I'm looking for a good resource (preferably a book) on . I am thorough with the basics (the mirror formula, lensmaker's formula, thin lens formula, etc). What I want is something that focuses specifically on various applications and extensions of these formulas to different situations. I would like a resource that supplements an introductory course in geometric optics while not venturing too much beyond the basic formulas.
I need to learn basic optics for a high school academic competition. Does anyone know any good places, preferably free and online, to learn the basics of optics, like lenses, angles of incidence, etc.? I can't seem to find an MIT OpenCourseWare course addressing the issue.
How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers? One of the basic (and frequently used) properties of cardinal exponentiation is that $(a^b)^c=a^{bc}$.  What is the proof of this fact?  As Arturo pointed out in his comment, in computer science this is called . Truncated alternating binomial sum Why GCP Kubernetes Node host location for europe-north1 seems to be in US? Proving a function is Lipschitz continuous Show that the following function is Lipschitz continuous and find a Lipschitz constant  $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{  , } |x-1|\leq\frac{1}{2}\text{  , } |y-e|\leq\frac{e}{2} $$  I have no clue on where to begin to prove this.   My first question is how I should interpret this function, is $x$ a sort of constant? When does HashSet 'add' method calls equals? How does the tester screwdriver work? How does the tester screwdriver work? If I put the tester screwdriver inside the "hot wire" of an electrical socket, it lits up if I press my finger against the metal cap on top of the screwdriver. This happens also if I stand on a surface of isolating material, such as wood. I read elsewhere that this happens due to stray capacitance formed by the "hot wire", the human body, and the ground. One has  $$ Z = R + \dfrac{1}{j \omega C}  $$  for the impedance, so if C  is high enough, the impedance should be close to r , the "effective resistance" of the formed circuit. Here I get lost; why r  is small enough to cause a current in the range of mA  even if I stand on an isolating surface?  So really what I am asking is how one may represent the system hot wire - screwdriver - human body - wooden floor - building - ground as an electrical circuit, and which parts of the physical system contribute to the resistance, capacitance (and inductance?) and in which proportion, even very approximately. prove that if $1 \le m \le n$ then $\left(1+\frac1n\right)^m < 1+\frac mn + \left(\frac mn\right)^2$ Ain't and negatives Adding script conditionally Flight ticket reimbursement after Travel Warning I've got a round-trip flight with Royal Air Maroc from Strasbourg, France (SXB) to Ouagadougou, Burkina Faso (OUA) on Monday, Nov 10. Return is on Dec 1. The recent turmoils in Ouagadougou have led the US Department of State for Travels to issue a , which is why I'd like to refrain from the trip.  Any chance I can get a reimbursement for my tickets? In Germany (where I live), an official travel warning can help you get a refund when you book a trip with a tour operator/travel agency, but in this case I booked the flight directly with a non-German airline, so German law surely doesn't apply here.  EDIT: Sorry for failing to state this earlier, but I had already called the airline as suggested in the answers. They had told me it's not possible to get a refund with my standard fare. However, I understand that sometimes you really have to bargain to get a refund, so I wanted to learn about similar experiences and possibly about some hard (legal) facts that I could bring forward. Use induction to prove a product of sums of squares is a sum of squares Beamer itemize not aligned when in blocks While making a poster, I've got two columns with blocks with itemize inside them. MWE:  \documentclass[final]{beamer} \mode&lt;presentation&gt;{}  \usepackage{times} \usepackage{amsmath,amssymb} \usepackage[english]{babel} \usepackage[latin1]{inputenc}  \begin{document} \begin{frame}[t]{}   %\leavevmode \begin{columns}[t, onlytextwidth] \begin{column}{.5\linewidth} \begin{block}{Contributions} \begin{itemize} \item First Item \item Second Item \end{itemize} \end{block} \end{column} \begin{column}{.5\linewidth} \begin{block}{Approach} \begin{itemize} \item First Item \item Second Item \end{itemize} \end{block} \end{column} \end{columns}  \end{frame} \end{document}   My local installation and ShareLaTeX would misalign the bullets(baselines added by me):   This problem seems similar to:        But this time it happens when itemize is inside the block. If one removes the blocks it works fine.  Am I missing something or does it look like a Beamer bug?  Thanks! I'm a LaTeX noob, sorry if I missed something stupid. 1.13 Detect if no mobs with a specific tag exist and execute commands My 1.12 solution can be found . When trying to update command blocks to 1.13, I have the following:  scoreboard objectives add invert dummy   (run at start) then:  scoreboard players set FakePlayer invert 1   (impulse unconditional repeat) chained into:  execute at @e[tag=summoned] run scoreboard players reset FakePlayer invert   (chain unconditional repeat) chained into:  execute if score FakePlayer invert = 1 invert run say "All summoned animals are dead!"   This should reset FakePlayers score continuously until all entities with tag "summoned" are dead, then it should run the final condition. This does not happen, and I can't see why. Cannot make a static reference to the non-static method Hiding reputation and encouraging thinking time would improve the quality of answers on SO I've only been a member of SO for a couple of months but I must say that the site can be a bit daunting even for an experienced developer.  Part of this is because when users with very high reputations answer questions then it seems that everyone assumes that they have the right answer.  I don't want to go into the self-fulfilling nature of reputation increase for high reputation people.  But I think it is clear that people with high reputation scores get more upvotes () because people assume (usually accurately) that they are correct.    But if reputation was hidden on SO, at least for, say, the first hour after a question is asked then the probability of the best answer being upvoted the most would surely increase.  This might also require the hiding of the names of the answerers if they have very high reps!  As a second method of getting improved answers I would correlate the value of an answer (in terms of reputation gained) with the difficulty of the question.  And the simplest way to do that would be based on the time required for the question to be answered.  So if a question is answered within a minute then it is a 20% question (2 points per upvote), within 2 minutes a 40% question, within 5 minutes a 60% question and so on, on to say a 200% answer (20 points per upvote) on questions that have been hanging around for months.  This would encourage considered answers to new questions and tempt people into reducing the backlog - rather than encouraging people to sit on the 'newest questions' hoping for a gimme.  These two measures would incentivise people to answer the questions that no-one likes the look of and reduce the relative value of answering easy questions. Setting Windows PowerShell environment variables Repeated squaring method How do I use the repeated squaring method to calcualte 2^176 (mod 177)? I'm not sure, but is there something about the fact that 177 is 1 greater than 176 that makes this a problem? As far as we know the gaze of the basilisk will instantly kill someone or eventually petrify him/her in case the gaze is somehow softened or reflected.   But why would it melt the film in Colin's camera? The film is just plastics with some specific chemicals.   A bit broader question would be - is the some information that the basilisk's gaze is somehow destructive to non-alive objects too and under what circumstances could it happen? Ownership of .Xauthority transferred to root Somehow, while playing around with LightDM and Webkit Greeter, the ownership of the .Xauthority file in my home dir was given to the root user and I couldn't login because I hadn't the privilegues to lock the file.  I was able to regain ownership of the file and I could login again. (After several hours of reinstalling LightDM and it's greeters)  So now everything's working fine again. But I'd like to know how this happened. Is this a bug in LightDM or Webkit Greeter or something else? Writing a list to a file with Python