A problem with de Vitali set in [0,1]
Vitali-type set with given outer measure
Isomorphism between dual space and bilinear forms Studying from Roman's Advanced Linear Algebra, I want to prove that $$U^* \otimes V^* \cong (U \otimes V)^* \cong \hom(U,V;\mathbb{F})$$ The author proves that $U^* \otimes V^* \cong (U \otimes V)^*$ by showing that there is an unique linear transformation   $$\theta:U^* \otimes V^* \to (U \otimes V)^*$$  defined by $\theta(f \otimes g)=f \odot g$ where $(f \odot g)(u \otimes v)=f(u)g(v)$  I want to prove that $U^* \otimes V^* \cong \hom(U,V;\mathbb{F})$. I tried to use the universal property by fixing $f \in U^*, g\in V^*$ and defining $S: U^*\times V^* \to \hom(U,V;\mathbb{F})$ as $S(f,g)=F_{f,g}$, where $F_{f,g}(u,v)=f(u)g(v)$, but I got stuck. What is Change Detection and performing such analysis with open source tools? Since multiple inheritance is bad (it makes the source more complicated) C# does not provide such a pattern directly. But sometimes it would be helpful to have this ability. For instance I'm able to implement the missing multiple inheritance pattern using interfaces and three classes like that: public interface IFirst { void FirstMethod(); } public interface ISecond { void SecondMethod(); }  public class First:IFirst  {      public void FirstMethod() { Console.WriteLine(&quot;First&quot;); }  }  public class Second:ISecond  {      public void SecondMethod() { Console.WriteLine(&quot;Second&quot;); }  }  public class FirstAndSecond: IFirst, ISecond {     First first = new First();     Second second = new Second();     public void FirstMethod() { first.FirstMethod(); }     public void SecondMethod() { second.SecondMethod(); } }  Every time I add a method to one of the interfaces I need to change the class FirstAndSecond as well. Is there a way to inject multiple existing classes into one new class like it is possible in C++? Maybe there is a solution using some kind of code generation? Or it may look like this (imaginary c# syntax): public class FirstAndSecond: IFirst from First, ISecond from Second { }  So that there won't be a need to update the class FirstAndSecond when I modify one of the interfaces.  EDIT Maybe it would be better to consider a practical example: You have an existing class (e.g. a text based TCP client based on ITextTcpClient) which you do already use at different locations inside your project.  Now you feel the need to create a component of your class to be easy accessible for windows forms developers. As far as I know you currently have two ways to do this:  Write a new class that is inherited from components and implements the interface of the TextTcpClient class using an instance of the class itself as shown with FirstAndSecond.  Write a new class that inherits from TextTcpClient and somehow implements IComponent (haven't actually tried this yet).   In both cases you need to do work per method and not per class. Since you know that we will need all the methods of TextTcpClient and Component it would be the easiest solution to just combine those two into one class. To avoid conflicts this may be done by code generation where the result could be altered afterwards but typing this by hand is a pure pain in the ass. How does one write on the outside of a circle?  I am looking for sample files/packages which imitate a magazine lay-out. Specifically a page with two columns having a circular picture between the two columns.    Example of the desired layout, created using Word: What is "%AppData%"? Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $? Why is ${x^{\frac{1}{2}}}$ the same as $\sqrt x $?   I'm currently studying indices/exponents, and this is a law that I was told to accept without much proof or explanation, could someone explain the reasoning behind this.  Thank you. What methods of suicide cause the least amounts of collateral damage? CK2 is a big game with lots of variables, so I'll stick to adding them upon being requested to minimize the hassle.  Anyway, let's say that you have a lot of children and the oldest one claims all the land according to laws of succession. After the father dies, the circumstances will obviously piss off a lot of people (in my case that would be 6 out of 8 vassals and 4 siblings with an opinion of -50 and below one of which is a sister married to the Basileus of the Byzantine friggin' Empire, a case of excommunication, 4 failed assassination attempts under 3 years... and lots and lots of vikings). Let's also say that the current playable character is completely incompetent compare to the next sibling in line.  So, what would be the most optimal methods to off myself without further screwing over the countryside so the next heir in line has to fix it (if we assume that said heir is actually capable of doing so)?  This is my list of DLCs (not including music and unit packs):   Charlemange Legacy of Rome Sword of Islam The Republic Way of Life In the comment field on StackOverflow (and other StackExchange sites as well, I presume) you're required to write 15 characters for the server to accept your post. If a shorter post is submitted, an information message states that the comment is too short. That is all good.  However, below the comment field is a counter, which counts up to 15 and then down from however many characters you're allowed to type minus fifteen. This counter doesn't always act the same as the actual validation: if I pad my message with whitespace, the validation notices and still says the message is too short, but the counter does not notice, and counts as usual.  Suggested fix: make the character counter trim the message before it counts, so the two messages say the same thing. I'm travelling to country X. I want to install an application which isn't available in my country, but is available in country X; but I prefer to download and install it in advance, while the data connection is still cheap.  