Understand Regular Expression
Reference - What does this regex mean?
Use of inequality $1 - \cos (x) \leq x^2 /2$ In  answer the value of $1 - \cos(x)$ has to be evaluated in order to find its upper limit, if it exists.  In particular, $x = 2 \pi / n$. The answer is related to the length of a side of a regular $n$-gon inscribed into a unit-radius circumference; because the perimeter of the $n$-gon is always less than $2 \pi$, the single side must always be less than $2 \pi / n$.  The inequality  $$1 - \cos (x) \leq \displaystyle \frac{x^2}{2}$$ (1)  is used and the proof is completed with   $$2(1 - \cos(x)) \leq (2 \pi / n)^2$$  $$\sqrt{2(1 - \cos(x))} \leq 2 \pi / n$$  But it is well known that the cosine is a function $f(x) \in [-1;1]$, so $1 - \cos (x) \in [0,2]$. By using this information, we would obtain  $$1 - \cos (x) \leq 2$$ (2)  The proof would provide  $$2(1 - \cos(x)) \leq 4$$  $$\sqrt{2(1 - \cos(x))} \leq 2$$  which is a completely different result.   Why in that case it is preferable to use (1) instead of (2)? How to choose when it is convenient to use (1) and when to use (2) in a proof? UPDATE statement to re-assign a column value according to a numeric formula Do redirection operators always open file descriptors in parallel? 1. Consider snippet#1:  $ cat test.txt &gt; test.txt cat: test.txt: input file is output file   It seems that cat makes its input file descriptor point to test.txt and then when it tries to set its output file descriptor to test.txt it throws the above error. Here it seems that cat is aware of the redirect operator and so proceeds to attempt to set the output file descriptor to test.txt  2. Consider snippet#2:  $ cat 1.txt 1:CAT 2:dog $ sed 's/cat/CAT/g' test.txt 1:CAT 2:dog $ sed 's/cat/CAT/g' test.txt &gt; test.txt $ cat test.txt # Note that test.txt is now empty $   Here we see that sed opens test.txt (last argument) in read mode and at the same time sets test.txt as its file output descriptor. Also the '&gt;' operator overwrites the contents of the file BEFORE sed starts to read from it.   I am aware that commands in a pipeline execute in parallel but have not come across any info on how redirection operators behave. Any supporting links would be helpful. Is $\mathbb{Z}[\sqrt{-7}]$ a UFD or not? Is $\mathbb{Z}[\sqrt{-7}]$ a UFD or not? I feel like there should be some easy example showing that it is not, but I can not think of one. Oracle 11g: increasing buffer cache with AMM On a 2-node RAC we have enabled Automatic Memory Management. The advisor views indicate that increasing the buffer cache by 4 GB will half the physical reads, and we have 4 GB to give.   How must we do this? By increasing memory_target and letting Oracle grab more memory for the buffer cache? Or by explicitly setting a value for db_cache_size (which now is zero)? i have a simple sort of a database file which consists only of entries in the following format     variable=value   i want to create a simple vim syntax highlighting for it and set it for specific file extension  for instance, variable part could be light blue, and value part light red  i googled it and came across things such as $vimruntime\syntax\, syntax set=, syntax match, and hi keywords, but couldn't set it up myself eventually  so i want a very simple vim code snippet that would realize it by matching the left and right hand sides and coloring them separately Why is wine made 'from' grapes, but tables are made 'of' wood? (1) Wine is an alcoholic beverage made from fermented grapes or other fruits. (Wikipedia) (2) Tables were made of marble or wood and metal (typically bronze or silver alloys), sometimes with richly ornate legs. (Wikipedia)  Why is 'from' used in (1), whereas 'of' is used  in (2)? Is it a stylistic choice whether to use 'of' or 'from'? Or, in case of chemical changing, differently from changing in shape only, 'from' is preferable? Which commands to convert a Ubuntu BIOS install to EFI/UEFI without boot-repair on single boot hardware? booting windows 10 spends too long I'm looking to book flights from) Philadelphia (PHL) to Detroit (DTW). Direct flights are quite expensive. I noticed, in expanding a search to surrounding airports, that I can book from Allentown (ABE) to Detroit for far less, and the connection is through Philadelphia.  