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Relationship between SVD and PCA. How to use SVD to perform PCA? List of CSS features not supported by IE6 How to see data from .RData file? How to do integer & float calculations, in bash or other languages/frameworks? Using echo "20+5" literally produces the text "20+5".  What command can I use to get the numeric sum, 25 in this case?  Also, what's the easiest way to do it just using bash for floating point? For example, echo $((3224/3807.0)) prints 0 :(.  I am looking for answers using either the basic command shell ('command line') itself or through using languages that are available from the command line. This is a very simple question but I believe it's nontrivial.  I would like to know if the following is true:      If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ are isomorphic.    Thanks!  If there isn't a proof (or disproof) of the general result, I would be interested to know if there are particular cases when this claim is true. pwd without symlinks In the series finale, Admiral Janeway travels back to 2378 to take Voyager back to Earth. Out-of-universe, of course, this point in time was chosen to fit in with the continuity of the rest of the series. In-universe, however, it seems that Janeway could have chosen any point in time, since she had the ability to travel to and from the Delta Quadrant at will. Is there a reason given, in-universe, for why she chose that particular incident to travel back to? Express the complex numbers given in Problem in the form $a+ib$ WiFi problems (Atheros driver) with FreeBSD 11 With FreeBSD 10.3 my Atheros card worked fine and showed up as /dev/ath0. Now with FreeBSD 11 this is no longer the case, though the module is loaded and it seems the system recognizes the card:  root@igloo:~ # sysctl net.wlan.devices net.wlan.devices: ath0   Still, ath0 cannot be used:  root@igloo:~ # ifconfig wlan0 create wlandevice ath0 ifconfig: must specify a parent device (wlandev) when creating   What must I do to use the card? Configure mouse speed (not pointer acceleration!) Is there way to increase the mouse speed in KDE4? I don't want any pointer acceleration, it's just the mouse speed that I want to change.  Edit: Unfortunately, editing the xorg.conf is not an option for me, because I want the users to be able to configure the mouse speed themselves and it is company policy do deny users permission to change the xorg.conf. What is a single word for a person who, as a habit, exaggerates a lot when relating information? What is a single word for a person who, as a habit, exaggerates a lot when relating information?   So for example, they relate something that happened but exaggerate so much that they are including clear falsehood and they are well known for doing that. Cooking away alcohol When you cook with wine or spirits, when does the alcohol cook away? Obviously high temperatures will do it, but how low of temperatures will work? Also, does it vary by the type of alcohol? When can a sum and integral be interchanged? Let's say I have $\int_{0}^{\infty}\sum_{n = 0}^{\infty} f_{n}(x)\, dx$ with $f_{n}(x)$ being continuous functions. When can we interchange the integral and summation? Is $f_{n}(x) \geq 0$ for all $x$ and for all $n$ sufficient? How about when $\sum f_{n}(x)$ converges absolutely? If so why? The more I read about this, the more confused I become. Illuminance, as I understand it,  measured in lux, is the perceived brightness of light on a surface. It's what you'd measure with an incident light sensor. Luminance, then, is how much reflected light you'd perceive looking at that surface. This is what the light meter in a camera measures. Am I correct so far?  In both cases, the word "perceived" is important, because the scales are weighted using a  to map wavelengths to . However, in the case of illuminance, there's only actual perception if your eye happens to be the surface in question.  I can basically grasp this, but then I come across charts saying things like "family living room: 50 lux". Wait, hold on! Does that really mean that the typical lights in a house are that bright, or is it just confused and wrong, or am I confused and wrong?  If you're not taking pictures of light sources directly, why would an incident light meter reading ever be useful in photography? Reflected light recorded on film or by a sensor is usually what makes a photograph. So, if I have an incident light meter, how does that reading relate meaningfully to my camera settings?  