We see Rainbows many times, but we always see them Horizontally or from Sideways. Why can't we see it from under it?/From the point where it starts or ends(end points)? I also want to stress on that, I am asking on viewing part and not about the formation, reason, causes of a rainbow.
A  is formed when a raindrop refracts light, but why then does the whole sky not become a huge rainbow when it rains? Would the light not be dispersed into ordinary white light? What causes it to look as if each end is nearly touching the ground?
evaluate the limit as $x$ approaches $0$ of $(\cos x-1)/x$ without l'Hopital's rule I know with l'Hopital's rule it becomes $-\sin(x)$ which has the limit $0$. However, I have been wondering how to evaluate this limit without l'Hopital's rule. How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$? Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$  Apart from induction, I tried with  to check the validity, but I can't think of an easy (manual) alternative.  Please suggest an intuitive/easy method. What are the zero divisors of $C[0,1]$? Suppose you have a ring $(C[0,1],+,\cdot,0,1)$ of continuous real valued functions on $[0,1]$, with addition defined as $(f+g)(x)=f(x)+g(x)$ and multiplication defined as $(fg)(x)=f(x)g(x)$. I'm curious what the zero divisors are.  My hunch is that the zero divisors are precisely the functions whose zero set contains an open interval. My thinking is that if $f$ is a function which is at least zero on an open interval $(a,b)$, then there exists some function which is nonzero on $(a,b)$, but zero everywhere else on $[0,1]\setminus(a,b)$. Conversely, if $f$ is not zero on any open interval, then every zero is isolated in a sense. But if $fg=0$ for some $g$, then $g$ is zero everywhere except these isolated points, but continuity would imply that it is also zero at the zeros of $f$, but then $g=0$, so $f$ is not a zero divisor.   I have a hard time stating this formally though, since I'm only studying algebra, and not analysis. Is this intuition correct, and if so, how could it be rigorously expressed? What is the "Wanna Cry" ransomware's possible impact on Linux users? It's just come to light that there's a $300 ransom you have to pay because ransomware targeting Microsoft Windows has encrypted your data. What steps do Linux users need to protect from this if for example they are using wine?  This ransomware is widely reported to be based on a tool developed by the NSA to hack into computers. The NSA tool was used by a hacker group called the Shadow Brokers. The code can be found in .  Microsoft released a patch (MS17-010) against this vulnerability on March 14, 2017. The mass infection is reported to have begun spreading on April 14th. This is discussed .  As I haven't booted Windows 8.1 in 6 to 8 weeks, can I apply this patch from Ubuntu without booting Windows first? (After research it may be possible ClamAV could report the vulnerability from the Linux side looking into Windows partition but it's unlikely it could apply the patch. The best method would be to reboot into Windows and apply patch MS17-010.)  Individuals and small companies who subscribe to Microsoft Automatic Updates are uninfected. Larger organizations who delay apply patches as they are tested against organization intranets are more likely to be infected.  On May 13, 2017, Microsoft took the extraordinary step of releasing a patch for Windows XP which has been unsupported for 3 years.  No word if wine is doing anything about a security update. It was reported in a comment below that Linux can be infected too when users run .  An  registered a domain name that acted as a kill-switch to the ransomware. I presume the non-existent domain was used by the hackers on their private intranet so they didn't infect themselves.  Next time they will be smarter so don't rely on this current kill-switch. Installing the Microsoft patch, which prevents exploiting a vulnerability in the SMBv1 protocol, is the best method.  On May 14, 2017 Red Hat Linux said they are not affected by "Wanna Cry" ransomware. This might mislead Ubuntu users along with Red Hat, CentOS, ArchLinux and Fedora users. Red Hat supports wine which answers below confirm can be effected. In essence Ubuntu and other Linux distro users googling this issue might be mislead by the Red Hat Linux Support answer .  