Prove that $\left\lfloor \lfloor x/2\rfloor/2 \right\rfloor=\lfloor x/4\rfloor$ for all $x$.
Prove that $\lfloor\lfloor x/2 \rfloor / 2 \rfloor = \lfloor x/4 \rfloor$
Writing my first script so I'm sure this is a basic question, but can someone please explain to me why I can:  cd ~ cd bin cd ~/bin cd 'bin'   But not  cd '~' cd '~/bin'   I need to cd to a directory path with a space in one of the directory names, so I need the quotes (it's Windows Program Files under wine). I can get around it with two cd commands, but why can't I put ~ in quotes?  If I type cd '~' (or cd "~") I get:  bash: cd: ~: No such file or directory A person that professes discrimination based on race is a racist.  What is the word for defining a person that discriminates based on culture (pretty sure it is not culturist) or social habits?  E.g., Eric Cartman doesn't like hippies, he discriminates against them. But a hippie is not a race, an ethnic, a religion or a nationality. It is a life style not related to any of these. Why isn't my Minecraft LAN server working? My LAN server won't let my cousin join me or me join my cousin. It won't show up on the multiplayer screen - it keeps saying "scanning for LAN servers" but it never finds it. I am at his house and we use different computers but it won't connect. How can I fix this? Does the sum of reciprocals of primes converge? Is this series known to converge, and if so, what does it converge to (if known)?  Where $p_n$ is prime number $n$, and $p_1 = 2$,  $$\sum\limits_{n=1}^\infty \frac{1}{p_n}$$ How do I cherry pick packages from a PPA? echo "-e" doesn't print anything I'm using GNU bash, version 3.00.15(1)-release (x86_64-redhat-linux-gnu). And this command:  echo "-e"    doesn't print anything. I guess this is because "-e" is one of a valid options of echo command because echo "-n" and echo "-E" (the other two options) also produce empty strings.  The question is how to escape the sequence "-e" for echo to get the natural output ("-e"). I have come across a question about determining possible minimal polynomials for a rank one linear operator and I am wondering if I am using the correct proof method.  I think that the facts needed to solve this problem come from the section on Nilpotent operators from Hoffman and Kunze's "Linear Algebra".     Question: Let $V$ be a vector space of dimension $n$ over the field $F$ and consider a linear operator $T$ on $V$ such that $\mathrm{rank}(T) = 1$.  List all possible minimal polynomials for $T$.   Sketch of Proof:   If $T$ is nilpotent then the minimal polynomial of $T$ is $x^k$ for some $k\leq n$.  So suppose $T$ is not nilpotent, then we can argue that $T$ is diagonalizable based on the fact that $T$ must have one nonzero eigenvalue otherwise it would be nilpotent (I am leaving details of the proof of diagonalization but it is the observation that the characteristic space of the nonzero eigenvalue is the range of T and has dimension $1$).  Thus the minimal polynomial of $T$ is just a linear term $x-c$.    Did I make a mistake assuming that $T$ can have only one nonzero eigenvalue?  Thanks for your help I want to type Arabic text in Photoshop, but the text appears disjointed and the letters do not connect at all. For example محمد appears as م ح م د.   I'm using Photoshop CS6 on OSX Mountain Lion. I'd like to add an invisible watermark to each PDF document that I produce so that in case it leaks, I know where it leaked from. Anyone knows what would a good way for an invisible watermark be in a PDF document and how to add one? Enlarged partition, filesystem still reports old size I resized the root partition with GParted, from 20 GB to 40 GB, but the filesystem still reports a size of 20 GB.  How can I use all 40 GB?    Update  The resized partition was an LVM Physical Volume (/dev/sda2), and the boot partition is ext4 (/dev/sda1). How do I resize root partition? How do I install Java? Book: Handful of children kept in a school as political mythological prisoners I've found plenty of posts about changing the visibility of a block programatically.  My problem is the opposite. Given a block's ID/delta, how do I find out if it should be displayed at the current path?  