Usage of articles with biological terminology
The definite article implying the generic idea of something
How to prove sum of powers property of roots of unity? We know that $1+\alpha_1+\alpha_2+...+\alpha_{n-1}=0$ where $\alpha_i$ are the roots of $z^n=1$. How can I prove that: $$1+\sum_{i=1}^{n-1}\alpha_i^m=\begin{cases}0\quad m\in Z,m\not\equiv0\pmod n\\n\quad m\in Z,m\equiv0\pmod n\\\end{cases}$$ What is the state-of-the-art method for automatically executing custom scripts upon USB device plug-in under current Linux distributions like Debian/CentOS/Fedora?  For example if you want to automatically mount/copy some files/umount a USB mass storage device based on its UUID (or device ID etc.). When is the formula $$S_{\infty} = \dfrac{a}{1-r}$$ valid?   Does |$r| &lt;1$? Why don't C++ compilers define operator== and operator!=? Help with completing the proof: Any unbounded sequence is either infinitely large or has a convergent subsequential limit Given $\{x_n\}$ is an unbounded sequence, prove:   $$ \lim_{n\to\infty}x_n = \infty\ \  \text{or}\ \ \lim_{n_k \to \infty} = A $$   In other words any unbounded sequence either infinitely large or has a convergent subsequence.     Everything before udpate section is wrong.  Part 1. Proving $|x_n| &gt; M \implies \lim_{n\to\infty} x_n = \infty$  $\Box$ Start with the definition of an unbounded sequence: $$ \forall M &gt; 0 \in \Bbb R \ \exists N \in \Bbb N: \forall n &gt; N\implies |x_n| &gt; M $$  Suppose that: $$ \lim_{n\to\infty}x_n \ne \infty $$ That means: $$ \forall n\in\Bbb N\ \exists C &gt;0 \in\Bbb R : |x_n| &lt; C $$  Now taking $M^\prime = \max\{M, C\}$ we have that: $$ \begin{cases} \begin{align} \exists N \in \Bbb N : &amp;\forall n&gt;N \implies &amp;|x_n| &gt; M^\prime \\ &amp;\forall n \in \Bbb N: &amp;|x_n| &lt; M^\prime \end{align} \end{cases} $$  So we have arrived to a contradiction which means the assumption is wrong and: $$ \lim_{n\to\infty}x_n = \infty $$ $\Box$     Part 2. It's not clear to me how to prove that if unboundedness does not imply infinity limit then it must imply the existence of a convergent subsequence.   Two questions is my mind:   Is part 1 correct?  How do I proceed with the second part?     I'm using the following definition. $$ \lim_{n\to\infty}x_n = +\infty \stackrel{\text{def}}{\iff} \forall \epsilon &gt; 0 \exists N \in \Bbb N :\forall n &gt; N \implies x_n &gt; \epsilon \\ \lim_{n\to\infty}x_n = -\infty \stackrel{\text{def}}{\iff} \forall \epsilon &gt; 0 \exists N \in \Bbb N :\forall n &gt; N \implies x_n &lt; -\epsilon \\ $$ A sequence is considered infinitely large when: $$ \lim_{n\to\infty}x_n = \infty\ \ \text{when}\ \ \lim_{n\to\infty}x_n = + \infty \ \ \text{or} \ \ \lim_{n\to\infty}x_n = - \infty $$    Update.   Looks like i've messed up a lot of things. I will try once again. As shown in comments and answers the statement holds only in case: $$ \lim_{n\to\infty} |x_n| = \infty \iff \forall \epsilon &gt; 0 \exists N \in \Bbb N: \forall n &gt; N \implies |x_n| &gt; \epsilon $$  Consider the following: $$ \lnot P = \lnot\left(\lim_{n\to\infty}|x_n| = \infty \right) \iff \exists \epsilon &gt; 0\ \forall N_1 \in \Bbb N : \exists n &gt; N_1 \land |x_n| &lt; \epsilon $$  But on the other hand it is given that $x_n$ is unbounded: $$ Q = \forall M &gt; 0\ \exists N_2 \in \Bbb N : |x_{N_2}| &gt; M $$  Construct a negative expression for boundedness: $$ \lnot P = \exists M &gt; 0\ \forall N_2 \in \Bbb N : |x_{N_2}| &lt; M $$  If $S = \lnot P \implies \lnot Q$ then $S = P \lor \lnot Q$  Let $\epsilon = M$, choose $N = \max\{N_1, N_2\}$ then both statements are true and: $$ \exists \epsilon &gt; 0\ \forall N = \max\{N_1, N_2\}: \exists n &gt; N \land |x_n| &lt; \epsilon $$  Now either $P$ is true, which would mean $\lim |x_n| = \infty$, or $\lnot Q$ is true which would mean the sequence is bounded and hence contains a convergent subsequence. I write financial applications where I constantly battle the decision to use a double vs using a decimal.  All of my math works on numbers with no more than 5 decimal places and are not larger than ~100,000. I have a feeling that all of these can be represented as doubles anyways without rounding error, but have never been sure.  