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How can I quickly know the rank of this / any other matrix? Prove that $\frac{1}{\sin^2 z } = \sum\limits_{n= -\infty} ^ {+\infty} \frac{1}{(z-\pi n)^2} $ Is there a formal proof for the superposition theorem? How does dark matter halo outside a galaxy help to explain galaxy rotation curve? How does a   outside a galaxy help to explain a ?  Suppose for simplicity we use a model of a star rotating about a more massive star in a fixed circular orbit. For another star at larger radius, the velocity should be smaller but observationally for spiral galaxies, the opposite happens (moves at even more speed or at least at the same speed as the closer star).  People hypothesize that dark matter halo may be present “surrounding” the system (i.e., halo exists outside the system) to account for such observation. I feel that exactly the opposite should be the case that is, dark matter halo should be inside the galaxy rather than outside. Reason being, if dark matter is outside the galaxy then the effective gravitational pull on the rotating star towards the center will be reduced due to outward attraction on the star exerted by dark matter existing outside. Thus, its speed of rotation should be reduced; but we observe the opposite. So how does dark matter halo "surrounding" a galaxy help to explain galaxy rotation curve? Where am I making mistake in argument? Showing that $G/(H\cap K)\cong (G/H)\times (G/K)$ How do I add a border around windows but keep the current theme? I seem to live my life in the terminal - many terminals in-fact.  I also love the Ambiance theme - but the new borderless windows does not provide the visual differentiation I'm used to (the same black windows appear to blend into each other).    Thus to my question - how do I add back the single pixel border for windows that is found in earlier Ubuntu versions?  I want to keep the Ambiance overall theme since this looks good and I dont want to swap to another theme.      Now I know that I can modify the Ubuntu Ambiance theme as follows:  mkdir ~/.themes cp -r /usr/share/themes/Ambiance .themes   Then flipping between the theme options in the Settings - Appearance picks up changes in the theme in now in the .themes folder.  I presume I need to alter one of the .css files but I dont know which one.  I was also looking at  but the solutions there doesnt work for 14.04.  I also looked at the  but got rather lost trying to figure out which one applies to my situation. Hazard Function - Survival Analysis I just started taking survival analysis class and I'm stumped on this question.  Let $T_1,...,T_n$ random independent continuous variables, with hazard function of $h_1(t),...,h_n(t)$. $T=min(T_1,...,T_n)$. And we need to show that the hazard function of T is $\sum_j{h_j(t)} $  Any help or direction are welcome :) What is the speed of sound in space? How does Cantor's diagonal argument work? Global setting of spacing between items in itemize environment for beamer Prevent Mac Terminal from reverting to default theme I can't find any way to change the Terminal theme and have that theme applied permanently.  I want to change to another darker theme other than the default 'basic' theme, and (most importantly) have it remain as the theme every time I open Terminal. How to use "`" in the same way it is used on the SE websites? Why does this stream return no element? Finding Zero Divisors and Units of Rings $\mathbb{Z}_n \times \mathbb{Z}_m$ How do I exactly find the zero divisors and units of a ring in the form of $\mathbb{Z}_n \times \mathbb{Z}_m$? For instance, how does one calculate zero divisors and units of $\mathbb{Z}_6 \times \mathbb{Z}_2$? I understand that a zero divisor is a non-zero element of $R$ such that $rs = 0$ for some other non-zero element $s$ in $R$. But I don't really quite understand how to apply that to a ring of the form $\mathbb{Z}_n \times \mathbb{Z}_m$. Also, the units of $\mathbb{Z}_6 \times \mathbb{Z}_2$ would be elements that send some element $s$ in $\mathbb{Z}_n \times \mathbb{Z}_m$ to the multiplicative identity, right? (which would be 1)? How do I apply these concepts to "product" rings? Thank you. How to make git mark a deleted and a new file as a file move? java swing background image Dictionary application needs to download dictionary "Apple Dictionary" before use I'm repeatedly being asked to download the "Apple Dictionary" when I open the dictionary application.     Dictionary application needs to download dictionary "Apple Dictionary" before use.   This doesn't happen every time I open it. I haven't pinned down specific usage cases where this message appears, but I think it may be every time after I reboot my computer.  Is there anyway for the Dictionary application to remember that I have "Apple Dictionary" permanently? This has only started happening after I upgraded from OS X Mountain Lion to macOS Sierra. Why was Snape called the Half-Blood Prince? My question comes from what I learned in the movies, considering I have not yet been able to read the books. I would just like to know why Severus Snape called himself the Half-Blood Prince? Did many people call him that or did he just think that it was a cool nickname? This question might be answered in the books so feel free to quote from it. Lets say I have something like:   \[ Let x = number of cats \]   (This is a contrived example, but pretend we are actually inside a long align environment block...) "Let" and "number of cats" need to be written like normal text, while x= needs to be math type text.  I've seen this:  \[ \mbox{Let } x = \mbox{ number of cats} \]   and this:  \[ \textrm{Let } x = \textrm{ number of cats} \]   They seem to produce equivalent output, but I'm not sure which is the "correct" one. Also, they are not ideal in that the extra space needs to be placed inside the text mode block, otherwise it is ignored as math mode and butts the text right up against the math.  Ideally there would be some command which allowed embedding text into an equation, which correctly inserted leading and trailing space around the block as required without adding space inside the text block.  How does one do that "correctly"? It seems that through various related questions here, there is consensus that the "95%" part of what we call a "95% confidence interval" refers to the fact that if we were to exactly replicate our sampling and CI-computation procedures  many times, 95% of thusly computed CIs would contain the population mean. It also seems to be the consensus that this definition does not permit one to conclude from a single 95%CI that there is a 95% chance that the mean falls somewhere within the CI. However, I don't understand how the former doesn't imply the latter insofar as, having imagined many CIs 95% of which contain the population mean, shouldn't our uncertainty (with regards to whether our actually-computed CI contains the population mean or not) force us to use the base-rate of the imagined cases (95%) as our estimate of the probability that our actual case contains the CI?   I've seen posts argue along the lines of "the actually-computed CI either contains the population mean or it doesn't, so its probability is either 1 or 0", but this seems to imply a strange definition of probability that is dependent on unknown states (i.e. a friend flips fair coin, hides the result, and I am disallowed from saying there is a 50% chance that it's heads).  Surely I'm wrong, but I don't see where my logic has gone awry...