Unfortunately, the Play Stores does not allow me to install this app. Is there any way to circumvent this limitation? Are there any versions of t-SNE for streaming data? Enabling GPU rendering for Cycles? I want to enable GPU rendering, but there is no option in User Preferences > System:    Why is this? How can I get cycles to render using my GPU? Sakurai QM section 5.8 If anyone is familiar with Sakurai's book, specifically section 5.8 on energy shift and decay width, I am stuck and could use some help. I can't see how he derives 5.8.9 (in the revised edition). He starts with (3) (given below), and derives (1) and (2). The equation is as follows:$$(1){\space}{\space}\space{\space}\space{\space}{\dot c_i\over c_i}\approx{{-i\over \hbar}V_{ii}+{\left(-i \over \hbar\right)}^2{{{\left|V_{ii}\right|}^2}\over \eta}+\left({-i\over \hbar}\right) {\sum_{m\not=i}}{{{\left|V_{mi}\right|}^2}\over {(E_i-E_m+i\hbar\eta)}} \over 1-{i \over \hbar}{V_{ii} \over \eta}} $$ $$(2)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\approx{-i\over \hbar}V_{ii}+\left({-i\over \hbar}\right) {\sum_{m\not=i}}{{{\left|V_{mi}\right|}^2}\over {(E_i-E_m+i\hbar\eta)}}$$ where $$(3)\,\,\,\,\,\,\,\,\,c_i(t)=\lim_{\eta\to 0}\left({1-{i\over \hbar\eta}V_{ii}}{e^{{\eta}t}}+{{\left({-i\over \hbar}\right)}^2}{{\left|V_{ii}\right|}^2}{e^{2{\eta}t}\over 2{\eta}^2}+\left({-i\over \hbar}\right){\sum_{m\not=i}{{{{\left|V_{mi}\right|}^2}}\over 2\eta(E_i-E_m+i{\hbar}\eta}}\right)$$First of all, he discards all kinds of terms in the denominator of the first expression, and I can't see how he justifies this, because as ${\eta}\to0$, every term in the expression for $c_i(t)$ blows up (besides the 1 of course). Then he seems to multiply top and bottom by the complex conjugate of the denominator in (1), arriving at (2). He seems to discard a term from the denominator here, which is of second order in V, and says the equation is formally correct up to second order. This doesn't make sense to me! Also, soon after, he has this strange expression with $$\lim_{x\to 0} {1\over(x+iε)}=Pr.(1/x)-iπδ(x)$$ What is this Pr symbol? Any help would be much appreciated! How to show wifi password? Do I need an airport transit visa through Switzerland if I have no Schengen visa? Am traveling from Cairo to Moscow via Swiss Air passthrough graphics card to virtualbox So I'm trying to get my laptops NVIDIA fx 880m to pass to a virtual box running window seven on a linux mint 17 install.  So far everything seems to be (maybe) heading in the right direction:  the device passed through happily, when I booted the virtual box it installed a bunch of new drivers, but when I try to install the NVIDIA driver on the guest it can't find the card. I looked under the device manager and there's no listing for the nvidia card under the pci bus, my guess would be that I need to disable it in the host so that it can be passed through to the guest (the virtual box manual said they can't be shared) but I'm not sure how to do so. Can anybody help with this? This question does not seem to me to be a duplicate of , because here the given solution was that it was not possible with a windows host, however in this case the host is linux Best way to make family photo album in LaTeX I use LaTeX for my research publishing and would like to make a Christmas photo album that I could send to a print shop. It occurred to me to try it with LaTeX this time (yes I know, the print version won't get delivered by Christmas). Among other reasons, it will look like I'm working on code! But I would benefit from some templates etc. and haven't had luck finding any.  Can anyone recommend resources for building a family photo album in LaTeX? Calculate $E(X^{2n})$ where $X$ is normal (0,1) I need help proving the following:  Let $X$ be normally distributed with parameters $\sigma=0$ and $\mu=1$. Let $n$ be a positive integer. Show that:  $$E(X^{2n})=\frac{(2n)!}{2^nn!}=:(2n-1)!!$$  I've tried the change of variables $Y=X^2$ and then calculating the integral, but got nowhere. Any hints?  Thanks in advance! The meaning of Bit-wise NOT in Python Ctrl+Alt+← can usually be set to restart X, however I can't find the option to allow this (it is disabled by default).  I have looked in System-&gt;Preferences-&gt;Keyboard and System-&gt;Preferences-&gt;Keyboard Shortcuts but I can't find an option to enable it.  Where can I change this setting? Solving $\textbf{r}''(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$ I have the following problem that I don't know how to solve:  Let $\textbf{r}(t)$ be a curve and let $r(t):= |\textbf{r}(t)|$. Let the curve $\textbf{r}(t)$ be defined by: $$\textbf{a}(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$$ Where $\textbf{a}(t) = \textbf{r}''(t)$ is the acceleration of the particle and $G,M \in \mathbb{R}$. Suppose that $\textbf{r}(t)$ and $\textbf{r}'(t)$ are not parallel. The objective is to determine the trajectory $\textbf{r}(t)$ in polar coordinates.  So basically we have the following equation: $$\textbf{r}''(t)=\frac{GM}{(r(t))^3}\textbf{r}(t)$$ The thing is that I don't know how I should proceed from now on. If we'd have just normal scalar function this would be a differential equation and I think I would be able to solve it, but like this I have no idea how to approach the problem. How can I solve this?  Edit: If $$\textbf{r}(t):=(x(t),y(t))$$ then the equation turns into the following system of equations: $$x''=\frac{GM}{(\sqrt{x^2 + y^2})^3}x$$ $$y''=\frac{GM}{(\sqrt{x^2 + y^2})^3}y$$ I don't know if this helps but I think that this turns thing easier perhaps.