Why doesn´t this new environment work?
enclosing a lstlisting environment inside my own environment using matlab-prettifier
Show that the conditional statement is a tautology without using a truth table I have been attempting to use identities to get to the answer but I am unable to get anywhere. Here is the equation I am trying to prove tautological without using truth tables:  $[(p\rightarrow q) \land (q \rightarrow r)] \rightarrow (p \rightarrow r )$  And here is my attempt at using identities to work my way up. (a link to a page with a table of logical equivalence identities: )  $\lnot [(p \rightarrow q) \land (q \rightarrow r)] \lor (p \rightarrow r)$  $\lnot [(\lnot p \lor q) \land (q \rightarrow r)] \lor (p \rightarrow r)$  $[\lnot (\lnot p \lor q) \land \lnot (q \rightarrow r)] \lor (p \rightarrow r)$  $[(p\land \lnot q) \land (q \land \lnot r)] \lor (p \rightarrow r)$  I don't know where to go from there. Euler-Lagrange equation in curved spacetime: how to calculate $\partial \mathcal{L}/\partial(\nabla_\nu A_\mu)$? My problem is mostly with notation. When we apply the Euler-Lagrange equation to the Lagrangian scalar $$\mathcal{L}= -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$$ Where $$F_{\mu\nu} = \partial_\mu A_\nu -\partial_\nu A_\mu= \nabla_\mu A_\nu -\nabla_\nu A_\mu$$ and I start with $$\frac{\partial\mathcal{L}}{\partial A_\mu}=\nabla_\nu\left(\frac{\partial \mathcal{L}}{\partial(\nabla_\nu A_\mu)}\right).$$   Now I rewrite the lagrangian as $\mathcal{L} = -\frac{1}{4}g^{\mu\alpha}g^{ \nu\beta}(\nabla_\mu A_\nu-\nabla_\nu A_\mu)(\nabla_\alpha A_\beta-\nabla_\beta A_\alpha)$ $$\mathcal{L}= -\frac{1}{4}g^{\mu\alpha}g^{\nu\beta}\left[(\nabla_\mu A_\nu)(\nabla_\alpha A_\beta)-(\nabla_\mu A_\nu)(\nabla_\beta A_\alpha)-(\nabla_\nu A_\mu)(\nabla_\alpha A_\beta)+(\nabla_\nu A_\mu)(\nabla_\beta A_\alpha) \right]$$ My problem is calculating $\frac{\partial \mathcal{L}}{\partial(\nabla_\nu A_\mu)}$, how are the rules of partial derivatives generalised to covariant ones? Also, is $\frac{\partial\mathcal{L}}{\partial A_\mu}=0$, is that correct ? Why is cold fusion considered bogus? Cold fusion is being mentioned a lot lately because of some new setup that apparently works. This is an unverified claim.  See for example:         ( of that last link in the Wayback Machine, given frequent  from that page.)   While we should give the scientific community time to evaluate the set up and eventually replicate the results, there is undoubtedly some skepticism that cold fusion would work at all, because the claim is quite extraordinary.  In the past, after Fleischmann and Pons announced their cold fusion results, in perfectly good faith, they were proven wrong by subsequent experiments.  What are the experimental realities that make Fleischmann and Pons style cold fusions experiments easy to get wrong?  Would the same risks apply to this new set up? Why is $a^n - b^n$ divisible by $a-b$? I did some mathematical induction problems on divisibility   $9^n$ $-$ $2^n$ is divisible by 7. $4^n$ $-$ $1$ is divisible by 3. $9^n$ $-$ $4^n$ is divisible by 5.   Can these be generalized as $a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer? But why is $a^n$ $-$ $b^n$$ = (a-b)N$ ?  I also see that $6^n$ $- 5n + 4$ is divisible by $5$ which is $6-5+4$ and $7^n$$+3n + 8$ is divisible by $9$ which is $7+3+8=18=9\cdot2$.  Are they just a coincidence or is there a theory behind?  Is it about modular arithmetic? Why do the $n \times n$ non-singular matrices form an "open" set? Display date/time in user's locale format and time offset I want the server to always serve dates in UTC in the HTML, and have JavaScript on the client site convert it to the user's local timezone.  Bonus if I can output in the user's locale date format. What would happen if two wizards made each other a Horcrux? Would they be invincible? How would you kill one of them? This applies to all d20 systems, but with an example from the from the Pathfinder  (emphasis mine):  Stonecunning: Dwarves receive a +2 bonus on Perception checks to potentially notice unusual stonework, such as traps and hidden doors located in stone walls or floors. They receive a check to notice such features whenever they pass within 10 feet of them, whether or not they are actively looking.  In this case, does the GM roll the check in secret or ask the player roll the check? If the GM asks the player to roll for the check, what stops situations like:  GM: (interrupting) &quot;Gimli, Roll a perception check.&quot; Dwarf player: &quot;Oh, it's a natural one, I guess I don't see anything.&quot; Human player: &quot;Okay, I'll roll a perception check for anything unusual in this room. Nice! 19. How about 19, does 19 work for you?&quot; GM: &quot;Gimli, everything looks kosher to you. Aragon, you notice a rock coloured tarp on the ground here, covering what looks like a pit trap.