Do we have $(z^r)^s=(z^s)^r$ for all $z\in\mathbb{C}-\{0\}$?
How to combine complex powers?
Center of mass in a straight rod I got an assignment to prove that in a straight homogeneous rod, you can always choose a coordinate system in such a way that  $$\int_S x_1 \, dx_1 \, dx_2=0 $$  $$\int_S x_2 \, dx_1 \, dx_2=0 $$  $$\int_S x_1x_2 \, dx_1 \, dx_2=0$$  where $S$ is the cross section of the rod  Now, intuitively, I can understand that the first two expressions basically state that if you find the mass center of the rod, then all you have to do is to put the origin of the coordinate system in the mass center, and the integrals turn out to be zero. What I'm curious about is whether or not this is a good enough of an explanation, and also, a hint on how to obtain the third equality. Python analog of PHP's natsort function (sort a list using a "natural order" algorithm) Managing Metered Connections on OSX How to differentiate formally? Should moderators be allowed to decline flags on their own comments? recursive statistics on file types in directory? I did a website scrape for a conversion project. I'd like to do some statistics on the types of files in there -- for instance, 400 .html files, 100 .gif, etc. What's an easy way to do this? It has to be recursive.   Edit: With the script that maxschelpzig posted, I'm having some problems due to the architecture of the site I've scraped. Some of the files are of the name *.php?blah=blah&amp;foo=bar with various arguments, so it counts them all as unique. So the solution needs to consider *.php* to be all of the same type, so to speak. Why does 1+++2 = 3? How does Python evaluate the expression 1+++2?  How many ever + I put in between, it is printing 3 as the answer. Please can anyone explain this behavior  And for 1--2 it is printing 3 and for 1---2 it is printing -1 Is it possible to animate the speed of a strip? I want to accelerate the speed of a video and decelerate some seconds before it ends. I wonder if it's possible to do it into the video editing. I know how to increase the speed of a strip but, what I need is a trasition between normal speed and fast speed. Let's say there is a user who has found a satisfactory answer to a common question asked on Stack Overflow (or other Stack Exchange website).  This answer may be a snippet of code, or an addon, or a framework, or something else.    Is it acceptable for this user to formulate an answer for one question, find every related question and then copy the exact answer to each of those questions?  And, if not, what should  users who finds themselves in this situation do? Should an image carousel on a mobile site automatically rotate? Finite ring has only zero divisors and units If a nonzero element $a$ of a ring $\mathbb{D}$ does not have a multiplicative inverse, then $a$ must be a zero divisor.   If $\mathbb{D}$ has finitely many elements, then the statement is true. However, I don't see why, as for the infinite case, 7 isn't a zero divisor only because every element $a &gt; 0$ when multiplied by a $b &gt; 0$ gives you $ab &gt; 0$ which has nothing to do with infinity or finite amounts of elements. Inductive proof of the closed formula for the Fibonacci sequence Fibonacci numbers $f(n)$ are defined recursively:  $f(n) = f(n-1) +f(n-2)$ for $n &gt; 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt 5}{2}\right)^n- \left(\dfrac{1 - \sqrt 5}{2}\right)^n \tag1$$  How to prove (1) using induction?  Remarks One could get (1) by the general method of solving recurrences: look for solutions of the form $f(n)=r^n$, then fit them to the initial values. But there should be a more concrete proof for this specific sequence, using the principle of mathematical induction. -I'm not a religious person so this is not a denial. I'm just trying to understand the most fundamental topic about Universe.  -I know the Big Bang cosmological model is not a law but it's a theory.  -I'm a non-Native English speaker so please let me know if the question is not clear for you.  -This question came to my mind after watching What are the powers shown by the Rings of Power? I asked this question  but I haven't gotten a useful answer. Could you tell me the difference in meaning between these sentences?        Do you think you will visit them next week?     Will you visit them next week?     Are you going to visit them next week?     Are you visiting them next week?     Are you going to be visiting them next week?     Are you planning to be visiting them next week?     Do you plan to visit them next week?      Will there be any difference in the answers?        I think I will visit them next week.     I will visit them next week.     I am going to visit them next week.     I am visiting them next week.     I am going to be visiting them next week.     I am planning to be visiting them next week.     I plan to visit them next week.      Can I use "planning to" instead of "going to"?  (E.g. "Are you planning to visit them next week? — Yes, I am planning to visit them next week.") Which of these sentences show that the subject of conversation has already been discussed? For example, I talked with a friend of mine about my plans to visit somebody. So my friend is aware of my plans.  And my last question. Which of these seven sentences express that the action will certainly take place? How do I resolve unmet dependencies after adding a PPA? Occasionally, when I'm installing stuff, I get an error like the following:  Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  package1 : Depends: package2 (&gt;= 1.8) but 1.7.5-1ubuntu1 is to be installed E: Unable to correct problems, you have held broken packages.   How can I resolve this? Why do high resolution cameras shoot stills at high resolutions, but only typically shoot video at 1080p which is about 2 megapixels?  For example I have a Sony-NEX 6 which can shoot stills at 16 megapixels but only 1080p by 1920p video.  Also how do digital cameras separate the frames in video if the lens shutter never closes?  And if you can shoot video at 30 fps or 60 fps then why can't cameras have a continuous drives at 30 to 60 fps? When conducting multiple regression, when should you center your predictor variables & when should you standardize them? How to calculate the following sum: How may one calculate  $$\lim_{n\to\infty} \ \left(\left(\sum_{k=1}^{n} \frac{1}{3k-1}\right) - \frac{\ln n}{3}\right) \ ?$$ Biometrika journal style file not getting along with TeXLive-2020 I cannot compile styleguide.tex in biometrika.zip (see ). Here is a MWE which leads to the same error message, though not sure yet that I've not tossed baby out with bath water. \documentclass{article} \RequirePackage[thmmarks]{ntheorem}  \makeatletter \def\arabic#1{{\rm\expandafter\@arabic\csname c@#1\endcsname}} \theoremnumbering{arabic} \newtheorem{definition}{Definition} \makeatother  \begin{document}  \begin{definition} This is a definition. \end{definition}  \end{document}  Here is the output of pdflatex: % pdflatex styleguide This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=pdflatex)  restricted \write18 enabled. entering extended mode (./styleguide.tex LaTeX2e &lt;2020-02-02&gt; patch level 5 L3 programming layer &lt;2020-03-06&gt; (/usr/local/texlive/2020/texmf-dist/tex/latex/base/article.cls Document Class: article 2019/12/20 v1.4l Standard LaTeX document class (/usr/local/texlive/2020/texmf-dist/tex/latex/base/size10.clo)) (./ntheorem.sty  Style `ntheorem', Version 1.24 &lt;2004/09/20&gt; ) (/usr/local/texlive/2020/texmf-dist/tex/latex/l3backend/l3backend-pdfmode.def ) (./styleguide.aux) ! Use of \@item doesn't match its definition. \update@series@target@value #1-&gt;\def \reserved@a {                                                   #1}\ifx \target@meta@famil... l.13 T       his is a definition. ? q OK, entering \batchmode