Calculating a matrix element, $\langle x|p\rangle$?
What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?
Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$. Let $A\in M_{n\times n}(\Bbb R)$ so that $I\notin span(A,A^2,...,A^n)$. Prove that $\det(A)=0$.   I was thinking of showing $A$ is not invertible, meaning it has an eigenvalue of $0$. Since no matter the power you give to $A$, it is still not the identity, you can deduce that at least one eigenvalue is indeed $0$. However this doesn't work if you have different eigenvalues in the Matrix, so I got stuck.  Help will be most appreciated. How long would a 150 Farad Capacitor light up an LED I have a bright white LED from a flashlight. Aproximatley How long will it light up with a 150 farad 2.5 volt capacitor. Do I need a resistor? And if so how many Ω? The capacitor is a maxwell 150 farad 2.7 volt boostcap . What is the difference between `grep`, `egrep`, and `fgrep`? Can anyone tell me the technical differences between grep, egrep, and fgrep and provide suitable examples?  When do I need to use grep over egrep and vice versa? Two finite abelian groups with the same number of elements of any order are isomorphic Suppose $G, H$ are finite abelian groups with the same number of elements of any given order. Show they are isomorphic.   Since finite abelian groups are isomorphic if and only if their Sylow subgroups are, we may restrict our attention to the case where $G, H$, are $p$-groups, but I can't quite make it past there... How do you clone a Git repository into a specific folder? Executing the command git clone git@github.com:whatever creates a directory in my current folder named whatever, and drops the contents of the Git repository into that folder:  /httpdocs/whatever/public   My problem is that I need the contents of the Git repository cloned into my current directory so that they appear in the proper location for the web server:  /httpdocs/public   I know how to move the files after I've cloned the repository, but this seems to break Git, and I'd like to be able to update just by calling git pull. How can I do this? How to generate a random alpha-numeric string Why was Snape called the Half-Blood Prince? My question comes from what I learned in the movies, considering I have not yet been able to read the books. I would just like to know why Severus Snape called himself the Half-Blood Prince? Did many people call him that or did he just think that it was a cool nickname? This question might be answered in the books so feel free to quote from it. Low Quality Posts you Have Already Flagged I just discovered that it is possible to get a post in the Low Quality Posts review queue that you have already flagged for closing, making reviewing that item impossible. Obviously, I don't want to 'Looks Good' something that I know should be closed, but at the same time I cannot 'Recommend Close' something I have already flagged leaving the only option being to skip the post.  Can the queue be changed so that it does not show posts which you have already flagged? Is it positive or negative for the average website to allow search engines to index user comments? I've been trying to answer this question myself by researching and searching on google but I haven't been able to come with a final answer. I have some hypothesis I would like share anyway:   The average website intends to reach the higher audience possible. User generated content by comments is an efficient way of increasing the relevance and the valuable information of the average website. Indexing comments will make any site attractive to spammers. Indexing comments is more convenient for new/low-traffic websites than it is for old/high-traffic websites. The web in general is currently choosing not to index comments in the average website because of fear to the SPAMMERS. The average author of a content-valuable comment would like his comment to be indexed Not indexing comments discourages the creation of content-valuable comments.   So, what do you think finally? Is it positive or negative for the average website to allow search engines to index user comments? Why? Does Wasserstein distance require the source and target distributions to have the same mass? If we minimize the Wasserstein loss, $$W_1 (P_S, P_T) = \underset{\gamma \in \Pi}{\text{min }} \sum_{x,y} |x-y|\gamma(x,y)$$ which means we are looking for the coupling $\gamma(x,y)$ that minimizes the cost of moving mass from the source distribution $P_S$ to the target distribution $P_T$,  Do the masses of the source and target distributions have to be identical? If so or not, how are the masses of $P_T$ and $P_S$ accounted for mathematically in, or when minimizing, the shown formula, or what might be missing/overlooked by the Wasserstein approach overall due to the mass differences? Calculate business days I need a method for adding "business days" in PHP. For example, Friday 12/5 + 3 business days = Wednesday 12/10.  