I just read on wikipedia that the the complex projective line is homeomorphic to the riemann sphere. How do I prove this? But, before that I have an extremely silly doubt that has been eating me. In the complex projective line, antipodal points are identified, but in the Riemann sphere they are clearly distinct. Then how can the two be homeomorphic.   I found out the following map from the complex projective line to $S^2$. But how do I prove this is a homeomorphism? Finding the inverse of this functions seems complicated, and then how do I prove that the inverse map of a open subset is a open subset, and vice versa?  $f:\mathbb{C}\mathbb{P}^1 \to S^2 = [z:w] \to \frac{(2Re(w\bar{z}), 2Im(w\bar{z}),|w|^2-|z|^2)}{(|w|^2+|z|^2)} $
Endow $S^3$ with its standard smooth structures from the stereographic projections, i.e. $$\chi_1:S^3 \setminus \{(0,0,0,1) \} \to \mathbb{R}^3, \chi(x,y,z,t) = \frac{1}{1-t}(x,y,z) $$ and $$\chi_2: S^3 \setminus \{(0,0,0,-1) \}   \to \mathbb{R}^3, \chi_2(x,y,z,t) = \frac{1}{1+t}(x,y,z).$$   Do the same for $S^2$, call the charts $\eta_1, \eta_2$ (in the same order as the ones for $S^3$). Now, consider the complex projective space $\mathbb{C} \mathbb{P}^1$ endowed with the smooth structure given by the charts $\{(U_0, \varphi_0), (U_1, \varphi_1)\},$ $$U_0 = \{[z_0:z_1] \ \mid \ z_0 \neq 0 \} \text{ and } U_1 = \{[z_0: z_1] \ \mid \ z_1 \neq 0 \},$$ where $[x:y]$ represents the complex line through the point $(x,y)$ and $0$ in $\mathbb{C}^2$, and $$\varphi_0([z_0: z_1]) = \frac{z_1}{z_0} \text{ and } \varphi_1([z_0: z_1]) = \frac{z_0}{z_1}. $$   Now define the Hopf maps: $$h: S^3 \to S^2, h(x,y,z,t) = (x^2+y^2-z^2-t^2, 2(yz-xt), 2(xz+yt)) $$ and $$H:S^3 \to \mathbb{C} \mathbb{P}^1, H(x,y,z,t) = [(x+iy):(z+it)]. $$  Is there an easier way to prove that $h$ is a well-defined smooth submersion and that $H$ is a smooth submersion (with respect to the respective smooth structures)? By this I mean we can of course compute $\eta_i \circ h \circ \chi_j^{-1}, \forall i,j \in \{1,2 \}$ and $\varphi_k \circ H \circ \chi_l^{-1}, \forall k,l \in \{1,2 \}$ and prove that they are smooth (in the classical real-analysis sense) and that they are submersions, but this is a very tedious process. So, are there easier ways to do this? Will this maybe be easier if we switch the smooth structure of the unit spheres from stereographic projections to, say, normal projections (though we increase the number of combinations of charts to verify from 4 to 48), since they induce the same smooth structure?
