Continuous function and induced homomorphism
3-manifold examples of homomorphisms between fundamental groups that are not induced by continuous maps
I have 41 GB of space on my E:/ drive. Is it sufficient to install Ubuntu? My Windows XP is installed in C:/ drive. Is it OK for dual boot? Should I install Ubuntu (I am installing using Live CD). I don't want to lose my Windows option. What is the probability of a biased coin coming up heads given that a liar is claiming that the coin came up heads? A biased coin is tossed.   Probability of Head - $\frac{1}{8}$  Probability of Tail - $\frac{7}{8}$  A liar watches the coin toss. Probability of his lying is $\frac{3}{4}$ and telling the truth is $\frac{1}{4}$. He says that that the outcome is Head. What is the probability that the coin has truly turned Head?  My Attempt:  I used the formula: $$P(A \mid B) = \frac{P(A\cap B)}{P(B)}$$  $\ \ \ \ \ \ $P(it is head GIVEN liar said it's head) = P(it's head AND liar said it's head) / P(liar said it's head)  or, P(it is head GIVEN liar said it's head) = $\frac{ \frac{1}{8} \frac{1}{4} }{ \frac{7}{8}\frac{3}{4} + \frac{1}{8}\frac{1}{4} }$ [using a probability tree will be helpful here]  or, P(it is head GIVEN liar said it's head) = $\frac{1}{22}$  The Question: Is the method I used wrong in any way? Some others I have talked to are saying the answer will be $\frac{1}{4}$. Their reasoning is this: since the liar lies 3 times out of 4 and he said it is head, then the probability of it being head is 1/4. So who is right? What will be the answer? so I stumbled upon this equation/formula, and I have no idea how to prove it. I don't know how should I approach it: $$ \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{\vphantom{\large A}2\,}\,}\,}\,}\ =\ 2\cos\left(\vphantom{\Large A}\pi \over 2^{n + 1}\right) $$  where $n\in\mathbb N$ and the square root sign appears $n$-times.  I thought about using sequences and limits, to express the LHS as a recurrence relation but I didn't get anywhere.  edit: Solved, thanks for your answers and comments. OS X lets me type special characters directly on the keyboard by holding the Option. For example: — is an em dash, “ and ” are curvy quotes, etc.  Is there a good cheat sheet listing all these keys somewhere? I tend to find myself holding option and running my hands up and down the keyboard looking for a specific character. Evaluate the integral $\int_{-1}^1\frac{dx}{(e^x+1)(x^2+1)}$. Evaluate:      $$\int_{-1}^1\frac{dx}{(e^x+1)(x^2+1)}$$ Google Chrome: Version 32.0.1700.107 m  This happens when displaying the profile and clicking on the number of visited days/consecutive.  . I just started blender and one of the main things you have to do is to render to make an animation, but I can't seem to do that. Whenever I click Render and then render image or render animation, at the top of the screen it says 'No camera found in Scene "Scene". I don't know why this is happening. Is "shined" correct? If so, is "he shined X on the tree" also correct? Recently, I overhead a former professor of mine use the word shined, a word that makes me grammatically uncomfortable.  She used it as following: "Then, after we shined a light on the other ball, what happened?..."  My first question is: Is using shined mainstream, over what sounds better to my ear, shone?  If so, is there any historical flip-flopping between the two usages?  Secondly, if shined is in fact legit, can it be used transitively?  Saying something like,      The light shined all throughout the night.   while odd, doesn't sound as bad to my ear as,     He shined the light on the ball throughout the night.   The latter sounds quite bad to my ear.  Indeed, à propos the original question, can shone &mdash; the better sounding alternative to my ear &mdash; be used transitively? Benefits & Practicality of Virtual Machines I have been living under a rock for the past 5 - 8 years, and am just now getting up to speed with virtualization, virtual machines and platforms like VMware or VirtualBox.  I think I am starting to get them, but there are two concepts about VMs that I'm just choking on for some reason; one has to do with their intrinsic benefit to the enterprise, and the other has to do with their practicality when being used.  So I understand that you can have multiple VMs running on the same physical machine, all doing their own "thing" as if they were separate machines. So let's say I want to make a nice little virtual network of servers in my home office so I can have a big sandbox to play inside of for all my projects. I set up 2 physical machines as servers, and on each of them I have, say, 4 virtual machines running (1 might be an app server, another might be a RDBMS, another might be a message broker, etc.).   Besides saving me money (as I only have to provide 2 physical machines instead of buying 8 servers), what instrinsic benefit/purpose do these VMs serve here? I don't care how clever virtual machines are, if my computer only has 8GB of RAM, and I have 4 VMs running on it, those VMs only get 8GB of RAM to share between them. Putting 4 VMs on my server doesn't endow it with 4 times that amount of RAM! So, as far as I see it, VMs save you money on hardware, but just end up bogging down that hardware as you have multiple machines competing over the same pool of resources. What am I missing here? Can VMs communicate with each other over your network using high-level protocols like TCP/IP, HTTP, FTP, etc.? Otherwise, it just doesn't make sense to implement all these VMs if you have to feed them separate and independent chunks of data/requests to process. Thunderbird - remove unused dictionaries from menu If $X,Y$ are equivalence relations, so is $X \times Y$ How do I do a search in File Explorer that includes subfolders of zip archives? In File Explorer, I can enter search terms in the upper-right search box to search the current folder and all subfolders. I can also navigate the folder tree of zip archives like any other folders. However, if I want my searches to include files in a zip archive I have to navigate to the specific folder in the zip archive where the file is located. My searches never include files in zip archives if I'm not in the archive, and when I am in the archive my searches don't include files within the archive's subfolders.  Is there a way for my searches to include zip archive subfolders? What is the use of a persistence layer in any application? How to cheat and avoid cheating in Flash games? Is it common for native speakers to put a question mark at the end of declarative sentence? How do I go about backing up/saving installed device drivers in Windows? I want to reinstall Windows on a machine which has some very obscure drivers. I don't much fancy having to hunt round for them (that's if I'm able to locate them at all). Is it possible to save or backup already installed drivers? If so, how do I do this? How do I then go about installing these 'saved' drivers? Where can I find information about upcoming special issues in a field of study? How can I find information about upcoming special issues related to a special field of study? There are good websites for upcoming conferences but I could not find any place to search for special issues. How can I prevent data theft from a company?  Are there any solutions to this problem?  Is there any software that encrypts important data (.doc, .xls, etc.) and prevents it from copying?  I need a solution where employees can read/write documents but can not copy then to a USB drive or e-mail them. With the last update of Fallout Shelter, Bethesda has added these "limited time" quests.  Question is, how limited is that time exactly? What happens when the quests "end", will it still be possible to get the benefits? counting number of specific string in each row I have a SNP table looks like this  A N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N N C N N N C N N N N N N N N N N N N N N N N N N C C N N N N N N N N N N C C N N N C C N N N N N N N N N N N N N N N N N   I want to count total number of N in each row? any idea how to do that