How to remove bad smell from fridge and refrigerator?
How to get rid of the smell from the fridge?
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$. Exercise 11, page 45 from Hungerford's book Algebra.     If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every   subgroup of $H$ is normal in $G$.   I am trying to show that $a^{-1}Ka\subset K$, but I got stuck. What I am supposed to do now?  Thanks for your kindly help. Prove that $g$ is tangent to the graph of $f$ So I had these two functions and the following exercise:  $f(x)=x^3-2x^2$ $g_p(x)=px$ Prove that $g_{-1}$ is a tangent line to the graph of $f$. How many points do $g_{-1}$ and $f(x)$ have in common?  So yeah, the points in common are fairly simple to figure out:  Points in common: $x=1$ and $x=0$  So to see if it's actually tangent to the graph, I took the derivative of both functions at the points where they meet. The result:  $f'(x)=3x^2-4x$ $g'_{-1}(x)=-1$  So $f'(1)=-1$ and $f'(0)=0$ So I'm not sure why I get this answer? By the question, we are supposed to prove that its tangent to the graph of $f$ but by my calculation, it seems like only $f(1)$ is tangent with $g_{-1}(1)$. Help is appreciated! (Please avoid hint answers). How do I deal with a runaway bride? I had always kind of wondered why we didn't see interference in things like windows -- we were taught that the condition is that the thickness of the film/slab/medium just has to be an integral number of wavelengths, $d=m\lambda$. My logic was, if visible light has wavelengths ~400-780nm, any window pane has got to be an integral number of wavelengths (or close to integral) for some of the wavelengths in the visible.  For example, if a window is, $6mm = 6\times10^6nm$ thick, then you should see interference for wavelengths (and some range around them) 200nm, 250nm, 300nm, 400nm, 500nm, 600nm... Because 6mm is an integral number of each of those.  Obviously that would be really annoying and isn't the case. I found  (warning: PDF) source that seems to explain it and I was wondering if what they're saying is sound. On the 10th page (p171 for them), it says that the reason we don't see this interference is that no window is that smooth, it has a variance in the thickness of at least several wavelengths, so the interference effect is effectively average out among all those thicknesses, which usually gets rid of it. They do the math and show that.  Is that the reason?  If so, that leads me to two questions: Could we, if we were really careful and used something like MBE, make a glass window with that sort of accuracy, and then see interference effects in the optical range? I know that quartz crystal monitors can easily give you nanometer accuracy for evaporators and sputterers, but they also have pretty limited growth rates.  My second question is, then, do we see interference effects in glass windows with electromagnetic radiation of a comparable wavelength, like microwaves?  Thanks! What are the options for managing multiple Ubuntu installations together and keeping them in same state Recently I got in charge of managing 2 computer labs (currently 35 workstations, could double in a month or two) in university with total control over software stack as long as I install a bunch of commonly used scientific software (Mathematica and friends).  I plan to use Ubuntu for various reasons.   Installing Ubuntu and managing updates etc by hand is certainly an option, But I want to know if there is a more efficient way to do this stuff in and a more "Don't Repeat Yourself" manner?  Tasks   Ubuntu installation Installation (duplication) of same software stack and configuration over each workstation Keeping them updated Syncing folders outside Dropbox using Symbolic Links and Junctions I've been reading a lot of information on this and yet I'm still unsure if its possible.  What I want to achieve is: sync a folder that is outside Dropbox and any changed to that folder or the files within get mirrored to the Dropbox folder and the Dropbox server.  This is what I've tried: I've install Link Shell Extension. Picked a folder with 10 pictures inside and created a junction for it inside the Dropbox folder. When I delete a pictures from inside the original folder (outside Dropbox) the change is reflected in the Junction folder inside Dropbox Folder, not on the server.  The exact same behaviour with the Symbolic Link option.  If I manually select Pause Syncing from Dropbox system tray icon and then Resume syncing = it uploads the changes to the server.  I know I can move all my files inside Dropbox folder and create the junctions/symlinks at their original locations but I want to do it the other way around.  Also what is the difference between choosing the Junction and Symbolic Link Option in regards to a folder?  Thank you. What is a current replacement of gnome-open command (generic open of files based on the type)?  Before:   gnome-open mydoc.pdf  # opened PDF in default application   Now:  gnome-open The program 'gnome-open' is currently not installed.  You can install it by typing: sudo apt-get install libgnome2-0 If in a UFD every maximal ideal is principal then it is a PID How do high traffic sites service more than 65535 TCP connections? Optimization when Cost Function Slow to Evaluate Spider-Man is the one franchise that is on course to a third reboot in a span of 15 years.   My question is what was the reason for these numerous reboots of Spider-Man which pretty much follow the same origin story for the Marvel superhero.  Has the comic creator or studio ever spoken out about the reasoning behind the same? Extracting extension from filename in Python Fix terminal after displaying a binary file How to create a "heat distortion" effect with a cycles procedural node set up without compositing I know how to do a heat distortion effect in compositing.   What I am looking for is a way to do the same effect with a material set up in cycles.   What would be the best way to create a procedural material that can be placed in front of any object and make it look like heat is rising in front of it (without compositing)?  Any ideas would be greatly appreciated. limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$ I am trying to solve the following problem at the level of a senior undergrad analysis level. So, the problem is as follows: We are given a function $f$ which is continuous on the interval $\left [ 0,1 \right ]$, and the question is to find the limit: $$\lim_{n\rightarrow \infty}\int_{0}^{1}x^{n}f(x)dx\;.$$ The second part of the problem is to deduce the following limit: $$\lim_{n\rightarrow \infty}n\int_{0}^{1}x^{n}f(x)dx\;.$$  For the first part: I just did the following: For every $0\leq x&lt; 1$: $x\leq M$, where $0&lt; M&lt; 1$. Then: $$\int_{0}^{1}x^{n}f(x)dx\leq M^{n}\int_{0}^{1}f(x)dx\;.$$ Then: $$\lim_{n\rightarrow \infty }\int_{0}^{1}x^{n}f(x)dx\leq \lim_{n\rightarrow \infty }M^{n}\int_{0}^{1}f(x)dx= 0.\int_{0}^{1}f(x)dx=0\;,$$ so $$\lim_{n\rightarrow \infty }\int_{0}^{1}x^{n}f(x)dx=\lim_{n\rightarrow \infty }f(1)\int_{0}^{1}1dx=f(1)\;.$$ Does that make sense? If not, please show me the correct one.  As for the second part, I have no idea what to do. Any help? How do I evaluate $\sum_{k=0}^{n} {n \choose k}{m \choose k}$ for a given $n$ and $m$.  I have tried to use binomial expansion and combine factorials, but I have gotten nowhere. I don't really know how to start this problem.  The answer is ${n+m \choose n}$. Any help is greatly appreciated.  EDIT: I'm looking for  a proof of this identity. I'm heading to India next week and have a connection in Munich, Germany. I made the mistake of booking a flight with only an hour and a half layover (freaking out!) and I'm wondering if I'll have to go through customs in Munich or if I can pass right on through. How do I get a DOI for a dataset? A DOI is a commonly used digital object identifier to refer to an electronic document. How do I secure a DOI for a research dataset, to help share and identify it? I have a new laptop which I am setting up as dual boot Windows/Linux.    I would like to access files from both Ubuntu and Windows 7.  Should I create a separate NTFS partition? What is the possessive of "you guys"?