Efficent way to copy and paste a line
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Whenever light rays are entering a denser medium they bend towards the normal.  Why do rays choose the path towards the normal? Why cannot they choose the path away form the normal? Redirect to SSL only if browser supports SNI Explicit formulas for Stirling numbers of second kind I have seen some recursive formulas for Stirling I and II kind but I am especially interested in explicit formulas.  I was thinking and get: $$ \left\{ {m \atop n} \right\} =  \sum_{k_i&gt;0 \wedge k_1+k_2+...+k_n = m} \frac{m!}{n!}\cdot \frac{1}{k_1! \cdot k_2! \cdot ... \cdot k_n!}$$ I am not sure if it is ok but supposedly it is (I created it with combinatorics interpretation)  Can somebody look at that and tell me if it is ok/wrong?  By the way,if you have some nice formulas, I will be grateful for sharing that. Windows Unable to Delete ._. File I have to do proof of $(a+b)^n$ and $(a+b)^{n+1}$ with mathematical induction.  I finished the first one $(a+b)^n = a^n + na^{n-1}b+\dots+b^n$.  I however have trouble with the second one, I don't know what to start exactly. Any help/hints? Can a spellcaster use spell slots above their normal limit if their ability scores allow bonus spells of that level? This question is inspired by discussion , about how Heighten Spell works with Reserve Feats.  The question that I am asking here deals with whether or not players can cast spells using spell slots for spell levels that they don't have access to yet.    I feel like an example would explain this concept best.  Take Xylitol, the first-level wizard with an intelligence of 18.  He has 5 0th-level spells per day and 3 1st level spells per day from his class.  He also has 1 bonus 1st level spell per day from his high intelligence.  The  table says that he also has 1 bonus spell of each 2nd, 3rd, and 4th levels.  The question is this:  Can Xylitol cast 1st level spells using the bonus slots for his 2nd, 3rd, and 4th level spells?  He clearly can't cast spells that are inherently higher level than 1st, since he can't learn them, but can he cast 1st level spells using bonus 2nd, 3rd, and 4th level slots?  This has come up on these boards before, .  What I'm looking for here is a clear rules citation that clarifies this one way or the other.  The wording in the section on ability scores makes it quite clear that characters cannot cast spells that are actually higher level than their class allows them to cast, but it is not clear that these higher level slots are barred until the appropriate spell level is unlocked.  If a character has access to a spell level that they can't use because their ability score isn't high enough, they can freely use those slots for lower level spells, so why can't they use their bonus spell slots for lower level spells?  As KRyan's answer says, the Basics section covers this.  Based on questions and comments both here and elsewhere, that section seems to be worded ambiguously enough that it's caused confusion.  I'd like to find another, more clear wording to this rule if at all possible.  Answers from either D&amp;D 3.5 or Pathfinder are acceptable, but please note which one your answer applies to.  N.B. I'd like to note that I personally don't feel that players should have access to higher level slots like this.  There are a large number of locations where it's heavily implied that this is not the case, but I feel that finding a clear rules citation here would be helpful.  I can't find one that is clear for either interpretation here. When a player joins the server, this command block (containing gamemode 2 @a) puts them into gamemode 2/adventure mode. How can I change this so that only non-admin/not "opped" players' gamemode gets changed? On a plane, there are two congruent circles, $O_1, O_2$. One line intersects $O _1$ and A and B and $O_2$ at C and D such that AB=BC=CD=14. Another line intersects $O _1$ and E and F and $O_2$ at G and H such that EF=FG=GH=6. What is the radius of the circle.   The configuration was hard to draw. . You can reflect the lines over the centers, but it is still the same. Cleary, $\widehat{AB}=\widehat{CD}$ and $\widehat{EF}=\widehat{GH}$. I have no idea how to proceed. Is there anything canon in the Terminator Universe that picks up where Terminator: The Sarah Connors Chronicles leaves off? So, Terminator canon is pretty twisted, given all that time-travel stuff.  But in the canon that includes the T:SCC, does anything pick up where the show left off that is considered canon? Showing a function on $[0,1]$ is the zero function How soon until I can take my newborn to take a walk outside? I understand that newborn's immune system is still weak and crowds/strangers are to be avoided. Knowing this, how soon can I take my newborn outside, for example to take ~1 hours walk in a non-crowded park? Any particular things to pay attention to when doing this? Why does an argument similiar to 0.999...