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Spaces in files names in \input or \includegraphics Normal subgroups of infinite symmetric group I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the set $\mathbb{N}$) has exactly 4 normal subgroups." Does anyone have any references or explanation for this? Is it possible to load some package from a .sty file only if some option is selected?  (Naive approach \DeclareOption{something}{\RequirePackage{something}} doesn't work.) Generator of group $D_n$ Microdisplacements with adaptive subdivision turning black On the grade of an ideal I need to prove the following statment (actually a special case of it).     Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Then   $\operatorname{grade}(I,M)\geq 2$ if and only if the homomorphism $M\rightarrow$Hom$_R(I,M)$ given by $m\mapsto(i\mapsto im)$ is an isomorphism.   This is Exercise 1.2.24 in Bruns and Herzog, Cohen-Macaulay Rings. Is it possible to send an email from my Java application using a GMail account?  I have configured my company mail server with Java app to send email, but that's not going to cut it when I distribute the application. Answers with any of using Hotmail, Yahoo or GMail are acceptable. Rank of a Matrix Sum I have $3\times3$ matrices such that S=A+B. I know there is an inequality connecting rank of the matrices A, B and its sum S. Could you write down that here? It will be a great help for me.  Means equation  or inequality connecting rank(S), rank(A) and rank(B). Does Earth really have two high-tide bulges on opposite sides? The bit that makes sense – tidal forces My physics teacher explained that most tidal effect is caused by the Moon rotating around the Earth, and some also by the Sun. They said that in the Earth and Moon system, the bodies are in freefall about each other. But that points on the surface of Earth, not being at Earth's centre of gravity, experience a slightly different pulls towards the Moon. The pull is a little greater if they are on the Moon's side, and slightly lesser if they are on the side away from the Moon. Once freefall is removed, on the Moon side this feels like a pull towards the Moon and on the the opposite side it feels like a repulsion from the Moon. This makes sense to me, and is backed up by other questions and answers here, like  and also  Phys.SE question. The bit that doesn't make sense – tidal bulges They also said that there are &quot;tidal bulges&quot; on opposite sides of the Earth caused by these forces. The bulges stay still relative to the Moon and Earth rotating through the bulges explains why we get two tides a day. They drew a picture like this one…  An image search for  finds hundreds of similar examples, and  from a scientist on Twitter. …But, if there is a tidal bulge on both sides of Earth, like a big wave with two peaks going round and around, how can an island, like Great Britain where I live, simultaneously have a high tide on one side and a low tide on the other? For example:   on the West coast  on the East  Two ports with tides 6 hours, or 180º apart. It's high tide at one while low tide at the other. But they are only  by road. Great Britain is much smaller than Earth. It's probably not even as big as the letter &quot;A&quot; in the word &quot;TIDAL&quot; in that picture.  To prove this isn't just Britain being a crazy anomaly, here is another example from New Zealand:     Two ports that are 180º (6 hours) apart, but separated by  through a national park. New Zealand, unlike the UK, is in fairly open ocean. I tried to update Grub so I run:  sudo update-grub  Generating grub configuration file ... Warning: Setting GRUB_TIMEOUT to a non-zero value when GRUB_HIDDEN_TIMEOUT is set is no longer supported.   What exactly this warning wants to be changed? (I upgraded from 12.04 to 14.04 before)  This is my /etc/default/grub file:  GRUB_DEFAULT="0" GRUB_HIDDEN_TIMEOUT="0" GRUB_HIDDEN_TIMEOUT_QUIET="true" GRUB_TIMEOUT="10" GRUB_DISTRIBUTOR="`lsb_release -i -s 2&gt; /dev/null || echo Debian`" GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX="" What would you call a word that doesn't exist in or translate well into another language? I've run into this situation several times, being a native Spanish speaker.   There are some words you just can't translate into another language. Is there a particular word to describe this?  I'm not refering to idioms.  Also, if its a noun, people tend to use them by surrounding them in quotes. Is this appropiate?  Example:     I arrived at around 5 pm and had some quesadillas.   Edit:  I also meant to cover words that exist in English, but cannot be translated into another language.   Example:      There isn't a word for "drive-thru" in your language. Can Windows Virtual PC run a 64-bit OS? Measuring accuracy of a logistic regression-based model I have command prompt in tcsh that shows the current working directory. The problem is that the command prompt becomes very big once I am in a deep hierarchy of folders. How do I set the command prompt to show, let's say, only the last 3 directory names? How do I suppress stderr warning messages from a command inside command substitution? touch ~/deleted/$(echo "directory_"$(readlink -f foo)|tr '/' '\') 2&gt; /dev/null   where foo is a directory  It's reading the full pathname of foo and creating it as a file replacing all '/' with '\' and putting a directory_ in front so the directory  /home/test/foo would create a file directory_\home\test\foo  It does exactly what I want except a warning keeps printing tr: warning: an unescaped backslash at end of string is not portable   a. I don't know what that means b. I would like it to not show  I tried to redirect stderr to /dev/null but it doesn't work. How do I suppress it? Delegates: Predicate vs. Action vs. Func Can someone provide a good explanation (hopefully with examples) of these 3 most important delegates:   Predicate Action Func When I right click the Filezilla icon, the only options are "launch" and "properties". How do I remove it from the top panel? I am trying to solve:  $$\int_{-\infty}^{\infty}\frac{x\sin{x}}{x^2 + a^2}dx$$  and show that it equals $\pi e^{-a}$ for all $a &gt; 0$.  I translate it to its complex equivalent:  $$lim_{R \rightarrow \infty} \int_{-R}^{R}\frac{z\sin{z}}{z^2 + a^2}dz = \int_{C}\frac{z\sin{z}}{z^2 + a^2}dz - \int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz$$  where C is a half circle with radius R, above the real axis, and $\gamma$ is only the curved (top part) of the half circle.  I've shown that $$\int_{\gamma}\frac{z\sin{z}}{z^2 + a^2}dz$$ tends to 0 as $R \rightarrow \infty$.  So now:  $$lim_{R \rightarrow \infty} \int_{-R}^{R}\frac{z\sin{z}}{z^2 + a^2}dz = \int_{C}\frac{z\sin{z}}{z^2 + a^2}dz$$  By residue theorem,   $$\int_{C}\frac{z\sin{z}}{z^2 + a^2}dz = 2\pi i \cdot res_{ai}f$$  as $ai$ is the pole of $f$ that lies above the real axis.  $$2\pi i \cdot res_{ai}f = 2\pi i \cdot lim_{z \rightarrow ai} (z-ai)\frac{z sin{z}}{z^2 + a^2}$$  By L'Hospital's rule, this is equivalent to:  $$2\pi i \cdot lim_{z \rightarrow ai} (z-ai)\frac{z^2\cos{z} + 2z\sin{z} - zai\cos{z} - ai\sin{z}}{2z} = 2\pi i \cdot \frac{\sin{ai}}{2} = \pi i \sin{ai}$$  But how is $\pi i \sin{ai}$ equivalent to $\pi e^{-a}$? Did I make a mistake? What characters/symbols are allowed within the CSS class selectors? I know that the following characters are invalid, but what characters are valid?  ~ ! @ $ % ^ &amp; * ( ) + = , . / ' ; : " ? &gt; &lt; [ ] \ { } | ` # How to install the latest stable version of Chromium?