Proof that $$\sum_{k=1}^n (-1)^kk^m{n \choose k} = 0$$ for every $n&gt;m$ and $m&gt;1$.  I tried to use properties of ${n \choose k}.\;$ I tried to use $${n \choose k} = {n-1 \choose k} + {n-1 \choose k-1}$$ and using induction got stuck here $$\sum_{k=1}^t (-1)^kk^m{t-1 \choose k-1} = 0$$, where $t = n - 1$.
Question: For every polynomial $P(x)$ of degree at n we have $$\sum_{i=0}^{n+1}(-1)^i\binom{n+1}{i}P(i)=0$$  Well I was inducting on degree of the polynomial, for $n=0$ its true and I got a bit confused in the inductive step, actually I cant figure the inductive step.   Is this property (if true) a famous one?  Thank you for your help.
Trying to find a Sci-Fi movie, probably from the 50's or 60's I’m trying to identify a movie I saw when I was very young.  It was a “Saturday Matinee” playing on one of the local stations, and I likely watched it in the late 60’s.  So I expect that the movie is from the 50’s or 60’s.  I watched it in black and white, but it could have been a color movie on a black and white set. I clearly recall a scene where scientists enter a lab and some see something move past them in a blur.  Later in the movie, they come upon themselves in the same lab where they appear to be frozen and not moving.  Then they notice that their “other selves” are moving, just almost inperceptibly slowly.  They are encountering themselves but at a completely different time scale. Unfortunately, I don’t recall much else about the movie, just the idea that they encounter themselves once with the others flashing past, and again seemingly not moving at all.  If this rings any bells with anyone, I’d love some pointers to try to track this movie down. In Visual Studio, I can cycle through my clipboard history by pressing CTRL+SHIFT+V rather than CTRL+V. I believe it's called a clipboard ring. But that feature doesn't seem to exist outside of Visual Studio, at least not with the same key combination. Is there a key stroke combination built in to windows to do the same, i.e. be able to cycle through your clipboard history while pasting? Cookie path and its accessibility to subfolder pages I'm in trouble with this homework.  Find the sum of the series $ 1 + \frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac19 + \frac1{12} +\cdots$, where the terms are the inverse of the positive integers whose only prime factors are 2 and 3.  Hint: write the series as a product of two geometric series.  Ok, I found some patterns in these numbers, but I can't find the two series. I recently discovered that my computer is 64-bit. I have backed up my Home folder, and reinstalled Ubuntu. The reinstall wasn't nearly as painful as I thought. There is one thing that I can't quite seem to figure out: how do I get the 64-bit Firefox Beta build? I always get the Beta builds, but I want to take advantage of the 64-bit architecture of my computer.  says that Mozilla has come out with a 64-bit version of Firefox, but I can't seem to find it.  I do understand  of using a 64-bit browser, but I've decided to jump right in and do it anyway. (Flash and Java are already 64-bit, and who cares about Silverlight, since it's not for Linux anyway?)  There's only one issue, and it's a big one: I can't find the 64-bit Beta PPA!!! (I really hate using .tar.gz files, but I'd be willing to do that as long as I could still access Firefox via the Launcher. Oh, speaking of which, I don't understand .tar.gz files. Once, I managed to run one (the Dropbox Beta build,) but I have no idea whatsoever on how to install them: as in, click on the icon and go.) Closure of a Ball is Contained in Closed Ball (usual metric) Let $\overline{B_d (x, r)}$ be the closure of an open ball, and let $\hat B_d (x, r)$ be a closed ball. Show that in any metric space $(X, d)$, $\overline{B_d (x, r)} \subseteq \hat B_d (x, r)$ for all $x \in X$ and $r &gt; 0$.    I used the definition of closure: $\overline{B}$ = $B \bigcup B'$. If $x \in \overline{B}$, then $x \in B$ or $x \in \overline{B}$, or both. If $x \in B$, then clearly $x \in \hat B$ since $B \subseteq \hat B$. But, if $x \in B'$, then $x$ is an accumulation point. Since $\hat B$ is closed, it contains all accumulation points of $B$. Thus, $x \in \hat B$.  Is this approach correct, or at least in the right direction? Should I be using the closure definition, or try another one? Thank you. This is a homework question of mine that Ive answered and I'd really like some feedback on my solution.       Prove or find a counter example to the claim that for all sets $A,B,C$, $$\text{if  }A\cap B=B\cap C=A\cap C=\emptyset \text{ then } A\cap B\cap C\neq \emptyset$$.   Since I can only either prove it is true, OR find a counter example I have gone with the latter because I think this statement is not true. My counter example is as follows:  Let $A = \{1,2,3 \}$ , $B = \{4,5,6 \}$ and $C = \{7,8,9 \}$. It is then clear that $A ∩ B =  B ∩ C =  A ∩ C = \emptyset$ .  