Why must unexposed and undeveloped film be handled in total darkness? Why can analogue, unexposed and undeveloped monochromatic photographic paper be handled in red/ yellow safelights without getting fogged?
I'm no expert in darkroom photography, but it seems a little odd that there is a type of light that doesn't affect film or developing paper etc. The only way that I could think to explain it is:   The low frequency red photons don't have enough energy to raise electron states in the film/paper, or Magic   either way, I'm curious to know how this handy tool is, as per title, safe.
"How are" or "How is" the wife and kid? Can the real and imaginary parts of $\dfrac{\sin z}z$ be simplified? I have calculated the real and imaginary parts of $\dfrac{\sin z}z.$ I've obtained  $$\begin{eqnarray} \frac{\sin z}z&amp;=&amp;\frac{\sin(x+iy)}{(x+iy)}\\ &amp;=&amp; \frac{\sin(x)\cosh(y)+i\cos(x)\sinh(y)}{x+iy}\\ &amp;=&amp; \frac{x\sin(x)\cosh(y)+y\cos(x)\sinh(y)}{x^2+y^2}+i\frac{x\cos(x)\sinh(y)-y\sin(x)\cosh(y)}{x^2+y^2}. \end{eqnarray}$$  Before calculating it, I hoped I could use this in a solution to a certain problem, but I would need to differentiate the real and imaginary parts of $\dfrac{\sin z}z$ with regard to $x$ and $y$. But what I'm getting is very complicated. I understand that it could be that that's just the way it is. But maybe there are some simplifications to be done either in the functions or in their derivatives? I've never really used hyperbolic functions before so I'm not sure I'm not missing something.  Added. The problem I'm working on is the following.     Prove that the solutions of the equation $$\tan z=z$$ are all real.   I will sketch the solution I know here.  It can be calculated that  $$\tan(x+iy)=\frac{\sin(2x)}{\cos(2x)+\cosh(2y)}+i\frac{\sinh(2y)}{\cos(2x)+\cosh(2y)}.$$  Plugging this to the equation, we get  $$\begin{eqnarray} x&amp;=&amp;\frac{\sin(2x)}{\cos(2x)+\cosh(2y)}\\ y&amp;=&amp;\frac{\sinh(2y)}{\cos(2x)+\cosh(2y)}. \end{eqnarray}$$  From this we can get   $$\sin(2x)\cdot2y=\sinh(2y)\cdot2x.$$  By plotting the functions $\sin,$ $\mathrm{id}$ and $\sinh$ (and proving the right inequalities we can see from the plots), we can see that this is only possible when $xy=0$. But when we plug $x=0$ to the second equation, we can show that the only solution is $y=0$, which ends the proof.  What strikes me in this solution is the fact that $\dfrac{\sin z}z$ is used. (Not explicitly in my statement.) This function has another connection to the problem:     The non-zero solutions of the equation $\tan z=z$ are exactly the zeros of the derivative of $\dfrac{\sin z}z.$   This is easy to prove. I wondered if there shouldn't be a solution to the problem which uses this fact, given that $\dfrac{\sin z}z$ appears in the solution I know. Integral of $ 1 \, / \, (1 + a \, \cos(x) )$ How to activate hyphenation in \ttfamily without explictly setting a font? I am suprised that in the code below no hyphenation is done. And I can not find something about that on the web. Maybe I used the wrong words for searching.  \documentclass{scrartcl} \usepackage{xltxtra} \usepackage{polyglossia} \setdefaultlanguage[spelling=new]{german} \usepackage{tabu}  \begin{document} \begin{ttfamily} \begin{tabu} spread \textwidth {X[l]|X[l]|X[l]|X[l]}     A&amp;     B&amp;     Sichtweise, Haltung, Akzeptanz, useability, Anwenderfreundlichkeit&amp;     D\\ \end{tabu} \end{ttfamily} \end{document} setfacl incorrectly changes group permissions Gauss's mean value theorem problem Suppose that  $f(z)$ is analytic and $f(z)\neq0$  on the disc $|z|&lt;1$ .  Show that for $0&lt;r&lt;1$,  $$ \exp\left[\frac{1}{2\pi}\int_{0}^{2\pi}{\log}|f(re^{i\theta})|\,d\theta\right] = |f(0)| $$  Any other theorems needed to solve this problem? I don't have any idea just with "Gauss's MVT"  Thank you in advanced. Any way to find out what US Customs & Border Protection has in their database on me? I travel from the US to Canada occasionally (about once per year on average) with my family. We are all US citizens and travel with US passports. Every time we re-enter the United States, I am given extra scrutiny.  Most recently, we used an automated check-in procedure at the airport. These are terminals where you scan your passport &amp; fill out a customs declaration online and it prints a receipt which you take to the customs agent. When I got my receipt, it had a big "X" printed on it and we were directed to a line to talk to an agent.  In the past, I've had questions directed at me or my family: "Have you ever been to New York?", "Do you own any firearms?", and once "Where is your daddy?" directed at my then 4 year old son. The agents also seem to examine my photo &amp; my face more closely than the other members of my family.  We get through customs OK, but it gives me a some anxiety. I suspect that there's someone out there with the same name (I have a pretty common name) who is on a "no fly" list or something.  My question is this: is there any way to find out what they're looking for when they scan my passport &amp; see my name? Like a Freedom of Information request or something? I know I could/should ask when I'm in the line, but as I said, it makes me pretty anxious. Leave newlines intact in bounty messages How do I know if my Australian ETA was approved? 