Travel within Schengen with Titre de Sejour, but expired passport
Can I travel within the Schengen area without a passport if I have my residency permit card?
Does $G\times K\cong H\times K$ imply $G\cong H$? A club has $12$ members. They need to choose a programming team of $3$ members and a hacking team of $4$ members. Members can be on both teams, but Jack only wants to be on at most one team. How many ways can both teams be chosen?  I did two different ways. I solved if Jack chooses programming, $C(12,3)=220$ $C(11,4)=330$. In total there are $550$ different ways both teams can be picked.  But, if Jack chooses hacking, I solved $C(11,3)=165$ and $C(12,4)=495$. In total there are $650$ ways both teams can be chosen.  I was wondering if this is the correct method. I got a flat/puncture on my bike tube, how can I fix it myself without going to the nearest bike shop? What kind of equipment would I need? Can somebody help me to prove this identity?     $$\prod_{k=1}^{n-1} \left(\;\cos\theta-\cos\frac{k\pi}{n}\;\right) = 2^{1-n}\sin(n\theta)\csc\theta$$   I was thinking of using the properties of the complex numbers and the binomial theorem, because I found this problem in the Complex Variables book from Schaum's, but it did not solve my problem.  Does anyone have an idea?  Thanks in advance! What is a good word to describe someone who is empathetic, quick, and witty in conversation What is the music on S3E16 when Todoroki was fighting group of ninjas? How to create id with AUTO_INCREMENT on Oracle? Can you contract the main verb '[I] have' in a sentence? Why is this markdown for a link not showing properly? The Stack Exchange (Personal Productivity) markdown  is showing literally rather than as a named link. Don't know why. Why should a CTE start with a semi-colon? Prove that $\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$ Problem  Suppose $f$ is a non-negative integrable function on a measure space $(X,\mathcal{A},\mu).$ Prove that $$\lim_{t\to \infty} t\mu(\{x:f(x)\geq t\})=0$$  Attempt  Let $E_t=\{x:f(x)\geq t\}$ Note that $f\chi_{E_t}\leq f$. Since $f$ is integrable, the Lebesgue dominated convergence theorem tells us that $$\lim_{t\to\infty} \int f\chi_{E_t}\rightarrow \int \lim_{t\to\infty} f\chi_{E_t}=0,$$ since $\lim_{t\to\infty} f\chi_{E_t}=0$ a.e. $\ ^{(1)}$ The result follows from the fact that $$t\mu(E_t)\leq\int f\chi_{E_t}.$$  Question  I am having trouble proving $(1)$. That is, how can I show that $f\chi_{E_t}\rightarrow 0$ a.e? The maximum energy stored by a capacitor is when it’s (theoretically) fully charged, or just before it reaches dielectric breakdown, if you will. It seems, then, that the maximum energy ($E$) $=QV$, where $Q$ is the maximum charge attained and $V$ is the maximum ‘voltage’. However, the correct equation for the maximum energy is $E=\frac{1}{2}QV$. Why? I’m looking for a simple, intuitive explanation because my reasoning that $E=QV$ seems sound to me. I know about , but I’m looking for a reason to disprove my specific argument that $E=QV$ and not the “Oh, it comes from integration.” response. Grub not showing on startup for Windows 8.1 Ubuntu 13.10 Dual boot Ok, so I'm such a newbie to Ubuntu and I bought a Windows 8 pre-installed laptop last month.  I updated to Windows 8.1 and then I thought about installing Ubuntu as a dual boot so I could mess around and learn more about it.   So I followed a Youtube tutorial (  ) and I got my stuff working fine.   The first few times I booted I got the GRUB menu instead of my default HP Boot OS Manager, and I was able to select my OS.   So I went to sleep and the next day I turned on my computer and the GRUB menu did not show up. I tried several times and it didn't automatically show up.  In order for me to see the GRUB menu I had to turn on my PC and on start had to press ESC to pause startup and press F9 to get boot options. Then from there I had to pick from OS Boot, Ubuntu, Ubuntu (Yes there were two Ubuntus available) and a default EFI file thingy.   When I click the first Ubuntu I get the GRUB Menu (I was too scared to try the second in case I screwed my laptop up) and I can safely load Ubuntu from there and use it (although I do have to increase my brightness every time I load Ubuntu because it somehow reduces my brightness to complete darkness on boot)  So my problem here is why isn't my GRUB showing on boot, after it worked on the first day?   My laptop is a HP TouchSmart j-078CA.  EDIT 1: FYI I just switched to Ubuntu LiveCD to check my Installation and found out that one of my partition is flagged as msftres. Here is a screenshot of my GParted on a LiveCD: Explanation behind Second Derivative of a Parametric Equation Formula What does Redis do when it runs out of memory? What type of solder is safest for home (hobbyist) use? What is virtual memory? I was double checking my notes for 'Virtual Memory' and the definition in my text book is:     Process of allocating a section of secondary storage to act as part of the main memory   Where as  says:     Virtual memory is a computer system technique which gives an application program the impression that it has contiguous working memory (an address space)   and (Wikipedia also says)     Note that "virtual memory" is more than just "using disk space to extend physical memory size"   Can anyone offer any clarification as to which is correct? Write access without read access Is selenium webdriver capable of database testing? After reading so many answers and forums it makes me confused. Some people say: Selenium webdriver is a browser automation tool and it has nothing to do with the database. And some other showing JARs and code that how we can connect selenium to the database and we can test database using selenium webdriver. So I would really like to know if we can do database testing using selenium? If yes then which databases tool it supports like Mysql, SQL server, Oracle... And if selenium is not suitable for database testing then please recommend to me the best tool for database testing. If $a$ and $b$ are consecutive integers, prove that $a^2 + b^2 + a^2b^2$ is a perfect square. Problem is as stated in the title. Source is Larson's 'Problem Solving through Problems'. I've tried all kinds of factorizations with this trying to get it to the form $$k^2l^2$$ but nothing's clicking. I tried Bézout but the same expression can be written as $$a^2 + (a^2 + 1)(a+1)^2$$ which would imply that there is no real root.  Would really appreciate some help, thanks.