Everyday we see ourselves in mirror and this is due to light which forms our image after reflection ( may be refraction sometimes) . But I wonder how can light carry the image. How is it possible that massless particles (or waves maybe, which is not an issue now ) can carry the image of the source from which they come or object by which they got reflected bat a tremendous speed ?   I know that things we perceive as they are because of our eyes but the point of that our eye must also perceive things which really occur. Like if the light rays are coming from an object  at infinity and incident on a concave mirror, then they get converged to points in the focal plane and we say that a real image has been formed there. Then the question is how light made that image  (which may be perceived by our eyes) ?
How do rays of light getting focused at a point create an image?  How are the information of object's shape and dimensions expressed as an image when light rays converge?  Is it understandable by solving wave equation for light?
GIS Point or discrete longitude and latitude columns? What are some good examples of a function  $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ and showing that $f^{-1}(x) = f'(x) \implies b-1=\frac{1}{b} \implies b=\phi$ and got $$f(x) = \frac{x^\phi}{\sqrt[\phi]{\phi}}$$ Which seems to work according to WolframAlpha, but I'm having trouble double-checking it. Any other ideas? Was this an oversight?  Or is it to do with the JVM? Let $a,b,c$ be positive real numbers satisfying $$(1+a+b+c)\left(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=16,$$ then $a+b+c=3$.  I thought about the application of the AM-GM-HM inequality, but in vain. I also thought about splitting 16 into factors and comparing but went nowhere. Any ideas. Thanks beforehand. How to change Power button shutdown action to run a script under systemd How does one change the power button shutdown action under say debian 10 systemd to run a custom script instead of shutting down the system?  Back when this was handled by acpi one could edit the scripts in /etc/acpi.  The only suggestion I have found thus far is to replace various power related binaries with my script.  Is there a way to do this without replacing binaries? I found this heart while exploring the crimson biome - what is it? I've placed a command block making it day all the time. I want to turn it off, but I've lost it. What's the quickest way to find it? I want to completely replace Ubuntu with Lubuntu I accidentally issued some commands I thought were safe in the terminal, and I ended up with no System Settings and lots of other missing functionality, but the OS still boots normally. I've decided I want to install Lubuntu instead of fixing Ubuntu for a fresh start and since my laptop is running on a 1GHz dual core AMD CPU.  I created a bootable USB as I usually do and placed Lubuntu 17.10 amd64 using Rufus 2.14.1086. I have GRUB that was installed with Ubuntu upon first install I still have Win8.1 which I still want to keep.  PS: I also tried creating a startup disk from Ubuntu  The question is: How can I boot from USB with GRUB2 so I can overwrite Ubuntu? Different ways to execute a shell script There are several ways to execute a script. The ones I know are: /path/to/script # using the path (absolute or relative) . script        # using the . (dot) source script   # using the `source` command  Are there any other way? What are the differences between them? Are there situations that I must use one and not another? Android Syntax Highlighting? java.lang.NumberFormatException: Invalid int: "" How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$? What does "65" mean? K-2SO beats up the troopers easily. Up in the sky, the bombers are preparing to attack. Just as they start the attack run, TIE fighters come in from the shield and swarm the rebel fleet.  JON VANDER (GOLD LEADER): Red Leader, this is Gold Leader, we’re starting our attack run on the  shield gate. Keep it tight and watch out for those towers. GOLD PILOT: More fighters. Form up, at 65.  What does &quot;65&quot; mean? My online friend is asking for money in order to visit my home country. Is this a legit request or a scam? On Travel.SE we often see questions about a person (usually a love interest) asking for money to pay for airfare, visas, permits and other expenses. For example        What general advice can be given in such situations to the person asking the question? Is it all a scam or can there be legitimate situations in which an online friend could be asking for money? Can anyone recommend a good exposition of the theory behind partial least squares regression (available online) for someone who understands SVD and PCA?  I have looked at many sources online and have not found anything that had the right combination of rigor and accessibility.    I have looked into The Elements of Statistical Learning, which was suggested in a comment on a question asked on Cross Validated, , but I don't think that this reference does the topic justice (it's too brief to do so, and doesn't provide much theory on the subject).  From what I've read, PLS exploits linear combinations of the predictor variables, $z_i=X \varphi_i$ that maximize the covariance $ y^Tz_i $ subject to the constraints $\|\varphi_i\|=1$ and $z_i^Tz_j=0$ if $i \neq j$, where the $\varphi_i$ are chosen iteratively, in the order in which they maximize the covariance.  But even after all I've read, I'm still uncertain whether that is true, and if so, how the method is executed. My Command is below and its print Free CPU and Used CPU below format.  I want to restrict awk output to 2 decimal place.  grep 'cpu ' /proc/stat | awk '{usage=($2+$4)*100/($2+$4+$5)} {free=($5)*100/($2+$3+$4+$5)} END {print "Used Cpu:",usage "%"} {print "Free Cpu:",free "%"}'   Output :   Free Cpu: 98.9588%  Used Cpu: 1.04125%   Desired output :  Free Cpu: 98.95%  Used Cpu: 1.04% How can I switch a 32-bit installation to a 64-bit one? I've got a 32-bit Ubuntu installation running on 64-bit hardware. Now that , I would like to switch to 64-bit without having to reinstall the OS.  This is one of the :  Shawn installed his system using the 32-bit version of Ubuntu, but his hardware is 64-bit and he wants to switch over. He manually installs the amd64 versions of dpkg and apt, replacing the i386 versions and changing which architecture is used as the default; then he installs the amd64 ubuntu-minimal package; then he installs the amd64 ubuntu-desktop package. Over time the remaining i386 packages are replaced automatically on upgrade.  However, when trying to follow the instructions in there, I cannot find any 64-bit version of dpkg or apt.  Did this user story got implemented in a different way in the final spec, or do I need to do something differently?  In short, how can I switch my 32-bit installation to 64-bit? I made something like this:    I did this with the Ellipse Tool and the Line Tool.  Now the circle is accurate, but the lines inside are very inaccurate. Is there another tool that does the same thing but with a lot more accuracy? Constructing a HoTT proof term of 1≠0 As an exercise in HoTT basics, I am trying to construct a term that has the type $Id_{Nat}(S(O),O)\to\bot$. If this were a Coq proof, I'd be done after a single inversion on the premise, as the impossible identity would leave zero cases to consider. I guess I could do something similar here, but I'm not sure how. What would be an HoTT term that has the type? Let $f$ be a complex function with the property that $f^{(n)} = 0$.  Does this imply that $f$ is a polynomial?  If so, why?  Upon thinking about this problem myself, I can easily observe that every $n$ degree polynomial satisfies $f^{(n+1)} = 0$.  Still, this doesn't show that some non-polynomial function couldn't also have this property.