Is there a short way to say $f(f(f(f(x))))$?  I know you can use recursion:  $g(x,y)=\begin{cases} f(g(x,y-1)) &amp; \text{if } y &gt; 0, \ \newline x &amp; \text{if } y = 0. \end{cases}$
If I have the function $f(x)$ and I want to apply it $n$ times, what is the notation to use?  For example, would $f(f(x))$ be $f_2(x)$, $f^2(x)$, or anything less cumbersome than $f(f(x))$? This is important especially since I am trying to couple this with a limit toward infinity.
What terminal command or program can list an SSD's Power On Hours? I'm looking to script using the unix shell the retrieval of SSD (and HDD) data that disk utility shows in the information window.    I'm immediately interested in the Power On Hours and Power Cycle, but if all the data comes from one data structure or tool, so much the better. The usual tools diskutil info disk0 and ioreg -l are not producing the "of interest" values listed in the screen shot above.  Is this information available from any of the built in OS X terminal tools or do I need to locate an open-source package that makes the same API calls to get this level of detail about a Mac drive? NRPE unable to read output, but why? Proof by induction of summation inequality: $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$ Prove by induction the summation of $\frac1{2^n}$ is greater than or equal to $1+\frac{n}2$.  We start with $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$$ for all positive integers.   I have resolved that the following attempt to prove this inequality is false, but I will leave it here to show you my progress. In my proof, I need to define P(n), work out the base case for n=1, and then follow through with the induction step. Strong mathematical induction may be used.  This is equivalent to $$\sum_{k=0}^n\frac1{2^k}\ge 1+\frac{n}2\;.$$  Let $P(n)$ be summation shown above.   Base case for $n=1$, the first positive integer,   $$\sum_{k=0}^1\frac1{2^k}=\frac1{2^0}+\frac1{2^1}=1+\frac12=\frac32\ge 1+\frac12=\frac32\;,$$  so base case is true.  Induction step: Assume $P(n)$ is true and implies $P(n+1)$. Thus   $$\sum_{k=0}^{n+1}\frac1{2^k}\ge\frac1{2^{n+1}}+\sum_{k=0}^n\frac1{2^k}\ge 1+\frac{n+1}2\;.$$  This can be written as   $$\sum_{k=0}^{n+1}\frac1{2^k}\ge \frac1{2^{n+1}}+1+\frac{n}2\ge 1+\frac{n+1}2\;.$$  I work the math out but I get stuck contradicting my statement. Please show your steps hereafter so I can correct my mistakes. Prove that the equation: $c_0+c_1x+\ldots+c_nx^n=0$ has a real solution between 0 and 1. Let $c_0,c_1,c_2,\ldots ,c_n$ be constants such that :  $$c_0+\frac{c_1}{2}+\ldots+\frac{c_{n-1}}{n}+\frac{c_n}{n+1}=0$$  I have to prove that the equation: $$c_0+c_1x+\ldots+c_nx^n=0$$  Has a real solution between 0 and 1.  Didn't know how to start...I thought that maybe I could use something about derivative...But I'm lost... Any help,much appreciated!!! Let $A$ and $B$ be two $n \times n$ symmetric and positive definite matrices.   If $A \prec B$, then is it true that $B^{-1} \prec A^{-1}$?  Here, $A \prec B$ means that $B-A$ is positive definite. Query works on Oracle 11g but fails on Oracle 8i $(\Bbb R,+)$ is a topological group. Is there any other group structure on $\Bbb R$ such that it is still a topological group and this group is not isomorphic to $(\Bbb R,+)$ ?  Refer to  for the general problem. In that problem I have conjectured that there is no such group however I couldn't prove it. Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $ Should I get the +2 bonus twice for suggesting a tag wiki? Redefining "cited on" string (and others) in biblatex I'm trying to modify the strings biblatex uses to set the "cited on" string used in the bibliography to identify the pages where a given references has been cited.    What I gather from the biblatex documentation (e.g., § 4.9.2.16) is that something like   \renewbibmacro{backrefpage}{&lt;newtext&gt;} \renewbibmacro{backrefpages}{&lt;newtext&gt;}   ought to work, but they have no effect for me (I'd also like to get this working for the "visited on" string, urlseen).  My MWE is unchanged from , with the addition of these (and related lines) just about anywhere. How to use induction to prove the product rule for higher derivatives? How do I show through mathematical induction that   \begin{equation} \frac{d^n}{dx^n}[f(x)\cdot g(x)] = \sum_{k = 0}^{n} \binom{n}{k} \frac{d^k}{dx^k} [f(x)] \cdot \frac{d^{n-k}}{dx^{n-k}} [g(x)] \end{equation}  which is the product rule for higher derivatives? How to fix "W: Duplicate sources.list entry"? polyglossia or bidi bug: [\d\.] gets inverted to [\.\d] After bidi was updated to v. 19.6 (2016-05-23), using at least one RTL language with polyglossia leads to number and following dot being inverted. Cf.  \documentclass{book} \usepackage{polyglossia} \setotherlanguage{hebrew} \begin{document} 6. \end{document}   with and without \setotherlanguage{hebrew}. Note that it is not necessary to actually use that language.  