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Why exactly does diffraction occur? Why do waves that were traveling in a straight direction change direction when passing through an opening?  I thought that the waves (red arrow) when colliding with the wall bounce in the opposite direction (green arrow).  And the waves that pass through the aperture follow its path normally as shown in the image on the right.   The waves that go in a straight direction should follow traveling straight line like a car that goes under a bridge the car is straight on the road. But this is not so.  Why does the direction of the waves change?  How is the direction of the waves calculated? Graph with $k$ components. Let $G$ be a simple graph on $n$ vertices. If $G$ has $k$ components and every component is complete prove that: $$|E| \le \frac{(n-k)(n-k+1)}{2}$$ I'm asking for advice. Inverse function of $x^x$ How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried using logs, differentiating, etc, etc, but to no avail. Why doesn't the height of a container element increase if it contains floated elements? Integral Inequality $|f''(x)/f(x)|$ Let $f$ be a $C^2$ function in $[0,1]$ such that $f(0)=f(1)=0$ and $f(x)\neq 0\,\forall x\in(0,1).$ Prove that  $$\int_0^1 \left|\frac{f{''}(x)}{f(x)}\right|dx\ge4$$ Suppress "In:" biblatex $x^{x^{x^{\dots}}}=2$, what is the value of $x$? I came up with this little simple exercise, stating:  $x^{x^{x^{\dots}}}$ infinite times is equal to $2$, find $x$.   As we're dealing with infinity, we can just separate the first $x$ and get $x^2=2 \implies x=\sqrt2$  But then I thought what if we change $2$ to $4$? Now we get $x^4=4 \implies x=\sqrt[4]4=\sqrt2$  Something went wrong, can someone explain? Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$. In Burton's Elementary Number Theory, when introducing absolute pseudoprimes (also called Carmichael numbers), the author states that they are composite numbers $n$ such that $a^{n-1} \equiv 1 \pmod n$ for all integers $a$ with $\gcd(a,n) = 1$.   When showing that 561 is an absolute pseudoprime, he then concludes that $a^{561} \equiv a \pmod{561}$ for all $a$.   I searched a bit on the internet, and it seems that these two are actually equivalent definitions of a Carmichael number. However I have not been able to find a proof; the fact that the implication in the title of the post is stated so plainly in my textbook makes me think it is quite simple to see, but I can't seem to understand the argument. Dual boot system Windows XP Pro and Ubuntu 12.04.  I have the bios set for the correct time and Ubuntu set for US Eastern time.  Ubuntu will boot up and the time will be off by -4 hours.  If I correct the time in Ubuntu then when I boot up in Windows XP the time will be off by + 4 hours.  This is a fresh install of 12.04. I did not have this problem before reinstalling Ubuntu. Return row only if value doesn't exist Does removing an iPhone from my icloud account also erase the data? Basic desktop actions are not available on Ubuntu 20.04 In Ubuntu 20.04, on the desktop, there is a number of actions I wish I could do, such as, by order of importance :  use keyboard shortcuts such as select all, copy / paste. use drag &amp; drop with icons deactivate the &quot;mouse over&quot; highlight, and restrict the clicking area to only the icon, not the surrounding square  To put it simply, I'd like to have the exact same features as offered by the desktop of Ubuntu 18.04 (probably related to a specific version of GNOME, and not Ubuntu itself). Am I missing something obvious here? I actually see no reason why those actions are removed or deactivated on the desktop, since they all work in any Nautilus window. How can I port forward with iptables? Horizontal scroll bar occludes text I've been running into this lately.  The scroll-bar gets stuck and makes it difficult to read the text.  I'm running Chrome Version 56.0.2924.87 (64-bit) on macos. Every once in a while after updating MiKTeX, the installation gets into an inconsistent state. This might be a problem with the packages (an example is described in ), but sometimes packages are split or renamed. The cases I remember are when  and when miktex-etex-base-2,9 was renamed to miktex-etex-base-2.9 (if I recall correctly; I am sure it was either that or something very similar). In each case the MiKTeX Update process suggested to remove the "old" packages but it didn't give the user any hints about him having to manually add the respective replacement packages back in the package manager to get one's MiKTeX installation back into a usable state.  What are best practices for keeping MiKTeX up to date? Note that I'm not asking about installing new packages or fonts in general; there are other questions covering this area.   In which order should one execute the Update process and the Synchronization of the repository? , it is recommended to check the MiKTeX homepage before updating. What else should one do? When and in which order (and why) should one execute the "Refresh FNDB" and "Update Formats" processes within the MiKTeX Options menu? What else can happen? Other advice? Connect MacBook Pro to two external monitors Is there a way to connect my MacBook Pro to two external monitors (either VGA or HDMI; no Thunderbolt) in such a way that the Mac screen serves as a third monitor?  What hardware do I need? What does a de Broglie wave look like? What does a  wave look like? Are de Broglie waves transverse or longitudinal?   Can they be polarized?    What about the de Broglie wave of a ground state neutral spin-zero Helium 4 atom?   What experimental evidence do we have that supports the detailed nature of a de Broglie wave?     I have always assumed that de Broglie waves were mathematically  identical to electromagnetic waves, but I just realized there is no basis for this assumption, and in fact it must be false, unless there is an analogue to both the magnetic and the electric components of the electromagnetic wave.  So what does a de Broglie wave "look like"? How can I get a specific line from a file? Eigenvalues of a matrix whose square equals its transpose Let $A$ be a $n\times n$ matrix with $A^2=A^t$. Show that every real eigen value of $A$ is either $ 0$ or $1$ Slit screen and wave-particle duality