Which option to choose in the advanced formatting?
How can I replace Windows with Ubuntu?
Can you attack with different weapons using Extra Attack? The fighter's  feature (PHB, p. 72) says:     Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.   Assuming I am holding two different weapons, and don't want to spend my bonus action on Two-Weapon Fighting, can I attack once with each weapon? Injection from $\mathbb{N}\to\{0,1\}$ to the set of all monotonic decreasing functions from $\mathbb{N}\to\mathbb{Z}$ The polar of a convex body $C$ is defined as:  For a convex body $C$, its polar $C^*$ is given by $$C^* = \{x\in\mathbb{R}^n: \langle x,c\rangle \le 1, \forall c\in C\}$$  If I start with an orthonormal basis $(e_i)_1^n$, positive constants $(\alpha_j)_1^n &gt; 0$ and the ellipsoid $$\varepsilon = \left\{x\in\mathbb{R}^n: \sum\limits_{j=1}^n \frac{\langle x,e_j\rangle^2}{\alpha_j^2} \leq 1  \right\}$$ how do I show that its polar ellipsoid is $$\varepsilon^* = S_1 = \left\{y\in\mathbb{R}^n: \sum_{j=1}^n \alpha_j^2 \langle y,e_j\rangle^2 \le 1\right\}$$ Using the definition, I saw that $$\varepsilon^* = S_2 = \{z\in\mathbb{R}^n: \langle z,x\rangle \le 1, \text{for all }x\in \varepsilon\}$$ Also, $\langle z,x\rangle = \sum_i\langle z,e_i\rangle\langle x,e_i\rangle$ I used Cauchy Schwartz to do one side of the proof, i.e. $S_1\subseteq S_2$: $$\left(\sum_i\langle y,e_i\rangle\langle x,e_i\rangle\right)^2 \leq \left(\sum_i\alpha_i^2 \langle y,e_i\rangle^2\right)\left(\sum_i \frac{\langle x,e_i\rangle^2}{\alpha_i^2}\right) \le 1 \implies \langle y,x\rangle \le 1$$ I need help with the other direction, i.e. $S_2\subseteq S_1$. Which other inequality might help? I'm reading a proof about Lagrangian => Hamiltonian and one part of it just doesn't make sense to me.  The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the Hamiltonian is defined via the Legendre transform  $$H(p,q,t) = \max_{\dot q} [p \cdot \dot q - L(q, \dot q, t)]$$  Under the right conditions there exists a function $\dot Q (p,q,t)$ such that  $$H(p,q,t) = p \cdot \dot Q(p,q,t) - L(q, \dot Q(p,q,t), t)$$  i.e. when some $\dot Q(p,q,t)$ satisfies $p = \frac{\partial L}{\partial \dot q}\rvert_{(q, \dot Q(p,q,t), t)} = \frac{\partial L}{\partial \dot q}(q, \dot Q(p,q,t), t)$  (Finding this function is usually called "inverting p")  By taking partials in the $p$ variable and using the relationship, we can obtain the relationship  $$\dot Q = \frac{\partial H}{\partial p}$$  Because of the notation I chose, I get the strong urge to say $\dot q = \frac{\partial H}{\partial p}$  , and in fact this is what the textbook does. But have we proved this?    In other words, how can we deduce that  $$q'(t) = \frac{\partial H}{\partial p}(p(t), q'(t), t)$$ for any differentiable vector valued function $q$? (or maybe there are more conditions we need on $q$? Here  $$p(t) = \frac{\partial L}{\partial \dot q}(q(t), q'(t), t)$$ according to Lagrange's equations. Where should I post my sum-up of the answers, as the OP? Sometimes as the OP (question asker), after reading through all the answers, I have my own sum-up about them. I want to tell people and all further readers about my feeling and comprehension about my desired answer and how all the answers helped towards solving my problem.  Where's the correct place to put that information? Things I've considered:   Edit into my question: No because that's not part of the question A comment under the question: I dislike that due to limited formatting in comments and its subject to removal (though the latter won't be the case) A comment under the highest-scored or the accepted answer: Still the same reason above. This is, for the time being, the place I'm posting that. A separate answer: Those thoughts probably don't qualify as an answer Are the Yongnuo clones of Canon flash equipment and lenses legal to sell/buy? A "friends list" on Stack Overflow would be nice I think it would be nice to have a "friends list" on Stack Overflow. This would allow us to be able to follow the questions and answers that are posted by users that we want to watch/monitor.  I suggest it be implemented in the same fashion as DotNetKicks.com; where you can add someone to your "friends list" without them needing to approve you the way that other social networks do it. It should be completely open for us to add anyone to our "friends list" that we like. Here are some options I thought of, not sure which is the right one.   