Can I book this flight but board in Philadelphia? I'll be traveling with carry-on luggage only, so on the return, it's obviously no problem. But would I be allowed through security and on to the plane in PHL with out having flown from Allentown? I have two files in a directory. One has correct permissions and the other has not. Is there a way I can "copy" the set of permissions from one file to another? What causes the texture of bread to be open, ie with lots of large holes, or close, ie a regular, uniform tender crumb with no large holes? How can I achieve either? How to move boot and root partitions to another drive I have two drives on my computer that have the following configuration:  Drive 1: 160GB, /home Drive 2: 40GB, /boot and /   Unfortunately, drive 2 seems to be dying, because trying to write to it is giving me errors, and checking out the SMART settings shows a sad state of affairs.  I have plenty of space on Drive 1, so what I'd like to do is move the / and /boot partitions to it, remove Drive 2 from the system, replace Drive 2 with a new drive, then reverse the process.  I imagine I need to do some updating to grub, and I need to move some things around, but I'm pretty baffled how to exactly go about this. Since this is my main computer, I want to be careful not to mess things up so I can't boot. What's the standard "roadmap" to learning quantum physics? Is $\bar X$ a minimum variance unbiased estimator of $\theta$ in an exponential distribution? I proceeded by finding $\operatorname E(\bar X).$ I considered $\bar X$ as a constant and simply got the term itself. This should suggest that $\bar X$ is not an unbiased estimator of $\theta$. That's obvious too cuz it's a constant(from what I assumed). However, I'm also aware that $\mu = \theta$ so that's probably not right.  I have to demonstrate first that $\bar X$ is an unbiased estimator of $\mu$. Recently, my iPhone 5 was stolen, and while I have a new iPhone 6, I received a message from a number of someone claiming to be in possession of the device, with a new number.   All images, contacts, Outlook mail, you name it, are still on the device.  While I realize it's my fault for not having it connected to iCloud to wipe manually, is there anything I can do to have this sensitive information wiped from the device? Count the number occurrences of a character in a string Let $G$ be a non-empty finite set with an associative binary operation so that cancellation law holds, i.e. $ab=ac$ or $ba=ca$ implies $b=c$, for any choices of $a,b,c$ in $G$. Assume that there is an identity element $e$ in $G$. Show that $G$ is a group.  Proof: To show $G$ is a group, conditions must hold. Suppose that $ab=ac$, then $b=eb=(a^{-1}a)b =a^{-1}(ab)=a^{-1}(ac) =(a^{-1}a)c=ec=c$. So left cancellation holds in $G$.  2) Suppose $ba=ca$, then $b=be=b(aa^{-1}) =(ba)a^{-1}=(ca)a^{-1} =c(aa^{-1})=ce=c$. So right cancellation holds in $G$.  3) If $a$ exists in $G$ then $aa^{-1}=a^{-1}a=e$, where $a$ is an inverse of $a^{-1}$. Since inverses are unique, $(a^{-1})^{-1}=a$.  4) Let $x$ be the inverse of $ab$. Then $(ab)x=e$. By associativity we have $a(bx)=aa^{-1}$. Through left cancellation we have  $bx=a^{-1}bx=ea^{-1}=b(b^{-1}a^{-1})$ and $x=b^{-1}a^{-1}$.  Thus $(ab)^{-1}=b^{-1}a^{-1}$. So all conditions hold, $G$ is a group.  Is this proof correct? I know to show $G$ is a group these conditions have to be met. This is all I have to show right? How does the standard error work? I have been looking into the inner-workings of the standard error recently, and I found myself unable to understand how it works. My understanding of the standard error is that it is the standard deviation of the distribution of sample means. My questions are:  • how do we know the standard error is the standard deviation of the sample means when we usually take just a single sample?  • why doesnt the equation to calculate the standard error mirror the standard deviation equation for a single sample? Prove $\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=$ $\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$ I'm trying prove $$\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$$  It is a part of solution of another problem I try a lot to prove it but I could not be succeed.  I tried using exponent manipulation in prime factorizations but it didn't help me.