My mother-in-law is currently taking care of my borther's new born baby. We applied for a 2 year Standard Visitor visa, assuming she would be stay for 6 months at a time, with a month or two break in-between coming back to South Africa.   Upon entry in the UK, border control informed her the limitation is not per visit, but she would be able to return again within the same 12 month cycle. In other words, she would have to return back to South Africa for at least 6 months before she would be able to return again to the UK.   The page explaining the visa and it's purposes, is not very clear and open to interpretation, which is why we chose the 2 year Standard Visitor visa in the first place. All information and documentation supplied supports out initial intent as well.     Question: would she be allowed back in, considering the situation, without a 6 month break?
A relative of mine has received a multiple-entry UK visa for 2 years. Is there a particular limit on how many months in total per year she can stay in UK? The way I understand the current rule is that such visitors can enter UK as many times as they want as long as they do not stay for longer than 6 months per single visit. However, does this mean that if she stayed with us for 5 months and went back home for 1-2 months, she could come and stay for another 5 months? I heard that it is advisable to spend only 180 day per calendar year in UK and not more, but I could not find that rule in any official UK Home Office documents. Thanks in advance for any info about this.
Can anyone hint me to prove:  $\forall n\in \mathbb{N}: \exists$ Fibonacci numbers $ F_{i_1},\ldots,F_{i_k}$ such that: $$\sum F_{i_k}=n$$    Note: Every Fibonacci number can appear only once. Thanks Did Stephen Baxter or Arthur C. Clarke's estate ever comment on continuing "A Time Odyssey"? 2007 book "Firstborn", the third and last book of "" trilogy by Stephen Baxter and Arthur C. Clarke - an "orthoquel" to "2001: A Space Odyssey" - ends with a cliffhanger which obviously implies continuation:      Myra said, “My God.Charlie? ”  Bisesa stared. “Who is ‘we’?”  “We call ourselves the Lastborn. We are at war. We are losing.” She held out her hands. “Please. Come with me now.”  Bisesa and Myra, still hugging each other, reached out their free hands. Their fingertips touched Charlie’s.  A clash of cymbals.   However, sadly, Sir Arthur died in 2008, with no further books forthcoming.  Did Stephen Baxter  or Arthur C. Clarke's estate ever comment on continuing "A Time Odyssey" after 2008?  Wikipedia has no info:     The 2008 Gollancz edition of the most recent book describes itself on the cover as the "Conclusion" of the series, but the plot is not concluded in the book. The immediate threat is averted but the enemy is not defeated, suggesting that further novels in the series were intended. With the death of Clarke, one of the two authors, it is not clear whether these will be written. Is there a clever solution to Arnold's "merchant problem"? There is a problem that appears in . The problem is also quoted .  You take a spoon of wine from a barrel of wine, and you put it into your cup of tea. Then you return a spoon of the (nonuniform!) mixture of tea from your cup to the barrel. Now you have some foreign substance (wine) in the cup and some foreign substance (tea) in the barrel. Which is larger: the quantity of wine in the cup or the quantity of tea in the barrel at the end of your manipulations?  Here's my solution: The key is to consider the proportions of wine and tea in the second spoonful (that is, the spoonful of the nonuniform mixture that is transported from the cup to the barrel). Let $s$ be the volume of a spoonful and $c$ be the volume of a cup. The quantity of wine in this second spoonful is $\frac{s}{s+c}\cdot s$ and the quantity of tea in this spoonful is $\frac{c}{s+c}\cdot s$. Then the quantity of wine left in the cup is $$s-\frac{s^2}{s+c}=\frac{sc}{s+c}$$ and the quantity of tea in the barrel now is also $\frac{cs}{s+c}.