This decision means what?      Sufficient justification for the purpose and circumstances of the   intended stay was not provided. The specific purpose for which the   visa was applied for is no longer valid or will cease to be valid   before you arrive in the member state of destination. No other   (secondary) purpose was stated at the time of the application, of if   it was, insufficient documentation was provided to demonstrate the   purpose of the stay. Therefore, your application for a visa is   refused.
Schengen refusal formulae can be difficult to understand sometimes.    When an applicant receives a refusal with "Justification for the purpose and conditions of the intended stay was not provided", what does it mean?
I recently read that:     What our current level of thermodynamics does tell us though is that if there is such a thing as a Big Crunch, the universe will look a lot different as it heads towards collapse than it did when expanding – the universe started in a state of low entropy and high order and will end in a state of high entropy and   low order.   But shouldn't the order be in a high state and hence entropy becomes a low state in the Big Crunch (assuming a closed universe)?  The reason I write this is because the gravitational forces in a closed universe are so strong they will cause gravitational collapse so the universe will shrink and go backwards, eventually through all the stages of the big bang and become a singularity once more.   It was my understanding that a singularity has a state of high order and low entropy. Have I got this the wrong way round?   If so; Can someone please explain why? formatting long numbers as strings in python What is an easy way in Python to format integers into strings representing thousands with K, and millions with M, and leaving just couple digits after comma?  I'd like to show 7436313 as 7.44M, and 2345 as 2,34K.  Is there some % string formatting operator available for that? Or that could be done only by actually dividing by 1000 in a loop and constructing result string step by step? What is the pressure at the center of the Earth? (Or a neutron star) The  has microgravity and is thought to be solid and almost 6000 degrees hot. What is the pressure like there? The confusing thing, for the common man I suppose, is that there's no gravity there, but lots of matter around it which does gravitate towards the center. How does that add up? Is the center just a gas cloud or is it highly compressed?  And I could add the aside question about what it is like in the microgravity center of a  or a black hole, but I don't expect anyone to know about that. Let $x*y=\gcd(x,y)$ on $D_n=\{x\in \mathbb N:x |n\}$. I have to prove that $(D_n,*)$ is a commutative monoid and I have a problem when finding the identity element.  $x*e=x$  So $gcd(x,e)=x \to e=xk,k\in \mathbb N^*$ and $e \leq n$  A good $e$ is $0$ but it's not in $D_n$ In statistical learning, implicitly or explicitly, one always assumes that the training set $\mathcal{D} = \{ \bf {X}, \bf{y} \}$ is composed of $N$ input/response tuples $({\bf{X}}_i,y_i)$ that are independently drawn from the same joint distribution $\mathbb{P}({\bf{X}},y)$ with  $$ p({\bf{X}},y) = p( y \vert {\bf{X}}) p({\bf{X}}) $$  and $p( y \vert {\bf{X}})$ the relationship we are trying to capture through a particular learning algorithm. Mathematically, this i.i.d. assumption writes:  \begin{gather} ({\bf{X}}_i,y_i) \sim \mathbb{P}({\bf{X}},y), \forall i=1,...,N \\ ({\bf{X}}_i,y_i) \text{ independent of } ({\bf{X}}_j,y_j), \forall i \ne j \in \{1,...,N\} \end{gather}  I think we can all agree that this assumption is rarely satisfied in practice, see this related  and the wise comments of @Glen_b and @Luca.  My question is therefore:      Where exactly does the i.i.d. assumption becomes critical in practice?    [Context]   I'm asking this because I can think of many situations where such a stringent assumption is not needed to train a certain model (e.g. linear regression methods), or at least one can work around the i.i.d. assumption and obtain robust results. Actually the results will usually stay the same, it is rather the inferences that one can draw that will change (e.g. heteroskedasticity and autocorrelation consistent HAC estimators in linear regression: the idea is to re-use the good old OLS regression weights but to adapt the finite-sample behaviour of the OLS estimator to account for the violation of the Gauss-Markov assumptions).   