intution behind homology
Intuition of the meaning of homology groups
There are many cards that can put multiple permanents onto the battlefield simultaneously, such as , , or .  How do these interact with the following types of abilities?  As [this permanent] enters the battlefield... You may have [this permanent] enter the battlefield as... [This permanent] enters the battlefield with... When [a permanent with certain characteristics] enters the battlefield...   Justification There have been many questions already about particular instances of this situation.  This question is intended to be a canonical question and answer to all questions such as:      Note to moderators: three of these five questions have already been marked as duplicates pointing to this one.  Can we do the same for the remaining two? Do I need to apply for a visa to enter South Korea as a tourist on an Australian passport? Do I need to apply for a visa to enter South Korea as a tourist for a few days when travelling on an Australian passport? Or will I be able to get one on arrival? How to output MySQL query results in CSV format? Is there an easy way to run a MySQL query from the Linux command line and output the results in  format?  Here's what I'm doing now:  mysql -u uid -ppwd -D dbname &lt;&lt; EOQ | sed -e 's/        /,/g' | tee list.csv select id, concat("\"",name,"\"") as name from students EOQ   It gets messy when there are a lot of columns that need to be surrounded by quotes, or if there are quotes in the results that need to be escaped. I'm a big fan of the votes tab in the profile page, as it allows me to keep track of my usage of the moderation tools. Or at least it would, if votes cast on deleted posts didn't unceremoniously disappear.  Although I don't like it, I can live with that for (up|down|close|reopen)-votes. In fact, I didn't even notice it before gaining 10k rep on SU. But it renders the deletion and undelete subtabs pretty much useless, since it hides – by design – all successful deletion and (yet) unsuccessful undeletion votes.  @slhck suggested in an answer to  the following rationale for this behavior:     While this sounds strange, it's just a result of the … let's call it "Stack Exchange policy" of not showing lists of deleted posts to non-moderators. In my opinion this policy is a little incoherent, because your flagging record shows deleted posts [...]   Actually, the flagging record shows deleted posts to pretty much all users. So does the reputation tab of the user profile (as long as you earned/lost reputation on that post).  If that is in fact the reason, why can't a 10k user see a list of posts he voted on (and thus, already saw)? Is there any practical difference between saving at 300ppi or 72ppi? Where should the period go when using parentheses? For example:     In sentence one, I use this example (which has a parenthesis at the end.)   Should the period be inside, or outside of the parentheses? What about if the entire sentence is a parenthetical (as below)?     (Where does the terminal punctuation go here?)   Is there a hard and fast rule? My pixel perfect picture gets blurry when compiled in LaTeX back commandbutton in jsf Matrix Group induction proof and order of elements question Add bst file with LaTeX / MiKTeX I'm trying to use elsarticle-num-ident.bst, an exact copy of  C:\Program Files (x86)\MiKTeX 2.9\bibtex\bst\elsarticle\elsarticle-num.bst    but every time I run LyX, I get the bibtex error  I couldn't open style file elsarticle-num-ident.bst   This is despite reconfiguring LyX and restarting, and despite running MixTex's Settings/Options' "Refresh FNDB" as Administrator.  It also doesn't work if I put the .bst file in the same folder as the .lyx file.  How should I help lyx/bibtex/Mixtex find the .bst file?  Once I get the identical copy to work, I plan to edit it to not print URLs, as suggested .  Related questions: If I seem tired, can I say: "I haven't slept last night"? If not, why have I been told that we use present perfect for actions that have present effects? Sudo is also user login password - why? how to change? When installing Ubuntu I can create a user with an encrypted folder, what I always do. Now after the installation the login password for this user is also the sudo password. Is that how it should be? I actually would like to have a different user login and sudo password, does is make sense? Should I do that, how? A particle experiences an acceleration described by  $$ a=kx^{-2} $$ where x is the displacement from the origin and k is an arbitrary constant.  To what value does the velocity v of the particle converge to as x approaches infinity if the particle starts at some point x0?  If I approach this problem with energy, then $$ W = \int F \mathrm{d}x $$ $$  = \int_{x_0}^\infty mkx^{-2} \mathrm{d}x $$ $$  = mk(-\infty^{-1}+{x_0}^{-1}) $$ $$ W = K = mk{x_0}^{-1} $$ $$ \frac{1}{2}mv^2 = mk{x_0}^{-1} $$ $$ v = \sqrt{2k{x_0}^{-1}} $$  How would I solve this problem with pure kinematics? (there appears to be some sort of cyclical dependency where acceleration affects velocity, velocity affects displacement, and displacement affects acceleration)  Likewise, two particles experience accelerations described by $$ a_1=k_1x^{-2} $$ $$and $$ $$ a_2=-k_2x^{-2} $$ where x is the distance between the two particles  What two velocities do the particles reach as x approaches infinity if the two particles are initially separated by some x0? I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain.       let $a,b\in R$ s.t. $ab\in P$ ,        $$ab+P=P\implies(a+P)(b+P)=P\implies\overline a \overline b=0$$  and as $R/P$ is an integral domain either $a=0$ or $b=0$ $\implies a+P=P$ or  $b+P=P$.        $\therefore$ either $a\in P$ or $b\in P$.   Hence, $P$ is a prime Ideal.    Am I correct in writing this or am I making some error... $$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5}  = \lim_{x\rightarrow 5} (x+5)$$  I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and simplified limit. I understand how to do this procedurally, but I'd like to know why this works. I've only been told the methodology of expanding the $x^2-25$ into $(x-5)(x+5)$, but I don't just want to understand the methodology which my teacher tells me to "just memorize", I really want to know what's going on. I've read about factoring in abstract algebra, and about irreducible polynomials (just an example...), and I'd like to get a bigger picture of the abstract algebra in order to see why we factor the limit and why the simplified is equal to the original if it's missing the $(x-5)$, which has been cancelled. I don't want to just memorize things, I would really like to understand, but I've been told that this is "just how we do it" and that I should "practice to just memorize the procedure." I really want to understand this in abstract algebra terms, please elaborate. Thank you very much. How can I obtain total number of points? Possible Duplicate:          Proof to the formula  $$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$ What's the meaning "them" here? I've confused about the word "them" in the middle of the sentence:     They know all about them   Westovers in Idaho.   Could you please tell me what the meaning of "them" here is?     Does it mean "they know about the oils which are made by Westovers in Idaho"?   The full text is here:      Dad had always said he wanted to build a room the size of a cruise ship   but I’d never thought he’d have the money. I looked to Mother for an   explanation but it was Dad who answered. The business was a roaring   success, he explained. Essential oils were popular, and Mother had the best   on the market. “Our oils are so good,” he said, “we’ve started eating into the   profits of the large corporate producers. They know all about them   Westovers in Idaho.”   Educated by Tara Westover Can't preview smoke simulation on Blender 2.82 Whenever I add a Quick Smoke effect, I can't preview it on the viewport unless I bake it.   And Yes, I added the Smoke Domain on the object.   I've seen tutorials like this  and this  that clearly shows the possibility of previewing the smoke before baking it.   I'm using Blender on a Mac with AMD GPU. I would guess that this is something related to OpenGL, as thoroughly explained here -  but that's for Windows.   Does anyone have any idea how to fix this? Rethrowing exceptions in Java without losing the stack trace