pow() function returning different result in C++
Is floating point math broken?
Proving that $\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$ converges jQuery/JavaScript: accessing contents of an iframe Let $ X \subset \mathbb{R}$ be a non-empty, open set and let $f: X \rightarrow \mathbb{R}$ be a continuous function. Show that the inverse image of an open set is open under f, i.e. show:  If $M \subset \mathbb{R}$ is open, then $f^{-1}(M)$ is open as well.  I think that this should basically follow from the definitions, but I'm still having some troubles with the proof.  Suppose $M \subset \mathbb{R}$ is open, then it follows that $\forall z \in M, \exists r &gt; 0: K_r(z) \subseteq M$  Since f is continuous it follows for each $x_0 \in X$ that for each $\epsilon &gt; 0 ,\exists \delta &gt; 0 : |f(x)-f(x_0)|&lt; \epsilon, \forall x \in X: |x-x_0|&lt;\delta$  Now I need to show that $f^{-1}(M)$ is open, so that for each $z \in M, \exists r&gt;0: K_r(f^{-1}(z_0)) \subseteq f^{-1}(M)$  I'm not sure how to prove this. I guess since $z = f(x)$ and since M is open we have that $\exists r&gt;0:\{z:|z-z_0|&lt;r\}\subseteq M$. And since f is continuous: $|f(x)-f(x_0)| = |z-z_0|&lt; \epsilon$ for all $|x-x_0|&lt;\delta$, but since $|x-x_0|=|f^{-1}(z)-f^{-1}(z_0)|&lt;\delta$ it somehow follows from that that $f^{-1}(M)$ is open?!  Which is just horrible and probably completely wrong. But I'm really confused from all the definitions right now and need some help please. Restored iOS 10 with broken Home button I restarted my iPhone 6 running iOS 10 and the home button is broken, so I was using Assistive Touch. But at the first launch, Assistive Touch is disabled. On iOS 10, we have to "Press home button to unlock".  What can I do? Show that an integer of the form $8k + 7$ cannot be written as the sum of three squares. I have figured out a (long, and tedious) way to do it. But I was wondering if there is some sort of direct correlation or another path that I completely missed.  My attempt at the program was as follows:   A number of the form, $8k + 7 = 7 (mod 8)$. That is, we are looking for integers a, b, c such that $a^2 + b^2 + c^2 = 7 (mod 8)$.   LONG and TEDIOUS way: $$(8k)^2 = 0 (mod 8)$$ $$(8k+1)^2 = 1 (mod 8)$$ $$(8k+2)^2 = 4 (mod 8)$$ $$(8k+3)^2 = 1 (mod 8)$$ $$(8k+4)^2 = 0 (mod 8)$$ $$(8k+5)^2 = 1 (mod 8)$$ $$(8k+6)^2 = 4 (mod 8)$$ $$(8k+7)^2 = 1 (mod 8)$$  That is, using three of these modulo there is no way to arrive at  $$a^2 + b^2 + c^2 = 7 (mod 8)$$ R code to solve for probability of normal distributions? I don't understand which R code I am supposed to be using to figure these problems out. I brief explanation of what the code is doing would be amazing. The problems below are two different applications I'm expected to understand and I don't get either. Thank you!  Problem 1. Let Z~N(0,1) For each of the following, find c such that the equation is true.   a) P(Z&lt;c) = 0.8413  b) P(0&lt;Z&lt;c) = 0.3051  c) P(-c&lt;Z&lt;c) = 0.8664   Problem 2. X~N(3,4) compute  a) P(X&gt;3)  b) P(1&lt;X&lt;8)  c) P(-1.5&lt;X&lt;1) Volume form on $(n-1)$-sphere $S^{n-1}$ Let $\omega$ the (n-1) form on $\mathbb{R}^n$ $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n}$$ Show that the restriction of $\omega$ to $S^{n-1}$ in precisely the volume for this sphere.  What I did was: $\omega$ never vanish on the sphere, because, defining $\sigma\in \Omega^{n-1}(S)$ for $$\sigma_{p}(v_{1},...,v_{n-1})=det(p,v_{1},...,v_{n-1})$$ and $i:S^{n-1}\rightarrow \mathbb{R}^{n}$ the inclusion function, then $\omega=i^{\ast}(\sigma)$ then $\omega\not=0$ and therfore is a volume form.   How proof that $\omega$ is the volume form?   The first thing that comes to mind is show that $\int_{S^{n-1}}\omega=Vol(S^{n-1})$ but I have serious problems with the definition, I think that is to much.   How see that $\omega$ is invariant on $\mathbb{R}^{n}$ under action of $O(n)$ I'm trying to create record Type selection screen in LWC, but getting this error  afterRender threw an error in 'c:createContact' [Cannot read property 'recordTypeInfos' of undefined]  HTML:  &lt;template&gt;     &lt;lightning-combobox name="recordType" label="Record Type" placeholder="Choose Status"                         value={value}                          options={recordTypeId}&gt;      &lt;/lightning-combobox&gt; &lt;/template&gt;   JS:  import { LightningElement, track, wire, api } from 'lwc'; import { getObjectInfo } from 'lightning/uiObjectInfoApi'; import CONTACT_OBJECT from '@salesforce/schema/Contact';  export default class CreateContact extends LightningElement {      @track statusOptions;     @track value;     @api objectApiName;      @track objectInfo;      @wire(getObjectInfo, { objectApiName: CONTACT_OBJECT,})     objectInfo;       get recordTypeId() {         // Returns a map of record type Ids          console.log(this.objectInfo);         const rtis = this.objectInfo.data.recordTypeInfos;         return rtis;     }  } Graphics horribly corrupted on Ubuntu 16.04 in VirtualBox, showing random strips of pixelated colors Inside a VirtualBox VM, I have installed myself an Ubuntu 16.04.1 LTS Server installation. However, when I installed the ubuntu-desktop package and rebooted, this happened:    Exact command I used: sudo apt-get update; sudo apt-get install ubuntu-desktop ..and then a reboot command.  