I may be wrong but I have always thought that the brain doesnt have any nerve endings, so why is it we get headaches and where does this pain come from?  I have heard lack of sleep, hydration or stress can causes headaches but I am interested in why and where the pain comes from this.
I've read many years ago in books, that the brain has no nerves on it, and if someone was touching your brain, you couldn't feel a thing.  Just two days before now, I had a very bad migraine, due to a cold. It's become better now, but when I had it I felt my head was going to literally split in half, as the pain was literally coming from my brain.  So it lead me to the question: How come people can get headaches if the brain has no nerves?
Single-word synonym for "immensely valuable"? I am looking for some (may be rarely know to nonnative English speakers) synonyms for  "immensely valuable" or "extremely valuable". Checking my understanding: $1 - 1 + 1 - 1 + 1 - ... = \frac{1}{2}$ I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...) = 1 - S$$ Therefore, $2S = 1$, so $S = \frac{1}{2}$. QED   I was under the impression that this sum doesn't converge, and therefore this step of the proof is invalid:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$   That is, it's not even valid to suppose that the sum is equal to some (implicitly real or complex) number $S$, and therefore the reasoning that follows is meaningless because $S$ doesn't exist and therefore has any properties we want it to have.  Is the reasoning I've given above correct? That is, is it correct for me to claim that the proof fails because $S$ doesn't exist?  (I've heard that there are techniques for evaluating divergent integrals by using complex analysis and Taylor series, so perhaps there's another way to prove that the summation is $\frac{1}{2}$; I just wanted to see whether my reasoning is sufficient to explain why this particular proof is incorrect. If the proof actually is correct, then I stand corrected!)  Thanks! Consider the Fibonacci numbers $F(0) = 0; F(1)=1; F(n) = F(n-1) + F(n-2)$.  Prove by induction that for all $n&gt;0$,  $$F(n-1)\cdot F(n+1)- F(n)^2 = (-1)^n$$  I assume $P(n)$ is true and try to show $P(n+1)$ is true, but I got stuck with the algebra. How do we reach $P(n+1)$ from $P(n)$? Also, strong induction may be used instead. Finding rank-nullity of a given matices Question about the proof of Ramsey's Theorem I was reading up on a proof of Ramsey's Theorem and I can't understand this part of the proof:      Pick a vertex $v$ from the graph, and partition the remaining vertices into two sets $M$ and $N$, such that for every vertex $w$, $w$ is in $M$ if $(v, w)$ is blue, and $w$ is in $N$ if $(v, w)$ is red. Because the graph has $R(r - 1, s) + R(r, s - 1) = |M| + |N| + 1$ vertices, it follows that either $|M| \geq R(r − 1, s)$ or $|N| \geq R(r, s − 1)$.   Why is that $|M| \geq R(r − 1, s)$ or $|N| \geq R(r, s − 1)$? I know from the proof of $R(3,3) = 6$ that given a vertex, at least $3$ are guaranteed to be either red or blue, so what is the pigeonhole argument for the general case? Integral solutions of hyperboloid $x^2+y^2-z^2=1$ Are there integral solutions to the equation $x^2+y^2-z^2=1$? Remember all opened files with split windows and their positions in Emacs How can I listen to communication between a process and a serial port? On a Gentoo Linux machine, a process is communicating with another computer via serial port (/dev/ttyS1). I want to listen to this communication (i.e. what is being sent/received by the process) without further interfering.   How can I achieve this with just basic Linux tools? I even can change the serial port the process is communicating with, so maybe I can 'shortcut' two devices (/dev/ttyS1 and /dev/xxx) listen to all communication, when all output to /dev/ttyS1 is forwarded to /dev/xxx and vice versa. But how do I do it? Something with socat? Detection of the Electric Charge of a Black Hole By the "No Hair Theorem", three quantities "define" a black hole; Mass, Angular Momentum, and Charge. The first is easy enough to determine, look at the radius of the event horizon and you can use the Schwarzschild formula to compute the mass. Angular Momentum can be found using the cool little ergosphere Penrose "discovered". However, I don't know how to determine the charge of the black hole.   