I'm affraid that $\cos(\sin x) &gt; \sin(\cos x)$ is not true   for $x\in\mathbb{R}$ , but it's seems works for some known values as :$\frac \pi 4$ and $\cdots$ , I have used standard method to compare them but i can't succed , Is there any simple method to show that if it true ?
The following problem was on a math competition that I participated in at my school about a month ago:      Prove that the equation $\cos(\sin x)=\sin(\cos x)$ has no real solutions.   I will outline my proof below.  I think it has some holes.  My approach to the problem was to say that the following equations must have real solution(s) if the above equation has solution(s):  $$ \cos^2(\sin x)=\sin^2(\cos x)\\ 1-\cos^2(\sin x)=1-\sin^2(\cos x)\\ \sin^2(\sin x)=\cos^2(\cos x)\\ \sin(\sin x)=\pm\cos(\cos x)\\ $$  I then proceeded to split into cases and use the identity $\cos t = \sin(\frac{\pi}{2} \pm t\pm y2\pi)$ to get   $$ \sin x=\frac{\pi}{2} \pm \cos x\pm y2\pi\\ $$  and the identity $-\cos t = \sin(-\frac{\pi}{2}\pm t\pm y2\pi) $ to get   $$ \sin x=- \frac{\pi}{2}\pm \cos x \pm y2\pi.\\ $$  where $y$ is any integer.  I argued that $y=0$ was the only value of $y$ that made any sense (since the values of sine and cosine remain between $-1$ and $1$).  Therefore, the above equations become   $$ \sin x=\frac{\pi}{2} \pm \cos x\implies \sin x \pm \cos x=\frac{\pi}{2}\\ $$  and   $$ \sin x=- \frac{\pi}{2}\pm \cos x\implies \cos x\pm\sin x= \frac{\pi}{2}.\\ $$  Then, by a short optimization argument, I showed that these last two equations have no real solutions.  First, does this proof make sense?  Second, if my proof makes sense, then I feel that it was not very elegant nor simple.  Is my approach the best, or is there a better (i.e., more elegant, shorter, simpler) proof?
Run a command that is shadowed by an alias Let's say I have the following alias in bash - alias ls='ls --color=auto' - and I want to call ordinary ls without options. Is the only way to do that is to unalias, do the command and then alias again? Or there is some nifty trick or workaround? I'm a bit confused on how extra damage works for critical hits. Let's take the following example: A thief with sneak attack rolls a crit. I'm under the impression he rolls double his normal attack damage dice AND double his Sneak Attack damage dice. Is this accurate? Another example, I finished DM-ing my first campaign recently. The Giant Spider enemy monster has an melee attack that causes a constitution saving throw. If the player fails, the enemy rolls an additional 2D8 of damage. In this situation, if there was a critical hit, and the player failed their saving throw, would I also roll double dice on the additional damage? Or is that considered a separate attack? I want to install the latest version of Eclipse but the Ubuntu Software Centre contains an older version.  Is there a PPA or some other way to install latest Eclipse? Please describe the steps for full installation. WinError 32 error when trying to delete shapefiles with QGIS python PHP validation/regex for URL I've been looking for a simple regex for URLs, does anybody have one handy that works well?  I didn't find one with the zend framework validation classes and have seen several implementations. I have some non-Gaussian distributed variable and I need to check if there are significant differences between the values of this variable in 5 different groups.  I have performed Kruskal-Wallis one-way analysis of variance (which came up significant) and after that I had to check which groups were significantly different. Since the groups are kind of sorted (the values of the variable in the first group are supposed to be lower than the values of the variable in the second group which are supposed to be lower than the values of the variable in the third group and so on) I only performed 4 tests:  Group 1 vs Group 2 Group 2 vs Group 3 Group 3 vs Group 4 Group 4 vs Group 5   I have performed this analysis with two different methods. I started by using Dunn's Multiple Comparison Test but nothing came up significant. On the other hand if I use Mann-Whitney test and correct for the number of tests (4) using Bonferroni, 3 tests come up significant.  What does it mean? Which results should I trust? I have a dual screen setup on my laptop (using 12.04 LTS) using a HDMI connected display. Everything works fine, but everytime I connect/disconnect the cable I have to go to Sound preferences and change the sound output device manually.  