Why does laser dot become bigger at bigger distance?   e.g: from millimeter to inches so that it distracts pilots in aircrafts. I knew that laser beam should't expand over distance, but remain stable. Is that true?
I was wondering why a laser beam diverges. If all the photons are in the same direction, I would imagine that it would stay that way over a long distance. I am aware that a perfectly collimated beam with no divergence cannot be created due to diffraction, but I am looking for an explanation based on  photons rather than wave physics.
Password Reset for Users on a Multisite Subsite How to use more than 10 tab stops in bmatrix or other amsmath matrix environments? I normally write using the American English forms, but for some particular words, I tend to naturally write it in the British manner (like with the word favourite rather than favorite).  Is it wrong to mix both styles? If not *wrong*, does it looks weird? Pivot rows into multiple columns I have a SQL Server instance that has a linked server to an Oracle server. There is a table on the Oracle server called PersonOptions which contains the following data:  ╔══════════╦══════════╗ ║ PersonID ║ OptionID ║ ╠══════════╬══════════╣ ║        1 ║ A        ║ ║        1 ║ B        ║ ║        2 ║ C        ║ ║        3 ║ B        ║ ║        4 ║ A        ║ ║        4 ║ C        ║ ╚══════════╩══════════╝   I need to pivot that data so the results are:  ╔══════════╦═════════╦══════════╦══════════╗ ║ PersonID ║ OptionA ║ Option B ║ Option C ║ ╠══════════╬═════════╬══════════╬══════════╣ ║        1 ║       1 ║        1 ║          ║ ║        2 ║         ║          ║        1 ║ ║        3 ║         ║        1 ║          ║ ║        4 ║       1 ║          ║        1 ║ ╚══════════╩═════════╩══════════╩══════════╝   Any suggestions? Is there an invalid pthread_t id? Try Ubuntu - Where does it store my files? I am wondering, when you select the option of just "Try Ubuntu" instead of installing, where does Ubuntu store the files that you download / copy / save?  I right click in the Home folder, and it specified I had 4GB free space. Where did it get this 4GB space from, and if I transition to another OS system, how can I access or delete all the previous data from the session of "Try Ubuntu"?  Thank you! It  to disable internet search results in start menu searches before the Windows 10 Creators Update, but I've just clean installed a new Windows 10 Pro image and that setting is gone. Is there some way to disable it through editing the registry/etc? How do you do a deep copy of an object in .NET? I want a true deep copy. In Java, this was easy, but how do you do it in C#? Prove $ \sin x + \frac{ \sin3x }{3} + ... + \frac{ \sin((2n-1)x) }{2n-1} >0 $ Prove that for $ 0&lt;x&lt; \pi $, $$ \quad S_n(x) = \sin x + \frac{ \sin3x }{3} + ... + \frac{ \sin((2n-1)x) }{2n-1} &gt;0 \quad \forall n = 1,2,... $$  Having trouble with this problem. This is an olympiad-style question, so an answer that doesn't use calculus or analysis would be preferred. A possible approach is induction, but for this we need to find a function in terms of $n$ and $x$ so that we can actually use the inductive step. If anyone has any ideas they would be appreciated.  If you really want to go down the calculus route (at this point I don't mind), then $ S_n' (x) = \cos x + \cos 3x + ... +\cos((2n-1)x) $ , which you can find a closed form for, but I don't know how useful that is. Proof of when is $A=X^TX$ invertible? Say we have an $n\times m$ matrix $X$. What are the specific properties that $X$ must have so that $A=X^TX$ invertible?  I know that when the rows and columns are independent, then matrix $A$ (which is square) would be invertible and would have a non-zero determinant. However, what confuses me is, what sort of conditions must we have on each row of $X$ such that $A$ would be invertible.   It would be very nice to have a solution of the form:   when $n &gt; m$ then $X$ must have... when $n &lt; m$ then $X$ must have... when $n = m$ then $X$ must have...   I think in the 3rd case we just need $X$ to be invertible but I was unsure of the other two cases. Using oscilloscope safely with AC mains I know there is quite few discussions already on this topic, but since English isn't my first language, I though it better to get some more information about this.  I have used oscilloscopes to troubleshoot electronics problems and measure AC engine current spikes at startup, but I have no experience using them to measure high voltage levels and my experience is with old analog scopes.  So now I have this new DSO and the input rating is 400V peak to peak AC, so it can handle about 140V AC RMS? Not that I'm going to input that much there.  Since I live in Europe the mains here are 230 V @ 50 Hz I can't measure them directly.  