Departing from airport before visa is valid, but arriving when it is
If my Schengen visa starts on the day of arrival which is one day ahead of departure, would they let me get into the plane?
Can principal component analysis be applied to datasets containing a mix of continuous and categorical variables? I have a dataset that has both continuous and categorical data. I am analyzing by using PCA and am wondering if it is fine to include the categorical variables as a part of the analysis. My understanding is that PCA can only be applied to continuous variables. Is that correct? If it cannot be used for categorical data, what alternatives exist for their analysis? Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$ This is half of Theorem 3.19 from Baby Rudin. Rudin claims the proof is trivial. What I've come up with so far doesn't seem trivial, however, and is probably also wrong (my problem with it is pointed out below). This makes me wonder whether I'm overlooking some useful fact and/or using an unprofitable approach.  Theorem. If $s_n \leq t_n$ for $n \geq N$, where $N$ is fixed, then  $$ \liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n. $$  Proof. Suppose that $s_n \leq t_n$ if $n \geq N$, but that  $$ \liminf_{n \rightarrow \infty} s_n &gt; \liminf_{n \rightarrow \infty} t_n. $$  Let $E_s$ denote the set of all subsequential limits of $\{s_n\}$, and let $E_t$ denote the set of all subsequential limits of $\{t_n\}$. Then  $$ \inf E_s &gt; \inf E_t. $$  This implies that there exists some $x \in E_t$ such that $\inf E_s &gt; x &gt; \inf E_t$, since otherwise $\inf E_t$ would not be the greatest lower bound of $E_t$. Hence some subsequence of $\{t_n\}$, say $\{t_{n_i}\}$, converges to $x &lt; \inf E_s$.  Lemma. (from Rudin) If $x &lt; \liminf_{n \rightarrow \infty} s_n$, then there exists an integer $N$ such that if $n \geq N$, then $s_n &gt; x$.   By the lemma, there exists an integer $N_0$ such that if $n \geq N_0$, then $s_n &gt; x$.   Now, let $\epsilon = \inf_{n \geq N_0} \{s_n - x\}$. This is where I think my proof breaks down. Can't $\epsilon$ be zero? Then, since $\{t_{n_i}\}$ converges to $x$, there exists an integer $N_1$ such that if $n_i \geq N_1$, then $|t_{n_i} - x| &lt; \epsilon$. But this means that, if $n_i \geq \max\{N, N_0, N_1\}$, we have both  $$ s_{n_i} &gt; t_{n_i}, $$  as well as $s_{n_i} \geq t_{n_i}$, a contradiction. Why does dpkg give me this parsing error? How to print "$" in here-document How can I echo "$" in a here-doc in bash?  For example, I want to have a file with the content on remote server like $ABC=home_dir.  $ ssh hostname sudo -s &lt;&lt;EOF echo "$ABC=home_dir" &gt; file EOF   But it would be treated as a variable. How can I print a literal $? Proving $\mathbb{Q}$ isn't homeomorphic to $\mathbb{Z}$ If an enemy is just below a 10-foot-high ceiling, are they in melee range of a creature on the ground? Edit a truetype font I'm trying to find a free tool (on windows ) that would enable me to change some settings on a TrueType font. Specifically, I'm trying to fix the following errors:  E1311 The macStyle bold bit doesn't match the name table's font subfamily string macStyle bold bit is set, but subfamily is Regular  W2000 The subfamily string is inconsistent with the style of the font OS/2.fsSelection bold bit is set, but subfamily string = 'Regular'  I can do that with fontographer, but would prefer to use something free.  Fontforge doesn't seem to work with this font, when saved it corrupts the cyrillic letters in the font. I would like to know if my brother can fly by United Airlines from USA to El Salvador, and my father, from El Salvador to USA using the same round trip ticket, both of them have the same name. With the release of iOS 6, the most commonly used mobile browser now supports standard generic image upload buttons. This is actually really handy.  I can take screenshots on my phone or iPad, and use them to answer questions about apps made to run on those devices (or ) to answer questions. Until the release of iOS 6 however, the only way to use these images in my answer was to either sync them over to my computer, and then answer from a desktop, or use a specialized app to upload the images, such as PicUp or DropBox, and then manually write out the markdown code.  After iOS 6, the Image Upload button works just fine, if I switch to viewing the full site from my iDevice, rather than the mobile view. I can upload images directly, and the markdown is handled for me. It's awesome, except for the fact that the non-mobile site is decidedly non-awesome on the small screen of my phone.  Which brings me to my feature request: Add the Image Upload button (and other markdown shortcuts, as deemed worthwhile) to the ask/answer editors on the mobile version of SE sites. "ignorance" can also mean a "willful disregard of something"? I try to use the 2nd or 3rd definition of words (to slow down speed readers). However, I am not sure if "ignorance" has a 2nd definition.  (common meaning)  is "a lack of knowledge". With this meaning, no syllable is stressed.   My ignorance of hardware means I cannot build my own pc.   (2nd meaning?) If pronounced differently, I think that "ignorance" can also mean "a willful and careless disregard of something". The "nor" syllable in "ig-NOR-ance" must be stressed.     The French were aware of the strategic weakness of the Maginot Line, and their ignorance of it enabled the German blitz through the Low Countries.   