How do I divide my root partition into two partitions?
How to reinstall Ubuntu keeping my data intact?
Break a previous commit into multiple commits Strong Induction Proof: Fibonacci number even if and only if 3 divides index The Fibonacci sequence is defined recursively by $F_1 = 1, F_2 = 1, \; \&amp; \; F_n = F_{n−1} + F_{n−2} \; \text{ for } n ≥ 3.$ Prove that $2 \mid F_n \iff 3 \mid n.$   Proof by Strong Induction : $\bbox[5px,border:1px solid green]{\color{green}{n = 1 }}$ $2 \mid F_1$ is false. Also, $3 \mid 1$ is false. The implication [False $\iff$ False] is vacuously true.  $\bbox[5px,border:1px solid green]{\color{green}{\text{Induction Hypothesis}}}$ Assume that $2 \mid F_i \iff 3 \mid n$ for every integer $i$ with $1 ≤ i ≤ k$.  $\bbox[5px,border:1px solid green]{\color{green}{k + 1 \text{th Case}}} \;$  To prove:  $\quad 2 \mid F_{k + 1} \iff 3 \mid k + 1.$   $\bbox[5px,border:1px solid green]{\color{green}{n = k + 1 = 2}} \;$ $2 \mid F_2$ is false. Also, $3 \mid 2$ is false. So [False $\iff$ False] is vacuously true.  Hence assume that $k + 1 ≥ 3.$ We now consider three cases:  $\bbox[5px,border:1px solid green]{\color{green}{\text{Case 1: } k + 1 = 3q}}$ Thus $3 \require{cancel}\cancel{\mid} k$ and $3 \require{cancel}\cancel{\mid} (k − 1)$. By the ind hyp, $3 \require{cancel}\cancel{\mid} k    \iff F_k$ odd &amp; $3 \require{cancel}\cancel{\mid} (k − 1) \iff F_{k - 1}$ odd. Since $F_{k+1} = F_k + F_{k−1}$, thus $F_{k+1}$ = odd + odd = even.  $\bbox[5px,border:1px solid green]{\color{green}{\text{Case 2: } k + 1 = 3q + 1}}$  Thus $3 | k$ and $3 \require{cancel}\cancel{\mid} (k − 1).$ By the ind hyp, $3 | k \iff F_k$ even &amp; $3 \require{cancel}\cancel{\mid} (k − 1) \iff F_{k - 1}$ odd. Thus $F_{k+1}$ odd.  $\bbox[5px,border:1px solid green]{\color{green}{{\text{Case 3: }} k + 1 = 3q + 2}}$ Thus $3 \require{cancel}\cancel{\mid} k$ and $3 | (k −1).$ By the ind hyp, $3 \require{cancel}\cancel{\mid} k \iff F_k$ odd and $3 \mid (k − 1) \iff F_{k - 1}$ even. Thus $F_{k+1}$ odd. $\blacksquare$  $\Large{1.}$ Does the proof clinch the $(\Leftarrow)$ of the $(k + 1)$th case?   $\Large{2.}$ Since the recursion contains $n, n - 1, n - 2$, thus the recursion "time lag" is $3$ here. So shouldn't $3$ base cases be checked?   $\Large{3.}$ Further to #2, shouldn't "assume $k + 1 \geq \cancel{3} 4$" instead?   $\Large{4.}$ Shouldn't the $n = k + 1 = 2$ case precede the induction hypothesis?  I referenced . Source: Exercise 6.35, P152 of Mathematical Proofs, 2nd ed. by Chartrand et al    Supplement to peterwhy's Answer:  $\Large{1.1.}$ I wrongly believed that all 3 Cases proved the $\Leftarrow$. I now see that Case 1 is $\Leftarrow$ via a Direct Proof. Cases 2 and 3 are $\Rightarrow$ via a Proof by Contraposition. Nonetheless, how would one foreknow/prevision to start from $3 \mid n$  for both directions of the proof? Problem with IPv6 sudo apt-get update/upgrade I am using Ubuntu 16.04 Beta 2 at the moment and every time I try to update at home via sudo apt-get update or upgrade trough sudo apt-get upgrade I wont get any feedback. Even installing of anything will have the same result. It will just "hang up" on 0% while fetching data.  $ sudo apt-get update [sudo] password for niclas:  0% [Connecting to archive.ubuntu.com (2001:67c:1560:8001::11)] [Connecting to archive.canonical.com (2001:67c:1360:8c01::1b)]   I have no idea how to solve the problem. I already tried to ping6 archive.ubuntu.com with 100% packet loss while just ping wont have any loss.  I changed the DNS-Server (IPv4 &amp; IPv6) to Google's one no solution either. how to prove $\mathbb Z_n$ has divisors of zero if and only if $n$ is not prime A non-zero number $a \in \mathbb Z_n$ is called a divisor of zero if there is a non-zero number $b \in \mathbb Z_n$ such that $ab\equiv 0\pmod n.$  How can I prove $\mathbb Z_n$ has divisors of zero if and only if $n$ is not prime.  I filled out the addition and multiplication tables for modulo 6 and 7 and tried to find out the relation, and it's definitely true. I know I need to prove it in both directions since it is an 'iff' question. But I still don't get it. Powershell Parameters How long can a flag remain active? I have flagged the following question as a duplicate.    Now I want to know: after how long time will it be stated as active?  I do not have problem with anything here. I just want to know after how long time the request (of a duplicate question) will automatically be rejected or approved.  