I would go ahead and make the switch from decimals to doubles for the obvious speed advantage, except that at the end of the day, I still use the ToString method to transmit prices to exchanges, and need to make sure it always outputs the number I expect. (89.99 instead of 89.99000000001)  Questions:   Is the speed advantage really as large as naive tests suggest? (~100 times) Is there a way to guarantee the output from ToString to be what I want? Is this assured by the fact that my number is always representable?   UPDATE: I have to process ~ 10 billion price updates before my app can run, and I have implemented with decimal right now for the obvious protective reasons, but it takes ~3 hours just to turn on, doubles would dramatically reduce my turn on time. Is there a safe way to do it with doubles? I followed the instructions and got success response but I didn't receive email.  Then I used the id to get Email Status and got 404 error.  Here is my json :  POST :   {     "To": {         "Address": "example@gmail.com",         "SubscriberKey": "example@gmail.com.tw"     },     "Options": {         "RequestType": "ASYNC"     } }   GET : {id}  Please let me know how I can resolve the issue. How to make a package manager wait if another instance of APT is running? I have seen many softwares such as Update Manager and Synaptic Package Manager, they wait if some other program is using the /var/lib/dpkg/lock and is locked. How can we do this through the Terminal? I saw apt-get's manual but didn't find anything useful. One click shutdown Ubuntu and load into alternative bootup Fatou's lemma and measurable sets I don't know how can I imply Fatou's lemma for any measurable sets $A_k$, that is, $$\lambda(\liminf A_k)\le\liminf\lambda(A_k).$$ How can I prove it? And is there any example in $\mathbb R$ of sequence of measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim\lambda(A_k)=1$, but $\liminf A_k=\varnothing$ ? Thanks for your help! I'm looking for a site, similar to StackOverflow, but with possibility to ask opinion based questions Since opinion based questions are not allowed neither here nor on stackoverflow, I would like to ask if any of you, know some other web page (outside Stack* ecosystem), where we can ask questions like:  "Which Java Web framework would you recommend to learn"   or any other, which by nature are opinion based.  EDIT1:  Allow me to explain a bit more - I know there are webgroups, forums and mailing lists BUT most of those are for single programming language or framework and I'm looking for something more generic. Stackoverflow is great because there are experts from all sort of programming languages and frameworks, so if it would be allowed to ask:   "Best web frameworks similar to ASP.NET MVC4"  I would get a lot of opinions, from different languages and frameworks, and that exactly what I'm looking for. When is assembly faster than C? One of the stated reasons for knowing assembler is that, on occasion, it can be employed to write code that will be more performant than writing that code in a higher-level language, C in particular.  However, I've also heard it stated many times that although that's not entirely false, the cases where assembler can actually be used to generate more performant code are both extremely rare and require expert knowledge of and experience with assembly.    This question doesn't even get into the fact that assembler instructions will be machine-specific and non-portable, or any of the other aspects of assembler.  There are plenty of good reasons for knowing assembly besides this one, of course, but this is meant to be a specific question soliciting examples and data, not an extended discourse on assembler versus higher-level languages.  Can anyone provide some specific examples of cases where assembly will be faster than well-written C code using a modern compiler, and can you support that claim with profiling evidence?  I am pretty confident these cases exist, but I really want to know exactly how esoteric these cases are, since it seems to be a point of some contention. Should laptops remain plugged in when their battery is 100% charged? When your laptop's battery is 100% charged, should you leave it plugged in so any battery power doesn't get used, or will that cause overcharging, overheating, etc.? Should the laptop be unplugged when the battery level is 100%?  I'm asking because my laptop's screen tends to get dim when unplugged, so I don't like to run it on battery. Add new JDBC Driver to connect Mathematica to SQL database Auto generate an Index Is there any way to auto-gererate an index for the makeidx package? I've written ~270 pages and didn't think of adding \index{} everywhere.  