On the last page of , the following statements are made (I'll jump right to around the point of interest):  Example: Consider $A=p_x$, $B=x$. Then  $$\langle A\rangle=\langle B\rangle=0;\,\,\,\Delta A=\sqrt{p_x^2},\,\,\,\Delta B=\sqrt{x^2}$$ $$[x,p]=i\hbar\implies C=\hbar=\langle C\rangle$$ $$\Delta p_x \Delta x \geq \frac{\hbar}{2}$$ For the minimum uncertainty state we have, $$p_x\psi = \frac{i\hbar}{2(\Delta x)^2}x\psi \tag{1}$$ With $p_x=i\hbar\frac{d}{dx}$, the solution to the differential equation gives $$\psi(x)=\frac{1}{\sqrt[4]{2\pi(\Delta x)^2}}e^{-\frac{x^2}{2(\Delta x)^2}}$$  It's clear that notation is being abused like Matilda, but putting that aside I can't see where equation $(1)$ came from. This looks like a good proof that Gaussians make HUP into an equality, so I would very much appreciate if you could help point me in the right direction.
Can anyone point me to a reference (preferably either something online or something a small liberal arts school would be likely to have in its library) that goes through a derivation of the minimum uncertainty wavefunction in more detail than in the Griffiths?  Edit I've moved the second part of my original question to a separate post:
1-1 correspondence between [0,1] and [0,1) Batch normalize audio volume on .ogg-files Are there any simple tools to normalize an .ogg-library of 10'000+ songs so that the volume is the same throughout all songs?  Terminal or GUI doesn't matter, it only need to be simple.  One caveat though, I don't want soft interludes/intermissions and ballads blown out of proportion.  Preferably the process should find the overall gain of the album (I have all my CD's ripped into separate folders) and normalize the level thereafter. How can I increase my disk space when Ubuntu is installed inside Windows? Open Mapping Theorem: counterexample The Open Mapping Theorem says that a linear continuous surjection between Banach spaces is an open mapping. I am writing some lecture notes on the Open Mapping Theorem. I guess it would be nice to have some counterexamples. After all, how can you appreciate it's meaning without a nice counterexample showing how the conclusion could fail and why the conclusion is not obvious at all.  Let $\ell^1 \subset \mathbb{R}^\infty$ be the set of sequences $(a_1, a_2, \dotsc)$, such that $\sum |a_j| &lt; \infty$. If we consider the $\ell^1$ norm $\|\cdot\|_1$ and the supremum norm $\|\cdot\|_s$, then, $(\ell^1, \|\cdot\|_1)$ is complete, while $(\ell^1, \|\cdot\|_s)$ is not complete.  In this case, the identity $$   \begin{array}{rrcl}     \mathrm{id}:&amp; (\ell^1, \|\cdot\|_1)&amp; \to &amp;(\ell^1, \|\cdot\|_s)     \\                 &amp; x &amp; \mapsto &amp; x   \end{array} $$ is a continuous bijection but it is not open.  I want a counterexample in the opposite direction. That is, I want a linear continuous bijection $T: E \to F$ between normed spaces $E$ and $F$ such that $F$ is Banach but $T$ is not open. This is equivalent to finding a vector space $E$ with non-equivalent norms $\|\cdot\|_c$ and $\|\cdot\|_n$, such that $E$ is complete when considered the norm $\|\cdot\|_c$, and such that $$   \|\cdot\|_c   \leq   \|\cdot\|_n. $$ The Open Mapping Theorem implies that $\|\cdot\|_n$ is not complete.  So, is anyone aware of such a counterexample? Finding what methods a Python object has In Spivak's Calculus 3rd Edition, there is an exercise to prove the following:  $$x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$$  I can't seem to get the answer. Either I've gone wrong somewhere, I'm overlooking something, or both. Here's my (non) proof:  $$\begin{align*} x^n - y^n &amp;= (x - y)(x^{n-1} + x^{n-2}y +\cdots+ xy^{n-2} + y^{n-1}) \\           &amp;= x \cdot x^{n-1} + x \cdot x^{n-2} \cdot y + \cdots + x \cdot x \cdot y^{n-2} + x \cdot y^{n-1}\\ &amp;\qquad + (-y) \cdot x^{n-1} + (-y) \cdot x^{n-2} \cdot y + \cdots + (-y) \cdot x \cdot y^{n-2} + (-y) \cdot y^{n-1}\\           &amp;= x^n + x^{n-1} y + \cdots + x^2 y^{n-2} + x y^{n-1} - x^{n-1}y - y^2 x^{n-2}  - \cdots- x y^{n-1} - y^n \\           &amp;= x^n + x^2 y^{n-2} - x^{n-2} y^2 - y^n  \\           &amp;\neq x^n - y^n \end{align*}$$  Is there something I can do with $x^n + x^2 y^{n-2} - x^{n-2} y^2 - y^n$ that I'm not seeing, or did I make a mistake early on?  EDIT:  I should have pointed out that this exercise is meant to be done using nine of the twelve basic properties of numbers that Spivak outlines in his book:     Associate law for addition   Existence of an additive identity   Existence of additive inverses   Commutative law for additions   Associative law for multiplication   Existence of a multiplicative identity   Existence of multiplicative inverses   Commutative law for multiplication   Distibutive law Is there a way to find review articles by the keyword? When I search for papers using some keywords in academic databases (like Google Scholar), there are lots of papers, but I would like to read literature reviews first because I’m new in that area. (By literature reviews, I mean article-length papers that analyze existing work on a specific topic.)  