My guess is therefore that the i.i.d. assumption is required not to be able to train a particular learning algorithm, but rather to guarantee that techniques such as cross-validation can indeed be used to infer a reliable measure of the model's capability of generalising well, which is the only thing we are interested in at the end of the day in statistical learning because it shows that we can indeed learn from the data. Intuitively, I can indeed understand that using cross-validation on dependent data could be optimistically biased (as illustrated/explained in ).   For me i.i.d. has thus nothing to do with training a particular model but everything to do with that model's generalisability. This seems to agree with a paper I found by Huan Xu et al, see "Robustness and Generalizability for Markovian Samples" .   Would you agree with that?  [Example]   If this can help the discussion, consider the problem of using the LASSO algorithm to perform a smart selection amongst $P$ features given $N$ training samples $({\bf{X}}_i,y_i)$ with $\forall i=1,...,N$ $$ {\bf{X}}_i=[X_{i1},...,X_{iP}] $$ We can further assume that:   The inputs ${\bf{X}}_i$ are dependent hence leading to a violation of the i.i.d. assumption (e.g. for each feature $j=1,..,P$ we observe a $N$ point time series, hence introducing temporal auto-correlation) The conditional responses $y_i \vert {\bf{X}}_i$ are independent. We have $P \gg N$.    In what way(s) does the violation of the i.i.d. assumption can pose problem in that case assuming we plan to determine the LASSO penalisation coefficient $\lambda$ using a cross-validation approach (on the full data set) + use a nested cross-validation to get a feel for the generalisation error of this learning strategy (we can leave the discussion concerning the inherent pros/cons of the LASSO aside, except if it is useful). OSX keeps switching to the first space Radius of Convergence of power series of Complex Analysis I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be greatly appreciated!  (i) $\displaystyle\sum_{n=0}^\infty \bigg(\dfrac{1}{2n}\bigg)z^{3n}$.  (ii) $\displaystyle\sum_{n=0}^\infty(2^n+i)(z-i)^n$.  (iii) Show that the radius of convergence of $\displaystyle\sum_{n=0}^\infty z^{n!}$ is $1$ and that there are infinitely many $z \in \mathbb{C}$ with $|z|=1$ for which the series diverges. How to install Skype 4.3? I'm about to travel to the US, where iPods / AirPods / iPhone are much cheaper than in my country.   I wonder if Apple's warranty is international, and/or what warranty coverage exists when I return to my country.  Where can I understand what Apple Warranty covers in the US when products leave the US? High Voltage Powerlines - clarification of energy loss Where do I find the client id and client secret of an existing connected app? Why do some websites show the company name next to the URL? Some websites show the company name next to the URL (in Chrome) and some don't.  How do I set this up for my website? Is there a reason why or why I wouldn't want to do this? What is the proper name for a "river plot" visualisation In a famous plot, Charles Minard visualised the losses of the French Army in the Russian campaign of Napoleon:    (another nice example is  xkcd plot)  Is there a canonical name for this type of visualisation? I'm actually looking for an R package to create such plots, but I don't even know how to look for it.  