I noticed that $$\frac{d^{n}}{dx^{n}} f(x)g(x)=\sum_{i=0}^n {n \choose i} f^{(i)}(x)g^{(n-i)}(x)$$ and it's had me scratching for a little bit. It's easy to see how the cross terms add up but can anyone draw a direct comparison between the behavior being exhibited here and the binomial expansion? It seems to work for the case equivalent to the multinomial theorem as well ($\frac{d^{m}}{dx^{m}} \prod_{k=1}^{n} f_k(x)$). Can someone produce a proof for this more general case (preferably by induction)? Any and all insights are welcome!
The binomial expansion:  $$(x+y)^{n} = \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k}$$  The General Leibniz rule (used as a generalization of the product rule for derivatives):  $$(fg)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} f^{(k)} g^{(n-k)}$$  Both formulas can be obtained simply by induction; Newton's binomial also has a combinatorial proof . It's striking how these formulas are similar; is there a possible connection between them? I was thinking that maybe the General Leibniz rule could be obtained using a combinatorial argument as well (hence the binomial coefficients)...
How to apply a bevel modifier, so that it affects only specific edges? What are the guidelines for reviewing? About reviewing The review queues contain posts that possibly need further action from the community, like improvement, closure or deletion. By presenting those posts for peer-review, the system aims for higher quality content. Some user actions, such as suggesting an edit, casting a flag, close vote or reopen vote, or posting for the first time, can trigger the inclusion of a post on a review queue. There are also posts included on the queues algorithmically, such as the contents of Late Answers and Low Quality Posts. General guidelines  Always read the full post you are reviewing. Don't rush. Take the necessary time to read the post carefully. If you don't care about a post, just click Skip or Not sure. When you need more context, open the post link to see the question and all answers. Most posts coming up for review can be improved. Use the Edit option, and edit thoroughly. If you're unsure how to review a post (perhaps it's outside your areas of expertise), skip it. Someone who understands it better will review it later.  Guidelines for specific review queues         (Stack Overflow only) SQlite: select into? How to find the id of the last added feature using pyQGIS? Is it possible to programmatically (in python) access the attribute info of the last added feature in QGIS 2.0 before edits are saved? I'm failing to find a way to get the ID and subsequently other attributes of my feature of interest (the one most recently added).  idx = layer.fieldNameIndex('myField') myList = [] lastEdited = ID THAT I'D LIKE TO RETRIEVE for feature in layer.getFeatures():     if feature.id() == lastEdited:         print feature.attributes()[idx]   From this I'd like to get 'myField' of my most recent addition.   UPDATE: Following lead of this , I was able to achieve my goals via the Python console, but am failing to get it to work for my plugin.  def updateTableWidget(self, fid):     idx = layer.fieldNameIndex('myField')     myList = []     for feature in layer.getFeatures():         if feature.id() == fid:             myList.append(feature.attributes()[idx])   Via the python console, I was able to connect to my layer like so:  layer.featureAdded.connect(updateTableWidget)   After adding a QgsMessageLog to my function, the attributes were correctly identified. I now have a beginner python issue, however. Calling the function in my plugin as shown below gives me TypeError: updateTableWidget() takes exactly 2 arguments (1 given):  self.updateTableWidget()   The function was called from feature form python file. Solve Euler Project #9 only mathematically - Pythagorean triplet The "Euler Project" problem 9 () asks to solve:  $a^2$ + $b^2$ = $c^2$ a + b + c = 1000  I find answers solving it with brute-force and programmatically, but is there a way to solve the problem ONLY mathematically? Can someone help, please?  Problem as explained in Project Euler website:  A Pythagorean triplet is a set of three natural numbers, a &lt; b &lt; c, for which,  a^2 + b^2 = c^2 For example, 3^2 + 4^2 = 9 + 16 = 25 = 52.  There exists exactly one Pythagorean triplet for which a + b + c = 1000.  Find the product abc. Show alert popup on standard page layout on click of standard button "Save" How to jusify {e,n} is a valid RSA public key? gdal_translate- why jpg compressed tif is 2 times greater than jpg file? mysqli_fetch_assoc() expects parameter / Call to a member function bind_param() errors. How to get the actual mysql error and fix it? Divisibility Proof with Induction - Stuck on Induction Step I'm working on a problem that's given me the run around for about a weekend. The statement:  For all $m$ greater than or equal to $2$ and for all $n$ greater than or equal to $0$, $m - 1$ divides $m^n - 1$.  Using induction over $n$, the base step comes easy since $m^n - 1$ is $0$ when $n = 0$.  