If I set the attenuation from the scope and probe to 10X, the voltage is 23 RMS and it is safe to measure? And yes the probes are rated for 1kV.  If that is true, it's also safe to measure voltage between two main lines (400V RMS), since the peak-to-peak would be about 112 volts?  SAFETY: When I am measuring AC, I will be using the scope from its internal battery or with a isolating transformer (1:1) so that it is floating, meaning that there is no connection to earth ground. Are there any other safety issues I have not realized? How to set a command block with a command inside using the setblock command Does someone know how to set a command block with a command inside using the setblock command?  I have created this code /setblock &lt;x&gt; &lt;y&gt; &lt;z&gt; command_block {TileEntityData:{Command:"testfor @p[_=1]"}} but it doesn't works.  Do someone know how to do that? Proving $\arg(zw)=\arg(z)+\arg(w)$ This is my attempt I know this is incomplete or may even be wrong.   Let $θ_1 \in \arg(z)$ and $θ_2 \in \arg(w)$. Then, $θ_1+θ_2 \in \arg(z)+\arg(w)$. Also, $θ_1+θ_2 \in \arg(zw)$. Is this sufficient for the proof or correct at all? Hope someone could help me out. Thanks.   EDIT: Just to avoid any confusions I will add how $\arg z$ is defined.   $\arg z=\{\text{Arg}\, z+2\pi k\mid k\text{ is an integer}\}$ Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic. I was thinking that the product of groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}$ is not cyclic, but $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/p\mathbb{Z}$ is cyclic if p is an odd prime. Then I found logic that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is certainly cyclic if $m$ and $n$ are coprimes.  I searched on google a proof of that, to compare with mine and the answer is that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is isomorphic to $\mathbb{Z}/mn\mathbb{Z}$ (which is clearly cyclic) $\iff$ $m$ and $n$ are coprimes.  But I got another proof and I want to know what you think about it :     I claim that $([1]_m,[1]_n)$ generates $\mathbb{Z}/m\mathbb{Z}$   $\times\mathbb{Z}/n\mathbb{Z}$, where $[a]_d$ is the canonical   projection of $a\in \mathbb{Z}$ into $\mathbb{Z}/d\mathbb{Z}$.       Let's take multiples of $([1]_m,[1]_n)$ until we get $m \cdot$   $([1]_m,[1]_n) = ([0]_m,[m]_n)$ (where I suppose $m=\min(m,n)$...   otherwise, consider $\mathbb{Z}/n\mathbb{Z}$   $\times\mathbb{Z}/m\mathbb{Z}$ and $\min(m,n)=n$).       The next step is $([1]_m,[m+1]_n)$. This pair is not equal to   $([1]_m,[1]_n)$, since $m+1 \not\equiv1$ mod $n$ ($m$ and $n$ are   supposed to be coprimes). We can now continue adding $([1]_m,[1]_n)$   to the results.      The last step is precisely reached when you got   $([mn]_m,[mn]_n)=([0]_m,[0]_n)$. At this point, the fact that $m$ and $n$ are coprimes is absolutely necessary because if they are not coprimes, $lcm(m,n)\ne m\cdot n$ and   $([lcm(m,n)]_m,[lcm(m,n)]_n)=([0]_m,[0]_n)$ but all the $m \cdot n$ possible pairs are not reached, only the $lcm(m,n)$ are ! Why the principal value of complex number argument is taken within $(-\pi,\pi]?$ Why the principal value of complex number argument is taken within $(-\pi,\pi]?$ What is the harm in considering similar other intervals like $[0,2\pi)$ for the purpose? How can I 'join' an array adding to the beginning of the resulting string the first character to join? Same mesh with different colors Displaying layout marks in a document set with different layouts Is there any package that allows for the display of layout marks within a document where the layout changes several times throughout? I'm using the geometry facilities to issuing \newgeometry commands throughout my document, but it doesn't update the displayed layout and neither does the showframe package or the facilities of the memoir class.  EDIT: As Werner has pointed out, the showframe option of the geometry package displays correctly any updated page layout set by issuing \newgeometry commands.   My trouble came out of the fact that I'm issuing also \pagecolor commands and the page color seems to be hiding the layout lines typeset according to the geometry package. My confusion only worsened because the showframe package layout lines were not hidden by the page color (the package got the layout wrong on some pages, though). So, it all boils down to a conflict with the page color.   Even though I'm using geometry's showframe option for "debugging" ends, and I don't mind to have no page color at that stage of development, is it possible to override this problem and have at the same time both the page layout displayed by geometry as well as the page colored?   Btw, should I open a new question given that the problem is actually different from the originally posted one? I had to immediate shut down, and I lost about 20 unsaved tabs, is there any way to recover these? "I couldn't but laugh" - correct or not?