There was an I/O error reading from the pipe. The process writing to the other end of the pipe died with a failure. All processes who could write to the pipe have closed it. The write buffer of the pipe is full. The peer has closed the other direction of the duplex pipe. Writing failed because there are no processes which could read from the pipe. A system call returned the EPIPE error, and there was no error handler installed. Can't boot Windows 7 after installing Ubuntu I installed Ubuntu yesterday alongside Windows 7 and whenever I start my computer I don't have the option to boot Windows. When the computer starts it goes straight to Ubuntu without giving me an OS selection menu. When it goes to Ubuntu it gives me Ubuntu repair options and Ubuntu normal start option. I still have all of the Windows 7 files on my hard drive, but Windows 7 has no way to boot. A high school Olympiad problem Order of general- and special linear groups over finite fields. No error message if the URL is very long I entered a URL with a very long parameter on Stack Exchange.  I have type this url    It shows a blank screen , does not shown any messages.    We need to show a “Page Not Found” message, same as what you did on other Stack Exchange pages. SIB 2009, Problem #2 Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2π]$ and $f&#39;&#39;(x)≥0\:\:∀\:x∈[0,2π]$. Show that$$ \int _0^{2\pi} f(x) \cos x dx \ge 0$$ I recently bought a Minolta film camera and it had some film with it. The film is dated 1991 - 1994. Can I use the film? Given $a,b,c$ are the sides of a triangle. Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$ Best WPF open source projects I have programmed with WPF for more than a year. When I started there weren't so many examples and guidelines to follow, now the situation is much more better.  I would like to ask you some help in listing the best WPF (open) source code examples or projects that you suggest to study in order to learn more about project structure, best practices, etc.  Thank you, Marco How Triangle Law and Parallelogram law of addition of Vectors are different?Ain't they.  Please don't tell me the things written in book......give me the appropriate reason.And how do i distinguish between the two while adding vectors,what am i trying to say is i get really confused where to use parallelogram law and where triangle law  P.s:my basics are really weak though!May be the question might not be right Simple Linq query has duplicated join against same table? Sci-fi story where all 'citizens' had shares in a global company Not everyone was a citizen though, there was a significant underclass who had no shares.  One character was a celebrity in some sort of combat game who was followed by camera drones &amp; used drugs to suppress puberty so he waa an eternal 'child', I have the impression that medical science allowed virtual immortality short of accidents.  One character (may have been the same one) had had a 'previous life' &amp; had committed personality suicide by some form of mind wipe bequeathing his worldly possessions to the new personality that would eventually develop in his old body.  I think the motive for this 'suicide' was essentially boredom, he'd lived a long time.  The mind wipe in question was pretty extensive &amp; all encompassing, he had to be cared for &amp; educated from scratch as if a newborn baby in an adult body, nappies &amp; all (but I'm pretty sure this was purely backstory in the book I read).  He also had miniature nano organisms (or bots?) like miniature worms or millipedes that complimented his immune system &amp; aided healing, they were self replicating, pretty sure they were some sort of bioengineered organism tailored to his DNA rather than bots  but not 100% on that.  There was an organism spanning a large area on the planet that absorbed (or consumed) anything that wandered into it but saved the DNA of anything it had previously absorbed &amp; might randomly reproduce any animal plant or individual it had previously absorbed fully formed.  One of the characters fell into this but got pulled out before the full effects took hold, as a result he grew leaves from his head &amp; his immune system was then constantly battling with the 'infection'.  When did I read this? probably some time in the 80's in the UK.  When was it written? I haven't a clue, sorry.  It was a paperback, probably something I had out of a library, because I never get rid of anything I've bought myself so I'd still have it &amp; be able to dig it out if it wasn't a library loan.  I have lots of fragments of world background &amp; 'colour' that seem to have stuck in my mind there, but a complete lack of any story arc to go with them. I want to find the conjugacy classes of this group: $$G=\{(a_1,a_2,a_3):a_i=\pm e_j\}\subset GL_3(\mathbb C)$$ I know that $S_3$ is one of the subgroups of this group, so I can check which elements of $S_3$ continues to be conjugate in our group, I know that in $Z(G)$ there is two elements $I,-I$. I know that I have to check the diagonal matrices on which ones are conjugate, but I don't know how to finish and find the others conjugacy class of this group, and I don't Know how to find the number of the elements of the conjugacy class. I'll be thankful for help