odd and even)  E.g., in the following code, if the first blog has an odd number of entries, then the first entry in a second blog will start as even, when I want it to start at odd.  {% for blog in blogs %}   {% for entry in blog.entries %}     &lt;div class="{% cycle 'odd' 'even' %}" id="{{entry.id}}"&gt;       {{entry.text}}     &lt;/div&gt;   {% endfor %} {% endfor %}   I've tried obviating this by patching with the resetcycle tag offered here:    to no avail.  (The code didn't work for me.)  I've also tried moving my inner loop into a custom tag, but this also did not work, perhaps because the compile/render cycle moves the loop back into the outer loop?  (Regardless of why, it didn't work for me.)  How can I accomplish this simple task!?  I'd prefer not to create a data structure in my view with this information pre-compiled; that seems unnecessary.  Thanks in advance. What is the geometric meaning of the determinant of a matrix? I know that "The determinant of a matrix represents the area of ​​a rectangle." Perhaps this phrase is imprecise, but I would like to know something more, please.  Thank you very much. I accidentally closed it without voting, how can I find it? When I suggest declaring war on the target it says there's already a proposition to do so. Short story (+radio play?) in which astronauts outside their ship slowly drift apart in space Looking for a short story in which the crew of a spaceship for some reason find themselves outside the ship, drifting apart in outer space. They can communicate via their radios but are individually drifting apart.  One by one, they lose communication as they drift further away from each other.  Might have also been adapted into a radio play? Trigger a button click with JavaScript on the Enter key in a text box Natural Sort Order in C# Anyone have a good resource or provide a sample of a natural order sort in C# for an FileInfo array?  I am implementing the IComparer interface in my sorts. How do you find the value of $\sum_{r=0}^{44} \tan^2(2r+1)$? Problem:   Find the value of $$\sum_{r=0}^{44} \tan^2(2r+1)$$  Note: The angles here are in degrees.    I don't know how to solve this question because trigonometric simplifications didn't get me anywhere. I think there was a method to solve this question using complex numbers which I no longer remember. Any hint/help will be appreciated. Proving that the function is differentiable and $f'(0)=0$ Consider a function $f (x) $ such that $f (x)=\begin {cases} x^2 &amp;  \text {x=irrational}     \\ 0   &amp;\text {x=rational} \end {cases}$ Then prove that $f $ is differentiable  at $x=0$ and $f'(0)=0$ $$\text {Attempt} $$ .                  .  Now I consider two sequences $x_n=\frac {\sqrt{2}}{n},y_n=\frac {1}{n} $ one irrational one rational both going to $0$ as $n\to \infty $. But now the problem is how to consider LHD,RHD.Do we have to consider irrational case then rational case and if both limits go to same value then say that function is differentiable.I can easily see the graph on graphical calculator and verify the result but I am more interested in analytical method. When I look at the definition given from the Mac OS X Dictionary (I have set American English as interface language, and the dictionary used is then the New Oxford American Dictionary), I read:     the /ði/ /ð(ə)/ /ði/ [called the definite article]   adjective   1. denoting one or more people or things already mentioned or assumed to be common knowledge: what's the matter? | call the doctor | the phone rang. Compare with a.   • used to refer to a person, place, or thing that is unique: the Queen | the Mona Lisa | the Nile.   • informal denoting a disease or affliction: I've got the flu.   • (with a unit of time) the present; the current: dish of the day | man of the moment.   • informal used instead of a possessive to refer to someone with whom the speaker or person addressed is associated: I'm meeting the boss | how's the family?   • used with a surname to refer to a family or married couple: the Johnsons were not wealthy.   • used before the surname of the chief of a Scottish or Irish clan: the O'Donoghue.   I have never thought of an article as adjective. Is normally an article defined as adjective?