I have a field collection that allows you to append a block using the Block Reference module (I preprocess the field collection and use my own template, so the logic has been taken away from the Block Reference module). It works, but I only want it to kick in if the "Show block on specific pages" rule says that block should be displayed. The adjective ones I have heard recently are    forward straightforward forthcoming frank   I was wondering if each of them has positive, negative or neutral meaning?  What are other similar terms that are used for positive, neutral and negative intention? Best lens for taking pictures of surfers This should be a very easy question but it might just be that I'm confusing myself. So we have the definition of a function $f$ on $S$ being continuous at $x_0$:     For any $\epsilon$>0, there exists $\delta&gt;0$ such that: whenever $|x-x_0|&lt;\delta$, we have $|f(x)-f(x_0)|&lt;\epsilon$   And I assume the negation is     There exists $\epsilon$>0 such that for all $\delta&gt;0$, $|x-x_0|&lt;\delta$ yet $|f(x)-f(x_0)|\ge \epsilon$.   Now I want to show that the function $f(x)=\sin(\frac{1}{x})$ together with $f(0)=0$ cannot be made into a continuous function at $x=0$. So I need to show that there exists $\epsilon&gt;0$ such that for all $\delta&gt;0$, $|x|&lt;\delta$ yet $|f(x)|\ge\epsilon$.  Let $\epsilon = \frac{1}{2}$. Then no matter what $\delta$ we choose, let $|x|&lt;\frac{1}{2}$. It is certainly possible that $|f(x)|\ge \frac{1}{2}$, because, well, $\frac{1}{x}$ can really take on arbitrarily large value as $x$ is small.  Now, what confuses me is that, as $x$ gets small, $f(x)$ can certainly be greater than $\frac{1}{2}$ for infinitely many times, but it will be less than that infinitely many times, too. But I suppose it doesn't really matter. So I think there's something wrong with my negation but I couldn't figure out where.  Update: The correct version can be found . Watch for Lemma 4.6 Use of "would" for subjunctive phrases This has been bugging me for some time; I tried to look for previous questions here but my language tools may not be sharp enough to phrase my query correctly so please forgive me if this has already been posted here.  Speakers of English as a foreign language tend to incorrectly use the word "would" when using some particular subjunctive constructs. Here is  (this time presumably by a native English speaker):     If that would happen, you won’t be alone.     (One correct version being "If that happens, you won't be alone".) I have detected this in speakers of a number of languages so I feel this is more likely to be a feature of the English language itself than a carry-over from their native language.  I have a faint intuition that this is because , and it certainly doesn't help that , but I can neither make that precise nor even really phrase a concrete question. Can someone shed some light on this?    Edit: just to provide some real-world examples of what I mean, here is one example taken from :     Publish or Perish is designed to empower individual academics to present their case for research impact to its best advantage. We would be concerned if it would be used for academic staff evaluation purposes in a mechanistic way.   This is obviously ungrammatical (to me), but I'm not sure how to characterize the mistake, or what the likely causes are (e.g. whether it follows from analogous constructions in other languages). Is my expansion of the state $| x \rangle$ correct? In my quantum mechanics textbook it says that the relation between the basis $|x\rangle$ and $|p\rangle$ is given by:  $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$  However, i'm not sure how to go about proving this relation. My idea was that the eigenstates of momentum can be written as below:  $\phi(x,p) = \Large \frac{1}{\sqrt{2\pi \hbar}} e^{-ipx/\hbar},$  which form an orthogonal basis.  But, we can expand any state, $|\text{state}\rangle$, in terms of the othorgonal basis right? So is my notation below correct for the expansion of the state $| x \rangle$ ?  $|x\rangle = \int_{-\infty}^{+\infty} \phi(p', x) |p'\rangle dp'$  If it were correct then the following would also be correct?  Using the fact that $\langle p | p' \rangle = \delta(p - p')$, $\langle p | x \rangle$ can be computed:  \begin{align} \langle p | x \rangle =&amp; \int_{-\infty}^{+\infty} \phi(p', x) \langle p | p' \rangle dp' \\ =&amp; \int_{-\infty}^{+\infty} \phi(p', x) \delta(p - p') dp' \\ =&amp; \phi(p, x) \\ =&amp; \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \end{align} as required.  Does the logic work here? Finitely generated monoids- axiomatization- compactness theorem.