Do I really need to go over everything and add \index{}? PhD without previous academic degree: truth or myth? Which are the spec bits in this cache? I have a direct mapped cache of size S with the line size L. The cache is physically indexed and tagged. The physical address is 50 bits, numbered from 0 to 49 (with 0 being the least significant bit). The machine has a word size of 4 bits and the memory is byte-addressable. Which bits of the 50 bits are compared with the address tag? I've had a couple of answers today where the OP indicated via a comment that I solved their problem, but the question wasn't upvoted or accepted.  In both cases the person probably wasn't able to upvote since they were new, but they could have accepted my answer. I felt a little guilty asking them to accept the answer, but since it solved their problem I felt that it would help others who stumbled on the question trying to solve their own problem to know that my solution really was the answer. Is asking the OP to accept your answer when they've said that it solved their problem a reasonable thing to do or does that feel like nagging for reputation? I have a multiboot system set up. The system has three drives. Multiboot is configured with Windows XP, Windows 7, and Ubuntu - all on the first drive. I had a lot of unpartitioned space left on the drive and was reserving it for adding other OSes and for storing files there in the future.  One day I went ahead and downloaded Partition Wizard and created a logical NTFS partition from within Windows 7, still some unpartitioned space left over. Everything worked fine, until I rebooted the computer a few days later.  Now I'm getting:    error: unknown filesystem.   grub rescue&gt;   First of all I was surprised not to find any kind of help command, by trying:    help, ?, man, --help, -h, bash, cmd, etc.  Now I'm stuck with non-bootable system. I have started researching the issue and finding that people usually recommend to boot to a Live CD and fix the issue from there. Is there a way to fix this issue from within grub rescue without the need for Live CD?  UPDATE  By following the steps from , I was able to boot to initramfs prompt. But not anywhere further than that.  So far from reading the manual on , I was able to see my drives and partitions using ls command. For the first hard drive I see the following: (hd0) (hd0,msdos6) (hd0,msdos5) (hd0,msdos2) (hd0,msdos1)  I now know that (hd0,msdos6) contains Linux on it, since ls (hd0,msdos6)/ lists directories. Others will give "error: unknown filesystem."  UPDATE 2  After the following commands I am now getting to the boot menu and can boot into Windows 7 and Ubuntu, but upon reboot I have to repeat these steps.    ls ls (hd0,msdos6)/ set root=(hd0,msdos6) ls / set prefix=(hd0,msdos6)/boot/grub insmod /boot/grub/linux.mod normal   UPDATE 3  Thanks Shashank Singh, with your instructions I have simplified my steps to the following. I have learned from you that I can replace msdos6 with just a 6 and that I can just do insmod normal instead of insmod /boot/grub/linux.mod. Now I just need to figure out how to save this settings from within grub itself, without booting into any OS.  set root=(hd0,6) set prefix=(hd0,6)/boot/grub insmod normal normal   UPDATE 4  Well, it seems like it is a requirement to boot into Linux. After booting into Ubuntu I have performed the following steps described in the :  sudo update-grub sudo grub-install /dev/sda   This did not resolve the issue. I still get the grub rescue prompt. What do I need to do to permanently fix it?  I have also learned that drive numbers as in hd0 need to be translated to drive letters as in /dev/sda for some commands. hd1 would be sdb, hd2 would be sdc, and so on. Partitions listed in grub as (hd0,msdos6) would be translated to /dev/sda6.  UPDATE 5  I could not figure out why the following did not fix grub:  sudo update-grub sudo grub-install /dev/sda   So I downloaded  based on an answer from  post. That seemed to do the trick after I picked the "Recommended Repair (repairs most frequent problems)" option. Suggested edits in top-bar put flags over the winterbash snowflake Suggested edits in the top bar put the flag count right over the winterbash snowflake. Looks pretty ugly if you ask me...