I found this differential equation $$ f''(x) = f^{-1}(x) $$ which is interesting to me. I wonder if there is a calculus oriented way to solve this differential equation. Is there some method to to solve differential equations containing inverse of a function ?
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?
Improve alignment of "Learn more" button in "Newest badge" popup The popup when you click on your newest badge in your user profile has a &quot;Learn more&quot; button, but its alignment is suboptimal; it's too close to the top of its container:  It looks better with a 4px top margin: How to handle fragmentation of auto_increment ID column in MySQL I'm curious of how illusions would affect spells or abilities that require a target to be looking at something. A good example of this would be Medusa's Petrifying Gaze which has the following effect:  Petrifying Gaze: When a creature that can see the medusa's eyes starts its turn within 30 feet of the medusa, the medusa can force it to make a DC 14 Constitution saving throw if the medusa isn't incapacitated and can see the creature. If the saving throw fails by 5 or more, the creature is instantly petrified. [...]  So if someone were to cast Minor Illusion to put a bag over Medusa's head, would other player/creatures who do not realize that the bag is an illusion be affected by the gaze if they were looking in the direction of it? Does the illusion block the effect? Inbox chops messages in the middle of an HTML character reference The messages in the inbox get chopped if they too long, and the chop can occur in the middle of an HTML character reference as can be seen in this screenshot:    The character being chopped is [ and it appears inside a code snippet, in case it makes a difference. What does  signify when using Vim LatexSuite? I have a LaTeX editing environment that does some assistive stuff such as auto-indenting, closing parentheses, etc.  However, one of the things that it does is unknown to me, and is difficult to search for. If I type an empty set of braces, it replaces them with  {}&lt;++&gt;   If I type two empty sets, for example when I'm setting up a fraction \frac{}{} what I actually get is  \frac\left\{ } \right\}&lt;++&gt;}&lt;++&gt;   Is this a problem? What does &lt;++&gt; do?  Edit: This is using Split List into Sublists with LINQ Is there any way I can separate a List&lt;SomeObject&gt; into several separate lists of SomeObject, using the item index as the delimiter of each split?  Let me exemplify:  I have a List&lt;SomeObject&gt; and I need a List&lt;List&lt;SomeObject&gt;&gt; or List&lt;SomeObject&gt;[], so that each of these resulting lists will contain a group of 3 items of the original list (sequentially).  eg.:   Original List: [a, g, e, w, p, s, q, f, x, y, i, m, c] Resulting lists: [a, g, e], [w, p, s], [q, f, x], [y, i, m], [c]   I'd also need the resulting lists size to be a parameter of this function. If $f$ is a continuous function with $\displaystyle \int_{0}^{x}f(t)dt\rightarrow \infty$ as $|x|\rightarrow \infty\;,$ Then show that every line $y=mx$ Intersect the curve $\displaystyle y^2+\int_{0}^{x}f(t)dt=2$  $\bf{My\; Try::}$ Put $y=mx$ in $$\displaystyle y^2+\int_{0}^{x}f(t)dt=2\;,$$ We get $$\displaystyle m^2x^2+y^2+\int_{0}^{x}f(t)dt=2.$$ We can write it as $$\displaystyle 2\int_{0}^{x}2m^2tdt+\int_{0}^{x}f(t)dt=2$$ So $$\displaystyle \int_{0}^{x}\left[2m^2t+f(t)\right]dt-2 =0$$ Now Let $$g(x) = \displaystyle \int_{0}^{x}\left[2m^2t+f(t)\right]dt-2.\;,$$ When $x\rightarrow \infty\;,$ Then $g(x)\rightarrow \infty.$ (Where $g(x)$ is an Continuous function.) Now I did not Understand How can i solve that, Help me Thanks How many people can the TARDIS transport? The interior of a TARDIS is much larger than its exterior, but how many occupants can it transport? Do I need a Kuwait transit visa as an Indian national? I am an Indian national and passport holder, traveling from New Delhi to Frankfurt (via Kuwait) and back with Kuwait Airways. I see no Terminal change for the onward and return journeys. I have no plans to leave the airport due to short layovers.  Do I need a Kuwait transit visa? delaying moldy bread I always keep bread around, home-made or otherwise.  Is there a way to keep it from molding, or at least delay the process?  I've tried the fridge with some success (extra day or two), but I don't like my bread that cold to eat.  I've tried zip-top bags with no success (seems to mold faster).  Bread-box didn't seem to make any difference in time.  Thanks A random variable with neither probability density function nor probability mass function... is this example wrong? This is from Shreve's Stochastic Calculus for Finance II, Appendix A.3. Consider infinitely many independent coin tosses with outcomes $\omega_n$, $n = 1, 2, \dots$ such that $\omega_n$ can be either $H$ = heads or $T$ = tails. Define \begin{equation} Y_n\left(\omega\right) = \begin{cases}1 &amp; \text{ if }\omega_n = H \\ 0&amp; \text{ if } \omega_n = T\text{.} \end{cases} \end{equation}  Define $\displaystyle Y = \sum\limits_{n=1}^{\infty}\dfrac{2Y_n}{3^{n}}$. Shreve shows that $Y$ can only take values in the Cantor set, and since that particular set has measure zero, its Lebesgue measure is zero as well, thus it does not have a probability density function (since integrating such a density function would give a value of $0$). This part I understand, as well as my professor.  Now Shreve claims that $Y$ does not have a probability mass function.      If it did, then for some number $x \in C$ [the Cantor set] we would have $\mathbb{P}\left(Y=x\right) &gt; 0$. But $x$ has a unique base-three expansion    \begin{equation} x = \sum\limits_{n=1}^{\infty}\dfrac{x_n}{3^{n}}\text{,} \end{equation}   where each $x_n$ is either 0, 1, or 2 unless $x$ is of the form $\dfrac{k}{3^{n}}$ for some positive integers $k$ and $n$. In the latter case, $x$ has two base-three expansions... In either case, there at most two choices of $w \in \Omega_{\infty}$ [that is, the infinite coin-toss space] for which $Y(\omega) = x$. In other words, the set $\{\omega \in \Omega; Y(\omega) = x\}$ has either one or two elements. The probability of a set with one element is zero, and the probability of a set with two elements is $0 + 0 = 0$. Hence $\mathbb{P}\left\{Y = x\right\} = 0$.   