With a UK 6-months visa, can I leave and re-enter before that?
Can I leave the UK and come back in, on a Tourist visa?
How can we prove that $D(f)\backsimeq\operatorname{Spec} A_f$? Let $$D(f)=\operatorname{Spec}A - V(f)=\{p\in \operatorname{Spec} A:f\notin p\}$$ be the principal open subsets of $\operatorname{Spec}A$ with the Zariski topology. I'm trying to prove that $D(f)\backsimeq \operatorname{Spec} A_f$, where $A_f$ is the localization of $A$ at $f$.  If we consider the canonical homomorphism $\varphi:A\to A_f$ given by $\varphi (a) = \frac {a}{1}$, we can define $\varphi':\operatorname{Spec} A_f\to \operatorname{Spec} A$, given by $\varphi'(p)=\varphi^{-1}(p),\forall p\in \operatorname{Spec}A_f$.  I'm having trouble proving that this map is surjective.  Thanks. Straight lines - product of slope of perpendicular lines. Let $l$ be a line with slope $\frac{1}{0}$, and $m$ a line with slope $0$.  Now, line $l$ is perpendicular to line $l$, therefore the product of the slope of line $l$ and the slope of line $m$ equals $-1$. This implies that $\frac{1}{0} \times 0 = -1$! When using sudo powers and calling gedit the top level menu with File Edit View Search Tools Documents Help is missing.  If I create user ID root and sign in once and call gedit will it create the necessary user configuration files that might fix this?  Will this also make the annoying error messages go away whenever I use sudo, gksu or pkexec as a regular user with elevated privileges togedit ?  Will there be other benefits with Nautilus and other Gnome derivative Ubuntu applications?  NOTE: by sudo powers you can google pkexec gedit 3.5k hits, gksu gedit 40k hits or sudo gedit 500k hits. I'm in the minority using the first method but I believe it will become the standard in Ubuntu 17.04.    June 12, 2017 Update Here are a list of errors you get when using pkexec gedit:  (gedit:13003): Gtk-WARNING **: Calling Inhibit failed: GDBus.Error:org.freedesktop.DBus.Error.ServiceUnknown: The name org.gnome.SessionManager was not provided by any .service files  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-spell-enabled not supported  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-encoding not supported  ** (gedit:13003): WARNING **: Set document metadata failed: Setting attribute metadata::gedit-position not supported   On this  there is a commitment to fix these distracting messages as of June 5, 2017. No indication is given for when the fix will be upstream nor if the functionality will be implemented or the error messages will simply disappear.  Note the bug report was filed over a year ago and over that time it only effects 18 people. This is a homework question. But I am completely stuck.  My only intuition was to go about it inductively from a "greedy algorithm" maybe know as the deletion-contraction algorithm. And to somehow use the information about the jth cycle to solve the j+1th. But I'm not sure how I'd do it. Thank you very much for looking this over. The advantages and disadvantages of using Kaiser Rule to select the number of principal components Serial protocol delimiting/synchronization techniques As asynchronous serial communication is widely spread among electronic devices even nowadays, I believe many of us have encountered such a question from time to time. Consider an electronic device D and a computer PC connected with serial line (RS-232 or similar) and required to exchange information continuously. I.e. PC is sending a command frame each X ms, and D is replying with status report/telemetry frame each Y ms (The report can be sent as response to requests or independently - doesn't really matter here). The communication frames can contain any arbitrary binary data. Assuming the communication frames are fixed-length packets. The problem: As the protocol is continuous, the receiving side might loose the synchronization or just &quot;join&quot; in the middle of an ongoing sent frame, so it just won't know where the start of frame (SOF) is. A the data has different meaning based on its position relatively to the SOF, the received data will become corrupted, potentially forever. The required solution Reliable delimiting/synchronization scheme to detect the SOF with short recovery time (i.e. it shouldn't take more than, say 1 frame to resynchronize). The existing techniques I am aware (and using some) of: 1) Header / checksum - SOF as predefined byte value. Checksum in the end of frame.  Pros: Simple. Cons: Not reliable. Unknown recovery time.  2) Byte stuffing:  Pros: Reliable, fast recovery, can be used with any hardware Cons: Not that suitable for fixed-size frame-based communication  3) 9th bit marking - prepend each byte with additional bit, while SOF marked with 1 and the data bytes are marked with 0:  Pros: Reliable, fast recovery Cons: Requires hardware support. Not directly supported by most of PC hardware and software.  