EDIT:  As I could not find a good package in R do do this type of plots, I have created my own, called "riverplot" -- you can download it . Here is a simplified version of the above diagram:    And an example of what other diagrams can be created with the package: How are Anonymous inner classes used in Java? Is $\int_{-\infty}^{\infty} \sin x \, \mathrm{dx}$ divergent or convergent? I was determining whether   $$\int_{-\infty}^{\infty} \sin x \, \mathrm{dx}$$  was divergent or convergent. So, I did the following steps:  $$\begin{align} \int_{-\infty}^{\infty} \sin x \, \mathrm{dx} &amp;= \int_{0}^{\infty}\sin x \, \mathrm{dx}+\int_{-\infty}^0\sin x \, \mathrm{dx} \\ &amp;=\lim_{t\rightarrow\infty} \left(-\cos x |^{t}_{0}\right) + \lim_{a\rightarrow-\infty} \left(-\cos x |^{0}_{a}\right)\\ &amp;=\lim_{t\rightarrow\infty} -\cos (t) + \cos 0 + \lim_{a\rightarrow-\infty} -\cos 0 + \cos a\\ &amp;=\lim_{t\rightarrow\infty}1 - \cos t + \lim_{a\rightarrow-\infty} -1+\cos a \end{align}$$  Now, at this point, it would be reasonable to say that both the limits are undefined and therefore, the integral is divergent but then if I try something like the following   \begin{align} \quad\quad&amp;=\lim_{t\rightarrow\infty}1 -\cos t + \lim_{a\rightarrow\infty} -1+\cos a \\ \quad\quad&amp;=\lim_{t\rightarrow\infty}-1 -\cos t + \lim_{a\rightarrow\infty} \cos a+1 \\ \quad\quad&amp;=\lim_{b\rightarrow\infty}-1 +\cos b - \cos b+1 \\ \quad\quad&amp;= 0 \end{align}  So, as you can see, it was shown before that the integral is divergent but with some manipulation, we came at an answer of $0$ but is that valid? I assume, a similar technique can be applied to $\int_{-\infty}^{\infty} \frac{1}{x} \, \mathrm{dx}$. Proof that every polynomial of odd degree has one real root I want to prove that every real polynomial of odd degree has at least one real root, using  the intermediate value theorem.  Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in \mathbb{R}$ and $n \in \mathbb{N}$.  By the fundamental theorem of algebra I know that $P(x)$ has exactly $2n+1$ complex roots, so  $P(x) = (x+r_1)(x+r_2) . . . (x+r_{2n+1})$ for each $r_i \in \mathbb{C}$  I do not know how to complete this but I do know that, at some point, I probably have to show that each root with imaginary part non zero has to come in conjugate pairs, and since $2n+1$ is odd there is at least $1$ root that is imaginary part $0$ and thus real. Of all the people we know, in the books, who were sent to Azkaban, the majority are Death Eaters (Dolohov, Lucius, Crouch Jr, Bellatrix and so on) or suspected to be involved with them (Sirius, Stan, et cetera). They were imprisoned for using the Unforgivable Curses, as well as for being accomplices of a seriously dark wizard. What other crimes are Azkaban-worthy, in normal circumstances? I'm looking for anything that's been stated in the books, or on Pottermore (basically, anything 'canon'), or anything one can conclude from information in the books. I think smaller crimes are punished by imposing a fine, but what about bigger ones? Surely murder and torture (i.e. crimes against humanity) aren't the only ones punishable by imprisonment? That certainly isn't the case in contemporary Muggle society. My thoughts Off the top of my head, I can name these crimes:  Being an unregistered Animagus1 That was what Hermione threatened Skeeter with.  Trespass and robbery2 That's how Sturgis Podmore was imprisoned.   Notes  Several of the arrests in the Half-Blood Prince and the Deathly Hallows were made by an authoritarian (if that's the right word on this case) government, in the time of war. They do not count as normal.  I don't mean that there should be an answer to this question, I was wondering if it was ever addressed in the books. I certainly don't expect a list of all people sent to Azkaban, and the reasons for doing so!     1 &quot;Well, yes,&quot; said Hermione calmly, taking a sip of her drink. &quot;Otherwise, as you very well know, I will inform the authorities that you are an unregistered Animagus. Of course, the Prophet might give you rather a lot for an insider's account of life in Azkaban.&quot;  The Beetle at Bay, Harry Potter and the Order of the Phoenix     2 Sturgis Podmore, 38, of number two, ... charged with trespass and robbery ... sentenced to six months in Azkaban.  Percy and Padfoot, Harry Potter and the Order of the Phoenix Recently, the titles to some questions have been displaying as "undefined":    "undefined" is replaced with the question's true title upon refreshing the page. Is this a bug? 9 rows is just way too small by default. Is this a text-editor for ants? Conditional distributions of the multivariate normal