So I think it would be helpful for me if there were a way to find literature reviews using the keyword or using the paper’s title. I want to categorize papers I’ve found by its document type (research, survey, review, etc).  What I've tried is using a specific academic database (ScienceDirect), which provides an advanced search, in which one can filter the paper by its type (original research article, review article, short survey, etc) – but I feel some limitations of this method. I think I can also do this by adding the word review to my keywords for searching or by reading a paper’s abstract, number of references and guessing whether it is review article or not.  Is there a more efficient way to find review articles by the keyword?  (My research area is computer science.)  UPDATE  "feel some limitations" in my post, I'm not sure but it seems that ScienceDirect only searches for literatures that belongs to ElseVier company. I think It doesn't cover the journal articles well, comparing to e.g) Google Scholar so I said I feel some limitations of it $\aleph_1$ is the cardinality of the countable ordinals. It is the least cardinal number greater than $\aleph_0$, and assuming the continuum hypothesis it's equal to $\mathfrak{c}$, the cardinality of the the real numbers.    My question is, is it possible for all $\aleph_1$ subsets of $\Bbb{R}$ to have Lebesgue measure $0$?  This is, of course, impossible assuming the continuum hypothesis, because then all sets of real numbers would have measure $0$, which is absurd.  But is $\mathsf{ZFC}$ + $\lnot\mathsf{CH}$ + "all subset of $\Bbb{R}$ of cardinality $\aleph_1$ have Lebesgue measure $0$" consistent?  If not, what if we replaced Lebesgue measure with some other measure?  It may be worth noting that, although there may be subsets of $\Bbb{R}$ with cardinality $\aleph_1$, there are no subsets of $\Bbb{R}$ which have the order-type $\omega_1$ (the order-type of the countable ordinals) under the usual ordering on $\Bbb{R}$.  Any help would be greatly appreciated.  Thank you in Advance. Possible Duplicate:           What book would you recommend for an undergraduate wanting to learn the basics of topology? I've come across several books that claim to give an introduction to the subject, but I wasn't sure which one(s) to pick.  Here are some of the books I came across:   Introduction to Topology - Bert Mendelson Basic Topology - M.A. Armstrong First Concepts of Topology - William G. Chinn Topology Without Tears - Sidney A. Morris Topology - James Munkres   They don't go over the exact same topics, some are more in-depth than others, some seem to target slightly different audiences etc. What would be right for me (not necessarily from that list)? To give some more background, I want to learn topology because I feel it will be somewhat helpful in the field of Computer Vision. How to show that $f'(x)<2f(x)$ This is question B4 in 1999 Putnam Exam. I would appreciate if somebody could help me with this.      Let $f(x),f'(x),f''(x),f'''(x)&gt;0$ , $f'''(x)$ is a continuous function and $f'''(x)&lt;f(x)$ on $\mathbb{R}$. Then show that   $$f'(x)&lt;2f(x),~ \forall x\in \mathbb{R}.$$ Reputation does not update after community wiki is removed from a post, and does not match the value displayed in the reputation audit If votes were cast on a post while it was community wiki, and a moderator removes wiki from that post, the votes should count towards reputation again. However, while this is visible in a reputation audit, , and the reputation score needs to be recalculated to reflect the change.  Can someone fix this? Empty Disk when trying to install dual-boot system I recently purchased an SSD to speed up my computer experience. Before, I had Windows 7 and Ubuntu in a dual-boot system. The plan was to install Windows 8 on the SSD and then set aside ~30GB for the system files of Ubuntu.  I installed Windows 8 just fine on the SSD, then I booted the Ubuntu install CD to make my partitions like always - but GParted tells me the entire SSD is unallocated. Now I am afraid I will lose my Windows installation if I try to do anything to it.  Why does GParted think there is no partition on the SSD? Shouldn't it be 4 Partitions, like the Windows installer told me?  