Can you integrate without a $dx$
What is $dx$ in integration?
Accessing a property of derived class from the base class in C# Smart Phone's camera socket I am just curious that the following format looks good to you or not? The preamble I used is just   \documentclass[12pt,a4paper]{article}  \usepackage{amsmath} \usepackage{amsthm} \usepackage{amssymb}   I don't feel it's so good because the first paragraph has no indent but the second does. If the first paragraph is much longer, than it's fine. But if not, it looks strange. Should I leave it, or you have suggestions for this? What are the main differences between VirtualBox networking types? I'm trying to use VirtualBox to host a couple of VMs.  Each VM should be able to connect to the Internet (this is needed primarily to install software) My Host browser should be able to connect to the Guest VMs.  Also I need to be able to ssh to the Guest VMs.     This is why I want each VM to have a static IP address.     This will also allow the VMs to connect to each other, say for database connectivity.  I have done this with VMware Fusion.  However, VMware has lately bitten me too often and destroyed so much work that I want to try VirtualBox.  But I'm not a network guru and don't know which type of networking I want.  So what I'm asking is simple:  What are the main differences between networking types. This could help me find out which type of networking I should use in my case.  And of course if anyone has a link to any documentation of how to do what I want, that would be greatly appreciated.  Thanks, ge Why does my hard drive LED light blink every second? The light blinks faintly about once a second.   It is slowly driving me insane.  How do I find out which program is causing this, so I can disable it?  Acer 5274 laptop/core i3 Windows 7 Home Premium Hitachi HTS545050B9A300 hard drive C# and USB HID Devices In Dummit and Foote, Exercise 15.18 (p. 727 3rd edition) one is asked to prove that for a commutative ring with unity $R$, and $f$ a non-nilpotent element of $R$, that the localization $R_f$ is isomorphic to $R[x]/(xf - 1)$.  I have done this, using the strategy outlined in the problem, using the universal property of $R_f$ proved in Theorem 36. The problem I have is this:  If $R_f \cong R[x]/(xf - 1)$ and $\phi:R[x] \to R_f$ is a surjective homomorphism, it stands to reason that $\text{ker}(\phi)$ should be $(xf - 1)$. I cannot show this directly. The surjective homomorphism I have in mind is, for:  $p(x) = a_0 + a_1x + \cdots + a_mx^m$  $\phi(p) = \dfrac{a_0}{1} + \dfrac{a_1}{f} + \cdots + \dfrac{a_m}{f^m}$  It is clear to me that $\phi$ annihilates $(xf - 1)$, so that ideal is contained in the kernel. What I cannot show is the other containment, or equivalently:  If $\phi(g) = 0_{R_f}$ then $g(x) \in (xf - 1)$. The trouble I am running into is that $R$ might be a "bad" ring, containing a lot of zero-divisors, $f$ might even be a zero-divisor (being non-nilpotent is a considerably weaker property), so the usual polynomial tricks of the division algorithm or using the irreducibility of $xf - 1$ aren't much use, here.  I feel I am missing something obvious....what is it? Movie Identification - 1990s VHS Sci-Fi/Horror Prove p-adic valuation is completely additive for prime p Does $\sum\limits_{k=1}^n 1 / k ^ 2$ converge when $n\rightarrow\infty$? I can prove this sum has a constant upper bound like this: $$\sum_{k=1}^n \frac1{k ^ 2} \lt 1 + \sum_{k=2}^n \frac 1  {k (k - 1)} = 2 - \frac 1 n \lt 2$$ And computer calculation shows that sum seems to converge to 1.6449.  But I still want to know:  Does this sum converge? Is there a name of this sum (or the series $\frac{1}{k^2} $)? How do I tell if ACR will work with my camera and how do I get the correct version for Photoshop? I am a beginner photographer with a Canon 7d and MacBook Pro. I'm not sure if Adobe camera Raw is compatible with either and really need advice on how to go about downloading the correct version. Why is the sky dark at night? The question is the well-known      If there are so many/infinite stars, in every direction, why is it dark at night? Where goes all the light we don't see?    Discussing with a friend we found different answers.   The first (my friend's answer) is that given the fact that in a finite amount of time (in this case, the age of the universe), light can travel only a certain, finite, distance, we cannot see, if they actually exist, stars farther than such distance (about 13 billions of years, I guess). Then he states (or better, I got him notice he implicitly does this)  that within this radius there aren't enough stars to fill the night sky, and that's why we have darkness without sun's light.   The second answer (mine, or anyway the one I heard from a minutephysics YouTube video) is that despite of the big size of the observable universe, cosmological expansion red shifts most of the light out of the visible radiation range, so that the majority of light can't be seen by our eyes at night, while if we use, for example, an infrared camera, we should see a brighter sky, and lenghtening more and more the wavelength at which we observe the sky, we come to the uniform background radiation.  