How does camera capture infrared rays?
Why is TV remote's infrared wave shown by a camera as purple light, why not red or something else?
Necessary and Sufficient conditions for $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$. I have a homework question that asks     Find necessary and sufficient conditions on $1 \leq i &lt; j \leq n$ so that $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$.   Here is what I have done so far.  Call $c = (1 \, 2 \, \dotsc \, n)$.  I made the observation that $$ \overbrace{c c \dotsb c}^{n - j + 1} (i \, j) \overbrace{c^{-1} c^{-1} \dotsb c^{-1}}^{n - j + 1} = (1 + i - j + n \, 1). $$ Thus, $\langle (i \, j), c \rangle = \langle (1 \, i - j + n + 1), c \rangle$, so it suffices to look at $\langle (1 \, k), c \rangle$ for positive integers $2 \leq k \leq n$.  I've checked that, for the value $k=2$, $\langle (1 \, 2), c \rangle = S_n$.  This immediately implies $\langle (1 \, n), c \rangle = S_n$.  I suspect that no other values of $k$ will work, but am not sure how to prove it. I have a range and need to sum it using SUM function but the result is N/A if there is any N/A value. How can I make the SUM function to treat the N/A value as 0 value?  Please help! Socket Tester all indicators light up What is the command to find out how much disk space is being used/remaining? Why doesn't Quicksilver create sonic booms? Under what conditions is the exponential map on a Lie algebra injective? Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\exp :\mathfrak{g}\rightarrow G$ be the exponential map.  In his blog,  that if a Lie group is not simply-connected, then $\exp$ will not be injective.  Conversely, is it true that if a Lie group is simply-connected, then $\exp$ is injective?  If not, what is a counter-example? I am lost.   So far...  If $\gcd (a,b) = 1$, by Bezout's Formula  $ax + by = 1$  If $c|(a+b)$, then $cf = a+b$  Then,  $a (x-y) + cfy = 1$   $b(yx) + cfx = 1$  Am I on the right track? Any suggestions? URL to create a new Google document What is the current URL to create a new Google Document please? Since Google last updated their Docs the old URL does not work. Is there a formula for $\sin(xy)$ Can we express a trigonometric function for the product of two angles as a function of trigonometric functions of its factors?  For example: Is there a formula for $\sin(xy)$ as a function of $\sin x$ and $\sin y$ or other trigonometric functions of $x$ and $y$. Issues with win key and right alt key after Windows Update (Polish diacritical signs not working) BJT in Reverse Active Mode of Operation What will happen if for a BJT transistor, it's emitter terminal is treated as collector and collector as emitter in a common emitter amplifier circuit? The rank of Jacobian matrix at a point of affine variety is independent of choice of generators What does NOEXEC flag mean when mounting directories on RHEL? I am trying to understand the NOEXEC flag when mounting.  I am having an execution issue within the /tmp directory on someone elses machine that I cannot access atm where the /tmp directory is mounted onto a different drive than '/' and NOEXEC is present.  I wanted to try and recreate this scenario on my machine, but I do not have a second hard drive.  I tried doing the following command:  mount --bind /test1 /test2   I then removed the bind flag and added NOEXEC in /etc/fstab.  Then, I created a file in /test2 called test.sh where it just echos 'hello world'.  I try and run it and it said 'permission denied'.  I then ran chmod 777 test.sh and was able to execute the file just fine.  I thought that the NOEXEC flag should not allow me to execute anything?  Is mount --bind /test1 /test2 not the same as mounting from a completely different physical drive? As in /test1 and /test2 are on different drives? I see a new logo with CA at youtube.com. Does anyone know what it means? Past Subjunctive vs Future Subjunctive form Can any one help me with this?  Determine all integer triples $(x,y,z)$ such that $1 ≤ x ≤ y ≤ z$ and $x + y + z + xy + yz + xz = xyz − 1$.  I thought of Vieta's formula but don't let me lead you into a dead end. Update: Failed to lock /var/cache/apt/archives/lock I am running the upgrade(12.10 to 13.04) through terminal and it keeps getting stuck at this:  Failed to lock /var/cache/apt/archives/lock   What do I do? I have root access. I have no idea what else to try other than a clean install.  