I have a bug where I am getting a current modification exception and I can't find it
Iterating through a Collection, avoiding ConcurrentModificationException when removing objects in a loop
How prove there is no continuous functions $f:[0,1]\to \mathbb R$, such that $f(x)+f(x^2)=x$. Prove that there is no continuous functions $f:[0,1]\to R$, such that   $$ f(x)+f(x^2)=x. $$   My try. Assume that there is a continuous function with this property. Thus, for any $n\ge 1$ and all $x\in [0,1]$, \begin{align*} f(x)&amp;=x-f(x^2)=x-\big(x^2-f(x^4)\big)=x-x^2+\big(x^4-f(x^8)\big)=\cdots\\ &amp;=x-x^2+x^4-\cdots+(-1)^n\left(x^{2^n}-f\big(x^{2^{n+1}}\big)\right) \end{align*}  since $f(0)=0$ and $\displaystyle\lim_{n\to \infty}x^{2^{n+1}}=0$ for any $x\in(0,1)$,it follows by the continuity of $f$ that $\displaystyle\lim_{n\to \infty}f\big(x^{2^{n+1}}\big)=0$, hence $$f(x)=x-x^2+x^4-x^8+\cdots+(-1)^nx^{2^n}+\cdots$$ for any $x\in (0,1)$  Why do I have prove that there is exists such a continuous functions $f$? Maybe  my example is wrong? Why?  if my   method is wrong,then How prove this problem ? Thank you. Who can see what in my profile? When editing my user profile (name, date of birth, etc.), how can I tell which information can be seen by other users (publicly visible) and what is only used by the system (and how it's used)? Harry Potter: Why 7? While rereading the entire Harry Potter series I noticed that the number 7 occurs quite often, for instance:    Seven books Seven Horcruxes planned by Tom Riddle Seven Weasley kids Seven years at Hogwarts Seven Harry Potters Seven people in a Quidditch team   Did J.K. Rowling ever mention why 7 was so important in Harry Potter? Fail of markdown parser detected if '_ 's are used in a naked URL Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$ Let $a\ge 2$ be an integer. Show that for positive integers $m,n$, we have $a^n - 1$ divides $a^m - 1$ if and only if $n$ divides $m$.   I am having trouble showing this. I've seen a similar problem on here but it only shows one direction for the proof. I would like to see both directions.  This is what I have so far: Let $m = nq + r$, then $a^m - 1 = a^{nq + r}- 1 = a^{nq}a^r - 1 = a^{nq}a^r - a^r + a^r - 1 = a^r(a^{nq} - 1) + a^r - 1$. Since $0\le r\lt n$, $r = 0$ then $a^m - 1$ = $a^{nq} - 1$ so $a^m$ = $a^{nq}$ . Thus $m = nq$ hence $n\mid m$. Makeindex error when ! used in \index in align environment See MWE below. The symbol ! in the \index causes an error. This is only the case in align environments (not in equation). So it seems to have to do something with amsmath. If ! is put in an \mbox there is no error. But then of course the font size is not adjusted to subindices. Curiously enough, \index{$C_{K!!}$} (two exclamation marks) doesn't cause any error.    \documentclass{article} \usepackage{amsmath} \usepackage{imakeidx} \makeindex \begin{document} \begin{align} y\index{$C_{K!}$} \end{align} \printindex \end{document}     Any solutions suggested? Is it a good approach to call return inside using {} statement? What is the fastest, easiest tool or method to convert text files between character sets?  Specifically, I need to convert from UTF-8 to ISO-8859-15 and vice versa.  Everything goes: one-liners in your favorite scripting language, command-line tools or other utilities for OS, web sites, etc.  Best solutions so far:  On Linux/UNIX/OS X/cygwin:   Gnu  suggested by  is best used as a filter. It seems to be universally available. Example:  $ iconv -f UTF-8 -t ISO-8859-15 in.txt &gt; out.txt   As pointed out by , there is an . Gnu  () suggested by  will convert one or several files in-place. Example:  $ recode UTF8..ISO-8859-15 in.txt   This one uses shorter aliases:  $ recode utf8..l9 in.txt   Recode also supports surfaces which can be used to convert between different line ending types and encodings:  Convert newlines from LF (Unix) to CR-LF (DOS):  $ recode ../CR-LF in.txt   Base64 encode file:  $ recode ../Base64 in.txt   You can also combine them.  Convert a Base64 encoded UTF8 file with Unix line endings to Base64 encoded Latin 1 file with Dos line endings:  $ recode utf8/Base64..l1/CR-LF/Base64 file.txt    On Windows with  ():    PS C:\&gt; gc -en utf8 in.txt | Out-File -en ascii out.txt  (No ISO-8859-15 support though; it says that supported charsets are unicode, utf7, utf8, utf32, ascii, bigendianunicode, default, and oem.)   Edit  Do you mean iso-8859-1 support? Using "String" does this e.g. for vice versa  gc -en string in.txt | Out-File -en utf8 out.txt   Note: The possible enumeration values are "Unknown, String, Unicode, Byte, BigEndianUnicode, UTF8, UTF7, Ascii".   CsCvt -  is another great command line based conversion tool for Windows. Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ Let $X=C[0,1]$ be the set of all continuous functions on $[0,1]$. For any two functions $f,g\in X$, set $$d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|.$$  Prove $(X,d)$ is a metric space.    