=1 show 999...=-1? I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the below argument that shows $999\ldots = -1$ is about? Here is the most usual proof I see that $0.999\ldots = 1$:  $x=0.999\ldots$ $10x=9.999\ldots$ $10x - x = 9$ $x=1$  Using this same argument template I can show $999\ldots=-1$:  $x= \ldots9999.0 $ $0.1x= \ldots9999.9$ $0.1x - x = 0.9$ $x=-1$  What might this mean? Edit from one of the comments: $$\sum_{k=0}^{\infty}{9 \cdot 10^k}=-1$$ Unitary map between sets of vectors I was doing a duplicate content search on the web for our site content and found that someone has downloaded/mirrored our entire website and is hosting it on a subdomain of their own main domain. When I check the WHOIS database, no details of the user are available.  Why do people do this when it is clearly a copyright infringement - is it an SEO thing?  What is the best way of dealing with these individuals? How to view the system console message after Blender crashes? I am testing a script in Blender game engine. I can view the helpful message on the external system console when the script throws exception. Sometimes the bugs of the script cause the Blender to crash and close the console immediately.    Is there any way to view the message after the Blender crashes? How to write the message to a text file for later viewing? Any suggestion? That car is way better than any car any of us have/has ever driven.      None of us know/knows which direction Ron's house is (in).    I have two questions here.   In both the cases, are we to use the singular verb, or the plural? In the end of the second sentence, is using the preposition where it's been used, necessary? Would it be okay if it had no prepositions in it? Give a sequence such that root test works while ratio test fails Remove Punctuation When Checking for Duplicate Titled Questions I recently saw a question titled "" (10k+ users only). I thought, "surely someone has asked this question before", and began looking for a duplicate.  is a question with the exact same title, just with a little different punctuation and casing. It was "Jquery vs Javascript" and "jQuery vs. Javascript?". The only difference being the ., ?, and the casing.  Since we now , I propose that when checking for questions with duplicate names, certain punctuation be striped out to prevent the duplicate from occurring in the first place.  As pointed out in the comments, if isn't already, white-space and non-printable characters may want to be removed for duplicate detection as well. So the question is really hard I think. I tried using a simple way by calculating the probability of each combination that makes a sum divisible by six, but it would take forever. Does anyone have any ideas?  Suppose that we roll a six-sided die ten times. What is the probability that the total of all ten rolls is divisible by six? I'm currently a Combinatorics student trying to parse through this solution. I do not understand the proof currently. Any help understanding it is greatly appreciated.    Question Let the number of partitions be given by the function $p(n)$, i.e. the number of ways of writing the integer $n$ as a sum of positive integers. Prove that that if $g$ is any polynomial function in $n$, then there exists an integer $N$ such that $g(n)&lt;p(n)$ for all $n&gt;N$. Do not use the asymptotic Hardy-Ramanujan formula.    Provided Proof.    Enough to show $n^a&lt;p(n)$ for fixed positive integer $a$. (This is just saying that we want to show $p(n)$ grows faster than any degree polynomial of $n$, right?) We have that $(i_1,i_2,\ldots,i_{a+1},1^{n-i_1-\cdots -i_{a+1}})$ is a partition of $n$ as long as $i_{a+1}\leq i_a\leq \cdots i_1 \leq \lfloor\frac{n}{a+1}\rfloor$. (I really don't understand this step. What is this partition? Of the coefficients or terms of polynomial $p(n)$? And how do we know this is $\leq \lfloor\frac{n}{a+1}\rfloor$?) The number of partitions $n$ of this form is multiset binomial coefficient $$\left(\dbinom{\lfloor \frac{n}{a+1}\rfloor}{a+1}\right)=\frac{\lfloor\frac{n}{a+1}\rfloor(\lfloor\frac{n}{a+1}\rfloor+1)\cdots(\lfloor\frac{n}{a+1}\rfloor+a-1)}{(a+1)!}$$ (I'm assuming that if I can understand (2.) then I can understand this line) This means $p(n)&gt;\left(\binom{\lfloor \frac{n}{a+1}\rfloor}{a+1}\right)$ for all $n&gt;0$. Since $a$ is constant, the given fraction grows with order $n^{a+1}$, which means $p(n)&gt;n^a$ for $n$ sufficiently large. (Why does it grow with order $n^{a+1}$?)   Thanks all again in advance!