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Why can we distinguish different pitches in a chord but not different hues of light? In music, when two or more pitches are played together at the same time, they form a chord. If each pitch has a corresponding wave frequency (a pure, or fundamental, tone), the pitches played together make a , which is obtained by simple addition. This wave is no longer a pure sinusoidal wave.  For example, when you play a low note and a high note on a piano, the resulting sound has a wave that is the mathematical sum of the waves of each note. The same is true for light: when you shine a 500nm wavelength (green light) and a 700nm wavelength (red light) at the same spot on a white surface, the reflection will be a superposition waveform that is the sum of green and red.  My question is about our perception of these combinations. When we hear a chord on a piano, we’re able to discern the pitches that comprise that chord. We’re able to “pick out” that there are two (or three, etc) notes in the chord, and some of us who are musically inclined are even able to sing back each note, and even name it. It could be said that we’re able to decompose a Fourier Series of sound.  But it seems we cannot do this with light. When you shine green and red light together, the reflection appears to be yellow, a “pure hue” of 600nm, rather than an overlay of red and green. We can’t “pick out” the individual colors that were combined. Why is this?  Why can’t we see two hues of light in the same way we’re able to hear two pitches of sound? Is this a characteristic of human psychology? Animal physiology? Or is this due to a fundamental characteristic of electromagnetism? While trying some problems along with my friends we had difficulty in this question.   True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$.   I couldn't do it and my friend justified it by saying that since the terms in the product have values less than $1$, so the value of the product can never be $1$. I don't know whether this justification is correct or not. But i referred to Tom Apostol's Mathematical Analysis book and found a theorem which states, that    The infinite product $\prod(1-a_{n})$ converges if the series $\sum a_{n}$ converges.   This assures that the above product converges. Could anyone help me in finding out where it converges to? And,   Does there exist a function $f$ in $\mathbb{N}$ ( like $n^{2}$, $n^{3}$) such that $\displaystyle \prod\limits_{n=1}^{\infty} \Bigl(1-\frac{1}{f(n)}\Bigr)$ has the value $1$? Proving the snake lemma without a diagram chase Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A&#39; \mathrel{\overset{f&#39;}{\to}} B&#39; \mathrel{\overset{g&#39;}{\to}} C&#39; \to 0 $$ and morphisms $a : A \to A&#39;, b : B \to B&#39;, c : C \to C&#39;$ making the obvious diagram commute. The snake lemma states that there is then an exact sequence $$0 \to \ker a \to \ker b \to \ker c \to \operatorname{coker} a \to \operatorname{coker} b \to \operatorname{coker} c \to 0$$ where the morphisms between the kernels are induced by $f$ and $g$ while the maps between the cokernels are induced by $f&#39;$ and $g&#39;$.  It is not hard to show that the morphisms induced by $f, g, f&#39;, g&#39;$ exist, are unique, and that the sequence is exact at $\ker a, \ker b, \operatorname{coker} b, \operatorname{coker} c$. With the use of a somewhat large diagram shown , we can even construct the connecting morphism $d : \ker c \to \operatorname{coker} a$. However, I'm stuck showing exactness at $\ker c$ and $\operatorname{coker} a$. I thought Freyd might have had an element-free proof in his book, but it turns out he proves it by diagram chasing and invoking the Mitchell embedding theorem [pp. 98–99]. Is there a direct proof? Functional Analysis. Continuous extensions of linear functionals. I would like to show that a continuous functional on a subspace $W$ of a normed space $(V,\|\cdot \|)$ has a unique continuous extension to $V$ iff $W$ is dense in $V$.  I have proved $(\Leftarrow$). But the converse is currently eluding me! Help would be much appreciated! How to apply CSS to iframe? I have a simple page that has some iframe sections (to display RSS links). How can I apply the same CSS format from the main page to the page displayed in the iframe? My friend and I have been wracking our heads with this one for the past 3 hours... We have 2 point masses, $m$ and $M$ in a perfect world separated by radius r.  Starting from rest, they both begin to accelerate towards each other. So we have the gravitational force between them as:  $$F_g ~=~ G\frac{Mm}{r^2}$$  How do we find out at what time they will collide?  What we're having trouble with is this function being a function of $r$, but I have suspected it as actually a function of $t$ due to the units of $G$ being $N(m/kg)^2$.  I've tried taking a number of integrals, which haven't really yielded anything useful.  