And is it possible to ignore this and install Ubuntu on the hard disk (where it, frankly, already is) and somehow make it possible to install GRUB on the SSD to revert my dual-boot system without damaging the installed Win8? How to align right within equation array How do I load an mp4 video with multiple audio tracks into blender? I've recently started using OBS multiplatform to record game play. I currently have it set to record sounds on 2 different tracks. The first for system sounds (like gamesound, mumble, etc.) and the second to record the mic.  I am able to switch between tracks when I play the unedited video using VLC, but when I load the mp4 into blender, I only get track 1.  Did I do something wrong, or does blender vse not support multiple tracks?  Blender after loading mp4:  VLC video information:   UPDATE  For now, it seems like the solution will be to use an external tool to get the second audio track and import it into blender afterwards. A  has been logged with the Blender bug reporting tool.  UPDATE 2  The bug report has been closed/archived with the gist being "not a bug". It has been added to the . If you are looking to have support for multiple audio tracks, I would suggest looking at other video editing software until Blender adds support for this feature. Find the minimum of $\sqrt{(x+1)^2+(y-1)^2}+\sqrt{(x+2)^2+(y+2)^2}+\sqrt{(x-1)^2+(y+1)^2}$ For real $x,y$ Find the minimum of $\sqrt{(x+1)^2+(y-1)^2}+\sqrt{(x+2)^2+(y+2)^2}+\sqrt{(x-1)^2+(y+1)^2}$ For real $x,y$. It is the distance of a point in a triangle from its vertexes which gives the minimum when $(x,y)$ is on the Fermat's point.  But now I am stuck in finding the location of Fermat's point.  $(1) \quad 4$ $(2) \quad 5$ $(3) \quad \sqrt{3}+2\sqrt{5}$ $(4) \quad 4\sqrt{2}$ So I had an exam today and one of the questions were: Give an example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points. At first I had no idea how to do it then I came up with this, even though I know it's wrong:  $$f(x)= \begin {cases} \ \sqrt{x}&amp;\text{if }x &lt;0\\ \sqrt{-x}\ &amp;\text{if }x &gt; 0\\ \end {cases}$$  Like I said I know it's wrong..but can someone give me an example of this? Thanks. Missing "References" heading I'm very new to classical propositional logic, and my lecturer is using a system with the following axioms:  A1. X→(Y→X)  A2. (X→(Y→Z))→((X→Y)→(X→Z))  A3. (¬Y→¬X)→(X→Y)  Use uniform substitution and Modes Ponens (MP) as rule of inference. Deduction Theorem also allowed.  I'd like to deduce the following:  (a) X, Y ⊢ ¬(X→(¬Y)) (conjunction, but expressed in terms of NOT and IMPLIES)  (b) X→((X→Y)→Y)  (c) (X→Y)→(((¬Y)→X)→Y)  (d) (((¬Y)→X)→Y)→(X→Y)  Remark: The last two are to show that the axiom (A3) can be replaced with a different axiom:  B3. (¬Y→¬X)→(((¬Y)→X)→Y).  Solution: Bram28 proves (b) and (c) below. I demonstrate (a) and (d).  (d) is alright; first, I claim that X, ((¬Y→X)→Y) ⊢ Y.  Axiom (A1) gives ⊢X→(¬Y→X) so that ((¬Y→X)→Y)⊢X→Y by Hypothetical Syllogism (HS), which Bram28 shows to be valid.  Now that we have ((¬Y→X)→Y)⊢X→Y just apply the deduction theorem to get what we want: ⊢((¬Y→X)→Y)→(X→Y).  (a) is slightly trickier. Axiom (A1) gives ⊢ Y→(X→Y) so that X, Y ⊢ (X→Y).  Next, by part (b), we have as a theorem that ⊢X→((X→¬Y)→¬Y).  Therefore X, Y ⊢(X→¬Y)→¬Y.  But ⊢(¬¬(X→¬Y))→(X→¬Y) by DN Elim, which Bram28 shows to be valid.  By HS, X, Y ⊢(¬¬(X→¬Y))→¬Y.  By Axiom (A3) and MP, X, Y ⊢(Y→¬(X→¬Y)).  Also, rather trivially X, Y ⊢Y.  Finally by MP we conclude X, Y ⊢¬(X→¬Y) as required. Alt-Tab shows two application switchers on Unity I started to have problems with my Ubuntu application switcher (Alt +  Tab) this morning. When I press Alt +  Tab, not only one, but two application switchers show up. Last night I customized some stuff using the Unity Tweak Tool. I thought I didn't touch the application switcher. So now it looks like this: Here's the problem: if $S = \mathbb{Q}[x]/(x^2)$, compute the group  of automorphisms $\mbox{Aut}(S)$.   First off, I have the solution and I know it's isomorphic to the multiplicative group $\mathbb{Q}\backslash\{0\}$, so I'm trying to understand it. It uses what is called the substitution principle and the fact that the restriction of any endomorphism $\phi: \mathbb{Q}[x]\to \mathbb{Q}[x]$ to $\mathbb{Q}$ is the identity. It says that these two facts implies that the endomorphisms of $\mathbb{Q}[x]$ are in 1-1 correspondence with $h \in \mathbb{Q}[x]$, where the endomorphism corresponding to $h$ is $\phi_h : \mathbb{Q}[x] \to \mathbb{Q}[x]$ given by $\phi_h(f) = f(h)$ for $f \in \mathbb{Q}[x]$. I guess I just don't understand how it applied the substitution principle? It's probably obvious, but for some reason I am not getting it.  An explanation would be greatly appreciated! Or if you have a different solution, that'd be cool too. Thanks!!