If we let $x ∈ A\cap B\cap C$ then $x$ must be in $A$ and $B$ and $C$. Since this cannot be possible because there are no common elements then this implies that $A∩B∩C$ has to be equal to a null set i.e $A∩B∩C = \emptyset$ and thereby showing that (at least for this scenario) if $A \cap  B =  B \cap  C =  A \cap  C = \emptyset$ then $A\cap B\cap C = \emptyset$.  Is this okay? Any feedback will be greatly appreciated thank you! I want to remove all the files from different directories, and want to keep only latest files 20 files. is this the correct command to do this?  ls -t1 /mnt/dwh/ftp/dwh_ftp_cbs/ARLOGS/ | tail -n +22 | xargs rm -f How does the Eco-Cooler air conditioner really work? describes a device for developing countries which apparently cools the air by 5 °C without electricity. According to , it works by lowering the pressure to drag in the ambient air. Wouldn't the ambient air be at the same temperature?  Does the constriction compress the air and temporarily increase the pressure and so temperature, which then dissipates heat because it is hotter than surroundings; then after passing the constriction, the pressure returns to almost ambient but the temperature drops to below ambient?  How does the Eco-Cooler air conditioner actually work, if at all? What is the difference between ubuntu-18.04-server and ubuntu-18.04-live-server? On the  there is a download link for ubuntu-18.04-live-server-amd64.iso, and on the  there is a download link for ubuntu-18.04-server-amd64.iso. Before 18.04, there is only a server version. What is the difference between these 2 images? Delegate OpenID to Google (NOT Google Apps) Cartoon in a world with magic stones wielded by four children which combine to become stronger, and can summon creatures Can you please help me find this animated show? There is a four member team with two boys and two girls. The boys can summon, with their stones, a rock golem (black wearing summoner) and a robot (orange wearing summoner). The two girls can summon something like a dragonfly (green wearing summoner, little girl) and the blue girl, I don't remember what she can summon.  I don't know where the animated show comes from.  By throwing the stone, they activate the stone's power. Each stone has different characters on it. When you don't match them correctly, you get a rebound effect. Pressure variation in a sound wave Often in my papers I want to mention theorems in the introduction that are introduced in more detail and proved in a later section.  I'd like to include the statement of these theorems twice over, each time numbered as the theorem would naturally appear in its later section.  My current solution (using amsthm) is, e.g.  %(in the introduction) \newtheorem*{thm:associativity}{Theorem \ref{thm:associativity}} \begin{thm:associativity} Lorem ipsum ... \end{thm:associativity}  %(in a later section) \begin{thm} \label{thm:associativity} Lorem ipsum ... \end{thm}      Can anyone think of a cleaner solution? How can i get currency symbols from currency code in iphone? A deck of 52 cards is shuffled, and we deal a bridge of 13 cards. Let $X$ be the number of aces and $Y$ be the number of spades. Show that $X$, $Y$ are uncorrelated.  Here is what I did:  $Cov(X,Y) = E[XY]-E[X]E[Y]$   uncorrelated means $Cov(X,Y) = 0$, hence  $E[XY]=E[X]E[Y]$   $E[X] =  \sum_{k=0}^{k=4} k \frac{\dbinom{4}{k} \dbinom{48}{13-k}}{\dbinom{52}{13}} $   $E[Y] =  \sum_{k=0}^{k=13} k \frac{\dbinom{13}{k} \dbinom{39}{13-k}}{\dbinom{52}{13}} $   Are the summations above correct? and how do I calculate $E[XY]$? Error while running Quickly Can I install any iOS version I wish by picking and choosing which firmware to restore? I have 100 files with 57,816 rows each.I want to do an outer join of these files by merging on basis of the common column.  I am using R programming for this but this is very slow.  fileList &lt;- list.files(, pattern=".txt") pm_list=lapply(fileList, read.table) merged_pm=merge_all(pm_list, by = "gene_short_name")   Is there any quick way to do this in bash? The other method that I can use is SQL but I have to create 100 tables first and then load them up before join,which is not very efficient way to do.  The number of rows are equal in each file, thats why I want to merge  based on common column and can't use cbind from R as the constants in the common column are a bit up and down in different files and not present at same location in each file.Below are two sample files.I want to join based on 'gene_short_name'  gene_short_name FPKM56  MT-TF   0.90 MT-TV   0 MT-RNR1 310.015 MT-TL1  0 MT-TM   0   File 2 is below:  gene_short_name FPKM53  MT-TF   0 MT-TV   0.344 MT-TM   0.10 MT-TL1  0 MT-RNR1 0 MT-ND2  158.332 Refereeing papers on behalf of a professor: should it be listed on a CV? My professor sometimes asks me to referee papers for him.  Should I list the corresponding journals in the professionals activity section of my CV?