Ubuntu Network Drivers
Installing Broadcom Wireless Drivers
How does one show that $$\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$$ for each nonnegative integer $n$?  I tried using the  but I guess I am applying it incorrectly. With the snake oil technique we have $$F(x)= \sum_{n=0}^{\infty}\left\{\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}\right\}x^{n}.$$ I think I have to interchage the summation and do something. But I am not quite comfortable in interchanging the summation. Like after interchaging the summation will $$F(x)=\sum_{k=0}^{n}\sum_{n=0}^{\infty}\binom{n+k}{k}\frac{1}{2^k}x^{n}?$$ Even if I continue with this I am unable to get the correct answer.  How does one prove this using the Snake oil technique?  A combinatorial proof is also welcome, as are other kinds of proofs. Does one wing of fly contain harmful microbes and the other contain only antibodies? There was  study but the conclusion and it's source's relability seemed a bit off. How come a house flies one wing contain all microbe and the other don't. Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$ Find all positive integers $a, b, c$ such that $1/a + 1/b + 1/c = 4/3999$.  The contest is just ended, so you may freely answer. (I did not attend the contest: it is an Italian contest for schools and the groups may ask other people for hints. This is how I got involved, but of course asking on MSE would have been cheating)  I found three solutions: I am explaining below what I did. I would like however to know if there is a more general approach. When I run sudo, what exactly happens to my environment?  When I run sudo command, it doesn't seem to see my or root's environment. For example, my path for both includes /usr/local/bin, but if I try to run one of the program's without the full path, it fails.  I thought sudo ran as root, and hence got root`s environment. Is there a different way that bash executes under sudo than under root or my normal user?  EDIT:  I have been using sudo -i lately, but recently it has been causing problems because my current working directory gets set to /root. This is as expected (sorta), but I still don't understand why sudo isn't recognizing my executables in /usr/local/bin.  EDIT:  I am running Fedora 15. Why do we need autoencoders? Recently, I have been studying autoencoders. If I understood correctly, an autoencoder is a neural network where the input layer is identical to the output layer. So, the neural network tries to predict the output using the input as golden standard.  What is the usefulness of this model? What are the benefits of trying to reconstruct some output elements, making them as equal as possible to the input elements? Why should one use all this machinery to get to the same starting point? Number of solutions to the congruence $x^q \equiv 1 \mod p$. Let $p, q$ be distinct odd primes, I would like to compute the number of solution $\mod p$ to the congruence $x^q \equiv 1 \mod p$. (Show that it's $gcd(q, p - 1))$  If $x^q \equiv 1 \mod p$, since $q$ is prime, this must mean that $x$ is either $1$ or has order $q$.   Hence, the problem reduces to finding the number of elements of $U_p$ which is either $1$ or has order $q$.  If $ q \nmid  p - 1$, then there is only one element, namely $1$.  If $q \mid p - 1$, then the number of elements of order $q$ is $\phi(q) = q - 1$, and so the number of solutions to the congruence $\mod p$ is $q - 1 + 1 = q$, and in either case the number of solutions is $\gcd(q, p - 1)$.  Does this look fine? I am looking for a way to convert a long string (from a dump), that represents hex values into a byte array.  I couldn't have phrased it better than the person that posted .  But to keep it original, I'll phrase it my own way: suppose I have a string "00A0BF" that I would like interpreted as the  byte[] {0x00,0xA0,0xBf}   what should I do?  