My ignorance of my girlfriend's hourly text messages quickly caused our break-up.   In a dictionary, "ignorance" is only defined as "lack of knowledge". But, I am not talking about that "ignorance". I am talking about the "ignorance" that is derived from "to ignore" and thus pronounced just like "to ignore", and thus has the meaning of "to ignore". There is precedence for this:  "to accept" --- derive ---> "acceptance" "to ignore" --- derive ---> "ignorance" ....  Anyone agree with me? Show $f$ is analytic if $f^8$ is analytic This is from Gamelin's book on Complex Analysis.  Problem: Show that if $f(z)$ is continuous on a domain $D$, and if $f(z)^8$ is analytic on $D$, then $f(z)$ is analytic on $D$. (I assume the intention is that $D$ is nice, i.e. open and connected)  I'm not exactly sure how to approach this. I'm guessing it has something to do with zeroes of $f$. For example, around non-zeroes, $f$ is analytic since $z^{1/8}$ is nonzero in a neighborhood. Or something along those lines. The details are eluding me.  Any help would be appreciated! I'm starting  and finally decided to learn how to write epsilon-delta proofs.  I have been working on chapter 5, number 3(ii). The problem, in essence, asks to prove that     $$\lim\limits_{x \to 1} \frac{1}{x} = 1.$$    Here's how I started my proof,     $$\left| f(x)-l \right|=\left| \frac{1}{x} - 1 \right| =\left| \frac{1}{x} \right| \left| x - 1\right| &lt; \epsilon \implies \left| x-1 \right| &lt; \epsilon |x|$$   I haven't made any further progress past this point. Is it possible to salvage this proof? Should I try an alternate approach? Icon missing when application is launched I've created an application launcher in /usr/share/applications that launches rdesktop. The file specifies an icon name and I've added the icons to:   /usr/share/icons/HighContrast/48x48/apps /usr/share/icons/HighContrast/22x22/apps /usr/share/icons/HighContrast/256x256/apps /usr/share/icons/HighContrast/32x32/apps /usr/share/icons/HighContrast/scalable/apps-extra /usr/share/icons/HighContrast/24x24/apps /usr/share/icons/HighContrast/16x16/apps /usr/share/icons/hicolor/scalable/apps /usr/share/icons/hicolor/128x128/apps /usr/share/icons/hicolor/16x16/apps   I then drug the launcher to the taskbar and the launcher with the correct logo is visible in the taskbar. However, when I start the task, a new icon is created in the taskbar, a square with a question mark in the center.  What icons do I need and where, so that the taskbar displays the correct icon for the application when it is launched?  The .desktop file contents:   [Desktop Entry] Version=1.0 Terminal=false Icon=mswindows Type=Application Categories=Network; Exec=rdesktop win7shared1 -g 1680x1000  Name=win7shared1 GenericName=rdesktop Comment=Open windows desktop  Keywords=Windows StartupNotify=false Position of a relative clause before/after a verb Proving $(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$ for any events $A$ and $B$ Show that for any two events $A$ and $B$, $$(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$$   Using the Inclusion-Exclusion formula, De Morgan's law and $P(AB^c)=P(A)-P(AB)$, I was able to expand the $\rm{l.h.s.}$ as $$(x+y)(x+y-4z)+2z(2z+1)+x(x-2)+y(y-2)+1\quad...\quad(*),$$ where $x=P(A),y=P(B)$ and $z=P(AB)$.  However I'm not being able to group the terms involved in the $\rm{l.h.s.}$ to form a 'suitable' expression and probably the way I have grouped them is not correct.  The expression $(*)$ is subject to the condition   $\max\{0,x+y-1\}\le z\le \min\{x,y\}$ and obviously $0\le x,y,z\le 1$, but I am unable to use this.      Is there an elegant/slick way to show the inequality? It probably can be shown without a lot of work and it's possible I'm missing something trivial here. Mutineers force captain to record instructions to spaceship's computer but he leaves out "please" The crew mutinies on a spaceship whose captain controls it with verbal orders. They test whether they can control the ship with recordings of the captain's voice with an order to increase orbital speed (currently orbiting a planet). It works, so they force the captain to record all the possible orders they may need, and execute him. Only then do they realize that the ship obeys only orders couched in polite language, and only the test recording was like this, but the other recordings are not. They will die once the ship's faster orbit somehow causes it to run out of fuel, or something like that. I seem to remember the ship's &quot;computer&quot; following the captain around on robotic legs. I read the story circa 1980, but it might have been published much earlier. Proving $\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$ without induction I was looking at: $$\sum_{k=1}^n{k^2}=\frac{n(n+1)(2n+1)}{6}$$  It's pretty easy proving the above using induction, but I was wondering what is the actual way of getting this equation? Why does PGFplots not recognize my expression? Order of calling base class constructor from derived class initialization list I'd like to use a background image for my world, but I'm not sure how to do it.   Right now I am using the sun as a lightpoint and it looks like sky and a flat plane (not realistic). I was thinking of just using a mountain background image for the world.  I've done that through textures etc. but I don't see ANYTHING in render even though it shows in worlds?  What am I doing wrong?    Also what's a cheap quick fix for cases like mine so it looks like the world isn't JUST blue sky?  Thanks guys!