Proof by Induction $\sum_{i=0}^{k-1} 2^i = 2^k-1$
Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$
Given a file like this:  1,768,12,46576457,7898 1,123,435,134,146 2,345,6756856,12312,1311 5,234,567465,12341,1341 1,3245,4356345,2442,13 9,423,2342,121,463 9,989,342,121,1212   I would like to list all rows (in bash terminal) such that the value in column 1 appears at least twice (in column 1). The result should be  1,768,12,46576457,7898 1,123,435,134,146 1,3245,4356345,2442,13 9,423,2342,121,463 9,989,342,121,1212 Effect Of Frequency on Photocurrent The situation is this.. I am performing a photoelectric effect experiment - just the usual - evacuated glass tube containing two metal electrodes connected to an external source of DC voltage (say an alkaline cell) and ammeter in series. I shine a light on one of the electrodes - the frequency is more than enough to cause eject photoelectrons, and the voltage of the cell opposes the electrons which are trying to get to the other electrode. The question is this - what happens to the current on increasing the intensity of light while keeping no. of photons falling on the electrode constant (in other words increasing the frequency)?   Does the current remain the same because the no. of photons falling on the electrode is constant and so no. of electrons being ejected is constant? Or does it increase because the electrons have more kinetic energy, ie, they are moving faster towards the other electrode so that the $t$ (time) term in $I$ = $q$/$t$ is less ($q$ remains constant because no. of electrons being ejected is still the same)?   This question appeared in one of my tests and the answer key says that the current remains constant but think that case 2 is correct and current would increase, which is why I asked this question.  Thanks. I seem to have screwed up something that allows you to inset evenly, I think I have tried every combination of settings in inset and pivot center and even is far from even.  I tried uninstalling and re installing blender 2.71 and get the same result.  The image attached is a simple default cube that I tried to inset evenly.  I have also attached the blend file link  help?? I couldn't find anything about Ubuntu acting as a  receiver or sender.   Can it work at all? Are there hardware prerequisites? Is WiFi a requirement or can it work over LAN or another kind of network connection? WiFi direct seems to be a necessary requirement, is it a sufficient one? (i.e. if a system supports WiFi direct does that mean it supports Miracast?) Are there differences in support between receiving/sending? How is the latency? (compared to the competition, i.e. VNC, commercial Miracast devices, etc.) How do I actually use it, if it's difficult?   Specifically, I plan to use it together with an Android phone (4.x Jelly Bean). I want to make some changes to Windows that require using the Group Policy Editor (gpedit.msc).  Unfortunately the Group Policy Editor is not included with the Starter Edition, Home and Home Premium editions of Windows.  How can I install it? How do you read CSS rule values with JavaScript? Proof that a sum of Bernoulli rvs has Binomial distribution Command to move a file to Trash via Terminal I would like to know if there is a command I can issue in a terminal so I do not classically remove (rm) the file, but instead move it to trash (i.e. Nautilus Move to Trash behavior).  In case there is such a command, I would also be interested in knowing what it is. @class vs. #import It is to my understanding that one should use a forward-class declaration in the event ClassA needs to include a ClassB header, and ClassB needs to include a ClassA header to avoid any circular inclusions. I also understand that an #import is a simple ifndef so that an include only happens once.  My inquiry is this: When does one use #import and when does one use @class? Sometimes if I use a @class declaration, I see a common compiler warning such as the following:     warning: receiver 'FooController' is a forward class and corresponding @interface may not exist.   Would really love to understand this, versus just removing the @class forward-declaration and throwing an #import in to silence the warnings the compiler is giving me. Does opacity:0 have exactly the same effect as visibility:hidden Why does friction play the role of centripetal force during the turning of a car? What makes frictional force the centripetal force of a car turning along a curve? As friction is the opposing force and acts anti-parallel so there is no component of frictional force towards the center,right? Then how can frictional force be centripetal force? We currently have two rewards checking accounts.  We are nearing the maximums permitted on each account, after which we receive .5% and .25% APY.  It's within our means to perhaps open another one, but at that point we are looking at using our debit cards 50 to 55 times a month.  I feel that running that number of debit transactions is getting out of hands as it's just the two of us.  I realize there are easy ways to game the system, but I want these things to stay around and another person doing just such that could lead to the ends of these accounts.  We are saving towards a new house down payment.  As such we don't require these funds to be fully liquid, but would still like to remain free of as much risk as possible -- however I realize we are unlikely to find anything near our rewards checking accounts (4% APY interest for meeting various rules: direct deposit, a certain number of debit card transactions, and not receiving a paper statement), but would hope we can get at least half of that.  We aren't fully sure when we will be buying, but we are probably a couple of years out.  For that reason I don't think CDs would work.  Even if we ladder them, when we need the money we will need it pretty much instantly.  Do I bite the bullet and just find the best available money market account?  What other options do I have that let me deposit monthly, and not have a timebomb associated to accessing that money that isn't in the initial lump sum? I am looking for a single word for a person who is able to explain himself very clearly and using the best possible words, etc. I am not looking for "perspicuous", etc. The focus is more on the language part: someone who uses the right language and words to describe himself. I know the word exists but its just skipping my head at the moment!!! Please help. How to understand the Ubuntu file system layout? Why didn't all the Jedi at the Jedi Temple sense Anakin and his legion of Clones’ thoughts? If you go back to Attack of the Clones, Anakin and Obi-Wan sensed that poisonous alien worms were about to kill Padme. Back in Revenge of the Sith, the Jedi could take on the legion or if there were too many, they could at least escape. MAC OS for Intel based PCs I have an Intel dual core PC with 4GB of RAM and Graphics card. For on of my student Assignment I need to install Latest possible MAC OS on my system as a secondary OS.  Like on boot it asks me that which OS i want to boot with Win 8.1 or Mac Os..  I searched on the internet and found two conflicting opinion some people said there are few MAC OS Version which can be directly installed on Intel PCs some says there aren't?   I am here to get rid of the confusion that is an official latest possible MAC OS version for Intel based PCs? If not what are my options if I want to run MAC OS on my PC.   Please not that Virtualization options won’t work for me, it has to be working as full OS not an os inside another. How can I solve the folowing Diophantine equation with two unknowns? Find one integer solution to the following Diophantine equation:   \begin{equation*} \\\forall \,x,y \in \mathbb{Z}\\ 6xy + x - y = 274 \end{equation*}   The resultat ist $x = 9$, $y = 5$ (Obtained with  )  I need to know a method that allows me to reach such a result. How can I calculate the value of these two variables in a Diophantine equation? For example: 6xy + x - y = 458 There is by now a captcha-problem Lost email settings and no icon for Mail I've lost my Settings → Mail, Contacts, and Calendars on my iPhone 5.  No icon for Mail and no Mail settings.  How do I get it back? Can I have my iPhone play sound for MacBook? Say I am playing a video on my MacBook Pro, and I don't want to distract others with the sound. The problem is, I don't have a 3.5 mm Headphone or AirPods at my hand, but I have an iPhone 8 Plus and its EarPods with Lightning Connector. Can I play the sound from the video playing on macOS with the iPhone, preferably without connecting the iPhone to the Mac so that I can plug in the EarPods to the iPhone's socket?  This is basically the reverse of