Solve Differential Equation $ f^{-1}(x)=f'(x) $ $$ f^{-1}(x)=f'(x)$$ Where $f^{-1}:$ Inverse of f   I know for a fact there is a specific solution that is demonstrated on this  and specifically:  $$ f(x) = x^{φ}\sqrt[φ]{\frac{1}{φ}} $$ Where $φ=\frac{1+\sqrt{5}}{2}$ is the golden ration.  What I wish to do, is to find (if they exist) the rest of the solutions to the equation. In case there is no other, I want to at least prove that there is no other solution. For the record, I seek a conventional solution with a mathematical proof backing it, and not just through making "educated" guesses.
Function whose inverse is also its derivative? What are some good examples of a function  $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ and showing that $f^{-1}(x) = f'(x) \implies b-1=\frac{1}{b} \implies b=\phi$ and got $$f(x) = \frac{x^\phi}{\sqrt[\phi]{\phi}}$$ Which seems to work according to WolframAlpha, but I'm having trouble double-checking it. Any other ideas?
I mean a average human can run at his full speed for 15 to 20 mins before getting tired. Trained athletes can go on running for much more time without tiring out.  Do Superheroes like Quicksilver or Flash ever get tired due to excessive running? Signed to unsigned conversion in C - is it always safe? Suppose I have the following C code.  unsigned int u = 1234; int i = -5678;  unsigned int result = u + i;   What implicit conversions are going on here, and is this code safe for all values of u and i? (Safe, in the sense that even though result in this example will overflow to some huge positive number, I could cast it back to an int and get the real result.) At the moment, in major part because they are two separate post objects in the system, the excerpt and the body of a tag wiki count as two separate posts with regards to suggested edits in the queue. Mechanically, this is sane, but when it comes to handling the actual queue, it can be confusing. A body edit might consist of little more than a bed of links, which looks bad. But then an excerpt is shown that includes the rest of the data, and since excerpts can be included on the actual wiki page, it makes a lot more sense when they're taken in the same space.  And this is how they are composed, too. The actual edit interface for working on tag wikis has both the excerpt and the body on the same page. In a sense, the excerpt behaves a lot like the "title" of the wiki, if we make a parallel to normal post edits.  Thus, my suggestion (which is to say a suggestion that I repossessed from Robert Cartaino) is that when suggested edits are reviewed for tag wikis, that they be presented in a single unit, treating the excerpt the same as how a title is treated in a question edit. This way, relevant information that is crossed between the two isn't lost because they are reviewed separately. Activate Rigid Body Physics at certain time? I've been trying to make a wrecking ball smash through a figure of sorts, not really sure what exactly but I made an example. The problem I'm having is that the figure does indeed fall apart, but I would like it to stay put together until the ball hits it. I achieved this by using Cell Fracture on the figure and just using Rigid Body. Again, I just need a way for the figure to stay still until hit by the ball, and then have all of its pieces come apart. Upper bound for norm of Hilbert space operator It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $\|T\|=\sup_{\|x\|=1}|\langle Tx,x\rangle|:=M(T)$, where $M(T)$ is called the  of $T$. It seems that for any bounded operator on $H$, we have $||T||\leq 2M$. Why is that so? I want to put a link to my SO profile on my homepage. I remember a codesnippet i saw somewhere that would basically display my Username and my reputation. Where was that? Arguably the most famous scene in Star Wars is when Darth Vader reveals to Luke Skywalker that      He is Luke's father.   This surprise was kept secret to most of the crew and even the cast , with Dave Prowse, the actor playing Darth Vader, delivering a fake line, the actual line being overdubbed. (Of course, Darth Vader wasn't voiced by him in the first place; all lines were overdubbed by James Earl Jones).   Like Mark Hamill relates in :     But, for example, your big scene, one of the classic cinematic moments when Darth Vader divulges his true identity, is no longer a revelation.       It's such a great moment! The fake line that was put in there just to try and keep the secret was "You don't know the truth: Obi-Wan killed your father!" But as much as I enjoyed leaking false information, it was a wonderfully hard secret to keep because (Irvin) Kershner, the director, brought me aside and said "Now I know this, and George knows this, and now you're going to know this, but if you tell anybody, and that means Carrie or Harrison, or anybody, we're going to know who it is because we know who knows."   