At a minimum I need the code to understand weekends, but ideally it should account for US federal holidays as well. I'm sure I could come up with a solution by brute force if necessary, but I'm hoping there's a more elegant approach out there. Anyone?  Thanks. We know that the addition and multiplication operators are both commutative, and the exponentiation operator is not.  My question is why.  As background there are plenty of mathematical schemes that can be used to define these operators. One of these is   where $H_0(a,b) = b+1$   (successor op) $H_1(a,b) = a+b$   (addition op) $H_2(a,b) = ab $   (multiplication op) $H_3(a,b) = a^b$   (exponentiation op) $H_4(a,b) = a\uparrow \uparrow b$   (tetration op: $a^{(a^{(...a)})}$ nested $b$ times ) etc.    Here it is not obvious to me why $H_1(a,b)=H_1(b,a)$ and $H_2(a,b)=H_2(b,a)$ but not $H_3(a,b)=H_3(b,a)$  Can anyone explain why this symmetry breaks, in a reasonably intuitive fashion?  Thanks. Prevent Use of the Back Button (in IE) Singular or plural verb form where subject includes a "parenthetical" element My question arises from , where OP asks whether he should use the singular or plural verb form after "the title, as well as the tone,"  As luck would have it, when I searched Google Books for , I found...        ...and...        At the risk of having this question closed as just peeving, I feel that the first one should have been plural "err", because without the commas it seems to me both "the title" and "the tone" have more or less equal status as "joint" subjects of the verb, making it plural.  On the other hand, I feel the commas in the second example effectively demote "the tone" to "parenthetical" status (the clause is almost incidental). Since this encourages us think of "the title" as the primary subject, it should have been singular "derives".  I'm no great fan of "grammatical rules", but it seems weird to me that the only two relevant examples for this specific wording should both (in opposite ways!) contradict my inclinations.  What's going on? In his lectures on String Theory (), Tong gives a proof of the Weyl anomaly, using equation $(4.36)$. It seems wrong to me.  Here he uses the OPE between the stress-energy tensors $T_{zz}T_{ww}$  obtained when trace vanishes, i.e. $T_{z \bar{z}} = 0$: this implies that they are holomorphic functions $T_{zz} = T_{zz}(z)$. But in this proof he starts from the fact that $T_{z \bar{z}} \neq 0$ (we want to proof this thing after all) and so $T_{zz}$ is not a holomorphic function anymore! In the OPE $(4.36)$ I should have also terms with $(\bar{z}- \bar{w})$.  I can't also understand why he uses in the rest of the proof only the singular term $(z-w)^{-4}$, neglecting the other ones $(z-w)^{-2}$, $(z-w)^{-1}$.  (The same proof is given in these lectures  on conformal field theory, equation $(6.9)$).  I'll be really thankful if someone could explain me this proof :) titlesec: loss of section numbering with the new update (2016/03/15) After reinstalling TeXLive distribution (install-tl-20160318, Ubuntu 14.04), a issue occurs with the updated version of the titlesec package: the section numbering is lost.  What could it be missing? Is there a modification I have not seen?  Below is a MWE comparison of the two versions. The prevtitlesec package is a reinstall of the previous version (2011/12/15).  \documentclass{book} \usepackage[margin=2cm,a4paper]{geometry} %\usepackage{prevtitlesec}% 2011/12/15 version \usepackage{titlesec}% 2016/03/15 version \usepackage{lipsum}  \begin{document}  \chapter{Title of the first chapter} \lipsum[1] \section{A test section} \lipsum[4] \subsection{A test subsection} \lipsum[4] \subsection{A test subsection} \lipsum[4]  \end{document}   titlesec package 2011/12/15:    titlesec package 2016/03/15: Is there a *simple* example of how the axiom of choice can lead to a counterintuitive result? How do I get label of the selected item from a lightning:select drop down list? I would like to get the value of the label of the selected drop down item.  In this case, the label is the Name of the Contact.  &lt;aura:component&gt;  &lt;aura:attribute name="contacts" type="Contact[]"/&gt;  &lt;lightning:select label="Contact" aura:id="contactList" name="contactList"&gt;     &lt;aura:iteration items="{!v.contacts}" var="c" indexVar="index"&gt;        &lt;option value="{!c.Id}"&gt;{!c.Name}&lt;/option&gt;    &lt;/aura:iteration&gt;  &lt;/lightning:select&gt;  &lt;/aura:component&gt;   I know how to get the value:  component.find("contactList").get("v.value");   But how do I get the label? "Forgot" vs "Forget" Is the following correct, or is there more to it?  "I forgot his name" — I knew his name, but I forgot it.  "I forget his name" — I keep forgetting his name. Where using "forget" basically means that you tried to remember this information before as well, but you couldn't then either. What is the difference between update-grub and update-grub2? I'm making some changes on my grub file under /etc/default/grub. In some tutorials I have seen sudo update-grub and others sudo update-grub2. What is the difference?