Finding an Inverse Function and Composition of Functions? I add for all y ∈ R
Finding an Inverse Function and Composition of Functions?
How do you prove that a harmonic planar mapping $f(x,y) = u(x,y) + i v(x,y)$ for real $u,v$ can be written as $f(x,y) = \phi(x,y) + \overline{\psi}(x,y)$ where $\phi$ is a holomorphic function, and $\overline{\psi}$ is an anti-holomorphic function (conjugate of a holomorphic function)? Was Elrond, in Tolkien's opinion, more inherently powerful than Galadriel? I always thought it was the other way around, but then I read this (Letter 246):     "Of the others only Gandalf might be expected to master him – being an emissary of the Powers and a creature of the same order, an immortal spirit taking a visible physical form. In the 'Mirror of Galadriel', 1381, it appears that Galadriel conceived of herself as capable of wielding the Ring and supplanting the Dark Lord. If so, so also were the other guardians of the Three, especially Elrond. But this is another matter. It was part of the essential deceit of the Ring to fill minds with imaginations of supreme power."   This "especially Elrond" confuses me. I don´t think that Elrond is inherently more powerful than Galadriel, who is described as the greatest of the Noldor, except for Fëanor (and possibly even his equal).  A quote from Unfinished Tales says:     "...In this he (Tolkien) emphasized the commanding stature of   Galadriel already in Valinor, the equal if unlike endowments of   Fëanor; and it is said here that so far from joining in Fëanor's   revolt she was in every way opposed to him."   Furthermore,      "A queen she was of the woodland Elves, the wife of Celeborn of   Doriath, yet she herself was of the Noldor and remember the Day before   days in Valinor, and she was the mightiest and fairest (so even   fairer than Arwen I would guess) of all the Elves that remained in   Middle-earth."   (The Silmarillion, of the Rings of Power)  And again...     "Galadriel was the greatest of the Noldor, except Fëanor maybe, though   she was wiser than he, and her wisdom increased with the long years."   (the Shibboleth of Fëanor)  Restating it:     "These two kinsfolk (Fëanor and Galadriel), the greatest of the   Eldar in Valinor, were unfriends for ever." "Who together with the   greatest of all the Eldar, Luthien Tinuviel, daughter of Elu Thingol,   are the chief matter of the legends and histories of the Elves."   (Unfinished Tales)  The latter means that Luthien, Fëanor and Galadriel are the chief matter of the history and legends. It seems to me that Tolkien thought of those three as the creme de la creme of the elven crop.  Reading all that I have serious problems believing that Elrond could be superior to her. As great as Elrond is, descending from Melian, he is no equal of Fëanor, who is said to be the mightiest of all elves, and if Galadriel is his equal, then there is no way that Elrond would be mightier than her.  Maybe I just read this "especially Elrond" wrong; it could as well mean that especially Elrond was deceived into thinking that he could master the ring, because that was the deceit of the ring. And Tolkien said explicitly that only Gandalf could possibly master it.  So, what do you think? Background image resize in Ubuntu 18.04.1 LTS Recently updated to 18.04.1 and now I cannot resize my background image from fill screen to center image.  How can I resize the image now that 18 doesnt have the quick resize feature? What are the “spam” and “rude or abusive” (offensive) flags, and how do they work? Why Doesn't a Bootable USB Boot This question is asked often in Ask Ubuntu, sometimes with few hints about the situation.   Please provide a list of possible reasons to help troubleshoot the problem. Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger? Given $a&gt;b&gt;2$ both positive integers, which of $a^b$ and $b^a$ is larger?  I tried an induction approach. First I showed that if $b=3$ then any $a \geq4$ satisfied $a^b&lt;b^a$.  Then using that as my base case I tried to show that given any pair of positive integers $a,b$ satisfying $a&gt;b&gt;2$ and $a^b&lt;b^a$, then $(a+1)^{b+1}&lt;(b+1)^{a+1}$ - but that is where I got stuck.  Any help would be appreciated. What do the subscripts in expectations mean in loss function Can someone explain me what this equation means? I read it from page 152 on . How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$ I need help to prove this in real analysis. I think it uses IMVT, but not sure how to do it.     Let $f(x)$ be a real valued continuous function on $[0,1]$. Show that   $$ \lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1) $$ launch a VM on Ubuntu On Ubuntu 14.04, I installed VirtualBox on which I have created 2 Ubuntu virtual machines.  Does Ubuntu provide a way to launch  the virtual machine I want at a particual moment ? Is there any solution for this problem by developing a .sh script for example or using any other programming language ? How do I recruit beta testers? We're a small company without enough employees to run a large scale beta in house.  The app is geo-location based so we're trying to recruits some beta testers from all over that aren't "friends and family".  Are there any sites or communities dedicated to something like this? How to change phone number in Google account? I gave Google my phone number to verify it, not knowing that it would change it, and now my phone number is completely different? What have I done, and how do I unverify it and change my phone number back? verbatiminput and fontsize Example of a recursive set $S$ and a total recursive function $f$ such that $f(S)$ is not recursive? Browsing wikipedia, I stumbled on the following: "The image of a computable set under a total computable bijection is computable." Given the form of the theorem, there must be some example of a example of a recursive set $S$ and a total recursive function $f$ such that $f(S)$ is not recursive (otherwise the statement would have presumably been stronger). But I'm having difficulty dreaming up a case where this happens. Does the alarm work when my phone is turned off? How can I replace end line with fixed text when the next line begins with a defined set of characters? I have several big files with some measurements.   It looks this way:  N 12344;PE 9.9999999;... #S 0 0 31 44 75 130 165 196... #S_+ "2 5 2 3 3 1 1 2 3 1 2 2...  N 12345;PE 9.9999999;... #S 0 0 34 57 84 133 152... #S_+ "1 0 1 1 2 3 0 0 0...  N 12346;PE 9.9999999;... #S 0 0 31 44 73 140 169... #S_+ "3 3 4 0 0 2 1 2 4...  N 25104;PE 9.9999999;... #S 0 0 36 52 102 108 145... #S_+ "1 1 0 1 0 0 3 0 1...  N 25105;PE 9.9999999;... #S 0 0 32 58 88 130 143...   Sample is here:   The file is much bigger but I replaced part of the data with "..." to make it shorter.  I need to somehow replace the line ends before "#S" - in fact simply merge the "N" line with the following two ones into one line (or with the following three ones so I can get rid of the blank lines). Expect output like this:  N 12344;PE 9.9999999; #S 0 0 31 44 75 130 165 196 #S_+ "2 5 2 3 3 1 1 2 3 1 2 2... N 12345;PE 9.9999999; #S 0 0 34 57 84 133 152 #S_+ "1 0 1 1 2 3 0 0 0... N 12346;PE 9.9999999; #S 0 0 31 44 73 140 169 #S_+ "3 3 4 0 0 2 1 2 4... N 25104;PE 9.9999999; #S 0 0 36 52 102 108 145 #S_+ "1 1 0 1 0 0 3 0 1... N 25105;PE 9.9999999; #S 0 0 32 58 88 130 143...   Is this possible to achieve using some command-line utility in linux?   My knowledge is quite limited in this area so I would appreciate any help.  thanks Wilson and Fermat's Theorem Question!     Show that the pair of positive integers $n$ and $n+2$ are twin primes if and only if $$4[(n-1)! + 1] + n = 0 \ (\text{mod} \ n(n+2)),$$ where $n$ does not equal $1$. Biblatex: submitting to a journal Transferring Money from India How should I understand "$A$ unless $B$"? In Markov inequality proof, why is $\int_a^\infty xp(x) \, dx \ge \int_a^\infty ap(x) \, dx$ Markov inequality,   $$\Pr(X \ge a) \le \frac{E[x]}{a}$$  Proof  $$\begin{aligned}      E(X) &amp;= \int_0^\infty xp(x)\,dx = \int_0^a xp(x)\,dx + \int_a^\infty xp(x)\,dx \\       &amp;\ge \int_a^\infty xp(x)\,dx \ge \int_a^\infty ap(x) \, dx  \end{aligned}$$  Why is the last inequality true? Why is it ok to change $x$ to $a$, and we know it is less than or equal to? How did they figure it out? Thanks!