Android Studio shortcuts are not working I use Android Studio in Ubuntu and everything was working fine. Since two weeks that shortcuts stopped working, Ctrl+C, Ctrl+V, Ctrl+Z. I tried to change Keymap in Settings but without success.  What could be the possible problem? Rectifiability of a curve Let $f$ be a function defined on $[0,1]$ by   $$f(x) = { 0, \text{ if } x = 0} $$ $$f(x) = { x \sin \frac 1 x , \text{ if } 0 &lt; x \leq 1} $$  Prove that the curve $\{(x, f(x)) : x \in [0,1]\}$ is not rectifiable.   I'm not sure how to approach this. The general idea seems logical, we're proving that the length of the curve is infinite, but the method seems difficult to find. I have just bought a Nikon 50mm f/1.8D prime lens.  I took this picture (leg of my glasses frame on a mouse, and a LED desk lamp is on top of it) : f1.8, 1/80s, ISO 200   The purple fringing is right in the middle of the picture. Does it happen very often to this lens? MacBook Pro 2020 Fan Issues I recently purchased a MacBook Pro 2020 13&quot; (1.4 GHz, 8GB Ram, i5). For some reason, when I open Chrome and watch Twitch my computers fans turn on and go crazy. After opening Activity Monitor, I noticed that Google Chrome Helper (Renderer) uses around 50-80% of my CPU. Also, sometimes when I run code on my MacBook Pro, the fans turn on when the program is compiling/running but slow down a few minutes after terminating the program. Also, when watching Twice on Safari, the fans don't come on. However, around 50-60% of the CPU is still being used. Should I be concerned about the CPU usage (I am nervous because I only recently purchased the laptop)? Inversion/non-inversion in wh-questions with long phrases after the wh-words Can a sentence like this:      "I don't know who the first man that made such and such thing in such and such place was,"    be grammatically correct if we don't put "was" at the end of the long phrase, that is, if we write:      "I don't know who was the first man that made such and such thing in such and such place"?    I can see in Google Books examples that in such cases the verb is often put after the wh-word, but I don't know if there is a grammar rule to support this. Some examples:  "We do not know who was the first man who ascended above a poor and humble people to become Egypt's first king ..."  "... we do not know what was the ultimate judgment of the various members of the community ..."  "I do not know who was the first to suggest a connection between the problem of free will and the breakdown ..."  "I do not know what was the date of this change in me, nor of the train of ideas ..."  "We do not know what was the primitive text from which Codex Bezae derived its Latin or its Greek ..."  "We do not know what was the practice in the days of the monarchy, but the story of Athaliah shows ..." How do I change JPanel inside a JFrame on the fly? To put it simple, there's a simple java swing app that consists of JFrame with some components in it. One of the components is a JPanel that is meant to be replaced by another JPanel on user action.  So, what's the correct way of doing such a thing? I've tried  panel = new CustomJPanelWithComponentsOnIt(); parentFrameJPanelBelongsTo.pack();   but this won't work. What would you suggest? What is the www-data user? I have tried using glm::rotate and such. glm::translate works fine, but rotate doesn't work (if I rotate before/after a translation).  Here is my current setup:  glm::vec4 Off = glm::vec4(0, 0, 0, 1); glm::vec4 result = Util::createTransform(offset) * Off;  for (int i = 0; i &lt; Tvertex.size(); i++) {     Tvertex[i] += glm::vec3(result.x, result.y, result.z);     Tcolors[i] = currentColor; }   And here is my Util::createTransform:  glm::mat4 Util::createTransform(glm::vec3 pos) {     glm::mat4 trans = glm::mat4(1.0);     trans = glm::translate(trans, pos);     return trans; }  glm::mat4 Util::createTransform(glm::vec3 pos, glm::vec3 rot) {     glm::mat4 trans = glm::mat4(1.0);     trans = glm::rotate(trans, glm::radians(rot.x), glm::vec3(1, 0, 0));     trans = glm::rotate(trans, glm::radians(rot.y), glm::vec3(0, 1, 0));     trans = glm::rotate(trans, glm::radians(rot.z), glm::vec3(0, 0, 1));     trans = glm::translate(trans, pos);     return trans; } Given two separate annulus with centers $[C_1,C_2]$ and their corresponding radii being $[R_1,r_1]$ and $[R_2,r_2]$ respectively, larger radius being $R$. There are methods to look at whether they are overlapping or not. If they overlap how to find the overlapping area. Is there any close form expression for it?  equation of first annulus   $r_1^2&lt;(x-x_{c1})^2 + (y-y_{c1})^2 &lt;R_1^2$   equation of second annulus  $r_2^2&lt;(x-x_{c2})^2 + (y-y_{c2})^2 &lt;R_2^2$  If there are three circles overlapping, we get overlapping area as solution to set of 4 in-equations by least squares approach(Though we don't end up getting total area but, area within tangents of circles ). My trackpad is broken so I can tap-to-click but i cannot drag and drop. This is really a problem when it comes to doing certain things.   