Since incident light meters are sold and used, this implies that there must be some useful conversion. But this is where my brain explodes. Google tells me that 1 lumen equals 1 candela, so therefore 1 lumen/m² (i.e. 1 lux) must equal 1 candela/m2 (i.e. 1 nit). But clearly something is missing from this. There's something called a "steradian". Cones are involved. I'd never heard of this before; how does it fit in? I can see how it might be determined when calculating the usefulness of LED home lighting, but for a photo, I'm at a loss.  Some part of my (exploded, from the last paragraph) brain is trying to relate this to the difference between flash metering with TTL and with an incident meter. But how can the incident meter work without knowing the reflectance properties of the objects in the scene? Is this what the "C" in the  is all about? Is it just an average-this-probably-will-work value, or is there more to it than that? And if it is just an average, what knowledge is necessary for compensating for off-averages scenes? (As with the , where the photographer simply judges if the scene should be rendered brighter or darker than the average given by the meter.)  So yeah, so much confusion. In short:   What's the difference? Can one meaningfully convert between the two? When and how are illuminance / incident light measurements useful for photography?     Update: I appreciate Stan's answer, which covers the third point of when and how pretty well. And I think I've basically got the first point figured out, as described above. But I'd appreciate some answers covering the issue of conversion as well, both in the mathematical abstract and as the practical for photography. And I wouldn't mind more why and how, either. Understanding the proof for showing that there are infinitely many $n\in\mathbb{Z}^+$ such that $n!+1$ is divisible by at least two distinct primes There is another post pertaining to this question; however, the other post does not address some specifics that concern me in the proof. Proof: Let $n=p-1, p \geq 7$. Then $p|(p-1)!+1$. We will show that $(p-1)!+1=p^n$ has no solutions. Suppose not. Suppose $(p-1)! + 1 = p^n$ has a solution for $p\geq 7$. Then $(p-1)! =p^n-1  = (p-1)(p^{n-1}+\cdots+p+1)$. So $(p-2)!\equiv n (\mod p-1)$. Now suppose that $(p-1)|n$. Then $p-1\leq n$. So $(p-1)!+1&lt;p^{p-1}\leq p^n$. Hence, there are no solutions. It remains to show that $(p-1)|(p-2)!$. But $p-1=2m$ means $2m|(p-2)!$. QED  Questions:  My understanding is that at the start of the proof, Wilson's theorem is used to show that at least one prime divides $n!+1$.  Why do we want to show that $(p-1)!+1=p^n$ has no solutions? The way I see it is that $p^n$ is composite, and we are basically saying that $p^n\nmid (p-1)!+1$, so no composite number divides $n! +1$. But how and where in the proof do we actually show that another prime $q \neq p$ divides $n!+1$?    Thanks in advance. I know that any subset of V with more than k vectors is always linearly dependent, but how do I include matrix A into the vectors given to prove the statement?   Also, for part (b), does the statement mean that Ax=0 only has the trivial solution since V intersects span{u1, u2, ..., uk}={0}?  Any help would be appreciated! Thank you! UBUNTU 14.04 cheese booth not detect webcam Cheese Booth shows an error:     There was an error playing video from the webcam     I test if webcam works fine or not by opening google hangout, its works fine.    Please tell, what is the issue in cheese booth.  Note: Sometimes the cheese booth works fine but most of time not If $x_1=5$, $x_{n+1}=x_n^2-2$, find $\lim x_{n+1}/(x_1\cdots x_n)$ How do I write an Apex unit test? This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, including more unit test resources, navigate to the  tag.  This canonical question is intended to address several classes of common questions by providing a quick summary and links to comprehensive resources:   How do I start writing my first unit test?  How do I unit test this code?  I need help writing this unit test.   Salesforce Stack Exchange looks for detailed, specific questions that the community can help you with, and can't write tests on your behalf. We feel that working with the resources below can help you get started, and we encourage you to make an attempt to write your test and return to SFSE with your specific questions when you encounter challenges you can't resolve. I am running Ubuntu 14.04 and Notepad++ 6.6.4 (through WINE)  Under the 'Run' tab, I clicked on "Launch in Firefox" , but nothing happened   I uninstalled Notepad++ and WINE, then shutdown. I re-installed Notepad++ and WINE and still have the same problem.   Does anyone know what the problem is?