May 15, 2017 Update. Over the last 48 hours Microsoft released patches called KB4012598 for  to protect against "Wanna Cry" ransomware. These Windows versions are no longer on automatic updates. Although I applied security update MS17-010 on my Windows 8.1 platform yesterday, my old Vista Laptop still needs patch KB4012598 downloaded and manually applied.       Moderator note: This question is not off topic - it asks about whether or not any Linux users need to do any steps for protecting against the risk.      It is perfectly on topic here, because it's relevant to Linux (which Ubuntu is), and it's also relevant for Ubuntu users running Wine or similar compatibility layers, or even VMs on their Ubuntu Linux machines. Why do ICs have NC (No Connection or No Function) pins? Integral of Sinc Function Squared Over The Real Line I am trying to evaluate $$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx $$ Would a contour work? I have tried using a contour but had no success. Thanks.  Edit: About 5 minutes after posting this question I suddenly realised how to solve it. Therefore, sorry about that. But thanks for all the answers anyways. Function whose image of every open interval is $(-\infty,\infty)$ How to find a function from reals to reals such that the image of every open interval is the whole of R?  Is there one which maps rationals to rationals? reformat date to have month in words Currently my text file looks like this..  David   Webb    Box 34 Rural Route 2    Nixa    MO  65714   (417)555-1478   555-66-7788 09-13-1970 Martha  Kent    1122 North Hwy 5    Smallville  KS  66789   (785)555-2322   343-55-8845 04-17-1965 Edward  Nygma   443 W. Broadway Gotham City NJ  12458   (212)743-3537   785-48-5524 08-08-1987 O'Reilly    Baba    123 Torch Ln    Joplin  MO  64801   (417)222-1234   456-65-3211 12-13-1999 Martin Bob  44 Boss Rd  Joplin  MO  64801   (417)888-4565   456-98-1111 01-01-2007   the dates are in the 9th field and I want to display them as January 7, 2017 instead of 01-07-2017 for example.  How should I do that? If using options please explain what they do briefly. Doing this in bash. Needing to put it in a script and output to a new file to preserve original. How do I print an IFrame from javascript in Safari/Chrome Can someone please help me out with printing the contents of an IFrame via a javascript call in Safari/Chrome.  This works in firefox:  $('#' + id)[0].focus(); $('#' + id)[0].contentWindow.print();   this works in IE:  window.frames[id].focus(); window.frames[id].print();   But I can't get anything to work in Safari/Chrome.  Thanks  Andrew Pivot in SQL 2008 R2 Subject–verb agreement not clear in "Is two bars OK?" Is the following correct?     Can I get some chocolate for myself? Yes, you can but not much.   Is two bars OK?   Is there any rule to explain the sentence above?  The dialogue's taken from Grammar Textbook Round-Up Level #3. Let: $P(x) = a_nx^n + a_{n-1}x^{n-1} + ..... + a_1x + a_0$ where $a_0a_n &lt; 0$ I have to prove that the Polynomial $P(x)$ has at least one positive root how can I prove it? Any ideas? How are the measurements used by Nielsen evolving to meet new methods of content delivery and viewing? I was interested to find out that in the 1950s, Nielsen used devices in the homes of the families that were surveyed to record their viewing preferences, and only later introduced their system of using a diary to record habits:  ...in 1950...This information was collected on a device that was attached to a television that recorded what was being watched.  