Is there any way to change the sound output device on connect/disconnection of cable, so when I connect my display the sound output is set to HDMI and when I disconnect it the sound goes back to laptop speakers? I'm not that into dynamic programming languages but I've written my fair share of JavaScript code. I never really got my head around this prototype-based programming, does any one know how this works? var obj = new Object(); obj.prototype.test = function() { alert('Hello?'); }; var obj2 = new obj(); obj2.test();  I remember a lot discussion I had with people a while back (I'm not exactly sure what I'm doing) but as I understand it, there's no concept of a class. It's just an object, and instances of those objects are clones of the original, right? But what is the exact purpose of this &quot;.prototype&quot; property in JavaScript? How does it relate to instantiating objects? Update: correct way var obj = new Object(); // not a functional object obj.prototype.test = function() { alert('Hello?'); }; // this is wrong!  function MyObject() {} // a first class functional object MyObject.prototype.test = function() { alert('OK'); } // OK  Also these  really helped a lot. Modified files in a git branch are spilling over into another branch What happens to the energy when waves perfectly cancel each other? What happens to the energy when waves perfectly cancel each other (destructive interference)? It appears that the energy "disappear" but the law of conservation of energy states that it can't be destructed. My guess is that the kinetic energy is transformed into potential energy. Or maybe it depends on the context of the waves where do the energy goes? Can someone elaborate on that or correct me if I'm wrong? Space after LaTeX commands I have a definition:  \newcommand{\arnold}{Arnold Schwarzenegger}   When I refer to it by:  \arnold is a   it is rendered as:   Arnold Schwarzeneggeris a    In order to have a space in front of "is" I would need to write  \arnold\ is a   Is there some other shorter way? Just don't show UI for actions that can't be actioned Recently I just deleted one of my answer in stackoverflow where there was one comment. After deleting the answer it also show for sometime that we all know but in the comment it still allow us to up to make it useful but as I clicked it it just showed me the message:  If we are not allowed to do then what was the meaning? Why don't it just be removed or make it disable if the answer is deleted. Just don't show UI for actions that can't be actioned Why is Euclid's proof on the infinitude of primes considered a proof? I've expressed Euclid's proof on the infinitude of primes on Mathematica:  f[x_] := Product[Prime[n], {n, 1, x}] + 1 TableForm[Table[{f[x], PrimeQ[f[x]]}, {x, 1, 20}]]   Which results in:  $\begin{array}{ll}  3 &amp; \text{True} \\  7 &amp; \text{True} \\  31 &amp; \text{True} \\  211 &amp; \text{True} \\  2311 &amp; \text{True} \\  30031 &amp; \text{False} \\  510511 &amp; \text{False} \\  9699691 &amp; \text{False} \\  223092871 &amp; \text{False} \\  6469693231 &amp; \text{False} \\  200560490131 &amp; \text{True} \\  7420738134811 &amp; \text{False} \\  304250263527211 &amp; \text{False} \\  13082761331670031 &amp; \text{False} \\  614889782588491411 &amp; \text{False} \\  32589158477190044731 &amp; \text{False} \\  1922760350154212639071 &amp; \text{False} \\  117288381359406970983271 &amp; \text{False} \\  7858321551080267055879091 &amp; \text{False} \\  557940830126698960967415391 &amp; \text{False} \\ \end{array}$  The proof flaws for all those values, how is it considered a proof then? I guess that there might be infinite prime numbers according to the proof, but what is the guarantee that at some point it won't fail indefinitely? The following is a rigorous proof of De Moivre's theorem by means of mathematical induction. The theorem put simply is that:  Any complex number, $z = a+bi$, on a cartesian plane can be expressed in polar form, where $a=r\cos\theta$ and $b=r\sin\theta$ and $r$ is the absolute distance from the origin to the point $z$. In light of this, the expression for the complex number $z$ in polar form is $z=r(\cos\theta+i\sin\theta)$ In order to find the $n^{th}$ power of $z$, the following rule applies: $$\displaystyle z^n=(r(\cos\theta+i\sin\theta))^n=r^n(\cos n\theta+i\sin n\theta)$$ where $n\in\mathbb N$ (I have confined $n$ to the natural numbers given that I intend on using mathematical induction to prove the theorem.  