I am Pakistani passport holder and I have a Schengen Visa and studying in Sweden going back to my country and the flight I booked has a 21-hour layover at Manchester do I need to apply for a transit visa.
Is there a website or some other way to find out if I need a transit visa for a short stop in the UK on my way somewhere else? Ideally this should give me the answer for all nationalities, and take into account any other visas I may have.
Question  Let $X_n\geq 0$ be independent for $n\geq 1$. The following are equivalent.  $\sum_{n=1}^\infty X_n&lt;\infty$ a.s. $\sum_{n=1}^\infty[P(X_n&gt;1)+E(X_nI(X_n\leq 1)]&lt;\infty$ $\sum_{n=1}^\infty E(X_n/(1+X_n))&lt;\infty$   My attempt I was able to prove $(1)$ implies $(2)$ and $(2)$ implies $(3)$ but not $(3)$ implies $(1)$. For $(1)$ implies $(2)$, observe that by Borel-Cantelli $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$, else $X_n&gt;1$ infinitely often w.p.1 so that $(1)$ is violated. To see that $\sum_{n=1}^\infty E(X_nI(X_n\leq 1)]&lt;\infty$, apply Kolmogorov three-series theorem (I think). For $(2)$ implies $(3)$ we can write that $$ \begin{align} \frac{X_n}{1+X_n}&amp; =\frac{X_n}{1+X_n}I(X_n\leq 1)+\frac{X_n}{1+X_n}I(X_n&gt;1)\\ &amp;\leq X_nI(X_n\leq 1)+ I(X_n&gt;1). \end{align} $$ Now take the expectation of both sides and sum on $n$ using $(2)$ to conclude $(3)$. For the last implication $(3)$ implies $(1)$, I tried to use Kolmogorov three series theorem. To this end put $Y_n=X_nI(X_n\leq 1)$. First note that $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$ since $$ P(X_n&gt;1)\leq P\left(\frac{X_n}{1+X_n}&gt;\frac{1}{2}\right)\leq 2E\frac{X_n}{1+X_n} $$ by Markov's inequality. But I am unable to show that $\sum EY_n&lt;\infty$ and $\sum\text{Var}(Y_n)&lt;\infty$. Any help is appreciated. I am looking for some (may be rarely know to nonnative English speakers) synonyms for  "immensely valuable" or "extremely valuable". Number of factors of a big number How to find the number of factors of $884466000$ without using a calculator? Evaluate $\lim_{x\rightarrow 0} \frac{\sin (6x)}{\sin(2x)}$ without L'Hopital Studying for CISSP these choices seemed limited:     What is the advantage of digital signatures over message authentication codes?         Digital Signature provides integrity verification while message authentication code does not   Digital Signature provides confidentiality while message authentication code does not   Digital Signatures provide authenticity while message authentication code does not   Digital Signatures works faster than message authentication codes      Using the following chart as a reference, I'm leaning towards #2, but none of the choices really fit properly…  Cryptographic primitive | Hash |    MAC    | Digital Security Goal           |      |           | signature ------------------------+------+-----------+------------- Integrity               |  Yes |    Yes    |   Yes Authentication          |  No  |    Yes    |   Yes Non-repudiation         |  No  |    No     |   Yes ------------------------+------+-----------+------------- Kind of keys            | none | symmetric | asymmetric                         |      |    keys   |    keys   Hopefully an expert here can clarify. Examples of patterns that eventually fail Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of number of edges on a complete graph starts $0,1,3,6,10$, so the next term must be 15 etc."  Granted, this second statement is less logically unsound than the first since it's not difficult to see the reason why the sequence must continue as such; nevertheless, the statement was made on a premise that boils down to "interesting patterns must always continue".  I try to counter this logic by creating a ridiculous argument like "the numbers $1,2,3,4,5$ are less than $100$, so surely all numbers are", but this usually fails to be convincing.  So, are there any examples of non-trivial patterns that appear to be true for a large number of small cases, but then fail for some larger case? A good answer to this question should:   be one which could be explained to the layman without having to subject them to a 24 lecture course of background material, and have as a minimal counterexample a case which cannot (feasibly) be checked without the use of a computer.   I believe conditions 1. and 2. make my question specific enough to have in some sense a "right" (or at least a "not wrong") answer; but I'd be happy to clarify if this is not the case. I suppose I'm expecting an answer to come from number theory, but can see that areas like graph theory, combinatorics more generally and set theory could potentially offer suitable answers. What's the difference between lists and tuples? What's the difference?  What are the advantages / disadvantages of tuples / lists? Durrett Example 1.9 - Pairwise independence does not imply mutual independence? Best solution to protect PHP code without encryption First of all, I'm not looking for miracle... I know how PHP works and that there's not really way to hide my code from the clients without using encryption. But that comes with the cost of an extension to be installed on the running server.  I'm looking for something different though... I'm not looking to encrypt my code or even obfuscate it. There are many PHP scripts without encrypted/obfuscated code but they are commercial applications. For instance, vBulletin and/or IP.Board forum applications.  I just want to know what approach do these guys use for their applications...  I'm also open to any other suggestions.  