My professor and I were able to follow this until the statement "The probability of a set with one element is zero." Is this a true statement? How so?   I thought it would be perhaps the fact that the Lebesgue measure of $[a, a] = \{a\}$ is zero, but I don't see any reason why Shreve would only use the Lebesgue measure, since it is not a general probability measure. How to get the list of properties of a class? Ok, after seeing , I have decided to skip to the end and go with .  So what I need is a beginner's practical guide to Git.  "Beginner" being defined as someone who knows how to handle their compiler, understands to some level what a  is, and has touched source control without understanding it very well.  "Practical" being defined as this person doesn't want to get into great detail regarding what Git is doing in the background, and doesn't even care (or know) that it's distributed.  Your answers might hint at the possibilities, but try to aim for the beginner that wants to keep a 'main' repository on a 'server' which is backed up and secure, and treat their local repository as merely a 'client' resource.  So:  Installation/Setup    How do you set up Git?  Try to cover Linux, Windows, Mac, think 'client/server' mindset.         Working with the code          How do you handle binary files (visio docs, for instance, or compiler environments)? How do you merge files changed at the "same time"?    Tagging, branching, releases, baselines    How do you pull a particular 'release'?   How do you resolve conflicts and complete the merge? How do you merge parts of one branch into another branch?        Other   Describe and link to a good GUI, IDE plugin, etc. that makes Git a non-command line resource, but please list its limitations as well as its good.    - Cross platform, included with Git  - Cross platform history viewer, included with Git  - Mac OS X  - Mac OS X history viewer  - Cross platform, commercial, beta  - console GUI for Linux  - GUI for Windows, Linux  - package for Windows, includes friendly GUI  Any other common tasks a beginner should know?     How do I work effectively with a subversion repository set as my source control source?   Other Git beginner's references                 Delving into Git     (and )    I will go through the entries from time to time and 'tidy' them up so they have a consistent look/feel and it's easy to scan the list - feel free to follow a simple "header - brief explanation - list of instructions - gotchas and extra info" template.  I'll also link to the entries from the bullet list above so it's easy to find them later. Order of elements is lcm-closed in abelian groups How can I prove that if $G$ is an Abelian group with elements $a$ and $b$ with orders $m$ and $n$, respectively, then $G$ contains an element whose order is the least common multiple of $m$ and $n$?  It's an exercise from Hungerford's book, but it's not homework. I could not solve it when I take the course on groups and I think it should be easy. There is a hint that says to first prove when $m$ and $n$ are coprimes. I did this part. But I have no idea how to solve the general case.  Thanks. When asking about , people sometimes refer to the strict aliasing rule. What are they talking about? Is there a system in place to prevent new users asking, answering and accepting answers to each others' off-topic questions? Normally new users are not allowed to up-vote until they gain a bit of reputation by receiving up-votes or accepts from the community of a site.  But this has a weak point that is being exploited:     A number of new users post, answer, and accept each others'    off-topic questions before we can close them (in less than 5 min).    Accepting each others' answers gains them enough reputation to    start up-vote each others' posts.   We are closing them and the moderators clean up afterwards but  this keeps happening over and over again and becoming frustrating  to the extent that I think a systematic solution might be required. Is there a system to prevent this kind of circumvention of  the reputation requirements using accepts by new users?  One possible solution might be to change the system so accepts from users who do not have enough reputation to up-vote  do not give any reputation (at least until the answer gets an up-vote). Stargate - Destiny's Gate Address from Icarus base Simple question - What is the address they dial to connect with the Ancient ship Destiny in Stargate Universe from Icarus base.  I have found the address on  - however they have an unconfirmed order, and not sure of the validity of this.  Any help deciphering the above would be great! After the upgrade of salesforce to Spring 15, I am getting this error "Error: You have exceeded the maximum number (5) of Apex scheduled jobs." when scheduling more than 5 batches. I have this issue in several environment. I used to more than 20 batches scheduled at once.   On the link below, it says that I can schedule up to 100 batches but I am being restricted to 5 only.   I can't have more than 5 jobs here. Can I add new methods to the String class in Java? I've got a problem using \label and \ref in Texmaker 3.1 (MiKTeX 2.9). When I put label somewhere, i got a frame with label name in it. How can I hide it?  \begin{figure}[h] \includegraphics[width=0.75\textwidth]{img/2}\caption{SomeCaption} \label{fig:4eb1} \end{figure} $\frac{w}{w_1}-1=4$ $4\leqslant n\leqslant3$, using (pic. \ref{fig:4eb1}),blah-blah.   PS: Sorry for my awesome English. :)