Does anyone know how to program a sticky key to turn a click into a click-select that can then be dragged about?   I can't resize my browser or open window and I can't drag applications into the folder when installing etc.  free solutions only please.  A mid 2009 macbook-pro 13". 2.53 Ghz Intel Core 2 Due.   Running os lion (grudgingly). Equivalence Relation problem Convert String to Integer in QGIS tables I want it to know how to convert string field values to integer values in QGIS Dufour... there are numbers and strings in the same field, when I use toreal(string) it stops and it won´t create the field. jQuery - auto size text input (not textarea!) Marking plural of code words In my blog (which is about programming) I often use reserved words from different programming languages.  Like this:     When column is nullable in both tables, this query won't return a match of two NULLs for the reasons described above: no NULLs are equal.   Here, NULL is a programming language keyword.  For numerous reasons, I mark them with &lt;code&gt; tag so they are rendered with a fixed width font, not translated into the foreign languages by Google Translate, can easily be seen on the page etc.  In the example above, which would be the best way to mark the plural?   NULLs NULL s NULL's NULLs NULLS   Option 1 looks ugly on the page (the word and the plural marker are rendered in different colors and fonts)  Option 2 adds a space between the word and the plural marker  Option 3 looks nicest but is not correct  Options 4 and 5 may leave the reader under impression that NULLS is a reserved word (which is not true) and this may be confusing. Also, Google Translate would leave the word as is which would be definitely wrong in other languages.  What would be the least of 5 evils? Temperature in space Temperature is a measure of kinetic energy transferred to particles, henceforth,  being vacuum, temperature cannot be measured. But then, there is cosmic background radiation. It is the leftover heat from the Big Bang, but then heat in space doesn't make sense, how does this radiation, heat up space time when it technically cannot be heated up? Moreover why is it referred to as heat when it is approximately just $2.5$ Kelvin, which is just above absolute zero? DES Initialization Vector in C# In doing instanton problems or when connecting quantum field theory to statistical mechanics, I often see people trying the Wick rotation trick by defining an imaginary time $\tau\equiv it$. So, in this case, we should make sure that $\tau$ is imaginary, right? However, I often see places where people seem to handle $\tau$ as a real number, for example setting $\tau$ to $\pm\infty$. Moreover, I remember in one of my quantum mechanics classes, the professor said the Euclidean time $\tau$ was real. I am so confused and need some good explanations. The rules for the monk ability Tongue of the Sun and Moon state:     Starting at 13th level, you learn to touch the ki of other minds so that you understand all spoken languages. Moreover, any creature that can understand a language can understand what you say.   PHB, pp.79   The text says that this ability is something you learn to do, not something that simply happens to you, so it's not necessarily always on. Further, it seems logical to say that this is something you would do actively. That is, you could choose not to touch someone's mind, thereby excluding them from understanding you. You could also choose not to understand someone yourself, just in case they're telling , or using a subliminal trigger.  Assuming you can choose not to touch someone's mind, let's reason further. Suppose you're speaking and don't know that there is someone listening in nearby. Would they understand you, i.e. is the default that you are or aren't touching someone's mind? Similarly, if you are speaking to a large group of people, would you have to focus on all of them at once for them to understand you? Would this take effort? Is there a range?  My questions are:   Can you exclude some characters from understanding you while allowing others to do so? Can you decide not to understand someone who is speaking?   Basically, these boil down to: Can you choose not to touch someone's mind?  Then, assuming that the answer to the above is yes:   Do you have to know someone is listening in order for them to understand you? Is there a limit to the number of people you can affect? A range? Will my software work on the next version of Windows I have some software and I'd like to know if it works on the next version of Windows (Windows 10). ..and I don't mean finding out why it doesn't compile, I mean serious debugging: breakpoints, figuring out value of variables at a given moment during run/compile, stepping, examining a stack (is there one?), etc. All the things that we take for granted when debugging a "real" programming language. Since TeX is "turing-complete", I would like to be able to really see what is going on internally.   I'm guessing that a GUI interface for the debugger would be too much to ask, but if it exists, I'd love to hear about it.