OK, I got the things, so it will remain active until moderators will have a look at it...  But don't you think we should have a limit that after a certain number of days this kind of request will be dropped (or cancelled)? How exactly are the beta and gamma distributions related? According to Wikipedia, the  is related to the gamma distribution by the following relation:  $$\lim_{n\to\infty}n B(k, n) = \Gamma(k, 1)$$  Can you point me to a derivation of this fact? Can it be generalized? For example, is there a similar relation that results in something other than a constant 1 for the Gamma second parameter? What if we have  $$\lim_{n\to\infty,m\to\infty,n=mb}n B(k, m) $$  That is, the two variables go to infinity while maintaining a constant ratio b.  The reason I'm asking is because I'm trying to figure out how to simplify a hieraerchical bayesian model involving the beta distribution.  (This is my first post; sorry for the math notation, the MathJaX syntax was too daunting, but I'll try to learn) I used a website called Seeking Arrangements and met a  sugar daddy. He asked for my email login, online banking login, routing, account number, Facebook, as well as my  address which is the only thing I gave him since it’s a PO. I’ve been extremely vague just to see if this is real.   I don’t plan to give him anything because I have only been talking to him for a few months and  I feel conflicted. I doubt that he is trying to scam me but I suspect that he has some kind of master plan. With things such as Venmo, Zelle, and Cashapp, what is the need for “put me on his payroll?”   He claims that he is in another country on a business trip.  He said that  he tried to send me a gift but he lost access to his card. He is now asking to deposit more than 10k into my account and have me wire it. Is this a l classic way of scamming people? If so, is it common that they spend months creating a relationship with you  to catfish you? How to make a file executable? How can I make my file so that I can double click on it, and it runs. It is a .sh script, but I also have files that say:  executable (application/x-executable)  in the description of what they are.  I can't run any of these from terminal, or by double clicking.  If possible, I would like a way using either the GUI or a Terminal, but not a combination of the two.  Here is a screenshot of what I see when I right click then go on properties. The file first:    And the shell script here:    NB: I accept that this is a duplicate (I was looking for it, and couldn't find it, so asked + answered it, hoping that I would find it) however, I don't think the question about .desktop files is a duplicate. Hilbert space of harmonic oscillator: Countable vs uncountable? Hm, this just occurred to me while answering another question:   If I write the Hamiltonian for a harmonic oscillator as $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ then wouldn't one set of possible basis states be the set of $\delta$-functions  $\psi_x = \delta(x)$, and that indicates that the size of my Hilbert space is that of $\mathbb{R}$.  On the other hand, we all know that we can diagonalize $H$ by going to the occupation number states, so the Hilbert space would be $|n\rangle, n \in \mathbb{N}_0$, so now the size of my Hilbert space is that of $\mathbb{N}$ instead.  Clearly they can't both be right, so where is the flaw in my logic? Value of $\frac{x^n -a^n}{ x-a}$? Pokemon Go login issues after update I downloaded the July 30th update for Pokemon Go and when I open the app I'm prompted to login, But once I login, it redirects me to this page on Cookies::     If I click "Done" in the top left, it prompts me to login again! I've tried reinstalling the game, restarting my phone, etc. I can't much enable Cookies on the inline Safari browser. It doesn't make sense, and cookies are on in Safari (because I don't use Safari and they're enabled by default).  How can I fix this? 408 Request Time-outs still continues I'm frequently getting 408 Request Time-out errors  when browsing through Stack Overflow and Meta. I used to get this four or five times a day before, but now it has become very frequent. I've read  in which Kyle Brandt ♦ says:     We send our web logs to SQL so we can query them. It turns out that we have a bug that prevents 408s from being logged to SQL. [...] I will do more extensive research when our 408s are part of queryable logs (should be a week or two [...]).   