4) 8th bit marking - kind of emulation of the above, while using the 8th bit instead of 9th, which is leaving only 7bits for each data word.  Pros: Reliable, fast recovery, can be used with any hardware. Cons: Requires an encoding/decoding scheme from/to the conventional 8-bit representation to/from 7-bit representation. Somewhat wasteful.  5) Timeout based - assume the SOF as the first byte coming after some defined idle time.  Pros: No data overhead, simple. Cons: Not that reliable. Won't work well with poor timing systems like, say, Windows PC. Potential throughput overhead.  Question: What are the other possible techniques/solutions exist to address the problem? Can you point to the cons in the above list which can be easily worked around, thus removing them? How do you (or would you) design your systems protocol? When will there be a Windows Phone app? Is it possible to generate transcendental numbers of arbitrary irrationality measure ? I am thinking of something like an infinite series which depends on a parameter fixing the irrationality measure.   All I have found so far is bounds on the irrationality measure of famous numbers, I have not been able to find something going the other way around. A person who cannot understand that there may be other perspectives I'm searching for a word to describe a person who cannot understand that there may be different perspectives. I initially thought of single-minded, but that pertains to determination.  One example of several perspectives could be the difference between a process and an action. Please assume that a process is composed of several actions. A process may be composed of other processes, but these processes would be actions in this greater process.  If you accept my definitions, then depending on the holistic or granularity level, we may consider something a process or an action, depending on whether we are considering its parts or how it's part of a greater thing.  A person may have trouble understanding that it's not true or false that something is a process or an action, it depends on how it relates with everything else from the perspective that we take. From one perspective it's a process, from another it's an action. For some people, things seem to be in one and only one absolute way, which is wrong, IMHO.   Is there a word to describe this kind of problem that some people seem to have?  PD: It's like . But I don't think most people know the word monist.  Here are some other, superficially similar terms that actually mean something quite different. A racist is (approximately) someone who considers other races inferior. A narrow-minded person considers other perspectives wrong or irrelevant, or may not consider them, but is aware that they exist. A monist would not even understand that there is more than one race or multiple perspectives; there is simply one reality, and that's it. Partial vs. Full Volume Wort Boil The starting position for most new extract brewers is to use a partial boil set up.  Defined as using a pot smaller than the intended batch size, all the ingredients are boiled in a smaller volume.  Then the wort is diluted to the final desired batch size.  (Often the dilution step is used as part of a cooling step)  What are the Ups and Downs of both Partial vs. Full boils? Assigning strings to arrays of characters I am a little surprised by the following.  Example 1:  char s[100] = "abcd"; // declare and initialize - WORKS   Example 2:  char s[100]; // declare s = "hello"; // initalize - DOESN'T WORK ('lvalue required' error)   I'm wondering why the second approach doesn't work. It seems natural that it should (it works with other data types)? Could someone explain me the logic behind this? After the DEM creation, I wanted to create slope layer.   However, the result has not satisfied me. I don't think the slope map is correct. Have you ever encountered false calculated terrain map?  This is the DEM:    And this is the slope layer. Label rows of a matrix by characters How to add label to index the rows and columns of the matrix ? because I want to number the line outside the matrix       \[     \begin{array}{lc}       \ Mat_{\varPhi to M} &amp; \left(\begin{array}{@{}ccccc@{}}             1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 \\             0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 \\             0 &amp; 0 &amp; 1 &amp; 0 &amp; 1 \\             0 &amp; 0 &amp; 0 &amp; 1 &amp; 1 \\             0 &amp; 0 &amp; 0 &amp; 0 &amp; 1               \end{array}\right) \\[15pt]     \end{array}     \] When I import text file into R, I get a special character appended to the first value of the first column Automatically use specific language in Notepad++ for files with no extension Long-distance movement arrow in gloss I have a question on drawing an arrow using gb4e. Currently, I am trying to use the method described in the link below. The problem takes place when I draw the long distance arrow like (5). As you can see the picture below, the line cross over the gloss. I would like to draw the horizontal line like the example (1). How can I evacuate the crossing?    \documentclass[11pt, dvipdfmx]{article} \usepackage{tikz} \usetikzlibrary{calc} \newcommand\tikzmark[1]{\tikz[remember picture, baseline=(#1.base)] \node[anchor=base,inner sep=0pt, outer sep=0pt] (#1) {#1};}  % This code from http://tex.stackexchange.com/q/55068/2693 \tikzset{     ncbar angle/.initial=90,     ncbar/.style={         to path=(\tikztostart)         -- ($(\tikztostart)!#1!\pgfkeysvalueof{/tikz/ncbar angle}:(\tikztotarget)$)         -- ($(\tikztotarget)!($(\tikztostart)!#1!\pgfkeysvalueof{/tikz/ncbar angle}:(\tikztotarget)$)!\pgfkeysvalueof{/tikz/ncbar angle}:(\tikztostart)$)         -- (\tikztotarget)     },     ncbar/.default=0.5cm, }  % Thanks to Paul Gessler adn Percusse for code improvement here \newcommand{\arrow}[2]{\begin{tikzpicture}[remember picture,overlay] \draw[-&gt;,shorten &gt;=3pt,shorten &lt;=3pt] (#1.base) to [ncbar=\arrowht] (#2.base); \end{tikzpicture} \setlength{\arrowht}{0ex} } \usepackage{gb4e} % The following code modified from  % http://permalink.gmane.org/gmane.comp.tex.linguistics/1036 % This adds some extra space after the first line \newlength{\arrowht} \setlength{\arrowht}{0ex} \newcommand*\cgdepthstrut{{\vrule height 0pt depth \arrowht width 0pt}} \renewcommand\eachwordone{\cgdepthstrut\rmfamily} \renewcommand\glt{\vskip -\topsep} \let\trans=\glt \newcommand\arrowex{\setlength{\arrowht}{2.5ex}\ex}   \begin{document} \begin{exe} \ex\label{scramble-det} \begin{xlist}         \arrowex          \gll          \ldots dat Jan \tikzmark{een} auto gisteren \tikzmark{t}  gekregen heeft.\\         \ldots that John a car  yesterday t gotten has \\         \ldots that John a car yesterday.         \arrow{t}{een} \end{xlist} \end{exe}  \begin{exe} \ex\label{scramble-dets} \begin{xlist}         \ex          \gll          \ldots dat Jan een auto gisteren t gekregen heeft.\\         \ldots that John a car  yesterday t gotten has \\         \ldots that John a car yesterday. \end{xlist} \end{exe}  \begin{exe} \ex[]{     \gll     John-wa totetumonaiukurai baka-da.    \\     John-Top very stupid-cop \\     `John is very studip.'     }  \ex[]{     \gll     John-wa totetumonaiukurai dousiyoumonai tondemonai sukuigatai baka-da.    \\     John-Top very very very very stupid-cop \\     `John is very studip.'     }  \arrowex[]{     \gll     \tikzmark{John}-wa totetumonaiukurai dousiyoumonai tondemonai sukuigatai  \tikzmark{t} baka-da.    \\     John-Top very very very very stupid-cop \\     `John is very stupid.'         \arrow{t}{John} }  \end{exe}  \end{document} Convert and redirect URL in uppercase to lowercase using .htaccess I would like to redirect a URL with uppercase characters http://localhost/A/B to a lowercase version http://localhost/a/b using the .htaccess file. Using regex I can capture A and B but is it possible to convert them to lowercase? I know how to solve "simple" recurrence relations. For instance, say you have:  $$c_0 = 20$$ $$c_1 = 30$$ $$c_n = 3 c_{n-1} - 2 c_{n-2}$$  We can write the characteristic equation as:  $$3x^{n-1} - 2x^{n-2} = x^n$$  Solving this with $n=2$, we get $x = 1$ or $x = 2$. This lets us write the relation $c_n = \alpha_1 1^n + \alpha_2 2^n$, and we can solve for $\alpha_1$ and $\alpha_2$ with the initial states $c_0$ and $c_1$.  However, this depends on the fact that $3x^{n-1} - 2x^{n-2} = x^n$ has two roots.  Now, I'm stuck on another problem where the characteristic equation has fewer roots than terms.  Say I have this recurrence relation instead:  $$a_0 = 0$$ $$a_1 = 2$$ $$a_2 = −1$$ $$a_n = 9a_{n-1} - 15a_{n-2} - 25a_{n-3}$$  The characteristic equation would be:  $$9x^{n-1} - 15x^{n-2} - 25x^{n-3} = x^n$$  However, solving with $n=3$, we only get two roots: $x=-1$ or $x=5$. There are not enough roots to write a relation in the form of $a_n = \alpha_1 r_1^n + \alpha_2r_2^n + \alpha_3r_3^n$. How do I proceed? Infinite Series $\sum\limits_{n=1}^\infty\left(\frac{H_n}n\right)^2$ How can I find a closed form for the following sum? $$\sum_{n=1}^{\infty}\left(\frac{H_n}{n}\right)^2$$ ($H_n=\sum_{k=1}^n\frac{1}{k}$). I need to find my external IP address from a shell script. At the moment I use this function:   myip () {      lwp-request -o text checkip.dyndns.org | awk '{ print $NF }' }   But it depends on perl-libwww, perl-html-format, perl-html-tree installed. What other ways can I get my external IP?