Does there exist a real number $r$ such that $\pi^r$ is a rational number? Clearly, $(\sqrt{2})^2=2$ etc. Is there any real number such that my question is true?
We're all familair with this beautiful proof whether or not an irrational number to an irrational power can be rational. It goes something like this:  Take $(\sqrt{2})^{\sqrt{2}}$  If it's rational, then you proved it, if it's irrational, take $((\sqrt{2})^{\sqrt{2}} ){^\sqrt{2}} = 2$ and you've proved it.  I'm wondering if you can raise $\pi$ or $e$ to a certain non-trivial real power to make it rational? And if not, where is the proof that it can't be done?   p.s. - I almost left out the real part, but then I realized that $e^{i\pi} = -1$.
Possessive form for a surname ending with "z" What is the proper possessive form for a surname that ends with “z”? Is it z’ or z’s? Is there a word for when an action has an effect opposite to the one intended? Is there a word or phrase to express the concept of an action having the opposite effect of the expected outcome?  For example, a drug taken to cure headaches that actually causes headaches, or an advertising campaign designed to deter smoking that leads to an increase in smoking. Where can I find a repository for older unsupported releases? I remember I saw a apt mirror somewhere containing older distributions but I cannot find it anymore. Does anybody know where to find it? I originally read this book in the mid- to late '70s.   The premise was a bureaucracy whose job it was to manipulate events in human history by making the smallest possible change that would effect the desired outcome. Not time cops in the sense of going after law breakers.  The intent was to maintain a policy of non-interference as much as possible but when interference was warranted then to make the least possible change and to do so invisibly.  For example, they might take someone's keys off the dresser and put them just out of sight underneath the dresser where they plausibly might have fallen.  The subject would never know of the manipulation but would then be delayed getting out of the house and the ripples grew from there to avert a war or otherwise affect events on a large scale.  After reading that, whenever I would lose something - especially in an improbable way such as dropping a screw at my feet and then finding it on the other side of the room half an hour later - I would take some consolation in the thought that I had just averted some global catastrophe.  I was pretty scatterbrained as a kid (we call that ADHD nowadays) so I experienced this a lot.  Given that I've spent so much time pondering the ideas from the book, I'd love to re-read it. Let’s take an example of a movie site. So a well structured set of URLs for this site can be:     Movie page - example.com/movies/the-matrix Review for movie - example.com/movies/the-matrix/reviews Trailers for movie - example.com/movies/the-matrix/trailers Specific trailer for movie - example.com/movies/the-matrix/trailers/trailer1   where it’s possible to navigate the resource even by directly deleting/appending resource name seamlessly.  For SEO purpose I read somewhere that for two given pages, Google will prefer the page which have search query up in the URL hierarchy among other things. So for the given 2 URLs and the query "the matrix trailer"    example.com/movies/the-matrix/trailers example.org/trailers/the-matrix/    `     Google might prefer the second URL as it has the keyword "trailer" at first level comparing to first URL which has "trailer" at 3rd level (a resource deeper in URL hierarchy may be less important).    If it is correct, one may be tempted to create URL hierarchy somewhat like      Movie page - example.com/movies/the-matrix Review for movie - example.com/reviews/the-matrix/ Trailers for movie - example.com/trailers/the-matrix/ Specific trailer for movie - example.com/trailers/the-matrix/trailer1     which will break the seamless hierarchy.    My questions are:   Is it true that resource deeper in URL hierarchy is tend to be less important when comparing to URLs of other site?   Is sacrificing meaningful, seamless hierarchy worth for this SEO optimization? What's usually the effective visual resolution of typical Hollywood Movies? I'm really surprised by the quality of the visuals in some new movies this year (2018). I'm curious to know in what resolution these cameras typically record to experience this quality on a cinema screen? bash chained logical operator execution order, lazy evaluation How do I make the Messages that are sent to me stop going to my Mac? I'm heading to India next week and have a connection in Munich, Germany. I made the mistake of booking a flight with only an hour and a half layover (freaking out!) and I'm wondering if I'll have to go through customs in Munich or if I can pass right on through. I know how to monitor the network interfaces on my local computer with Wireshark. But, how do I monitor the traffic flowing through my wireless access point from connected computers?  