Any guidance?  No, this is not an actual homework problem, we're just 2 math/physics/computer people who are very bored at work :) How to know if Mac OS X machine is being remotely viewed? Currently I have no way to know if my machine is being remotely viewed by our tech support, and would prefer to know, given the massive privacy breach that would entail should this be happening. I am running Mac OS X Snow Leopard.  Running  ps -A | grep Remote   I know this service is running:  /System/Library/CoreServices/RemoteManagement/ARDAgent.app/Contents/MacOS/ARDAgent   My knowledge of administration of Macs on a network is limited, so I don't know what tools allow this. Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$ Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on  $\mathcal{C}^0(\mathbb{R},\mathbb{R})$?  A related question is: Can we proved that $\mathcal{C}^0(\mathbb{R},\mathbb{R})$ has a norm without the axiom of choice? Exclude property from serialization via custom attribute (json.net) Output for Print Is there a way to output for Print. I want to save in a higher DPI than the default 72dpi say 300dpi that's a good quality for printed pieces. I tried downloading an add-on but don't know if I am installing it correctly it isn't showing up but the bar in python is green. I'm using the 2.69 version of Blender. If no such feature exists what size have any of you found best for print quality. Thank you Can't Open files from a directory in python Internet Accounts spontaneously forgets passwords and I can't reauthorize them I've had this problem before, but I was able to resolve it that time. macOS 10.13.4, everything up to date. Yesterday, Internet Accounts forgot my LinkedIn password and would not reauthenticate. Today, it forgot my Facebook password.   I've tried:   App specific password. Removing the account and re-adding it. Turning off 2-factor authentication. Deleting associated entries from Keychain. Logging out of iCloud, deleting Keychain entries, removing and re-adding the account, then logging back in.   First of all, what the hell—why does this keep happening?! Secondly, what else can I do to fix this short of reinstalling macOS? What is the life expectancy of a Wookiee? At lunch, my coworkers and I were discussing The Force Awakens, and started questioning Chewbacca's age.  A quick check of  and it says he is possibly about 230 years old as of Episode VII.  Following the  link on that page, Wookieepedia claims that they live to be 400 years old.  Do we know from canon what the average life expectancy of a Wookiee is? Let $a_n$ be the number of those permutation $\sigma $ on $\{1,2,...,n\}$  such that $\sigma $ is a product of exactly two disjoint cycles. Then find $a_4$ and $a_5$.   Calculating $a_4$: Possible cases which can happen are $(12)(34),(13)(24),(14)(23)$, any cycle of the form $(123)$ or $(12)$ i.e. two-cycles and three cycles thus we have in total $3+\frac{1}{3}4P_3+\frac{1}{2}4P_2=3+8+6=17$ but the correct answer is given to be either $11$ or $14$.  Where am I wrong? Please help. Log4Net: Programmatically specify multiple loggers (with multiple file appenders) How to (programmatically, without xml config) configure multiple loggers with Log4Net? I need them to write to different files. Changing permissions on serial port I'm using the Arduino IDE in Ubuntu, and am having issues with the serial port. It has worked in the past, but for reasons that may be unnecesary, I felt the need to change the ownership of some of the files from root ownership to my users ownership.  This made the IDE work correctly, but I lost the ability to use the correct serial port.  In the dev folder, the port I need is listed as permission 166. Someone (who is no longer in the area to help me) swapped the permissions to 666, which made it all work gloriously.  However, it reverted back as soon as I restarted my computer, and if I now try to use the command:  sudo chmod 666 ttyACM0   nothing happens. No error messages, but no permission change either.    How can I change it, and how can I get it to change permanently.  I apologize if this question is overly simplistic or unclear, I'm an ubuntu noob, and I wouldn't begrudge feedback! I am having trouble to set up my Epson printer on Ubuntu 16.04. How can I install the drivers? What are the basics needed to learn quantum computing? Why does one get an illusion as moon following him? When you run or ride bike at night if you observe the moon you feel like he moves along with you as the same speed you are going. Why? On a "1-D line", the point that minimizes the sum of the distances is the median, WHY? There is a set A which contains some discrete points (1-dimension), for example {1, 3, 37, 59}. And I want to pick one point from A which can minimize the sum of distances between this point and others.  For this "1-D line", many people say the median is the right point, but could anybody give me a concrete proof on it?