$ So the quantities that we are asked to compare are the same. However, Arnol'd also says  Children five to six years old like them very much and are able to solve them, but they may be too difficult for university graduates, who are spoiled by formal mathematical training.  Given the simple nature of the solution, I'm going to guess that there is a trick to it. How would a six year old solve this problem? My university education is interfering with my thinking. Very simply put:  I have a class that consists mostly of static public members, so I can group similar functions together that still have to be called from other classes/functions.  Anyway, I have defined two static unsigned char variables in my class public scope, when I try to modify these values in the same class' constructor, I am getting an "unresolved external symbol" error at compilation.  class test  { public:     static unsigned char X;     static unsigned char Y;      ...      test(); };  test::test()  {     X = 1;     Y = 2; }   I'm new to C++ so go easy on me. Why can't I do this? Proving $\lim_{n\to\infty} a^{\frac{1}{n}}=1$ by definition of limit given a sequence $a_n=a^{\frac{1}{n}}$ for $n\in\mathbb{N}^*$, $a\in\mathbb{R},a&gt;1$ then proof that $\lim\limits_{n\to+\infty}a_n=1$ by definition.   proof:  given $a_n=a^{\frac{1}{n}}$ for $a\in\mathbb{R},a&gt;1$.  for $n\in\mathbb{N}^*,n+1&gt;n\Rightarrow \frac{1}{n+1}&lt;\frac{1}{n}$ and because $a&gt;1$ we gets $a^{\frac{1}{n+1}}&lt;a^{\frac{1}{n}}$, since $\frac{1}{n+1}&gt;0$, then $1=a^{0}&lt;a^{\frac{1}{n+1}}&lt;a^{\frac{1}{n}}\Rightarrow 1&lt;a_{n+1}&lt;a_{n}$.  then we proof that $\lim\limits_{n\to+\infty}a_n=1$ we need to show that $\forall\epsilon&gt;0,\exists N,\forall n&gt;N,|a_n-1|&lt;\epsilon$  then for $\epsilon&gt;0$, choose $N=\frac{1}{\log_a(\epsilon+1)}$, since $a&gt;1$ we gets that $$\forall n&gt;N,1&lt;a^{\frac{1}{n}}&lt;a^\frac{1}{N}\Rightarrow0&lt;a^{\frac{1}{n}}-1&lt;a^{\frac{1}{N}}-1\Rightarrow |a^{\frac{1}{n}}-1|&lt;|a^{\frac{1}{N}}-1|=|a^{\log_a(\epsilon+1)}-1|=|\epsilon+1-1|=\epsilon\Rightarrow |a^{\frac{1}{n}}-1|&lt;\epsilon$$  wich implies that $\lim\limits_{n\to+\infty}a_n=1\square$  my proof to the limit is correct? If G is not commutative, then is there always a subgroup that is not a normal subgroup? How can I get this 'Version: 0.1.0 - 94" value from the  tag in selenium webdriver? How can I get this Version: 0.1.0 - 94 value from the &lt;div&gt; tag in selenium webdriver? and here is my code:  &lt;div _ngcontent-qbk-c3="" style="text-align: center;"&gt;Version: 0.1.0 - 94&lt;/div&gt; What is the VC dimension of a decision tree? What is the  of a decision tree with k splits in two dimensions? Let us say the model is CART and the only allowed splits are parallel to the axes.  So for one split we can  and then for any labeling of the points we could get perfect prediction (i.e.: shattered points)  But what about 2 splits, or any general k? No notification (+15) for accepted answer I'm referring to the colored circles with numbers next to them on the top of the page, to the right of my reputation counter. What do they represent? What are acceptable reasons for flagging as "Requires Moderator attention"? After reading , I feel I don't quite understand this function. What are the guidelines for flagging a question as &quot;Requires Moderator attention&quot;? Vandermonde identity corollary $\sum_{v=0}^{n}\frac{(2n)!}{(v!)^2(n-v)!^2}={2n \choose n}^2$ I am trying to prove this identity:  $$\sum_{v=0}^{n}\frac{(2n)!}{(v!)^2(n-v)!^2}={2n \choose n}^2$$ I think this identity (corollary of Vandermonde identity): $${n\choose 0}^2+{n\choose 1}^2+{n\choose 2}^2+\cdots+{n\choose n}^2={2n \choose n}$$ is applicable for solving it. Please give me some hints. thank you Questions in titles need to be repeated, why?  To me this pattern seems ridiculous. But I frequently encounter the "not a real question" situation, when I put my question in the title, then the context for that specific question in the 'question body'.     So:  I am to take it that the title is almost completely irrelevant for establishing subsequent question body is asking? I choose my titles to expose the actual question so it need not be repeated below. If questions need to be repeated in the body, could we simply title it "A question, click here and read below" ?   