Unable to resize the partition because the disk was locked in Ubuntu 18.X
How do I join my partitions?
Proof of triangle inequality I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :) Is the axiom of choice needed in proving that metric spaces in which every infinite subset has a limit point are compact? Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$. I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it? Can I install a ROM made for a different device? How to pass a list of columns to data.table where some are predetermined Time Dilation When In Orbit Imagine a space ship orbiting the sun at near relativistic speeds, when compared to Earth (say from the reference frame of the sun) there would be x amount of time dilation.  This would mean people on the ship could live for 100's of Earth years.  My question is what would happen if a second ship flew directly away from the Sun and back (ignoring acceleration) at the same speed the orbiting ship is travelling, would the time dilation for the ship that travels in a straight line be the same as that moving in a circle?  The reason I ask is that I was imagining a scenario where intergalactic travel is possible as time dilation means even a journey of light years only ages the crew a fraction of that time, with space stations orbiting certain star systems at similar speeds so that people that don't work on ships can live a similar length of time. I understand that time dilation is all linked to a certain reference frame but working with a reference frame of the sun what would the time dilation be for the different objects taking different paths but at the same speed relative to the sun? If the ship docked back with the space station would they both say the same amount of time had passed? Submit Buttons on the Left or the Right? We have been puzzling over where to put the submit button, on the left or the right. In researching, we noticed that many sites put buttons on the bottom right in dialogue boxes, and on the bottom left in forms.  It makes sense: in a dialogue box it seems to denote finality, being in the endpoint of the window for left–right readers; in a form, the bottom right could be in a different position relative to the rest of the form if the window is resized.  It seems to be a convention, but I don't see anyone talking about this specific phenomenon in the following questions, nor the articles referenced therein:        So, is this level of button position optimization worth it if it creates an inconsistency in button placement? Is this a good way to go? Dirac Delta function inverse Fourier transform We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-i\omega t} dt = 1,$$  and if I were to reconstruct the function back in time domain, the inverse Fourier transform is defined as  $$\delta(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{i\omega t} d\omega.$$  How do I compute this integral analytically? I tried this in mysql:  mysql&gt; alter table region drop column country_id;   And got this:  ERROR 1025 (HY000): Error on rename of './product/#sql-14ae_81' to './product/region' (errno: 150)   Any ideas? Foreign key stuff? Product of invertibles in $\Bbb Z_n$ [Wilson's Theorem generalization] How can we compute the product of all invertibles in $\Bbb Z_n$?  In the special case $n=p$ where $p$ is a prime it is Wilson's theorem. By pairing inverses it reduces to computing the product of all $a$ such that $a^2\equiv 1\pmod{n}.\,$  How can we do that? How long was Luke Skywalker's training with Yoda? Prior to the spell cast on Mjolnir, did Thor control lightning on his own? In the film Thor, Odin casts a spell on Mjolnir and strips Thor of his power.  He then banishes Thor to Midgard (Earth).  On Earth, Thor is seemingly just a strong and durable human.  Later in the film, he becomes worthy, and regains control of Mjolnir.  This then grants him his full power set and control over Mjolnir.  My question is, prior to this, could Thor control lightning all on his own?  Or is this purely an ability of Mjolnir? Understanding dummy (manual or automated) variable creation in GLM How would I go about finding $\prod_{n=1}^\infty (1+\frac{1}{n^2})$ ? Setting $a_k = \prod_{n=1}^k (1+\frac{1}{n^2})$, the first few partial products are $a_1 = 2$, $a_1 = 2.5$, $a_3 \approx 2.78$, $a_4 \approx 2.591$, $a_{20} \approx 3.50$, and they eventually appear to converge at around $3.639$. (These values were found with a quick python program. So much for my initial conjecture that the product converged to $e$.)   