In 1953, the company began sending out diaries to a smaller sample of homes (“Nielsen families”) within the survey to have them record what they had watched from   So, presumably as multiple different viewers in the home were considered in the sample, the means through which the ratings were recorded changed. Given that so many people are able to record shows digitally and watch when convenient and that so many view their favorites on outlets such as iTunes, how have the Nielsen ratings systems adapted (or how does the company plan to adapt them) to capture our modern viewing preferences? While a DVR-recorded show still technically makes its original &quot;timeslot&quot; relevant to the equation, will Nielsen consider timeslot-specific data a hindrance to comparisons between those shows viewed on cable and the same show viewed on iTunes? Integral of Wiener Squared process I don't have a background of stochastic calculus.   It is known  that definite integral of standard  from $0$ to $t$ results in another Gaussian process with slice distribution that is normal distributed with mean equal to $0$ and variance $\frac{T^3}{3}$ i-e  $$ \int_0^{t} W_s ds \sim \mathcal{N}(0,\frac{t^3}{3})    $$  Question:  What if we square the standard Wiener process and then integrate i-e $$ \int_0^{t} W_s^2 ds \sim ?    $$  Would that be scaled Chi-square distributed ? Why do adult insects have 6 legs? Apparently, there is an advantage to having 6 legs in the insect world. What is that advantage, if anything? Why would such an advantage exist for insects, but not for other, larger land animals?  What do those middle legs do that the front and hind legs cannot, if anything? Are they really essential, or do they just generate more muscle power? Show that $(2,0,4) , (4,1,-1) , (6,7,7)$ form a right triangle What I tried:  Let $A(2,0,4)$, $B(4,1,-1)$, $C(6,7,7)$ then  $$\vec{AB}=(2,1,-5), \vec{AC}=(4,7,3), \vec{BC}=(2,6,8)$$  Then I calculated the angle between vectors:  $$\begin{aligned} \alpha_1 &amp;= \cos^{-1}\left(\frac{(2,1,-5)(4,7,3)}{\sqrt{2^2+1^2+(-5)^2}\sqrt{4^2+7^2+3^2}}\right) \\ &amp;= \cos^{-1}(0)=90° \\ \alpha_2 &amp;= \cos^{-1}\left(\frac{(4,7,3)(2,6,8)}{\sqrt{4^2+7^2+3^2}\sqrt{2^2+6^2+8^2}}\right) \\ &amp;= \cos^{-1}\left(\frac{74}{\sqrt{74}\sqrt{104}}\right)=32.49\\ \alpha_3 &amp;= \cos^{-1}\left(\frac{(2,6,8)(2,1,-5)}{\sqrt{2^2+6^2+8^2}\sqrt{2^2+1^2+(-5)^2}}\right) \\ &amp;= \cos^{-1}\left(\frac{-30}{\sqrt{104}\sqrt{30}}\right)=122.5° \end {aligned}$$  As you can see, these angles don't even form a triangle, what am I doing wrong, any thoughts? Note: This is an attempt to create a canonical question that covers all instances of &quot;low-graphics mode&quot; error that occurs to a user, including but not limited to installation of wrong drivers, incorrect or invalid lightdm greeters, low disk space, incorrect installation of graphics card like ATI and Nvidia, incorrect configuration of xorg.conf file while setting up multiple monitors among others. If you are experiencing the &quot;low-graphics mode&quot; error when trying to login but none of the following answers work for you, please do ask a new question and then update the answers of this canonical question as and when your new question gets answered.   When I try to boot into my computer, I am getting this error:  The system is running in low-graphics mode Your screen, graphics cards, and input device settings could not be detected correctly. You will need to configure these yourself.   How do I fix the failsafe X mode and login into my computer?  Answer index: I'm writing a LINQ to SQL statement, and I'm after the standard syntax for a normal inner join with an ON clause in C#.  How do you represent the following in LINQ to SQL:  select DealerContact.* from Dealer  inner join DealerContact on Dealer.DealerID = DealerContact.DealerID Uniformly convergence of $f_{n+1}(x)= \int^x_0 f_n(t)dt$ $\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected? If I want to prove that $\mathbb{R} \setminus \mathbb{Q}$ is disconnected, does it suffice to say that there are two open disjoint sets that cover $\mathbb{R}\setminus\mathbb{Q}$, namely:  $$(- \infty, 0), (0, \infty)\text{ ?}$$  Along the same lines, I want to prove or disprove that $S = \mathbb{R}^2 \setminus \mathbb{Q}^2$ (points $(x, y) \in S$ that have at least one irrationais connected. I feel that it is also disconnected; does it suffice to say that there are two open disjoint sets that cover $S = \mathbb{R}^2 \setminus \mathbb{Q}^2$, namely $((- \infty, - \infty), (0, 0))$ and $((0, 0), (\infty, \infty))$? (Very iffy on my assertion and my notation, sorry.)  Thanks!