My induction hypothesis is letting $k \geq 0$ and assuming that $m - 1$ divides $m^k - 1$. In order to show that $m - 1$ divides $m^{k+1} - 1$, I obviously need to use the induction hypothesis. However, no matter where I try to use the fact that $m^k - 1 = (m-1)a$ for some $a$ in the integers, the expression $m^{k+1} - 1$ always becomes more difficult to get to $m^{k+1} - 1$ being equal to $(m-1)b$ for some $b$ in the integers.   In other words, I can't figure out any actually helpful way to apply the induction hypothesis with the goal of proving the next step.  Any help would be appreciated! I have a manuscript that is under review in a major journal which is a main result out of a few recent years of my research. When applying for faculty positions, I would like to send the draft as a part of my writing sample. But since the material is not yet published, I am not sure if it is a good idea.  What are the foreseeable disadvantages of sending an article in review to the faculty search committee?  My area is neuroscience / biomedical engineering.  Update: The paper got into a high-profile journal, and I got a job at a different place which did not require a writing sample. Euclidean domain in which the quotient and remainder are always unique Let $R$ be a Euclidean domain in which the quotient and remainder are always unique. Does it follow that the ring $R$ is either a field or a polynomial ring $F[X]$ for some field $F$? ATI Radeon HD 7770 no GPU rendering? How do you use the Dungeon Defender's Portal Gun? The topic pretty much sums up my question. I have not yet figured out how the gun operates aside from the fact that it creates Portal portals. The weapon description indicates that it deals 250+ base damage, but how is that damage applied to the monsters? Symbols instead of numbers as footnote markers How can I make LaTeX use symbols (*, †, ‡, and so on) instead of numbers to mark footnotes?  (The numbers are confusing because I use superscripted numbers for citations.) What is Deviance? (specifically in CART/rpart) What is "Deviance," how is it calculated, and what are its uses in different fields in statistics?  In particular, I'm personally interested in its uses in CART (and its implementation in rpart in R).   I'm asking this since the  seems somewhat lacking and your insights will be most welcomed. I've previously used the default Email app on my CyanogenMod 7 EVO because I had an Exchange account at work. However, I switched jobs recently and my new place just uses GMail corporate accounts (which I enjoy). Since I don't have the Exchange account anymore, I'd like to move over to the GMail app because I prefer the UI and whatnot, but I can't for the life of me find any kind of "Select All" functionality in it.  The Email app allows me to do this from the options menu (Menu-&gt;Select All selects everything in the current folder). In GMail I noticed that if I select one or more items, then "Deselect All" becomes available in the options menu, but not "Select All". I've also tried various forms of long-pressing on messages, accounts, and folders to no avail.  Is there any way to do this in the GMail app, or am I just stuck without it? I searched the issue tracker and didn't find anything relevant, which surprised me a bit, to be honest. Obviously I can do this from a computer if I need to empty a folder/label, but I'd kind of like to retain the functionality on my phone, too. Am I missing some non-obvious (or perhaps even obvious) menu item somewhere? How to search for all questions with bounties across the network? I wanted to search for questions on all SE sites that had a bounty. I am not sure if it is possible, and if it is how can I?   I have seen other questions that show how to find questions on each individual site, but how can I find all of them in one search from the homepage stackexchange.com? If this has been asked already, please let me know.  It has been suggested that the question was already asked (click  for the question), but that question only showed how to find those questions on each separate site or some complex way to search. I am hoping to find a way to search for bountied questions easily or for someone to create a way to do this. Run cmd.exe in Administrator mode without explorer.exe I've recently had trouble with explorer.exe not working.  I found a possible solution , using .  The problem is that I cannot "right-click Command Prompt, and then click Run as administrator", since explorer.exe does not work for even a second without crashing.  I tried to use the  to open cmd as administrator, but I get an error 1327 because I have a blank password.    My question is, is there any other way to run cmd.exe with admin privileges?  If not, is there any way to run sfc.exe without admin privileges?  If not, does anyone have solution to my original problem of explorer.exe stopping and restarting ad nauseam? Site Analytics header does not show the link to "traffic sources" when in that section I was seeing the Site Analytics page (/site-analytics)  (>25K users) and also  and found out that the header is not consistent:   When in history, the header shows "history | traffic sources" When in traffic sources, the header just shows "history"   Graphically: