Why does the visible light spectrum appear to be circular
Why does light of high frequency appear violet?
prove that if $E$ is connected and $E \subseteq F \subseteq \overline{E}$, then $F$ is connected. Define a set $A$ to be disconnected iff there exist nonempty relatively open sets $U$ and $W$ in $A$ with $U\cap W = \emptyset$ and $A = U\cup W.$ Define a set $A$ to be connected iff it is not disconnected.(there are many equivalent definitions, but I want to prove this lemma using this one). Prove that if $E$ is connected and $E\subseteq F \subseteq \overline{E},$ then $F$ is connected.  Let $U, W$ be a separation for $F$. Find open sets $O_U$ and $O_W$ so that $U = F \cap O_U$ and $W = F\cap O_W.$ I claim that $E\cap O_U, E\cap O_W$ separate $E$. However, I'm unable to show that $U' = E\cap O_U, W' =E\cap O_W \neq \emptyset$ (I think this should be straightforward, but for some reason, I can't figure this out). Suppose $U' = \emptyset.$ Then $E\cap O_U = \emptyset.$ Since $E \subseteq F = U\cup W = F\cap (O_U \cup O_W)\subseteq O_U\cup O_W,$ we have that $E \subseteq O_W,$ so $E\cap O_W = E\subseteq F\cap O_W = W\subseteq F\subseteq \overline{E}.$ Observe that since $E\cap O_U = \emptyset, F\cap O_U = (F\backslash E)\cap O_U\subseteq F\backslash E.$ Similarly, $W'\neq \emptyset.$ Clearly, $U', W'$ are relatively open in $E$. Suppose $U'\cap W' \neq \emptyset.$ Let $x\in U'\cap W'.$ Then $x\in E\cap O_U\cap O_W \subseteq F\cap O_U\cap O_W = U\cap W = \emptyset,$ a contradiction. So $U'\cap W' = \emptyset.$ Also, $U'\cup W' = (E\cap O_U)\cup (E\cap O_W) = E\cap (O_U \cup O_W)$ and $E\subseteq (O_U \cup O_W),$ so $U'\cup W' = E.$ Show that $S + T$ and $aT$ are linear Let  $S:  V \to W$ and $T: V \to W$ be linear transformations. Given $a$ in $\mathbb{R}$, define functions $(S+T): V \to W$ and $(aT): V \to W$ by $(S+T)(v)=S(v) + T(v)$ and $(aT)(v)= aT(v)$ for all $v$ in $V$. Show that $S+T$ and $aT$ are linear transformations. Why "would" instead of "will" in this sentence? The goal for these negotiations is to reach a mutually-agreed long-term comprehensive solution that would ensure Iranˈs nuclear program will be exclusively peaceful.    what does would imply here? If it implies future, why will has not been used?  A good link teaching would thoroughly would be really appreciated. Specially, distinguishing between the senses of would is not always easy. How do I get my Epson Scanner working on Ubuntu 13.10 I have the Epson SX218 with a built-in scanner. The printer works fine but if I want to scan something simple scan says no scanner found.  Any ideas what I can do?  By the way, how can I check the ink level of the printer?? String formatting in Python I want to do something like String.Format("[{0}, {1}, {2}]", 1, 2, 3) which returns:  [1, 2, 3]   How do I do this in Python? Remove empty directory from CVS? I'm not quite sure how this happened, but somehow a completely empty hierarchy of directories has ended up in my repository:  com/ com/companyname/ com/companyname/blah/ com/sun/ com/sun/java/ com/sun/java/jax_rpc_ri/   I think what happened was that these directories did have files in them, but then a developer realized he/she shouldn't have checked them in in the first place since these are by-products of the build process, so he/she removed the files but somehow the empty directories are left in the repository as ancient relics.  How can I remove this from CVS? The only results I seem to be able to find on google say that there shouldn't be a need to remove empty directories as CVS won't keep them around in the first place, and that the -P (prune) options to cvs update should remove them from the working directory  - which is zero help if you actually have empty directories in your repository.  A cvs remove and cvs commit doesn't seem to take care of this situation:  $ cvs remove -Rf com cvs remove: Removing com cvs remove: Removing com/companyname cvs remove: Removing com/companyname/blah cvs remove: Removing com/sun cvs remove: Removing com/sun/java cvs remove: Removing com/sun/java/jax_rpc_ri $ cvs commit com cvs commit: Examining com cvs commit: Examining com/companyname cvs commit: Examining com/companyname/blah cvs commit: Examining com/sun cvs commit: Examining com/sun/java cvs commit: Examining com/sun/java/jax_rpc_ri $ ls -l com total 24 drwxrwxr-x  2 matt matt 4096 Oct 15 14:38 CVS drwxrwxr-x  9 matt matt 4096 Oct 15 14:38 companyname drwxrwxr-x  4 matt matt 4096 Oct 15 14:38 sun   It's still there!  Does SVN have this weird behavior too? I've been Googling around with no luck.  I'm looking for a way to do getc() or gets() or whatever that does not echo to the terminal.  I saw kbhit() but this doesn't appear to be part of ANSI.  Ideally, I'd like code that looks like  char s[100]; no_echo_gets(s); /* Won't echo to screen while being typed */ printf("%s\n", s);   Does anybody know a good ANSI compliant way to do this? When closing a question as a duplicate there is a text box which says: "search, or enter a question link or numeric id". You can copy and paste a question link into here, but this fails if the URL starts with https:// instead of http://, with the message "Your search returned no matches; please try a different search". C# Mono - Low Level Keyboard Hook Getting "'file' is different from the same file on the system" when I attempt to update through synaptic Encountered an error while updating through Synaptic Manager. Subsequent updates gave the error at the bottom of the output. I'm a noob so any help is appreciated.  al@al:~$ sudo apt-get -f upgrade Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following NEW packages will be installed:   libjack-jackd2-0:i386 The following packages have been kept back:   linux-headers-generic skype 0 upgraded, 1 newly installed, 0 to remove and 2 not upgraded. Need to get 0 B/202 kB of archives. After this operation, 493 kB of additional disk space will be used. Do you want to continue [Y/n]? y (Reading database ... 310056 files and directories currently installed.) Unpacking libjack-jackd2-0:i386 (from .../libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb) ... dpkg: error processing /var/cache/apt/archives/libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb (--unpack):  './usr/share/doc/libjack-jackd2-0/buildinfo.gz' is different from the same file on the system dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Errors were encountered while processing:  /var/cache/apt/archives/libjack-jackd2-0_1.9.8~dfsg.2-1precise1_i386.deb E: Sub-process /usr/bin/dpkg returned an error code (1) Obtain screen status using ADB VHDL Transition Function I'm writing a LINQ to SQL statement, and I'm after the standard syntax for a normal inner join with an ON clause in C#.  How do you represent the following in LINQ to SQL:  select DealerContact.* from Dealer  inner join DealerContact on Dealer.DealerID = DealerContact.DealerID Having line color vary with data index for line graph in matplotlib? I read here at  that holding Shift during boot can bring up the GRUB menu. However I try holding one or the other Shift keys, tapping, tapping then holding, nothing works. Am I missing something? How does one access the GRUB menu during boot time?   No dual boot here, just standard Lubuntu 14.04 install. Live notification shown for own edits When editing an answer, the &quot;new revision&quot; message is shown for a short moment. This makes no sense because the revision you just created is automatically displayed anyway. Here $n$ belongs to natural numbers. Firstly, I proved the relation by putting $n = 1$ . Then, taking $$|\sin(mx)| \leq m|\sin x|$$ true, I had to prove $$|\sin(m + 1)x| \leq (m + 1)|\sin x|$$  Now, here I got stuck. How to prove it?? Please help. If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative. I was told that the following does not work:     Employees having children will be granted a day off.   However, this sounds better:     Having children, those employees will be granted a day off.   What is wrong with the first clause? I've heard that i can travel to the US any time before my tourist visa expiry.  Is that true?  (please advise)