How can an electromagnetic field escape the event horizon of a Reissner-Nordström black hole? Is there any experiment we could theoretically do to a black hole to determine its charge? In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions means "iff".  For example I am reading "Essential topology" and the following definition is written:     In a topological space $T$, a collection $B$ of open subsets of $T$ is said   to form a basis for the topology on $T$ if every open subset of $T$ can be written as a union of sets in $B$.   Should I assume the converse in such a case? Should I assume that given a basis $B$ for a topological space, every open set can be written as a union of sets in $B$?  This is just an example, I am not asking specifically about this sentence. find a $\int_{x}^{g(x)} f(t)\,dt = 1$ 'pyrcc5' is not recognized as an internal or external command I installed QGIS 3.0 on Windows 10. I get this error everytime i run the pyrcc4 or pyrcc5 command:  'pyrcc4' is not recognized as an internal or external command  Even if I find a place with a pyrcc4.exe file or a place with apyrcc5.bat file and add the path to the environment variables I get the same error. In the previous installation of osgeo4w i had a pyqtdev folder or something similar appear I think it was on C/. Was there supposed to be such a folder, do I need to download pyqt separately? Why does this identity hold for Fejér Kernels? How can I access Apache (on VirtualBox guest) from host? I installed Apache on an Ubuntu VM. When I go into the guest and load up Firefox I can see that Apache is working fine when I browse to localhost.  I would like to access this same page through the host. I've tried using the IP address of the guest but nothing comes up. Ideally I would love to set up my Windows 7 (host) to request http://guestserver.com of the Apache server on the guest. How can I do this? Show: $\sum_{k=1}^{\infty}\mu(\left\{f\geq k\right\})\leq\int f\, d\mu\leq\sum_{k=0}^{\infty}\mu(\left\{f>k\right\})$ Show that für $f\colon (\Omega,\mathcal{A})\to\mathbb{R}_{\geq 0}$ measurable, it is     $$ \sum_{k=1}^{\infty}\mu(\left\{f\geq k\right\})\leq\int f\, d\mu\leq\sum_{k=0}^{\infty}\mu(\left\{f&gt;k\right\}). $$      I do not know if one needs it here, but we defined the integral of a non-negative measurable function $f\colon (\Omega,\mathcal{A})\to (\overline{\mathbb{R}},\overline{\mathcal{B}})$ by $$ \int_{\Omega}f\, d\mu=\sup\left\{\sum_{k=1}^{n}(\inf\left\{f(x): x\in A_k\right\})\mu(A_k): n\in\mathbb{N}, A_1,\ldots,A_n\in\mathcal{A}\mbox{ disjoint },\Omega=\bigcup_{k=1}^{n}A_k\right\}. $$    My first idea was to use this definition, but I did not come along with it...  Maybe you can help me to show that?  (I think, maybe it has something to do with monotone convergence theorem or majorated convergence?) How do I concatenate const/literal strings in C? I'm working in C, and I have to concatenate a few things.  Right now I have this:  message = strcat("TEXT ", var);  message2 = strcat(strcat("TEXT ", foo), strcat(" TEXT ", bar));   Now if you have experience in C I'm sure you realize that this gives you a segmentation fault when you try to run it. So how do I work around that? What if a light clock travels perpendicular to mirrors that make up the clock? How to map stock ticker symbols to ISIN (International Securities Identification Number)? How can I can convert ticker symbols (which include information on the exchange) to an International Securities Identification Number (ISIN) (which contains no information on the exchange)? Also I am interested in the other way round — given an ISIN and a market identification code (MIC) how can I determine the ticker symbol? Any databases around? Why is.vector on a data-frame doesn't return TRUE? What is the joint distribution of $Z=\min(X,Y)$ and $I_{Z=X}$?