This just seems like a corrupted top-left portion of the screen, as my cursor can appear in view (but corrupted):    This is a purely graphical issue because I can use the keyboard to successfully progress to a desktop:    Note: On startup it shows correct graphics for a split-second then reverts to glitch.    Any help on this? I am familiar with what partitions are, but I saw , and realised I didn't know what "aligned partitions" are.  So what does it mean to "align" partitions? What are the benefits, and the downsides? ufw: how to forward single port to another port on different interface I have two interfaces eth0 (192.168.1.2) and eth1 (192.168.3.2).  I receive data from 8080 at eth0. I want to forward it to eth1 port 80.  I'm using UFW, because i don't understand iptables and how to add it at the beggining of /etc/ufw/before.rules.  I've already edited DEFAULT_FORWARD_POLICY="ACCEPT" on /etc/default/ufw and already edited net.ipv4.ip_forward=1 on /etc/ufw/sysctl.conf Is there a streaming API for JSON? What rules say that summoning undead is evil, and what rules say why? Is the orbit in Schwarzshild metric a path with maximal proper time? In curved spacetime particles follow timelike geodesics, which should have maximal proper time (at least locally). I thought this path usually corresponds to a global maximum, and there are only strange exceptions. Propably I was wrong, or I really don't understand something, because it doesn't seem to be true for the case of orbitals in Schwarzschild geometry.  The Schwarzschild metric is (with +--- signature)  $$ g_{\mu\nu} = \textrm{diag}\left(1-\frac{2GM}{r},-\left(1-\frac{2GM}{r}\right)^{-1},-r^2,-r^2\sin^2\theta\right) $$  An observer with fixed $r,\theta,\phi$ coordinates will age according to $d\tau_1 = \sqrt{g_{tt}}dt$. An observer, which orbits the center at the same constant radius with $\theta=\pi/2$ will age according to $d\tau_2 = \sqrt{g_{tt}-\omega^2r^2\sin^2\theta}dt$, so he ages slower during the same amount of coordinate time $dt$. If the two observers start from the same point, they will meet again after a full rotation of the orbiting observer, and the orbiting observer will age less. What I don't understand is that the orbit is the geodesic (if the angular velocity is set properly), so why didn't he age more? He should have been on the path with maximal proper time. Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse:   let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$,  then as $R$ is closed under multiplication $\prod_{n=1}^k\ x_i=x_j$,  therefore by canceling $x_j$ we get $x_1x_2\cdots x_{j-1}x_{j+1}\cdots x_k=1 $,  by commuting any of these elements to the front we find an inverse for first term, e.g. for $x_m$ we have $x_m(x_1\cdots x_{m-1}x_{m+1}\cdots x_{j-1}x_{j+1}\cdots x_k)=1$, where $(x_m)^{-1}=x_1\cdots x_{m-1}x_{m+1}\cdots x_{j-1}x_{j+1}\cdots x_k$.   As far as I can see this is correct, so we have found inverses for all $x_i\in R$ apart from $x_j$ if I am right so far. How would we find $(x_{j})^{-1}$? What's the standard "roadmap" to learning quantum physics? I'm really interested in quantum physics and would like to learn more. However, I don't know where to start and in what order I should learn things. So, ideally I'm looking for some sort of roadmap of what to learn. What physics topics do I need to know to start learning about quantum mechanics? (In addition to the mathematical topics mentioned at )  My current knowledge is mostly popular science stuff, like tv shows on Discovery Science and National Geographic Channel. So I have a basic understanding of some basic principals. There's also a recent lecture from Brian Cox that I have watched which gave a bit more in-depth information. Sequence of rationals with an irrational limit have denominators going to infinity What are the general rules regarding when can you refreeze food that was frozen and then thawed in the fridge? and what are the reasons behind them? (Hygiene? Food quality?). Let's assume for the sake of the argument that the items were in a non-frozen state only a few hours.  To be more specific, I'm interested in the application of these rules for  - Ginger garlic paste - Dough - Indian Curries - Green Chilly paste - Frozen green peas I lost my Italian residence permit abroad, how can I obtain a duplicate so that I can fly back to Italy? Sweden is going to ban me from flying if I can't show them my lost permesso di soggiorno from Italy. It was issued in Florence because my old one expired while I was in school. How do I get a copy?