Please note that I'm a single person and not working for a company. My product is also very specific, it won't sell that much. I just want you guys to know that I can't afford to consult a legal professional either to sue someone or prepare a commercial license. I'm just looking for a simple way to protect my simple product, if it's indeed possible, somehow... Always wanting more , of anything, never has enough,never contented with what they have How to get a list of the filenames of the figures embedded in a document I have a TeX document that is embedding a huge amount of figures from a fig folder that is full of another humongous amount of other figures.  I would like to move the document along with its figures, but I don't want to take the whole fig folder, just the figures in there that are embedded by the document.  Is there a package (or any other way) that allows me to, for example, print out the filenames of all the figures embedded by the document? Something like:  fig/figurename1.eps fig/figurename2.eps ... etc   I could then just copy-paste that list and add some cp's in a plain text file and automate the copy of those files to my desired location by running the file in my shell:  $ cp fig/figurename1.eps wherever/fig/ $ cp fig/figurename2.eps wherever/fig/ ... etc   Of course this is just an idea, if somebody has an alternative to the "file listing and copy-pasting in a shell file" I would be happy to listen =) I thought I remembered hearing the food agency  say that you should never reheat chicken meat after the initial cooking. However, I cannot find the statement of it anymore. Now I'm starting to suspect my memory.  Could this statement be true?  Is it unsafe to reheat chicken? How do I set an environment variable on the command line and have it appear in commands? If I run  export TEST=foo echo $TEST   It outputs foo.  If I run  TEST=foo echo $TEST   It does not. How can I get this functionality without using export or a script? Article "communicated by" What does it mean when it says under a journal article that it has been communicated by "XYZ" where XYZ is not the author but some other scholar with a very strong reputation? What is the relationship to the actual author and/or the content? Is this some sort of seal of approval to get results out and known quickly? (I am specifically wondering in the context of mathematics and mathematical physics.) Why did Slytherin House not get disbanded? YA book about a young powerful female caster attending a school that trains in magical arts and duels I read a young adult novel around the turn of the century. This story features a young female protagonist that is very naturally gifted in magic. She is attending a  to learn mastery and control over her gifts.  A lot of her school revolves around magic dueling. The participants are taught to extend their inner aura like a second magical skin, possibly referred to as creating a shield, or . The user can increase the strength of the shield by focusing harder on it, but doing so means you have less ability to be offensive. The amount of energy needed to create a shield was half the energy a foe would need to focus at you to break it.  Ex: Mage A and B are equally matched. Mage A focuses half his power into a shield. Mage B would then need to focus 100% of his power to break A’s shield.  Another aspect I remember is that when dueling, a school master would extend a personal shield on to one of the combatants as a sort of secondary shield that goes underneath the combatants. When one mage broke the others mages personal shield, the school masters shield protects the losers life from harm, and the breaker of the shield is declared the winner.  I distinctly remember one match in which the protagonist is having a duel with another strong student. She creates a wind storm around herself that picks up the sand from the arena floor and makes her impossible to see and thus makes it hard for her opponent to focus on her shield to break it. She then lashes out at her opponent with incredibly powerful strikes with the final strike being so hard it shatters the opponents shield and then breaks the school masters shield as well and stops just shy of killing her dueling partner.  Any assistance on this one? I think my daughter would love to read it. How to create a good glass reflection with Principled BSDF Performance differences between visibility:hidden and display:none I need to get rid of the GNOME Desktop Environment which has been installed by default when I updated my pc from 15.10 to 16.04.  After a while, I found myself not only with GNOME DE, but also with Cinnamon alongside (in the login screen, I could choose at least five or six DE!)  Now I am using Unity, and I'm happy with that, and I successfully uninstalled Cinnamon, but now I'm afraid that after uninstalling GNOME I will mess up the Unity DE, since I see many packages running in unity with the packagename starting with gnome-... Would someone mind to help me? Thx in advance Bash shell script output alignment My script:  date echo -e "${YELLOW}Network check${NC}\n\n"  while read hostname do  ping -c 1 "$hostname" &gt; /dev/null 2&gt;&amp;1 &amp;&amp;  echo -e "Network $hostname : ${GREEN}Online${NC}" || echo -e "${GRAY}Network $hostname${NC} : ${RED}Offline${NC}"  done &lt; list.txt         sleep 30 clear done   Is outputting info like this:    Network 10.x.xx.xxx : Online    Network 10.x.xx.xxx : Offline    Network 10.x.xx.xxx : Offline    Network 10.x.xx.xxx : Offline    Network 10.x.xx.x : Online    Network 139.xxx.x.x : Online    Network 208.xx.xxx.xxx : Online    Network 193.xxx.xxx.x : Online   which I'd like to clean up to get something like this:    Network 10.x.xx.xxx       : Online   Network 10.x.xx.xxx       : Offline    Network 10.x.xx.xxx       : Offline     Network 10.x.xx.x         : Online     Network 139.xxx.x.x       : Online   Network 208.xx.xxx.xxx    : Online     Network 193.xxx.xxx.x     : Online   Network 193.xxx.xxx.xxx   : Offline