Cast shatter on the other side of a wall
Do you need line of sight to cast spells on someone?
From this  by orlp:     If I strap a rocket booster to a rocket in space and fire it for one second, then the work provided is much higher when the rocket is flying fast compared to when the rocket was stationary. In both cases the rocket fires for the same duration but in the former case the rocket travels a much greater distance during this period. What gives? Power of a generator generates the group iff this power is coprime to the group order. Let $\langle g\rangle$ be a cyclic group of order $n$. Suppose $1\leq q \leq n-1$, I want to show that $g^q$ generates $\langle g\rangle$ if and only if $\gcd(n,q)=1$.  Suppose $g^q$ generates $\langle g\rangle$, then   $$ 1,g^q,g^{2q},...,g^{(n-1)q} $$  are all distinct and $g^{nq}=1$, since $n$ is the order of the group. Note that this is the lowest multiple of $q$ for which this is the case. Therefore $\operatorname{lcm}(q,n)=qn$ and it follows that $q$ and $n$ are coprime.  On the other hand, if $\gcd(q,n)=1$, then $\operatorname{lcm}(q,n)=qn$, we have $g^{nq}=1$ and hence all of  $$ 1,g^q,g^{2q},...,g^{(n-1)q} $$  are distinct again and $g^q$ is a generator.  Is my proof correct at all and is there maybe a more elegant argument? Adding CSS Reference on Site Page I have a group $G$ with $p^2$ elements, where $p$ is a prime number. Some (potentially) useful preliminary information I have is that there are exactly $p+1$ subgroups with $p$ elements, and with that I was able to show $G$ has a normal subgroup $N$ with $p$ elements.   My problem is showing that $G$ is abelian, and I would be glad if someone could show me how.   I had two potential approaches in mind and I would prefer if one of these were used (especially the second one).   First: The center $Z(G)$ is a normal subgroup of $G$ so by Langrange's theorem, if $Z(G)$ has anything other than the identity, it's size is either $p$ or $p^2$. If $p^2$ then $Z(G)=G$ and we are done. If $Z(G)=p$ then the quotient group of $G$ factored out by $Z(G)$ has $p$ elements, so it is cylic and I can prove from there that this implies $G$ is abelian. So can we show theres something other than the identity in the center of $G$?  Second: I list out the elements of some other subgroup $H$ with $p$ elements such that the intersection of $H$ and $N$ is only the identity (if any more, due to prime order the intersected elements would generate the entire subgroups). Let $N$ be generated by $a$ and $H$ be generated by $b$. We can show $NK= G$, i.e every element in G can be wrriten like $a^k b^l $. So for this method, we just need to show $ab=ba$ (remember, these are not general elements in the set, but the generators of $N$ and $H$).   Do any of these methods seem viable? I understand one can give very strong theorems using Sylow theorems and related facts, but I am looking for an elementary solution (no Sylow theorems, facts about p-groups, centrailzers) but definitions of centres and normalizers is fine. Unable to correct problems, you have held broken packages PDF export on macOS Mojave produces bold fonts Understanding Group Permissions Why shouldn't I care what model/make/year my bicycle is? We often get questions on bicycles.stackexchange about identifying a bicycles manufacturer, model, make, and year.  Often these questions are downvoted or closed -- usually with a comment attached to them that it's not necessary to know the exact model/make/year of a bicycle if you want to fix, repair, or ride your bike. Why should owners not care about the model/make/year of their bicycle? Especially when it comes to less expensive bikes (aka BSOs) and BMX bicycles? This is intended as a canonical question that we can point closed questions to. Also see      (generally not helpful for identifying a brand/model) When a body lies on the surface of the Earth it is under the influence of gravity. The force on the body due to gravity causes it to exert a force on the ground and the normal reaction acts in the opposite direction causing the resultant force on the body to be zero.  However, how can the body exert a force on the ground when it does not have any acceleration? Since force equals mass times acceleration how does a body without acceleration experience a force? Background: I have the opportunity to pursue a Computer Science graduate degree at a well respected program (around top 20). I spoke with the professors at the university and I feel like I am a good fit with their research program. My ultimate pursuits are to complete a PhD in CS. I've done research as an undergraduate and it has only confirmed my notion that I want to pursue a research track career.  More background: I applied to jobs (in case I was rejected to all the programs I applied for) prior to hearing back from my respective graduate programs. I accepted a position for an industry job (this was to secure I wouldn't be both unemployed &amp; not in school).  Dilemma: Turns out my top graduate school is very interested in me, and I'm very interested in them. I would rather go to graduate school than work in industry forever. The industry job pays very well. I am split between A) working for a maximum of 1 year(It would only be 1 year, seriously I do not care about the money enough to work past a year) and B) going straight to graduate school.   My concerns are as follows, if I choose A):    Could I potentially defer my admissions? If I am not allowed to defer my admissions, would I have a good chance of re-applying and being accepted a year later? Would my potential advisers look down on me for deciding to work a year?   I already made my mind up it would be a 1 year gig if I decide A). I know some people say once you make money, you may not be able to readjust to the graduate salary pay. But I don't think that will apply to because I'm going to live very frugally with or without industry pay. It has been some months since I've turned on my Playstation 3. Today, when I turned it on with my controllers, it didn't work. I thought "Hey, the battery surely ran out", so I tried to charge them but they don't seem to be charging at all. Could it be that they're officially weight papers by now? Prevent iTunes from starting when iPhone/iPod/iPad is plugged in on Windows Is there any way to prevent iTunes from starting under XP when an iPhone is plugged in?  Using iTunes v 9.  Usually I just want to charge my phone and don't want iTunes to start.  Note: Exact  of this question - except this is for v9 on Windows.  The answer in the related question doesn't work for me. I'm trying to include a link to some code on OneDrive; the link includes an underscore, which is causing a lot of trouble.  I tried adding the entry like this:  @misc{code,     title = {Code},     url = {https://1drv.ms/link_link} }   I use \usepackage{url} and \bibliographystyle{IEEEtran}. Everything compiles fine and the bibliography shows the correct entry. However, if you click the link, it opens a page at https://1drv.ms/link - so ignoring the underscore and everything after it, which leads to a wrong page. Even copy and paste from the PDF doesn't work since the underscore is copied as space, which again is a broken link.  How can I add the link such that it can be clicked and copied properly? I tried \_, \usepackage{underscore}, {{link}}, different bibtex entries but nothing worked. Also all the related questions/answers don't answer mine. Are there any reputable studies on spanking? Recently I needed to do weighted random selection of elements from a list, both with and without replacement. While there are well known and good algorithms for unweighted selection, and some for weighted selection without replacement (such as modifications of the resevoir algorithm), I couldn't find any good algorithms for weighted selection with replacement. I also wanted to avoid the resevoir method, as I was selecting a significant fraction of the list, which is small enough to hold in memory.  Does anyone have any suggestions on the best approach in this situation? I have my own solutions, but I'm hoping to find something more efficient, simpler, or both. Performance differences between equal (=) and IN with one literal value I'd like to use a different + symbol in addition to + in math mode, preferably one which is just bolder than the standard one.  I'm currently using \pmb{+} from the amsbsy package, but it's a bit too lean for my taste, and it looks a bit pointy to the ends.   I am aware of \ding{58} from the pifont package, as well as \Plus from bbding, both of which seem to be non-math mode symbols, and look really bad next to math symbols.  Am I missing some other option to produce a bold +, apart from drawing the symbol myself? I can get the element like this $("#txtEmail") but I'm not sure how to get the actual value. How to read a file line-by-line into a list? Why do hard links exist?