Base step: We must prove that what is stated is true for $n=1$, hence $$z^1=r^1(\cos 1\cdot\theta+i\sin1\cdot \theta)$$ $$=r(\cos\theta+i\sin\theta)$$ which is true. Next we assume it is true for $n=k$, and therefore proceed to the inductive step. Inductive step: When calculating $z^{n+1}$, it's the same as calculating $z^nz$, hence $$z^{n+1}=(r^n(\cos n\theta + i\sin n\theta))(r(\cos \theta+i\sin \theta))$$ $$=r^nr(\cos n\theta+i\sin n\theta)(\cos\theta +i\sin \theta)$$ $$=r^{n+1}(\cos n\theta\cdot\cos\theta+\cos n\theta \cdot i\sin\theta+i\sin n\theta\cdot\cos\theta+i^2\sin n\theta\cdot\sin\theta)$$ from here, $i^2=-1$, and therefore $+i^2\sin n\theta\cdot\sin\theta$ becomes $-\sin n\theta\cdot\sin\theta$. In addition to this, the following trigonometric identities will be used: $$\sin(n\theta + \theta) = \sin n\theta\cdot\cos\theta+\cos n\theta\cdot\sin\theta$$ $$or$$ $$k\sin(n \theta+\theta)=k\sin n \theta\cdot\cos \theta + \cos n\theta\cdot k\sin\theta$$ $$and$$ $$\cos(n \theta + \theta)=\cos n\theta\cdot\cos\theta-\sin n\theta\cdot\sin\theta$$ by using these, the original equation has now become $$=r^{n+1}(\cos(n \theta + \theta) + i\sin(n \theta + \theta))$$ then, after factoring out $\theta$, the theorem becomes $$=r^{n+1}(\cos((n+1)\theta)+i\sin((n+1)\theta))$$ which completes the proof. Question: is this proof correct all the way through? Or have I missed any conditions? If all is well in the case of this proof, could somebody aid me in proving it for $\{n|n\in\mathbb R,n\geq1\}$ Is there a way to get correctly aligned print export with OpenLayers plugin? I am a beginner using QGIS (1.8.0 Lisboa) on a Mac (OS X 10.6.8 Snow Leopard). I work on a file with two layers in the following sequence (top to bottom):    a vector polygon layer with a few outlines of various buildings in France and Italy (CRS: EPSG:4326 - WGS 84 - "on the fly" re projection is activated ) a Google satellite layer uploaded via open layer plugin.   Sometimes when changing the scale (scrolling in and out) the Google layer updates a little wrong in size or position compared to the polygon layer. But as soon as I "pan" the map everything reconstructs right. Now the problem I have concerns the print composer.  The result of any kind of export - pdf, or image or svg - of my map (vector layer + Google layer in the background) differs from what is shown in the preview of the print composer: the vector layer and the Google layer positions are mismatching and in a different scale. I can send some pictures showing screenshots of the print composer preview and the final results if required.  I tried almost everything:    new files with different CRS settings "cache" update preview "render" update preview different paper sizes,  different scales,   The problem appears only using an open layer map (Google, OpenStreetMap...). If I try the same using two vector layer (shapefile) - one with the buildings, another one with the city in background - everything works out great.  ...exasperating! Since it is a real important research project I would be very grateful for some help and tips. Are there wizarding schools in the rest of the "magical" world? Are there other schools for witches and wizards in the rest of the world? In the books they only speak of 3 schools, but what about Asian or American people?  And even more, were they affected by the events happening in Europe? Why Process Builder doesn't work on Delete Scenario? As most of the time we have to think of requirement to accomplished using out-of-Box feature like e.g process builder which we think of an alternative to trigger, but suppose I have to handle scenario of Delete using Process Builder then there is limitation!(e.g calculation of roll-up on Delete) so I am curious about what is that limitation because of which Delete scenario is not handled in Process Builder. Quadratic Formula in Complex Variables I have a Redstone Power loop that connects to a Command Block that is set to kill all entities except for myself (/kill @e[type=!Player]), because I'm building a creative map, but don't want slimes all over the place. Problem is if I want to place an Item Frame than the Command Block kills it.  So, is there anyway to exclude items from getting deleted, or make it so it will only target mobs? 14.04.5 to 16.04 Update failed - xenial.tar.gz.gpg missing The update from Trusty to Xenial is officially available via the software updater. After updating all packages, upgrading through the software updater yields:  Fetching the upgrade failed. There may be a network problem.   Doing a sudo do-release-upgrade yields:      Checking for a new Ubuntu release Err Upgrade tool signature                                                        403  Sorry, not allowed to fetch that type of file: xenial.tar.gz.gpg         Err Upgrade tool                                                                  403  Sorry, not allowed to fetch that type of file: xenial.tar.gz             Fetched 0 B in 0s (0 B/s)                                                       WARNING:root:file 'xenial.tar.gz.gpg' missing Failed to fetch Fetching the upgrade failed. There may be a network problem.