I am a Java novice and ended up using BigInteger and watching out for leading hex zeros. But I think it is ugly and I am sure I am missing something simple. On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $ The well-known fable of an astronaut sending signals out to an external observer while falling toward an event horizon states that the time lapse between such signals becomes greater even if in the astronaut sends them out periodically (as judged in his inertial frame). When viewed from earth and weighing time-dilation due to the gravitational field of the collapsing star, how is it possible that a black hole can form in finite time (for any external observer) if it takes an infinite amount of time to "see" events occurring at the event horizon? How to deal with "java.lang.OutOfMemoryError: Java heap space" error? Does visiting Canada from USA under ESTA reset the visa waiver? Spring Data JPA Update @Query not updating? terminal won't launch after install python 3.5 After I installed Python 3.5 and ran the commands below, my terminal and many other programs won't launch any more.  cd /usr/bin sudo rm python3 sudo ln -s python3.5 python3    How can I restore the default Python version again to fix these problems? Last night I was grilling chicken thighs on a barbecue and I was told that if there was any pink inside them, they are not safe to eat. My question is twofold:  (a) Is this true?  (b) Is there a way to tell that doesn't involve cutting the thighs open to have a look at the colour? How to Create a Half-Turn Staircase? I would like to create a half-turn staircase without a landing. Is there an easy way to accomplish this without having to manually calculate and create each step individually?  Here’s an image similar to what I would like to achive:    I have found the Archimesh add-on including . While I can bend the created stairs, I haven’t found any way to get anywhere close to the image above. My stairs just always look out of shape and I don’t know how to get the outline of the staircase into a square.  I should mention that I only have very little experience with Blender modeling, so it’s well possible that I’m just missing some curve modification basics. Perhaps Archimesh is also the wrong approach for this in the first place? How can you accelerate without moving? I have a document I am trying to copy to learn TeX. Here is what I have encountered: How can I have the "or" in the equation. This is what I have right now:  \[ f = ma; \]  But $a$ is the change in velocity,  \[ f = m \frac{dv}{dt};\] \[ f = m  \frac{d^2y}{dt^2};\]   Edit: Trying out Stefan's answer Docker ppa package for Ubuntu 19.10 Does someone knows if it exists some ppa packages source for the newest Ubuntu version 19.10 i had problem with that when i upgraded from Ubuntu 19.04 to 19.10 i didn't work? I've been using Ubuntu for a few years now, but I just installed Version 16.04, and I've been having a lot of issues. I'm not a command line expert, but I tried to install some software last night, and it is creating issues (and also not working properly).   When I run sudo apt-get -f install, I see the following message:  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following additional packages will be installed:   lua-event lua-expat lua-filesystem lua-sec lua-socket lua5.1 prosody Suggested packages:   lua-zlib lua-dbi-postgresql lua-dbi-mysql lua-dbi-sqlite3 The following packages will be REMOVED:   jitsi-meet-tokens The following NEW packages will be installed:   lua-event lua-expat lua-filesystem lua-sec lua-socket lua5.1 prosody 0 upgraded, 7 newly installed, 1 to remove and 22 not upgraded. 3 not fully installed or removed. Need to get 0 B/413 kB of archives. After this operation, 2,296 kB of additional disk space will be used. Do you want to continue? [Y/n] Y (Reading database ... 176266 files and directories currently installed.) Removing jitsi-meet-tokens (1.0.962-1) ... dpkg: error processing package jitsi-meet-tokens (--remove):  subprocess installed post-removal script returned error exit status 10 Errors were encountered while processing:  jitsi-meet-tokens E: Sub-process /usr/bin/dpkg returned an error code (1)   Can someone help me figure out what I need to do to get this resolved? I'm fine with deleting jitsi, but I'm not sure if that will correct the problem.   Currently there is a red circle with a line through it in the upper right corner that indicates there may be installed packages with unmet dependencies. "Thus" vs. "Thusly" I read an article that used "thusly" and was wondering if there is any grammatical credence to it.   The :     The issue started when Sokolowski quickly ran out of storage capacity in his 32GB Surface after adding media files. After poking around, he realized that while the device has a 32GB drive, only 16GB of that capacity is available for users’ data. Thusly Microsoft is being sued for false advertisement, with the suit seeking class action status.   Is thusly an outmoded form of thus, or does it have a different meaning altogether?