Using a cue stick, I strike a pool ball on a pool table vertically downwards, slightly off centre. What is the motion of the ball?
Direction of friction for object that rolls with slipping
Singular or plural verb after a series connected by "or" Collapsing a stream of multiple lines into one line I have a file containing a a large number of lines, each of which contains a bunch of numbers which are separated by spaces. I process this data in a pipe in some way, and then I want to collapse the multiple lines into a single line of all the numbers separated by spaces.  Is there a standard command-line utility I can use to do this? It seems like most line-by-line utilities won't mess with the newlines... Proving $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)<3$ for all positive integers $n$ Prove that $\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)&lt;3$   for all positive integers $n$   This problem is copied from Math Olympiad Treasures by Titu Andreescu and Bogdan Enescu.They start by stating that induction wouldn't directly work here since the right hand side stays constant while the left increases.They get rid of this problem by strengthening the hypothesis.$$\left(1+\dfrac{1}{1^3}\right)\left(1+\dfrac{1}{2^3}\right)\cdots\left(1+\dfrac{1}{n^3}\right)\le3-\dfrac{1}{n}$$ and then proceed by induction.  The problem is that I can't find a motivation for the above change.I mean,we could have subtracted a lot of things from the RHS but what should nudge us to try $\dfrac{1}{n}$?The rest of the proof is quite standard,but I can't see how I am supposed to have thought of it.Is it just experience?Or is it a standard technique?A little guidance and motivation will be appreciated. Pass shell variable as a /pattern/ to awk Having the following in one of my shell functions:  function _process () {   awk -v l="$line" '   BEGIN {p=0}   /'"$1"'/ {p=1}   END{ if(p) print l &gt;&gt; "outfile.txt" }   ' }   , so when called as _process $arg, $arg gets passed as $1, and used as a search pattern. It works this way, because shell expands $1 in place of awk pattern! Also l can be used inside awk program, being declared with -v l="$line". All fine.  Is it possible in same manner give pattern to search as a variable?  Following will not work,  awk -v l="$line" -v search="$pattern" '   BEGIN {p=0}   /search/ {p=1}   END{ if(p) print l &gt;&gt; "outfile.txt" }   '   ,as awk will not interpret /search/ as a variable, but instead literally. Is "shined" correct? If so, is "he shined X on the tree" also correct? Recently, I overhead a former professor of mine use the word shined, a word that makes me grammatically uncomfortable.  She used it as following: "Then, after we shined a light on the other ball, what happened?..."  My first question is: Is using shined mainstream, over what sounds better to my ear, shone?  If so, is there any historical flip-flopping between the two usages?  Secondly, if shined is in fact legit, can it be used transitively?  Saying something like,      The light shined all throughout the night.   while odd, doesn't sound as bad to my ear as,     He shined the light on the ball throughout the night.   The latter sounds quite bad to my ear.  Indeed, à propos the original question, can shone &mdash; the better sounding alternative to my ear &mdash; be used transitively? How to learn commutative algebra? I want to learn commutative algebra from scratch. I was wondering, as you are experts in mathematics, what you think is the best way to learn commutative algebra? Is there any video course available for commutative algebra? Will there be some online course for commutative algebra on some website like Coursera, etc?  I know noncommutative algebra up to the Artin-Wedderburn Theorem. Also, I know group theory up to the Sylow theorems and Galois Theory. I also know some basic topology.  I'm new to this site so I don't know what tags I should add for this question. Please feel free to edit my question.  Edit 1: I want to learn commutative algebra for learning Algebraic Geometry. Is it OK to leave "What have you tried?" comments? What is a session 0? What is a "Session 0"?   When is it done? What does it contain? Is it recommended, if not mandatory? What is each person's role (DM/GM, players, etc.)? What topics are best discussed there?   Usually with my groups, we go directly to the meat of the game right after the character creation, but I see a lot of answers that basically say "you should have a session 0", but it's never explained what it is made of, so it's kind of an enigma for me. Need a proofreading why all the units are satisfied $a^2-2b^2 =\pm1$ for $\mathbf{Z}[\sqrt{2}]$ All the units are satisfied Pell's equation $a^2-2b^2=\pm1$  for $\mathbf{Z}[\sqrt{2}]$, $a,b\in\mathbf{Z}$.  Here is my proof:  Let $a+b\sqrt{2}$ be a unit $\in\mathbf{Z}[\sqrt{2}]$. This implies $$(a+b\sqrt{2})(c+d\sqrt{2})=1, c+d\sqrt{2}\in\mathbf{Z}\sqrt{2}$$ $$\implies \mathrm{norm}((a+b\sqrt{2})(c+d\sqrt{2}))=1$$ $$\implies \mathrm{norm}(a+b\sqrt{2})\cdot \mathrm{norm}(c+d\sqrt{2})=1$$ $$\implies (a+b\sqrt{2})(a-b\sqrt{2})(c+d\sqrt{2})(c-d\sqrt{2})=1$$ $$\implies (a^2-2b^2)(c^2-2d^2)=1$$ $$\implies \left[ a^2-2b^2=1\text{ and } {c}^2-2{d}^2=1\right]\text{ or } \left[ a^2-2{b}^2=-1\text{ and }{c}^2-2{d}^2=-1\right]$$ $$\implies {a}^2-2{b}^2=\pm1$$ Strange Recurrence: What is it asymptotic to? So I have the following recurrence relation for the growth rate of an algorithm:  $T(n)$ = time taken by algorithm to solve problem of size n:  $$T(n) = T(n-1) + T(\lceil(n/2)\rceil)$$  Clearly this does not obey any polynomial law but now my question is what does it obey?  I merely want an asymptotic equivalence.  My guess is subexponential since this is faster than polynomial but clearly slower than any exponential form.  What though is the question.... Solve for $f(x)$ if $f(f(x))=6x-f(x)$ How will people upgrade from 12.10 to later releases after 13.04 is EOL? The Medibuntu Project has come to an end - what do I do now? How do I disable auto-maximizing of newly launched windows in GNOME? Issue with particles in blender donut tutorial How to download Ubuntu over an unreliable connection UTF-8 -> ASCII in C language Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded?  Is there a good reason for this or is it simply "because it works". I have a NetGear WGR614 wireless router.  There is a 'region' setting in the wireless options.  I'm wondering what technical purpose of this setting is?  The admin help states:     It may not be legal to operate the router in a region other than the region shown here.    Which makes me wonder if it could be adjusting it's max power output based on regional radio laws? Convergence of $\sum_{n=1}^{\infty}{\sqrt[n]{n}-1 \over n}$ I am trying to conclude about the convergence of $$\sum_{n=1}^{\infty}{\sqrt[n]{n}-1 \over n}$$ If I take ${\sqrt[n]{{\sqrt[n]{n}-1 \over n}}}= {\sqrt[n]{\sqrt[n]{n}-1} \over \sqrt[n]{n}}$ this looks like it converges to $0$ but I am suspicious about the numerator. At the same time, other techniques I know like the ratio test, $n(\sqrt[n]{n}-1)$ and comparisons don't yield meaningful results. Any hints?