So my question is, what is the right answer? Is it one of the above-mentioned or even a third one?  EDIT: I've seen both answer may be acceptable, but what is the main contributing phenomenon? Way to list "Contact Us" email address on web site, yet reduce likelihood of spam? What's a good way to list a "Contact Us" email address on a web site, while reducing the likelihood it will get spammed?  Is putting the email address in an image the best technique, or are there others? Yesterday, my laptop caught a virus which caused massive damage. Since them, I have been trying to recover important files before reformatting my computer, a task the virus has not made easy.   Restoration points predating the attack have been deleted.  Most of my folders show empty.  My Start menu is essentially empty, with the exception of Trillian and Mirror's Edge. The same goes for my Desktop, which only has programs which were installed after the attack.  Searching for files through my computer is pretty much useless, as it only rarely brings up anything. I suspect most of my files have not been deleted. While my folders show empty, uTorrent still does display them and I can open them from here. Unfortunately, when I select Open Containing Folder, the folder still shows as completely empty even if I'm currently watching a video from that very folder. Further adding evidence to the not-deleted-just-missing theory, the data recovery software I'm using (Restoration) cannot find only find an handful of the missing files.   If they were deleted, I could do a forensic recovery to get them back but since they're probably still somewhere on my computer, just out out of my reach, I can't find them.   Under those circumstances, is there a way I can recover those files? Let $ G $ be an abelian group of order 6.  Then $ G $ has one element of order 6.  And so, $ G $ is cyclic and isomorphic to $ \mathbb Z _6 $.  In general, It is not true that an abelian group of order $n$ has an element of order $n$.  However in the case of $n=6$, it is true. I'm wondering there is any rule of the case (for example, a rule of $n$ that makes an abelian group cyclic). Which "mean" to use and when? So we have arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM). Their mathematical formulation is also well known along with their associated stereotypical examples (e.g., Harmonic mean and it's application to 'speed' related problems).  However, a question that has always intrigued me is "how do I decide which mean is the most appropriate to use in a given context?" There must be at least some rule of thumb to help understand the applicability and yet the most common answer I've come across is: "It depends" (but on what?).  This may seem to be a rather trivial question but even high-school texts failed to explain this -- they only provide mathematical definitions!  I prefer an English explanation over a mathematical one -- simple test would be "would your mom/child understand it?" Do photons have acceleration? Selecting features with NULL attributes in QGIS How do I identify which features have NULL attributes in the attribute table of a shapefile?  When i try [field]= 'NULL' it selects the features with value = 0 (see picture) $$f(x,y) = \left\{\begin{array}{cc} \frac{xy}{x^2+y^2} &amp; (x,y)\neq(0,0) \\ f(x,y) = 0 &amp; (x,y)=(0,0) \end{array}\right.$$  In order to verify if this function is differentiable, I tried to prove it by the theorem that says that if $\frac{∂f}{∂x}$ and $\frac{∂f}{∂y}$ exist and are continuous at the point $(x_0,y_0)$, then the function is differentiable at this point. So I did:  $$\frac{\partial f}{\partial x}(0,0) = \lim_{h\to 0}\frac{f(0+h,0)-f(0,0)}{h} = 0$$ $$\frac{\partial f}{\partial y}(0,0) = \lim_{h\to 0}\frac{f(0,0+h)-f(0,0)}{h} = 0$$  so we have that the partial derivatives at point $(0,0)$ is $0$. Now, if we take the derivative at $(x,y)\neq (0,0)$ and then take the limit of it as $(x,y)\to(0,0)$, we can see if the derivatives are continuous or not. So here it is:  $$\frac{\partial f}{\partial x}(x,y) = \frac{y(y^2-x^2)}{(x^2+y^2)}$$  but   $$\lim_{(x,y)\to(0,0)} \frac{y(y^2-x^2)}{(x^2+y^2)} $$ does not exist (by wolfram alpha... but can anybody tell me an easy way to prove this limit does not exist? easier than taking the limit in different directions?), therefore the derivative is not continuous at $(0,0)$, so we can't say $f$ is differentiable at $(0,0)$, but for $(x,y)\neq (0,0)$ the function is continuous, as it is a quotient of continuous functions. So $f$ is at least differentiable at $(x,y)\neq (0,0)$.   Now, to verify differentiability at $(0,0)$ I think we must use the limit definition of differentiablity: A function is differentiable at $(0,0)$ iff: $$\lim_{(h,k)\to (0,0)} \frac{f(0+h,0+k)-f(0,0)-\frac{\partial f}{\partial x}(0,0)-\frac{\partial f}{\partial y}(0,0)}{\|(h,k)\|} = 0$$  Let's calculate this limit:  $$\lim_{(h,k)\to (0,0)} \frac{f(0+h,0+k)-f(0,0)-\frac{\partial f}{\partial x}(0,0)-\frac{\partial f}{\partial y}(0,0)}{\|(h,k)\|} = \\ \lim_{(h,k)\to (0,0)} \frac{\frac{hk}{h^2+k^2}}{\sqrt{h^2+k^2}} = \\ \lim_{(h,k)\to (0,0)} \frac{hk}{(h^2+k^2)\sqrt{h^2+k^2}}$$  which I think, it's a limit that does not exist, therefore the function isn't differentiable at $(0,0)$ Finding specific records of US WW2 veterans?