How can I swap the Esc and the Caps Lock keys?
How can I remap Caps lock to both Escape and Control?
I am having the following equation:  \begin{equation}   Q(\lambda,\hat{\lambda}) = -\frac{1}{2} P(O \mid \lambda ) \sum_s \sum_m \sum_t \gamma_m^{(s)} (t) \left( n \log(2 \pi ) + \log \left| C_m^{(s)} \right| + \left( \mathbf{o}_t - \hat{\mu}_m^{(s)} \right) ^T C_m^{(s)-1} \left(\mathbf{o}_t - \hat{\mu}_m^{(s)}\right) \right) \end{equation}   which does not very well fit on one line. How can I split this over two lines? What I have in mind is that I specify the splitting place, and that the first line is left aligned and the second line right aligned to make clear that it is still the same equation.  The linebreak \\ does not work. The square of an integer is congruent to 0 or 1 mod 4 This is a question from the free Harvard online abstract algebra .  I'm posting my solutions here to get some feedback on them.  For a fuller explanation, see  post.  This problem is from assignment 6.  The notes from this lecture can be found .     a) Prove that the square $a^2$ of an integer $a$ is congruent to 0 or 1 modulo 4.   b) What are the possible values of $a^2$ modulo 8?   a) Let $a$ be an integer. Then $a=4q+r, 0\leq r&lt;4$ with $\bar{a}=\bar{r}$.  Then we have $a^2=a\cdot a=(4q+r)^2=16q^2+8qr+r^2=4(4q^2+2qr)+r^2, 0\leq r^2&lt;4$ with $\bar{a^2}=\bar{r^2}$.  So then the possible values for $r$ with $r^2&lt;4$ are 0,1.  Then $\bar{a^2}=\bar{0}$ or $\bar{1}$.  b) Let $a$ be an integer. Then $a=8q+r, 0\leq r&lt;8$ with $\bar{a}=\bar{r}$.  Then we have $a^2=a\cdot a=(8q+r)^2=64q^2+16qr+r^2=8(8q^2+2qr)+r^2, 0\leq r^2&lt;8$ with $\bar{a^2}=\bar{r^2}$.  So then the possible values for $r$ with $r^2&lt;8$ are 0,1,and 2.  Then $\bar{a^2}=\bar{0}$, $\bar{1}$ or $\bar{4}$.  Again, I welcome any critique of my reasoning and/or my style as well as alternative solutions to the problem.  Thanks. Add a transparent image on top of a material How can I force a program to open in a specific desktop? I've created an environment where I use 3 different desktop configurations, but every time I reboot, I have to manually open each program in that target desktop to make it look like it did before.   So, I'd like to write a .bat file that can get all my programs running in their appropriate desktops. I'm wondering if something like this is possible:  "C:\Program Files (x86)\Mozilla Firefox\firefox.exe" -desktop:2 "C:\Program Files\My\Other\Software.exe" -desktop:3 ...etc.   Obviously, this example is not real, but I hope there's something similar. I don't care if it's hacky :) What does it mean to accept an answer? How do I accept an answer, and what are the rules? Can I accept my own self-answer? What are the reputation benefits? Which answer should I accept? Should I upvote the accepted answer? What should I do if none of the answers are suitable?      For more information, see "" in the . What is aperture, and how does it affect my photographs? Add a UIButton as a subview to a UITabBar Integral Representation of the Zeta Function: $\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$ How does one get from this $$\zeta(s)=\sum_{k=1}^{\infty}\frac1{k^s}$$ to the integral representation $$\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$ of the Riemann Zeta function?  I can see that it can be rewritten as  $$\Gamma(s)\zeta(s)=\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$ and the Gamma function as an integral yields  $$\zeta(s)\int_{0}^\infty \frac{x^{s-1}}{e^x}dx=\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$  But this approach does not work as the right integral does not converge.  So how does one go from the summation to the integral representation? How do you properly use namespaces in C++? Since many users are facing the NullReferenceException: Object reference not set to an instance of an object error in Unity, I thought that it would be a good idea to gather from multiple source some explanation and ways to fix this error.  Symptoms I am getting the error below appearing in my console, what does it mean and how do I fix it?  NullReferenceException: Object reference not set to an instance of an object Mach-O 64-bit format does not support 32-bit absolute addresses. NASM Accessing Array Using fflush(stdin) Shortcut for "null if object is null, or object.member if object is not null" Factorial of $i$ With $i^2=-1$, what is $i!$ ?  I'm not sure if it is $i$ or if it is an infinite product $i(i-1)(i-2)(i-3)(i-4)(i-5)\dots$  It would make sense either way.  Is it not defined? Is there a cross-browser onload event when clicking the back button? For all major browsers (except IE), the JavaScript onload event doesn’t fire when the page loads as a result of a back button operation — it only fires when the page is first loaded.  Can someone point me at some sample cross-browser code (Firefox, Opera, Safari, IE, …) that solves this problem? I’m familiar with Firefox’s pageshow event but unfortunately neither Opera nor Safari implement this. For one of the clients, I have spin up a modern communication site collection and on the landing page of the same, I am seeing 'Get the mobile app' showing up, I was able to hide the 'feedback' button using css but no luck with Get the mobile app button.  Can someone please share css to hide 'get mobile app' button, appreciate the help. How does Windows know whether it has internet access or if a Wi-Fi connection requires in-browser authentication? In Windows 7, the notification area networking icon will show an error indicator if there is no internet access , and the error icon goes away once there is a successful connection to the internet . Sometimes, if the WiFi connection requires an in-browser authentication step, like on many guest networks in hotels or universities, then the following pop-up bubble appears, saying as much:    How does Windows know whether or not it has a successful internet connection? Presumably it is checking some online Microsoft service to see whether it has a successful connection, gets redirected to some other page, or doesn't get any response at all, but I haven't seen anywhere that this process or the services used are documented. Can anybody explain how this works? I would prefer answers that refer to facts, rather than just guessing, but if you have a really good guess, then go for it.     This question was a .   Read the May 16th, 2011  for more details or  Question of the Week. Is the product of symmetric positive semidefinite matrices positive definite? I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices?  My proof of the positive definite case falls apart for the semidefinite case because of the possibility of division by zero... How can I ensure that old documents will compile in the future? Determine which circuits are parallel and which are in series I need to find the Equivalent Resistance across A and B. The problem is, I don't know which one's are in series and which ones are in parallel. How do I identify which ones are parallel or series?