So how do we explain ?       And [David Prowse] offered a glimpse of a possible plot for the second sequel. Darth Vader and Luke Skywalker, the young hero in the first film played by Mark Hamill, are hooked up in a do-or-die light saber duel when Luke learns that Darth is, in fact, his long-lost father.   So exactly how secret was this secret anyway?    As DVK points out , the newspaper article is probably authentic. This is a repost with additional info of an older question that was never answered. I have stumbled upon this old question while also trying to find the name of the book series myself, as I dearly wish to read it again but for the life of me, I can't remember the name of it.      This is a long shot but I'm really in need of help for this because      it was an amazing book. It was either a kid's or YA book. A guess      from what year it was published would be 90s-mid 2000s, as I read it      around 2008-2011.         The book cover was in first person view, with the main character   holding a weapon, and was a 3D rendered image inside a dungeon or   cave (that was very reminiscent of early 90s CGI animation), though   there may be variants.    It would have been set around when computer   games were mainstream (maybe around the same time as when it might've   been published) It's about a teenage boy who is bored at home, so he   looks through an old family computer to play a game and finds one   that he has never seen before. When he plays it he gets transported   into the game which is a fantasy-medieval setting.    All that I can   remember about the game and story is that the main character is on a   quest, he rides horses a lot, (to the point where he can ride horses   perfectly in real life), at one point he eats food at an ingame   tavern/restaurant? and he tosses the ingame coin currency at the   server because he's in a hurry, (don't know why I remember that bit   specifically haha), and he meets a gnome/dwarf character who is   helping him on his journey, who has a home in an ingame mountain, and   that time goes faster in the game. (For example, if he spent 5   minutes in the real world, 3 in-game days would have passed)   Sometimes as he's playing something stops him from being able the   play the game for a few weeks, maybe a concerned parent because he's   playing too much or there's a family vacation. I remember during the   time when he's not playing the game he's on a farm and knew how to   ride the horses there.    There was a character in the real world who   was the main character's sibling, relative or friend. They were like   a rival and was also very whiny (to some extent). They were on the   farm as well with him and questioned how the main character knew how   to ride a horse when he had never done it in real life.   This is all I can remember but if you need more specific information I can try and answer. It is definitely not Space Demons or Demons Don't Dream. If anyone knows anything remotely similar please mention it, I would really appreciate it. Thank you!!            Additional info:    It is not an online game or virtual reality game. It was a normal PC game, and all other characters are (I think), just NPCs.  He woke up for the first time in a field/farm and was super confused about what happened and where he was, as he had never played the game before. He is asked a super serious question before starting the game up for the first time, like "are you sure you wish to play ___?", and then he is transported into the game.  He frequently exited the game, to continue his IRL life. And sometimes, he would lose access to the family desktop PC for a couple of days at a time, and when he went back everyone in the game (NPCs), thought he had disappeared because he was gone for ages. I think once, his family when on holidays and he couldn't take the PC, for example.  It was a book series, not a single book. In later books he would play the game again months later in the real world, and years and years would have passed in the game world.  The plot involved 2 main warring sides/factions/countries of humans, but there were fantasy/magical races and characters involved. I vaguely remember something about each side having official colours, one red and one blue. I think the main character was on the blue side.   The original question was never correctly answered, so I'm posting again to see if anyone can remember.   