&quot;  In this case, by telling the dwarf to roll his &quot;free&quot; perception check, he's telegraphed to the players that there is something to notice. How can this be fixed, without bluffing checks for nothing in particular to throw off the signal to noise ratio of similar checks? This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse:   let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$,  then as $R$ is closed under multiplication $\prod_{n=1}^k\ x_i=x_j$,  therefore by canceling $x_j$ we get $x_1x_2\cdots x_{j-1}x_{j+1}\cdots x_k=1 $,  by commuting any of these elements to the front we find an inverse for first term, e.g. for $x_m$ we have $x_m(x_1\cdots x_{m-1}x_{m+1}\cdots x_{j-1}x_{j+1}\cdots x_k)=1$, where $(x_m)^{-1}=x_1\cdots x_{m-1}x_{m+1}\cdots x_{j-1}x_{j+1}\cdots x_k$.   As far as I can see this is correct, so we have found inverses for all $x_i\in R$ apart from $x_j$ if I am right so far. How would we find $(x_{j})^{-1}$? In comments, moderators have their diamonds in the OP rectangle:        And in the review (suggested edits), mods have their diamonds outside the blue OP background highlight rectangle:        Of course, it Doesn't Really Matter At All, but it is a small discrepancy (I am not saying that in the review, OP rectangles have a bigger padding). Why does magnetic force vary in proportion to the cube of the distance instead of the square? Magnetic forces vary from gravity and electromagnetic radiation (such as light) in that gravity and radiation diminish by the square of the distance from the source, but magnetism diminishes by the cube of the distance.  Why is this? Can every even integer be expressed as the difference of two primes? If $0 &lt; \alpha &lt; 1$, then       $$ (x + y)^{\alpha} &lt; x^{\alpha} + y^{\alpha} $$      for $x$, $y$ positive.   Is this inequality true in general?  I tried using Young's Inequality: For $z,t &gt; 0$, and for $n$, $m$ such that $n+m=1$, then   $$ z^n + t^m \leq nz + mt $$  So, using this we have   $$ (x+y)^{\alpha} \cdot 1^{1 - \alpha} \leq \alpha(x+y) + (1-\alpha) = \alpha x + \alpha y + (1-\alpha)$$  which is not as tight as I want. Touch enabled monitors and Android How do I round a number in JavaScript? While working on a project, I came across a JS-script created by a former employee that basically creates a report in the form of  Name : Value Name2 : Value2   etc.  The peoblem is that the values can sometimes be floats (with different precision), integers, or even in the form 2.20011E+17. What I want to output are pure integers. I don't know a lot of JavaScript, though.  How would I go about writing a method that takes these sometimes-floats and makes them integers? I was looking for proofs using Calculus for the area of a circle and come across this one $$\int 2 \pi r \, dr = 2\pi \frac {r^2}{2} = \pi r^2$$  and it struck me as being particularly easy. The only other proof I've seen was by a teacher and it involved integrating $x = \sqrt{r^2 - y^2}$ from $-1$ to $1$, using trig substitutions and then doubling the area to get $\pi r^2$ but the above proof seemed much more straight forward.  Is it a valid proof, or is it based on circular logic or some other kind of fallacy? Getting an "Authentication token manipulation" error when trying to change my user password I am logging in to my Ubuntu Server using my username. Once I am logged in I am typing passwd command. Entering a new password but a second after getting following error messages:  passwd: Authentication token manipulation error passwd: password unchanged   What is wrong here? How can I change my password otherwise if I don't have access to that server physically, i.e. I am connecting remotely with ssh using terminal. How to properly secure a wireless home network I'm soon going to be setting up a wireless network for me and some buddies and (hopefully) making it secure. I have some questions and concerns regarding this and figured someone here could give me a hand and steer me in the right direction.   I have a WRT55AG (version 2) loaded with the factory firmware. I've heard that it's possible to load your own firmware onto routers like these. Is it worth it? What can be achieved with it? I was wondering what kind of security I should use on my wireless. Ideally I would want all devices to be 'accepted', but I'm not sure how to accomplish this. I'm guessing it has something to do with MAC addresses, but I don't know the first thing about them. It would be great that ALONG WITH encryption (I'm told WPA2 is the best), every device would need to be added to a list of some sort before even being able to access the AP. Another feature I was thinking of would be completely hiding my wireless from showing up on other people's computer. Is this possible (like an SSID broadcast thing). Is it worth the trouble? One more option I was looking to implement would be bandwidth calculators. Ideally something in the router to measure the amount of bandwidth being used and how much has been used for that month (this has to do with bills and limits, obviously). Is it possible to throttle down speed when a certain limit is reached?   These are alot of points but I was hoping someone who has more knowledge than me in this subject could give me a hand and a few pointers. Hopefully all this is possible and I'm not dreaming :) Failed to download repository information due to missing CDROM Unexpected AdjacencyMatrix output