Sun-path on the sky
Why do all the planets of the solar system orbit in roughly the same 2D plane?
$n$ Lines in the Plane How am I to "[u]se induction to show that $n$ straight lines in the plane divide the plane into $\frac{n^2+n+2}{2}$ regions"? It is assumed here that no two lines are parallel and that no three lines have a common point. Further, this is not a non-Euclidean problem, but I wouldn't mind a discussion on the non-Euclidean nature of the problem.  I was thinking it might be easier to show using the unit sphere.    A GOOD REFERENCE I FOUND THAT WILL BE USEFUL TO ANYONE COMING ACROSS THIS PROBLEM IS Concrete Mathematics, Graham, Knuth, Patashnik pp. 4–8. App Store is the wrong language Where can I have a genuine Ainu cultural experience in Hokkaido? During my next trip to Japan I may end up in Hokkaido. I've read something about this mysterious indigenous group called Ainu/Aynu/Aino/Ezo, but I must admit my understanding of the whole thing is rather limited right now.   What I wish to know is: is there such a thing like an authentic Ainu village/town/whole-region where Ainu people live according to their tradition, as opposed to something staged for tourists?  Edit: given the comments below I wish to specify better what I mean with the word 'authentic'. Of course, I don't expect pristine landscapes and Ainu people living in blissful harmony with fluffy penguins (semi-quote) but I won't be surprised to find out there still are authentic villages in remote areas of Hokkaido where mass tourism is not an issue at all.  People are subject to tourist's attention (tourist coming from different countries, at least) everywhere, not only when they "made a conscious decision to live a traditional life". This doesn't actually mean anything. Indian people are surprisingly differents (in local customs) from Italians and Italians take a lot of pictures during their travels there, still the tourism isn't banned in India. Question about a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$ About a bijection between $\mathbb R$ and $\mathbb R \cup \{\infty\}$  If we'll take a line $f(x)=\begin{cases} ax &amp;, x\in\mathbb R\\ 0 &amp;, x=\infty  \end{cases}$ then we'll have a bijection.   But about this function: $g(x)=\begin{cases} x &amp;, x\in\mathbb R\setminus \{0\}\\ 0 &amp;, x=\infty  \end{cases}$ ? it appears to be a bijection and it's even simpler. Some questions might be asked on three different sites, each site needing three different answers. For example, a question about a text editor.  So perhaps there should be a way to link the questions on the three different sites. Adding a link manually in the questions is certainly possible, but perhaps a feature officially supporting this will be beneficial. Error while running Quickly What is the secret of making a really juicy burger? As the title says; What is the secret of making a really juicy burger? How do I hide javascript code in a webpage? Install software from packages, or build it from source? This is a  about when to use packaged software and when to build and install it yourself.   When should I install software from packages provided by my OS distribution, instead of building and installing it myself from source?  When is it better to build from source? Why is this sequence of functions not uniformly convergent? For the following sequence of functions and its limit function, we can see that $f_n(x)$ is clearly pointwise convergent  $$f_n(x) = x^n\text{ }\forall x\in[0,1]\text{ and }\forall n\in\mathbb N^*\\ f(x) = \begin{cases}0&amp;\text{if } x\in[0,1)\\1&amp;\text{if } x=1\end{cases}$$  However, I was wondering why this is not uniformly convergent. The condition for uniform convergence is:  $$|f_n(x) - f(x)| &lt; \epsilon,\ \ \ \forall x \text{ when } n &gt; N$$  Now most sources present an argument along the lines of: assume that $f_n(x)$ is uniformly convergent and that $0 &lt; x &lt; 1$, this means that $x^n&lt;\epsilon$ whenever $n&gt;N$. Specifically, this would mean $x^{N+1}&lt;\epsilon$ for some fixed $N$. But if we now pick $x$ such that $1 &gt; x &gt; ε^{\frac{1}{N+1}}$, then this would lead to a contradiction, therefore $f_n(x)$ is not uniformly convergent.  However, I was wondering why couldn't we take $n$ to infinity. If $0 &lt; x &lt; 1$, then $\lim_{n\rightarrow \infty} |f_n(x) - f(x)| = 0$ (which is less than $\epsilon$). Now since $|f_n(x) - f(x)|$ will always be $0$ if $n &gt; \infty$, then wouldn't this be uniformly convergent? So this is the situation:   My brother received a PS4 on his birthday. He doesn't have a PS+ account, and I don't have a PS4, so I activated it as my primary PS4. Based on what I've read on the FAQ, any user can play my digital games and do online gaming since I activated the system as my primary PS4. A friend offered his account so we can play his digital games. He has a primary PS4 activated.   The question is, can we really play his games without any issues? For example, can we download GTAV from his account and play it using any account on our PS4? How can I automatically enable large text when logging in to the ubuntu session? Let's say that I have a folder, within my Documents folder, that has files I want no one to access without a password.  Is there a way to lock down that folder so that it's password protected / encrypted?  In fact, is it possible to password-protect any individual folder? I used to use a lot ctrl+alt+numpad to move windows, how can I use this feature on ubuntu 18 like it was default before?  Thank you in advance Can you make a bonus-action attack from Two-Weapon Fighting before you take your action? The rules on  state:     When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand.   However, the description of  states:     You choose when to take a bonus action during your turn, unless the bonus action’s timing is specified   So, can you attack with your bonus action first? $$\int_0^{2\pi} \frac{1}{2+\sin(x)} \ dx$$  $$\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$  $$z=\alpha(x)=e^{ix}$$   $$\sin(x)=\frac{z^2-1}{2zi}$$ $$2+\sin(x)=\frac{4zi+z^2-1}{2zi}$$ $$\frac{1}{2+\sin(x)}=\frac{2zi}{4zi+z^2-1}$$    $$\int_\alpha \frac{2zi}{4zi+z^2-1} \ \ \frac{1}{zi}$$    $$4zi+z^2-1=0$$ $$z_1=\frac{-4i+2\sqrt{3}i}{2}$$ $$z_2=\frac{-4i-2\sqrt{3}i}{2}$$    $$\rvert z_1 \rvert &lt;1$$ $$\rvert z_2 \rvert &gt;1$$   Using Residue theorem:   $$\int_\alpha \frac{2}{4zi+z^2-1}=2 \pi i \ \lim_{z\rightarrow-2i+\sqrt{3}i} \ \frac{2}{z+2i+\sqrt{3}i}=\frac{2 \pi}{\sqrt{3}}$$    Is it correct?  Thanks! Solve simple trig equation When we measure the speed of a moving element we do it with the help of a reference frame. Now if we need to measure the speed of time, is it possible? Does time actually have a speed? How can I get the page numbers only of a pattern in a pdf file, regardless if the pattern is multiline? Using Rule of Three to obtain confidence interval for a binomial population I have a large population of data instances (say, 1000 instances) that are either of class1 or of class2. I would like to obtain a confidence interval for how many instances are of class1 without exhaustively checking all instances. I have randomly sampled 50 instances, and all 50 were of class1. I used the  to determine that a 95% confidence interval for the percentage that an instance is of class1 is  [0.94, 1].  From my sampling, I know that at least 50 instances are of class1. For the remaining 1000 – 50 = 950 instances whose classes are unknown, I assume I can apply the [0.94, 1] confidence interval found above. Therefore, can I conclude that, with a 95% confidence, there are at least 50 + (1000 – 50)(0.94) = 943 instances from the population of 1000 that are of class1?  If this conclusion isn’t statistically sound, how can I obtain a confidence interval for class1?