Invariance of maximally entangled state How can the equality below hold? If I have a maximally entangled qudit state $\lvert \Phi\rangle_{AB}$ and any $d\times d$ matrix $M$, then it is claimed that  $$ (M_A\otimes I_B)\lvert \Phi\rangle_{AB} = (I_A\otimes M_B^T)\lvert\Phi\rangle_{AB}$$  I can see that this works for qubits and using the Pauli spin operators, but I am not seeing it in the general case.
Invariance of a maximally entangled state under unitary operation $U \otimes U^\dagger$ Apparently the (d-dimensional) maximally entangled state, $|E \rangle = \sum_{i} |ii\rangle /\sqrt{d}$ is invariant under operations of the form $U \otimes U^\dagger$. I want to prove this result, which amounts to showing that  $$\left(\sum_i U | i \rangle \otimes V |i \rangle = \sum_i | ii \rangle \right)\Rightarrow (V = U^\dagger )$$  I have no idea how to even start. I suppose it's some simple linear algebra result, but I don't see it. A hint would be appreciated.
Tricks and tips for finding information in man pages Does anyone have any tricks and tips for finding information in man pages? How to go with proving this : Let $R$ be a ring and let $I$ be an Ideal of $R$.  Prove that the factor ring $R/I$ is commutative iff $rs-sr \in I$ $\forall r$ and $s$ in $R$  any hint how to go with this... Followup to "".  It is argued that even a tiny residual charge would result in huge amounts of electricity in bulk matter, everything would be different, etc. I do not find that a convincing answer: suppose $n$ protons plus $n+1$ electrons are neutral. Why wouldn't we also expect there to be $n$ protons to every $n+1$ electrons? That is, there is no bulk matter problem if every $n$'th atom is a negative ion (for hydrogen).  Now, what empirical lower bound can we give for $n$ in that kind of scenario? Proof by induction that if $n \in \mathbb N$ then it can be written as sum of different Fibonacci numbers Proof that every natural number $n\in \mathbb N$, can be written as the sum of different Fibonacci numbers between $F_2,F_3,\ldots,F_k,\ldots$.  For example: $32 = 21 + 8+3 = F_8+F_6+F_4$  Research effort:  The base step it's simple:  Let $k=1$ it can be britten as $k=1=F_2$  For the inductive step I considered:  Let $k = k  F_2 =F_2+F_2+\cdots+F_2 = \sum_{i=2}^w a_iF_i, a_i =\{0,1\} $  Then if $k$ suffice I want to see if $k+1$ suffices too... But I'm not realy seeing how to use the inductive hypothesis, so I assume It's wrong.  Any thoughts on which can the inductive step be? Which is the best way to put function plots into a LaTeX document? Converting ISO 8601-compliant String to java.util.Date Restate theorem in separate file I state a theorem in the main body of a document and prove it in the appendix. I use thmtools's restatable environment to restate the theorem there. However, I now need to have my appendix in a separate pdf file (hence a separate tex document). I'm using xr and xcite to get the cross-references and bibliographical references to play nicely.  But the commands created by restatable are not carried. Is there any way to to the restating of the theorem in the appendix in a separate document?  Here's a MWE:  main document: main.tex  \documentclass{article} \usepackage{thmtools} \newtheorem{theorem}{Theorem}  \begin{document}  \begin{restatable}{theorem}{thmcmd}\label{thmlabel} Here is the statement of the theorem. \end{restatable}  The above is Theorem~\ref{thmlabel}.  In this way I can restate it: \thmcmd*  \end{document}   appendix: appendix.tex  \documentclass{article} \usepackage{xr} \externaldocument{main} \usepackage{thmtools} \newtheorem{theorem}{Theorem}  \begin{document}  I can refer to Theorem~\ref{thmlabel}.  But I cannot restate it like this: % \thmcmd*  \end{document} Best methods to fix broken tracks I've made a couple of PCBs which need the odd few 10 mil tracks, but are mostly 20/30 mil. Unfortunately the etching process isn't fantastic and there are numerous breaks in the traces, most of them small.  I've managed to convince solder lumps to go over the top of the smaller gaps, and have used components legs to bridge some of the larger ones. The solder lumps don't feel particularly reliable though, and there are some gaps which are too small and awkward to get a piece of wire across but too big to be bridged with just the iron.  Are there any standard ways to reliably fix up broken traces?  Update: Thanks for the suggestions everyone!  I found and tried out a silver conductive pen. As a first time user I found it difficult to dispense, and obviously I didn't use enough of it or mix it properly because it didn't even end up being conductive.  It seems like the main problem was that I was trying to use pieces of wire which are too short. I found some small gauge tinned wire and used longer lengths, running it over significant lengths of good track on either side of the fault. That makes it much easier to position, and you can tack them down one end at a time. What does "all the more" mean and how is it used? Here is a sample context:     As you might expect, the framework also provides some convenience   classes to make the task of writing custom validators all the more agreeable.   (Emphasis added.) How do I drag and drop files into an application? I've seen this done in Borland's  environment, but I'm not sure how to go about it for a C# application I'm working on. Are there best practices or gotchas to look out for? Contextually grey out panel element in python 2.8 I have a panel with split column.  