Can I travel to other Schengen countries with a single-entry visa type D?
Can I visit Schengen countries on a Type-D Schengen Visa?
Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$? I would like to enable the root account on my Ubuntu installation.  I realize the security implications of this. Showing that $G/(H\cap K)\cong (G/H)\times (G/K)$ Suppose that $H$ and $K$ are normal subgroups of a group $G$ such that $HK=G$. I need to show that  $G/(H\cap K)\cong (G/H)\times (G/K)$.  So from the second isomorphism theorem we have that:  $HK/H\cong H/(H\cap K)$ which gives that $G/H\cong H/(H\cap K)$  $HK/K\cong K/(H\cap K)$ which gives that $G/K\cong K/(H\cap K)$  So we must now show that:  $H/(H\cap K)\times K/(H\cap K)\cong G/(H\cap K)$  So consider the map $\theta:H/(H\cap K)\times K/(H\cap K)\rightarrow G/(H\cap K)$ such that $\theta(h(H\cap K)\times k(H\cap K))=hk(H\cap K)$  Now this is a homomorphism and is surjective. To check injectiviy we see that if:  $\theta(h_1(H\cap K)\times k_1(H\cap K))=\theta(h_2(H\cap K)\times k_2(H\cap K))$ then we have that:  $(h_1k_1(H\cap K))=(h_2k_2(H\cap K))$ so that $(h_1k_1)(h_2k_2)^{-1}(H\cap K)=(H\cap K)$ which gives that $h_1k_1(H\cap K)=h_2k_2(H\cap K)$ So we are done.  Is the above correct?  Thanks for any help Which metric to use to evaluate highly imbalance classification model performance I have to do classification model to predict the possibilities of person getting cancer based on certain attributes. The data is highly imbalanced. As per client requirement I have to report model performance based on any one of these 4 metrics AUC_Weighted normalized root mean square log loss Accuracy  Can you suggest me which one should I use? How to make a HTML Page in A4 paper size page(s)? Forcing linebreaks in \url Verification that $\lim_{x \to a}\frac{f(x)-f(a)}{x-a}$ and $\lim_{h\to 0}\frac{f(a+h) -f(a)}{h}$ are equivalent definitions of the derivative. I wanted to verify that for definition of the derivative it is true that:  $$ \lim_{x \to a}\frac{f(x)-f(a)}{x-a}= \lim_{h\to 0}\frac{f(a+h) -f(a)}{h}$$  If I denote $h=x-a$,  we can  let $x\to a$, this means that $h \to 0$. Also notice that $f(x)=f(a+h)$ $$ \lim_{x \to a}\frac{f(x)-f(a)}{x-a}=\lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$$ As desired. Is the main idea here really just to use a variable substitution? I suppose there is some formal theorem I also use here that when $f(x) \to y$  we can let $g(f(x)) \to g(y)$ if the function $g$ is continuous. I'm having trouble writing a regular expression that matches valid IPv6 addresses, including those in their compressed form (with :: or leading zeros omitted from each byte pair).    Can someone suggest a regular expression that would fulfill the requirement?  I'm considering expanding each byte pair and matching the result with a simpler regex. How to setup Java-Home and Path in Ubuntu 12.04 Storing Images in DB - Yea or Nay? proof that on a compact manifold a vector field is complete I would like to know if my proof of the following statement is correct     If $M$ is a compact manifold, then every vector field $X$ over $M$ is complete.   Proof:  I take $p\in M$ and $(\gamma_p,I_p)$ a maximal integral curve through $p$. $I_p$ is open and non empty so if $I_p$ is closed, it is equal to $\mathbb{R}$.  So take $s\in \text{Adh}(I_p)$. There exists $t_i\rightarrow s, t_i\in I_p$. Define $q$ as $\text{lim}(\gamma_p(t_i))$ and take a maximal integral curve passing by $q$ as $(\gamma_q,I_q)$. Then since the vector fields are $C^\infty$, $\gamma_q(I_q)\cap\gamma_p(I_p)$ is nonempty (intuitively, this is clear but don't really know how to show it).  On the intersection, $\gamma_p$ and $\gamma_q$ coincide. So we can extend the maximal curve $\gamma_p$ to include $\gamma_q$. This implies that $q\in\gamma_p(I_p)$ and is actually equal to $\gamma_p(s)$  I think the main idea is there even though it probably lacks a bit of rigour. If $S$ is a finitely generated graded algebra over $S_0$, $S_{(f)}$ is finitely generated algebra over $S_0$? Let $S = \bigoplus_{n\ge 0} S_n$ be a graded commutative ring. Let $f$ be a homogeneous element of $S$ of degree $&gt; 0$. Let $S_{(f)}$ be the degree $0$ part of the graded ring $S_f$, where $S_f$ is the localization with respect to the multiplicative set $\{1, f, f^2,\dots\}$. Suppose $S$ is finitely generated algebra over $S_0$. Then $S_{(f)}$ is a finitely generated algebra over $S_0$. What is the "wheel" user in OS X? Implement ```-style (fenced) Markdown code blocks Currently, Stack Exchange’s Markdown parser only allows four-space indents to represent code blocks:      // some code     // another line of code    and other Markdown implementations allow for an alternative syntax, that doesn’t require indenting each code line:  ``` // some code // another line of code ```   This is much more convenient to type out. It would be super useful if Stack Exchange could support this syntax.    By extension, this syntax also allow you to specify the source language right after the opening ```:  ```js // some code // another line of code ```   …which would then enable syntax highlighting for that specific language. Although it’s interesting metadata, I don’t think this feature is needed on Stack Overflow, as the syntax highlighting library works pretty well for various languages.  So, even if you would allow this syntax but ignore the ```language, this would greatly improve my productivity on Stack Exchange.  What do you think?  The moderators are currently collecting feedback regarding this feature, so please post a comment with your thoughts. Good idea? Bad idea? Don’t really care? What do you think are the benefits/drawbacks? Experiences? Let them know! Thanks! Solution for PHP / MySQL website loading fast in multiple regions Does "She had a brother" mean that her brother is dead? My Friend has sent me this riddle to solve:     "If you solve this, then you are a genius. Mr.Smith has 4   daughters🙍‍♀🙍‍♀🙍‍♀🙍‍♀ and each daughter had a brother.  How many   children does Mr. Smith have now?"   He said the answer is not 5, but 4 only as in the riddle it is mentioned as each daughter had a brother. So their only brother is dead.     Is he right? Types of polynomial functions. Quadratic, cubic, quartic, quintic, ...,? I would very much like to have a complete list of the types of polynomial functions. I know that theres:  Quadratic :                      (AX^2 + BX + C) Cubic     :               (AX^3 + BX^2 + CX + D) Quartic   :        (AX^4 + BX^3 + CX^2 + DX + E) Quintic   : (AX^5 + BX^4 + CX^3 + DX^2 + EX + F)   What are the names of polynomial functions to the further powers? Sexstic? Septic? Octic? I'm not sure, so if someone could enlighten me, then that would be great. Please provide a list that goes at least to the seventh power, but it would be nice if you could go further. Thanks. Layover at Narita, Japan. Do I need a transit visa? Why can't the "Launcher" Icons be made smaller? At the smallest setting, they are still 4 times bigger than they need to be on a 10" Netbook. The Icons in the "Top Panel" are much smaller and we users don't have a problem selecting them. 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.