16.04 LTS upgrade alert- How to handle it? I am running Ubuntu 14.04 LTS. Today when I logged into my computer I got an update alert saying that I could upgrade to Ubuntu 16.04 LTS. The other options were not now or do not upgrade. If I chose to upgrade will it delete my /home directories and the installed software under /opt and /usr/local ? Second mean value theorem proof I am asked to prove the second mean value theorem:  Let $f$ and $g$ be defined on $[a,b]$ with $g$ continuous, $f\ge 0$, and $f$ integrable. Then there is a point $x_0 \in (a,b)$ such that $$ \int_{a}^{b}f(x)g(x)dx = g(x_0) \int_{a}^{b}f(x)dx. $$  My (incomplete) proof goes as follows:  Since $ [a,b] $ is compact and $ g $ is continuous, we know there exist $ x_1, x_2 \in [a,b] $ such that $ g(x_1) \le g(x) \le g(x_2) $ for any $ x \in [a,b] $. Thus, we have the following string of inequalities $$ f(x)g(x_1) \le f(x)g(x) \le f(x)g(x_2), \, \, \forall x \in [a,b], $$ $$ \Rightarrow g(x_1)\int_{a}^{b} f(x)dx \,\le \, \int_{a}^{b} f(x)g(x) dx \, \le  \, g(x_2)\int_{a}^{b}f(x)dx. $$  My problem now is to show the final equality. It seems to me the intermediate value theorem is somehow involved, but I cannot see how to use it. Furthermore, how do I make use of the $f\ge 0$ condition in the hypothesis? Intuitive interface for Composing Boolean Logic? I'm interested to know how people have, or would construct an interface which simplifies constructing logical boolean conditions? (Yes, it's a database query builder) In my mind a bad interface has them typing lots of AND, NOT IN, OR keywords etc. Or else has a million combo boxes.  I'm wondering if anyone has a good example I could glean ideas from? Or some suggestions?  At the moment I'm considering a split screen with a grid for composing the query and a canvas to display a visual representation of the sets of conditions and how they overlap etc. Like a Venn Diagram. This still gives me a problem with the usability/intuitiveness of the controls on the grid.  EDIT:  I'm very interested in ideas which simplify it for less technical users. However, ideas for a version of the UI for advanced users are very helpful as well. I would like to parse a string such as p1=6&amp;p2=7&amp;p3=8 into a NameValueCollection.  What is the most elegant way of doing this when you don't have access to the Page.Request object? What is the meaning of modular fractions? What is the meaning of $$ \\\frac{m}{n}\equiv k\pmod l $$ and $$ \\\frac{m}{n} \mod l $$ $m, n, k, l \in\mathbb N$ and $\gcd(m, n)=1$. Why is this simple Excel calculation not netting Zero? It's supposed to be zero but for some reason the result produces a peculiar series of digits at the end.  Also, here's the .xlsx of it: http://83.212.111.36/Book1.xlsx I am taking an abstract algebra course and I am really interested about the topic.  So much so that I spend most of my time reading supplementary materials.  I consequently know a lot more theorems than the ones covered in class.    In a recent quiz, I used a theorem in one of my proofs that was not mentioned in the lecture since.  My professor gave me partial marks for my answer for the reason that we didn't cover this theorem in class even though the proof was completely valid!  I can't really see where he is coming from.  How would someone be able to use a theorem correctly if he doesn't know its proof?  I was quite baffled by his comment.   Do you agree with this?  EDIT: The professor just Emailed me and said that after thinking for a while, he decided to award me the full mark for the question. He also mentioned that he didn't want discourage me from studying the subject (since he saw most of my previous quiz grades were full marks) and I was passionate about it, but kindly requested that I include any proof for a theorem that I cite from now on in his exams. He also sent a broadcast message to the whole class indicating this. For $f$ Riemann integrable prove $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$ Suppose $f$ is a Riemann integrable function on $[0,1]$. Prove that $\lim_{n\to\infty} \int_0^1x^nf(x)dx=0.$   This is what I am thinking: Fix $n$. Then by Jensen's Inequality we have $$0\leq\left(\int_0^1x^nf(x)dx\right)^2 \leq \left(\int_0^1x^{2n}dx\right)\left(\int_0^1f^2(x)dx\right)=\left(\frac{1}{2n+1}\right)\left(\int_0^1f^2(x)dx\right).$$Thus, if $n\to\infty$ then $$0\leq \lim_{n\to \infty}\left(\int_0^1x^nf(x)dx\right)^2 \leq 0$$ and hence we get what we want. How correct (or incorrect) is this? "contribute to investigating" or "contribute to investigate" I'm trying to find, which of the following sentences is correct?  A) We try to highlight how they contribute to investigate their importance. B) We try to highlight how they contribute to investigating their importance.  I found that both cases were used frequently in formal writing. Edited: Sources for each case: For A: &quot;Their multifaceted nature means that there are very many routes to investigating their importance for understanding social change&quot; in &quot;A lineage for contemporary inter-city studies&quot; book. B: &quot;More factors are analysed to investigate their importance in scoring from shots and shot probabilities&quot; in &quot;Applications of logistic regression to shots at goal at association football&quot; book. Why are our planets in the solar system all on the same disc/plane/layer? I am asked to solve $$u_x+u_y=1$$ If is was homogeneous i.e., $u_x+u_y$ the answer would be $u(x,y)=f(y-x)$ where $f$ is an arbitrary function. I have found the following set of solutions: $$u(x,y)=\lambda x +(1-\lambda)y$$ where $\lambda$ is an arbitrary constant(real or imaginary). I just have no idea what method other then trial and error would have lead me here. Any ideas? Thanks!