My internet connection is somewhat stable and I haven't experienced this issue on other sites before. I don't think it's a browser-specific issue as it is reproducible on latest versions of Google Chrome and Firefox (on Ubuntu 13.10). However, it is specific to Stack Overflow and Meta. I'm unable to reproduce this on other SE sites.  Am I the only one facing this issue? Could it be something wrong with the regional CDN or something like that? Could someone please take a look at this issue? ()  This appears when I try to close-vote questions, too: How can I make ffmpeg be quieter/less verbose? By default ffmpeg sends a whole lot of messages to stderr: when built, how it was built, codecs, etc, etc, etc.  How can I make it quieter?  I've tried -v 0 (and -v 10 since the documentation just coyly says Set the logging verbosity level. with no indication of what the range of inputs is) -- still not quiet.  I've tried -loglevel quiet -- still not quiet.  I should mention, I'm looking for "quieter," not "no output ever".  If there's an error I want to see it, but I don't need to hear about ffmpeg's configuration every. single. time. Conjugate subgroup strictly contained in the initial subgroup? Fantasy piece involving different worlds and a crippled protagonist This is a novel that I think I remember reading somewhere in the 1990s courtesy of my brother (who does not remember the book). Only a few details are coming to mind at the moment. It's a male protagonist. Something was not quite right with his legs, which led to him being nicknamed Frog or Leapfrog due to his usual method of locomotion, which consisted of pushing off from his legs to his hands and vice versa. Something happens and he winds up moving between a number of worlds (I want to say that he fell into a well or a hole in a field). There was a part which stuck with me where he's in a bottomless hole and finds he keeps falling faster and faster without end with the rushing wind becoming actively painful (I don't remember if he actually reaches a terminal velocity or if some aspect of the world prevented it while still allowing the rushing air to affect him physically).  Unfortunately, that's all I can remember. I don't remember whether he eventually gains companions on the voyage or how it ended. The most vivid memory is that of the nonstop falling. Clipping a GEE Image with Python API I am looking for a way to clip image collection by a given polygon geometry with Python API, The GEE JavaScript equivalent is simply ImageCollection.clip(geom) () With the python API I get the error: 'ImageCollection' object has no attribute 'clip'.  I'm adding my code after using .mean() as a reducer My code: set the boundary polygon my_boundary = ee.Geometry.Polygon([ [[-5, 40],[65, 40],[65, 60],[-5, 60],[-5, 60]]] ,proj=None) #set the ImageCollection and Image img_Collection = ee.ImageCollection('COPERNICUS/S5P/NRTI/L3_NO2').filterDate('2020-03-01', '2020-03-27').select('NO2_column_number_density') my_Image=img_Collection.mean().clip(my_boundary) #Create Ipyleaflet map my_map = ipyleaflet.Map(zoom=7) my_map.add_layer(layer=my_Image) my_map error: AttributeError: 'Image' object has no attribute 'model_id' Is there a place to dispute duplicate markings? Problem: If   $$C_0+\frac{C_1}{2}+\cdots + \frac{C_{n-1}}{n}+\frac{C_n}{n+1} =0,$$  where $C_0,...,C_n$ are real constants, prove that the equation  $$C_0+C_1x+\cdots +C_{n-1}x^{n-1}+C_nx^n=0$$  has at least one real root between $0$ and $1$.  Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exercise 4. The div containing tags for a question doesn't reflow, causing layout issues for questions with long/many tags This appears to be a css bug which (I think) has recently been introduced. It's most visible when viewing questions on Stack Exchange sites using mobile devices (the full site with responsiveness, not the mobile site) but can easily be reproduced by resizing a browser window on a question where the combined width of the tags is large. A couple of example questions show this behaviour:  On Meta Stack Overflow:   On Meta Stack Exchange:     Using the Meta Stack Overflow question to demonstrate (as the total width of the tags is larger):  At 1100px, the tags just about fit into their containing div.     At 800px, the tags appear on top of related questions:     At 450px, the tags are now wider than the view (the last tag is completely off the screen), causing scroll bars to appear:      This doesn't happen to lists of questions - the tags flow onto multiple lines:     Reproduced on: { Firefox 81, Chrome 84, Edge 84 } on Windows 10, and Chrome 85 on Android 11.