I do not know whether this is a bug in bidi or an incompatible change which polyglossia needs to adapt to. Because of this I have not opened an issue on polyglossia's github page. Sadly, bidi's contact information are outdated – the sourceforge referred to on CTAN is unused and the github page that is mentioned in the package's readme and PDF does not seem to exist any more either.  The problem might be something along the lines of     Does anybody have the previous version of bidi at hand?  TIA, Dario How to search something on across multiple sites with a single Google query? I tried   (search term site:stackoverflow.com) OR (search term site:superuser.com) OR (search term site:stackexchange.com) search term site:stackoverflow.com,superuser.com,stackexchange.com search term site:stackoverflow.com site:superuser.com site:stackexchange.com   None of them works. What is the correct way to search? At what times before $1{:}30$ in the afternoon do all three doctors schedule their appointments to begin at the same time? In a certain medical group, Dr. Schwartz schedules appointments to begin $30$ minutes apart, Dr. Ramirez schedules appointments to begin $25$ minutes apart, and Dr. Wu schedules appointments to begin $50$ minutes apart. All three doctors schedule their first appointments to begin at $8{:}00$ in the morning, which are followed by their successive appointments throughout the day without breaks. Other than at $8{:}00$ in the morning, at what times before $1{:}30$ in the afternoon do all three doctors schedule their appointments to begin at the same time?  I'm not sure how to approach this question. They all begin at the same time at $8{:}00$. Also doesn't the overlapping of appointments depend on how long their respective appointments take? Totally ordering the power set of a well ordered set. Let's say I take a set $S$, where $S$ can be well ordered. From what I understand, one can use that well ordering to totally order $\mathscr{P}(S)$.  How does a body actually use the well ordering of $S$ to construct a total ordering of $\mathscr{P}(S)$?  Perhaps construct is too strong a word. Is there a way to prove that a total ordering of $\mathscr{P}(S)$ just exists, no need to explicitly state what it is, since that might be very difficult. How to get access to USB-storage from an application installed as Snap? I have Ubuntu 18.04 LTS installed. I inserted USB-flash to my laptop. It was automounted to /media/username/DISKLABEL and opened in Nautilus. I can read and write objects to it. How can I access it from the application, which is installed as Snap? I tried snaps of the following applications such as gedit, arduino-mhall119, organize-my-files. All them show me errors if I navigate to /media folder:    Could not read the contents of Media Error opening the directory of '/media': permission denied  Output of active Snap  is below: $ snap interfaces  | grep -v &quot;.*-$&quot; Slot                             Plug :browser-support                 organize-my-files :core-support                    core:core-support-plug :cups-control                    gedit :desktop                         gedit,organize-my-files :desktop-legacy                  gedit,organize-my-files :gsettings                       gedit,organize-my-files :home                            arduino-mhall119,gedit,organize-my-files :network                         arduino-mhall119,gedit,organize-my-files,pulsemixer :opengl                          organize-my-files :pulseaudio                      organize-my-files,pulsemixer :unity7                          arduino-mhall119,gedit,organize-my-files :wayland                         gedit :x11                             arduino-mhall119,organize-my-files,pulsemixer gnome-3-26-1604:gnome-3-26-1604  gedit gtk-common-themes:gtk3-themes    gedit gtk-common-themes:icon-themes    gedit gtk-common-themes:sound-themes   gedit -                                arduino-mhall119:serial-port -                                gedit:avahi-observe -                                gedit:mount-observe  Updates:  connecting mount-observe to Gedit with snap connect gedit:mount-observe does not help. it seems that we need special Snap with support of  interface. Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$ I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$.  I already know the logical Proof:  $${n \choose k}^2 = {n \choose k}{ n \choose n-k}$$  Hence summation can be expressed as:  $$\binom{n}{0}\binom{n}{n} + \binom{n}{1}\binom{n}{n-1} + \cdots + \binom{n}{n}\binom{n}{0}$$  One can think of it as choosing $n$ people from a group of $2n$  (imagine dividing a group of $2n$ into $2$ groups of $n$ people each. I can get $k$ people from group $1$ and another $n-k$ people from group $2$. We do this from $k = 0$ to $n$. Proving $9^n - 8n - 1$ is divisible by $8^2$ for $n\ge 0$? The main doubt one can have about mirrorless is, I think that they do not provide ttl viewfinder. Is this a big disadvantage? In what way is it worst (or best) than classical reflex system?  What are the others advantages and disadvantages of a mirrorless compared to a reflex?  Have mirrorless a chance to replace reflex (at least for non pros) in the future?