Checking package manager Reading package lists... Done     Building dependency tree           Reading state information... Done Building data structures... Done   Calculating the changes Calculating the changes  Do you want to start the upgrade?   3 packages are going to be removed. 155 new packages are going to be  installed. 1204 packages are going to be upgraded.   You have to download a total of 595 M. This download will take about  1 hour 15 minutes with a 1Mbit DSL connection and about 23 hours with  a 56k modem.   Fetching and installing the upgrade can take several hours. Once the  download has finished, the process cannot be canceled.    Continue [yN]  Details [d]y  Fetching  Could not download the upgrades   The upgrade has aborted. Please check your Internet connection or  installation media and try again. All files downloaded so far have  been kept.   Failed to lock /var/cache/apt/archives/lock    Restoring original system state  Aborting Reading package lists... Done     Building dependency tree           Reading state information... Done Building data structures... Done  root@chaitanya-XPS-Ubuntu:~# Apt "could not find a distribution template" error When I attempt to install anything using apt-get I get the following error:  Traceback (most recent call last):   File "/usr/bin/add-apt-repository", line 60, in &lt;module&gt;     sp = SoftwareProperties()      File "/usr/lib/python2.6/dist-packages/softwareproperties/SoftwareProperties.py", line 90, in __init__     self.reload_sourceslist()   File "/usr/lib/python2.6/dist-packages/softwareproperties/SoftwareProperties.py", line 538, in reload_sourceslist     self.distro.get_sources(self.sourceslist)       File "/usr/lib/python2.6/dist-packages/aptsources/distro.py", line 90, in get_sources     raise NoDistroTemplateException("Error: could not find a " aptsources.distro.NoDistroTemplateException: Error: could not find a distribution template   Any idea what this means and how to resolve it? Decode escaped Url without using HttpUtility.UrlDecode Is there any function that converts an escaped Url string to its unescaped form? System.Web.HttpUtility.UrlDecode() can do that job but I don't want to add a reference to System.Web.dll. Since my app is not a web application, I don't want to add a dependency for only using a function in an assembly.  UPDATE: Check  about the same issue. Why do some materials reflect (metals) and other materials reflect and refract (glass) from the quantum perspective? Recently I was asked to explain the difference between reflection and total internal reflection from a purely conceptual standpoint (no math).    Let me explain what I already know.  Reflection and refraction at the quantum level are the same thing.  Light is a photon.  A photon is a discrete particle that has wave characteristics (a finite wave traveling like a bullet).  As the photon travels it collides with electrons in the matter of the medium it is traveling in.  Depending on the energy of the photon and the allowed energy bands of the medium the photon cause the electron to jump up a level.  If the photon is absorbed then the medium will increase its motion (at the macro scale increasing its temperature).  If the photon is not absorbed it will be re-emitted (really as a new photon). I have read and watched Feynman's QED lectures and book and have a pretty good understanding of his process for determining how all these paths come together to give the net path of the photon.  The general rule of thumb is that the photon wants to take the path which requires the least amount of time.  I understand how this principle goes to explain refraction and reflection.    What I don't seem to understand is why does one material seem to cause a higher percentage of refraction compared to another (metal vs. glass).  What about the electron configuration of a the medium changes the net effect of the absorption and re-emissions of the photons?  Is there a change in the probability of the photon being re-emitted in a reverse direction?  Is there a farther distance the photon can travel before being incident onto an electron?  This is the part where my understand breaks down.    When you have hit the critical angle in a medium that refracts and the light completely reflects, are the photons moving is the same manner as they would be in a material that always reflects? How does this connect to the question in the previous paragraph?  I know I have a bunch of mini questions embedded in answering this one larger question.  Any help on any of the parts would be greatly appreciated?