In the 2nd row is 3 boolean variables. I want to grey out the 2nd and 3rd option by default. I want the 2nd to be changeable if 1st is true, and 3rd to be changeable if both 1 and two are true. I've read something about "poll()" but not how to actually use it, or set conditions. Physics should only be enablable if singleMesh, and destructable only if physics enabled.      # Create two columns, by using a split layout.     split = layout.split()      # First column     col = split.column()     col.label(text="Building Size:")     col.prop(bpy.context.scene.user_vars, 'buildingHeight')     col.prop(bpy.context.scene.user_vars, 'buildingWidth')     col.prop(bpy.context.scene.user_vars, 'buildingDepth')      # Second column, aligned     col = split.column()     col.label(text="Dynamics:")     col.prop(bpy.context.scene.user_vars, 'singleMesh')     col.prop(bpy.context.scene.user_vars, 'enablePhysics')     col.prop(bpy.context.scene.user_vars, 'destructable_Building')  #EDIT This is what the specific solution to my example looks like. col = split.column() col.label(text="Dynamics:") col.prop(vars, 'singleMesh') sub = col.row() sub.enabled = vars.singleMesh sub.prop(vars, 'enablePhysics') sub = col.row() sub.enabled = vars.enablePhysics sub.prop(vars, 'destructable_Building') Inversion in "only [adverb] have they" I have seen this construction quite often:      Online ads have been around since the dawn of the Web, but only in   recent years have they become the rapturous life dream of Silicon   Valley.   What is the rule there?. When your sentence doesn't start with pronoun + verb, invert them as verb + pronoun?. I know it sounds awkward but is it possible (grammatically correct) to use something similar to:      Online ads have been around since the dawn of the Web, but only in   recent years they have become...   And in any case, does this only work with have (or has)? Maybe it works fine with 'had' but I can't think of an example right now. What do the nomodeset, quiet and splash kernel parameters mean? What do Kernel command line parameters (also also known as boot options) mean?  When you go to Ubuntu (or better, GRUB) options, you have things like nomodeset, quiet and splash. What do these options mean?  I am also interested in other parameters and would be interested to see descriptions of any kernel parameter (like acpi), but these are not essential. $1,000 is pronounced as "one thousand dollars". Reading from left to right, it seems like it would make more sense to write the value as: 1,000$. This way the pronunciation of the value follows the way it's written. Every other numerical value I can think of places the unit after the value:   1 kg 1 L 1 m, etc...   I'm not sure if this question fits in with the rest of the questions on this site. But I'd like to know why currencies are so different from other quantities. Not to mention the need to separate groups of three digits with commas. Don't certain processes decrease entropy? Text spirals with TikZ I just ran across this, and I'm curious about whether it can be replicated in TeX...    Ideally one would have the spiral getting tighter and tighter as one got to the centre, as in    I know TikZ has the ability to fit text along a path, but I'm not sure how one might make text get smaller and smaller.  The closest I can find is , but that uses manually inserted font commands.  Edit: I got what I wanted using mwibrow's answer: I have created multiple objects, while testing I applied a drop shadow style on one object and its looking very good. Now I want to apply the same style to all other objects (around 20). Is there any way like global color to create a style once and apply to other objects with single click automatically?  Or I have to apply that drop shadow effect on each object one by one manually? Is temperature a Lorentz invariant in relativity? I am trying to understand the permission usage for the cp command in unix.   What are the permissions required for copying a file into a directory What are the permissions required for copying out of a directory?   I'm assuming for my first question, the directory will need execute permissions to perform the operation and the file will need read permissions to read the binary of the file and then the output file will need write permissions.  For the second the directory itself would need execute permissions to perform the copy operation and the file would need read permissions to extract the data from the file.  Can anyone tell me if my logic is flawed? How to prove that any (integer)$^{1/n}$ that isn't an integer, is irrational? Is my proof beneath perfect and complete?  I wanted to prove that for any nth root of an integer, if it's not an integer, than it's irrational: $$\begin{cases} m,n\in \mathbb{N}\\\sqrt[n]{m}\notin \mathbb{N} \end{cases}\implies \sqrt[n]{m}\notin \mathbb{Q}.$$  I start by assuming that $m^{\frac 1n}$ is rational and non-integer. So there exist  co-prime integers $a,b$ so that $$\sqrt[n]{m}=\frac{a}{b}$$ $$\implies  m=\frac{a^n}{b^n}\in\mathbb{N}.$$ But since $a$ and $b$ have no common factor, $a^n$ and $b^n$ also have no common factor. So: $$\frac{a^n}{b^n}\notin\mathbb{N},$$ a contradiction.