Tagged bug, as I have found this "problem" to be quite reproducible.  Steps to reproduce:   Ask a question with the clear, simple question in the title, and the evidence/context/example in the question body. Await close votes. A Regular Tetrahedron is a cool Polyhedron. A regular tetrahedron has this property:  For any two of its vertices exists a third vertex, which forms a regular triangle with these 2 vertices.(It doesn't necessarily have to be a face of it).  Are there any other polyhedrons that have the same property?  I think there isn't such a polyhedron. But not sure how to prove it. I've tried proving that there isn't an irregular tetrahedron with this property, assuming that there is a pair of unequal sides.   Current answers are a little confused with the conditions of the problem. Fantasy novel about kid who controls magic through coin changer device I read this book in high school, and I can't remember what it's called. Some kid learns he has magical ability, but only finds out later that he controls it through some kind of coin changer he's been tinkering with the whole book. Given a joint density function, what is the conditional expectation $E[Y|x]$? The random variables $X$ and $Y$ have the joint density $$f_{X,Y}(x,y)= \left\{\begin{matrix}e^{-y}, \mbox{ } 0\leq x \leq y \le \infty \\ 0,  \mbox{   otherwise}\end{matrix}\right.$$ Evaluate the conditional expectation $E[Y|x]$.    I'm not sure if I'm doing this correctly. Please let me know if something is off in my solution. Thanks for reading.  My solution: (My expected value is negative, which I'm pretty sure can never be the case. Also in line 4 and 8, are the domains correct?)  $$E[Y|x] = \int_{-\infty}^{\infty}yf_{Y|X}(y|x)dy.$$  $$f_{Y|X}(y|x) = \frac{f_{X,Y}(x,y)}{f_{X}(x)}.$$    $$f_{X}(x) = \int_{-\infty}^{\infty}f_{X,Y}(x,y)dy$$   $$= \int_{0}^{\infty}e^{-y}dy $$  $$=\left [ -e^{-y} \right]\Big|_0^\infty $$  $$ = -e^{-\infty}-(-e^{0})=1. $$ $$f_{X}(x) = 1, x \geq 0.$$ Thus,   $$E[Y|x] = \int_{0}^{\infty}-ye^{-y}dy$$  $$= -\int_{0}^{\infty}ye^{-y}dy$$ After integration by parts,  $$E[Y|x] = \left [ ye^{-y}+e^{-y} \right]\Big|_0^\infty$$ $$=0-[e^{0} + 0]=-1$$ Therefore, $E[Y|x] = -1.$  Thanks in advance for reading this long problem and responding! When a Battle Master fighter gets a critical hit and uses a maneuver that adds a superiority die to the damage, is that damage die doubled? The Battle Master fighter has access to a number of maneuvers that add a superiority die to the damage roll. For instance, the Trip Attack maneuver says (PHB, p. 74):     When you hit a creature with a weapon attack, you can expend one   superiority die to attempt to knock the target down. You add the   superiority die to the attack’s damage roll, and if the target is   Large or smaller, it must make a Strength saving throw. On a failed   save, you knock the target prone.   The Critical Hits rule states (PHB, p. 196):     When you score a critical hit, you get to roll extra dice for the   attack’s damage against the target. Roll all of the attack’s damage   dice twice and add them together. Then add any relevant modifiers as   normal. [...] If the attack involves other damage dice, such as from   the rogue’s Sneak Attack feature, you roll those dice twice as well.   During our game, the fighter PC rolled a critical hit. He then wanted to use a superiority die to use the Trip Attack maneuver to trip the enemy. Based on the above rules, is the damage from the superiority die also doubled?  I'm assuming the answer is yes based on the last statement in the Critical Hits section, but I wanted to confirm. PrintWriter append method not appending Remove array element based on object property How to Prevent the System.QueryException I am getting "Error System.QueryException: List has no rows for assignment to SObject".   Line 1: Account ac = [select id from Account where name='testaccount' limit 1] Line 2: if(ac != null) {           //do something         }   I got the above error in line 2 if there is no record.   Any help?