My attempt:  This question comes down to showing that $d$ is a metric, which means that:   $d(f,g)\geq 0$ $d(f,g)=0\iff f=g$ $d(f,g)=d(g,f)$ Triangular sadness.     1) Nonnegativity is guaranteed by taking the supremum of the absolute value of a sum, since the absolute value of a sum is non-negative, the supremum of non-negative numbers is non-negative.  $\square$    2) For the supremum of $|f(x)-g(x)|$ to be equal to $0$, the greatest number for $|f(x)-g(x)|$ from $0\leq x\leq 1$ must be $0$, hence $|f(x)-g(x)|=0$ for all values on this range, and thus $f(x)=g(x)$ is the constant zero map.  $\square$    3) $|f(x)-g(x)|=|g(x)-f(x)|$ implies that $\sup \limits_{0\leq x \leq 1} |f(x)-g(x)|=\sup \limits_{0\leq x \leq 1} |g(x)-f(x)|$ $\square$    4) Triangular sadness:  We need to prove that $d(f,g)\leq d(f,h) + d(g,h)$  I.e.  $\sup \limits_{0\leq x \leq 1} |f(x)-g(x)|\leq \sup \limits_{0\leq x \leq 1} |f(x)-h(x)|+\sup \limits_{0\leq x \leq 1} |g(x)-h(x)|$  Now I am not sure how I can 'behave' with the sup 'function'. Can I do the following?  $$\sup|f(x)-g(x)| = \sup|f(x)-g(x)+h(x)-h(x)|=\sup|f(x)-h(x)+(h(x)-g(x))|$$ Then use normal triangular inequality:  $$\sup|f(x)-h(x)+(h(x)-g(x))|\leq \sup(|f(x)-h(x)|+|h(x)-g(x)|)=\sup(|f(x)-h(x)|+|g(x)-h(x)|)$$   This is obviously less than $\sup \limits_{0\leq x \leq 1} |f(x)-h(x)|+\sup \limits_{0\leq x \leq 1} |g(x)-h(x)|$, but I don't know how to prove it. How do I finish that one off. Generating a unique machine id I am trying to create a beamer animation using tikz, but I can't even get basic rotating of a tikz image working as desired.  I have a rectangle specified by  \draw[rounded corners,dotted] (-3.5,0) rectangle (-0.5,2);   I would like to rotate this rectangle about the point (-1,.5) (which is near the bottom right corner of the rectangle).  I've tried:  {\rotatebox[origin=c,x=-1,y=.5]{130}{             \draw[rounded corners,dotted] (-3.5,0) rectangle (-0.5,2);}}   But this produces an error.  If I just use the origin=c option, it compiles, but it rotates the rectangle about the point (0,0).  I've also tried it with just the x=-1,y=.5 (without the origin option), and it fails to compile.  I have searched the documentation, which is where I learned about the options x=xdim and y=ydim.  However, I can't find a single example demonstrating how these options should be used.  Any suggestions would be greatly appreciated.    Also, if there is a better command for rotating things that will be more suitable for use in a beamer animation, please let me know.  Thanks! How do you prove the  from statistical mechanics? To prove entropy will only increase with time? How to prove? Please guide. I have a parent theme that uses correctly load_theme_textdomain() to load all the translated strings in many languages.  Then I created a child theme that uses load_child_theme_textdomain() to achieve the same thing for its strings.  There are certain translated strings for a particular language on the parent theme that I'd like to replace/override in the child theme.  I know if they were on a template file I could replace the file and simply change the textdomain for those strings, but unfortunately the ones I'm talking about are used in many places and also in the dashboard (so inside some filter/action functions).  So my question is: is there a way to replace those translated strings inside the child theme without having to replace parent template files or functions?  I don't know, maybe adding a parent-theme.mo file inside the languages folder of the child theme with only those strings translated, or something like that. How can I pass a command line argument into a shell script? I know that shell scripts just run commands as if they were executed in at the command prompt.  I'd like to be able to run shell scripts as if they were functions... That is, taking an input value or string into the script.  How do I approach doing this? Combinations of selecting $n$ objects with $k$ different types Suppose that I am buying cakes for a party. There are $k$ different types and I intend to buy a total of $n$ cakes. How many different combinations of cakes could I possibly bring to the party? Computation of likelihood when $n$ is very large, so likelihood gets very small? How to transfer a DVD to my hard drive How can I buy apps in the App Store without a credit card? While I'm waiting for my credit card to arrive, I still want to be able to buy some apps for my iPad!  Are there alternative ways to pay for apps, without resorting to someone else's credit card? What to do about rep inflation? Overall reputation scores are headed nowhere but up.  Pretty soon there will be thousands of mods.  What to do?  Thoughts on possibilities:   Thousands of mods is fine.  Carry on. Adjust the threshold values for high-level SO privileges. Do a straight-up inflation adjustment.  You had x rep, then at the turn of the year we applied the Consumer Rep Index to all scores, now you have (x / inflation) rep. Implement a rep decay mechanism, so that inactivity sheds some of the overall rep that's built up. Disk space usage doesn't add up with df & du