It seems like this might not be possible without some kind of special software on the wireless access point itself. When to use RSA and when ElGamal asymmetric encryption Show that $\mathbb{Z}/n^2\mathbb{Z}$ is not isomorphic to other group I'm currently working on a proof, but didn't get the right idea, a hint would be appreciated:  Show that $\mathbb{Z}/n^2\mathbb{Z}$ is not isomorphic to $\mathbb{Z}/n \times \mathbb{Z}/n$  So, lets assume there is an isomorphism between these groups, my first idea to was to construct a contradiction with the homomorphism property. But I don't know how, there should be an element in  $\mathbb{Z}/n \times \mathbb{Z}/n$ which is not in $\mathbb{Z}/n^2$ to reach this kind of contradiction. Of course there is $n \in \mathbb{Z}/n^2$ which is not in the other group but I don't know if this helps.  Thanks "There are several reasons proposed for the collapse of the bridge." There are several reasons proposed for the collapse of the bridge.   Is this present simple tense or the present perfect tense? I thought it might be the latter since there is a retrospective aspect to the sentence. However, I am quite unsure. Registry key / GPO to disable and block Windows 10 upgrade? How can I suspend/hibernate from command line? How can I suspend or hibernate my laptop using command line, without installing additional software? Showing a function on $[0,1]$ is the zero function Suppose $f$ is continuous on $[0,1]$ and   $$ \int\limits_0^1 f(x) x^n d x = 0 $$   for all $n=0,1,2,...$. Then, $f(x) = 0$  I was trying to use by parts  $$ 0 = f(x) \frac{x^{n+1}}{n+1} \bigg|_0^1 - \int\limits_0^1 \frac{x^{n+1}}{n+1} f(x) dx $$  Thus,  $$ \frac{f(1)}{n+1} = \int\limits_0^1 \frac{ x^{n+1} f(x) dx }{n+1} \implies f(1) = \int\limits_0^1 x^{n+1} f(x) dx $$  but here I am stuck. Maybe my idea is not the best one? How would you approach this problem? Straighten Vertices Along a Line On a line, I need to line up the bottom vertex points in relation to a top reference vertex point. How could I reset the X Y and Z coordinates of these vertex points to essentially line them up all through use of the vector of the reference point vertex.  I do not have access to Mesh -> Transform -> Scale on Axis for aligning vertices along a line. UK visa refusal on V 4.2 a + c (and sometimes 'e') Many of the UK visa refusals we see here share a common pattern and the prevailing reasons refer to V 4.2 (a) and (c).    I do understand that, while the applicants may describe very different individual circumstances, there's a consistent pattern and, broadly, fall into specific categories:    Credibility (lifestyle, lack of ties, visit history, lies &amp; omissions) Funding (insufficient funds, provenance of funds, funds parking) Sponsorship (family, friends, employer)   Question: What most commonly triggers a UK visa refusal where V 4.2 (a) and (c) are given as the grounds?  Secondarily: Given that there is a clear pattern, to what extent are these refusals predictable? Does sponsorship make a difference? Is there a set of personal circumstances, however abstract, where a refusal is all but guaranteed? For example, why do those we see here on TSE who sought entry for the PLAB or British Army appear to be refused with a common theme? Is there a uniform shortcoming or is it just discrimination?   PLAB: What is about applications for a visa, for the purpose of sitting the Professional and Linguistic Assessments Board (PLAB) exam, that seem to invite refusals. After all, the test is given so that international medical graduates can show that they qualify to practise medicine in the UK. The first part, PLAB 1, can be taken in centres outside of the UK. However, PLAB 2 can be taken only in the UK. Why can't you can't get a visa just to sit the exam, promising that you'll leave immediately after?    British Army: Since Commonwealth citizens are eligible to apply online to join, even those who don't reside in the UK, why is it so difficult to get a visa just to attend an interview to see whether you are suitable? Even with an invitation from the Army, such visa applications seem to be unsuccessful. Isn't a career in the British Army a valid reason?     Lastly:  After such a refusal, what approaches would increase the chance of a successful application? An explanation of Hawking Radiation Galois extension is (not) transitive Let $ K/ L /F $ be fields. If $K / L$ is Galois and $ L / F $ is Galois, then $ K / F$ is Galois. We mentioned this very quickly in today's class without justifying. But I have trouble seeing this. Any help is appreciated.  Our definition for $ K / F$ being Galois is the $F$-automorphism group $G = G( K / F) : = \{ \sigma \in aut (G) : \sigma_F = id_F \}$ fixes only $F$, i.e. $ \{ x \in K : \sigma (x) = x \text { for all } \sigma \in G  \} = F$.  But we also showed that $ K /F$ is Galois if and only if $ K/F$ is normal and separable.