I found hard (for me at least) example, in fact it just would be nice to prove.  $$\binom{2n}{n} = \sum_{k=0}^{n} \binom{n}{k}\binom{n}{n-k}$$
How can we prove that     $$\sum_{k=0}^{n}\binom{R}{k}\binom{M}{n-k}=\binom{R+M}{n}?$$   (Presumptive) Source: Theoretical Exercise 8, Ch 1, A First Course in Probability, 8th ed by Sheldon Ross.
Salesforce lightning get current users username How I would go about getting the current users username or first and last name?  I have been looking at this for the last couple of hours - I was just wondering how I would go about getting their name and then displaying it on the page.  For example the current user is 'John Smith' and in the lightning component I want to output John Smith's name like:     Welcome to our community John Smith! Renting a car in Ireland when under 25? My friend and I are planning a vacation to Ireland later this year. We'll be there for a week, flying into Dublin and then renting a car and driving around the country.  Both of us are 23. Are there any age restrictions on renting cars in Ireland? What would be the cheapest/best option for us? I just read a description of CVE-2020-1971, which indicates that a patch was released today for openSSL, the patched version is 1.1.1i -- my version of Ubuntu 20.04.1 LTS is running openSSL version 1.1.1f.  From last March. Software Updater says that my system is up to date.  I am hoping that this openSSL update is released by Canonical by tomorrow... can anyone confirm they are working on this? Consider a partial function $f$ that is defined only for a few values of its domain (my exact use case is $\delta$ transition functions in automata). One can 'complete' it by saying $$g(x)=0\iff f(x) \text{ is not defined.}$$  Is there a symbol to mean "undefined"? Would it be correct, or accurate, to write $\nexists f(x)$? Why does DNS work the way it does? This is a  about DNS (Domain Name Service).   If my understanding of the DNS system is correct, the .com registry holds a table that maps domains (www.example.com) to DNS servers.   What is the advantage? Why not map directly to an IP address? If the only record that needs to change when I am configuring a DNS server to point to a different IP address, is located at the DNS server, why isn't the process instant? If the only reason for the delay are DNS caches, is it possible to bypass them, so I can see what is happening in real time? How do I disable a bottom panel with the list of open windows on GNOME 3? I have a Schengen visa valid from 19/07/2016 to 10/08/2016. But the duration of stay is 7 days only.  I want to stay for 6 days more, for a total of 14 days. And my flight includes travel from a Schengen state to another. Where did folder "merge" or "keep both" go in OS X 10.9 Mavericks? Before Mavericks one could hold the ALT key and move a folder "A" onto a folder "B" where already an "A" existed. In this case Mac OS X asked in Lion if I want to merge - or on Mountain Lion if I want to keep both (which was also a merge).  I can't find this option in Mavericks.  Did Apple remove this useful feature?  Any hints?  (And yes: I own ForkLift, which can do it - I just sometimes still prefer the Finder). I have seen many ways to make a USB an installer for about any Windows OS. But how can I (using grub or something like that) make it bootable to all editions on Windows XP and 7?   I have tried Googling it and researching, I've even tried to do it, But I dont understand much about GRUB or linux at all.  Please keep in mind that I am not very good with linux, so please use as many details as possible. Helical motion of charged particle in external magnetic field Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$ I've been trying to install 12.04 for a while now, I'm running on Windows 7 and I already made the partition (it's dynamic). I've tried booting from a disc and also the Windows installer. I haven't tried used any repair-boot programs or anything. Every time I tried to load from either, I get an error:  command=C:\windows\sysnative\bcdedit.exe /set {03f9077f-661f-11e1-845a-d08b36406c3d} device partition=U;  retval=1  stderr=An error has occured setting the element data. The request isn't supported  stdout=   Then after that, it makes a log file with information that I don't understand.  If it is simply due to the fact of it being a dynamic partition, someone please help. If it is more extensive, anyone with details on possible solutions or definite ones please express them. I'd like to use a background image for my world, but I'm not sure how to do it.   Right now I am using the sun as a lightpoint and it looks like sky and a flat plane (not realistic). I was thinking of just using a mountain background image for the world.  I've done that through textures etc. but I don't see ANYTHING in render even though it shows in worlds?  What am I doing wrong?    Also what's a cheap quick fix for cases like mine so it looks like the world isn't JUST blue sky?  Thanks guys! Change the URL in the browser without loading the new page using JavaScript How would I have a  action that may have some effects on the current page but would also change the URL in the browser so if the user hits reload or bookmark, then the new URL is used?    It would also be nice if the back button would reload the original URL.  I am trying to record JavaScript state in the URL. In my textbook it is given that      'The wave function describes the position and state of the electron and its square gives the probability density of electrons.'    Can someone give me a very simple example of a wave function with explanation?  (Note: This question is not a duplicate. I have searched for other questions of this type but the answers were overwhelmingly difficult to understand.) I am studying English in Korea.   All my friends and I are studying about 'sentence patterns'  Sentence patterns, said in here, are S+V, S+V+C, S+V+O, S+V+I.O+D.O, S+V+O+O.C.  But my teacher said these 5-sentence patterns are not perfect in grammar. But  because we don't use English by mother language, they will be helpful for understanding English.  Then, I came up with something.   Is sentence patterns used everywhere? (like U.S.A, U.K, Australia, anyother country where people use english by their mother language?)  I really wonder. I would like to look at my referral traffic over time on a per domain-basis in Google Analytics. This is what I have done:   Set New visits as advanced segment  Navigated to traffic-sources -> Referrals Checked a coulple of domains Clicked plot rows   The individual domains now show up in the graph, but the problem is that "all traffic" is also visible. All traffic is much bigger than the graphs for individual domains. This makes it very hard to see any changes as they look all values are Zero.  Is there any way to hide "Visists" from the graph in order to zoom in on the vertical axis? Here is my problem: I've got a dialog with some parameters that the user can change (via a spinner for example). Each time one of these parameters is changed, I launch a thread to update a 3D view according to the new parameter value.  If the user changes another value (or the same value again by clicking many times on the spinner arrow) while the first thread is working, I would like to abort the first thread (and the update of the 3D view) and launch a new one with the latest parameter value.  How can I do something like that?  PS: There is no loop in the run() method of my thread, so checking for a flag is not an option: the thread updating the 3D view basically only calls a single method that is very long to execute. I can't add any flag in this method asking to abort either as I do not have access to its code. How does the Rectified Linear Unit (ReLU) activation function produce non-linear interaction of its inputs? When used as an activation function in deep neural networks The ReLU function outperforms other non-linear functions like tanh or sigmoid . In my understanding the whole purpose of an activation function is to let the weighted inputs to a neuron interact non-linearly. For example, when using $sin(z)$ as the activation, the output of a two input neuron would be:  $$ sin(w_0+w_1*x_1+w_2*x_2) $$   which would approximate the function  $$  (w_0+w_1*x_1+w_2*x_2) - {(w_0+w_1*x_1+w_2*x_2)^3 \over 6} + {(w_0+w_1*x_1+w_2*x_2)^5 \over 120} $$  and contain all kinds of combinations of different powers of the features  $x_1$ and $x_2$.   Although the ReLU is also technically a non-linear function, I don't see how it can produce non-linear terms like the $sin(), tanh()$ and other activations do.  Edit: Although my question is similar to , I'd like to know how even a cascade of ReLUs are able to approximate such non-linear terms. Floor function in complex plane My question is very simple. How would one take the floor of a complex number and what is the floor of i?