Books that were ruled out in original post: Eden's Gate &amp; Heir Apparent. Add source to figure caption How can I add a source to a figure? I need to add \ref, \cite or just free text. Something like that: \begin{figure} [ht]   \centering   \includegraphics[width=0.95\textwidth]{res/figure.pdf}     \caption{Caption}     \source{\ref{},\cite{} or free Text}   \label{fig:gliederung} \end{figure}  Should give me:  The source shouldn't appear in \listoffigures. "Closed" (square) root symbol I found a cool trick in the  which allows the square root sign to be altered such that it &quot;closes&quot; over its contents. Here's an example:  (&quot;closed&quot; root on the left, normal one on the right) The way they do this is as follows: % New definition of square root: % it renames \sqrt as \oldsqrt \let\oldsqrt\sqrt % it defines the new \sqrt in terms of the old one \def\sqrt{\mathpalette\DHLhksqrt} \def\DHLhksqrt#1#2{% \setbox0=\hbox{$#1\oldsqrt{#2\,}$}\dimen0=\ht0 \advance\dimen0-0.2\ht0 \setbox2=\hbox{\vrule height\ht0 depth -\dimen0}% {\box0\lower0.4pt\box2}} The cool thing is that (as far as I've tested it) this works regardless of the math font and with pdftex as well as xetex. However what is not possible is mulitple roots, such as \sqrt[3]{a}. Those produce incorrect output when this &quot;trick&quot; is being used. So my question is, is somebody able to tweak this code so that it works for multiple roots as well? How to add a display mode to a VirtualBox VM? I am using my first VirtualBox VM in order to run Windows 8 Preview on my Windows 7 machine.  It looks like only a few resolutions are available for me to choose.  My physical display is 1600x900, so I'd like to be able to set my Windows 8 VM to that in order to have a fullscreen experience.  How do I add a 1600x900 to the choices I have in the "Screen Resolution" dialog? Tikz, Tree without overlaps Find index of nested item in python How do I wrap a macro definition in an environment? I have a default Blender cylinder positioned upright atop a ground plane:    How can I rig the cylinder so that can be animated to wiggle/jitter (perhaps procedurally with noise f-curve modifiers on two of its rotation axis) without penetrating the floor?    A simple parent-child relationship as in the image above is sufficient to rotate (and translate) the cylinder on 1 axis in either a positive OR negative direction.   I'm looking for a setup that would allow me easily to rotate the cylinder on two or even three axis in both a positive and negative direction.  I imagine this would be possible with some drivers applied to the XYZ coordinates using trigonometric equations but the math escapes me. It might also be possible using a particular bone setup with the correct constraints but my attempts to find something that works so far have not succeeded.  Thanks for your assistance. Which words in a title should be capitalized? Are there any concrete rules that say which words (parts of speech) in a title should start with a capital letter? What would be a correct capitalization for the title of this question? $-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1\pmod{4}$ I came across this problem and I believe Lagrange's theorem is the key to its solution.  The question is:     Let $p$ be an odd prime. Prove that there is some integer $x$ such that $x^2 \equiv −1 \pmod p$ if and only if $p \equiv 1 \pmod 4$.   I appreciate any help.  Thanks. According to , $[0,1]$ cannot be written as union of countable disjoint closed sets, is the same true about (uncountable) family of disjoint closed intervals ? My PC has a build-in VT1828S chip. The connection between PC and monitor (which has built-in speakers) is through an HDMI cable.  Audio preferences shows 2 devices for audio output: analog stereo and RS880 Audio Device [Radeon HD 4200] digital stereo (HDMI).  When I plug a headphone in the rear jack and I choose the former device, everything works. When I choose the latter device, instead, no sound is heard, and video playback is too fast. It is as if video frames want to synchronise with the audio timing, but the audio timing information is wrong.  HDMI audio worked perfectly with Ubuntu 11.04, but stopped after 11.10 upgrade. Can "but also" be split in "not only ... but also"? I think I generally agree with the statement  When using not only . . . but also in a sentence, parallelism should be the goal. It means that the words following both parts of this correlative conjunction (i.e., not only and but also) should belong to the same parts of speech.   However, the Cambridge dictionary has an example that is not 100% parallel in my eyes, as &quot;but also&quot; is split:  Not only did she forget my birthday, but she also didn’t even apologise for forgetting it.   What I would expect, with maximum parallelism, is  Not only did she forget my birthday, but also she didn’t even apologise for forgetting it.  Is my expectation wrong? Is my formulation acceptable?