I tried writing $1+\frac{1}{n^2}$ as $\frac{n^2 + 1}{n^2}$, but it wasn't immediately obvious how to prove convergence / find the limit because the product doesn't really factor or telescope. Is there a way to calculate the limit? What are the alternatives for storing client side encrypted data? I want to store client side encrypted data in Drupal 7, so that it is encrypted on the client end, and stored and transmitted in encrypted form. The password is never transmitted or stored on the server. The only modules I have found which do this are  and  but they are for Drupal 6. I need something that is user specific, so that different users can use different keys, rather than a single key or a set of keys used on the server end. Are there any similar modules for Drupal 7? I have scene that looks like this:   And I want to add volumetric lighting to it so that light forms a rays of light when it go through the trees like on this example that I found in youtube:   But the problem is that all of these tutorials show how to make this lighting coming from windows, but when I try to replicate it in my forest scene it looks like big ugly fog with cubical shape. Also the light is coming from the back of the camera (as you can see) and there are some spots that needs this special lighting (like places where light goes through trees) and some that don't (like big bright are on the bottom left corner) so I can't just add a giant cube to all of the scene. Let $B$ be a commutative domain and $A \subseteq B$ a subring. Suppose that the extension $A \to B$ is finite, i.e. $B$ is finitely generated as an $A$-module. If $I \cap A = \lbrace 0 \rbrace$ for some ideal $I$ of $B$, can we conclude that $I = \lbrace 0 \rbrace$?  I know that if $I$ contains a nonzero prime ideal, the statement holds since $A \to B$ is an integral extension, whereby no two comparable prime ideals of $B$ can have the same intersection with $A$. But does the statement still hold if $I$ does not contain a nonzero prime ideal? And what if $A \to B$ is only assumed to be an integral extension? I live in a town where fresh or whole milk is not readily available. I was wondering if I could make cheese from UHT Milk. If so, what kind of cheese and what method I should use? Or rather, Should I? If $f:E\rightarrow S$ is an elliptic curve over a scheme $S$ (so $f$ is proper and smooth of relative dimension one with geometrically connected fibers of genus one, equipped with a section $0:S\rightarrow E$), then is the sheaf $\underline{\omega}_{E/S}:=f_*\Omega_{E/S}^1$ actually free of rank one? According to the statement of Grothendieck-Serre duality in Hida's book Geometric Modular Forms and Elliptic Curves, there should be a canonical isomorphism $\underline{\omega}_{E/S}\cong\mathcal{Hom}_{\mathscr{O}_S}(R^1f_*\mathscr{O}_E,\mathscr{O}_S)$, and the first result on elliptic curves in this book is that $R^1f_*\mathscr{O}_E\cong\mathscr{O}_S$. It's also not clear to me whether $S$ is being assumed (locally) Noetherian, but if that's necessary for Grothendieck-Serre duality to hold, then I'm fine with assuming it. I can't find another reference with a statement of Grothendieck-Serre duality in this generality which does not use the language of derived categories (which I unfortunately don't understand).  The reason I'm kind of skeptical about this is that in Hida's book, as well as in Katz-Mazur, it is said that $\underline{\omega}_{E/S}$ is invertible, so that an $\mathscr{O}_S$-basis $\omega$ for $\underline{\omega}_{E/S}$ can be found locally on $S$. If the invertible sheaf in question were really trivial then one would be able to choose a global $\mathscr{O}_S$-basis for $\underline{\omega}_{E/S}$, and there would be no reason to talk about doing so locally. Hida goes on to say that, choosing an $\mathscr{O}_S$-basis $\omega$ locally on $S$ allows one to regard $(\Omega_{E/S}^1,\omega)$ as a relative effective Cartier divisor in $E/S$, which also doesn't make complete sense to me because if we can only find $\omega$ locally, how are we getting a global section $\omega\in H^0(E,\Omega_{E/S}^1)=H^0(S,\underline{\omega}_{E/S})$ (unless $\underline{\omega}_{E/S}$ really is trivial)?